Research Article Theoretical Analysis of Effects of Wall Suction...

Research ArticleTheoretical Analysis of Effects ofWall Suction on Entropy Generation Rate in LaminarCondensate Layer on Horizontal Tube

Tong-Bou Chang12

1Department of Mechanical and Energy Engineering National Chiayi University No 300 Syuefu Road Chiayi City 600 Taiwan2Center of Energy Research amp Sensor Technology National Chiayi University No 300 Syuefu Road Chiayi City 600 Taiwan

Correspondence should be addressed to Tong-Bou Chang tbchangmailncyuedutw

Received 28 August 2014 Revised 11 November 2014 Accepted 17 November 2014 Published 4 December 2014

Academic Editor Hang Xu

Copyright copy 2014 Tong-Bou Chang 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

The effects of wall suction on the entropy generation rate in a two-dimensional steady film condensation flow on a horizontal tubeare investigated theoretically In analyzing the liquid flow the effects of both the gravitational force and the viscous force are takeninto account In addition a film thickness reduction ratio 119878119891 is introduced to evaluate the effect of wall suction on the thicknessof the condensate layer The analytical results show that the entropy generation rate depends on the Jakob number Ja the Rayleighnumber Ra the Brinkman number Br the dimensionless temperature difference 120595 and the wall suction parameter 119878119908 In additionit is shown that in the absence of wall suction a closed-form correlation for the Nusselt number can be derived Finally it is shownthat the dimensionless entropy generation due to heat transfer119873119879 increases with an increasing suction parameter 119878119908 whereas thedimensionless entropy generation due to liquid film flow friction119873119865 decreases

1 Introduction

Film condensation on a horizontal tube has many thermalengineering applications including chemical vapor deposi-tion distillation and heat exchange The problem of laminarfilm condensation flow was first investigated by Nusselt [1]who considered the case of film condensation on a verticalplateThe results showed that a local balance existed betweenthe viscous force and the weight of the condensate filmprovided that three simplifying assumptions were satisfiednamely (1) the condensate film was very thin (2) the convec-tive and inertial effects were very small and (3) the tempera-ture within the condensate layer varied linearly over the filmthickness In later studies researchers investigated the lami-nar film condensation of quiescent vapors under more real-istic assumptions [2ndash5] Yang and Chen [6] investigated theeffects of surface tension and ellipticity on laminar film con-densation on a horizontal elliptical tube The results showedthat the heat transfer coefficient increased with an increasing

surface tension force and tube ellipticity Hu and Chen [7]investigated the problem of turbulent film condensation onan inclined elliptical tube and found that the heat transferperformance improved as the vapor velocity increased Addi-tionally it was shown that a circular tube resulted in a higherheat transfer coefficient than an elliptical tube

Irreversibilities due to heat transfer and friction inevitablyexist in practical thermal systemsThis phenomenon referredto as entropy generation reduces the energy available toperform work and should therefore be minimized Bejan [8]proposed a method for minimizing entropy generation insuch key applications as power generation refrigeration andenergy conservation Bejan [9] also conducted a second lawthermodynamics analysis of the entropy generation mini-mization problem for single-phase convection heat transferThe same author [10] devised effective methods for minimiz-ing entropy generation in heat transfer systems consistingof flat plates or cylinders placed in a crossflow Saouli andAıboud-Saouli [11] performed a second law thermodynamic

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 172605 8 pageshttpdxdoiorg1011552014172605

2 Mathematical Problems in Engineering

analysis of the convection heat transfer problem in single-phase falling liquid film flow along an inclined heated plateIn investigating the problem of heat transfer with phase-change Adeyinka and Naterer [12] analyzed the entropy gen-eration and energy availability in vertical film condensationheat transfer The results indicated that entropy generationprovides a useful parameter for optimizing two-phase sys-tems Dung and Yang [13] applied the entropy generationminimization method proposed by Bejan in [8] to optimizefilm condensation heat transfer on a horizontal tube Theirresults showed that the optimal group Rayleigh parameterexists over the parametric range investigated for horizontaltube at which the entropy is generated at a minimum rate Liand Yang [14] applied the entropy generation minimizationmethod to optimize the heat transfer performance of a hori-zontal elliptical cylinder in a saturated vapor flow In a recentstudy Chang and Wang [15] investigated the heat transfercharacteristics and entropy generation rate of a condensatefilm on a horizontal plate and found that the overall entropygeneration rate induced by the heat transfer irreversibilityeffect is equivalent to the Nusselt number

The problem of laminar film condensation with wallsuction effects has beenwidely discussed in the literature [16ndash18] In general the results have shown that wall suction signif-icantly enhances the condensation heat transfer performanceHowever the literature lacks a systematic investigation intothe effects of wall suction on the entropy generation rate inlaminar film condensation on a horizontal tube Accordinglythe present study performs an analytical investigation intothe effects of the Jakob number Rayleigh number Brinkmannumber dimensionless temperature difference and suctionforce on the dimensionless entropy generation rate for ahorizontal tube in a stationary saturated vapor Notably inperforming the analysis the effects of both the gravitationalforce and the viscous force are taken into explicit account

2 Analysis

Consider a pure vapor in a saturated state condensing on ahorizontal and permeable tube (see Figure 1) Assume thatthe vapor has a uniform temperature 119879sat and the tube (withdiameter 119863) has a constant wall temperature 119879119908 If 119879sat ishigher than 119879119908 a thin liquid condensate-layer is formed onthe surface of the tube In analyzing the heat transfer charac-teristics of the condensate film the same assumptions as thoseused byRohsenow [19] are applied namely (1) the condensatefilm flow is steady and laminar and thus the effects of inertiaand convection can be ignored (ie a creeping film flowis assumed) (2) the wall temperature vapor temperatureand properties of the dry vapor and condensate respectivelyare constant and (3) the condensate film has negligiblekinetic energy Consequently the governing equations for thecondensate layer can be formulated as follows

continuity equation

120597119906

120597119909+120597V120597119910= 0 (1)

Vapor temperature

Wall temperature

Liquid-vapor interface

Wall suction

Ein

EsuckEcond

Eout

x

Vw

y

120575

120575

D

Tsat

x120579

g

Tw

x + dx

Figure 1 Physical model

momentum equation

0 = 1205831205972119906

1205971199102+ (120588 minus 120588V) 119892 sin 120579 (2)

energy equation

0 = 1205721205972119879

1205971199102 (3)

In (1)ndash(3) the (119909 119910) coordinates are local coordinates and areused specifically to simplify the governing equations Thelocal coordinate frame (119909 119910) and cylindrical coordinateframe (119903 120579) are related as 119909 = (1198632)120579 and 119910 = 119903 minus (1198632)The boundary conditions are given as follows

at the tube surface that is 119910 = 0

V = V119908 119879 = 119879119908 (4)

at the liquid-vapor interface that is 119910 = 120575

119879 = 119879sat (5)

