International Journal of Heat and Mass Transfer 63 (2013) 91–100

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International Journal of Heat and Mass Transfer

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Fluid flow and forced convection heat transfer around a solid cylinderwrapped with a porous ring

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.03.006

⇑ Corresponding author. Tel.: +44 161 3063980.E-mail address: nima.shokri@manchester.ac.uk (N. Shokri).

Saman Rashidi a, Ali Tamayol b,c, Mohammad Sadegh Valipour a, Nima Shokri d,⇑a Department of Mechanical Engineering, Semnan University, P.O. Box 35196-45399, Semnan, Iranb Biomedical Engineering Department, McGill University, Montreal, Canada H3A 0G1c Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, MA 02139, USAd School of Chemical Engineering and Analytical Science, University of Manchester, Manchester M13 9PL, UK

a r t i c l e i n f o

Article history:Received 20 September 2012Received in revised form 4 March 2013Accepted 5 March 2013

Keywords:Cylinder embedded in porous mediaDarcy numbersVolume averaged equationsOptimization process

a b s t r a c t

Convective heat transfer from cylinders embedded in porous media is important for many engineeringapplications. In the present study, flow-field and heat transfer around a cylinder embedded in a layerof homogenous porous media is investigated numerically. The range of Reynolds and Darcy numbersare chosen to be 1–40 and 1 � 10�8–1 � 10�1, respectively. Volume averaged equations are used for mod-eling transport phenomena within the porous layer and conservation laws of mass, momentum, anenergy are applied in the clear region. A comprehensive parametric study is carried out and effects of sev-eral parameters, such as porous layer thickness and permeability as well as the Darcy and Reynolds num-bers on flow-field and heat transfer characteristics are studied. Finally an optimization process isconducted in order to determine the optimal thickness and porosity of the porous layer resulting inthe lowest heat transfer from the cylinder. The numerical results indicate that, in the presence of a porouslayer around the cylinder, the wake length increases with decreasing the Darcy number while the criticalradius of insulation increases.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Fluid flow and heat transfer across a cylinder embedded inside aporous layer has various environmental and engineering applica-tions such as insulation of heat pipes [1], compact heat exchangersdesign [2], reactor safety analysis and combustion [3] and control-ling the spread of petroleum-based products in groundwater andsoil. In addition, micro-cylinders covered by a porous structureembedded within channels can be used for designing novel com-pact micro-heat exchangers for various applications such as elec-tronics cooling [4].

Depending upon the thermophysical properties of the porousmedium, the effective solid and fluid interface area and the overallheat and mass transfer rates from the base cylinder may change.Researchers have employed a variety of techniques for characteriz-ing the convection heat transfer across a cylinder embedded in aporous layer [5–7].

Layeghi and Nouri-Borujerdi [7] studied convective heat trans-fer from an array of circular cylinders in direct contact with passingflow or surrounded by porous media. Their numerical simulationssuggested a heat transfer enhancement (over 80%) due to the

presence of a highly conductive porous medium. However, the por-ous medium layer increased the overall pressure drop of thesystem.

Bhattacharyya et al. [8] presented numerical simulations on thefluid motion around and through a porous cylinder. Their resultsshowed that the drag experienced by the porous cylinder decreaseswith the increase of the Reynolds number and reduce in the Darcynumber. Bhattacharyya and Singh [6] investigated the effect of theporous layer properties on the convective heat transfer augmenta-tion from an isothermal circular tube. The porous layer in theirstudy was of a foam material with high porosity and thermal con-ductivity. They studied effects of salient parameters such as thepermeability and thermal conductivity of the porous layer, flowReynolds and Grashof numbers, and the porous layer thicknesson the mixed convection. For example, they showed that a thinporous layer with high thermal conductivity can significantly en-hance the heat transfer even at low permeability. Rong et al. [9]performed numerical simulations of flow around a square cross-section cylinder covered by a porous layer. Their results revealedthat the drag and lift coefficient increases with increase of porouslayer thickness when Da P 10�4 and have little changes whenDa < 10�4. In a notable study, Al-Sumaily et al. [10] numericallystudied the time-dependent forced convection heat transfer froma single circular cylinder embedded in a horizontal channel filled

Fig. 1. Computational domain and geometry of cylinder.

