effect of thermal asymmetry on laminar forced convection heat transfer in a porous annular channel

Effect of Thermal Asymmetry on Laminar Forced ConvectionHeat Transfer in a Porous Annular Channel

By J. Mitrovic* and B. Maletic

DOI: 10.1002/ceat.200600069

The effect of thermal asymmetry on laminar forced convection heat transfer in an annular porous channel with a Darcy dissi-pation of fluid kinetic energy was investigated numerically. The cylindrical surfaces making the channel boundaries were keptat constant but different temperatures. The thermal asymmetry thus imposed on the system results in an asymmetric tem-perature field and different heat fluxes across the channel boundaries. Depending on the Darcy, Péclet and Reynolds num-bers, the thermal asymmetry may lead to a reversal of the heat flux along the channel at least at one of the channel walls. Thecorresponding Nusselt number becomes zero and subsequently experiences a discontinuity, thereby jumping from infinite ne-gative to infinite positive, or vice versa. This feature is observed in the region of thermal development. In the fully developedheat transfer region, the Nusselt numbers can be positive or negative for the same inlet conditions, depending on the heatsource strength. In the case of a plug flow, the analytical expressions for the Nusselt numbers have been obtained.

1 Introduction

The heat transfer with forced convection in porous mediais an interesting and challenging physical problem the solu-tion of which is important in several areas of engineeringpractice [1]. It has, therefore, been extensively studied in thepast, and various fluid flow and heat transfer arrangementshave been treated both analytically and numerically [2–5].However, the problem is still far away from being complete-ly solved, even the governing equations are the subject mat-ter of scientific debates [6–9]. Nevertheless, the mathemati-cal models used so far account for different effects and thesolutions obtained are adapted to various boundary condi-tions. For instance, Kaviany [10] studied laminar forced con-vection in a porous channel bounded by isothermal parallelplates, adopting the Brinkman-extended Darcy model. Vafaiand Kim [11] arrived at a closed form solution with fully de-veloped forced convection in a porous plane channel ex-posed to a symmetric heating at constant heat flux. Nield etal. [12] analyzed the fully developed forced convection in afluid-saturated porous medium channel with isothermal orisoflux boundaries. Nield et al. [13] investigated the heattransfer in the thermally developing region of a hydrodyna-mically developed flow in a plane porous channel boundedby isothermal plates. The energy equation they used ac-counts for viscous dissipation and axial heat conduction. Thesolutions reported illustrate the effects of the Brinkman,Péclet and Darcy numbers on heat transfer for different dis-sipation models. Mohamad [14] investigated the flow fieldand heat transfer with laminar forced convection in conduitsat various filling with a porous material. As far as the fullyfilled channel is concerned, the effect of the Darcy number

on heat transfer in the fully developed flow region may belargely neglected for Da � 1. Haji-Sheikh and Vafai [15]performed heat transfer analyses for various cross sectionsof the conduits without a heat source, giving detailed insightsinto the effect of the Darcy number on the thermal perfor-mance of porous inserts. For this model, Haji-Shekh [16]provided an approximate expression for the Nusselt number.Haji-Shekh et al. [17] used the Green’s function to treat thefluid flow and heat transfer in porous conduits demonstrat-ing the results for a parallel plate channel.

While dealing with the transport processes in porous me-dia the so-called local thermal equilibrium (LTE) model isfrequently adopted. By this model, the fluid and the porousmedium are considered as a single phase having physicalproperties of the actual phases mostly weighted by the vol-ume fractions occupied by these phases. The applicability ofthe LTE model is confined to a certain range of process andsystem parameters like fluid velocity and transport proper-ties of the phases. Contrary to this model, the model of localthermal nonequilibrium (LTNE) accounts for thermal inter-action among the phases within the porous system [18–25].In this model the thermal interaction is based on a heattransfer coefficient at the phase interface within the poroussystem which is a priori unknown. The LTNE two-equationmodel is usually considered to be more adequate than theone-equation LTE model. The boundary between thesemodels regarding their applicability has been discussed inseveral papers, see, e.g., [20], and the references therein.

