Research ArticleMathematical and Numerical Analysis ofHeat Transfer Enhancement by Distribution ofSuction Flows inside Permeable Tubes
A-R A Khaled
Mechanical Engineering Department King Abdulaziz University PO Box 80204 Jeddah 21589 Saudi Arabia
Correspondence should be addressed to A-R A Khaled akhaled4yahoocom
Received 1 February 2015 Revised 24 March 2015 Accepted 27 March 2015
Academic Editor Rama S R Gorla
Copyright copy 2015 A-R A Khaled This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Heat transfer enhancement in permeable tubes subjected to transverse suction flow is investigated in this work Both momentumand energy equations are solved analytically and numerically Both solutions based on negligible entry regions are well matchedTwo different suction velocity distributions are considered A parametric study including the influence of the average suctionvelocity and the suction velocity profile is conducted for various Peclet numbers It is found that enhancement of heat transfer overthat in impermeable tubes is only possible with large Peclet numbers This enhancement increases as suction velocities towardsthe tube outlet increase and as those towards the tube inlet decrease simultaneously The identified enhancement mechanisms areexpanding the entry regions increasing the transverse advection and increasing the downstream excess temperatures under sametransverse advection The average suction velocity that produces maximum enhancement increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at large Peclet numbers The maximum reported enhancement ratios for theexponential and linear suction velocity distributions are 1762-fold and 1467-fold above those for impermeable tubes respectivelyThis work demonstrates that significant heat transfer enhancement is attainable when the suction flow inside the permeable tubesis distributed properly
1 Introduction
The study of fluid flow and heat transfer inside permeabletubes or channels exposed to surface suction is importantto many industrial applications These applications includetranspiration cooling where channel surfaces are cooled bypassing cooled fluid through the pores of these surfacescontrolling boundary layers over surfaces of airplane wingsor turbine blades by injection or suction of fluid at theses sur-faces lubrication of permeable bearings fluid filtration pro-cesses cooling of combustion chambers exhaust ports andcooling of porous walled reactors [1] Accordingly the aimof the present work is to investigate heat transfer enhance-ment inside permeable tubes subjected to nonuniform sur-face suction
Bergles [2] indicated that surface suction is an effectivetechnique that can be used to enhance heat transfer [2 3]He indicated that improvement in heat transfer coefficient is
expected to reach several hundred percent for laminar flowwith suction at the solid boundary [2ndash5] He pointed outthat this enhancement is due to reduction in the boundarylayer thickness [2] This can be clearly seen for externalflows [4ndash8] However surface suction within internal flowstends to reduce the mean velocity inside the tube or thechannel This effect may thicken the boundary layer andcauses impediments of both flows near the boundary andheat transfer rate Therefore the novelty of the present workis to explore the various conditions that may reveal heattransfer enhancement inside permeable tubes exposed tosurface suction
Among initial works that analyzed the problem of fluidflow and heat transfer inside permeable tubes or channelswith surface suction are the works of Kinney [9] Pedersonand Kinney [10] Raithby [11] and Sorour and Hassab [12]These works illustrated the variations of the temperatureprofile and Nusselt number with wall Reynolds number in
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 953604 11 pageshttpdxdoiorg1011552015953604
2 Mathematical Problems in Engineering
porous tubes or channelsThese works andmany others werethe motivations behind recent works that accounted for allpossible hydrodynamic conditions on heat transfer insidechannels subjected to wall suction For example Sorour etal [13] and Bubnovich et al [14] analyzed dynamically andthermally the developing flow inside a channel subjected tononuniform suction at onewall Hwang et al [15] investigatednumerically forced laminar convection in the entrance regionof a square duct subjected to uniform mass transpirationMakinde et al [1] analyzed heat transfer in channels exposedto wall suction in presence of nanofluids with both wall slipand viscous dissipation effects All of the aforementionedworks and many similar ones in the literature did not explorethe heat transfer enhancement due to surface suction insidepermeable tubes Thus the motivation behind the presentwork is to enrich the literature with a study about the role ofsurface suction and its profile inside permeable tubes on heattransfer enhancement
Promoting flow close to the energy exchanging bound-ary usually results in heat transfer enhancement [2 16]Meanwhile the internal flow impedance near this boundaryincreases as suction Reynolds number (ReV
119908
) increases [911] For permeable circular tubes the internal flow starts toseparate from the boundary when ReV
119908
= 45978 due to theadverse pressure gradient caused by reduction in the meanfluid velocity To avoid this instability condition and to sustainmaximumvelocity close to tube surface the suctionReynoldsnumber must be ReV
119908
le 104 This constraint can be shownusing Kinney [9] and Raithby [11] works to yield universalsurface friction coefficient to be at least equal to 95 of itsmaximum value when ReV
119908
= 0 Accordingly the flow closeto the energy exchanging boundary can be kept maximallypromoted under this constraint Therefore the present workis concerned with heat transfer enhancement inside perme-able tubes exposed to surface suction with 0 lt ReV
119908
le 104In the next sections flow and heat transfer inside a
preamble tube subjected to internal suction flow aremodelledand analyzed The surface suction velocity is considered tohave either linear or exponential profile distributions Bothmomentum and thermal energy transfer equations are solvedusing various analytical and numerical methods Differentheat transfer enhancement indicators are computed Bothanalytical and numerical computations of these indicatorsare validated under an applicable constraint and using earlystudies An extensive parametric study has been conducted inorder to identify and explore the influence of average suctionvelocity suction velocity profile and Peclet number on theheat transfer enhancement indicators
2 Problem Formulation
21 Modeling of Flow and Heat Transfer inside the PermeableTube Consider a tube of length 119871 and inner diameter 119863The tube wall is permeable so as to allow fluid suction at itsboundary as shown in Figure 1 Both the flow inside the tubeand that through the permeable boundary are consideredto be laminar flows The density specific heat thermalconductivity and the dynamic viscosity of the fluid are120588 119888119901 119896 and 120583 respectively The dimensionless continuity
(a)
(b)
L
w(x)
w(x)
r
x u
Fluid flow
D
(c)
Figure 1 (a) 3D viewof the tubewith suction passages embedded onits material volume (b) cross section of the tube and (c) schematicprofile of the tube and the coordinates system
momentum and energy equations of the fluid are given by[17ndash19]
120597119906
120597119909
+
1
119903
120597
120597119903
(119903 V) = 0 (1)
119886Re119903 (119906120597119906
120597119909
+ V120597119906
120597119903
) = 16
119889119901
119889119909
+
2
119903
120597
120597119903
(119903
120597119906
120597119903
) (2)
119886Pe119903 (119906120597120579
120597119909
+ V120597120579
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579
120597119903
) (3)
Mathematical Problems in Engineering 3
where 119906 and V are the dimensionless axial and radialvelocities respectively 119909 and 119903 are the dimensionless axialand radial positions respectively 119886 and Pe119903 are the tubeaspect ratio and the reference Peclet number respectively119901 and 120579 are the dimensionless pressure and dimensionlesstemperature fields respectively The dimensionless variablesand parameters used in (1)ndash(3) are given by
119886 =
2119863
119871
(4a)
119903 =
2119903
119863
(4b)
119909 =
119909
119871
(4c)
119906 (119909 119903) =
119906
119906119900
(5a)
V (119909 119903) =V119886119906119900
(5b)
Re119903 =120588119906119900119863
120583
(6a)
Pe119903 =120588119888119901119906119900119863
119896
= Re119903 Pr (6b)
119906119900 =
1199011 minus 1199012
32120583119871119863
2 (6c)
119901 (119909) =
119901 minus 1199011
1199012 minus 1199011
(7a)
120579 (119909 119903) =
119879 minus 1198791
119902
10158401015840119904119863 (2119896)
(7b)
where 119906119900 1199011 1199012 and 1198791 are reference axial velocity inletpressure outlet pressure and inlet temperature respectively119902
10158401015840
119904is the constant heat flux applied at the inner surface of the
tube The boundary conditions of (1)ndash(3) are given by
119909 = 0 119906avg = 1 (8a)
119909 = 0 120579 = 0 (8b)
119909 = 1 119901 = 1 (9)
119903 = 0 120597119906
120597119903
= 0 (10a)
119903 = 0 120597V120597119903
= 0 (10b)
119903 = 0 120597120579
120597119903
= 0 (10c)
119903 = 1 119906 = 0 (11a)
119903 = 1 V = V119908 (119909) (11b)
119903 = 1 120597120579
120597119903
= 1 (11c)
where V119908(119909) = V119908(119909)(119886119906119900) V119908(119909) is the dimensional localsuction velocity at the tube inner surface 119906avg is the meandimensionless axial velocity at any given cross section UsingKinney [9] and Raithby [11] works it can be shown that with0 le 119886Re119903V119908 le 104 the convective terms in (2) can beneglectedThe aforementioned range results in less than 50relative error associated with calculating the universal wallfriction coefficient by neglecting the convective terms Theprevious constraint can practically be satisfied for high aspectratio tubes (1119886 ≫ 1) viscous fluids or with small suctionvelocities Accordingly the solution of (2) in absence of theconvective terms with boundary conditions given by (10a)and (11a) is the following
119906 (119909 119903) = 2119901
1015840(1 minus 119903
2) (12)
where 1199011015840 = 119889119901119889119909 Substituting (12) in (1) and solving theresulting equation yield to the following distribution of thedimensionless radial velocity
V (119909 119903) = minus119901
10158401015840(119903 minus
119903
3
2
) (13)
where 119901
10158401015840= 119889
2119901119889119909
2 Applying the boundary conditiongiven by (11b) results into the following differential equation
119901
10158401015840= minus2V119908 (119909) (14)
The mass flow rate (119909) and mean axial velocity 119906avg(119909)can be calculated from the following expressions
119872(119909) =
(119909)
119900
= 2int
1
0
119903 119906 119889119903 = 119901
1015840(119909) (15)
119906avg (119909) equiv119906avg (119909)
119906119900
=
(119909) (120588120587119863
24)
119906119900
= 119901
1015840(119909)
(16)
where 119900 is the reference mass flow rate 119900 = 12058811990611990012058711986324
TheNusselt number at the tube inner boundary is definedas
Nu equiv
ℎ119863
119896
=
2
120579119882 minus 120579119898
(17)
where ℎ 120579119898 and 120579119882 are the convection heat transfer coeffi-cient between the tube inner boundary and the fluid flow (ℎ =
119902
10158401015840
119904[119879119882 minus 119879119898]) dimensionless mean bulk temperature and
the dimensionless temperature of the tube inner boundary(120579119882 = 120579(119909 119903 = 1)) respectively 119879119882 and 119879119898 are the tem-perature at the tube inner boundary and the mean bulktemperature of the fluid respectively 120579119898 can be computedfrom the following equation
120579119898 (119909) equiv119879119898 (119909) minus 1198791
119902
10158401015840119904119863 (2119896)
= 4int
1
0
119903 (1 minus 119903
2) 120579 (119909 119903) 119889119903 (18)
The Integral Energy EquationThe integral form of the energyequation can be formed using (3) It can be expressed in thefollowing form
119906avg119889120579119898
119889119909
+ 4(
V119908Nu
) =
4
119886Pe119903 (19)
4 Mathematical Problems in Engineering
The Fully Developed Nusselt Number Under fully developedcondition 120597120579120597119909 = 119889120579119898119889119909 As such (3) can be reduced tothe following using (19)
8 (1 minus 119903
2) 1 minus
119886Pe119903V119908Nufd
+ 2119886Pe119903V119908 (119903 minus119903
3
2
)
120597120579fd120597119903
=
2
119903
120597
120597119903
(119903
120597120579fd120597119903
)
(20)
where Nufd is the fully developed Nusselt number Equation(20) can be solved analytically and Nufd expression can bearranged in the following form
Nufd119885
= (81 minus exp(31198858
)
+ 2119885 exp(31198858
)int
0
minus34
exp (1198851199102)radic119910 + 1
119889119910)
sdot (119885 + 8 1 minus exp(31198858
)
+ 2119885 exp(31198858
)int
0
minus34
exp (1198851199102)radic119910 + 1
119889119910)
minus1
(21)
where 119885 = 119886Pe119903V119908(119909) The following approximation can beused
1
radic119910 + 1
cong 1198861 exp (1198862119910) + 1198863 exp (1198864119910) minus
3
4
le 119910 le 0
(22)
This approximation has maximum relative error less than03 when the coefficients are equal to
1198861 = 98075 times 10
minus3 1198862 = minus53965
1198863 = 09874 1198864 = minus049834
(23)
By substituting (22) in (21) Nufd can be approximated by thefollowing expression
Nufd119885
= (8 1 minus exp(31198858
) minus 1198601 exp (1198865) minus exp(31198858
)
minus1198602 exp (1198866) minus exp(31198858
))
sdot (8 1 minus exp(31198858
) minus 1198601 exp (1198865) minus exp(31198858
)
minus1198602 exp (1198866) minus exp(31198858
) + 119885)
minus1
(24)
where
1198865 = minus(
3
4
) 1198862 1198866 = minus(
3
4
) 1198864
1198601 =41198861119885
21198862 + 119885
1198602 =41198863119885
21198864 + 119885
(25)
211 Case I Linear Distribution of the Suction Velocity Forthis case the dimensionless suction velocity denoted by V119908119860has the following linear distribution
where V119900 = V119900(119886119906119900) V119900 is the average suction velocity overthe length of tube inner surface 119861 is an arbitrary controllingparameter
212 Case II Exponential Distribution of the Suction VelocityFor this case the dimensionless suction velocity denoted byV119908119861 has the following exponential distribution
V119908119861 (119909) = V119900119862 exp (119862119909)exp (119862) minus 1
(27)
where 119862 is an arbitrary controlling parameter
213 The Pressure Gradient for Both Cases Substituting (26)and (27) in (14) and solving for the dimensionless pressuregradient the following expressions can be obtained
119889119901
119889119909
=
1 minus 2V1199001199092+ 2119861 (119909
2minus 119909) Case I
1 minus 2V119900 [exp (119862119909) minus 1
exp (119862) minus 1
] Case II(28)
The ranges of 119861 and V119900 that result in having both 119906avg119860 andV119908119860 larger than zero can be shown to be
0 le V119900 lt1
2
0 le 119861 le 2V119900 (29)
The ranges of 119862 and V119900 that result in having both 119906avg119861 andV119908119861 larger than zero can be shown to be
0 le V119900 lt1
2
minusinfin lt 119862 lt infin (30)
22 Perfect Fluid Slip Case (Ideal Case) The ideal case of thepresent problem is constructed when the fluid is subjectedto perfect slip condition at the solid boundary For this idealcase the conservation of mass and the continuity equationreveal the following expressions
119889119906119904
119889119909
= minus2V119908 (31)
119906119904 =
1 minus 2V1199001199092+ 2119861 (119909
2minus 119909) Case I
1 minus 2V119900 exp (119862119909) minus 1
exp (119862) minus 1
Case II(32)
V = V119908119903 (33)
Mathematical Problems in Engineering 5
where 119906119904 = 119906s119906119900 119906119904 is the fluid velocity for this ideal caseUnder this ideal condition the energy equation reduces tothe following
119886Pe119903 (119906119904120597120579119904
120597119909
+ V119908119903120597120579119904
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904
120597119903
) (34)
where 120579119904 is the dimensionless temperature for the ideal caseFor fully developed condition where 120597120579119904120597119909 = 119889120579119898119889119909 (34)can be reduced to the following when (19) is implemented
41 minus
119885
Nu119904fd + 119885(119903
120597120579119904fd
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904fd
120597119903
) (35)
where Nu119904fd is the fully developed value of the Nusseltnumber for the ideal case By solving (35) and application ofthe boundary condition given by (10c) Nu119904fd can be foundIt is equal to
Nu119904fd119885
=
4 1 minus exp (1198854)4 1 minus exp (1198854) + 119885
(36)
For this case the dimensionless mean bulk temperature canbe found to be equal to
120579119898119904 (119909) equiv
119879119904119898 (119909) minus 1198791
119902
10158401015840119904119863 (2119896)
= 2int
1
0
119903120579119904119889119903 (37)
23 Heat Transfer Enhancement Indicators Let the heattransfer enhancement indicator 120582 be defined as ratio of thetube inner boundary excess temperature at the exit sectionfor the reference case (V119908 = 0) to that quantity when V119908 gt 0Mathematically the enhancement indicator 120582 is written as
120582 =
120579 (119909 = 1 119903 = 1)|V119908=0
120579 (119909 = 1 119903 = 1)
(38)
When V119908 = 0 the system becomes an impermeable tubeconfining an internal flow and subjected to uniform heat fluxFor this case the Nusselt number at the exit can be shown tobe correlated to 119886Pe119903 according to the following expression
Nu (119909 = 1 V119908 = 0)
= (438 minus 13 times 10
minus3119886Pe119903 + 39 times 10
minus4(119886Pe119903)
2
+ 65 times 10
minus7(119886Pe119903)
3)
sdot (1 minus 46 times 10
minus5119886Pe119903 + 86 times 10
minus5(119886Pe119903)
2
+ 43 times 10
minus8(119886Pe119903)
3)
minus1
(39)
with maximum relative error less than 03 when 1 le 119886Pe119903 le1000
The second performance indicator 120578119904 is defined as ratioof the tube inner boundary excess temperature at the exitsection for the perfect fluid slip case to that quantity underno-slip condition Mathematically 120578119904 is written as
24 Enhancement Indicators for Fully Developed Flow withUniform Suction For uniform suction case the mean veloc-ity inside the tube can be found using (32) by setting 119861 = V119900It is equal to
119906119904 = 119906avg = 1 minus 2V119900119909 V119900 lt1
2
(41)
By substituting (41) in (19) and solving the resulting equationthe mean bulk temperature distribution can be obtained It isgiven by
120579119898 (119909) = ln [1 minus 2V119900119909]2([1Nufd]minus1[119886Pe119903V119900])
(42)
Accordingly the dimensionless temperature at the exit isequal to
120579 (119909 = 1 119903 = 1)
= 120579119882 (119909 = 1) = ln [1 minus 2V119900]2([1Nufd]minus1[119886Pe119903V119900])
+
2
Nufd
(43)
Therefore the performance indicators 120582 and 120578119904 for this caseare equal to
120582fd =96 + 11119886Pe119903
ln [1 minus 2V119900]48([119886Pe
119903Nufd]minus1V119900)
+ 48 (119886Pe119903Nufd)
(44)
120578119904fd =ln [1 minus 2V119900]
2([119886Pe119903Nu119904fd]minus1V119900)
+ 2 (119886Pe119903Nu119904fd)
ln [1 minus 2V119900]2([119886Pe
119903Nufd]minus1V119900)
+ 2 (119886Pe119903Nufd)
(45)
The plots of 120579119898(119909 = 1) 120579119904119898(119909 = 1) Nufd Nu119904fd and 120582fdfor various 119886Pe119903 and V119900 values are seen in Figures 2ndash4 Theupper value of V119900 that makes 120582fd = 1 can be obtained usingnumerical solving techniques [20] It is denoted by V119900119901 V119900119901can be correlated to 119886Pe119903 through the following correlation
V119900119901 = (minus28076 times 10
6+ 820470 (119886Pe119903)
minus 83102 (119886Pe119903)2+ 18754 (119886Pe119903)
3)
sdot (1 + 713480 (119886Pe119903) minus 123760 (119886Pe119903)2
+ 37172 (119886Pe119903)3)
minus1
(46)
The percentage error associated with (46) is less than 062when 119886Pe119903 ge 4 The plot of 119886Pe119903 versus V119900119901 is shown in Fig-ure 3 Surprisingly the value of V119900 making 120578119904fd = 1 is foundto be independent on 119886Pe119903 This value is denoted by V119900119890 andit is equal to
V119900119890 = 031606 (47)
The plots of 120578119904fd for various 119886Pe119903 and V119900 are seen in Figure 4120578119904fd ge 1 when V119900 gt V119900119890
6 Mathematical Problems in Engineering
000 010 020 030 040 05020E minus 03
20E minus 02
20E minus 01
20E + 00
20E + 01
Fully developed flow condition
120579m(x
=1)120579
sm(x
=1)
120579m(x = 1)
120579sm(x = 1)
aPer = 1 2 4 8 16 32
w = o
o
Figure 2 Effects of V119900 on outlet mean bulk temperature under fullydeveloped flow condition
3 Numerical Methodology and Results
Equations (3) and (35) are solvable numerically without iter-ations using the implicit finite difference method discussedby Khaled [21] and Blottner [22]They were discretized usingtwo-point backward and three-point central differencingquotients for the first derivative with respect to 119909-direc-tion and both first and second derivatives with respect to119903-direction respectively The finite difference equation of(3) is given by
The pair (119894 119895) represents the location of the discretized pointin the numerical grid of the fluid domain The number 119899represents the total number of 119895-nodes along each 119894-sectionThe tube inner surface is located in the numerical grid at(119894 119895 = 119899) while the center of the tube is located in thenumerical grid at (119894 119895 = 1) The resulting 119899 minus 1 (119899 = 401)tridiagonal systemof algebraic equations obtained by (48) at agiven 119894-section was solved usingThomas algorithm [22] Theprevious procedure step was repeated for the consecutive 119894-values until 119894 reached the value 119898 (119898 = 2001) at which 119909 =
1 Numerical investigations were performed using differentmesh sizes to assess and ascertain grid-size independentresults Furthermore doubling the values of119898 and 119899 resultedin less than 04 error in the calculated parameters formoderate 119886Pe119903 values
The results of the present work are shown in Figures 2ndash14 Those obtained using the described numerical method
000
009
018
027
036
045
054
10E minus 01
10E + 02
10E + 02
10E + 01
10E + 01
10E + 00
10E + 00
aPerw aPer
op
Nu f
dN
u sfd
Nufd minus aPerwNusfd minus aPerwop minus aPer
Figure 3 Effects of 119886Pe119903 on fully developed Nusselt numbers andV119900119901
060
074
088
102
116
130
144
000
115
230
345
460
575
690
000 010 020 030 040 050
w = o
Fully developed flow condition
aPer = 1 2 4 8 16 32 50
120582fd
120582fd
120578s
fd
120578sfd
o
Figure 4 Effects of V119900 on the first and second performance indica-tors under fully developed flow condition
are shown in Figures 5ndash14 The numerical results for the casewith 119886Pe119903 = 1 shown in Figures 4ndash8 were compared withthe analytical solution utilizing (17) (24) (26) (27) (38) and(40) The numerical results match well with the analyticalsolutions since the case with 119886Pe119903 = 1 results in fullydeveloped flow condition at the tube exit Also it is shownfrom Figure 3 that the fully developed Nusselt number when119886Pe119903V119908 = 10 is equal to Nufd = 10725 This quantity canbe shown from Figure 9 of Raithby [11] work to be equal toNufd = 1083 when 119886Re119903V119908 rarr 0 The relative error betweenthese values is 098 which is very small These validationsincreased the confident levels in the obtained results
4 Discussion of the Results
41 The System Thermal Performance for Uniformly Dis-tributed Suction Flow As suction velocity V119908 increases theoutlet mean axial velocity 119906119898(119909 = 119871) decreases causing
Mathematical Problems in Engineering 7
010
045
080
115
150
185
220
000 020 040 060 080 10010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
B
o = 025 033 049
Figure 5 Effects of 119861 on performance indicator 120582 and outlet meanbulk temperature for Case I
085
095
105
115
125
135
145
000 020 040 060 080 100
120578s
B
o = 025 033 049
o = 025
o = 049
3o = 1
aPer = 1 numericalaPer = 1 analyticalaPer = 10
Figure 6 Effects of 119861 on performance indicator 120578119904 for Case I