Nomenclature

cp specific heat (J/kg K)Cp pressure coefficient (–)CF Forchheimer coefficient (-)d porous layer thickness (m)D cylinder diameter (m), D = 2RDa Darcy number (–), Da = K/D2

Dp characteristic diameter of a particle (m)FD total drag force (N)h heat transfer coefficient (W/m2 K)k thermal conductivity (W/m K)K permeability (m2)Nu Nusselt number (–)P pressure (Pa)Pr Prandtel number, (–), t/ar radial coordinate (m)R cylinder radius (m)Rc thermal conductivity ratio, keff/kf

Re Reynolds number (–), qU1D/lT temperature (K)u, v velocity component in r, h direction respectively (m s�1)

Greek symbolsl dynamic viscosity (kg m�1 s�1)

t fluid kinematic viscosity (m2 s�1), t = l/qq fluid density (kg m�3)e porosity (–)a thermal diffusivity of fluid (m2/s), k/qcp

h cross-radial coordinate (m)d non-dimensional porous layer thickness (m), d = d/Dx vorticity (1/s)K viscosity ratio (–), K = leff/lf

Subscriptsave averagecr criticaleff effectivef fluidp pressure forces solidv viscous forcew wall1 free stream1 clear fluid domain2 porous domain

92 S. Rashidi et al. / International Journal of Heat and Mass Transfer 63 (2013) 91–100

by spherical particles under local thermal non-equilibrium condi-tion and constant porosity. They investigated the influence of por-ous materials and its characteristics on the heat transfer andthermal responses. Similar to Bhattacharyya and Singh [6], theyfound that the presence of the porous particles enhances the heattransfer rate. Their results suggested that an increase in Biot num-ber reduces fluid Nusselt number while it increases the Nusseltnumber in the solid phase. In addition, it was observed thatincreasing the ratio of solid to fluid thermal conductivity reducedthe deviations from thermal equilibrium condition in the porousbed. Hooman and his co-workers modeled metalfoam heatexchangers as a solid surface surrounded by a porous material[11,12]. They employed analytical and numerical methods forstudying their thermal performance and reported a model for pre-dicting the thermal properties of metalfoam heat exchangers. Re-cently, they investigated the effect of dust deposition on thehydrodynamic and thermal performance of metalfoam heatexchangers [13,14]. They showed that a thin layer of fouling depos-ited on the surface of fibers of the metalfoam could significantly af-fect the thermal performance and pressure drop of the heatexchanger. Valipour and Zare Ghadi [15] numerically studied theforced convective heat transfer around and through a porous circu-lar cylinder.

Along with the numerical studies, experiments have been alsoconducted to study the effects of a porous layer wrapped arounda solid cylinder on heat transfer and thermal responses. For exam-ple, Ahmed [16] experimentally studied forced convection about ahorizontal cylinder embedded in a porous medium for2000 < Re < 3000. Their experimental results revealed that theaverage heat transfer increases as the Peclet and Reynolds numberincreased for steady state condition.

Al-Salem et al. [17] experimentally investigated the effects ofthe porosity and thickness of porous layer on heat transferenhancement in a cross flow over a heated cylinder. They couldexperimentally demonstrate a heat transfer augmentation as a re-sult of the porous layer presence.

Our literature review showed that the pertinent literature isrich due to diverse applications of flow around a cylinder covered

by a porous layer. However, we noticed a lack of information anddata in the targeted range of Reynolds number which is importantfor designing compact micro heat exchangers. As a result, we fo-cused on this range of Reynolds number.

