simulation of thermally developing laminar ow in partially lled...

Scientia Iranica B (2016) 23(6), 2641{2649

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Simulation of thermally developing laminar ow inpartially �lled porous pipes under wall suction

A. Nouri-Borujerdi� and M.H. Seyyed-Hashemi

School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran.

Received 28 January 2015; received in revised form 3 October 2015; accepted 4 January 2016

KEYWORDSPartially �lled porouspipe;Developing forced-convection laminar ow;Entrance length;Wall suction;Numerical method.

Abstract. This study numerically investigates uid ow and heat transfer enhancementof a two-dimensional developing laminar ow in an axisymmetric pipe with partially �lledporous material attached to the wall. The e�ects of porous layer in the range of 0 � �=R < 1and Darcy number in the range of 10�6 � Da � 10�4 are investigated. It is found thatthermal entrance length increases with porous layer thickness and is longer than a plainpipe. In addition, the dependence of Nusselt number on the thickness of the porous layeris not monotonic. A critical value of the porous layer thickness exists at which the Nusseltnumber value reaches the minimum. As the porous layer is thickening, more uid is pushedtowards the middle of the pipe and less uid goes through the porous layer in comparisonwith the case that the porous layer is located in the center. In this case, the heat convectionis reduced relative to heat conduction due to the physical contact between the solid matrixand the wall surface.© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

Heat transfer and transport phenomena in porousmedia are important processes in many engineeringapplications such as heat exchanger, electronic cooling,oil extraction, chemical catalytic reactors, and ther-mal insulations. It is generally known that partially�lling a conduit with saturated porous media leadsto the conclusion that higher heat transfer rate in aforced convection can be achieved at the expense ofa reasonable pressure drop. However, the penalty ofthe pressure drop will be signi�cant if the conduitis fully �lled with porous media. Recently, porousmedia have been employed to promote heat transferin thermal systems and this has received considerableattention [1-3]. The e�ects of a random porositymodel on heat transfer performance of porous mediaare numerically demonstrated by Fu and Huang [4].

*. Corresponding author.E-mail address: [email protected] (A. Nouri-Borujerdi)

Deng and Martinez [5] numerically and analyticallyconsidered two-dimensional ow for partially �lledporous channel under wall suction. The numericalinvestigation of Yucel and Guven [6] on convectioncooling enhancement of heated elements in a parallel-plate channel using porous inserts shows that heattransfer can be enhanced by using high thermal con-ductivity of porous medium. Also, the pressure dropincreases rapidly along the channel with increase in theReynolds number. Erdogan and Imrak [7] consideredfully laminar ow of an incompressible viscous uid in auniformly porous pipe with suction and injection. Theyexpressed the velocity �led in a series form in terms ofthe modi�ed Bessel function of the �rst kind of order n.They found that ow properties depend on the suctionvelocity and for large values of suction velocity, the ow near the region of the suction shows a boundary-layer character. In this region, the velocity and thevortices vary sharply. Jiang and Ren [8] numericallyinvestigated forced-convection heat transfer in porousmedia using a thermal non-equilibrium model. Their