01
10
100
10E + 0210E + 0110E + 00
aPerB = 0B = o
B = 2o
120582
o = 025 033 049
o = 025 033 049
(44)
Figure 7 Effects of 119886Pe119903 on performance indicators 120582 for various 119861values for Case I
066
080
094
108
122
136
150
10E + 0210E + 0110E + 00
aPer
120578s
B = 0B = o
B = 2o
o = 025 033 049
(45)
Figure 8 Effects of 119886Pe119903 on performance indicators 120578119904 for various 119861values for Case I
010
048
085
123
160
198
235
minus200 minus120 minus040 040 120 20010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
C
o = 025 033 049
o = 025 033 049
Figure 9 Effects of 119862 on performance indicator 120582 and outlet meanbulk temperature for Case II
087
097
107
117
127
137
147
minus200 minus120 minus040 040 120 200C
120578s
aPer = 1 numericalaPer = 1 analyticalaPer = 10
o = 025 033 049
o = 025
o = 049
3o = 1
Figure 10 Effects of 119862 on performance indicator 120578119904 for Case II
8 Mathematical Problems in Engineering
01
10
100
10E + 0210E + 0110E + 00
aPer
120582
C = 20
C = 0
C = minus20
o = 025 033 049
o = 025 033 049
(44)
Figure 11 Effects of 119886Pe119903 on performance indicators 120582 for various119862values for Case II
075
088
101
114
127
140
153
10E + 0210E + 0110E + 00
aPerC = 20
C = 0
C = minus20
120578s
o = 025 033 049
(45)
Figure 12 Effects of 119886Pe119903 on performance indicators 120578119904 for various119862 values for Case II
the axial convection which is proportional to 119886Pe119903119906avg119889120579119898119909to decrease This may cause an increase in the outlet meanbulk temperature 119879119898(119909 = 119871) as shown in Figure 2 forsmall 119886Pe119903 values 119886Pe119903 le 4 When 119886Pe119903 ge 8 the trans-verse convection becomes significant as it is proportionalto 2119886Pe119903V119908(120579119882 minus 120579119898) The latter expression is equivalent to4119886Pe119903V119908Nu This expression increases as 119886Pe119903V119900 increasesThis increase causes significant reduction in the temperatureboundary layer thicknesswhich causes reduction in the outletmean bulk temperature as shown in Figure 2 The sum of theaxial and transverse convection is constant which is equalto the total convection heat transfer rate given by 119902conv =
119902
10158401015840
119904120587119863119871 When V119900 approaches V119900 = 05 the axial advection
which is proportional to 119906avg approaches zero at the tubeoutlet This causes 119879119898(119909 = 119871) to increase apparently as V119900approaches V119900 = 05 asymptotically as can be seen fromFigure 2
Linear suction
Exponential suction
Linear suctionprofile
Exponential suctionprofile
0000
0085
0170
0255
0340
0425
0510
10
100
oc
oc
120582m
ax
120582max
10E + 0210E + 0110E + 00
aPer
profile
profile
Figure 13 Effects of 119886Pe119903 on 120582max and V119900119888 for both Case I and CaseII
Figure 14 Effects of 119886Pe119903 on 120578119904119888 for both Case I and Case II
The Nusselt number for perfect fluid slip at the solidboundary (36) case is larger than that for the no-slip condi-tion case (24) as shown in Figure 3 that is Nu119904 gt Nu Thisindicates that the tube excess temperature at the boundary forthe perfect fluid slip condition is smaller than that for the no-slip condition case That excess temperature is proportionalto 120579119882 minus 120579119898 Consequently the transverse convection due tofluid slip is smaller than that due to no-slip condition as thisconvection is proportional to both 119886Pe119903V119908 and 120579119882 minus 120579119898 As aresult the axial convection due to fluid slip condition is largerthan that due to no-slip condition as the total convectionheat transfer rate is constant Therefore 119879119898(119909 = 119871) for thefluid slip condition is always larger than that for the no-slipcondition as shown in Figure 2 It is shown from Figure 3 thatthe asymptotic value of Nufd is equal to Nufd = 119886Pe119903V119908 Thisfinding is shown in the works of Kinney [9] and Raithby [11]
Figure 4 shows that the first heat transfer performanceindicator 120582fd is larger than one when 0 lt V119900 lt V119900119901
Mathematical Problems in Engineering 9
where the variation of V119900119901 with 119886Pe119903 is shown in Figure 3and given by (46) It can be shown from Figure 3 that when119886Pe119903 ge 978 V119900119901 ge 0495 It should be mentioned herethat having 120582fd gt 1 indicates that the heat transfer to thepermeable tube exposed to suction flow results in havinglower tube outlet temperature than that for the impermeabletube Consequently more thermal energy can be transferredfrom the permeable tube to the fluid in order to raise itsoutlet temperature to reach that of the impermeable tubeFurthermore Figure 4 shows that the second heat transferperformance indicator 120578119904fd becomes above one when V119900 gt
V119900119890 = 031606 Having 120578119904fd gt 1 indicates that thepermeable tube subjected to no-slip condition has smalleroutlet boundary temperature than that for the permeable tubesubjected to perfect fluid slip at the solid boundary This isnot the case for impermeable tubes as 120578119904fd lt 1 when V119900 = 0
as shown in Figure 4
42 The SystemThermal Performance for Linearly DistributedSuction Flow When 119861 = 0 the suction flow increaseslinearly from V119908 = 0 at the inlet (119909 = 0) to maximumvalue of V119908 = 2V119900 at the tube outlet (119909 = 1) For thiscase the thermal entry region effect is maximum as 119906avgis maximum at inlet In this region the axial convectioncoefficient is substantial as it is the coldest fluid region insidethe tube The thermal entry region influences the flow bycausing reductions in the boundary excess temperatures nearthe inlet Also the transverse convection is maximum at thetube outlet when 119861 = 0 as the suction velocity is maximumthere This decreases the boundary excess temperatures nearthe tube outlet As 119861 increases 119906avg close to the tube inletdecreases due to the increase in V119908 and V119908 close to thetube outlet decreases Accordingly both thermal entry regioneffect and transverse convection at the tube outlet decrease as119861 increases Both effects tend to increase both the boundaryexcess temperature and outlet mean bulk temperature As aresult 120579119898(119909 = 1) increases and 120582 decreases as 119861 increases asshown in Figure 5
For large 119886Pe119903 values both thermal entry region effectand the transverse convection at the tube outlet are largerthan those for smaller 119886Pe119903 values Accordingly 120582 valueswhen 119886Pe119903 = 10 are larger than those when 119886Pe119903 = 1
as shown in Figure 5 When 119886Pe119903 = 10 and 119861 = 0 it isnoticed that 120582 increases as V119900 increases This is because theoutlet mean bulk temperature decreases as V119900 increases forlarge 119886Pe119903 values as seen from Figure 2 Figure 5 shows that120582 values for V119900 = 13 plot are larger than those for V119900 = 049
plot when 119861 gt 046 This indicates that the convection dueto thermal entrance region effect as compared to transverseconvection increases as V119900 decreases when 119861 gt 046 and119886Pe119903 = 10 When 119886Pe119903 = 10 120578119904 is noticed from Figure 6to decrease as 119861 increases It is because that both thermalentry region effect and V119908 at the tube outlet decrease as 119861increases The former effect is due to the increase in V119908 nearthe tube inlet as 119861 increases Both effects tend to increase theexcess of Nu119904 above Nu Accordingly transverse convectiondue to fluid slip condition decreases due to the decrease inthe boundary excess temperature Consequently the axialconvection due to the fluid slip condition is increased more
than that due to the no-slip condition This causes 120579119904119898 toincrease as compared to 120579119898Thus 120578119904 decreases as 119861 increases
Figure 7 shows that 120582 increases as 119886Pe119903 increases At small119886Pe119903 values the transverse convection is negligible and Nu119904approaches Nu for large 119886Pe119903 values These lead to have 120578119904approaching one either as 119886Pe119903 decreases towards 119886Pe119903 =
0 or as 119886Pe119903 increases above 119886Pe119903 = 100 These trendscan be seen from Figure 8 For moderate 119886Pe119903 values theincrease in 119886Pe119903 not only causes an increase in the transverseconvection but it also results in reducing the boundary excesstemperature due to the increases in Nu This reduction ismore distinct for the case of the perfect fluid slip condition asNu119904 gt Nu The combined aforementioned effects can resultin reduction of transverse convection as 119886Pe119903 increases for theperfect fluid slip case It is because the transverse convectionis proportional to 119886Pe119903V119908(120579119882 minus 120579119898) as mentioned beforeAccordingly 120578119904 may have one local maximum at a specific119886Pe119903 value as clearly shown in Figure 8 for V119900 = 049 plots
43 The System Thermal Performance for Exponentially Dis-tributed Suction Flow For the upper 119862 value (119862 = 2)the thermal entry region effect is maximum since 119906avg ismaximum at tube inlet because V119908 is minimum there Alsothe transverse convection at the tube outlet is maximumas V119908 is maximum there These effects tend to decreasethe boundary excess temperatures at the tube outlet As 119862decreases 119906avg close to the tube inlet decreases due to theincrease in V119908 and V119908 close to the tube outlet decreasesAccordingly both thermal entry region effect and trans-verse convection at the tube outlet decrease as 119862 decreasesBoth effects increase both the boundary excess temperatureand outlet mean bulk temperature As a result the 120579119898(119909 =
1) increases and 120582 decreases as 119862 decreases as shown inFigure 9
For large 119886Pe119903 values both thermal entry region effectand outlet transverse convection are larger than those forsmaller 119886Pe119903 values Accordingly 120582 values when 119886Pe119903 = 10
are larger than those when 119886Pe119903 = 1 as shown in Figure 9When 119886Pe119903 = 10 and 119862 = 2 it is noticed that 120582 increases asV119900 increases It is because the outlet mean bulk temperaturedecreases as V119900 increases for large 119886Pe119903 values as shown fromFigure 2 Figure 9 shows that 120582 values for V119900 = 13 plot arelarger than those for V119900 = 049 plot when 119862 lt 06 Thisindicates that the flow gets more dominated by the thermalentry region effect than by the transverse convection as V119900decreases when 119862 lt 06 and 119886Pe119903 = 10 Figure 11 showsthat 120582 increases as 119886Pe119903 increases When 119886Pe119903 = 10 120578119904is noticed from Figure 10 to decrease as 119862 decreases It isbecause both thermal entry region effect and V119908 at the outletdecrease as 119862 decreases As indicated previously both effectstend to increase the excess of Nu119904 above Nu Accordingly thetransverse convection due to fluid slip condition decreasesdue to the decrease in the boundary excess temperatureConsequently the axial convection due to fluid slip conditionis increased more than that due to no-slip condition causing120579119904119898 to increase Thus 120578119904 decreases as 119862 decreases Figure 12shows that for moderate 119886Pe119903 values 120578119904 may have one localmaximumat a specific 119886Pe119903 valueThe physical interpretationfor this phenomenon is discussed in Section 42
10 Mathematical Problems in Engineering
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
porous tubes or channelsThese works andmany others werethe motivations behind recent works that accounted for allpossible hydrodynamic conditions on heat transfer insidechannels subjected to wall suction For example Sorour etal [13] and Bubnovich et al [14] analyzed dynamically andthermally the developing flow inside a channel subjected tononuniform suction at onewall Hwang et al [15] investigatednumerically forced laminar convection in the entrance regionof a square duct subjected to uniform mass transpirationMakinde et al [1] analyzed heat transfer in channels exposedto wall suction in presence of nanofluids with both wall slipand viscous dissipation effects All of the aforementionedworks and many similar ones in the literature did not explorethe heat transfer enhancement due to surface suction insidepermeable tubes Thus the motivation behind the presentwork is to enrich the literature with a study about the role ofsurface suction and its profile inside permeable tubes on heattransfer enhancement
Promoting flow close to the energy exchanging bound-ary usually results in heat transfer enhancement [2 16]Meanwhile the internal flow impedance near this boundaryincreases as suction Reynolds number (ReV
119908
) increases [911] For permeable circular tubes the internal flow starts toseparate from the boundary when ReV
119908
= 45978 due to theadverse pressure gradient caused by reduction in the meanfluid velocity To avoid this instability condition and to sustainmaximumvelocity close to tube surface the suctionReynoldsnumber must be ReV
119908
le 104 This constraint can be shownusing Kinney [9] and Raithby [11] works to yield universalsurface friction coefficient to be at least equal to 95 of itsmaximum value when ReV
119908
= 0 Accordingly the flow closeto the energy exchanging boundary can be kept maximallypromoted under this constraint Therefore the present workis concerned with heat transfer enhancement inside perme-able tubes exposed to surface suction with 0 lt ReV
119908
le 104In the next sections flow and heat transfer inside a
preamble tube subjected to internal suction flow aremodelledand analyzed The surface suction velocity is considered tohave either linear or exponential profile distributions Bothmomentum and thermal energy transfer equations are solvedusing various analytical and numerical methods Differentheat transfer enhancement indicators are computed Bothanalytical and numerical computations of these indicatorsare validated under an applicable constraint and using earlystudies An extensive parametric study has been conducted inorder to identify and explore the influence of average suctionvelocity suction velocity profile and Peclet number on theheat transfer enhancement indicators
2 Problem Formulation
21 Modeling of Flow and Heat Transfer inside the PermeableTube Consider a tube of length 119871 and inner diameter 119863The tube wall is permeable so as to allow fluid suction at itsboundary as shown in Figure 1 Both the flow inside the tubeand that through the permeable boundary are consideredto be laminar flows The density specific heat thermalconductivity and the dynamic viscosity of the fluid are120588 119888119901 119896 and 120583 respectively The dimensionless continuity
(a)
(b)
L
w(x)
w(x)
r
x u
Fluid flow
D
(c)
Figure 1 (a) 3D viewof the tubewith suction passages embedded onits material volume (b) cross section of the tube and (c) schematicprofile of the tube and the coordinates system
momentum and energy equations of the fluid are given by[17ndash19]
120597119906
120597119909
+
1
119903
120597
120597119903
(119903 V) = 0 (1)
119886Re119903 (119906120597119906
120597119909
+ V120597119906
120597119903
) = 16
119889119901
119889119909
+
2
119903
120597
120597119903
(119903
120597119906
120597119903
) (2)
119886Pe119903 (119906120597120579
120597119909
+ V120597120579
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579
120597119903
) (3)
Mathematical Problems in Engineering 3
where 119906 and V are the dimensionless axial and radialvelocities respectively 119909 and 119903 are the dimensionless axialand radial positions respectively 119886 and Pe119903 are the tubeaspect ratio and the reference Peclet number respectively119901 and 120579 are the dimensionless pressure and dimensionlesstemperature fields respectively The dimensionless variablesand parameters used in (1)ndash(3) are given by
119886 =
2119863
119871
(4a)
119903 =
2119903
119863
(4b)
119909 =
119909
119871
(4c)
119906 (119909 119903) =
119906
119906119900
(5a)
V (119909 119903) =V119886119906119900
(5b)
Re119903 =120588119906119900119863
120583
(6a)
Pe119903 =120588119888119901119906119900119863
119896
= Re119903 Pr (6b)
119906119900 =
1199011 minus 1199012
32120583119871119863
2 (6c)
119901 (119909) =
119901 minus 1199011
1199012 minus 1199011
(7a)
120579 (119909 119903) =
119879 minus 1198791
119902
10158401015840119904119863 (2119896)
(7b)
where 119906119900 1199011 1199012 and 1198791 are reference axial velocity inletpressure outlet pressure and inlet temperature respectively119902
10158401015840
119904is the constant heat flux applied at the inner surface of the
tube The boundary conditions of (1)ndash(3) are given by
119909 = 0 119906avg = 1 (8a)
119909 = 0 120579 = 0 (8b)
119909 = 1 119901 = 1 (9)
119903 = 0 120597119906
120597119903
= 0 (10a)
119903 = 0 120597V120597119903
= 0 (10b)
119903 = 0 120597120579
120597119903
= 0 (10c)
119903 = 1 119906 = 0 (11a)
119903 = 1 V = V119908 (119909) (11b)
119903 = 1 120597120579
120597119903
= 1 (11c)
where V119908(119909) = V119908(119909)(119886119906119900) V119908(119909) is the dimensional localsuction velocity at the tube inner surface 119906avg is the meandimensionless axial velocity at any given cross section UsingKinney [9] and Raithby [11] works it can be shown that with0 le 119886Re119903V119908 le 104 the convective terms in (2) can beneglectedThe aforementioned range results in less than 50relative error associated with calculating the universal wallfriction coefficient by neglecting the convective terms Theprevious constraint can practically be satisfied for high aspectratio tubes (1119886 ≫ 1) viscous fluids or with small suctionvelocities Accordingly the solution of (2) in absence of theconvective terms with boundary conditions given by (10a)and (11a) is the following
119906 (119909 119903) = 2119901
1015840(1 minus 119903
2) (12)
where 1199011015840 = 119889119901119889119909 Substituting (12) in (1) and solving theresulting equation yield to the following distribution of thedimensionless radial velocity
V (119909 119903) = minus119901
10158401015840(119903 minus
119903
3
2
) (13)
where 119901
10158401015840= 119889
2119901119889119909
2 Applying the boundary conditiongiven by (11b) results into the following differential equation
119901
10158401015840= minus2V119908 (119909) (14)
The mass flow rate (119909) and mean axial velocity 119906avg(119909)can be calculated from the following expressions
119872(119909) =
(119909)
119900
= 2int
1
0
119903 119906 119889119903 = 119901
1015840(119909) (15)
119906avg (119909) equiv119906avg (119909)
119906119900
=
(119909) (120588120587119863
24)
119906119900
= 119901
1015840(119909)
(16)
where 119900 is the reference mass flow rate 119900 = 12058811990611990012058711986324
TheNusselt number at the tube inner boundary is definedas
Nu equiv
ℎ119863
119896
=
2
120579119882 minus 120579119898
(17)
where ℎ 120579119898 and 120579119882 are the convection heat transfer coeffi-cient between the tube inner boundary and the fluid flow (ℎ =
119902
10158401015840
119904[119879119882 minus 119879119898]) dimensionless mean bulk temperature and
the dimensionless temperature of the tube inner boundary(120579119882 = 120579(119909 119903 = 1)) respectively 119879119882 and 119879119898 are the tem-perature at the tube inner boundary and the mean bulktemperature of the fluid respectively 120579119898 can be computedfrom the following equation
120579119898 (119909) equiv119879119898 (119909) minus 1198791
119902
10158401015840119904119863 (2119896)
= 4int
1
0
119903 (1 minus 119903
2) 120579 (119909 119903) 119889119903 (18)
The Integral Energy EquationThe integral form of the energyequation can be formed using (3) It can be expressed in thefollowing form
119906avg119889120579119898
119889119909
+ 4(
V119908Nu
) =
4
119886Pe119903 (19)
4 Mathematical Problems in Engineering
The Fully Developed Nusselt Number Under fully developedcondition 120597120579120597119909 = 119889120579119898119889119909 As such (3) can be reduced tothe following using (19)
8 (1 minus 119903
2) 1 minus
119886Pe119903V119908Nufd
+ 2119886Pe119903V119908 (119903 minus119903
3
2
)
120597120579fd120597119903
=
2
119903
120597
120597119903
(119903
120597120579fd120597119903
)
(20)
where Nufd is the fully developed Nusselt number Equation(20) can be solved analytically and Nufd expression can bearranged in the following form
Nufd119885
= (81 minus exp(31198858
)
+ 2119885 exp(31198858
)int
0
minus34
exp (1198851199102)radic119910 + 1
119889119910)
sdot (119885 + 8 1 minus exp(31198858
)
+ 2119885 exp(31198858
)int
0
minus34
exp (1198851199102)radic119910 + 1
119889119910)
minus1
(21)
where 119885 = 119886Pe119903V119908(119909) The following approximation can beused
1
radic119910 + 1
cong 1198861 exp (1198862119910) + 1198863 exp (1198864119910) minus
3
4
le 119910 le 0
(22)
This approximation has maximum relative error less than03 when the coefficients are equal to
1198861 = 98075 times 10
minus3 1198862 = minus53965
1198863 = 09874 1198864 = minus049834
(23)
By substituting (22) in (21) Nufd can be approximated by thefollowing expression
Nufd119885
= (8 1 minus exp(31198858
) minus 1198601 exp (1198865) minus exp(31198858
)
minus1198602 exp (1198866) minus exp(31198858
))
sdot (8 1 minus exp(31198858
) minus 1198601 exp (1198865) minus exp(31198858
)
minus1198602 exp (1198866) minus exp(31198858
) + 119885)
minus1
(24)
where
1198865 = minus(
3
4
) 1198862 1198866 = minus(
3
4
) 1198864
1198601 =41198861119885
21198862 + 119885
1198602 =41198863119885
21198864 + 119885
(25)
211 Case I Linear Distribution of the Suction Velocity Forthis case the dimensionless suction velocity denoted by V119908119860has the following linear distribution
where V119900 = V119900(119886119906119900) V119900 is the average suction velocity overthe length of tube inner surface 119861 is an arbitrary controllingparameter
212 Case II Exponential Distribution of the Suction VelocityFor this case the dimensionless suction velocity denoted byV119908119861 has the following exponential distribution
V119908119861 (119909) = V119900119862 exp (119862119909)exp (119862) minus 1
(27)
where 119862 is an arbitrary controlling parameter
213 The Pressure Gradient for Both Cases Substituting (26)and (27) in (14) and solving for the dimensionless pressuregradient the following expressions can be obtained
119889119901
119889119909
=
1 minus 2V1199001199092+ 2119861 (119909
2minus 119909) Case I
1 minus 2V119900 [exp (119862119909) minus 1
exp (119862) minus 1
] Case II(28)
The ranges of 119861 and V119900 that result in having both 119906avg119860 andV119908119860 larger than zero can be shown to be
0 le V119900 lt1
2
0 le 119861 le 2V119900 (29)
The ranges of 119862 and V119900 that result in having both 119906avg119861 andV119908119861 larger than zero can be shown to be
0 le V119900 lt1
2
minusinfin lt 119862 lt infin (30)
22 Perfect Fluid Slip Case (Ideal Case) The ideal case of thepresent problem is constructed when the fluid is subjectedto perfect slip condition at the solid boundary For this idealcase the conservation of mass and the continuity equationreveal the following expressions
119889119906119904
119889119909
= minus2V119908 (31)
119906119904 =
1 minus 2V1199001199092+ 2119861 (119909
2minus 119909) Case I
1 minus 2V119900 exp (119862119909) minus 1
exp (119862) minus 1
Case II(32)
V = V119908119903 (33)
Mathematical Problems in Engineering 5
where 119906119904 = 119906s119906119900 119906119904 is the fluid velocity for this ideal caseUnder this ideal condition the energy equation reduces tothe following
119886Pe119903 (119906119904120597120579119904
120597119909
+ V119908119903120597120579119904
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904
120597119903
) (34)
where 120579119904 is the dimensionless temperature for the ideal caseFor fully developed condition where 120597120579119904120597119909 = 119889120579119898119889119909 (34)can be reduced to the following when (19) is implemented
41 minus
119885
Nu119904fd + 119885(119903
120597120579119904fd
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904fd
120597119903
) (35)