The objective of the present research is to further investigateand provide additional insights about the effects of the presenceof porous layers on the thermal insulation or heat transfer aug-mentation. Besides, we study the potential influences of a porouslayer wrapped around a solid cylinder on several hydrodynamicsparameters such as drag coefficient, pressure coefficient, lengthof recirculation region, streamlines and determine the critical ra-dius of porous layer insulation for a cylindrical body in laminar re-gime. To achieve the objectives of this study, we performed acomprehensive numerical study to investigate the laminar forcedconvection flow around a solid cylinder wrapped with porous ringand reveal the effects of Darcy and Reynolds numbers, thermal

S. Rashidi et al. / International Journal of Heat and Mass Transfer 63 (2013) 91–100 93

conductivity of porous substrate and porous layer thickness on theflow pattern and heat transfer characteristics.

2. Analysis

2.1. Problem statement

Fig. 1 shows the geometry of the studied problem, which com-prises a long circular cylinder with the diameter of D embeddedwithin a layer of porous medium of the known thickness (d). Theside effects are neglected and the problem is modeled two dimen-sionally. Laminar flow with the temperature of T1 flowing acrossthe isothermal cylinder at a constant temperature of Tw is consid-ered. In addition, the following assumptions are made:

� The porous matrix around the cylinder is assumed homoge-neous and isotropic with uniform porosity and tortuosity.� All fluid properties are considered to be constant.� The solid phase temperature is equal to that of the fluid phase

(local thermal equilibrium (LTE)). Local thermal equilibrium isoften used when studying heat transfer in porous media andthis assumption holds when the temperature differencebetween the solid and fluid phases is not significant. Thisassumption breaks down during rapid heating or cooling orproblems with significant temperature variation across the por-ous media [18].

2.2. Governing equations

Equations that govern the two zones described in the previoussection, i.e., the clear fluid and the porous medium zones, areNavier–Stokes and volume averaged equations, respectively. Itshould be noted that the clear fluid and porous medium domainsare indicated by subscripts (1) and (2), respectively. To generalizethe analysis, the following dimensionless parameters are substi-tuted in the governing equations [6].

r ¼ r�

R; h ¼ h�; u ¼ u�

U1; v ¼ v�

U1; p ¼ p�

qU21; T ¼ T� � T1

Tw � T1

ð1Þ

The variable with superscript ⁄ denotes dimensional variables.Following the above assumptions and by substituting the

dimensionless numbers, the mass conservation equation in region1 (clear fluid) becomes:

@

@rðru1Þ þ

@

@hðv1Þ ¼ 0 ð2Þ

And momentum equations in r and h direction reads:

v1

r@u1

@hþu1

@u1

@r�v2

1

r

� �¼�@p1

@r

þ 2Re

@2u1

@r2 þ1r@u1

@rþ 1

r2

@2u1

@h2 �2r2

@v1

@h�u1

r2

!

ð3Þ

v1

r@v1

@hþu1

@v1

@rþu1v1

r

� �¼�1

r@p1

@h

þ 2Re

@2v1

@r2 þ1r@v1

@rþ 1

r2

@2v1

@h2 þ2r2

@u1

@h�v1

r2

!

ð4Þ

The energy equation in region 1 can be written as:

u1@T1

@rþ v1

r@T1

@h

� �¼ 2

RePr@2T1

@r2 þ1r@T1

@rþ 1

r2

@2T1

@h2

!ð5Þ

In region 2, i.e., the porous medium region, the volume-aver-aged equations are used.

The volume-averaged mass conservation equation becomes:

@

@rðru2Þ þ

@

@hðv2Þ ¼ 0 ð6Þ

And the momentum equations in r and h direction can be writ-ten as:

1e

v2

r@u2

@hþu2

@u2

@r�v2

2

r

� �¼�e

@p2

@r

þK2eRe

@2u2

@r2 þ1r@u2

@rþ 1

r2

@2u2

@h2 �2r2

@v2

@h�u2

r2

!