2642 Nouri-Borujerdi and Seyyed-Hashemi/Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 2641{2649

results indicated that the convection heat transfer inporous media can be predicted numerically by usingthe thermal non-equilibrium model with the idealconstant-wall heat- ux boundary condition. Alazmiand Vafai [9] numerically studied constant-wall heat- ux boundary conditions in porous media under lo-cal thermal non-equilibrium conditions. Tzeng [10]experimentally and numerically examined the forced-convection heat transfer in a rectangular channel �lledwith sintered bronze beads and periodically spacedheat blocks. The result indicated that such arrange-ment can be used to improve the cooling performanceof heated blocks, where the blocks are used to simulateheated electronic components, by sintered metallicporous media. Aguilar-Madera [11] presented themodeling of momentum and heat transfer in parallel-plate channels partially �lled with a porous insert usinga one-domain approach. With this methodology, theyavoided specifying the boundary conditions at the uidporous insert boundary at the cost of solving morecomplicated transport models. Wu et al. [12] tookadvantage of longitudinal vortex generator and slit tosimulate heat transfer and uid ow characteristics ofcomposite �n numerically. Their computational resultsshow that composite �n can improve the synergy oftemperature gradient and velocity �elds, and its equiv-alent thermal resistance is smaller and its irreversibilityof heat transfer is lower. Eiamsa-ard et al. [13] and Basand Ozceyhan [14] enhanced heat transfer in tubes byinserting twisted tapes. They stated that the optimumtradeo� between the enhancement of heat transfer andincreasing of friction are obtained from a critical valuebetween their ratio. Yang et al. [15] performed the heattransfer performance assessment for forced convectionin a heated tube partially �lled with a porous materialin the core and attached to the wall. It was found thatthe local thermal non-equilibrium analysis was essentialfor the case of forced convection in a tube with a heatedwall surface covered with a porous medium layer. Onthe other hand, the local thermal equilibrium analysissu�ced to capture transport phenomena for the caseof forced convection in a tube with a porous mediumcore. In a comparatively low range of pumping power,the heat transfer performance of the tube with a porousmedium core is higher than that of the tube with a wallcovered with a porous medium layer. However, in ahigh range of pumping power, the latter performanceexceeds the former. Poulikakos and Kazmierczak [16]presented a theoretical study of fully developed forcedconvection in a channel and a circular pipe partially�lled with a porous matrix attached to the wall. Bothcases of constant-wall heat ux and constant-wall tem-perature were studied. Of particular importance is the�nding that the dependence of Nu on the thickness ofthe porous layer is not monotonic. A critical thicknessexists at which the value of Nu reaches the minimum.

The present work numerically investigates a devel-oping laminar forced-convection heat transfer in a pipewith partially �lled porous medium. The pipe wall isunder suction and at constant temperature. The e�ectsof thermal entrance length, thickness of the porouslayer, and Darcy number on the rate of heat transferand on the pressure drop are discussed.

2. Problem statement and formulation

Let us assume that a laminar air ow enters a partially�lled porous pipe with radius R and length L atuniform velocity uin and temperature Tin (Figure 1).The wall of the pipe is reticulated at temperature Twand under suction. The air ow can leave the pipeboth at the outlet and through the wall at atmosphericpressure. The porosity and permeability of the porouslayer are assumed to be � and K, respectively, withthickness � in the range of 0 < � < R. The porouslayer can be placed at the center or on the wall andextends inward, toward the center line.

2.1. Mass and momentum equationsTo present the model equations in one-domain ap-proach, a single set of unsteady transport equationsincluding mass and momentum applicable for the entiredomain comprising the clear uid and porous layer isgiven as follows [17]:@u@z

+1r@(rv)@r

= 0; (1)

1�@�u@t

+1�2

�@(�u2)@z

+1r@(�ruv)@r

�= �dp

dz

+ ~��

1r@@r

�r@u@r

�+@2u@z2

�� Ku� �CFp

Kjuju; (2)

1�@�v@t

+1�2

�@(�uv)@z

+1r@(�rv2)@r

�= �dp

dr

+ ~��

1r@@r

�r@v@r

�+@2v@z2

�� Kv � �CFp

Kjvjv: (3)

Figure 1. Schematic of the porous pipe with distributionof suction velocity through the wall.

Nouri-Borujerdi and Seyyed-Hashemi/Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 2641{2649 2643

The corresponding momentum equations of the clear uid are determined from Eqs. (2) and (3) when � = 1and K ! 1, and the corresponding governing equa-tions of the uid phase in the porous layer, referred toas Brinkman-Forchheimer of Darcy's law, are obtainedif 0 < � < 1 and K = �nite.