where Nu119904fd is the fully developed value of the Nusseltnumber for the ideal case By solving (35) and application ofthe boundary condition given by (10c) Nu119904fd can be foundIt is equal to
Nu119904fd119885
=
4 1 minus exp (1198854)4 1 minus exp (1198854) + 119885
(36)
For this case the dimensionless mean bulk temperature canbe found to be equal to
120579119898119904 (119909) equiv
119879119904119898 (119909) minus 1198791
119902
10158401015840119904119863 (2119896)
= 2int
1
0
119903120579119904119889119903 (37)
23 Heat Transfer Enhancement Indicators Let the heattransfer enhancement indicator 120582 be defined as ratio of thetube inner boundary excess temperature at the exit sectionfor the reference case (V119908 = 0) to that quantity when V119908 gt 0Mathematically the enhancement indicator 120582 is written as
120582 =
120579 (119909 = 1 119903 = 1)|V119908=0
120579 (119909 = 1 119903 = 1)
(38)
When V119908 = 0 the system becomes an impermeable tubeconfining an internal flow and subjected to uniform heat fluxFor this case the Nusselt number at the exit can be shown tobe correlated to 119886Pe119903 according to the following expression
Nu (119909 = 1 V119908 = 0)
= (438 minus 13 times 10
minus3119886Pe119903 + 39 times 10
minus4(119886Pe119903)
2
+ 65 times 10
minus7(119886Pe119903)
3)
sdot (1 minus 46 times 10
minus5119886Pe119903 + 86 times 10
minus5(119886Pe119903)
2
+ 43 times 10
minus8(119886Pe119903)
3)
minus1
(39)
with maximum relative error less than 03 when 1 le 119886Pe119903 le1000
The second performance indicator 120578119904 is defined as ratioof the tube inner boundary excess temperature at the exitsection for the perfect fluid slip case to that quantity underno-slip condition Mathematically 120578119904 is written as
24 Enhancement Indicators for Fully Developed Flow withUniform Suction For uniform suction case the mean veloc-ity inside the tube can be found using (32) by setting 119861 = V119900It is equal to
119906119904 = 119906avg = 1 minus 2V119900119909 V119900 lt1
2
(41)
By substituting (41) in (19) and solving the resulting equationthe mean bulk temperature distribution can be obtained It isgiven by
120579119898 (119909) = ln [1 minus 2V119900119909]2([1Nufd]minus1[119886Pe119903V119900])
(42)
Accordingly the dimensionless temperature at the exit isequal to
120579 (119909 = 1 119903 = 1)
= 120579119882 (119909 = 1) = ln [1 minus 2V119900]2([1Nufd]minus1[119886Pe119903V119900])
+
2
Nufd
(43)
Therefore the performance indicators 120582 and 120578119904 for this caseare equal to
120582fd =96 + 11119886Pe119903
ln [1 minus 2V119900]48([119886Pe
119903Nufd]minus1V119900)
+ 48 (119886Pe119903Nufd)
(44)
120578119904fd =ln [1 minus 2V119900]
2([119886Pe119903Nu119904fd]minus1V119900)
+ 2 (119886Pe119903Nu119904fd)
ln [1 minus 2V119900]2([119886Pe
119903Nufd]minus1V119900)
+ 2 (119886Pe119903Nufd)
(45)
The plots of 120579119898(119909 = 1) 120579119904119898(119909 = 1) Nufd Nu119904fd and 120582fdfor various 119886Pe119903 and V119900 values are seen in Figures 2ndash4 Theupper value of V119900 that makes 120582fd = 1 can be obtained usingnumerical solving techniques [20] It is denoted by V119900119901 V119900119901can be correlated to 119886Pe119903 through the following correlation
V119900119901 = (minus28076 times 10
6+ 820470 (119886Pe119903)
minus 83102 (119886Pe119903)2+ 18754 (119886Pe119903)
3)
sdot (1 + 713480 (119886Pe119903) minus 123760 (119886Pe119903)2
+ 37172 (119886Pe119903)3)
minus1
(46)
The percentage error associated with (46) is less than 062when 119886Pe119903 ge 4 The plot of 119886Pe119903 versus V119900119901 is shown in Fig-ure 3 Surprisingly the value of V119900 making 120578119904fd = 1 is foundto be independent on 119886Pe119903 This value is denoted by V119900119890 andit is equal to
V119900119890 = 031606 (47)
The plots of 120578119904fd for various 119886Pe119903 and V119900 are seen in Figure 4120578119904fd ge 1 when V119900 gt V119900119890
6 Mathematical Problems in Engineering
000 010 020 030 040 05020E minus 03
20E minus 02
20E minus 01
20E + 00
20E + 01
Fully developed flow condition
120579m(x
=1)120579
sm(x
=1)
120579m(x = 1)
120579sm(x = 1)
aPer = 1 2 4 8 16 32
w = o
o
Figure 2 Effects of V119900 on outlet mean bulk temperature under fullydeveloped flow condition
3 Numerical Methodology and Results
Equations (3) and (35) are solvable numerically without iter-ations using the implicit finite difference method discussedby Khaled [21] and Blottner [22]They were discretized usingtwo-point backward and three-point central differencingquotients for the first derivative with respect to 119909-direc-tion and both first and second derivatives with respect to119903-direction respectively The finite difference equation of(3) is given by
The pair (119894 119895) represents the location of the discretized pointin the numerical grid of the fluid domain The number 119899represents the total number of 119895-nodes along each 119894-sectionThe tube inner surface is located in the numerical grid at(119894 119895 = 119899) while the center of the tube is located in thenumerical grid at (119894 119895 = 1) The resulting 119899 minus 1 (119899 = 401)tridiagonal systemof algebraic equations obtained by (48) at agiven 119894-section was solved usingThomas algorithm [22] Theprevious procedure step was repeated for the consecutive 119894-values until 119894 reached the value 119898 (119898 = 2001) at which 119909 =
1 Numerical investigations were performed using differentmesh sizes to assess and ascertain grid-size independentresults Furthermore doubling the values of119898 and 119899 resultedin less than 04 error in the calculated parameters formoderate 119886Pe119903 values
The results of the present work are shown in Figures 2ndash14 Those obtained using the described numerical method
000
009
018
027
036
045
054
10E minus 01
10E + 02
10E + 02
10E + 01
10E + 01
10E + 00
10E + 00
aPerw aPer
op
Nu f
dN
u sfd
Nufd minus aPerwNusfd minus aPerwop minus aPer
Figure 3 Effects of 119886Pe119903 on fully developed Nusselt numbers andV119900119901
060
074
088
102
116
130
144
000
115
230
345
460
575
690
000 010 020 030 040 050
w = o
Fully developed flow condition
aPer = 1 2 4 8 16 32 50
120582fd
120582fd
120578s
fd
120578sfd
o
Figure 4 Effects of V119900 on the first and second performance indica-tors under fully developed flow condition
are shown in Figures 5ndash14 The numerical results for the casewith 119886Pe119903 = 1 shown in Figures 4ndash8 were compared withthe analytical solution utilizing (17) (24) (26) (27) (38) and(40) The numerical results match well with the analyticalsolutions since the case with 119886Pe119903 = 1 results in fullydeveloped flow condition at the tube exit Also it is shownfrom Figure 3 that the fully developed Nusselt number when119886Pe119903V119908 = 10 is equal to Nufd = 10725 This quantity canbe shown from Figure 9 of Raithby [11] work to be equal toNufd = 1083 when 119886Re119903V119908 rarr 0 The relative error betweenthese values is 098 which is very small These validationsincreased the confident levels in the obtained results
4 Discussion of the Results
41 The System Thermal Performance for Uniformly Dis-tributed Suction Flow As suction velocity V119908 increases theoutlet mean axial velocity 119906119898(119909 = 119871) decreases causing
Mathematical Problems in Engineering 7
010
045
080
115
150
185
220
000 020 040 060 080 10010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
B
o = 025 033 049
Figure 5 Effects of 119861 on performance indicator 120582 and outlet meanbulk temperature for Case I
085
095
105
115
125
135
145
000 020 040 060 080 100
120578s
B
o = 025 033 049
o = 025
o = 049
3o = 1
aPer = 1 numericalaPer = 1 analyticalaPer = 10
Figure 6 Effects of 119861 on performance indicator 120578119904 for Case I
01
10
100
10E + 0210E + 0110E + 00
aPerB = 0B = o
B = 2o
120582
o = 025 033 049
o = 025 033 049
(44)
Figure 7 Effects of 119886Pe119903 on performance indicators 120582 for various 119861values for Case I
066
080
094
108
122
136
150
10E + 0210E + 0110E + 00
aPer
120578s
B = 0B = o
B = 2o
o = 025 033 049
(45)
Figure 8 Effects of 119886Pe119903 on performance indicators 120578119904 for various 119861values for Case I
010
048
085
123
160
198
235
minus200 minus120 minus040 040 120 20010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
C
o = 025 033 049
o = 025 033 049
Figure 9 Effects of 119862 on performance indicator 120582 and outlet meanbulk temperature for Case II
087
097
107
117
127
137
147
minus200 minus120 minus040 040 120 200C
120578s
aPer = 1 numericalaPer = 1 analyticalaPer = 10
o = 025 033 049
o = 025
o = 049
3o = 1
Figure 10 Effects of 119862 on performance indicator 120578119904 for Case II
8 Mathematical Problems in Engineering
01
10
100
10E + 0210E + 0110E + 00
aPer
120582
C = 20
C = 0
C = minus20
o = 025 033 049
o = 025 033 049
(44)
Figure 11 Effects of 119886Pe119903 on performance indicators 120582 for various119862values for Case II
075
088
101
114
127
140
153
10E + 0210E + 0110E + 00
aPerC = 20
C = 0
C = minus20
120578s
o = 025 033 049
(45)
Figure 12 Effects of 119886Pe119903 on performance indicators 120578119904 for various119862 values for Case II
the axial convection which is proportional to 119886Pe119903119906avg119889120579119898119909to decrease This may cause an increase in the outlet meanbulk temperature 119879119898(119909 = 119871) as shown in Figure 2 forsmall 119886Pe119903 values 119886Pe119903 le 4 When 119886Pe119903 ge 8 the trans-verse convection becomes significant as it is proportionalto 2119886Pe119903V119908(120579119882 minus 120579119898) The latter expression is equivalent to4119886Pe119903V119908Nu This expression increases as 119886Pe119903V119900 increasesThis increase causes significant reduction in the temperatureboundary layer thicknesswhich causes reduction in the outletmean bulk temperature as shown in Figure 2 The sum of theaxial and transverse convection is constant which is equalto the total convection heat transfer rate given by 119902conv =
119902
10158401015840
119904120587119863119871 When V119900 approaches V119900 = 05 the axial advection
which is proportional to 119906avg approaches zero at the tubeoutlet This causes 119879119898(119909 = 119871) to increase apparently as V119900approaches V119900 = 05 asymptotically as can be seen fromFigure 2
Linear suction
Exponential suction
Linear suctionprofile
Exponential suctionprofile
0000
0085
0170
0255
0340
0425
0510
10
100
oc
oc
120582m
ax
120582max
10E + 0210E + 0110E + 00
aPer
profile
profile
Figure 13 Effects of 119886Pe119903 on 120582max and V119900119888 for both Case I and CaseII
Figure 14 Effects of 119886Pe119903 on 120578119904119888 for both Case I and Case II
The Nusselt number for perfect fluid slip at the solidboundary (36) case is larger than that for the no-slip condi-tion case (24) as shown in Figure 3 that is Nu119904 gt Nu Thisindicates that the tube excess temperature at the boundary forthe perfect fluid slip condition is smaller than that for the no-slip condition case That excess temperature is proportionalto 120579119882 minus 120579119898 Consequently the transverse convection due tofluid slip is smaller than that due to no-slip condition as thisconvection is proportional to both 119886Pe119903V119908 and 120579119882 minus 120579119898 As aresult the axial convection due to fluid slip condition is largerthan that due to no-slip condition as the total convectionheat transfer rate is constant Therefore 119879119898(119909 = 119871) for thefluid slip condition is always larger than that for the no-slipcondition as shown in Figure 2 It is shown from Figure 3 thatthe asymptotic value of Nufd is equal to Nufd = 119886Pe119903V119908 Thisfinding is shown in the works of Kinney [9] and Raithby [11]
Figure 4 shows that the first heat transfer performanceindicator 120582fd is larger than one when 0 lt V119900 lt V119900119901
Mathematical Problems in Engineering 9
where the variation of V119900119901 with 119886Pe119903 is shown in Figure 3and given by (46) It can be shown from Figure 3 that when119886Pe119903 ge 978 V119900119901 ge 0495 It should be mentioned herethat having 120582fd gt 1 indicates that the heat transfer to thepermeable tube exposed to suction flow results in havinglower tube outlet temperature than that for the impermeabletube Consequently more thermal energy can be transferredfrom the permeable tube to the fluid in order to raise itsoutlet temperature to reach that of the impermeable tubeFurthermore Figure 4 shows that the second heat transferperformance indicator 120578119904fd becomes above one when V119900 gt
V119900119890 = 031606 Having 120578119904fd gt 1 indicates that thepermeable tube subjected to no-slip condition has smalleroutlet boundary temperature than that for the permeable tubesubjected to perfect fluid slip at the solid boundary This isnot the case for impermeable tubes as 120578119904fd lt 1 when V119900 = 0
as shown in Figure 4
42 The SystemThermal Performance for Linearly DistributedSuction Flow When 119861 = 0 the suction flow increaseslinearly from V119908 = 0 at the inlet (119909 = 0) to maximumvalue of V119908 = 2V119900 at the tube outlet (119909 = 1) For thiscase the thermal entry region effect is maximum as 119906avgis maximum at inlet In this region the axial convectioncoefficient is substantial as it is the coldest fluid region insidethe tube The thermal entry region influences the flow bycausing reductions in the boundary excess temperatures nearthe inlet Also the transverse convection is maximum at thetube outlet when 119861 = 0 as the suction velocity is maximumthere This decreases the boundary excess temperatures nearthe tube outlet As 119861 increases 119906avg close to the tube inletdecreases due to the increase in V119908 and V119908 close to thetube outlet decreases Accordingly both thermal entry regioneffect and transverse convection at the tube outlet decrease as119861 increases Both effects tend to increase both the boundaryexcess temperature and outlet mean bulk temperature As aresult 120579119898(119909 = 1) increases and 120582 decreases as 119861 increases asshown in Figure 5
For large 119886Pe119903 values both thermal entry region effectand the transverse convection at the tube outlet are largerthan those for smaller 119886Pe119903 values Accordingly 120582 valueswhen 119886Pe119903 = 10 are larger than those when 119886Pe119903 = 1
as shown in Figure 5 When 119886Pe119903 = 10 and 119861 = 0 it isnoticed that 120582 increases as V119900 increases This is because theoutlet mean bulk temperature decreases as V119900 increases forlarge 119886Pe119903 values as seen from Figure 2 Figure 5 shows that120582 values for V119900 = 13 plot are larger than those for V119900 = 049
plot when 119861 gt 046 This indicates that the convection dueto thermal entrance region effect as compared to transverseconvection increases as V119900 decreases when 119861 gt 046 and119886Pe119903 = 10 When 119886Pe119903 = 10 120578119904 is noticed from Figure 6to decrease as 119861 increases It is because that both thermalentry region effect and V119908 at the tube outlet decrease as 119861increases The former effect is due to the increase in V119908 nearthe tube inlet as 119861 increases Both effects tend to increase theexcess of Nu119904 above Nu Accordingly transverse convectiondue to fluid slip condition decreases due to the decrease inthe boundary excess temperature Consequently the axialconvection due to the fluid slip condition is increased more
than that due to the no-slip condition This causes 120579119904119898 toincrease as compared to 120579119898Thus 120578119904 decreases as 119861 increases
Figure 7 shows that 120582 increases as 119886Pe119903 increases At small119886Pe119903 values the transverse convection is negligible and Nu119904approaches Nu for large 119886Pe119903 values These lead to have 120578119904approaching one either as 119886Pe119903 decreases towards 119886Pe119903 =
0 or as 119886Pe119903 increases above 119886Pe119903 = 100 These trendscan be seen from Figure 8 For moderate 119886Pe119903 values theincrease in 119886Pe119903 not only causes an increase in the transverseconvection but it also results in reducing the boundary excesstemperature due to the increases in Nu This reduction ismore distinct for the case of the perfect fluid slip condition asNu119904 gt Nu The combined aforementioned effects can resultin reduction of transverse convection as 119886Pe119903 increases for theperfect fluid slip case It is because the transverse convectionis proportional to 119886Pe119903V119908(120579119882 minus 120579119898) as mentioned beforeAccordingly 120578119904 may have one local maximum at a specific119886Pe119903 value as clearly shown in Figure 8 for V119900 = 049 plots
43 The System Thermal Performance for Exponentially Dis-tributed Suction Flow For the upper 119862 value (119862 = 2)the thermal entry region effect is maximum since 119906avg ismaximum at tube inlet because V119908 is minimum there Alsothe transverse convection at the tube outlet is maximumas V119908 is maximum there These effects tend to decreasethe boundary excess temperatures at the tube outlet As 119862decreases 119906avg close to the tube inlet decreases due to theincrease in V119908 and V119908 close to the tube outlet decreasesAccordingly both thermal entry region effect and trans-verse convection at the tube outlet decrease as 119862 decreasesBoth effects increase both the boundary excess temperatureand outlet mean bulk temperature As a result the 120579119898(119909 =
1) increases and 120582 decreases as 119862 decreases as shown inFigure 9
For large 119886Pe119903 values both thermal entry region effectand outlet transverse convection are larger than those forsmaller 119886Pe119903 values Accordingly 120582 values when 119886Pe119903 = 10
are larger than those when 119886Pe119903 = 1 as shown in Figure 9When 119886Pe119903 = 10 and 119862 = 2 it is noticed that 120582 increases asV119900 increases It is because the outlet mean bulk temperaturedecreases as V119900 increases for large 119886Pe119903 values as shown fromFigure 2 Figure 9 shows that 120582 values for V119900 = 13 plot arelarger than those for V119900 = 049 plot when 119862 lt 06 Thisindicates that the flow gets more dominated by the thermalentry region effect than by the transverse convection as V119900decreases when 119862 lt 06 and 119886Pe119903 = 10 Figure 11 showsthat 120582 increases as 119886Pe119903 increases When 119886Pe119903 = 10 120578119904is noticed from Figure 10 to decrease as 119862 decreases It isbecause both thermal entry region effect and V119908 at the outletdecrease as 119862 decreases As indicated previously both effectstend to increase the excess of Nu119904 above Nu Accordingly thetransverse convection due to fluid slip condition decreasesdue to the decrease in the boundary excess temperatureConsequently the axial convection due to fluid slip conditionis increased more than that due to no-slip condition causing120579119904119898 to increase Thus 120578119904 decreases as 119862 decreases Figure 12shows that for moderate 119886Pe119903 values 120578119904 may have one localmaximumat a specific 119886Pe119903 valueThe physical interpretationfor this phenomenon is discussed in Section 42
10 Mathematical Problems in Engineering
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
where 119906 and V are the dimensionless axial and radialvelocities respectively 119909 and 119903 are the dimensionless axialand radial positions respectively 119886 and Pe119903 are the tubeaspect ratio and the reference Peclet number respectively119901 and 120579 are the dimensionless pressure and dimensionlesstemperature fields respectively The dimensionless variablesand parameters used in (1)ndash(3) are given by
119886 =
2119863
119871
(4a)
119903 =
2119903
119863
(4b)
119909 =
119909
119871
(4c)
119906 (119909 119903) =
119906
119906119900
(5a)
V (119909 119903) =V119886119906119900
(5b)
Re119903 =120588119906119900119863
120583
(6a)
Pe119903 =120588119888119901119906119900119863
119896
= Re119903 Pr (6b)
119906119900 =
1199011 minus 1199012
32120583119871119863
2 (6c)
119901 (119909) =
119901 minus 1199011
1199012 minus 1199011
(7a)
120579 (119909 119903) =
119879 minus 1198791
119902
10158401015840119904119863 (2119896)
(7b)
where 119906119900 1199011 1199012 and 1198791 are reference axial velocity inletpressure outlet pressure and inlet temperature respectively119902
10158401015840
119904is the constant heat flux applied at the inner surface of the
tube The boundary conditions of (1)ndash(3) are given by
119909 = 0 119906avg = 1 (8a)
119909 = 0 120579 = 0 (8b)
119909 = 1 119901 = 1 (9)
119903 = 0 120597119906
120597119903
= 0 (10a)
119903 = 0 120597V120597119903
= 0 (10b)
119903 = 0 120597120579
120597119903
= 0 (10c)
119903 = 1 119906 = 0 (11a)
119903 = 1 V = V119908 (119909) (11b)
119903 = 1 120597120579
120597119903
= 1 (11c)
where V119908(119909) = V119908(119909)(119886119906119900) V119908(119909) is the dimensional localsuction velocity at the tube inner surface 119906avg is the meandimensionless axial velocity at any given cross section UsingKinney [9] and Raithby [11] works it can be shown that with0 le 119886Re119903V119908 le 104 the convective terms in (2) can beneglectedThe aforementioned range results in less than 50relative error associated with calculating the universal wallfriction coefficient by neglecting the convective terms Theprevious constraint can practically be satisfied for high aspectratio tubes (1119886 ≫ 1) viscous fluids or with small suctionvelocities Accordingly the solution of (2) in absence of theconvective terms with boundary conditions given by (10a)and (11a) is the following
119906 (119909 119903) = 2119901
1015840(1 minus 119903
2) (12)
where 1199011015840 = 119889119901119889119909 Substituting (12) in (1) and solving theresulting equation yield to the following distribution of thedimensionless radial velocity
V (119909 119903) = minus119901
10158401015840(119903 minus
119903
3
2
) (13)
where 119901
10158401015840= 119889
2119901119889119909
2 Applying the boundary conditiongiven by (11b) results into the following differential equation
119901
10158401015840= minus2V119908 (119909) (14)
The mass flow rate (119909) and mean axial velocity 119906avg(119909)can be calculated from the following expressions
119872(119909) =
(119909)
119900
= 2int
1
0
119903 119906 119889119903 = 119901
1015840(119909) (15)
119906avg (119909) equiv119906avg (119909)
119906119900
=
(119909) (120588120587119863
24)
119906119900
= 119901
1015840(119909)
(16)
where 119900 is the reference mass flow rate 119900 = 12058811990611990012058711986324
TheNusselt number at the tube inner boundary is definedas
Nu equiv
ℎ119863
119896
=
2
120579119882 minus 120579119898
(17)
where ℎ 120579119898 and 120579119882 are the convection heat transfer coeffi-cient between the tube inner boundary and the fluid flow (ℎ =
119902
10158401015840
119904[119879119882 minus 119879119898]) dimensionless mean bulk temperature and
the dimensionless temperature of the tube inner boundary(120579119882 = 120579(119909 119903 = 1)) respectively 119879119882 and 119879119898 are the tem-perature at the tube inner boundary and the mean bulktemperature of the fluid respectively 120579119898 can be computedfrom the following equation
120579119898 (119909) equiv119879119898 (119909) minus 1198791
119902
10158401015840119904119863 (2119896)
= 4int
1
0
119903 (1 minus 119903
2) 120579 (119909 119903) 119889119903 (18)
The Integral Energy EquationThe integral form of the energyequation can be formed using (3) It can be expressed in thefollowing form
119906avg119889120579119898
119889119909
+ 4(
V119908Nu
) =
4
119886Pe119903 (19)
4 Mathematical Problems in Engineering
The Fully Developed Nusselt Number Under fully developedcondition 120597120579120597119909 = 119889120579119898119889119909 As such (3) can be reduced tothe following using (19)
8 (1 minus 119903
2) 1 minus
119886Pe119903V119908Nufd
+ 2119886Pe119903V119908 (119903 minus119903
3
2
)
120597120579fd120597119903
=
2
119903
120597
120597119903
(119903
120597120579fd120597119903
)
(20)
where Nufd is the fully developed Nusselt number Equation(20) can be solved analytically and Nufd expression can bearranged in the following form
Nufd119885
= (81 minus exp(31198858
)
+ 2119885 exp(31198858