Mathematical Problems in Engineering 3

Integrating the momentum equation given in (2) usingthe boundary conditions given in (4) and (5) the velocity dis-tribution within the condensate film is obtained as

119906 =(120588 minus 120588V) 119892 sin 120579

120583(120575119910 minus

1

21199102) (6)

Meanwhile integrating the energy equation given in (3)using the boundary conditions given in (4) and (5) the temp-erature profile is obtained as

119879 = 119879119908 + Δ119879119910

120575 (7)

Utilizing the second law approach proposed by Bejan [8]the local entropy generation equation for the convection heattransfer within the condensate film is given as

11987810158401015840=119896

1198792119908

[(120597119879

120597119909)

2

+ (120597119879

120597119910)

2

] +120583

119879119908

(120597119906

120597119910)

2

= 11987810158401015840119879 + 11987810158401015840119865

(8)

where 119896 is the thermal conductivity of the liquid condensateThe local entropy generation equation contains two com-

ponents namely the entropy generation due to heat transfer11987810158401015840119879 and the entropy generation due to fluid friction 119878

10158401015840119865 From

(8) 11987810158401015840119879 and 11987810158401015840119865 are defined respectively as

11987810158401015840119879 =

119896

1198792119908

[(120597119879

120597119909)

2

+ (120597119879

120597119910)

2

] (9)

11987810158401015840119865 =

120583

119879119908

(120597119906

120597119910)

2

(10)

Substituting the velocity distribution in (6) and the lineartemperature profile in (7) into (9) and (10) the local entropygeneration due to heat transfer and fluid friction can berewritten respectively as

11987810158401015840119879 =

119896

1198792119908

(Δ119879

120575)

2

11987810158401015840119865 =

[(120588 minus 120588V) 119892 sin 120579]2

119879119908120583(120575 minus 119910)

2

(11)

Integrating (11) with respect to 119910 the local entropygeneration across the film thickness is obtained as

1198781015840= int

120575

011987810158401015840119889119910 = 119878

1015840119879 + 1198781015840119865

1198781015840119879 = int

120575

011987810158401015840119879119889119910 =

119896

1198792119908

Δ1198792

120575

1198781015840119865 = int

120575

011987810158401015840119865119889119910 =

[(120588 minus 120588V) 119892 sin 120579]2

31198791199081205831205753

(12)

Integrating (12) with respect to 120579 the total entropygeneration within the condensate film can be derived asfollows

119878 = int

120587

01198781015840119889120579 = 119878119879 + 119878119865 (13)

119878119879 = int

120587

0

119896

1198792119908

Δ1198792

120575119889120579 =

119896Δ1198792

1198792119908

int

120587

0

1

120575119889120579 (14)

119878119865 = int

120587

0

[(120588 minus 120588V) 119892 sin 120579]2

31198791199081205831205753119889120579

=[(120588 minus 120588V) 119892]

2

3119879119908120583int

120587

01205753sin2120579119889120579

(15)

Let the characteristic entropy generation rate be definedas

1198780 =119896

119863(Δ119879

119879119908

)

2

(16)

The dimensionless total entropy generation can then becalculated as

119873 =119878

1198780

= 119873119879 + 119873119865 (17)

Substituting (16) and (17) into (13)ndash(15) the dimension-less entropy generation numbers for the heat transfer andfluid friction irreversibilities are obtained respectively as fol-lows

119873119879 = int

120587

0

119863

120575119889120579

119873119865 =119863119879119908 [(120588 minus 120588V) 119892]

2

3119896120583Δ1198792int

120587

01205753sin2120579119889120579

(18)

Let the following dimensionless parameters be intro-duced

1199060 =(120588 minus 120588V) 119892

1205831198632 Br =

12058311990620

119896Δ119879 Ψ =

Δ119879

119879119908

(19)

Furthermore let the dimensionless liquid film thicknessbe defined as

120575lowast=120575

119863 (20)

Substituting (19) into (18) and using (20) the dimension-less entropy generation number equations can be rewritten as

119873119879 = int

120587

0

1

120575lowast119889120579 (21)

119873119865 =Br3Ψ

int

120587

0(120575lowast)3 sin2120579119889120579 (22)

However 119873119879 and 119873119865 cannot yet be derived since 120575lowast isunknown


To solve 120575lowast assume that the first law of thermodynamicsand the mass conservation equation are coupled in thegoverning equations The schematic presented at the top ofFigure 1 shows the energy balance within a small controlvolume of liquid condensate extending from 119909 to 119909+119889119909 Theenergy flow entering the control volume is given as in =

int120575

0120588119906(ℎ119891119892 +Cp(119879sat minus119879))119889119910|119909 while that exiting the control

volume is given as out = int120575

0120588119906(ℎ119891119892 +Cp(119879sat minus119879))119889119910|119909+119889119909

Furthermore the net energy sucked out of the condensatelayer is equal to suck = [120588(ℎ119891119892 +CpΔ119879)V119908]119889119909 while the heattransferred into the condensate layer as a result of conductionis equal to cond = minus119896(120597119879120597119910)119889119909 Therefore the overallenergy balance in the liquid film that is in minus out minus suck =cond can be expressed as

119889

119889119909int

120575

0120588119906 (ℎ119891119892 + Cp (119879sat minus 119879)) 119889119910119889119909

+ [120588 (ℎ119891119892 + CpΔ119879) V119908] 119889119909 = 119896120597119879

120597119910119889119909

(23)

where the first term on the left-hand side of (23) representsthe net energy flux across the liquid film (ie from 119909 to 119909 +119889119909) while the second term represents the net energy suckedout of the condensate layer

Substituting (6) and (7) into (23) andusing the correlation119889119909 = (1198632)119889120579 (23) can be rewritten as

120588 (120588 minus 120588V) 119892 (ℎ119891119892 + (38)CpΔ119879)3120583

2119889

1198631198891205791205753 sin 120579

+ 120588 (ℎ119891119892 + CpΔ119879) V119908 = 119896Δ119879

120575

(24)

For analytical convenience let the following dimensionlessparameters be introduced

Ja = CpΔ119879ℎ119891119892 + (38)CpΔ119879

Pr =120583Cp119896

Ra =120588 (120588 minus 120588V) 119892Pr119863

3

1205832

Re119908 =120588V119908119863120583

119878119908 = (1 +5

8Ja)Re119908

PrRa

(25)

Substituting (25) and (20) into (24) yields the following

120575lowast 119889

119889120579(120575lowast3 sin 120579) + 3

2119878119908120575lowast=3

2

JaRa (26)