The coe�cient �̂f = �f=� is an e�ective liquidviscosity. Brinkman [18] set �f and ~�f equal to eachother; but, in general, that is not true. Bear andBachmat [19] suggested a detailed averaging processthat led to the result: ~�f=�f = 1=�� , for an isotropicporous medium, where � was a quantity called thetortuosity of the medium. On the other hand, straightvolume averaging as presented by Ochoa-Tapia andWhitaker [20] gives ~�f=�f = 1=�. u and v are averagedvelocity of the uid taken with respect to a volume ele-ment of the medium (incorporating both solid and uidmaterials) in z- and r-direction, respectively. Thesequantities have been given various names, by di�erentauthors, such as Darcy velocity and super�cial velocity.P is pressure and denotes an intrinsic quantity and thatDarcy's equation is not a balance of forces averagedover a representative elementary volume. Also, CF is adimensionless form-drag constant and according to theresults of Handley and Heggs [21], its value approacheszero when the porous Reynolds number is of order oneor less. Beavers et al. [22] showed that the boundingwalls could have a substantial e�ect on the value of CFand found that their data correlated fairly well withthe expression CF = 0:55(1�5:5d=De), in which d wasthe diameter of their sphere and De was the equivalentdiameter of the bed de�ned in terms of the height,h, and width, w, of the bed by De = 2hw=(h + w).The coe�cient, K, is called the speci�c or intrinsicpermeability and is independent of the nature of the uid, but it depends on the geometry of the medium.It is in general a second-order tensor. For the case of anisotropic medium, it is a scalar quantity and its valuefor natural materials varies widely with dimension.If K is determined by the geometry of the medium,then, clearly, it is possible to calculate K in terms ofthe geometrical parameters. For example, in the caseof beds of particles or �bbers, one can introduce ane�ective average particle or �ber diameter dpe�. Thehydraulic radius theory of Carman-Kozey leads to therelationship:

K =d2

pe��3

150(1� �)2 ; (4)

where:

dpe� =

1R0d3pf(dp)d(dp)

1R0d2pf(dp)d(dp)

: (5)

f(dp) is the density function of the distribution ofdiameter dp. The constant 150 in Eq. (4) was obtainedby seeking the best �t with experimental results.

2.2. Energy equationWe start with a simple situation in which the mediumis isotropic and where radiation e�ect, viscous dissipa-tion, and the work done by pressure changes are absent.Here, we assume that heat conduction in the uid andsolid phases takes place in parallel so that there is nonet heat transfer from one phase to the other. Usually,it is a good approximation to assume that the solid and uid phases are in thermal equilibrium; but there aresituations, such as highly transient problem and somesteady-state problems, in which this is not so [23].

(�Cp)stag@T@t

+ �fCpf�@uT@z

+1r@rvT@r

�= [kstag+�kdis]

�1r@@r

�r@T@r

�+@2T@z2

�:

(6)

The corresponding energy equation of the clear uid isdetermined from Eq. (6), where the porosity is � = 1and the corresponding energy equation of the porouslayer is obtained if 0 < � < 1. Meanwhile, kstag =�kf +(1��)ks and (�Cp)stag = ��fCpf +(1��)�sCps.The e�ective thermal conductivity of the uid consistsof the stagnant and dispersion components as kfe� =kf +kdis, which is expressed by empirical correlation ofWakao and Kaguei [24].

kfe� = kf�1 + 0:1

RepPr�

�: (7)

Also, Calmidi and Mahajan [25,26] proposed the fol-lowing correlations:

kstag = �kf + 0:19ks(1� �)0:763; (8)

kdis = 0:06�fCpfuD

pK

�; (9)

in which uD is Darcian velocity (uniform inlet velocity).The Nusselt number is:

Nu =_q00wD

(Tb � Tw)(kstag + �kdis); (10)

where, Tb is the bulk or mean temperature of thestream across the pipe and is calculated by:

Tb =1

Auin

ZA

uTdA: (11)


3. Boundary conditions

At the interface between the wall and the porous layeror between a porous layer and a clear uid, we canimpose continuity of the normal component of the heat ux and continuity of the temperature because of theassumption of local thermal equilibrium.