)int
0
minus34
exp (1198851199102)radic119910 + 1
119889119910)
sdot (119885 + 8 1 minus exp(31198858
)
+ 2119885 exp(31198858
)int
0
minus34
exp (1198851199102)radic119910 + 1
119889119910)
minus1
(21)
where 119885 = 119886Pe119903V119908(119909) The following approximation can beused
1
radic119910 + 1
cong 1198861 exp (1198862119910) + 1198863 exp (1198864119910) minus
3
4
le 119910 le 0
(22)
This approximation has maximum relative error less than03 when the coefficients are equal to
1198861 = 98075 times 10
minus3 1198862 = minus53965
1198863 = 09874 1198864 = minus049834
(23)
By substituting (22) in (21) Nufd can be approximated by thefollowing expression
Nufd119885
= (8 1 minus exp(31198858
) minus 1198601 exp (1198865) minus exp(31198858
)
minus1198602 exp (1198866) minus exp(31198858
))
sdot (8 1 minus exp(31198858
) minus 1198601 exp (1198865) minus exp(31198858
)
minus1198602 exp (1198866) minus exp(31198858
) + 119885)
minus1
(24)
where
1198865 = minus(
3
4
) 1198862 1198866 = minus(
3
4
) 1198864
1198601 =41198861119885
21198862 + 119885
1198602 =41198863119885
21198864 + 119885
(25)
211 Case I Linear Distribution of the Suction Velocity Forthis case the dimensionless suction velocity denoted by V119908119860has the following linear distribution
where V119900 = V119900(119886119906119900) V119900 is the average suction velocity overthe length of tube inner surface 119861 is an arbitrary controllingparameter
212 Case II Exponential Distribution of the Suction VelocityFor this case the dimensionless suction velocity denoted byV119908119861 has the following exponential distribution
V119908119861 (119909) = V119900119862 exp (119862119909)exp (119862) minus 1
(27)
where 119862 is an arbitrary controlling parameter
213 The Pressure Gradient for Both Cases Substituting (26)and (27) in (14) and solving for the dimensionless pressuregradient the following expressions can be obtained
119889119901
119889119909
=
1 minus 2V1199001199092+ 2119861 (119909
2minus 119909) Case I
1 minus 2V119900 [exp (119862119909) minus 1
exp (119862) minus 1
] Case II(28)
The ranges of 119861 and V119900 that result in having both 119906avg119860 andV119908119860 larger than zero can be shown to be
0 le V119900 lt1
2
0 le 119861 le 2V119900 (29)
The ranges of 119862 and V119900 that result in having both 119906avg119861 andV119908119861 larger than zero can be shown to be
0 le V119900 lt1
2
minusinfin lt 119862 lt infin (30)
22 Perfect Fluid Slip Case (Ideal Case) The ideal case of thepresent problem is constructed when the fluid is subjectedto perfect slip condition at the solid boundary For this idealcase the conservation of mass and the continuity equationreveal the following expressions
119889119906119904
119889119909
= minus2V119908 (31)
119906119904 =
1 minus 2V1199001199092+ 2119861 (119909
2minus 119909) Case I
1 minus 2V119900 exp (119862119909) minus 1
exp (119862) minus 1
Case II(32)
V = V119908119903 (33)
Mathematical Problems in Engineering 5
where 119906119904 = 119906s119906119900 119906119904 is the fluid velocity for this ideal caseUnder this ideal condition the energy equation reduces tothe following
119886Pe119903 (119906119904120597120579119904
120597119909
+ V119908119903120597120579119904
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904
120597119903
) (34)
where 120579119904 is the dimensionless temperature for the ideal caseFor fully developed condition where 120597120579119904120597119909 = 119889120579119898119889119909 (34)can be reduced to the following when (19) is implemented
41 minus
119885
Nu119904fd + 119885(119903
120597120579119904fd
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904fd
120597119903
) (35)
where Nu119904fd is the fully developed value of the Nusseltnumber for the ideal case By solving (35) and application ofthe boundary condition given by (10c) Nu119904fd can be foundIt is equal to
Nu119904fd119885
=
4 1 minus exp (1198854)4 1 minus exp (1198854) + 119885
(36)
For this case the dimensionless mean bulk temperature canbe found to be equal to
120579119898119904 (119909) equiv
119879119904119898 (119909) minus 1198791
119902
10158401015840119904119863 (2119896)
= 2int
1
0
119903120579119904119889119903 (37)
23 Heat Transfer Enhancement Indicators Let the heattransfer enhancement indicator 120582 be defined as ratio of thetube inner boundary excess temperature at the exit sectionfor the reference case (V119908 = 0) to that quantity when V119908 gt 0Mathematically the enhancement indicator 120582 is written as
120582 =
120579 (119909 = 1 119903 = 1)|V119908=0
120579 (119909 = 1 119903 = 1)
(38)
When V119908 = 0 the system becomes an impermeable tubeconfining an internal flow and subjected to uniform heat fluxFor this case the Nusselt number at the exit can be shown tobe correlated to 119886Pe119903 according to the following expression
Nu (119909 = 1 V119908 = 0)
= (438 minus 13 times 10
minus3119886Pe119903 + 39 times 10
minus4(119886Pe119903)
2
+ 65 times 10
minus7(119886Pe119903)
3)
sdot (1 minus 46 times 10
minus5119886Pe119903 + 86 times 10
minus5(119886Pe119903)
2
+ 43 times 10
minus8(119886Pe119903)
3)
minus1
(39)
with maximum relative error less than 03 when 1 le 119886Pe119903 le1000
The second performance indicator 120578119904 is defined as ratioof the tube inner boundary excess temperature at the exitsection for the perfect fluid slip case to that quantity underno-slip condition Mathematically 120578119904 is written as
24 Enhancement Indicators for Fully Developed Flow withUniform Suction For uniform suction case the mean veloc-ity inside the tube can be found using (32) by setting 119861 = V119900It is equal to
119906119904 = 119906avg = 1 minus 2V119900119909 V119900 lt1
2
(41)
By substituting (41) in (19) and solving the resulting equationthe mean bulk temperature distribution can be obtained It isgiven by
120579119898 (119909) = ln [1 minus 2V119900119909]2([1Nufd]minus1[119886Pe119903V119900])
(42)
Accordingly the dimensionless temperature at the exit isequal to
120579 (119909 = 1 119903 = 1)
= 120579119882 (119909 = 1) = ln [1 minus 2V119900]2([1Nufd]minus1[119886Pe119903V119900])
+
2
Nufd
(43)
Therefore the performance indicators 120582 and 120578119904 for this caseare equal to
120582fd =96 + 11119886Pe119903
ln [1 minus 2V119900]48([119886Pe
119903Nufd]minus1V119900)
+ 48 (119886Pe119903Nufd)
(44)
120578119904fd =ln [1 minus 2V119900]
2([119886Pe119903Nu119904fd]minus1V119900)
+ 2 (119886Pe119903Nu119904fd)
ln [1 minus 2V119900]2([119886Pe
119903Nufd]minus1V119900)
+ 2 (119886Pe119903Nufd)
(45)
The plots of 120579119898(119909 = 1) 120579119904119898(119909 = 1) Nufd Nu119904fd and 120582fdfor various 119886Pe119903 and V119900 values are seen in Figures 2ndash4 Theupper value of V119900 that makes 120582fd = 1 can be obtained usingnumerical solving techniques [20] It is denoted by V119900119901 V119900119901can be correlated to 119886Pe119903 through the following correlation
V119900119901 = (minus28076 times 10
6+ 820470 (119886Pe119903)
minus 83102 (119886Pe119903)2+ 18754 (119886Pe119903)
3)
sdot (1 + 713480 (119886Pe119903) minus 123760 (119886Pe119903)2
+ 37172 (119886Pe119903)3)
minus1
(46)
The percentage error associated with (46) is less than 062when 119886Pe119903 ge 4 The plot of 119886Pe119903 versus V119900119901 is shown in Fig-ure 3 Surprisingly the value of V119900 making 120578119904fd = 1 is foundto be independent on 119886Pe119903 This value is denoted by V119900119890 andit is equal to
V119900119890 = 031606 (47)
The plots of 120578119904fd for various 119886Pe119903 and V119900 are seen in Figure 4120578119904fd ge 1 when V119900 gt V119900119890
6 Mathematical Problems in Engineering
000 010 020 030 040 05020E minus 03
20E minus 02
20E minus 01
20E + 00
20E + 01
Fully developed flow condition
120579m(x
=1)120579
sm(x
=1)
120579m(x = 1)
120579sm(x = 1)
aPer = 1 2 4 8 16 32
w = o
o
Figure 2 Effects of V119900 on outlet mean bulk temperature under fullydeveloped flow condition
3 Numerical Methodology and Results
Equations (3) and (35) are solvable numerically without iter-ations using the implicit finite difference method discussedby Khaled [21] and Blottner [22]They were discretized usingtwo-point backward and three-point central differencingquotients for the first derivative with respect to 119909-direc-tion and both first and second derivatives with respect to119903-direction respectively The finite difference equation of(3) is given by
The pair (119894 119895) represents the location of the discretized pointin the numerical grid of the fluid domain The number 119899represents the total number of 119895-nodes along each 119894-sectionThe tube inner surface is located in the numerical grid at(119894 119895 = 119899) while the center of the tube is located in thenumerical grid at (119894 119895 = 1) The resulting 119899 minus 1 (119899 = 401)tridiagonal systemof algebraic equations obtained by (48) at agiven 119894-section was solved usingThomas algorithm [22] Theprevious procedure step was repeated for the consecutive 119894-values until 119894 reached the value 119898 (119898 = 2001) at which 119909 =
1 Numerical investigations were performed using differentmesh sizes to assess and ascertain grid-size independentresults Furthermore doubling the values of119898 and 119899 resultedin less than 04 error in the calculated parameters formoderate 119886Pe119903 values
The results of the present work are shown in Figures 2ndash14 Those obtained using the described numerical method
000
009
018
027
036
045
054
10E minus 01
10E + 02
10E + 02
10E + 01
10E + 01
10E + 00
10E + 00
aPerw aPer
op
Nu f
dN
u sfd
Nufd minus aPerwNusfd minus aPerwop minus aPer
Figure 3 Effects of 119886Pe119903 on fully developed Nusselt numbers andV119900119901
060
074
088
102
116
130
144
000
115
230
345
460
575
690
000 010 020 030 040 050
w = o
Fully developed flow condition
aPer = 1 2 4 8 16 32 50
120582fd
120582fd
120578s
fd
120578sfd
o
Figure 4 Effects of V119900 on the first and second performance indica-tors under fully developed flow condition
are shown in Figures 5ndash14 The numerical results for the casewith 119886Pe119903 = 1 shown in Figures 4ndash8 were compared withthe analytical solution utilizing (17) (24) (26) (27) (38) and(40) The numerical results match well with the analyticalsolutions since the case with 119886Pe119903 = 1 results in fullydeveloped flow condition at the tube exit Also it is shownfrom Figure 3 that the fully developed Nusselt number when119886Pe119903V119908 = 10 is equal to Nufd = 10725 This quantity canbe shown from Figure 9 of Raithby [11] work to be equal toNufd = 1083 when 119886Re119903V119908 rarr 0 The relative error betweenthese values is 098 which is very small These validationsincreased the confident levels in the obtained results
4 Discussion of the Results
41 The System Thermal Performance for Uniformly Dis-tributed Suction Flow As suction velocity V119908 increases theoutlet mean axial velocity 119906119898(119909 = 119871) decreases causing
Mathematical Problems in Engineering 7
010
045
080
115
150
185
220
000 020 040 060 080 10010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
B
o = 025 033 049
Figure 5 Effects of 119861 on performance indicator 120582 and outlet meanbulk temperature for Case I
085
095
105
115
125
135
145
000 020 040 060 080 100
120578s
B
o = 025 033 049
o = 025
o = 049
3o = 1
aPer = 1 numericalaPer = 1 analyticalaPer = 10
Figure 6 Effects of 119861 on performance indicator 120578119904 for Case I
01
10
100
10E + 0210E + 0110E + 00
aPerB = 0B = o
B = 2o
120582
o = 025 033 049
o = 025 033 049
(44)
Figure 7 Effects of 119886Pe119903 on performance indicators 120582 for various 119861values for Case I
066
080
094
108
122
136
150
10E + 0210E + 0110E + 00
aPer
120578s
B = 0B = o
B = 2o
o = 025 033 049
(45)
Figure 8 Effects of 119886Pe119903 on performance indicators 120578119904 for various 119861values for Case I
010
048
085
123
160
198
235
minus200 minus120 minus040 040 120 20010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
C
o = 025 033 049
o = 025 033 049
Figure 9 Effects of 119862 on performance indicator 120582 and outlet meanbulk temperature for Case II
087
097
107
117
127
137
147
minus200 minus120 minus040 040 120 200C
120578s
aPer = 1 numericalaPer = 1 analyticalaPer = 10
o = 025 033 049
o = 025
o = 049
3o = 1
Figure 10 Effects of 119862 on performance indicator 120578119904 for Case II
8 Mathematical Problems in Engineering
01
10
100
10E + 0210E + 0110E + 00
aPer
120582
C = 20
C = 0
C = minus20
o = 025 033 049
o = 025 033 049
(44)
Figure 11 Effects of 119886Pe119903 on performance indicators 120582 for various119862values for Case II
075
088
101
114
127
140
153
10E + 0210E + 0110E + 00
aPerC = 20
C = 0
C = minus20
120578s
o = 025 033 049
(45)
Figure 12 Effects of 119886Pe119903 on performance indicators 120578119904 for various119862 values for Case II
the axial convection which is proportional to 119886Pe119903119906avg119889120579119898119909to decrease This may cause an increase in the outlet meanbulk temperature 119879119898(119909 = 119871) as shown in Figure 2 forsmall 119886Pe119903 values 119886Pe119903 le 4 When 119886Pe119903 ge 8 the trans-verse convection becomes significant as it is proportionalto 2119886Pe119903V119908(120579119882 minus 120579119898) The latter expression is equivalent to4119886Pe119903V119908Nu This expression increases as 119886Pe119903V119900 increasesThis increase causes significant reduction in the temperatureboundary layer thicknesswhich causes reduction in the outletmean bulk temperature as shown in Figure 2 The sum of theaxial and transverse convection is constant which is equalto the total convection heat transfer rate given by 119902conv =
119902
10158401015840
119904120587119863119871 When V119900 approaches V119900 = 05 the axial advection
which is proportional to 119906avg approaches zero at the tubeoutlet This causes 119879119898(119909 = 119871) to increase apparently as V119900approaches V119900 = 05 asymptotically as can be seen fromFigure 2
Linear suction
Exponential suction
Linear suctionprofile
Exponential suctionprofile
0000
0085
0170
0255
0340
0425
0510
10
100
oc
oc
120582m
ax
120582max
10E + 0210E + 0110E + 00
aPer
profile
profile
Figure 13 Effects of 119886Pe119903 on 120582max and V119900119888 for both Case I and CaseII
Figure 14 Effects of 119886Pe119903 on 120578119904119888 for both Case I and Case II
The Nusselt number for perfect fluid slip at the solidboundary (36) case is larger than that for the no-slip condi-tion case (24) as shown in Figure 3 that is Nu119904 gt Nu Thisindicates that the tube excess temperature at the boundary forthe perfect fluid slip condition is smaller than that for the no-slip condition case That excess temperature is proportionalto 120579119882 minus 120579119898 Consequently the transverse convection due tofluid slip is smaller than that due to no-slip condition as thisconvection is proportional to both 119886Pe119903V119908 and 120579119882 minus 120579119898 As aresult the axial convection due to fluid slip condition is largerthan that due to no-slip condition as the total convectionheat transfer rate is constant Therefore 119879119898(119909 = 119871) for thefluid slip condition is always larger than that for the no-slipcondition as shown in Figure 2 It is shown from Figure 3 thatthe asymptotic value of Nufd is equal to Nufd = 119886Pe119903V119908 Thisfinding is shown in the works of Kinney [9] and Raithby [11]
Figure 4 shows that the first heat transfer performanceindicator 120582fd is larger than one when 0 lt V119900 lt V119900119901
Mathematical Problems in Engineering 9
where the variation of V119900119901 with 119886Pe119903 is shown in Figure 3and given by (46) It can be shown from Figure 3 that when119886Pe119903 ge 978 V119900119901 ge 0495 It should be mentioned herethat having 120582fd gt 1 indicates that the heat transfer to thepermeable tube exposed to suction flow results in havinglower tube outlet temperature than that for the impermeabletube Consequently more thermal energy can be transferredfrom the permeable tube to the fluid in order to raise itsoutlet temperature to reach that of the impermeable tubeFurthermore Figure 4 shows that the second heat transferperformance indicator 120578119904fd becomes above one when V119900 gt
V119900119890 = 031606 Having 120578119904fd gt 1 indicates that thepermeable tube subjected to no-slip condition has smalleroutlet boundary temperature than that for the permeable tubesubjected to perfect fluid slip at the solid boundary This isnot the case for impermeable tubes as 120578119904fd lt 1 when V119900 = 0
as shown in Figure 4
42 The SystemThermal Performance for Linearly DistributedSuction Flow When 119861 = 0 the suction flow increaseslinearly from V119908 = 0 at the inlet (119909 = 0) to maximumvalue of V119908 = 2V119900 at the tube outlet (119909 = 1) For thiscase the thermal entry region effect is maximum as 119906avgis maximum at inlet In this region the axial convectioncoefficient is substantial as it is the coldest fluid region insidethe tube The thermal entry region influences the flow bycausing reductions in the boundary excess temperatures nearthe inlet Also the transverse convection is maximum at thetube outlet when 119861 = 0 as the suction velocity is maximumthere This decreases the boundary excess temperatures nearthe tube outlet As 119861 increases 119906avg close to the tube inletdecreases due to the increase in V119908 and V119908 close to thetube outlet decreases Accordingly both thermal entry regioneffect and transverse convection at the tube outlet decrease as119861 increases Both effects tend to increase both the boundaryexcess temperature and outlet mean bulk temperature As aresult 120579119898(119909 = 1) increases and 120582 decreases as 119861 increases asshown in Figure 5
For large 119886Pe119903 values both thermal entry region effectand the transverse convection at the tube outlet are largerthan those for smaller 119886Pe119903 values Accordingly 120582 valueswhen 119886Pe119903 = 10 are larger than those when 119886Pe119903 = 1
as shown in Figure 5 When 119886Pe119903 = 10 and 119861 = 0 it isnoticed that 120582 increases as V119900 increases This is because theoutlet mean bulk temperature decreases as V119900 increases forlarge 119886Pe119903 values as seen from Figure 2 Figure 5 shows that120582 values for V119900 = 13 plot are larger than those for V119900 = 049
plot when 119861 gt 046 This indicates that the convection dueto thermal entrance region effect as compared to transverseconvection increases as V119900 decreases when 119861 gt 046 and119886Pe119903 = 10 When 119886Pe119903 = 10 120578119904 is noticed from Figure 6to decrease as 119861 increases It is because that both thermalentry region effect and V119908 at the tube outlet decrease as 119861increases The former effect is due to the increase in V119908 nearthe tube inlet as 119861 increases Both effects tend to increase theexcess of Nu119904 above Nu Accordingly transverse convectiondue to fluid slip condition decreases due to the decrease inthe boundary excess temperature Consequently the axialconvection due to the fluid slip condition is increased more
than that due to the no-slip condition This causes 120579119904119898 toincrease as compared to 120579119898Thus 120578119904 decreases as 119861 increases
Figure 7 shows that 120582 increases as 119886Pe119903 increases At small119886Pe119903 values the transverse convection is negligible and Nu119904approaches Nu for large 119886Pe119903 values These lead to have 120578119904approaching one either as 119886Pe119903 decreases towards 119886Pe119903 =
0 or as 119886Pe119903 increases above 119886Pe119903 = 100 These trendscan be seen from Figure 8 For moderate 119886Pe119903 values theincrease in 119886Pe119903 not only causes an increase in the transverseconvection but it also results in reducing the boundary excesstemperature due to the increases in Nu This reduction ismore distinct for the case of the perfect fluid slip condition asNu119904 gt Nu The combined aforementioned effects can resultin reduction of transverse convection as 119886Pe119903 increases for theperfect fluid slip case It is because the transverse convectionis proportional to 119886Pe119903V119908(120579119882 minus 120579119898) as mentioned beforeAccordingly 120578119904 may have one local maximum at a specific119886Pe119903 value as clearly shown in Figure 8 for V119900 = 049 plots
43 The System Thermal Performance for Exponentially Dis-tributed Suction Flow For the upper 119862 value (119862 = 2)the thermal entry region effect is maximum since 119906avg ismaximum at tube inlet because V119908 is minimum there Alsothe transverse convection at the tube outlet is maximumas V119908 is maximum there These effects tend to decreasethe boundary excess temperatures at the tube outlet As 119862decreases 119906avg close to the tube inlet decreases due to theincrease in V119908 and V119908 close to the tube outlet decreasesAccordingly both thermal entry region effect and trans-verse convection at the tube outlet decrease as 119862 decreasesBoth effects increase both the boundary excess temperatureand outlet mean bulk temperature As a result the 120579119898(119909 =
1) increases and 120582 decreases as 119862 decreases as shown inFigure 9
For large 119886Pe119903 values both thermal entry region effectand outlet transverse convection are larger than those forsmaller 119886Pe119903 values Accordingly 120582 values when 119886Pe119903 = 10
are larger than those when 119886Pe119903 = 1 as shown in Figure 9When 119886Pe119903 = 10 and 119862 = 2 it is noticed that 120582 increases asV119900 increases It is because the outlet mean bulk temperaturedecreases as V119900 increases for large 119886Pe119903 values as shown fromFigure 2 Figure 9 shows that 120582 values for V119900 = 13 plot arelarger than those for V119900 = 049 plot when 119862 lt 06 Thisindicates that the flow gets more dominated by the thermalentry region effect than by the transverse convection as V119900decreases when 119862 lt 06 and 119886Pe119903 = 10 Figure 11 showsthat 120582 increases as 119886Pe119903 increases When 119886Pe119903 = 10 120578119904is noticed from Figure 10 to decrease as 119862 decreases It isbecause both thermal entry region effect and V119908 at the outletdecrease as 119862 decreases As indicated previously both effectstend to increase the excess of Nu119904 above Nu Accordingly thetransverse convection due to fluid slip condition decreasesdue to the decrease in the boundary excess temperatureConsequently the axial convection due to fluid slip conditionis increased more than that due to no-slip condition causing120579119904119898 to increase Thus 120578119904 decreases as 119862 decreases Figure 12shows that for moderate 119886Pe119903 values 120578119904 may have one localmaximumat a specific 119886Pe119903 valueThe physical interpretationfor this phenomenon is discussed in Section 42
10 Mathematical Problems in Engineering
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
The Fully Developed Nusselt Number Under fully developedcondition 120597120579120597119909 = 119889120579119898119889119909 As such (3) can be reduced tothe following using (19)
8 (1 minus 119903
2) 1 minus
119886Pe119903V119908Nufd
+ 2119886Pe119903V119908 (119903 minus119903
3
2
)
120597120579fd120597119903
=
2
119903
120597
120597119903
(119903
120597120579fd120597119903
)
(20)
where Nufd is the fully developed Nusselt number Equation(20) can be solved analytically and Nufd expression can bearranged in the following form
Nufd119885
= (81 minus exp(31198858
)
+ 2119885 exp(31198858
)int
0
minus34
exp (1198851199102)radic119910 + 1
119889119910)
sdot (119885 + 8 1 minus exp(31198858
)
+ 2119885 exp(31198858
)int
0
minus34
exp (1198851199102)radic119910 + 1
119889119910)
minus1
(21)
where 119885 = 119886Pe119903V119908(119909) The following approximation can beused
1
radic119910 + 1
cong 1198861 exp (1198862119910) + 1198863 exp (1198864119910) minus
3
4
le 119910 le 0
(22)
This approximation has maximum relative error less than03 when the coefficients are equal to
1198861 = 98075 times 10
minus3 1198862 = minus53965
1198863 = 09874 1198864 = minus049834
(23)
By substituting (22) in (21) Nufd can be approximated by thefollowing expression