� e2ReDa

u2�eCF

2ffiffiffiffiffiffiDap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

2þv22

qu2

ð7Þ

1e

v2

r@v2

@hþu2

@v2

@rþu2v2

r

� �¼�e

r@p2

@h

þK2eRe

@2v2

@r2 þ1r@v2

@rþ 1

r2

@2v2

@h2 þ2r2

@u2

@h�v2

r2

!

� e2ReDa

v2�eCF

2ffiffiffiffiffiffiDap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

2þv22

qv2

ð8Þ

The energy equation with the local thermal equilibriumassumption can be written as:

u2@T2

@rþ v2

r@T2

@h

� �¼ 2Rc

RePr@2T2

@r2 þ1r@T2

@rþ 1

r2

@2T2

@h2

!ð9Þ

Where Rc, Da, Re and Pr are thermal conductivity ratio, Darcy num-ber, Reynolds number and Prandtel number, respectively expressedas:

Rc ¼keff

kf; Da ¼ K

D2 ; Re ¼ qU1Dl

; Pr ¼ ma

ð10Þ

The volume-averaged fluid velocity V!

inside the porous regionwith porosity e is related to the fluid velocity v! through theDupuit–Forchheimer relationship as v!¼ eV

!. Forchheimer coeffi-

cient CF [19] can be calculated depending on the microstructureof the porous medium. Here we used the widely accepted Erguncorrelation expressed as [20]:

CF ¼1:75ffiffiffiffiffiffiffiffiffiffiffiffiffi150e3p ð11Þ

There are various models for calculating the permeability of dif-ferent porous materials [21–24]. Our analysis is generalized andcan be applied to any type of porous media. However, as an exam-ple, we used the Carman–Kozeny model for calculation of flowcoefficients. This model was originally proposed for packed bedof spherical particles (such as sandstone), however, Tamayol andBahrami [25] have shown that this model can also be applied tovarious types of fibrous materials (such as foams). As a result weused this relationship for both fibrous and packed bed porousstructures.

Permeability is calculated from the following relationship [26]:

K ¼ 1180

e3D2p

ð1� eÞ2ð12Þ

where K is the permeability and Dp is the average particle size of theporous bed. In our calculations, we consider (Dp) to be equal to100 lm.

94 S. Rashidi et al. / International Journal of Heat and Mass Transfer 63 (2013) 91–100

2.3. Boundary conditions

The dimensions of the computational domain in the vertical andhorizontal directions should be selected in a way that minimizesthe effects of the outer boundaries. Based on the results reportedby others in the literature, computational domain is defined tobe 50 times of the cylinder diameter, i.e., 50D � 50D.

The governing equations (2)–(9) are subjected to the followingboundary conditions:

On the surface of the solid cylinder:

u2 ¼ v2 ¼ 0; T2 ¼ 1 ð13Þ

Along the upstream boundary (uniform flow):

For 0 < h <p4) u1 ¼ � cos h; v1 ¼ sin h; T1 ¼ 0 ð14Þ

At the upper and downer boundaries (infinity boundarycondition):

@u1

@r¼ 0;

@v1

@r¼ 0; T1 ¼ 0 ð15Þ

There are several boundary conditions in the literature for thevelocity, temperature, and shear stress at the interface region suchas jump and no-jump in the velocity and temperature at the inter-face of the porous medium and clear fluid. In the present study weassume continuity of velocity, shear stress, temperature and heatflux here at the interface between the porous and the fluid region.These conditions are given by Alazmi and Vafai [27]:

u1 ¼ u2; v1 ¼ v2 ð16Þ

lf@v1

@r¼ leff

@v2

@rð17Þ

T1 ¼ T2 ð18Þ

@T1

@r¼ Rc

@T2

@r;

@T1

@h¼ Rc

@T2

@hð19Þ

In porous media regions, an effective dynamic viscosity andthermal conductivity are applied in shear stress and conductionflux expressions [27]:

leff ¼lf

eð20Þ

keff ¼ ekf þ ð1� eÞks ð21Þ

2.4. Solution technique

The set of governing equations (2)–(9) with the relevant bound-ary conditions are solved numerically using the Finite VolumeMethod. SIMPLE algorithm is used for pressure and velocity cou-pling [28]. Second order differencing scheme has been used forall equations. It should be noted that for the convergence criterion,maximum relative error in the values of the dependent variablesbetween two successive iterations, in all runs was set to 10�5.