The boundary conditions for the liquid film thickness aregiven as

119889120575lowast

119889120579= 0 at 120579 = 0 (27a)

120575lowast997888rarr infin at 120579 = 120587 (27b)

Assuming that the wall suction effect is ignored (ie 119878119908 = 0)(26) can be expressed as

120575lowast1003816100381610038161003816119878119908=0

119889

119889120579((120575lowast1003816100381610038161003816119878119908=0)3sin 120579) = 3

2

JaRa (28)

where 120575lowast|119878119908=0 is the dimensionless local liquid film thicknessin the absence of wall suction

Using the separation of variables method the analyticalsolution for the dimensionless local film thickness can bederived as

120575lowast1003816100381610038161003816119878119908=0 = sinminus13120579 (2 Ja

Ra)

14

(int

120579

0sin13120579119889120579)

14

(29)

The value of 120575lowast|119878119908=0 along the surface of the horizontal

tube can then be calculated by integrating int1205790sin13120579119889120579

Let the effect of wall suction on the thickness of the con-densate layer be characterized by the following film thicknessreduction ratio

119878119891 = 1 minus120575lowast

120575lowast|119878119908=0

(30)

Since 120575lowast le 120575lowast|119878119908=0 the value of 119878119891 falls within the range of

0 le 119878119891 le 1 Substituting (30) into (28) the following equationfor 119878119891 is obtained

3 (120575lowast1003816100381610038161003816119878119908=0)4sin 120579 (1 minus 119878119891)

3 119889119891

119889120579

+ 120575lowast1003816100381610038161003816119878119908=0

119889

119889120579((120575lowast1003816100381610038161003816119878119908=0)3sin 120579)

times (1198784119891 minus 4119878

3119891 + 6119878

2119891 minus 4119878119891)

+3

2120575lowast1003816100381610038161003816119878119908=0 (1 minus 119878119891) times 119878119908 = 0

(31)

Setting 120579 to 0 the initial condition 119878119891 (0) is obtained as

1198601198784119891 (0) + 119861119878

3119891 (0) + 119862119878

2119891 (0) + 119863119878119891 (0) + 119864 = 0 (32)

where

119860 = (120575lowast1003816100381610038161003816119878119908=0120579=0)

4

119861 = minus4119860

119862 = 6119860

119863 = minus4119860 minus3

2(120575lowast1003816100381610038161003816119878119908=0120579=0) times 119878119908

119864 =3

2(120575lowast1003816100381610038161003816119878119908=0120579=0) times 119878119908

(33)

The exact value of 119878119891 (0) in (32) can be determinedusing the bisection method [20] The variation of 119878119891 in the 120579direction can then be obtained by substituting 119878119891 (0) and (29)


into (31) Furthermore the dimensionless local liquid filmthickness can be derived as

120575lowast(120579) = (1 minus 119878119891) times 120575

lowast1003816100381610038161003816119878119908=0

= (1 minus 119878119891) sinminus13

120579 (2JaRa)

14

(int

120579

0sin13120579119889120579)

14

(34)

In general the local Nusselt number is given by

Nu120579 =ℎ120579119863

119896 (35)

where

ℎ120579 =119896

120575 (36)

Substituting (34) into (35) the local Nusselt number can berewritten as

Nu120579 =1

120575lowast (120579)

= (RaJa)

14 sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

(37)

The mean Nusselt number can then be derived as

Nu = 1120587int

120587

0

1

120575lowast (120579)119889120579

=1

120587(RaJa)

14

int

120587

0

sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

119889120579

(38)

Comparing (21) and (38) the dimensionless entropygeneration due to heat transfer119873119879 can be obtained as

119873119879 = int

120587

0

1

120575lowast119889120579 = 120587 timesNu (39)

3 Results and Discussion

In the present study the working fluid was assumed to bewater vapor (one of the most commonly used liquids inengineering applications) Moreover for the case where thewall section effect was ignored (ie the suction Reynoldsnumber velocity Re119908 was set equal to zero) the mean Nusseltnumber was derived by substituting 119878119891 = 0 into (38) that is

Nu10038161003816100381610038161003816119878119908=0=1

120587(RaJa)

14

int

120587

0sin13120579(4int

120579

0sin13120579119889120579)

minus14

119889120579

(40)

In addition an explicit formulation for the mean Nusseltnumber was obtained by using a simple numerical method

[20] to deal with the integration term int120587

0sin13120579(4 int120579

0sin13

120579119889120579)minus14

119889120579 in (32) yielding

Nu10038161003816100381610038161003816119878119908=0= 1224 times (

RaJa)

14

(41)

Yang and Chen [6] used a novel transformation methodto investigate the problem of film condensation on a hori-zontal elliptical tube in the absence of wall suction Howeverthe parameters defined in [6] differ from those used in thecurrent analysis Therefore to enable a direct comparison tobe made with the present results the formulations presentedin [6] should be transformed from their original formats andexpressed in terms of the current parameters Based on thederivations presented in [6] the mean Nusselt number for acircular tube should be transformed as

Nu = 1225 times (RaJa)

14

(42)

It is evident that a good agreement exists between (41) and(42)Thus the basic validity of the analytical model proposedin the present study is confirmed

As shown in (17) the dimensionless entropy generationin the condensate layer on the horizontal tube is induced byboth heat transfer and liquid film flow friction In (19) 1199060 Brand Ψ are defined as 1199060 = ((120588 minus 120588V)119892120583)119863

2 Br = 12058311990620119896Δ119879and Ψ = Δ119879119879119908 respectively It thus follows that (Br120595) =((120588 minus 120588V)

211989221198634120583119896119879119908) For the water vapor considered in the

present study Br120595 has a value of 5 given a tube diameter of119863 = 38 inches and a temperature of 100∘C Figure 2 showsthat the dimensionless entropy generation caused by heattransfer119873119879 increases with an increasing suction parame-ter119878119908 This finding is reasonable since the thickness of the liquidfilm reduces with an increasing suction effect and there-fore improves the heat transfer performance Moreover thefinding is consistent with (38) and (39) which show that119873119879is equal to 120587 times the mean Nusselt number Nu and Nuincreases with increasing 119878119891 In addition Figure 2 shows thatthe dimensionless entropy generation due to heat transfer119873119879 is proportional to (RaJa)

14 as predicted by both (41)and (42)

Figure 3 shows that the dimensionless entropy generationdue to liquid film flow friction119873119865 decreases with an increas-ing suction parameter 119878119908 Again this finding is reasonablesince as the suction parameter increases a greater amountof liquid is sucked into the porous tube Consequently thequantity of liquid condensate on the tube surface is reducedand thus the entropy generation caused by liquid film flowfriction also reduces Figure 3 shows that the dimensionlessentropy generation due to liquid film flow friction 119873119865reduces as (RaJa)14 increases (ie an opposite tendency tothat observed for119873119879 (or Nu)) This result is reasonable sincea larger value of119873119879 (or Nu) implies the existence of a thinnercondensate film on the tube surface and hence a lower liquidfilm flow friction