Nufd119885
= (8 1 minus exp(31198858
) minus 1198601 exp (1198865) minus exp(31198858
)
minus1198602 exp (1198866) minus exp(31198858
))
sdot (8 1 minus exp(31198858
) minus 1198601 exp (1198865) minus exp(31198858
)
minus1198602 exp (1198866) minus exp(31198858
) + 119885)
minus1
(24)
where
1198865 = minus(
3
4
) 1198862 1198866 = minus(
3
4
) 1198864
1198601 =41198861119885
21198862 + 119885
1198602 =41198863119885
21198864 + 119885
(25)
211 Case I Linear Distribution of the Suction Velocity Forthis case the dimensionless suction velocity denoted by V119908119860has the following linear distribution
where V119900 = V119900(119886119906119900) V119900 is the average suction velocity overthe length of tube inner surface 119861 is an arbitrary controllingparameter
212 Case II Exponential Distribution of the Suction VelocityFor this case the dimensionless suction velocity denoted byV119908119861 has the following exponential distribution
V119908119861 (119909) = V119900119862 exp (119862119909)exp (119862) minus 1
(27)
where 119862 is an arbitrary controlling parameter
213 The Pressure Gradient for Both Cases Substituting (26)and (27) in (14) and solving for the dimensionless pressuregradient the following expressions can be obtained
119889119901
119889119909
=
1 minus 2V1199001199092+ 2119861 (119909
2minus 119909) Case I
1 minus 2V119900 [exp (119862119909) minus 1
exp (119862) minus 1
] Case II(28)
The ranges of 119861 and V119900 that result in having both 119906avg119860 andV119908119860 larger than zero can be shown to be
0 le V119900 lt1
2
0 le 119861 le 2V119900 (29)
The ranges of 119862 and V119900 that result in having both 119906avg119861 andV119908119861 larger than zero can be shown to be
0 le V119900 lt1
2
minusinfin lt 119862 lt infin (30)
22 Perfect Fluid Slip Case (Ideal Case) The ideal case of thepresent problem is constructed when the fluid is subjectedto perfect slip condition at the solid boundary For this idealcase the conservation of mass and the continuity equationreveal the following expressions
119889119906119904
119889119909
= minus2V119908 (31)
119906119904 =
1 minus 2V1199001199092+ 2119861 (119909
2minus 119909) Case I
1 minus 2V119900 exp (119862119909) minus 1
exp (119862) minus 1
Case II(32)
V = V119908119903 (33)
Mathematical Problems in Engineering 5
where 119906119904 = 119906s119906119900 119906119904 is the fluid velocity for this ideal caseUnder this ideal condition the energy equation reduces tothe following
119886Pe119903 (119906119904120597120579119904
120597119909
+ V119908119903120597120579119904
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904
120597119903
) (34)
where 120579119904 is the dimensionless temperature for the ideal caseFor fully developed condition where 120597120579119904120597119909 = 119889120579119898119889119909 (34)can be reduced to the following when (19) is implemented
41 minus
119885
Nu119904fd + 119885(119903
120597120579119904fd
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904fd
120597119903
) (35)
where Nu119904fd is the fully developed value of the Nusseltnumber for the ideal case By solving (35) and application ofthe boundary condition given by (10c) Nu119904fd can be foundIt is equal to
Nu119904fd119885
=
4 1 minus exp (1198854)4 1 minus exp (1198854) + 119885
(36)
For this case the dimensionless mean bulk temperature canbe found to be equal to
120579119898119904 (119909) equiv
119879119904119898 (119909) minus 1198791
119902
10158401015840119904119863 (2119896)
= 2int
1
0
119903120579119904119889119903 (37)
23 Heat Transfer Enhancement Indicators Let the heattransfer enhancement indicator 120582 be defined as ratio of thetube inner boundary excess temperature at the exit sectionfor the reference case (V119908 = 0) to that quantity when V119908 gt 0Mathematically the enhancement indicator 120582 is written as
120582 =
120579 (119909 = 1 119903 = 1)|V119908=0
120579 (119909 = 1 119903 = 1)
(38)
When V119908 = 0 the system becomes an impermeable tubeconfining an internal flow and subjected to uniform heat fluxFor this case the Nusselt number at the exit can be shown tobe correlated to 119886Pe119903 according to the following expression
Nu (119909 = 1 V119908 = 0)
= (438 minus 13 times 10
minus3119886Pe119903 + 39 times 10
minus4(119886Pe119903)
2
+ 65 times 10
minus7(119886Pe119903)
3)
sdot (1 minus 46 times 10
minus5119886Pe119903 + 86 times 10
minus5(119886Pe119903)
2
+ 43 times 10
minus8(119886Pe119903)
3)
minus1
(39)
with maximum relative error less than 03 when 1 le 119886Pe119903 le1000
The second performance indicator 120578119904 is defined as ratioof the tube inner boundary excess temperature at the exitsection for the perfect fluid slip case to that quantity underno-slip condition Mathematically 120578119904 is written as
24 Enhancement Indicators for Fully Developed Flow withUniform Suction For uniform suction case the mean veloc-ity inside the tube can be found using (32) by setting 119861 = V119900It is equal to
119906119904 = 119906avg = 1 minus 2V119900119909 V119900 lt1
2
(41)
By substituting (41) in (19) and solving the resulting equationthe mean bulk temperature distribution can be obtained It isgiven by
120579119898 (119909) = ln [1 minus 2V119900119909]2([1Nufd]minus1[119886Pe119903V119900])
(42)
Accordingly the dimensionless temperature at the exit isequal to
120579 (119909 = 1 119903 = 1)
= 120579119882 (119909 = 1) = ln [1 minus 2V119900]2([1Nufd]minus1[119886Pe119903V119900])
+
2
Nufd
(43)
Therefore the performance indicators 120582 and 120578119904 for this caseare equal to
120582fd =96 + 11119886Pe119903
ln [1 minus 2V119900]48([119886Pe
119903Nufd]minus1V119900)
+ 48 (119886Pe119903Nufd)
(44)
120578119904fd =ln [1 minus 2V119900]
2([119886Pe119903Nu119904fd]minus1V119900)
+ 2 (119886Pe119903Nu119904fd)
ln [1 minus 2V119900]2([119886Pe
119903Nufd]minus1V119900)
+ 2 (119886Pe119903Nufd)
(45)
The plots of 120579119898(119909 = 1) 120579119904119898(119909 = 1) Nufd Nu119904fd and 120582fdfor various 119886Pe119903 and V119900 values are seen in Figures 2ndash4 Theupper value of V119900 that makes 120582fd = 1 can be obtained usingnumerical solving techniques [20] It is denoted by V119900119901 V119900119901can be correlated to 119886Pe119903 through the following correlation
V119900119901 = (minus28076 times 10
6+ 820470 (119886Pe119903)
minus 83102 (119886Pe119903)2+ 18754 (119886Pe119903)
3)
sdot (1 + 713480 (119886Pe119903) minus 123760 (119886Pe119903)2
+ 37172 (119886Pe119903)3)
minus1
(46)
The percentage error associated with (46) is less than 062when 119886Pe119903 ge 4 The plot of 119886Pe119903 versus V119900119901 is shown in Fig-ure 3 Surprisingly the value of V119900 making 120578119904fd = 1 is foundto be independent on 119886Pe119903 This value is denoted by V119900119890 andit is equal to
V119900119890 = 031606 (47)
The plots of 120578119904fd for various 119886Pe119903 and V119900 are seen in Figure 4120578119904fd ge 1 when V119900 gt V119900119890
6 Mathematical Problems in Engineering
000 010 020 030 040 05020E minus 03
20E minus 02
20E minus 01
20E + 00
20E + 01
Fully developed flow condition
120579m(x
=1)120579
sm(x
=1)
120579m(x = 1)
120579sm(x = 1)
aPer = 1 2 4 8 16 32
w = o
o
Figure 2 Effects of V119900 on outlet mean bulk temperature under fullydeveloped flow condition
3 Numerical Methodology and Results
Equations (3) and (35) are solvable numerically without iter-ations using the implicit finite difference method discussedby Khaled [21] and Blottner [22]They were discretized usingtwo-point backward and three-point central differencingquotients for the first derivative with respect to 119909-direc-tion and both first and second derivatives with respect to119903-direction respectively The finite difference equation of(3) is given by
The pair (119894 119895) represents the location of the discretized pointin the numerical grid of the fluid domain The number 119899represents the total number of 119895-nodes along each 119894-sectionThe tube inner surface is located in the numerical grid at(119894 119895 = 119899) while the center of the tube is located in thenumerical grid at (119894 119895 = 1) The resulting 119899 minus 1 (119899 = 401)tridiagonal systemof algebraic equations obtained by (48) at agiven 119894-section was solved usingThomas algorithm [22] Theprevious procedure step was repeated for the consecutive 119894-values until 119894 reached the value 119898 (119898 = 2001) at which 119909 =
1 Numerical investigations were performed using differentmesh sizes to assess and ascertain grid-size independentresults Furthermore doubling the values of119898 and 119899 resultedin less than 04 error in the calculated parameters formoderate 119886Pe119903 values
The results of the present work are shown in Figures 2ndash14 Those obtained using the described numerical method
000
009
018
027
036
045
054
10E minus 01
10E + 02
10E + 02
10E + 01
10E + 01
10E + 00
10E + 00
aPerw aPer
op
Nu f
dN
u sfd
Nufd minus aPerwNusfd minus aPerwop minus aPer
Figure 3 Effects of 119886Pe119903 on fully developed Nusselt numbers andV119900119901
060
074
088
102
116
130
144
000
115
230
345
460
575
690
000 010 020 030 040 050
w = o
Fully developed flow condition
aPer = 1 2 4 8 16 32 50
120582fd
120582fd
120578s
fd
120578sfd
o
Figure 4 Effects of V119900 on the first and second performance indica-tors under fully developed flow condition
are shown in Figures 5ndash14 The numerical results for the casewith 119886Pe119903 = 1 shown in Figures 4ndash8 were compared withthe analytical solution utilizing (17) (24) (26) (27) (38) and(40) The numerical results match well with the analyticalsolutions since the case with 119886Pe119903 = 1 results in fullydeveloped flow condition at the tube exit Also it is shownfrom Figure 3 that the fully developed Nusselt number when119886Pe119903V119908 = 10 is equal to Nufd = 10725 This quantity canbe shown from Figure 9 of Raithby [11] work to be equal toNufd = 1083 when 119886Re119903V119908 rarr 0 The relative error betweenthese values is 098 which is very small These validationsincreased the confident levels in the obtained results
4 Discussion of the Results
41 The System Thermal Performance for Uniformly Dis-tributed Suction Flow As suction velocity V119908 increases theoutlet mean axial velocity 119906119898(119909 = 119871) decreases causing
Mathematical Problems in Engineering 7
010
045
080
115
150
185
220
000 020 040 060 080 10010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
B
o = 025 033 049
Figure 5 Effects of 119861 on performance indicator 120582 and outlet meanbulk temperature for Case I
085
095
105
115
125
135
145
000 020 040 060 080 100
120578s
B
o = 025 033 049
o = 025
o = 049
3o = 1
aPer = 1 numericalaPer = 1 analyticalaPer = 10
Figure 6 Effects of 119861 on performance indicator 120578119904 for Case I
01
10
100
10E + 0210E + 0110E + 00
aPerB = 0B = o
B = 2o
120582
o = 025 033 049
o = 025 033 049
(44)
Figure 7 Effects of 119886Pe119903 on performance indicators 120582 for various 119861values for Case I
066
080
094
108
122
136
150
10E + 0210E + 0110E + 00
aPer
120578s
B = 0B = o
B = 2o
o = 025 033 049
(45)
Figure 8 Effects of 119886Pe119903 on performance indicators 120578119904 for various 119861values for Case I
010
048
085
123
160
198
235
minus200 minus120 minus040 040 120 20010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
C
o = 025 033 049
o = 025 033 049
Figure 9 Effects of 119862 on performance indicator 120582 and outlet meanbulk temperature for Case II
087
097
107
117
127
137
147
minus200 minus120 minus040 040 120 200C
120578s
aPer = 1 numericalaPer = 1 analyticalaPer = 10
o = 025 033 049
o = 025
o = 049
3o = 1
Figure 10 Effects of 119862 on performance indicator 120578119904 for Case II
8 Mathematical Problems in Engineering
01
10
100
10E + 0210E + 0110E + 00
aPer
120582
C = 20
C = 0
C = minus20
o = 025 033 049
o = 025 033 049
(44)
Figure 11 Effects of 119886Pe119903 on performance indicators 120582 for various119862values for Case II
075
088
101
114
127
140
153
10E + 0210E + 0110E + 00
aPerC = 20
C = 0
C = minus20
120578s
o = 025 033 049
(45)
Figure 12 Effects of 119886Pe119903 on performance indicators 120578119904 for various119862 values for Case II
the axial convection which is proportional to 119886Pe119903119906avg119889120579119898119909to decrease This may cause an increase in the outlet meanbulk temperature 119879119898(119909 = 119871) as shown in Figure 2 forsmall 119886Pe119903 values 119886Pe119903 le 4 When 119886Pe119903 ge 8 the trans-verse convection becomes significant as it is proportionalto 2119886Pe119903V119908(120579119882 minus 120579119898) The latter expression is equivalent to4119886Pe119903V119908Nu This expression increases as 119886Pe119903V119900 increasesThis increase causes significant reduction in the temperatureboundary layer thicknesswhich causes reduction in the outletmean bulk temperature as shown in Figure 2 The sum of theaxial and transverse convection is constant which is equalto the total convection heat transfer rate given by 119902conv =
119902
10158401015840
119904120587119863119871 When V119900 approaches V119900 = 05 the axial advection
which is proportional to 119906avg approaches zero at the tubeoutlet This causes 119879119898(119909 = 119871) to increase apparently as V119900approaches V119900 = 05 asymptotically as can be seen fromFigure 2
Linear suction
Exponential suction
Linear suctionprofile
Exponential suctionprofile
0000
0085
0170
0255
0340
0425
0510
10
100
oc
oc
120582m
ax
120582max
10E + 0210E + 0110E + 00
aPer
profile
profile
Figure 13 Effects of 119886Pe119903 on 120582max and V119900119888 for both Case I and CaseII
Figure 14 Effects of 119886Pe119903 on 120578119904119888 for both Case I and Case II
The Nusselt number for perfect fluid slip at the solidboundary (36) case is larger than that for the no-slip condi-tion case (24) as shown in Figure 3 that is Nu119904 gt Nu Thisindicates that the tube excess temperature at the boundary forthe perfect fluid slip condition is smaller than that for the no-slip condition case That excess temperature is proportionalto 120579119882 minus 120579119898 Consequently the transverse convection due tofluid slip is smaller than that due to no-slip condition as thisconvection is proportional to both 119886Pe119903V119908 and 120579119882 minus 120579119898 As aresult the axial convection due to fluid slip condition is largerthan that due to no-slip condition as the total convectionheat transfer rate is constant Therefore 119879119898(119909 = 119871) for thefluid slip condition is always larger than that for the no-slipcondition as shown in Figure 2 It is shown from Figure 3 thatthe asymptotic value of Nufd is equal to Nufd = 119886Pe119903V119908 Thisfinding is shown in the works of Kinney [9] and Raithby [11]
Figure 4 shows that the first heat transfer performanceindicator 120582fd is larger than one when 0 lt V119900 lt V119900119901
Mathematical Problems in Engineering 9
where the variation of V119900119901 with 119886Pe119903 is shown in Figure 3and given by (46) It can be shown from Figure 3 that when119886Pe119903 ge 978 V119900119901 ge 0495 It should be mentioned herethat having 120582fd gt 1 indicates that the heat transfer to thepermeable tube exposed to suction flow results in havinglower tube outlet temperature than that for the impermeabletube Consequently more thermal energy can be transferredfrom the permeable tube to the fluid in order to raise itsoutlet temperature to reach that of the impermeable tubeFurthermore Figure 4 shows that the second heat transferperformance indicator 120578119904fd becomes above one when V119900 gt
V119900119890 = 031606 Having 120578119904fd gt 1 indicates that thepermeable tube subjected to no-slip condition has smalleroutlet boundary temperature than that for the permeable tubesubjected to perfect fluid slip at the solid boundary This isnot the case for impermeable tubes as 120578119904fd lt 1 when V119900 = 0
as shown in Figure 4
42 The SystemThermal Performance for Linearly DistributedSuction Flow When 119861 = 0 the suction flow increaseslinearly from V119908 = 0 at the inlet (119909 = 0) to maximumvalue of V119908 = 2V119900 at the tube outlet (119909 = 1) For thiscase the thermal entry region effect is maximum as 119906avgis maximum at inlet In this region the axial convectioncoefficient is substantial as it is the coldest fluid region insidethe tube The thermal entry region influences the flow bycausing reductions in the boundary excess temperatures nearthe inlet Also the transverse convection is maximum at thetube outlet when 119861 = 0 as the suction velocity is maximumthere This decreases the boundary excess temperatures nearthe tube outlet As 119861 increases 119906avg close to the tube inletdecreases due to the increase in V119908 and V119908 close to thetube outlet decreases Accordingly both thermal entry regioneffect and transverse convection at the tube outlet decrease as119861 increases Both effects tend to increase both the boundaryexcess temperature and outlet mean bulk temperature As aresult 120579119898(119909 = 1) increases and 120582 decreases as 119861 increases asshown in Figure 5
For large 119886Pe119903 values both thermal entry region effectand the transverse convection at the tube outlet are largerthan those for smaller 119886Pe119903 values Accordingly 120582 valueswhen 119886Pe119903 = 10 are larger than those when 119886Pe119903 = 1
as shown in Figure 5 When 119886Pe119903 = 10 and 119861 = 0 it isnoticed that 120582 increases as V119900 increases This is because theoutlet mean bulk temperature decreases as V119900 increases forlarge 119886Pe119903 values as seen from Figure 2 Figure 5 shows that120582 values for V119900 = 13 plot are larger than those for V119900 = 049
plot when 119861 gt 046 This indicates that the convection dueto thermal entrance region effect as compared to transverseconvection increases as V119900 decreases when 119861 gt 046 and119886Pe119903 = 10 When 119886Pe119903 = 10 120578119904 is noticed from Figure 6to decrease as 119861 increases It is because that both thermalentry region effect and V119908 at the tube outlet decrease as 119861increases The former effect is due to the increase in V119908 nearthe tube inlet as 119861 increases Both effects tend to increase theexcess of Nu119904 above Nu Accordingly transverse convectiondue to fluid slip condition decreases due to the decrease inthe boundary excess temperature Consequently the axialconvection due to the fluid slip condition is increased more
than that due to the no-slip condition This causes 120579119904119898 toincrease as compared to 120579119898Thus 120578119904 decreases as 119861 increases
Figure 7 shows that 120582 increases as 119886Pe119903 increases At small119886Pe119903 values the transverse convection is negligible and Nu119904approaches Nu for large 119886Pe119903 values These lead to have 120578119904approaching one either as 119886Pe119903 decreases towards 119886Pe119903 =
0 or as 119886Pe119903 increases above 119886Pe119903 = 100 These trendscan be seen from Figure 8 For moderate 119886Pe119903 values theincrease in 119886Pe119903 not only causes an increase in the transverseconvection but it also results in reducing the boundary excesstemperature due to the increases in Nu This reduction ismore distinct for the case of the perfect fluid slip condition asNu119904 gt Nu The combined aforementioned effects can resultin reduction of transverse convection as 119886Pe119903 increases for theperfect fluid slip case It is because the transverse convectionis proportional to 119886Pe119903V119908(120579119882 minus 120579119898) as mentioned beforeAccordingly 120578119904 may have one local maximum at a specific119886Pe119903 value as clearly shown in Figure 8 for V119900 = 049 plots
43 The System Thermal Performance for Exponentially Dis-tributed Suction Flow For the upper 119862 value (119862 = 2)the thermal entry region effect is maximum since 119906avg ismaximum at tube inlet because V119908 is minimum there Alsothe transverse convection at the tube outlet is maximumas V119908 is maximum there These effects tend to decreasethe boundary excess temperatures at the tube outlet As 119862decreases 119906avg close to the tube inlet decreases due to theincrease in V119908 and V119908 close to the tube outlet decreasesAccordingly both thermal entry region effect and trans-verse convection at the tube outlet decrease as 119862 decreasesBoth effects increase both the boundary excess temperatureand outlet mean bulk temperature As a result the 120579119898(119909 =
1) increases and 120582 decreases as 119862 decreases as shown inFigure 9
For large 119886Pe119903 values both thermal entry region effectand outlet transverse convection are larger than those forsmaller 119886Pe119903 values Accordingly 120582 values when 119886Pe119903 = 10
are larger than those when 119886Pe119903 = 1 as shown in Figure 9When 119886Pe119903 = 10 and 119862 = 2 it is noticed that 120582 increases asV119900 increases It is because the outlet mean bulk temperaturedecreases as V119900 increases for large 119886Pe119903 values as shown fromFigure 2 Figure 9 shows that 120582 values for V119900 = 13 plot arelarger than those for V119900 = 049 plot when 119862 lt 06 Thisindicates that the flow gets more dominated by the thermalentry region effect than by the transverse convection as V119900decreases when 119862 lt 06 and 119886Pe119903 = 10 Figure 11 showsthat 120582 increases as 119886Pe119903 increases When 119886Pe119903 = 10 120578119904is noticed from Figure 10 to decrease as 119862 decreases It isbecause both thermal entry region effect and V119908 at the outletdecrease as 119862 decreases As indicated previously both effectstend to increase the excess of Nu119904 above Nu Accordingly thetransverse convection due to fluid slip condition decreasesdue to the decrease in the boundary excess temperatureConsequently the axial convection due to fluid slip conditionis increased more than that due to no-slip condition causing120579119904119898 to increase Thus 120578119904 decreases as 119862 decreases Figure 12shows that for moderate 119886Pe119903 values 120578119904 may have one localmaximumat a specific 119886Pe119903 valueThe physical interpretationfor this phenomenon is discussed in Section 42
10 Mathematical Problems in Engineering
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
where 119906119904 = 119906s119906119900 119906119904 is the fluid velocity for this ideal caseUnder this ideal condition the energy equation reduces tothe following
119886Pe119903 (119906119904120597120579119904
120597119909
+ V119908119903120597120579119904
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904
120597119903
) (34)
where 120579119904 is the dimensionless temperature for the ideal caseFor fully developed condition where 120597120579119904120597119909 = 119889120579119898119889119909 (34)can be reduced to the following when (19) is implemented
41 minus
119885
Nu119904fd + 119885(119903
120597120579119904fd
120597119903
) =
2
119903
120597
120597119903
(119903
120597120579119904fd
120597119903
) (35)
where Nu119904fd is the fully developed value of the Nusseltnumber for the ideal case By solving (35) and application ofthe boundary condition given by (10c) Nu119904fd can be foundIt is equal to
Nu119904fd119885
=
4 1 minus exp (1198854)4 1 minus exp (1198854) + 119885
(36)
For this case the dimensionless mean bulk temperature canbe found to be equal to
120579119898119904 (119909) equiv
119879119904119898 (119909) minus 1198791
119902
10158401015840119904119863 (2119896)
= 2int
1
0
119903120579119904119889119903 (37)
23 Heat Transfer Enhancement Indicators Let the heattransfer enhancement indicator 120582 be defined as ratio of thetube inner boundary excess temperature at the exit sectionfor the reference case (V119908 = 0) to that quantity when V119908 gt 0Mathematically the enhancement indicator 120582 is written as
120582 =
120579 (119909 = 1 119903 = 1)|V119908=0
120579 (119909 = 1 119903 = 1)
(38)
When V119908 = 0 the system becomes an impermeable tubeconfining an internal flow and subjected to uniform heat fluxFor this case the Nusselt number at the exit can be shown tobe correlated to 119886Pe119903 according to the following expression