The corresponding boundary conditions of theabove governing equations are speci�ed as follows:

u(r; 0) = uin; vin(r; 0) = 0; Tf (r; 0) = Tin;(12a)

@u(r; L)@z

= 0;@v(r; L)@z

= 0;

@Tf (r; L)@z

= 0;@Ts(r; L)

@z= 0;

P (r; L) = P1; (12b)

@u(0; z)@r

= 0;@Tf (0; z)

@r= 0;

v(0; z) = 0; p(R; z) = patm; (12c)

u(R; z) = 0; Tf (R; z) = Ts(R; z) = Tw: (12d)

Indices s and f refer to solid and uid, respectively.Since the outside of the pipe is exposed to the atmo-spheric pressure and is constant along the boundary forall values of z, @p=@z = 0 and, hence, u = 0 for all z.Therefore, based on the continuity equation, @u=@z = 0at r = R. We conclude that:

@v(R; z)@r

= 0: (13)

4. Numerical method with one-domainapproach

To determine the ow variables in the clear uid andin the porous layer, Eqs. (1)-(5) have been solvednumerically in one-domain approach with an interfacezone across which the porosity and permeability arecontinuous. In the one-domain approach, the interfacecan be either a continuous transition zone across whichthe physical variables encounter possibly strong butnevertheless continuous variations (e.g., Goyeau etal. [27] and Valdes-Parada et al. [28]) or a discontinuousinterface where the physical variables are possiblydiscontinuous (e.g., Silva and de Lemos [29]). Jametand Goyeau [30] show that the discontinuous one-domain approach and the two-domain approach aremathematically strictly equivalent, provided that thediscontinuous one-domain approach is in the sense ofdistributions. On the other hand, the experimentalresults of Basu and Khalili [31] show that a one-domain

approach can provide good predictions of interfacial ow. In this regard, Arquiset et al. [32] used aharmonic mean formulation of the permeability andPoulikakos [33] also used a hyperbolic tangent functionto handle the abrupt change in governing parameters.

In this work, a �nite-volume method in one-domain approach is used to discretize the governingequations based on a staggered grid with uniform gridspacing in r- and z-direction. We also used a hyperbolictangent function to handle the abrupt change in theporosity and permeability in the interface zone. The uid-porous interface is supposed to be located atthe center of the cell interface. Scalars variables arestored at the center of the mesh cells and variablesof the vectors are stored at the mesh faces. Thesecond-order upwind di�erencing scheme is used forconvection terms and the central di�erencing schemeis used for di�usion terms. The pressure and velocitylinked in the momentum equations are solved on thestaggered grid arrangement. Subsequently, the set ofthe discretized linear algebraic equations for u; v, andp are solved using successive line-by-line relaxation bytri-diagonal matrix algorithm. Either of the algebraicequations includes only three non-zero coe�cients.The convergence criterion for solution of equations isalso established by local and global mass and energyconservations in which their corresponding residualsare set equal to 10�6.

5. Results and discussion

It is assumed that air ow enters a partially �lledporous pipe with radius R = 2 cm and length L =50 cm at uniform inlet velocity uin = 0:805 m/s andtemperature Tin = 25�C (see Figure 1). The air owexits through both the outlet and the permeable wallof the pipe at temperature Tw = 35�C and atmosphericpressure. The porous layer is placed in the center orattached to the pipe wall and extends inward, towardthe centerline. The porous material is made of acommercial aluminum with physical properties of � =2770 kg/m3, Cp = 875 J/kg.�K, k = 177 W/m.K, andporosity of � = 0:7.