As discussed above the dimensionless entropy genera-tion due to heat transfer increases with increasing (RaJa)whereas the dimensionless entropy generation due to liquid


10

20

8

6

4

21 1003 5 7 9 20 40 60 80

RaJa

NT

BrΨ = 50

Sw = 02

Sw = 01

Sw = 005

Sw = 00

Figure 2 Dimensionless entropy generation due to heat transferversus RaJa as function of 119878119908

1 1003 5 7 9 20 40 60 80001

01

1

10

002

005

02

05

2

5

NF

BrΨ = 50

RaJaSw = 02

Sw = 01

Sw = 005

Sw = 00

Figure 3 Dimensionless entropy generation due to liquid film flowfriction versus RaJa as function of 119878119908

film flow friction decreases As a result it follows that thereshould exist a minimum value of the dimensionless totalentropy generation119873 at a certain value of (RaJa) Figure 4shows the variation of119873with (RaJa) as a function of the wallsuction parameter 119878119908 for a constant Br120595 = 5 It is seen thatthe minimum value of 119873 occurs at (RaJa)opt = 856 670

1 102 3 4 5 6 7 8 9 20 30 40

10

9

8

7

6

5

N

Minimum point

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00

Figure 4 Dimensionless total entropy generation versus RaJa asfunction of 119878119908

525 and 392 given wall suction parameter values of 119878119908 = 0005 01 and 02 respectively The corresponding minimumvalues of119873 are determined from (17) to be 5233 5226 5220and 5208 respectively

Let the respective effects of the liquid film flow frictionirreversibility and heat transfer irreversibility on the entropygeneration rate be quantified by an irreversibility ratio 119873119865119873119879 Clearly the entropy generation rate is dominated by theliquid film flow friction irreversibility when119873119865119873119879 gt 1 butby the heat transfer irreversibility when119873119865119873119879 lt 1 Figure 5shows that for wall suction parameters of 119878119908 = 0 005 01and 02 the contribution of the heat transfer irreversibilityto the entropy generation rate is greater than that of theliquid film flow friction irreversibility (ie 119873119865119873119879 lt 1)when RaJa ge 290 254 228 and 188 respectively In otherwords a higher value of 119878119908 results in a higher heat transferperformance and therefore broadens the range of RaJa overwhich the heat transfer irreversibility dominates In practicalapplications RaJa has a value of more than 10 Thus asshown in Figure 5 the total entropy generation rate is domi-nated by the heat transfer process between the saturatedvapor and the wall

4 Conclusion

This study has examined the entropy generation rate in alaminar condensate film on a horizontal tube with wall suc-tion effects It has been shown that the mean Nusselt numbervaries as a function of RaJa Moreover the dimensionlessentropy generation number induced by heat transfer irre-versibility is equal to 120587 times themeanNusselt number while


0 4 8 12 16 202 6 10 14 180

04

08

12

16

2

02

06

1

14

18

NFN

T

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00

Figure 5 Irreversibility ratio versus RaJa as function of 119878119908

the dimensionless entropy generation number induced byfilm flow friction irreversibility is equal to (Br3Ψ) int120587

0(120575lowast)3

sin2120579119889120579 Finally it has been shown that the presence of awall suction effect reduces the thickness of the liquid filmthereby increasing the heat transfer coefficient and entropygeneration due to heat transfer but decreasing the entropygeneration due to liquid film flow friction

Nomenclature

Br Brinkman number defined in (19)Cp Specific heat at constant pressure119863 Diameter of circular tube119892 Acceleration of gravityℎ Heat transfer coefficientℎ119891119892 Heat of vaporizationJa Jakob number defined in (25)119896 Thermal conductivity119873 Dimensionless overall entropy generation number

defined in (17)Nu Nusselt number defined in (35)Pr Prandtl number defined in (25)Ra Rayleigh number defined in (25)Re119908 Suction Reynolds number defined in (25)11987810158401015840 Local entropy generation rate defined in (8)119878 Overall entropy generation rate defined in (13)119878119891 Film thickness reduction ratio defined in (30)1198780 Characteristic entropy generation rate defined in (16)119878119908 Suction parameter defined in (25)119879 TemperatureΔ119879 Saturation temperature minus wall temperature

119906 Velocity component in 119909-directionV Velocity component in 119910-direction

Greek Symbols

120575 Condensate film thickness120583 Liquid viscosity120588 Liquid density120579 Angle measured from top of tube120595 Dimensionless temperature difference defined in (19)

Superscripts

mdash Average quantitylowast Dimensionless variable

Subscripts

min Minimum quantitysat Saturation property119908 Quantity at wall

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This study was supported by the National Science Council ofTaiwan under Grant no NSC 101-2221-E-218-016

References

[1] WNusselt ldquoDie oberflachenKondensation desWasserdampesrdquoZeitsehrift des Vereines Deutscher Ingenieure vol 60 no 2 pp541ndash546 1916

[2] V D Popov ldquoHeat Transfer during vapor condensation on ahorizontal surfacesrdquo TrudyKiev TeknolInstPishch Prom vol 11pp 87ndash97 1951

[3] G Leppert and B Nimmo ldquoLaminar film condensation onsurface normal to body or inertial forcesrdquo Transactions of theASME Journal of Heat Transfer vol 80 pp 178ndash179 1968

[4] B Nimmo and G Leppert ldquoLaminar film condensation on afinite horizontal surfacerdquo in Proceedings of the 4th InternationalHeat Transfer Conference pp 402ndash403 1970

[5] T Shigechi N Kawae Y Tokita and T Yamada ldquoFilm con-densation heat transfer on a finite-size horizontal plate facingupwardrdquo JSME Series B vol 56 pp 205ndash210 1990

[6] S A Yang and C K Chen ldquoRole of surface tension and elliptic-ity in laminar film condensation on a horizontal elliptical tuberdquoInternational Journal of Heat and Mass Transfer vol 36 no 12pp 3135ndash3141 1993

[7] H-P Hu and C-K Chen ldquoSimplified approach of turbulentfilm condensation on an inclined elliptical tuberdquo InternationalJournal of Heat andMass Transfer vol 49 no 3-4 pp 640ndash6482006


[8] A Bejan ldquoEntropy generation minimization the method andits applicationsrdquo in Proceedings of the ASME-ZSITS Interna-tional Thermal Science Seminar pp 7ndash17 Bled Slovenia June2000

[9] A Bejan Entropy Generation Minimization chapter 4 CRCPress Boca Raton Fla USA 1996