The analysis is carried out to determine the hydrodynamics ofthe flow, velocity and temperature distribution and then otherparameters such as Nusselt number, drag and pressure coefficientsand streamlines are determined.

2.5. Grid independence study

For a certain range of Reynolds numbers a recirculating wake isformed at the rear of the cylinder. The geometrical parameter con-sidered here is the length of the re-circulating wake that is shownin Fig. 2(a). A simple, two-dimensional and square mesh clustered

near the walls and at the interface between fluid and surface of thecylinder is used. This mesh is refined near the interface betweenfluid and solid region where the velocity and temperature gradientis large compared to other regions. The schematic of grid-meshused for the present computations is shown in Fig. 2(b). Several dif-ferent grid sizes are tested to ensure that the calculated results aregrid independent. Table 1 shows the effect of mesh size on thelength of recirculation region (LR) and average Nusselt numberfor solid cylinder (without porous ring) at fixed Re = 40. Thenumerical results are obtained for four different mesh sizes (in Ta-ble 1, n �m refers to the number of mesh points in the radial andcross radial directions, respectively). Comparing the computedlength of recirculation region and average Nusselt number underdifferent grid sizes, it is found the difference between the calcu-lated LR and Nusselt number between the two grids namely460 � 270 and 480 � 281 is 0.31% and 0.39%, respectively. Thus,our grid sensitivity test proves that the grid-independent resultscould be achieved using 480 � 281 mesh size for the simulation.

2.6. Validation

For demonstrating the validity and precision of the model andnumerical solution used in the present study, we compared our re-sults with data reported in Bhattacharyya et al. [8]. Fig. 3 presentsthe separation angle measured from the upstream stagnation point(for a single cylinder) as a function of Reynolds number comparedwith the results presented in Bhattacharyya et al. [8]. It was foundthat at Re = 20 the separation occurs at angle 137� which is veryclose to 136.5� reported by Bhattacharyya et al. [8]. Furthermore,Fig. 3 shows that separation angle decreases with increasing Rey-nolds number.

3. Results and discussion

The main objective of the present study is to determine the ef-fect of salient geometrical and flow parameters on the velocity andtemperature distribution as well as the heat transfer. Simulationsare performed in the ranges of 1 < Re < 40 and 1 � 10�8

< Da < 1 � 10�1 with constant Prandtl number equal to 7.2. In thefollowing subsections, the obtained results are presented anddiscussed.

3.1. Hydrodynamics results

The effect of Reynolds number on the streamlines around cylin-der is illustrated in Fig. 4 for the case of (a) the cylinder without aporous layer; (b) Da = 1 � 10�2 and (c) Da = 1 � 10�6 (d = 0.1). It canbe seen that at a small Reynolds number (Re = 1), the flow is fullyattached to the cylinder surface for all cases. For Re = 10, a re-cir-culating wake clearly appears because the pressure starts to in-crease in the rear of the cylinder and the particle nowexperiences an adverse pressure gradient [29]. Consequently, theflow separates from the cylinder surface. Also the wake length in-creases with a further increase in Reynolds number.