Model tests: In order to validate the numericalresults and to show that the results are independentof the grid size, a number of grids with di�erent sizesare used. To compare the present results with thepreviously published results in the open literature,Figures 2 and 3 are plotted for the case in which theporous layer is placed in the center of the pipe withoutwall suction. These �gures respectively show the meshre�nement and axial velocity at the outlet cross sectionof a pipe with radius R = 3:2 cm and length L = 50 cmfor four grid points 8�13, 17�26, 33�51, and 65�101.It is found that the velocity will be independent grid


Figure 2. Mesh re�nement for the axial velocity at theoutlet cross section of the pipe and the results ofMohammad [1] without wall suction.

Figure 3. Axial velocity for di�erent thicknesses of theporous layer located in the pipe center and the results ofMohammad [1] without wall suction.

size when the grid points are equal to or greater than65 � 101. In this �gure, the data of Mohammad [1] isalso given for comparison. The other cases where theporous layer is attached to the wall have been plottedin Figures 3-9.

Model results: Figure 3 depicts three axial velocitypro�les at the outlet cross section of the pipe (z = L)for two di�erent porous layers (�=R = 0:1; 0:2) locatedin the center of the pipe. The third one (�=R = 0)corresponds to the empty pipe or no porous layer. Thedata of Mohammad [1] is also shown for three layers,i.e. �=R = 0; 0:1 and 0.2. The comparison shows agood agreement between these three cases. It is worthnoting that the velocity pro�le in the porous layers for�=R = 0:1, and 0:2 is Darcy ow and for the clear pipe(�=R = 0) is Poiseuille ow with maximum speed atthe center of the pipe.

Figure 4 represents the fully developed axial ve-

Figure 4. Pro�les of the developed axial velocity fordi�erent thicknesses of porous layer in the vicinity of thewall with suction.

Figure 5. Distribution of suction velocity through thewall for di�erent thicknesses of the porous layer.

locity pro�les for four di�erent porous layer thicknesses(�=R = 0:2; 0:4; 0:6; 0:8) at the outlet cross section ofthe pipe with wall suction. In this case, the porouslayer is attached to the wall. As the porous layer isthickening, more uid is pushed towards the middle ofthe pipe and the maximum speed occurs at the center.In other words, less mass ow rate of air goes throughthe porous layer in comparison with the case that theporous layer is located in the central region.

Figure 5 depicts distribution of suction velocitythrough the wall for four di�erent thicknesses of porouslayer in the vicinity of the wall (�=R = 0:2; 0:4; 0:6 and0:8). The coordinates axes have been dimensionlessby the inlet velocity and the length of the pipe. Thedischarge velocity increases sharply near the intakepipe; then, it gradually decreases and eventuallyreaches zero. Furthermore, the thicker layer makes theemptying rate increase. It can be deduced from Darcylaw that emptying rate is proportional to the pressuregradient.

Figure 6 represents the pressure distribution in


Figure 6. Distribution of radial pressure in the middlelength of the pipe for di�erent thicknesses of porous layerattached to the wall with suction.

Figure 7. Thermal entrance length versus the porouslayer thickness with wall suction.

the r-direction and in the middle length of the pipefor di�erent thicknesses of porous layer attached to thewall with suction. Each curve consists of two parts,a vertical part that belongs to the clear uid regionwith constant pressure and the other part belongs tothe porous layer with a pressure reduction. In theporous layer, the pressure starts from the atmosphericpressure at the wall and reaches the pressure of air ow at the interface between two regions. For instance,when �=R = 0:6, air pressure in the middle of the pipeis (p � p1)=�u2

in = 0:19; then, it gradually decreasesacross the porous layer and reaches the atmosphericpressure at the wall.

Figure 7 exhibits the thermal entrance lengthagainst the porous layer thickness. It is found thatthe entrance length increases somehow linearly with

Figure 8. Developing local Nusselt number along thepipe for di�erent porous layer thicknesses attached to thewall with suction.

porous layer thickness and is approximated by functionof zent=L = 0:135�=R. Thickening of the porous layerpushes the uid towards the middle of the pipe andmore uid goes through the clear uid region. In otherwords, the corresponding Reynolds number increasesand thermal entrance length and, consequently, heattransfer coe�cient and pressure drop increase as well.