[10] A Bejan ldquoA study of entropy generation in fundamentalconvective heat transferrdquo Journal of Heat Transfer vol 101 no4 pp 718ndash725 1979

[11] S Saouli and S Aıboud-Saouli ldquoSecond law analysis of laminarfalling liquid film along an inclined heated platerdquo InternationalCommunications in Heat and Mass Transfer vol 31 no 6 pp879ndash886 2004

[12] O B Adeyinka and G F Naterer ldquoOptimization correlation forentropy production and energy availability in film condensa-tionrdquo International Communications in Heat andMass Transfervol 31 no 4 pp 513ndash524 2004

[13] S-C Dung and S-A Yang ldquoSecond law based optimization offree convection film-wise condensation on a horizontal tuberdquoInternational Communications in Heat and Mass Transfer vol33 no 5 pp 636ndash644 2006

[14] G-C Li and S-A Yang ldquoThermodynamic analysis of freeconvection film condensation on an elliptical cylinderrdquo Journalof the Chinese Institute of Engineers vol 29 no 5 pp 903ndash9082006

[15] T B Chang and F J Wang ldquoAn analytical investigation into theNusselt number and entropy generation rate of film condensa-tion on a horizontal platerdquo Journal of Mechanical Science andTechnology vol 22 no 11 pp 2134ndash2141 2008

[16] T-B Chang W-Y Yeh and G-L Tsai ldquoFilm condensation onhorizontal tube with wall suction effectsrdquo Journal of MechanicalScience and Technology vol 23 no 12 pp 3399ndash3406 2010

[17] T-B Chang and W-Y Yeh ldquoTheoretical investigation intocondensation heat transfer on horizontal elliptical tube instationary saturated vapor with wall suctionrdquo Applied ThermalEngineering vol 31 no 5 pp 946ndash953 2011

[18] T B Chang ldquoEffects of surface tension on laminar filmwisecondensation on a horizontal plate in a porous medium withsuction at the wallrdquoChemical Engineering Communications vol195 no 7 pp 721ndash737 2008

[19] W M Rohsenow ldquoHeat transfer and temperature distributionin laminar film condensationrdquo Transactions of the ASMEJournal of Heat Transfer vol 78 pp 1645ndash1648 1956

[20] M L James G M Smith and J C Wolford Applied NumericalMethods for Digital Computation Happer amp Row New YorkNY USA 3rd edition 1985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


analysis of the convection heat transfer problem in single-phase falling liquid film flow along an inclined heated plateIn investigating the problem of heat transfer with phase-change Adeyinka and Naterer [12] analyzed the entropy gen-eration and energy availability in vertical film condensationheat transfer The results indicated that entropy generationprovides a useful parameter for optimizing two-phase sys-tems Dung and Yang [13] applied the entropy generationminimization method proposed by Bejan in [8] to optimizefilm condensation heat transfer on a horizontal tube Theirresults showed that the optimal group Rayleigh parameterexists over the parametric range investigated for horizontaltube at which the entropy is generated at a minimum rate Liand Yang [14] applied the entropy generation minimizationmethod to optimize the heat transfer performance of a hori-zontal elliptical cylinder in a saturated vapor flow In a recentstudy Chang and Wang [15] investigated the heat transfercharacteristics and entropy generation rate of a condensatefilm on a horizontal plate and found that the overall entropygeneration rate induced by the heat transfer irreversibilityeffect is equivalent to the Nusselt number

The problem of laminar film condensation with wallsuction effects has beenwidely discussed in the literature [16ndash18] In general the results have shown that wall suction signif-icantly enhances the condensation heat transfer performanceHowever the literature lacks a systematic investigation intothe effects of wall suction on the entropy generation rate inlaminar film condensation on a horizontal tube Accordinglythe present study performs an analytical investigation intothe effects of the Jakob number Rayleigh number Brinkmannumber dimensionless temperature difference and suctionforce on the dimensionless entropy generation rate for ahorizontal tube in a stationary saturated vapor Notably inperforming the analysis the effects of both the gravitationalforce and the viscous force are taken into explicit account

2 Analysis

Consider a pure vapor in a saturated state condensing on ahorizontal and permeable tube (see Figure 1) Assume thatthe vapor has a uniform temperature 119879sat and the tube (withdiameter 119863) has a constant wall temperature 119879119908 If 119879sat ishigher than 119879119908 a thin liquid condensate-layer is formed onthe surface of the tube In analyzing the heat transfer charac-teristics of the condensate film the same assumptions as thoseused byRohsenow [19] are applied namely (1) the condensatefilm flow is steady and laminar and thus the effects of inertiaand convection can be ignored (ie a creeping film flowis assumed) (2) the wall temperature vapor temperatureand properties of the dry vapor and condensate respectivelyare constant and (3) the condensate film has negligiblekinetic energy Consequently the governing equations for thecondensate layer can be formulated as follows

continuity equation

120597119906

120597119909+120597V120597119910= 0 (1)

Vapor temperature

Wall temperature

Liquid-vapor interface

Wall suction

Ein

EsuckEcond

Eout

x

Vw

y

120575

120575

D

Tsat

x120579

g

Tw

x + dx

Figure 1 Physical model

momentum equation

0 = 1205831205972119906

1205971199102+ (120588 minus 120588V) 119892 sin 120579 (2)

energy equation

0 = 1205721205972119879

1205971199102 (3)

In (1)ndash(3) the (119909 119910) coordinates are local coordinates and areused specifically to simplify the governing equations Thelocal coordinate frame (119909 119910) and cylindrical coordinateframe (119903 120579) are related as 119909 = (1198632)120579 and 119910 = 119903 minus (1198632)The boundary conditions are given as follows

at the tube surface that is 119910 = 0

V = V119908 119879 = 119879119908 (4)

at the liquid-vapor interface that is 119910 = 120575

119879 = 119879sat (5)



119906 =(120588 minus 120588V) 119892 sin 120579

120583(120575119910 minus

1

21199102) (6)


119879 = 119879119908 + Δ119879119910

120575 (7)


11987810158401015840=119896

1198792119908

[(120597119879

120597119909)

2

+ (120597119879

120597119910)

2

] +120583

119879119908

(120597119906

120597119910)

2

= 11987810158401015840119879 + 11987810158401015840119865

(8)



10158401015840119865 From


11987810158401015840119879 =

119896

1198792119908

[(120597119879

120597119909)

2

+ (120597119879

120597119910)

2

] (9)

11987810158401015840119865 =

120583

119879119908

(120597119906

120597119910)

2

(10)


11987810158401015840119879 =

119896

1198792119908

(Δ119879

120575)

2

11987810158401015840119865 =

[(120588 minus 120588V) 119892 sin 120579]2

119879119908120583(120575 minus 119910)