Effect of Darcy number on the streamlines around cylinder ford = 0.5 is illustrated in Fig. 5. It indicates that the flow can easilypenetrate the porous layer with the increase of Darcy number. Alsothis figure shows that at high Darcy numbers (10�2), the stream-lines pass through the porous layer with a small deviation sincehighly permeable porous media exerts little resistance againstthe passing free stream. Thus, the effect of the porous medium be-comes insignificant and the streamline approaches the case of acylinder without a porous layer (for high Darcy number). It is note-worthy that separation is delayed with the increase of Darcy num-

Fig. 2. (a) Schematic of the recirculation zone for cylinder; (b) typical example of grid near the wall.

Table 1Effect of grid size on the average Nusselt number and the overall wake length for solidcylinder (without porous ring) and Re = 40.

Cases Grid size (n �m) LR Nuave

1 300 � 194 2.169 7.0242 400 � 251 2.214 7.1603 460 � 270 2.230 7.2144 480 � 281 2.237 7.242

Fig. 3. Variation of separation angle with Reynolds number for a single cylinder.

S. Rashidi et al. / International Journal of Heat and Mass Transfer 63 (2013) 91–100 95

ber; therefore, there is a reverse relationship between the Darcynumber and the wake length.

By comparing the results shown in Figs. 4 and 5 for Re = 20 andDa = 10�2, we can conclude that with the increase of porous layer

thickness, the wake behind the cylinder further penetrates the por-ous layer.

The effects of the Reynolds and Darcy numbers on the wakelength are shown in Fig. 6(a) indicates variation of the wake lengthversus Reynolds number for d = 0.1 and different Darcy numbers. Itshows that the wake length increases with increasing Reynoldsnumber. Moreover, the presence of a porous layer and a reductionin the Darcy number around the cylinder increase the wake length.

Fig. 6(b) indicates variation of the wake length versus Darcynumber for (d = 0.3), Re = 20 and Re = 40. It shows that the wakelength decreases with increasing Darcy number. As marked onthe figure, there are two limit states (upper and lower limits).Upper limit is for lower Darcy numbers (outer solid cylinder) andlower limit is for higher Darcy numbers (inner solid cylinder). Toinvestigate the effect of Darcy number on velocity profile, the tan-gential and normal velocities along the porous cylinder surface(d = 1) are presented in Fig. 7. It shows that for the flow passing apermeable surface, the fluid velocity at the surface is nonzeroand the magnitudes of the velocity components u and v, reducesas Darcy number decreases.

Tangential and normal velocities have different effects on theflow patterns. The normal velocity resembles the effect of basebleed (secondary flow that is injected at the base of the truncatedplug [30]) and when the Darcy number is very small, normal veloc-ity is nearly zero on the porous surface due to no-penetrationboundary conditions.

The tangential velocity resembles the effect of semi-slip bound-ary condition (an intermediate condition between slip conditionusually experienced by a void body and no-slip condition experi-enced by a solid surface) and when the Darcy number is very small,tangential velocity is nearly zero on the porous cylinder due to no-slip boundary conditions [31].

Fig. 8 shows the distribution of pressure coefficient around sur-face of porous layer (d = 10) for various Darcy number at Re = 10and Re = 20. The pressure coefficient Cp is obtained as [8]:

Fig. 4. Configuration of streamlines of cases: (a) a single cylinder; (b) Da = 1 � 10�2; (c) Da = 1 � 10�6 and d = 0.1.

96 S. Rashidi et al. / International Journal of Heat and Mass Transfer 63 (2013) 91–100

Cp ¼ðp� � p�0Þ þ 1

2 qU21

12 qU2

1ð22Þ

p0⁄ denotes the front stagnation point pressure and p⁄ is the

wall pressure.It can be seen that when Darcy number goes to a small value,

the pressure coefficient approaches to an asymptote which equalsto that of a cylinder without porous layer and when Darcy numberis relatively large, the pressure coefficient number is much largerthan that of a solid cylinder. These changes are more obvious alongthe rear part of the cylinder where fluid flows out and adversepressure gradient decreases. For Da 6 1 � 10�3, Darcy numberdoes not have a significant effect on the pressure coefficient alongthe front of the cylinder which is the region where fluid flows intothe porous sheath.