Nu (119909 = 1 V119908 = 0)
= (438 minus 13 times 10
minus3119886Pe119903 + 39 times 10
minus4(119886Pe119903)
2
+ 65 times 10
minus7(119886Pe119903)
3)
sdot (1 minus 46 times 10
minus5119886Pe119903 + 86 times 10
minus5(119886Pe119903)
2
+ 43 times 10
minus8(119886Pe119903)
3)
minus1
(39)
with maximum relative error less than 03 when 1 le 119886Pe119903 le1000
The second performance indicator 120578119904 is defined as ratioof the tube inner boundary excess temperature at the exitsection for the perfect fluid slip case to that quantity underno-slip condition Mathematically 120578119904 is written as
24 Enhancement Indicators for Fully Developed Flow withUniform Suction For uniform suction case the mean veloc-ity inside the tube can be found using (32) by setting 119861 = V119900It is equal to
119906119904 = 119906avg = 1 minus 2V119900119909 V119900 lt1
2
(41)
By substituting (41) in (19) and solving the resulting equationthe mean bulk temperature distribution can be obtained It isgiven by
120579119898 (119909) = ln [1 minus 2V119900119909]2([1Nufd]minus1[119886Pe119903V119900])
(42)
Accordingly the dimensionless temperature at the exit isequal to
120579 (119909 = 1 119903 = 1)
= 120579119882 (119909 = 1) = ln [1 minus 2V119900]2([1Nufd]minus1[119886Pe119903V119900])
+
2
Nufd
(43)
Therefore the performance indicators 120582 and 120578119904 for this caseare equal to
120582fd =96 + 11119886Pe119903
ln [1 minus 2V119900]48([119886Pe
119903Nufd]minus1V119900)
+ 48 (119886Pe119903Nufd)
(44)
120578119904fd =ln [1 minus 2V119900]
2([119886Pe119903Nu119904fd]minus1V119900)
+ 2 (119886Pe119903Nu119904fd)
ln [1 minus 2V119900]2([119886Pe
119903Nufd]minus1V119900)
+ 2 (119886Pe119903Nufd)
(45)
The plots of 120579119898(119909 = 1) 120579119904119898(119909 = 1) Nufd Nu119904fd and 120582fdfor various 119886Pe119903 and V119900 values are seen in Figures 2ndash4 Theupper value of V119900 that makes 120582fd = 1 can be obtained usingnumerical solving techniques [20] It is denoted by V119900119901 V119900119901can be correlated to 119886Pe119903 through the following correlation
V119900119901 = (minus28076 times 10
6+ 820470 (119886Pe119903)
minus 83102 (119886Pe119903)2+ 18754 (119886Pe119903)
3)
sdot (1 + 713480 (119886Pe119903) minus 123760 (119886Pe119903)2
+ 37172 (119886Pe119903)3)
minus1
(46)
The percentage error associated with (46) is less than 062when 119886Pe119903 ge 4 The plot of 119886Pe119903 versus V119900119901 is shown in Fig-ure 3 Surprisingly the value of V119900 making 120578119904fd = 1 is foundto be independent on 119886Pe119903 This value is denoted by V119900119890 andit is equal to
V119900119890 = 031606 (47)
The plots of 120578119904fd for various 119886Pe119903 and V119900 are seen in Figure 4120578119904fd ge 1 when V119900 gt V119900119890
6 Mathematical Problems in Engineering
000 010 020 030 040 05020E minus 03
20E minus 02
20E minus 01
20E + 00
20E + 01
Fully developed flow condition
120579m(x
=1)120579
sm(x
=1)
120579m(x = 1)
120579sm(x = 1)
aPer = 1 2 4 8 16 32
w = o
o
Figure 2 Effects of V119900 on outlet mean bulk temperature under fullydeveloped flow condition
3 Numerical Methodology and Results
Equations (3) and (35) are solvable numerically without iter-ations using the implicit finite difference method discussedby Khaled [21] and Blottner [22]They were discretized usingtwo-point backward and three-point central differencingquotients for the first derivative with respect to 119909-direc-tion and both first and second derivatives with respect to119903-direction respectively The finite difference equation of(3) is given by
The pair (119894 119895) represents the location of the discretized pointin the numerical grid of the fluid domain The number 119899represents the total number of 119895-nodes along each 119894-sectionThe tube inner surface is located in the numerical grid at(119894 119895 = 119899) while the center of the tube is located in thenumerical grid at (119894 119895 = 1) The resulting 119899 minus 1 (119899 = 401)tridiagonal systemof algebraic equations obtained by (48) at agiven 119894-section was solved usingThomas algorithm [22] Theprevious procedure step was repeated for the consecutive 119894-values until 119894 reached the value 119898 (119898 = 2001) at which 119909 =
1 Numerical investigations were performed using differentmesh sizes to assess and ascertain grid-size independentresults Furthermore doubling the values of119898 and 119899 resultedin less than 04 error in the calculated parameters formoderate 119886Pe119903 values
The results of the present work are shown in Figures 2ndash14 Those obtained using the described numerical method
000
009
018
027
036
045
054
10E minus 01
10E + 02
10E + 02
10E + 01
10E + 01
10E + 00
10E + 00
aPerw aPer
op
Nu f
dN
u sfd
Nufd minus aPerwNusfd minus aPerwop minus aPer
Figure 3 Effects of 119886Pe119903 on fully developed Nusselt numbers andV119900119901
060
074
088
102
116
130
144
000
115
230
345
460
575
690
000 010 020 030 040 050
w = o
Fully developed flow condition
aPer = 1 2 4 8 16 32 50
120582fd
120582fd
120578s
fd
120578sfd
o
Figure 4 Effects of V119900 on the first and second performance indica-tors under fully developed flow condition
are shown in Figures 5ndash14 The numerical results for the casewith 119886Pe119903 = 1 shown in Figures 4ndash8 were compared withthe analytical solution utilizing (17) (24) (26) (27) (38) and(40) The numerical results match well with the analyticalsolutions since the case with 119886Pe119903 = 1 results in fullydeveloped flow condition at the tube exit Also it is shownfrom Figure 3 that the fully developed Nusselt number when119886Pe119903V119908 = 10 is equal to Nufd = 10725 This quantity canbe shown from Figure 9 of Raithby [11] work to be equal toNufd = 1083 when 119886Re119903V119908 rarr 0 The relative error betweenthese values is 098 which is very small These validationsincreased the confident levels in the obtained results
4 Discussion of the Results
41 The System Thermal Performance for Uniformly Dis-tributed Suction Flow As suction velocity V119908 increases theoutlet mean axial velocity 119906119898(119909 = 119871) decreases causing
Mathematical Problems in Engineering 7
010
045
080
115
150
185
220
000 020 040 060 080 10010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
B
o = 025 033 049
Figure 5 Effects of 119861 on performance indicator 120582 and outlet meanbulk temperature for Case I
085
095
105
115
125
135
145
000 020 040 060 080 100
120578s
B
o = 025 033 049
o = 025
o = 049
3o = 1
aPer = 1 numericalaPer = 1 analyticalaPer = 10
Figure 6 Effects of 119861 on performance indicator 120578119904 for Case I
01
10
100
10E + 0210E + 0110E + 00
aPerB = 0B = o
B = 2o
120582
o = 025 033 049
o = 025 033 049
(44)
Figure 7 Effects of 119886Pe119903 on performance indicators 120582 for various 119861values for Case I
066
080
094
108
122
136
150
10E + 0210E + 0110E + 00
aPer
120578s
B = 0B = o
B = 2o
o = 025 033 049
(45)
Figure 8 Effects of 119886Pe119903 on performance indicators 120578119904 for various 119861values for Case I
010
048
085
123
160
198
235
minus200 minus120 minus040 040 120 20010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
C
o = 025 033 049
o = 025 033 049
Figure 9 Effects of 119862 on performance indicator 120582 and outlet meanbulk temperature for Case II
087
097
107
117
127
137
147
minus200 minus120 minus040 040 120 200C
120578s
aPer = 1 numericalaPer = 1 analyticalaPer = 10
o = 025 033 049
o = 025
o = 049
3o = 1
Figure 10 Effects of 119862 on performance indicator 120578119904 for Case II
8 Mathematical Problems in Engineering
01
10
100
10E + 0210E + 0110E + 00
aPer
120582
C = 20
C = 0
C = minus20
o = 025 033 049
o = 025 033 049
(44)
Figure 11 Effects of 119886Pe119903 on performance indicators 120582 for various119862values for Case II
075
088
101
114
127
140
153
10E + 0210E + 0110E + 00
aPerC = 20
C = 0
C = minus20
120578s
o = 025 033 049
(45)
Figure 12 Effects of 119886Pe119903 on performance indicators 120578119904 for various119862 values for Case II
the axial convection which is proportional to 119886Pe119903119906avg119889120579119898119909to decrease This may cause an increase in the outlet meanbulk temperature 119879119898(119909 = 119871) as shown in Figure 2 forsmall 119886Pe119903 values 119886Pe119903 le 4 When 119886Pe119903 ge 8 the trans-verse convection becomes significant as it is proportionalto 2119886Pe119903V119908(120579119882 minus 120579119898) The latter expression is equivalent to4119886Pe119903V119908Nu This expression increases as 119886Pe119903V119900 increasesThis increase causes significant reduction in the temperatureboundary layer thicknesswhich causes reduction in the outletmean bulk temperature as shown in Figure 2 The sum of theaxial and transverse convection is constant which is equalto the total convection heat transfer rate given by 119902conv =
119902
10158401015840
119904120587119863119871 When V119900 approaches V119900 = 05 the axial advection
which is proportional to 119906avg approaches zero at the tubeoutlet This causes 119879119898(119909 = 119871) to increase apparently as V119900approaches V119900 = 05 asymptotically as can be seen fromFigure 2
Linear suction
Exponential suction
Linear suctionprofile
Exponential suctionprofile
0000
0085
0170
0255
0340
0425
0510
10
100
oc
oc
120582m
ax
120582max
10E + 0210E + 0110E + 00
aPer
profile
profile
Figure 13 Effects of 119886Pe119903 on 120582max and V119900119888 for both Case I and CaseII
Figure 14 Effects of 119886Pe119903 on 120578119904119888 for both Case I and Case II
The Nusselt number for perfect fluid slip at the solidboundary (36) case is larger than that for the no-slip condi-tion case (24) as shown in Figure 3 that is Nu119904 gt Nu Thisindicates that the tube excess temperature at the boundary forthe perfect fluid slip condition is smaller than that for the no-slip condition case That excess temperature is proportionalto 120579119882 minus 120579119898 Consequently the transverse convection due tofluid slip is smaller than that due to no-slip condition as thisconvection is proportional to both 119886Pe119903V119908 and 120579119882 minus 120579119898 As aresult the axial convection due to fluid slip condition is largerthan that due to no-slip condition as the total convectionheat transfer rate is constant Therefore 119879119898(119909 = 119871) for thefluid slip condition is always larger than that for the no-slipcondition as shown in Figure 2 It is shown from Figure 3 thatthe asymptotic value of Nufd is equal to Nufd = 119886Pe119903V119908 Thisfinding is shown in the works of Kinney [9] and Raithby [11]
Figure 4 shows that the first heat transfer performanceindicator 120582fd is larger than one when 0 lt V119900 lt V119900119901
Mathematical Problems in Engineering 9
where the variation of V119900119901 with 119886Pe119903 is shown in Figure 3and given by (46) It can be shown from Figure 3 that when119886Pe119903 ge 978 V119900119901 ge 0495 It should be mentioned herethat having 120582fd gt 1 indicates that the heat transfer to thepermeable tube exposed to suction flow results in havinglower tube outlet temperature than that for the impermeabletube Consequently more thermal energy can be transferredfrom the permeable tube to the fluid in order to raise itsoutlet temperature to reach that of the impermeable tubeFurthermore Figure 4 shows that the second heat transferperformance indicator 120578119904fd becomes above one when V119900 gt
V119900119890 = 031606 Having 120578119904fd gt 1 indicates that thepermeable tube subjected to no-slip condition has smalleroutlet boundary temperature than that for the permeable tubesubjected to perfect fluid slip at the solid boundary This isnot the case for impermeable tubes as 120578119904fd lt 1 when V119900 = 0
as shown in Figure 4
42 The SystemThermal Performance for Linearly DistributedSuction Flow When 119861 = 0 the suction flow increaseslinearly from V119908 = 0 at the inlet (119909 = 0) to maximumvalue of V119908 = 2V119900 at the tube outlet (119909 = 1) For thiscase the thermal entry region effect is maximum as 119906avgis maximum at inlet In this region the axial convectioncoefficient is substantial as it is the coldest fluid region insidethe tube The thermal entry region influences the flow bycausing reductions in the boundary excess temperatures nearthe inlet Also the transverse convection is maximum at thetube outlet when 119861 = 0 as the suction velocity is maximumthere This decreases the boundary excess temperatures nearthe tube outlet As 119861 increases 119906avg close to the tube inletdecreases due to the increase in V119908 and V119908 close to thetube outlet decreases Accordingly both thermal entry regioneffect and transverse convection at the tube outlet decrease as119861 increases Both effects tend to increase both the boundaryexcess temperature and outlet mean bulk temperature As aresult 120579119898(119909 = 1) increases and 120582 decreases as 119861 increases asshown in Figure 5
For large 119886Pe119903 values both thermal entry region effectand the transverse convection at the tube outlet are largerthan those for smaller 119886Pe119903 values Accordingly 120582 valueswhen 119886Pe119903 = 10 are larger than those when 119886Pe119903 = 1
as shown in Figure 5 When 119886Pe119903 = 10 and 119861 = 0 it isnoticed that 120582 increases as V119900 increases This is because theoutlet mean bulk temperature decreases as V119900 increases forlarge 119886Pe119903 values as seen from Figure 2 Figure 5 shows that120582 values for V119900 = 13 plot are larger than those for V119900 = 049
plot when 119861 gt 046 This indicates that the convection dueto thermal entrance region effect as compared to transverseconvection increases as V119900 decreases when 119861 gt 046 and119886Pe119903 = 10 When 119886Pe119903 = 10 120578119904 is noticed from Figure 6to decrease as 119861 increases It is because that both thermalentry region effect and V119908 at the tube outlet decrease as 119861increases The former effect is due to the increase in V119908 nearthe tube inlet as 119861 increases Both effects tend to increase theexcess of Nu119904 above Nu Accordingly transverse convectiondue to fluid slip condition decreases due to the decrease inthe boundary excess temperature Consequently the axialconvection due to the fluid slip condition is increased more
than that due to the no-slip condition This causes 120579119904119898 toincrease as compared to 120579119898Thus 120578119904 decreases as 119861 increases
Figure 7 shows that 120582 increases as 119886Pe119903 increases At small119886Pe119903 values the transverse convection is negligible and Nu119904approaches Nu for large 119886Pe119903 values These lead to have 120578119904approaching one either as 119886Pe119903 decreases towards 119886Pe119903 =
0 or as 119886Pe119903 increases above 119886Pe119903 = 100 These trendscan be seen from Figure 8 For moderate 119886Pe119903 values theincrease in 119886Pe119903 not only causes an increase in the transverseconvection but it also results in reducing the boundary excesstemperature due to the increases in Nu This reduction ismore distinct for the case of the perfect fluid slip condition asNu119904 gt Nu The combined aforementioned effects can resultin reduction of transverse convection as 119886Pe119903 increases for theperfect fluid slip case It is because the transverse convectionis proportional to 119886Pe119903V119908(120579119882 minus 120579119898) as mentioned beforeAccordingly 120578119904 may have one local maximum at a specific119886Pe119903 value as clearly shown in Figure 8 for V119900 = 049 plots
43 The System Thermal Performance for Exponentially Dis-tributed Suction Flow For the upper 119862 value (119862 = 2)the thermal entry region effect is maximum since 119906avg ismaximum at tube inlet because V119908 is minimum there Alsothe transverse convection at the tube outlet is maximumas V119908 is maximum there These effects tend to decreasethe boundary excess temperatures at the tube outlet As 119862decreases 119906avg close to the tube inlet decreases due to theincrease in V119908 and V119908 close to the tube outlet decreasesAccordingly both thermal entry region effect and trans-verse convection at the tube outlet decrease as 119862 decreasesBoth effects increase both the boundary excess temperatureand outlet mean bulk temperature As a result the 120579119898(119909 =
1) increases and 120582 decreases as 119862 decreases as shown inFigure 9
For large 119886Pe119903 values both thermal entry region effectand outlet transverse convection are larger than those forsmaller 119886Pe119903 values Accordingly 120582 values when 119886Pe119903 = 10
are larger than those when 119886Pe119903 = 1 as shown in Figure 9When 119886Pe119903 = 10 and 119862 = 2 it is noticed that 120582 increases asV119900 increases It is because the outlet mean bulk temperaturedecreases as V119900 increases for large 119886Pe119903 values as shown fromFigure 2 Figure 9 shows that 120582 values for V119900 = 13 plot arelarger than those for V119900 = 049 plot when 119862 lt 06 Thisindicates that the flow gets more dominated by the thermalentry region effect than by the transverse convection as V119900decreases when 119862 lt 06 and 119886Pe119903 = 10 Figure 11 showsthat 120582 increases as 119886Pe119903 increases When 119886Pe119903 = 10 120578119904is noticed from Figure 10 to decrease as 119862 decreases It isbecause both thermal entry region effect and V119908 at the outletdecrease as 119862 decreases As indicated previously both effectstend to increase the excess of Nu119904 above Nu Accordingly thetransverse convection due to fluid slip condition decreasesdue to the decrease in the boundary excess temperatureConsequently the axial convection due to fluid slip conditionis increased more than that due to no-slip condition causing120579119904119898 to increase Thus 120578119904 decreases as 119862 decreases Figure 12shows that for moderate 119886Pe119903 values 120578119904 may have one localmaximumat a specific 119886Pe119903 valueThe physical interpretationfor this phenomenon is discussed in Section 42
10 Mathematical Problems in Engineering
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
Figure 2 Effects of V119900 on outlet mean bulk temperature under fullydeveloped flow condition
3 Numerical Methodology and Results
Equations (3) and (35) are solvable numerically without iter-ations using the implicit finite difference method discussedby Khaled [21] and Blottner [22]They were discretized usingtwo-point backward and three-point central differencingquotients for the first derivative with respect to 119909-direc-tion and both first and second derivatives with respect to119903-direction respectively The finite difference equation of(3) is given by
The pair (119894 119895) represents the location of the discretized pointin the numerical grid of the fluid domain The number 119899represents the total number of 119895-nodes along each 119894-sectionThe tube inner surface is located in the numerical grid at(119894 119895 = 119899) while the center of the tube is located in thenumerical grid at (119894 119895 = 1) The resulting 119899 minus 1 (119899 = 401)tridiagonal systemof algebraic equations obtained by (48) at agiven 119894-section was solved usingThomas algorithm [22] Theprevious procedure step was repeated for the consecutive 119894-values until 119894 reached the value 119898 (119898 = 2001) at which 119909 =
1 Numerical investigations were performed using differentmesh sizes to assess and ascertain grid-size independentresults Furthermore doubling the values of119898 and 119899 resultedin less than 04 error in the calculated parameters formoderate 119886Pe119903 values
The results of the present work are shown in Figures 2ndash14 Those obtained using the described numerical method
000
009
018
027
036
045
054
10E minus 01
10E + 02
10E + 02
10E + 01
10E + 01
10E + 00
10E + 00
aPerw aPer
op
Nu f
dN
u sfd
Nufd minus aPerwNusfd minus aPerwop minus aPer
Figure 3 Effects of 119886Pe119903 on fully developed Nusselt numbers andV119900119901
060
074
088
102
116
130
144
000
115
230
345
460
575
690
000 010 020 030 040 050
w = o
Fully developed flow condition
aPer = 1 2 4 8 16 32 50
120582fd
120582fd
120578s
fd
120578sfd
o
Figure 4 Effects of V119900 on the first and second performance indica-tors under fully developed flow condition
are shown in Figures 5ndash14 The numerical results for the casewith 119886Pe119903 = 1 shown in Figures 4ndash8 were compared withthe analytical solution utilizing (17) (24) (26) (27) (38) and(40) The numerical results match well with the analyticalsolutions since the case with 119886Pe119903 = 1 results in fullydeveloped flow condition at the tube exit Also it is shownfrom Figure 3 that the fully developed Nusselt number when119886Pe119903V119908 = 10 is equal to Nufd = 10725 This quantity canbe shown from Figure 9 of Raithby [11] work to be equal toNufd = 1083 when 119886Re119903V119908 rarr 0 The relative error betweenthese values is 098 which is very small These validationsincreased the confident levels in the obtained results
4 Discussion of the Results
41 The System Thermal Performance for Uniformly Dis-tributed Suction Flow As suction velocity V119908 increases theoutlet mean axial velocity 119906119898(119909 = 119871) decreases causing
Mathematical Problems in Engineering 7
010
045
080
115
150
185
220
000 020 040 060 080 10010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
B
o = 025 033 049
Figure 5 Effects of 119861 on performance indicator 120582 and outlet meanbulk temperature for Case I
085
095
105
115
125
135
145
000 020 040 060 080 100
120578s
B
o = 025 033 049
o = 025
o = 049
3o = 1
aPer = 1 numericalaPer = 1 analyticalaPer = 10
Figure 6 Effects of 119861 on performance indicator 120578119904 for Case I
01
10
100
10E + 0210E + 0110E + 00
aPerB = 0B = o
B = 2o
120582
o = 025 033 049
o = 025 033 049
(44)
Figure 7 Effects of 119886Pe119903 on performance indicators 120582 for various 119861values for Case I
066
080
094
108
122
136
150
10E + 0210E + 0110E + 00
aPer
120578s
B = 0B = o
B = 2o
o = 025 033 049
(45)
Figure 8 Effects of 119886Pe119903 on performance indicators 120578119904 for various 119861values for Case I
010
048
085
123
160
198
235
minus200 minus120 minus040 040 120 20010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
C
o = 025 033 049
o = 025 033 049
Figure 9 Effects of 119862 on performance indicator 120582 and outlet meanbulk temperature for Case II
087
097
107
117
127
137
147
minus200 minus120 minus040 040 120 200C
120578s
aPer = 1 numericalaPer = 1 analyticalaPer = 10
o = 025 033 049
o = 025
o = 049
3o = 1
Figure 10 Effects of 119862 on performance indicator 120578119904 for Case II
8 Mathematical Problems in Engineering
01
10
100
10E + 0210E + 0110E + 00
aPer
120582
C = 20
C = 0
C = minus20
o = 025 033 049
o = 025 033 049
(44)
Figure 11 Effects of 119886Pe119903 on performance indicators 120582 for various119862values for Case II
075
088
101
114
127
140
153
10E + 0210E + 0110E + 00
aPerC = 20
C = 0
C = minus20
120578s
o = 025 033 049
(45)
Figure 12 Effects of 119886Pe119903 on performance indicators 120578119904 for various119862 values for Case II
the axial convection which is proportional to 119886Pe119903119906avg119889120579119898119909to decrease This may cause an increase in the outlet meanbulk temperature 119879119898(119909 = 119871) as shown in Figure 2 forsmall 119886Pe119903 values 119886Pe119903 le 4 When 119886Pe119903 ge 8 the trans-verse convection becomes significant as it is proportionalto 2119886Pe119903V119908(120579119882 minus 120579119898) The latter expression is equivalent to4119886Pe119903V119908Nu This expression increases as 119886Pe119903V119900 increasesThis increase causes significant reduction in the temperatureboundary layer thicknesswhich causes reduction in the outletmean bulk temperature as shown in Figure 2 The sum of theaxial and transverse convection is constant which is equalto the total convection heat transfer rate given by 119902conv =
119902
10158401015840
119904120587119863119871 When V119900 approaches V119900 = 05 the axial advection
which is proportional to 119906avg approaches zero at the tubeoutlet This causes 119879119898(119909 = 119871) to increase apparently as V119900approaches V119900 = 05 asymptotically as can be seen fromFigure 2