2

(11)


1198781015840= int

120575

011987810158401015840119889119910 = 119878

1015840119879 + 1198781015840119865

1198781015840119879 = int

120575

011987810158401015840119879119889119910 =

119896

1198792119908

Δ1198792

120575

1198781015840119865 = int

120575

011987810158401015840119865119889119910 =

[(120588 minus 120588V) 119892 sin 120579]2

31198791199081205831205753

(12)


119878 = int

120587

01198781015840119889120579 = 119878119879 + 119878119865 (13)

119878119879 = int

120587

0

119896

1198792119908

Δ1198792

120575119889120579 =

119896Δ1198792

1198792119908

int

120587

0

1

120575119889120579 (14)

119878119865 = int

120587

0

[(120588 minus 120588V) 119892 sin 120579]2

31198791199081205831205753119889120579

=[(120588 minus 120588V) 119892]

2

3119879119908120583int

120587

01205753sin2120579119889120579

(15)


1198780 =119896

119863(Δ119879

119879119908

)

2

(16)


119873 =119878

1198780

= 119873119879 + 119873119865 (17)


119873119879 = int

120587

0

119863

120575119889120579

119873119865 =119863119879119908 [(120588 minus 120588V) 119892]

2

3119896120583Δ1198792int

120587

01205753sin2120579119889120579

(18)


1199060 =(120588 minus 120588V) 119892

1205831198632 Br =

12058311990620

119896Δ119879 Ψ =

Δ119879

119879119908

(19)


120575lowast=120575

119863 (20)


119873119879 = int

120587

0

1

120575lowast119889120579 (21)

119873119865 =Br3Ψ

int

120587

0(120575lowast)3 sin2120579119889120579 (22)




int120575



0120588119906(ℎ119891119892 +Cp(119879sat minus119879))119889119910|119909+119889119909


119889

119889119909int

120575

0120588119906 (ℎ119891119892 + Cp (119879sat minus 119879)) 119889119910119889119909

+ [120588 (ℎ119891119892 + CpΔ119879) V119908] 119889119909 = 119896120597119879

120597119910119889119909

(23)



120588 (120588 minus 120588V) 119892 (ℎ119891119892 + (38)CpΔ119879)3120583

2119889

1198631198891205791205753 sin 120579

+ 120588 (ℎ119891119892 + CpΔ119879) V119908 = 119896Δ119879

120575

(24)


Ja = CpΔ119879ℎ119891119892 + (38)CpΔ119879

Pr =120583Cp119896

Ra =120588 (120588 minus 120588V) 119892Pr119863

3

1205832

Re119908 =120588V119908119863120583

119878119908 = (1 +5

8Ja)Re119908

PrRa

(25)


120575lowast 119889

119889120579(120575lowast3 sin 120579) + 3

2119878119908120575lowast=3

2

JaRa (26)


119889120575lowast

119889120579= 0 at 120579 = 0 (27a)



120575lowast1003816100381610038161003816119878119908=0

119889

119889120579((120575lowast1003816100381610038161003816119878119908=0)3sin 120579) = 3

2

JaRa (28)




Ra)

14

(int

120579

0sin13120579119889120579)

14

(29)




119878119891 = 1 minus120575lowast

120575lowast|119878119908=0

(30)



3 (120575lowast1003816100381610038161003816119878119908=0)4sin 120579 (1 minus 119878119891)

3 119889119891

119889120579

+ 120575lowast1003816100381610038161003816119878119908=0

119889

119889120579((120575lowast1003816100381610038161003816119878119908=0)3sin 120579)

times (1198784119891 minus 4119878

3119891 + 6119878

2119891 minus 4119878119891)

+3


(31)


1198601198784119891 (0) + 119861119878

3119891 (0) + 119862119878

2119891 (0) + 119863119878119891 (0) + 119864 = 0 (32)

where

119860 = (120575lowast1003816100381610038161003816119878119908=0120579=0)

4

119861 = minus4119860

119862 = 6119860

119863 = minus4119860 minus3

2(120575lowast1003816100381610038161003816119878119908=0120579=0) times 119878119908

119864 =3

2(120575lowast1003816100381610038161003816119878119908=0120579=0) times 119878119908

(33)





lowast1003816100381610038161003816119878119908=0

= (1 minus 119878119891) sinminus13

120579 (2JaRa)

14

(int

120579

0sin13120579119889120579)

14

(34)


Nu120579 =ℎ120579119863

119896 (35)

where

ℎ120579 =119896

120575 (36)


Nu120579 =1

120575lowast (120579)

= (RaJa)

14 sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

(37)


Nu = 1120587int

120587

0

1

120575lowast (120579)119889120579

=1

120587(RaJa)

14

int

120587

0

sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

119889120579

(38)


119873119879 = int

120587

0

1

120575lowast119889120579 = 120587 timesNu (39)



Nu10038161003816100381610038161003816119878119908=0=1

120587(RaJa)

14

int

120587

0sin13120579(4int

120579

0sin13120579119889120579)

minus14

119889120579

(40)



0sin13120579(4 int120579

0sin13

120579119889120579)minus14

119889120579 in (32) yielding

Nu10038161003816100381610038161003816119878119908=0= 1224 times (

RaJa)

14

(41)



14

(42)










10

20

8

6

4

21 1003 5 7 9 20 40 60 80

RaJa

NT

BrΨ = 50

Sw = 02

Sw = 01

Sw = 005

Sw = 00


1 1003 5 7 9 20 40 60 80001

01

1

10

002

005

02

05

2

5

NF

BrΨ = 50

RaJaSw = 02

Sw = 01

Sw = 005

Sw = 00



1 102 3 4 5 6 7 8 9 20 30 40

10

9

8

7

6

5

N

Minimum point

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00




4 Conclusion



0 4 8 12 16 202 6 10 14 180

04

08

12

16

2

02

06

1

14

18

NFN

T

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00



0(120575lowast)3


Nomenclature




Greek Symbols


Superscripts


Subscripts




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 =(120588 minus 120588V) 119892 sin 120579

120583(120575119910 minus

1

21199102) (6)


119879 = 119879119908 + Δ119879119910

120575 (7)


11987810158401015840=119896

1198792119908

[(120597119879

120597119909)

2

+ (120597119879

120597119910)

2

] +120583

119879119908

(120597119906

120597119910)

2

= 11987810158401015840119879 + 11987810158401015840119865

(8)



10158401015840119865 From


11987810158401015840119879 =

119896

1198792119908

[(120597119879

120597119909)

2

+ (120597119879

120597119910)

2

] (9)

11987810158401015840119865 =

120583

119879119908

(120597119906

120597119910)

2

(10)