Figure 8 illustrates the developing local Nusseltnumber along the pipe for four di�erent porous layerthicknesses (0:2; 0:4; 0:6; 0:8) and Darcy number ofDa = 10�5. Each curve, after passing a distancefrom the inlet pipe, reaches a constant value. Thisdistance corresponds to the thermal entrance lengthand depends on the thickness of the porous layer.

Figure 9 illustrates the fully developed Nusseltnumber versus the porous layer thickness without con-sideration of convective terms in momentum equationsas well as without wall suction. A surprising �ndingis that dependence of the Nusselt number on thethickness of the porous layer is not monotonic. Thatis, there exists a critical value of the porous layerthickness at which the value of the Nusselt numberreaches the minimum. The physical interpretation isthat thickening of the porous layer pushes the uidtowards the middle of the pipe and less mass owrate goes through the porous layer. In this event,the rate of convective heat transfer between the wallsurface and the uid saturated porous layer is reducedin comparison with the conduction part due to thecontact between the solid matrix and the wall surface.The �gure also includes the data of Poulikakos andKazmierczak [16] for comparison.

6. Conclusion

This study has numerically considered a developinglaminar forced-convection heat transfer in a partially�lled porous pipe with wall suction for the range of


Figure 9. Fully developed Nusselt number versus theporous layer thickness and in comparison with the resultsof Popukakous [16] without wall suction.

Darcy number, 10�6 � Da � 10�4, and porous layerthickness, 0 � �=R < 1. It is found that thermalentrance length increases with porous layer thicknessand is longer than that of the pipe with no porouslayer. Further, the local Nusselt numbers approachthe developed value after a distance from the inlet ofthe pipe and depend on the porous layer thickness.Due to the resistance of the porous matrix to the uid ow, less mass ow rate goes through the porouslayer in comparison with the case that the porous layeris located in the center. In this case, the rate ofconvection heat transfer between the wall and the uidsaturated porous region is reduced relative to the heatconduction between them.

Nomenclature

A Surface areaCp Speci�c heatCF Inertial coe�cientD Pipe diameterDa Darcy number K=D2

dp Particle diameterh Heat transfer coe�cientk Thermal conductivityK PermeabilityL LengthNu Nusselt number, hd=k_q00 Heat uxp Pressure

Pr Prantel numberr Radial coordinateR Pipe radiusRe Reynolds number, vD=�t TimeT Temperatureu Axial velocityv Radial coordinatez Axial coordinate

Greek letters

� Thermal di�usivity, k=(�Cp)� Porous wall or porous layer thickness� Dynamic viscosity� Kinematic viscosity� Density� Porosity

Subscripts

atm Atmospheredis Dispersione� E�ectivef Fluidin Inlet conditionsp Porous, particles Solidstag Stagnationw Wall

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Biographies

Ali Nouri-Borujerdi is Professor of Mechanical En-gineering at Sharif University of Technology, Tehran,Iran, where he teaches undergraduate- and graduate-level courses in the Thermal/Fluids Sciences Groupof Mechanical Engineering Department. His teachingfocuses on heat transfer, computational uid dynamics,

and two-phase ows, including boiling and condensa-tion. His current research programs include numericalsimulation of two-phase ow, compressible turbulent ows, and porous media. Professor Nouri has publishedmore than 140 articles in international journals andconferences.

Mohammad-Hossein Seyyed-Hashemi receivedhis BS degree in Mechanical Engineering from IranUniversity of Sciences and Technology. He receivedMS degree in Thermal Fluid Sciences from MechanicalEngineering Department at Sharif University of Tech-nology, Tehran, Iran.

simulation of thermally developing laminar ow in partially lled...

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