Fig. 9 exhibit variation of drag coefficient on porous layer(d = 10) versus Darcy numbers for Re = 10, 20 and 40. Drag coeffi-cient is defined as [29]:

CD ¼ CDv þ CDp ¼Fd

0:5qU21D

ð23Þ

The drag force per unit length on the cylinder is calculated as[32]:

Fd ¼D2

Z 2p

0ð�p� cos h� 2mx sin hÞdh ð24Þ

Fig. 9 shows when 10�6 < Da < 10�4, CD is close to 2.91, 1.98,1.49 for Re = 10, 20 and 40, respectively indicating that the dragcoefficient approaches to an asymptote which equals to that of acylinder without porous layer [31]. This is an expectable resultsince a porous layer with very low permeability (correspondingto low Darcy numbers) will allow little or no fluid to pass throughit. But, the drag coefficient begins to decrease with increase inDarcy number, because a layer with a high permeability allowsthe fluid to flow through it with a low resistance. Therefore thedrag coefficient decreases in the presence of highly permeablelayers.

3.2. Heat transfer results

The heat transfer rate from the isothermal cylinder at Tw cov-ered by a porous layer to a passing flow at T1 has been computed

Fig. 5. Configuration of streamlines of cases: at different Darcy numbers, Re = 20and d = 0.5.

Fig. 6. Variation of wake length: (a) with Reynolds number at different Darcynumbers and d = 0.1; (b) with Darcy numbers and d = 0.3 at Re = 20, 40.

Fig. 7. (a) Tangential; (b) normal velocities along the porous cylinder surface forRe = 20 and d = 1.

S. Rashidi et al. / International Journal of Heat and Mass Transfer 63 (2013) 91–100 97

for different values of the Darcy and Reynolds numbers and porouslayer thicknesses. Depending on the ratio of the thermal conductiv-ities of the porous material to the passing fluid, the porous layercan serve as an insulation or an extended surface for heat transferaugmentation.

The local and average Nusselt numbers are defined as [33]:

Nu ¼ �Rc@T@r

� �r¼1

ð26Þ

Nuave ¼1

2p

Z 2p

0NuðhÞdh ð27Þ

Fig. 10 shows the variation of the average Nusselt number fordifferent Darcy number and porous layer thickness where the re-sults are reported for Re = 40 and the cylinder is covered by aninsulating porous material, i.e., ks < kf (kf (salt water) = 0.596; ks

(foam) = 0.092). This figure reveals that the convective heat trans-fer coefficient decreases as porous layer thickness increases orDarcy number decreases. However, it should be noted the totalheat transfer rate also depends on the surface area which changesby variation of the porous layer thickness. While the additionalinsulation to a cylinder increases the heat conduction resistanceof the insulation layer, it increases the interface area thus decreas-ing the convection resistance of the surface. Therefore, there is a

Fig. 8. Distribution of pressure coefficient around surface of porous cylinder forvarious Darcy numbers, d = 10 and at (a) Re = 10; (b) Re = 20.

Fig. 9. Variation of drag coefficient on porous cylinder versus Darcy numbers ford = 10, Re = 10, 20 and 40.

Fig. 10. Variation of average Nusselt number versus Darcy numbers for differentporous layer (foam) thickness and Re = 40, ks/kf = 0.15.

98 S. Rashidi et al. / International Journal of Heat and Mass Transfer 63 (2013) 91–100

competing behavior between the two phenomena. The critical ra-dius of insulation for an isothermal cylindrical body can be ex-pressed as rcr = keff/h [34].