11987810158401015840119879 =

119896

1198792119908

(Δ119879

120575)

2

11987810158401015840119865 =

[(120588 minus 120588V) 119892 sin 120579]2

119879119908120583(120575 minus 119910)

2

(11)


1198781015840= int

120575

011987810158401015840119889119910 = 119878

1015840119879 + 1198781015840119865

1198781015840119879 = int

120575

011987810158401015840119879119889119910 =

119896

1198792119908

Δ1198792

120575

1198781015840119865 = int

120575

011987810158401015840119865119889119910 =

[(120588 minus 120588V) 119892 sin 120579]2

31198791199081205831205753

(12)


119878 = int

120587

01198781015840119889120579 = 119878119879 + 119878119865 (13)

119878119879 = int

120587

0

119896

1198792119908

Δ1198792

120575119889120579 =

119896Δ1198792

1198792119908

int

120587

0

1

120575119889120579 (14)

119878119865 = int

120587

0

[(120588 minus 120588V) 119892 sin 120579]2

31198791199081205831205753119889120579

=[(120588 minus 120588V) 119892]

2

3119879119908120583int

120587

01205753sin2120579119889120579

(15)


1198780 =119896

119863(Δ119879

119879119908

)

2

(16)


119873 =119878

1198780

= 119873119879 + 119873119865 (17)


119873119879 = int

120587

0

119863

120575119889120579

119873119865 =119863119879119908 [(120588 minus 120588V) 119892]

2

3119896120583Δ1198792int

120587

01205753sin2120579119889120579

(18)


1199060 =(120588 minus 120588V) 119892

1205831198632 Br =

12058311990620

119896Δ119879 Ψ =

Δ119879

119879119908

(19)


120575lowast=120575

119863 (20)


119873119879 = int

120587

0

1

120575lowast119889120579 (21)

119873119865 =Br3Ψ

int

120587

0(120575lowast)3 sin2120579119889120579 (22)




int120575



0120588119906(ℎ119891119892 +Cp(119879sat minus119879))119889119910|119909+119889119909


119889

119889119909int

120575

0120588119906 (ℎ119891119892 + Cp (119879sat minus 119879)) 119889119910119889119909

+ [120588 (ℎ119891119892 + CpΔ119879) V119908] 119889119909 = 119896120597119879

120597119910119889119909

(23)



120588 (120588 minus 120588V) 119892 (ℎ119891119892 + (38)CpΔ119879)3120583

2119889

1198631198891205791205753 sin 120579

+ 120588 (ℎ119891119892 + CpΔ119879) V119908 = 119896Δ119879

120575

(24)


Ja = CpΔ119879ℎ119891119892 + (38)CpΔ119879

Pr =120583Cp119896

Ra =120588 (120588 minus 120588V) 119892Pr119863

3

1205832

Re119908 =120588V119908119863120583

119878119908 = (1 +5

8Ja)Re119908

PrRa

(25)


120575lowast 119889

119889120579(120575lowast3 sin 120579) + 3

2119878119908120575lowast=3

2

JaRa (26)


119889120575lowast

119889120579= 0 at 120579 = 0 (27a)



120575lowast1003816100381610038161003816119878119908=0

119889

119889120579((120575lowast1003816100381610038161003816119878119908=0)3sin 120579) = 3

2

JaRa (28)




Ra)

14

(int

120579

0sin13120579119889120579)

14

(29)




119878119891 = 1 minus120575lowast

120575lowast|119878119908=0

(30)



3 (120575lowast1003816100381610038161003816119878119908=0)4sin 120579 (1 minus 119878119891)

3 119889119891

119889120579

+ 120575lowast1003816100381610038161003816119878119908=0

119889

119889120579((120575lowast1003816100381610038161003816119878119908=0)3sin 120579)

times (1198784119891 minus 4119878

3119891 + 6119878

2119891 minus 4119878119891)

+3


(31)


1198601198784119891 (0) + 119861119878

3119891 (0) + 119862119878

2119891 (0) + 119863119878119891 (0) + 119864 = 0 (32)

where

119860 = (120575lowast1003816100381610038161003816119878119908=0120579=0)

4

119861 = minus4119860

119862 = 6119860

119863 = minus4119860 minus3

2(120575lowast1003816100381610038161003816119878119908=0120579=0) times 119878119908

119864 =3

2(120575lowast1003816100381610038161003816119878119908=0120579=0) times 119878119908

(33)





lowast1003816100381610038161003816119878119908=0

= (1 minus 119878119891) sinminus13

120579 (2JaRa)

14

(int

120579

0sin13120579119889120579)

14

(34)


Nu120579 =ℎ120579119863

119896 (35)

where

ℎ120579 =119896

120575 (36)


Nu120579 =1

120575lowast (120579)

= (RaJa)

14 sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

(37)


Nu = 1120587int

120587

0

1

120575lowast (120579)119889120579

=1

120587(RaJa)

14

int

120587

0

sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

119889120579

(38)


119873119879 = int

120587

0

1

120575lowast119889120579 = 120587 timesNu (39)



Nu10038161003816100381610038161003816119878119908=0=1

120587(RaJa)

14

int

120587

0sin13120579(4int

120579

0sin13120579119889120579)

minus14

119889120579

(40)



0sin13120579(4 int120579

0sin13

120579119889120579)minus14

119889120579 in (32) yielding

Nu10038161003816100381610038161003816119878119908=0= 1224 times (

RaJa)

14

(41)



14

(42)










10

20

8

6

4

21 1003 5 7 9 20 40 60 80

RaJa

NT

BrΨ = 50

Sw = 02

Sw = 01

Sw = 005

Sw = 00


1 1003 5 7 9 20 40 60 80001

01

1

10

002

005

02

05

2

5

NF

BrΨ = 50

RaJaSw = 02

Sw = 01

Sw = 005

Sw = 00



1 102 3 4 5 6 7 8 9 20 30 40

10

9

8

7

6

5

N

Minimum point

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00




4 Conclusion



0 4 8 12 16 202 6 10 14 180

04

08

12

16

2

02

06

1

14

18

NFN

T

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00



0(120575lowast)3


Nomenclature




Greek Symbols


Superscripts


Subscripts




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int120575



0120588119906(ℎ119891119892 +Cp(119879sat minus119879))119889119910|119909+119889119909


119889

119889119909int

120575

0120588119906 (ℎ119891119892 + Cp (119879sat minus 119879)) 119889119910119889119909

+ [120588 (ℎ119891119892 + CpΔ119879) V119908] 119889119909 = 119896120597119879

120597119910119889119909

(23)



120588 (120588 minus 120588V) 119892 (ℎ119891119892 + (38)CpΔ119879)3120583

2119889

1198631198891205791205753 sin 120579

+ 120588 (ℎ119891119892 + CpΔ119879) V119908 = 119896Δ119879

120575

(24)