In summary, the treatments in all of the above-mentionedreferences are restricted to thermally symmetric boundaryconditions. In practice, however, it is hardly possible toaccomplish such conditions, and a thermal asymmetry willperhaps be the rule rather than exception. Thermal asymme-try is shown by Mitrovic and Maletic [26] to materially affectthe heat transfer in laminar forced convection in a conduitof annular cross section without a porous insert, and similareffects of thermal asymmetry may qualitatively also be

750 © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Chem. Eng. Technol. 2006, 29, No. 6

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[*] J. Mitrovic (author to whom correspondence should be addressed,[email protected]), B. Maletic, Institut für Energie- und Verfahrens-technik, Thermische Verfahrenstechnik und Anlagentechnik, UniversitätPaderborn, D-33095 Paderborn, Germany.

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expected in the case of porous channels. Mahmud and Fraser[27] studied the heat transfer and entropy generation withlaminar fully developed flow in a porous channel boundedby parallel plates, which were kept isothermal at differenttemperatures thus imposing a thermal asymmetry on thesystem. The inertia in the momentum equation was disre-garded, while a volumetric source term was included intothe energy equation. The physical properties were taken asconstant, so the flow field is decoupled from the temperaturefield, and the one-dimensional transport equations weresolved analytically. The thermal asymmetry results in anasymmetric temperature distribution in the porous gap theshape of which depends on the Eckert, the Prandtl and theDarcy number. It, consequently, also affects the heat transferacross the porous insert. The analysis by Mahmud and Fra-ser [27] is restricted to the cold plate. Mitrovic and Maletic[28] extended their work by including also the hot plate intothe considerations. As was shown, not only the intensity, butalso the direction of the heat flux at the hot plate dependson the Eckert, Prandtl and Darcy number, that is, on thethermal asymmetry, and the fluid in the porous channel iscooled by the hot plate in one case, but heated in the other.Under particular conditions, this plate becomes adiabatic.

Reviewing the literature did not reveal any study of fluidflow and heat transfer in a porous annular channel that isexposed to a thermal asymmetry. In the present paper, theauthors report on some representative results on heat trans-fer with a steady-state laminar flow in porous annuli. Theythus extend their considerations [26, 29] confined to anempty annular channel by inserting a porous medium thatfills the whole annular space. The porous insert of a homoge-neous permeability is sandwiched between two cylindricalsurfaces that are at constant, but different temperatures. Theheat source corresponds to the Darcy model, the axial heatconduction being neglected. The problem is treated numeri-cally on the basis of the LTE model by using Mathematica[30]. An analytical solution of the energy equation was ob-tained in the fully developed plug flow region. As is demon-strated in the paper, the thermal asymmetry substantiallyaffects the heat transfer. Depending on the Darcy, Eckertand Péclet numbers, the thermal asymmetry can lead to areversal of the heat flux with discontinuities of the Nusseltnumbers on both cylindrical channel surfaces.

2 Physical Model and Governing Equations

The physical model is illustrated in Fig. 1. Two concentri-cally arranged cylindrical surfaces of constant temperatures,TW1 and TW2 ≠ TW1 bind a porous layer of a thicknessd = r2 – r1 saturated with a Newtonian fluid flowing alongthe z-coordinate. The flow and temperature fields first de-pend on r and z, but after a sufficiently large flow lengthdownstream the channel inlet, in the fully developed region,these fields become independent of z.