By considering the following dimensionless parameters:

r ¼ r�

R; T ¼ T� � T1

Tw � T1ð28Þ

Convective heat transfer coefficient on the porous layer (r is theinsulation radius) is calculated as:

hðT�2��r�¼r� T1Þ ¼ �kf

@T�1@r�

����r�¼r

) hðTw � T1ÞT2jr¼rR

¼ � kf

RðTw � T1Þ

@T1

@r

����r¼r

R

) h ¼ � kf

R1

T2jr¼rR

@T1

@r

����r¼r

R

ð29Þ

At the interface between the porous and the fluid region, T1 = T2,so:

h ¼ � kf

R1

T1jr¼rR

@T1

@r

����r¼r

R

ð30Þ

For r < rcr (r is the insulation radius), the rate of heat transfer fromthe cylinder increases with the addition of insulation, reaches amaximum when r ¼ rcr , and starts to decrease for r > rcr . Thus whenr < rcr , insulating the cylinder may actually increase the rate of heattransfer from the cylinder instead of decreasing it.

Fig. 11 depicts how the porosity and Darcy number affects thecritical radius of insulation. As indicated in this figure, the criticalradius of insulation decreases with increasing Darcy number inall cases; in particular, the decrease is more obvious at a thick insu-lation ring. Also shown in this figure is that the rate of heat transferfrom the cylinder increases in the domains located above the solidline. Fig. 11 also suggests that for a thick and low permeable insu-lation ring, insulating the cylinder will increase the rate of heattransfer from the cylinder instead of decreasing it. Fig. 12 illus-trates variation of average Nusselt number versus Darcy numberswhere the results are reported for Re = 40 and different porouslayer thickness. Porous ring is filled with sand stone material, i.e.,ks > kf (kf (salt water) = 0.596; ks (sand stone) = 2.327). In this fig-ure, the horizontal line refers to the average Nusselt number fora cylinder without porous layer. It is notable that at non-dimen-sional porous layer thickness equal to 1 for Da > 2.2 � 10�5, theaverage Nusselt number increases with adding porous layer to cyl-inder due to relative higher thermal conductivity of porous mate-rial. However, for Da < 2.2 � 10�5, the average Nusselt number

Fig. 11. Variation of critical radius of insulation with Darcy numbers and fordifferent porous layer (foam) thickness and Re = 40, ks/kf = 0.15.

Fig. 12. Variation of average Nusselt number versus Darcy numbers for differentporous layer (sand stone) thickness and Re = 40, ks/kf = 3.9.

S. Rashidi et al. / International Journal of Heat and Mass Transfer 63 (2013) 91–100 99

decreases with adding porous layer to cylinder because the veloc-ity and rate of convection heat transfer considerably reduces withdecrease of permeability. Therefore, for heat transfer enhancementtechniques, it is more suitable to choice permeable porous layerwith high thermal conductivity.

4. Conclusion

In the present work, fluid motion and heat transfer around a so-lid circular cylinder wrapped with porous layer has been studied.Also, the effects of Darcy and Reynolds numbers and porous layerthickness on the flow patterns and heat transfer characteristicswere investigated. The highlights of our parametric study are asfollows:

� The presence of a porous layer around a cylinder increases thewake length. Consequently, a decrease in Darcy number leadsto an increase in the wake length.� The drag coefficient decreases with increasing Darcy number.

Because a highly permeable layer allows the fluid to flowthrough it with the least resistance and this resistance increasesas the permeability decreases.

� The magnitude of velocity components decreases as Darcynumber decreases because with decreasing Darcy number, theno-slip and no-penetration conditions are dominant.� For relatively large Darcy numbers (Da = 1 � 10�2), the pressure

coefficient is significantly larger than that of a single cylinderand these changes are more obvious along the rear part of thecylinder where fluid flows out and adverse pressure gradientdecreases .� For heat exchanger applications, a porous medium with high

permeability (since the fluid flows faster through porous layerwith high permeability, so the rate of convection heat transferare considerably increased with increase of Darcy number)and high thermal conductivity improves the thermalperformance.� The critical radius of insulation decreases with increasing Darcy

number because the convection heat transfer coefficientincreases with increase in Darcy number for porous materialwith low thermal conductivity.

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