Linear suction
Exponential suction
Linear suctionprofile
Exponential suctionprofile
0000
0085
0170
0255
0340
0425
0510
10
100
oc
oc
120582m
ax
120582max
10E + 0210E + 0110E + 00
aPer
profile
profile
Figure 13 Effects of 119886Pe119903 on 120582max and V119900119888 for both Case I and CaseII
Figure 14 Effects of 119886Pe119903 on 120578119904119888 for both Case I and Case II
The Nusselt number for perfect fluid slip at the solidboundary (36) case is larger than that for the no-slip condi-tion case (24) as shown in Figure 3 that is Nu119904 gt Nu Thisindicates that the tube excess temperature at the boundary forthe perfect fluid slip condition is smaller than that for the no-slip condition case That excess temperature is proportionalto 120579119882 minus 120579119898 Consequently the transverse convection due tofluid slip is smaller than that due to no-slip condition as thisconvection is proportional to both 119886Pe119903V119908 and 120579119882 minus 120579119898 As aresult the axial convection due to fluid slip condition is largerthan that due to no-slip condition as the total convectionheat transfer rate is constant Therefore 119879119898(119909 = 119871) for thefluid slip condition is always larger than that for the no-slipcondition as shown in Figure 2 It is shown from Figure 3 thatthe asymptotic value of Nufd is equal to Nufd = 119886Pe119903V119908 Thisfinding is shown in the works of Kinney [9] and Raithby [11]
Figure 4 shows that the first heat transfer performanceindicator 120582fd is larger than one when 0 lt V119900 lt V119900119901
Mathematical Problems in Engineering 9
where the variation of V119900119901 with 119886Pe119903 is shown in Figure 3and given by (46) It can be shown from Figure 3 that when119886Pe119903 ge 978 V119900119901 ge 0495 It should be mentioned herethat having 120582fd gt 1 indicates that the heat transfer to thepermeable tube exposed to suction flow results in havinglower tube outlet temperature than that for the impermeabletube Consequently more thermal energy can be transferredfrom the permeable tube to the fluid in order to raise itsoutlet temperature to reach that of the impermeable tubeFurthermore Figure 4 shows that the second heat transferperformance indicator 120578119904fd becomes above one when V119900 gt
V119900119890 = 031606 Having 120578119904fd gt 1 indicates that thepermeable tube subjected to no-slip condition has smalleroutlet boundary temperature than that for the permeable tubesubjected to perfect fluid slip at the solid boundary This isnot the case for impermeable tubes as 120578119904fd lt 1 when V119900 = 0
as shown in Figure 4
42 The SystemThermal Performance for Linearly DistributedSuction Flow When 119861 = 0 the suction flow increaseslinearly from V119908 = 0 at the inlet (119909 = 0) to maximumvalue of V119908 = 2V119900 at the tube outlet (119909 = 1) For thiscase the thermal entry region effect is maximum as 119906avgis maximum at inlet In this region the axial convectioncoefficient is substantial as it is the coldest fluid region insidethe tube The thermal entry region influences the flow bycausing reductions in the boundary excess temperatures nearthe inlet Also the transverse convection is maximum at thetube outlet when 119861 = 0 as the suction velocity is maximumthere This decreases the boundary excess temperatures nearthe tube outlet As 119861 increases 119906avg close to the tube inletdecreases due to the increase in V119908 and V119908 close to thetube outlet decreases Accordingly both thermal entry regioneffect and transverse convection at the tube outlet decrease as119861 increases Both effects tend to increase both the boundaryexcess temperature and outlet mean bulk temperature As aresult 120579119898(119909 = 1) increases and 120582 decreases as 119861 increases asshown in Figure 5
For large 119886Pe119903 values both thermal entry region effectand the transverse convection at the tube outlet are largerthan those for smaller 119886Pe119903 values Accordingly 120582 valueswhen 119886Pe119903 = 10 are larger than those when 119886Pe119903 = 1
as shown in Figure 5 When 119886Pe119903 = 10 and 119861 = 0 it isnoticed that 120582 increases as V119900 increases This is because theoutlet mean bulk temperature decreases as V119900 increases forlarge 119886Pe119903 values as seen from Figure 2 Figure 5 shows that120582 values for V119900 = 13 plot are larger than those for V119900 = 049
plot when 119861 gt 046 This indicates that the convection dueto thermal entrance region effect as compared to transverseconvection increases as V119900 decreases when 119861 gt 046 and119886Pe119903 = 10 When 119886Pe119903 = 10 120578119904 is noticed from Figure 6to decrease as 119861 increases It is because that both thermalentry region effect and V119908 at the tube outlet decrease as 119861increases The former effect is due to the increase in V119908 nearthe tube inlet as 119861 increases Both effects tend to increase theexcess of Nu119904 above Nu Accordingly transverse convectiondue to fluid slip condition decreases due to the decrease inthe boundary excess temperature Consequently the axialconvection due to the fluid slip condition is increased more
than that due to the no-slip condition This causes 120579119904119898 toincrease as compared to 120579119898Thus 120578119904 decreases as 119861 increases
Figure 7 shows that 120582 increases as 119886Pe119903 increases At small119886Pe119903 values the transverse convection is negligible and Nu119904approaches Nu for large 119886Pe119903 values These lead to have 120578119904approaching one either as 119886Pe119903 decreases towards 119886Pe119903 =
0 or as 119886Pe119903 increases above 119886Pe119903 = 100 These trendscan be seen from Figure 8 For moderate 119886Pe119903 values theincrease in 119886Pe119903 not only causes an increase in the transverseconvection but it also results in reducing the boundary excesstemperature due to the increases in Nu This reduction ismore distinct for the case of the perfect fluid slip condition asNu119904 gt Nu The combined aforementioned effects can resultin reduction of transverse convection as 119886Pe119903 increases for theperfect fluid slip case It is because the transverse convectionis proportional to 119886Pe119903V119908(120579119882 minus 120579119898) as mentioned beforeAccordingly 120578119904 may have one local maximum at a specific119886Pe119903 value as clearly shown in Figure 8 for V119900 = 049 plots
43 The System Thermal Performance for Exponentially Dis-tributed Suction Flow For the upper 119862 value (119862 = 2)the thermal entry region effect is maximum since 119906avg ismaximum at tube inlet because V119908 is minimum there Alsothe transverse convection at the tube outlet is maximumas V119908 is maximum there These effects tend to decreasethe boundary excess temperatures at the tube outlet As 119862decreases 119906avg close to the tube inlet decreases due to theincrease in V119908 and V119908 close to the tube outlet decreasesAccordingly both thermal entry region effect and trans-verse convection at the tube outlet decrease as 119862 decreasesBoth effects increase both the boundary excess temperatureand outlet mean bulk temperature As a result the 120579119898(119909 =
1) increases and 120582 decreases as 119862 decreases as shown inFigure 9
For large 119886Pe119903 values both thermal entry region effectand outlet transverse convection are larger than those forsmaller 119886Pe119903 values Accordingly 120582 values when 119886Pe119903 = 10
are larger than those when 119886Pe119903 = 1 as shown in Figure 9When 119886Pe119903 = 10 and 119862 = 2 it is noticed that 120582 increases asV119900 increases It is because the outlet mean bulk temperaturedecreases as V119900 increases for large 119886Pe119903 values as shown fromFigure 2 Figure 9 shows that 120582 values for V119900 = 13 plot arelarger than those for V119900 = 049 plot when 119862 lt 06 Thisindicates that the flow gets more dominated by the thermalentry region effect than by the transverse convection as V119900decreases when 119862 lt 06 and 119886Pe119903 = 10 Figure 11 showsthat 120582 increases as 119886Pe119903 increases When 119886Pe119903 = 10 120578119904is noticed from Figure 10 to decrease as 119862 decreases It isbecause both thermal entry region effect and V119908 at the outletdecrease as 119862 decreases As indicated previously both effectstend to increase the excess of Nu119904 above Nu Accordingly thetransverse convection due to fluid slip condition decreasesdue to the decrease in the boundary excess temperatureConsequently the axial convection due to fluid slip conditionis increased more than that due to no-slip condition causing120579119904119898 to increase Thus 120578119904 decreases as 119862 decreases Figure 12shows that for moderate 119886Pe119903 values 120578119904 may have one localmaximumat a specific 119886Pe119903 valueThe physical interpretationfor this phenomenon is discussed in Section 42
10 Mathematical Problems in Engineering
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
Figure 5 Effects of 119861 on performance indicator 120582 and outlet meanbulk temperature for Case I
085
095
105
115
125
135
145
000 020 040 060 080 100
120578s
B
o = 025 033 049
o = 025
o = 049
3o = 1
aPer = 1 numericalaPer = 1 analyticalaPer = 10
Figure 6 Effects of 119861 on performance indicator 120578119904 for Case I
01
10
100
10E + 0210E + 0110E + 00
aPerB = 0B = o
B = 2o
120582
o = 025 033 049
o = 025 033 049
(44)
Figure 7 Effects of 119886Pe119903 on performance indicators 120582 for various 119861values for Case I
066
080
094
108
122
136
150
10E + 0210E + 0110E + 00
aPer
120578s
B = 0B = o
B = 2o
o = 025 033 049
(45)
Figure 8 Effects of 119886Pe119903 on performance indicators 120578119904 for various 119861values for Case I
010
048
085
123
160
198
235
minus200 minus120 minus040 040 120 20010E minus 01
10E + 02
10E + 01
10E + 00
120582
aPer = 1 numericalaPer = 1 analyticalaPer = 10
120582
120579m(x = 1)
120579m(x
=1)
C
o = 025 033 049
o = 025 033 049
Figure 9 Effects of 119862 on performance indicator 120582 and outlet meanbulk temperature for Case II
087
097
107
117
127
137
147
minus200 minus120 minus040 040 120 200C
120578s
aPer = 1 numericalaPer = 1 analyticalaPer = 10
o = 025 033 049
o = 025
o = 049
3o = 1
Figure 10 Effects of 119862 on performance indicator 120578119904 for Case II
8 Mathematical Problems in Engineering
01
10
100
10E + 0210E + 0110E + 00
aPer
120582
C = 20
C = 0
C = minus20
o = 025 033 049
o = 025 033 049
(44)
Figure 11 Effects of 119886Pe119903 on performance indicators 120582 for various119862values for Case II
075
088
101
114
127
140
153
10E + 0210E + 0110E + 00
aPerC = 20
C = 0
C = minus20
120578s
o = 025 033 049
(45)
Figure 12 Effects of 119886Pe119903 on performance indicators 120578119904 for various119862 values for Case II
the axial convection which is proportional to 119886Pe119903119906avg119889120579119898119909to decrease This may cause an increase in the outlet meanbulk temperature 119879119898(119909 = 119871) as shown in Figure 2 forsmall 119886Pe119903 values 119886Pe119903 le 4 When 119886Pe119903 ge 8 the trans-verse convection becomes significant as it is proportionalto 2119886Pe119903V119908(120579119882 minus 120579119898) The latter expression is equivalent to4119886Pe119903V119908Nu This expression increases as 119886Pe119903V119900 increasesThis increase causes significant reduction in the temperatureboundary layer thicknesswhich causes reduction in the outletmean bulk temperature as shown in Figure 2 The sum of theaxial and transverse convection is constant which is equalto the total convection heat transfer rate given by 119902conv =
119902
10158401015840
119904120587119863119871 When V119900 approaches V119900 = 05 the axial advection
which is proportional to 119906avg approaches zero at the tubeoutlet This causes 119879119898(119909 = 119871) to increase apparently as V119900approaches V119900 = 05 asymptotically as can be seen fromFigure 2
Linear suction
Exponential suction
Linear suctionprofile
Exponential suctionprofile
0000
0085
0170
0255
0340
0425
0510
10
100
oc
oc
120582m
ax
120582max
10E + 0210E + 0110E + 00
aPer
profile
profile
Figure 13 Effects of 119886Pe119903 on 120582max and V119900119888 for both Case I and CaseII
Figure 14 Effects of 119886Pe119903 on 120578119904119888 for both Case I and Case II
The Nusselt number for perfect fluid slip at the solidboundary (36) case is larger than that for the no-slip condi-tion case (24) as shown in Figure 3 that is Nu119904 gt Nu Thisindicates that the tube excess temperature at the boundary forthe perfect fluid slip condition is smaller than that for the no-slip condition case That excess temperature is proportionalto 120579119882 minus 120579119898 Consequently the transverse convection due tofluid slip is smaller than that due to no-slip condition as thisconvection is proportional to both 119886Pe119903V119908 and 120579119882 minus 120579119898 As aresult the axial convection due to fluid slip condition is largerthan that due to no-slip condition as the total convectionheat transfer rate is constant Therefore 119879119898(119909 = 119871) for thefluid slip condition is always larger than that for the no-slipcondition as shown in Figure 2 It is shown from Figure 3 thatthe asymptotic value of Nufd is equal to Nufd = 119886Pe119903V119908 Thisfinding is shown in the works of Kinney [9] and Raithby [11]
Figure 4 shows that the first heat transfer performanceindicator 120582fd is larger than one when 0 lt V119900 lt V119900119901
Mathematical Problems in Engineering 9
where the variation of V119900119901 with 119886Pe119903 is shown in Figure 3and given by (46) It can be shown from Figure 3 that when119886Pe119903 ge 978 V119900119901 ge 0495 It should be mentioned herethat having 120582fd gt 1 indicates that the heat transfer to thepermeable tube exposed to suction flow results in havinglower tube outlet temperature than that for the impermeabletube Consequently more thermal energy can be transferredfrom the permeable tube to the fluid in order to raise itsoutlet temperature to reach that of the impermeable tubeFurthermore Figure 4 shows that the second heat transferperformance indicator 120578119904fd becomes above one when V119900 gt
V119900119890 = 031606 Having 120578119904fd gt 1 indicates that thepermeable tube subjected to no-slip condition has smalleroutlet boundary temperature than that for the permeable tubesubjected to perfect fluid slip at the solid boundary This isnot the case for impermeable tubes as 120578119904fd lt 1 when V119900 = 0
as shown in Figure 4
42 The SystemThermal Performance for Linearly DistributedSuction Flow When 119861 = 0 the suction flow increaseslinearly from V119908 = 0 at the inlet (119909 = 0) to maximumvalue of V119908 = 2V119900 at the tube outlet (119909 = 1) For thiscase the thermal entry region effect is maximum as 119906avgis maximum at inlet In this region the axial convectioncoefficient is substantial as it is the coldest fluid region insidethe tube The thermal entry region influences the flow bycausing reductions in the boundary excess temperatures nearthe inlet Also the transverse convection is maximum at thetube outlet when 119861 = 0 as the suction velocity is maximumthere This decreases the boundary excess temperatures nearthe tube outlet As 119861 increases 119906avg close to the tube inletdecreases due to the increase in V119908 and V119908 close to thetube outlet decreases Accordingly both thermal entry regioneffect and transverse convection at the tube outlet decrease as119861 increases Both effects tend to increase both the boundaryexcess temperature and outlet mean bulk temperature As aresult 120579119898(119909 = 1) increases and 120582 decreases as 119861 increases asshown in Figure 5
For large 119886Pe119903 values both thermal entry region effectand the transverse convection at the tube outlet are largerthan those for smaller 119886Pe119903 values Accordingly 120582 valueswhen 119886Pe119903 = 10 are larger than those when 119886Pe119903 = 1
as shown in Figure 5 When 119886Pe119903 = 10 and 119861 = 0 it isnoticed that 120582 increases as V119900 increases This is because theoutlet mean bulk temperature decreases as V119900 increases forlarge 119886Pe119903 values as seen from Figure 2 Figure 5 shows that120582 values for V119900 = 13 plot are larger than those for V119900 = 049
plot when 119861 gt 046 This indicates that the convection dueto thermal entrance region effect as compared to transverseconvection increases as V119900 decreases when 119861 gt 046 and119886Pe119903 = 10 When 119886Pe119903 = 10 120578119904 is noticed from Figure 6to decrease as 119861 increases It is because that both thermalentry region effect and V119908 at the tube outlet decrease as 119861increases The former effect is due to the increase in V119908 nearthe tube inlet as 119861 increases Both effects tend to increase theexcess of Nu119904 above Nu Accordingly transverse convectiondue to fluid slip condition decreases due to the decrease inthe boundary excess temperature Consequently the axialconvection due to the fluid slip condition is increased more
than that due to the no-slip condition This causes 120579119904119898 toincrease as compared to 120579119898Thus 120578119904 decreases as 119861 increases
Figure 7 shows that 120582 increases as 119886Pe119903 increases At small119886Pe119903 values the transverse convection is negligible and Nu119904approaches Nu for large 119886Pe119903 values These lead to have 120578119904approaching one either as 119886Pe119903 decreases towards 119886Pe119903 =
0 or as 119886Pe119903 increases above 119886Pe119903 = 100 These trendscan be seen from Figure 8 For moderate 119886Pe119903 values theincrease in 119886Pe119903 not only causes an increase in the transverseconvection but it also results in reducing the boundary excesstemperature due to the increases in Nu This reduction ismore distinct for the case of the perfect fluid slip condition asNu119904 gt Nu The combined aforementioned effects can resultin reduction of transverse convection as 119886Pe119903 increases for theperfect fluid slip case It is because the transverse convectionis proportional to 119886Pe119903V119908(120579119882 minus 120579119898) as mentioned beforeAccordingly 120578119904 may have one local maximum at a specific119886Pe119903 value as clearly shown in Figure 8 for V119900 = 049 plots
43 The System Thermal Performance for Exponentially Dis-tributed Suction Flow For the upper 119862 value (119862 = 2)the thermal entry region effect is maximum since 119906avg ismaximum at tube inlet because V119908 is minimum there Alsothe transverse convection at the tube outlet is maximumas V119908 is maximum there These effects tend to decreasethe boundary excess temperatures at the tube outlet As 119862decreases 119906avg close to the tube inlet decreases due to theincrease in V119908 and V119908 close to the tube outlet decreasesAccordingly both thermal entry region effect and trans-verse convection at the tube outlet decrease as 119862 decreasesBoth effects increase both the boundary excess temperatureand outlet mean bulk temperature As a result the 120579119898(119909 =
1) increases and 120582 decreases as 119862 decreases as shown inFigure 9
For large 119886Pe119903 values both thermal entry region effectand outlet transverse convection are larger than those forsmaller 119886Pe119903 values Accordingly 120582 values when 119886Pe119903 = 10
are larger than those when 119886Pe119903 = 1 as shown in Figure 9When 119886Pe119903 = 10 and 119862 = 2 it is noticed that 120582 increases asV119900 increases It is because the outlet mean bulk temperaturedecreases as V119900 increases for large 119886Pe119903 values as shown fromFigure 2 Figure 9 shows that 120582 values for V119900 = 13 plot arelarger than those for V119900 = 049 plot when 119862 lt 06 Thisindicates that the flow gets more dominated by the thermalentry region effect than by the transverse convection as V119900decreases when 119862 lt 06 and 119886Pe119903 = 10 Figure 11 showsthat 120582 increases as 119886Pe119903 increases When 119886Pe119903 = 10 120578119904is noticed from Figure 10 to decrease as 119862 decreases It isbecause both thermal entry region effect and V119908 at the outletdecrease as 119862 decreases As indicated previously both effectstend to increase the excess of Nu119904 above Nu Accordingly thetransverse convection due to fluid slip condition decreasesdue to the decrease in the boundary excess temperatureConsequently the axial convection due to fluid slip conditionis increased more than that due to no-slip condition causing120579119904119898 to increase Thus 120578119904 decreases as 119862 decreases Figure 12shows that for moderate 119886Pe119903 values 120578119904 may have one localmaximumat a specific 119886Pe119903 valueThe physical interpretationfor this phenomenon is discussed in Section 42
10 Mathematical Problems in Engineering
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
Figure 11 Effects of 119886Pe119903 on performance indicators 120582 for various119862values for Case II
075
088
101
114
127
140
153
10E + 0210E + 0110E + 00
aPerC = 20
C = 0
C = minus20
120578s
o = 025 033 049
(45)
Figure 12 Effects of 119886Pe119903 on performance indicators 120578119904 for various119862 values for Case II
the axial convection which is proportional to 119886Pe119903119906avg119889120579119898119909to decrease This may cause an increase in the outlet meanbulk temperature 119879119898(119909 = 119871) as shown in Figure 2 forsmall 119886Pe119903 values 119886Pe119903 le 4 When 119886Pe119903 ge 8 the trans-verse convection becomes significant as it is proportionalto 2119886Pe119903V119908(120579119882 minus 120579119898) The latter expression is equivalent to4119886Pe119903V119908Nu This expression increases as 119886Pe119903V119900 increasesThis increase causes significant reduction in the temperatureboundary layer thicknesswhich causes reduction in the outletmean bulk temperature as shown in Figure 2 The sum of theaxial and transverse convection is constant which is equalto the total convection heat transfer rate given by 119902conv =
119902
10158401015840
119904120587119863119871 When V119900 approaches V119900 = 05 the axial advection
which is proportional to 119906avg approaches zero at the tubeoutlet This causes 119879119898(119909 = 119871) to increase apparently as V119900approaches V119900 = 05 asymptotically as can be seen fromFigure 2
Linear suction
Exponential suction
Linear suctionprofile
Exponential suctionprofile
0000
0085
0170
0255
0340
0425
0510
10
100
oc
oc
120582m
ax
120582max
10E + 0210E + 0110E + 00
aPer
profile
profile
Figure 13 Effects of 119886Pe119903 on 120582max and V119900119888 for both Case I and CaseII
Figure 14 Effects of 119886Pe119903 on 120578119904119888 for both Case I and Case II
The Nusselt number for perfect fluid slip at the solidboundary (36) case is larger than that for the no-slip condi-tion case (24) as shown in Figure 3 that is Nu119904 gt Nu Thisindicates that the tube excess temperature at the boundary forthe perfect fluid slip condition is smaller than that for the no-slip condition case That excess temperature is proportionalto 120579119882 minus 120579119898 Consequently the transverse convection due tofluid slip is smaller than that due to no-slip condition as thisconvection is proportional to both 119886Pe119903V119908 and 120579119882 minus 120579119898 As aresult the axial convection due to fluid slip condition is largerthan that due to no-slip condition as the total convectionheat transfer rate is constant Therefore 119879119898(119909 = 119871) for thefluid slip condition is always larger than that for the no-slipcondition as shown in Figure 2 It is shown from Figure 3 thatthe asymptotic value of Nufd is equal to Nufd = 119886Pe119903V119908 Thisfinding is shown in the works of Kinney [9] and Raithby [11]
Figure 4 shows that the first heat transfer performanceindicator 120582fd is larger than one when 0 lt V119900 lt V119900119901
Mathematical Problems in Engineering 9