The following assumptions are adopted:

– the permeability of the porous medium is homogeneous,– the physical properties are constant,– free convection effects are neglected,– the fluid temperature and velocity are constant in the inlet

cross section,– the phases are locally at thermal equilibrium,– the heat source term corresponds to the Darcy flow model,– the system is at steady state,– the fluid flow is taken as one-dimensional and developed,– the axial heat conduction and the radial heat convection

are neglected.With these assumptions, the fluid flow is described by the

Brinkman momentum equation1):

leff∂2w∂r2 � 1

r∂w∂r

� �� l

Kw � ∂p

∂z� 0 (1)

(see [13]), whereas the heat transfer obeys the following en-ergy equation:

w∂T∂z

� j∂2T∂r2 � 1

r∂T∂r

� �� e

qcp(2)

Here, leff denotes an effective viscosity, w the axial velo-city, l the fluid viscosity, K the permeability of the porousmedium, p the pressure, T the temperature and e a volu-metric heat source; j, q, and cp are the usual fluid properties,r and z follow from Fig. 1.

In the present considerations, the simplest expression forthe heat source, e, is:

e � lK

w2 (3)

the so-called Darcy dissipation of the kinetic energy of fluid,will be used. It ignores viscous dissipation, which may be-come important in certain ranges of process parameters [13].

Eqs. (1) and (2) can be written dimensionless as follows:

∂2W∂R2 � 1

R∂W∂R

� lleff

1Da

W � lleff

Re∂P∂Z

� 0 (4)

Pe W∂h∂Z

� ∂2h∂R2 �

1R

∂h∂R

� �� Ec Pr

DaW2 (5)

where the dimensionless quantities are defined by:

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Figure 1. Physical system.

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1) List of symbols at the end of the paper.

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R � rr2

, Z � z

r2

, W � wwIN

,qwINr2

l� Re

Da � Kr2

2, P � p

qw2IN

, Ec � w2IN

cp TW2 � TIN� �, Pr � mj

Pe � wINr2

j, h � T � TIN

TW2 � TIN(6)

The index IN refers to the channel inlet. Note that thedefinitions of the Eckert number Ec and the temperature hare inappropriate for TW2 = TIN

3 Flow Field and Pressure Drop

Setting:

lleff

1Da

� m2,l

leffRe

∂P∂Z

� U (7)

W � ReDa ∂P

∂Z� N (8)

f � mR (9)

the momentum Eq. (4) becomes a modified Bessel equation:

∂2N

∂f2 � 1f

∂N∂f

� N � 0 (10)

Its solution can be written as:

N = C1I0(f) + C2K0(f) (11)

where I0(f) and K0(f) represent the zero-order modifiedBessel functions of the first kind and of the second kind,respectively; C1 and C2 are the constants of integration.

With the boundary conditions:

R = R1 = r1/r2; f = f1 = R1m: W = 0R = R2 = r2/r2 = 1, f = f2 = m: W = 0 (12)

one obtains:

C1 � D1

DReDa

∂P∂Z

, C2 � D2

DReDa

∂P∂Z

(13)

D � I0 f1� � K0 f1� �I0 f2� � K0 f2� ��

�� I0 mR1� � K0 mR1� �I0 m� � K0 m� �

�� (14)

D1 � 1 K0 f1� �1 K0 f2� ��

�� 1 K0 mR1� �1 K0 m� ��

�� (15)

D2 � I0 f1� � 1I0 f2� � 1

�� I0 mR1� � 1

I0 m� � 1

�� (16)

Averaging the velocity w in the cross section of the annu-lar channel results in an expression for the pressure dropfunction, U:

�w � 1A

�A

wdA � wIN1A

�A

WdA (17)

W � D1 I0 f� � � D2 K0 f� �D

� 1� �

Um2 (18a)

W � D1 I0 mR� � � D2 K0 mR� �D

� 1� �

Um2 (18b)

1A

�A

WdA � �wwIN

� 1,2

1 � R21

� R2

R1

W R dR � 1 (19)

Um2 � 1 � R2

1

� �m2

2 f2

f1

D1 I0 f� � � D2 K0 f� �D

fdf � 1 � R21

� �m2

(20)

Combining Eqs. (7) and (20), one readily obtains the pres-sure drop ∂P/∂Z as a function of the process and system pa-rameters.