Ja = CpΔ119879ℎ119891119892 + (38)CpΔ119879

Pr =120583Cp119896

Ra =120588 (120588 minus 120588V) 119892Pr119863

3

1205832

Re119908 =120588V119908119863120583

119878119908 = (1 +5

8Ja)Re119908

PrRa

(25)


120575lowast 119889

119889120579(120575lowast3 sin 120579) + 3

2119878119908120575lowast=3

2

JaRa (26)


119889120575lowast

119889120579= 0 at 120579 = 0 (27a)



120575lowast1003816100381610038161003816119878119908=0

119889

119889120579((120575lowast1003816100381610038161003816119878119908=0)3sin 120579) = 3

2

JaRa (28)




Ra)

14

(int

120579

0sin13120579119889120579)

14

(29)




119878119891 = 1 minus120575lowast

120575lowast|119878119908=0

(30)



3 (120575lowast1003816100381610038161003816119878119908=0)4sin 120579 (1 minus 119878119891)

3 119889119891

119889120579

+ 120575lowast1003816100381610038161003816119878119908=0

119889

119889120579((120575lowast1003816100381610038161003816119878119908=0)3sin 120579)

times (1198784119891 minus 4119878

3119891 + 6119878

2119891 minus 4119878119891)

+3


(31)


1198601198784119891 (0) + 119861119878

3119891 (0) + 119862119878

2119891 (0) + 119863119878119891 (0) + 119864 = 0 (32)

where

119860 = (120575lowast1003816100381610038161003816119878119908=0120579=0)

4

119861 = minus4119860

119862 = 6119860

119863 = minus4119860 minus3

2(120575lowast1003816100381610038161003816119878119908=0120579=0) times 119878119908

119864 =3

2(120575lowast1003816100381610038161003816119878119908=0120579=0) times 119878119908

(33)





lowast1003816100381610038161003816119878119908=0

= (1 minus 119878119891) sinminus13

120579 (2JaRa)

14

(int

120579

0sin13120579119889120579)

14

(34)


Nu120579 =ℎ120579119863

119896 (35)

where

ℎ120579 =119896

120575 (36)


Nu120579 =1

120575lowast (120579)

= (RaJa)

14 sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

(37)


Nu = 1120587int

120587

0

1

120575lowast (120579)119889120579

=1

120587(RaJa)

14

int

120587

0

sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

119889120579

(38)


119873119879 = int

120587

0

1

120575lowast119889120579 = 120587 timesNu (39)



Nu10038161003816100381610038161003816119878119908=0=1

120587(RaJa)

14

int

120587

0sin13120579(4int

120579

0sin13120579119889120579)

minus14

119889120579

(40)



0sin13120579(4 int120579

0sin13

120579119889120579)minus14

119889120579 in (32) yielding

Nu10038161003816100381610038161003816119878119908=0= 1224 times (

RaJa)

14

(41)



14

(42)










10

20

8

6

4

21 1003 5 7 9 20 40 60 80

RaJa

NT

BrΨ = 50

Sw = 02

Sw = 01

Sw = 005

Sw = 00


1 1003 5 7 9 20 40 60 80001

01

1

10

002

005

02

05

2

5

NF

BrΨ = 50

RaJaSw = 02

Sw = 01

Sw = 005

Sw = 00



1 102 3 4 5 6 7 8 9 20 30 40

10

9

8

7

6

5

N

Minimum point

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00




4 Conclusion



0 4 8 12 16 202 6 10 14 180

04

08

12

16

2

02

06

1

14

18

NFN

T

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00



0(120575lowast)3


Nomenclature




Greek Symbols


Superscripts


Subscripts




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












lowast1003816100381610038161003816119878119908=0

= (1 minus 119878119891) sinminus13

120579 (2JaRa)

14

(int

120579

0sin13120579119889120579)

14

(34)


Nu120579 =ℎ120579119863

119896 (35)

where

ℎ120579 =119896

120575 (36)


Nu120579 =1

120575lowast (120579)

= (RaJa)

14 sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

(37)


Nu = 1120587int

120587

0

1

120575lowast (120579)119889120579

=1

120587(RaJa)

14

int

120587

0

sin13120579 (4 int1205790sin13120579119889120579)

minus14

(1 minus 119878119891)

119889120579

(38)


119873119879 = int

120587

0

1

120575lowast119889120579 = 120587 timesNu (39)



Nu10038161003816100381610038161003816119878119908=0=1

120587(RaJa)

14

int

120587

0sin13120579(4int

120579

0sin13120579119889120579)

minus14

119889120579

(40)



0sin13120579(4 int120579

0sin13

120579119889120579)minus14

119889120579 in (32) yielding

Nu10038161003816100381610038161003816119878119908=0= 1224 times (

RaJa)

14

(41)



14

(42)










10

20

8

6

4

21 1003 5 7 9 20 40 60 80

RaJa

NT

BrΨ = 50

Sw = 02

Sw = 01

Sw = 005

Sw = 00


1 1003 5 7 9 20 40 60 80001

01

1

10

002

005

02

05

2

5

NF

BrΨ = 50

RaJaSw = 02

Sw = 01

Sw = 005

Sw = 00



1 102 3 4 5 6 7 8 9 20 30 40

10

9

8

7

6

5

N

Minimum point

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00




4 Conclusion



0 4 8 12 16 202 6 10 14 180

04

08

12

16

2

02

06

1

14

18

NFN

T

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00



0(120575lowast)3


Nomenclature




Greek Symbols


Superscripts


Subscripts




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

20

8

6

4

21 1003 5 7 9 20 40 60 80

RaJa

NT

BrΨ = 50

Sw = 02

Sw = 01

Sw = 005

Sw = 00


1 1003 5 7 9 20 40 60 80001

01

1

10

002

005

02

05

2

5

NF

BrΨ = 50

RaJaSw = 02

Sw = 01

Sw = 005

Sw = 00



1 102 3 4 5 6 7 8 9 20 30 40

10

9

8

7

6

5

N

Minimum point

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00




4 Conclusion



0 4 8 12 16 202 6 10 14 180

04

08

12

16

2

02

06

1

14

18

NFN

T

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00



0(120575lowast)3


Nomenclature




Greek Symbols


Superscripts


Subscripts




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 4 8 12 16 202 6 10 14 180

04

08

12

16

2

02

06

1

14

18

NFN

T

BrΨ = 50

RaJa

Sw = 02

Sw = 01

Sw = 005

Sw = 00



0(120575lowast)3


Nomenclature




Greek Symbols


Superscripts


Subscripts




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Date post:	09-May-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Research Article Theoretical Analysis of Effects of Wall Suction...

Documents