where the variation of V119900119901 with 119886Pe119903 is shown in Figure 3and given by (46) It can be shown from Figure 3 that when119886Pe119903 ge 978 V119900119901 ge 0495 It should be mentioned herethat having 120582fd gt 1 indicates that the heat transfer to thepermeable tube exposed to suction flow results in havinglower tube outlet temperature than that for the impermeabletube Consequently more thermal energy can be transferredfrom the permeable tube to the fluid in order to raise itsoutlet temperature to reach that of the impermeable tubeFurthermore Figure 4 shows that the second heat transferperformance indicator 120578119904fd becomes above one when V119900 gt
V119900119890 = 031606 Having 120578119904fd gt 1 indicates that thepermeable tube subjected to no-slip condition has smalleroutlet boundary temperature than that for the permeable tubesubjected to perfect fluid slip at the solid boundary This isnot the case for impermeable tubes as 120578119904fd lt 1 when V119900 = 0
as shown in Figure 4
42 The SystemThermal Performance for Linearly DistributedSuction Flow When 119861 = 0 the suction flow increaseslinearly from V119908 = 0 at the inlet (119909 = 0) to maximumvalue of V119908 = 2V119900 at the tube outlet (119909 = 1) For thiscase the thermal entry region effect is maximum as 119906avgis maximum at inlet In this region the axial convectioncoefficient is substantial as it is the coldest fluid region insidethe tube The thermal entry region influences the flow bycausing reductions in the boundary excess temperatures nearthe inlet Also the transverse convection is maximum at thetube outlet when 119861 = 0 as the suction velocity is maximumthere This decreases the boundary excess temperatures nearthe tube outlet As 119861 increases 119906avg close to the tube inletdecreases due to the increase in V119908 and V119908 close to thetube outlet decreases Accordingly both thermal entry regioneffect and transverse convection at the tube outlet decrease as119861 increases Both effects tend to increase both the boundaryexcess temperature and outlet mean bulk temperature As aresult 120579119898(119909 = 1) increases and 120582 decreases as 119861 increases asshown in Figure 5
For large 119886Pe119903 values both thermal entry region effectand the transverse convection at the tube outlet are largerthan those for smaller 119886Pe119903 values Accordingly 120582 valueswhen 119886Pe119903 = 10 are larger than those when 119886Pe119903 = 1
as shown in Figure 5 When 119886Pe119903 = 10 and 119861 = 0 it isnoticed that 120582 increases as V119900 increases This is because theoutlet mean bulk temperature decreases as V119900 increases forlarge 119886Pe119903 values as seen from Figure 2 Figure 5 shows that120582 values for V119900 = 13 plot are larger than those for V119900 = 049
plot when 119861 gt 046 This indicates that the convection dueto thermal entrance region effect as compared to transverseconvection increases as V119900 decreases when 119861 gt 046 and119886Pe119903 = 10 When 119886Pe119903 = 10 120578119904 is noticed from Figure 6to decrease as 119861 increases It is because that both thermalentry region effect and V119908 at the tube outlet decrease as 119861increases The former effect is due to the increase in V119908 nearthe tube inlet as 119861 increases Both effects tend to increase theexcess of Nu119904 above Nu Accordingly transverse convectiondue to fluid slip condition decreases due to the decrease inthe boundary excess temperature Consequently the axialconvection due to the fluid slip condition is increased more
than that due to the no-slip condition This causes 120579119904119898 toincrease as compared to 120579119898Thus 120578119904 decreases as 119861 increases
Figure 7 shows that 120582 increases as 119886Pe119903 increases At small119886Pe119903 values the transverse convection is negligible and Nu119904approaches Nu for large 119886Pe119903 values These lead to have 120578119904approaching one either as 119886Pe119903 decreases towards 119886Pe119903 =
0 or as 119886Pe119903 increases above 119886Pe119903 = 100 These trendscan be seen from Figure 8 For moderate 119886Pe119903 values theincrease in 119886Pe119903 not only causes an increase in the transverseconvection but it also results in reducing the boundary excesstemperature due to the increases in Nu This reduction ismore distinct for the case of the perfect fluid slip condition asNu119904 gt Nu The combined aforementioned effects can resultin reduction of transverse convection as 119886Pe119903 increases for theperfect fluid slip case It is because the transverse convectionis proportional to 119886Pe119903V119908(120579119882 minus 120579119898) as mentioned beforeAccordingly 120578119904 may have one local maximum at a specific119886Pe119903 value as clearly shown in Figure 8 for V119900 = 049 plots
43 The System Thermal Performance for Exponentially Dis-tributed Suction Flow For the upper 119862 value (119862 = 2)the thermal entry region effect is maximum since 119906avg ismaximum at tube inlet because V119908 is minimum there Alsothe transverse convection at the tube outlet is maximumas V119908 is maximum there These effects tend to decreasethe boundary excess temperatures at the tube outlet As 119862decreases 119906avg close to the tube inlet decreases due to theincrease in V119908 and V119908 close to the tube outlet decreasesAccordingly both thermal entry region effect and trans-verse convection at the tube outlet decrease as 119862 decreasesBoth effects increase both the boundary excess temperatureand outlet mean bulk temperature As a result the 120579119898(119909 =
1) increases and 120582 decreases as 119862 decreases as shown inFigure 9
For large 119886Pe119903 values both thermal entry region effectand outlet transverse convection are larger than those forsmaller 119886Pe119903 values Accordingly 120582 values when 119886Pe119903 = 10
are larger than those when 119886Pe119903 = 1 as shown in Figure 9When 119886Pe119903 = 10 and 119862 = 2 it is noticed that 120582 increases asV119900 increases It is because the outlet mean bulk temperaturedecreases as V119900 increases for large 119886Pe119903 values as shown fromFigure 2 Figure 9 shows that 120582 values for V119900 = 13 plot arelarger than those for V119900 = 049 plot when 119862 lt 06 Thisindicates that the flow gets more dominated by the thermalentry region effect than by the transverse convection as V119900decreases when 119862 lt 06 and 119886Pe119903 = 10 Figure 11 showsthat 120582 increases as 119886Pe119903 increases When 119886Pe119903 = 10 120578119904is noticed from Figure 10 to decrease as 119862 decreases It isbecause both thermal entry region effect and V119908 at the outletdecrease as 119862 decreases As indicated previously both effectstend to increase the excess of Nu119904 above Nu Accordingly thetransverse convection due to fluid slip condition decreasesdue to the decrease in the boundary excess temperatureConsequently the axial convection due to fluid slip conditionis increased more than that due to no-slip condition causing120579119904119898 to increase Thus 120578119904 decreases as 119862 decreases Figure 12shows that for moderate 119886Pe119903 values 120578119904 may have one localmaximumat a specific 119886Pe119903 valueThe physical interpretationfor this phenomenon is discussed in Section 42
10 Mathematical Problems in Engineering
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
where the variation of V119900119901 with 119886Pe119903 is shown in Figure 3and given by (46) It can be shown from Figure 3 that when119886Pe119903 ge 978 V119900119901 ge 0495 It should be mentioned herethat having 120582fd gt 1 indicates that the heat transfer to thepermeable tube exposed to suction flow results in havinglower tube outlet temperature than that for the impermeabletube Consequently more thermal energy can be transferredfrom the permeable tube to the fluid in order to raise itsoutlet temperature to reach that of the impermeable tubeFurthermore Figure 4 shows that the second heat transferperformance indicator 120578119904fd becomes above one when V119900 gt
V119900119890 = 031606 Having 120578119904fd gt 1 indicates that thepermeable tube subjected to no-slip condition has smalleroutlet boundary temperature than that for the permeable tubesubjected to perfect fluid slip at the solid boundary This isnot the case for impermeable tubes as 120578119904fd lt 1 when V119900 = 0
as shown in Figure 4
42 The SystemThermal Performance for Linearly DistributedSuction Flow When 119861 = 0 the suction flow increaseslinearly from V119908 = 0 at the inlet (119909 = 0) to maximumvalue of V119908 = 2V119900 at the tube outlet (119909 = 1) For thiscase the thermal entry region effect is maximum as 119906avgis maximum at inlet In this region the axial convectioncoefficient is substantial as it is the coldest fluid region insidethe tube The thermal entry region influences the flow bycausing reductions in the boundary excess temperatures nearthe inlet Also the transverse convection is maximum at thetube outlet when 119861 = 0 as the suction velocity is maximumthere This decreases the boundary excess temperatures nearthe tube outlet As 119861 increases 119906avg close to the tube inletdecreases due to the increase in V119908 and V119908 close to thetube outlet decreases Accordingly both thermal entry regioneffect and transverse convection at the tube outlet decrease as119861 increases Both effects tend to increase both the boundaryexcess temperature and outlet mean bulk temperature As aresult 120579119898(119909 = 1) increases and 120582 decreases as 119861 increases asshown in Figure 5
For large 119886Pe119903 values both thermal entry region effectand the transverse convection at the tube outlet are largerthan those for smaller 119886Pe119903 values Accordingly 120582 valueswhen 119886Pe119903 = 10 are larger than those when 119886Pe119903 = 1
as shown in Figure 5 When 119886Pe119903 = 10 and 119861 = 0 it isnoticed that 120582 increases as V119900 increases This is because theoutlet mean bulk temperature decreases as V119900 increases forlarge 119886Pe119903 values as seen from Figure 2 Figure 5 shows that120582 values for V119900 = 13 plot are larger than those for V119900 = 049
plot when 119861 gt 046 This indicates that the convection dueto thermal entrance region effect as compared to transverseconvection increases as V119900 decreases when 119861 gt 046 and119886Pe119903 = 10 When 119886Pe119903 = 10 120578119904 is noticed from Figure 6to decrease as 119861 increases It is because that both thermalentry region effect and V119908 at the tube outlet decrease as 119861increases The former effect is due to the increase in V119908 nearthe tube inlet as 119861 increases Both effects tend to increase theexcess of Nu119904 above Nu Accordingly transverse convectiondue to fluid slip condition decreases due to the decrease inthe boundary excess temperature Consequently the axialconvection due to the fluid slip condition is increased more
than that due to the no-slip condition This causes 120579119904119898 toincrease as compared to 120579119898Thus 120578119904 decreases as 119861 increases
Figure 7 shows that 120582 increases as 119886Pe119903 increases At small119886Pe119903 values the transverse convection is negligible and Nu119904approaches Nu for large 119886Pe119903 values These lead to have 120578119904approaching one either as 119886Pe119903 decreases towards 119886Pe119903 =
0 or as 119886Pe119903 increases above 119886Pe119903 = 100 These trendscan be seen from Figure 8 For moderate 119886Pe119903 values theincrease in 119886Pe119903 not only causes an increase in the transverseconvection but it also results in reducing the boundary excesstemperature due to the increases in Nu This reduction ismore distinct for the case of the perfect fluid slip condition asNu119904 gt Nu The combined aforementioned effects can resultin reduction of transverse convection as 119886Pe119903 increases for theperfect fluid slip case It is because the transverse convectionis proportional to 119886Pe119903V119908(120579119882 minus 120579119898) as mentioned beforeAccordingly 120578119904 may have one local maximum at a specific119886Pe119903 value as clearly shown in Figure 8 for V119900 = 049 plots
43 The System Thermal Performance for Exponentially Dis-tributed Suction Flow For the upper 119862 value (119862 = 2)the thermal entry region effect is maximum since 119906avg ismaximum at tube inlet because V119908 is minimum there Alsothe transverse convection at the tube outlet is maximumas V119908 is maximum there These effects tend to decreasethe boundary excess temperatures at the tube outlet As 119862decreases 119906avg close to the tube inlet decreases due to theincrease in V119908 and V119908 close to the tube outlet decreasesAccordingly both thermal entry region effect and trans-verse convection at the tube outlet decrease as 119862 decreasesBoth effects increase both the boundary excess temperatureand outlet mean bulk temperature As a result the 120579119898(119909 =
1) increases and 120582 decreases as 119862 decreases as shown inFigure 9
For large 119886Pe119903 values both thermal entry region effectand outlet transverse convection are larger than those forsmaller 119886Pe119903 values Accordingly 120582 values when 119886Pe119903 = 10
are larger than those when 119886Pe119903 = 1 as shown in Figure 9When 119886Pe119903 = 10 and 119862 = 2 it is noticed that 120582 increases asV119900 increases It is because the outlet mean bulk temperaturedecreases as V119900 increases for large 119886Pe119903 values as shown fromFigure 2 Figure 9 shows that 120582 values for V119900 = 13 plot arelarger than those for V119900 = 049 plot when 119862 lt 06 Thisindicates that the flow gets more dominated by the thermalentry region effect than by the transverse convection as V119900decreases when 119862 lt 06 and 119886Pe119903 = 10 Figure 11 showsthat 120582 increases as 119886Pe119903 increases When 119886Pe119903 = 10 120578119904is noticed from Figure 10 to decrease as 119862 decreases It isbecause both thermal entry region effect and V119908 at the outletdecrease as 119862 decreases As indicated previously both effectstend to increase the excess of Nu119904 above Nu Accordingly thetransverse convection due to fluid slip condition decreasesdue to the decrease in the boundary excess temperatureConsequently the axial convection due to fluid slip conditionis increased more than that due to no-slip condition causing120579119904119898 to increase Thus 120578119904 decreases as 119862 decreases Figure 12shows that for moderate 119886Pe119903 values 120578119904 may have one localmaximumat a specific 119886Pe119903 valueThe physical interpretationfor this phenomenon is discussed in Section 42
10 Mathematical Problems in Engineering
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
44 The System Maximum Thermal Performance IndicatorFigure 13 shows that the maximum first performance indica-tor 120582max increases as 119886Pe119903 increases Also the critical suctionparameter V119900119888 that produces 120582max is seen from this figure toincrease as 119886Pe119903 increases Moreover Figure 13 demonstratesthat 120582max for the exponential suction flow distribution isalways larger than that for the linear suction flowdistributionTwo interesting findings can be withdrawn from Figure 14The first one is that when 119886Pe119903 = 11 both suction flow dis-tributions will have their maximum 120578119904119862 values where 120578119904119862 isthe value of 120578119904 at the conditions that produce120582maxThe secondone is that when 119886Pe119903 = 4 both suction flow distributionswill have 120578119904 = 120578119904119862 = 1 It is shown from Figure 13 thatthe maximum enhancement ratios are 1762-fold and 1467-fold above those for impermeable tubes for the exponentialand linear suction velocity distributions respectively Thesevalues are obtained at 119886Pe119903 = 100 Finally the phenomenonthat there is proper flux distribution that maximizes the heattransfer enhancement indicator agrees with the findings ofKhaled [23] andWang et al [24]The flux in the present workis considered to be the mass flux of the suction flow while itis the heat flux in Khaledrsquos [23] work and it is a localized massflux close to the boundary in Wang et alrsquos [24] work
5 Conclusions
Flow and heat transfer in permeable tubes subjected totransverse suction flow were analyzed in this work Thecontinuity momentum and energy equations of the internalfluid were solved analytically and numericallyThe numericaland the analytical results based on negligible combined entryregions were well matched Two different suction velocitydistributions were considered They are the linear and expo-nential distributions The influence of the average suctionvelocity the suction velocity profile and the Peclet numberon the heat transfer enhancement indicators was studiedIt was found that heat transfer enhancement over that inimpermeable tubes is only attainable if large Peclet numbersare encountered This enhancement is further increasedas suction velocities towards the tube outlet increase andas those towards the tube inlet decrease simultaneouslyThe enhancement mechanisms were identified and they areexpanding the entry regions increasing the transverse advec-tion and increasing the downstream excess temperaturesunder same transverse advection The maximum reportedenhancement ratios in this work are 1762-fold and 1467-foldabove those for impermeable tubes for the exponential andlinear suction velocity distributions respectivelyThe averagesuction velocity that maximizes the heat transfer enhance-ment indicator increases as the Peclet number increasesuntil it reaches asymptotically its uppermost value at largePeclet numbers Finally this work reveals that significant heattransfer enhancement is attainable when the suction flowinside the permeable tube is managed properly
Nomenclature119886 Aspect ratio (119886 = 119863[2119871])119861 Controlling parameter for suction flow
with linear profile (26)
119862 Controlling parameter for suction flowwith exponential profile (27)
119888119901 Fluid specific heat (JkgK)119863 Tube inner diameter (m)ℎ Convection heat transfer coefficient
(Wm2K)119896 Fluid thermal conductivity (WmsdotK)119871 Tube length (m) Mass flow rate at given section (15)119900 Mass flow rate at inlet section (kgs)Nu Nusselt number (Nu = ℎ119863119896)Pe119903 Reference Peclet number
distance (119903 = 2119903119863)119879 Fluid temperature field (K)1198791 Inlet fluid temperature (K)119879119898 Fluid mean bulk temperature field (K)119879119882 Tube inner boundary temperature (K)119906 119906 (Dimensional dimensionless) axial
velocity field (119906 = 119906119906119900)119906119904 Axial velocity field under perfect slip
condition (ms)119906119904 Dimensionless axial velocity field under
velocity (V = V119886119906119900)V119908 V119908 (Dimensional dimensionless) local
suction velocity (ms)V119900 V119900 (Dimensional dimensionless) average
suction velocity (ms)119909 119909 Dimensional and dimensionless axial
distances (119909 = 119909119871)
Greek Symbols
120578119904 Second heat transfer enhancementindicator (40)
120582 First heat transfer enhancementindicator (38)
120583 Fluid dynamic viscosity (Nsm2)120579 Dimensionless temperature field (7b)120579119904 Dimensionless temperature field
under perfect slip condition (34)120579119898 Dimensionless mean bulk
temperature (18)120579119882 Tube dimensionless inner boundary
temperature (17)120588 Fluid density (kgm3)
Mathematical Problems in Engineering 11
Subscripts
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
avg Average value of the quantityfd Fully developed value of the quantity119904 Quantity under perfect slip flow at
the solid boundary
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was funded by the Deanship of Scientific research(DSR) King Abdulaziz University JeddahThe author there-fore acknowledges with thanks DSR technical and financialsupport
References
[1] O D Makinde S Khamis M S Tshehla and O Franks ldquoAnal-ysis of heat transfer in Berman flow of nanofluids with Navierslip viscous dissipation and convective coolingrdquo Advances inMathematical Physics vol 2014 Article ID 809367 13 pages2014
[2] A E Bergles ldquoTechniques to enhance heat transferrdquo in Hand-book of Heat Transfer W M Rohsenow J P Hartnett and Y ICho Eds chapter 11 McGraw-Hill New York NY USA 3rdedition 1998
[3] D Westphalen K W Roth and J Brodrick ldquoHeat transferenhancementrdquoASHRAE Journal vol 48 no 4 pp 68ndash71 2006
[4] I Antonir and A Tamir ldquoThe effect of surface suction on con-densation in presence of a non-condensable gasrdquo Journal ofHeatTransfermdashTransactions of the ASME vol 99 no 3 pp 496ndash4991977
[5] J Lienhard and V Dhir ldquoA simple analysis of laminar filmcondensation with suctionrdquo Transactions ASMEmdashJournal ofHeat Transfer vol 94 no 3 pp 334ndash336 1972
[6] A-RAKhaledAMRadhwan and SAAl-Muaike ldquoAnalysisof laminar falling film condensation over a vertical plate with anaccelerating vapor flowrdquo Journal of Fluids Engineering vol 131no 7 Article ID 0713041 2009
[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002
[8] S-C Wang C-K Chen and Y-T Yang ldquoNatural convectionof non-Newtonian fluids through permeable axisymmetric andtwo-dimensional bodies in a porous mediumrdquo InternationalJournal of Heat and Mass Transfer vol 45 no 2 pp 393ndash4082001
[9] R B Kinney ldquoFully developed frictional and heat-transfer char-acteristics of laminar flow in porous tubesrdquo International Jour-nal of Heat and Mass Transfer vol 11 no 9 pp 1393ndash1401 1968
[10] R J Pederson and R B Kinney ldquoEntrance-region heat transferfor laminar flow in porous tubesrdquo International Journal of Heatand Mass Transfer vol 14 no 1 pp 159ndash161 1971
[11] G Raithby ldquoLaminar heat transfer in the thermal entranceregion of circular tubes and two-dimensional rectangular ductswith wall suction and injectionrdquo International Journal of Heatand Mass Transfer vol 14 no 2 pp 223ndash243 1971
[12] M M Sorour andM A Hassab ldquoEffect of sucking the hot fluidfilm on the performance of flat plate solar energy collectorsrdquoApplied Energy vol 14 no 3 pp 161ndash173 1983
[13] M M Sorour M A Hassab and S Estafanous ldquoDevelopinglaminar flow in a semiporous two-dimensional channel withnonuniform transpirationrdquo International Journal of Heat andFluid Flow vol 8 no 1 pp 44ndash54 1987
[14] V I Bubnovich N O Moraga and C E Rosas ldquoNumericalforced convection in a circular pipe with nonuniform blowingor suction through the porous wallrdquo Numerical Heat TransferPart A Applications vol 33 no 8 pp 875ndash890 1998
[15] G J Hwang Y C Cheng and M L Ng ldquoDeveloping laminarflow and heat transfer in a square ductwith one-walled injectionand suctionrdquo International Journal of Heat and Mass Transfervol 36 no 9 pp 2429ndash2440 1993
[16] A-R A Khaled M Siddique N I Abdulhafiz and A Y Bou-khary ldquoRecent advances in heat transfer enhancements areview reportrdquo International Journal of Chemical Engineeringvol 2010 Article ID 106461 28 pages 2010
[17] A Bejan Convective Heat Transfer Wiley New York NY USA2nd edition 1995
[18] P H Oosthuizen andD Naylor Introduction to Convective HeatTransfer Analysis McGraw-Hill New York NY USA 1999
[19] A-R A Khaled ldquoModeling and computation of heat transferthrough permeable hollow-pin systemsrdquo Advances in Mechani-cal Engineering vol 2012 Article ID 587165 12 pages 2012
[20] S C Chapra and P R CanaleNumerical Methods for EngineersMcGraw-Hill New York NY USA 2009
[21] A-R A Khaled ldquoHeat transfer enhancement in a verticaltube confining two immiscible falling co-flowsrdquo InternationalJournal of Thermal Sciences vol 85 pp 138ndash150 2014
[22] F G Blottner ldquoFinite difference methods of solution of theboundary-layer equationsrdquo American Institute of Aeronauticsand Astronautics vol 8 pp 193ndash205 1970
[23] A-R A Khaled ldquoHeat transfer enhancement due to properlymanaging the distribution of the heat flux exact solutionsrdquoEnergy Conversion and Management vol 53 no 1 pp 247ndash2582012
[24] J Wang CWu and K Li ldquoHeat transfer enhancement throughcontrol of added perturbation velocity in flow fieldrdquo EnergyConversion and Management vol 70 pp 194ndash201 2013