4 General Expressions for Temperature Fieldand Heat Transfer

4.1 Temperature Field

Inserting the expression (18b) for the velocity W in Eq. (5)gives:

PeUm2 f m�R1�R� � ∂h

∂Z� ∂2h

∂R2 �1R

∂h∂R

� ��

Ec Pr U2

m2

leff

lf 2 m�R1�R� � (21)

f m�R1�R� � � D1 I0 mR� � � D2 K0 mR� �D

� 1 (22)

Introducing:

X � Ec Pr U2

m2

leff

l(23)

w � hX

, n � ZPe

(24)

Eq. (21) takes the form:

f m�R1�R� � Um2

∂w∂n

� ∂2w∂R2 �

1R

∂w∂R

� �� f 2 m�R1�R� � (25)

This equation could be basically solved by the method ofseparation of variables. The steps would be analogous tothose for a parallel plate channel at thermal symmetry[13, 31, 32]. However, because of the complexity of the ex-pected solution (a series of Bessel functions), this way wasnot taken in the present paper. Instead, Eq. (25) is inte-grated numerically by using Mathematica [30].

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The boundary conditions to be satisfied are:

Z = 0, f = 0, R1 ≤ R ≤ 1

h � TIN � TIN

TW2 � TIN� hIN � 0, w = 0, R = R1, 0 ≤ f ≤ ∞:

T = TW1, h � TW1 � TIN� �TW2 � TIN� � � hW1, w � hW1

X� w1

R = R2 =1, 0 ≤ f ≤ ∞:

T = TW2, h � TW2 � TIN� �TW2 � TIN� � � hW2 � 1,

w � hW2

X� 1

X� w2 (26)

The quantity, w, represents the dimensionless temperaturedifference scaled by the parameter X. As the parameter X isa measure of the temperature, h, the field of the quantity, w,coincides with the temperature field for X = 1. For othervalues of this parameter, the temperature field will corre-spondingly be stretched or compressed.

4.2 Heat Transfer Coefficients and Nusselt Numbers

The Fourier heat flux, q, is defined by:

q � �k∂T∂r

� �kTW2 � TIN

r2

∂h∂R

(27a)

qr2

k TW2 � TIN� � � � ∂h∂R

� �X∂w∂R

(27b)

giving the local heat transfer coefficient, a:

a � � kTW � �T

∂T∂r

� �W

(28)

where the index W refers to the wall. Thus:

a1 � � kr2

1hW1 � �h

∂h∂R

� �W1

(29)

a2 � � kr2

11 � �h

∂h∂R

� �W2

(30)

The bulk mean fluid temperature �T is obtained from:

�T ��

ATwdA

�A

wdA (31)

or:

�h ��T � TIN

TW2 � TIN�� 1

R1

hWdR� 1

R1

WdR (32)

�w � �h�

X �� 1

R1

wWdR� 1

R1

WdR (33)

Introducing the hydraulic diameter, d:

d � 4AU

� 2 r2 � r1� � � 2r2 1 � R1� � (34)

Eqs. (29) and (30) yield:

Nu1 � a1dk

� �21 � R1

hW1 � �h

∂h∂R

� �W1

� �21 � R1

w1 � �w∂w∂R

� �W1

(35)

Nu2 � a2dk

� �21 � R1

1 � �h

∂h∂R

� �W2

� �21 � R1

w2 � �w∂w∂R

� �W2

(36)

Given the quantities hW1, m, X, and R1 or m, w1, w2, andR1, the Nusselt numbers will depend only on the axial di-mensionless coordinate n.

5 Representative Results

As is obvious from the momentum and energy equationswith the boundary conditions (see Eqs. (10), (12), (25), and(26)) the independent parameters affecting the dimension-less velocity and temperature fields are R1, m, X, and hW1. Acombination and variation of these parameters would resultin an almost infinite series of fluid flow and heat transfer sit-uations. In the following, some representative results will begiven, thereby using, instead of h, the scaled temperaturew = h/X, where appropriate. The radius R1 of the inner cy-linder and the parameter X will be largely kept constantthroughout the numerical examples.

5.1 Velocity Distribution and Pressure Drop

The velocity distribution according to Eq. (18) is illustrat-ed in Fig. 2 for the selected values of the parameters R1 andm. The parameter m is observed to flatten the velocity pro-file, resulting in a smaller velocity variation in the cross sec-tion of the annulus at larger m, which is rooted in a smallerpermeability K of the porous medium that stronger sup-presses the boundary layer effect near the walls on the fluidflow. At a small m, the velocity profile makes the impressionof a laminar flow in an empty annulus, which approachesmore and more the profile of a turbulent flow as the param-eter m increases.

The analysis of the flow velocity and pressure drop is basi-cally possible in four directions determined by the limits ofthe parameters m and R1. Namely, letting R1 increase fromzero to one, the annular channel changes from a circulartube to a flat channel, the latter extending infinitely in onedirection orthogonal to the fluid flow. On the other hand,for m = 0, the Darcy term in the momentum equation (1)disappears, resulting in a flow situation without a porous in-sert, whereas for very large m, m → ∞, a piston-like flowwith a constant velocity in the channel cross section is ex-pected, W → 1 (see Fig. 2 for m = 100).


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For m → 0, the velocity profile coincides with the one ofthe empty annulus according to the expression:

W � 2R2 � 1 � 1 � R2

1

lnR1lnR

R21 � 1lnR1

� 1 � R21

� � (37a)

The other limiting case, a circular tube filled with a porousmedium, gives the following expression for the velocity inthe fully developed flow region satisfying the usual bound-ary conditions:

W � I0 n� �I0 m� � � 1� �

Um2 (37b)

where n = mR, R = r/r2, r2 being the tube radius.At a smaller R1, that is, at a larger annulus width d = r2 –

r1, the velocity profile becomes more flattened, particularlyat larger m, tending to the limit W → 1 at m → ∞. Both theradial position and the magnitude of the velocity maximumdepend on m, as follows from Eq. (18) with dW/dR = 0:

D1I1(mR) – D2K1(mR) = 0 (38)

where I1(mR) and K1(mR) are, respectively, the first-ordermodified Bessel functions of the first kind and of the secondkind.

This behavior has a strong impact on the pressure drop.Fig. 3 illustrates the pressure drop characteristic U in Eq. (20)as a function of the parameter m for the selected values of theradius R1. As may be inferred from the figure, the quantity Uassumes very large values at large m, that is, at small Darcynumber Da; a decrease in R1 shifts the curve downward. ForR1 → 0, the function U approaches the limit given by:

U R1→0� � � � I0 m� �I2 m� �m2 (39a)

where I2(m) is the second-order modified Bessel function ofthe first kind. The lowermost curve in Fig. 3 corresponds tothis expression, which is valid for a circular tube, thus:

∂P∂Z

� � m2

ReI0 m� �I2 m� �

leff

l(39b)

representing the pressure drop in a circular tube with aporous insert in the laminar developed flow.

Scaling the function U(m,R1) in Eq. (20) by the limitU(R1 → 0) in Eq. (39a), one obtains at very large Darcynumber Da the following relationship:

U m�R1� �U R1→0� � �

lnR1

1 � lnR1 � 1 � lnR1� �R21

, 1�

m2→∞ (40)

The applicability range of this expression can be obtainedfrom Fig. 3.

5.2 Temperature Distribution and Heat Transfer

5.2.1 Thermally Developing Flow with Thermal Symmetry

For TW1 = TW2 = TW, that is, for hW1 = hW2 = 1 andw1 = w2 = w, the thermal boundary conditions are sym-metric, and the fluid is cooled or heated by both cylindricalsurfaces of the annulus in each cross-sectional flow area.

Fig. 4 illustrates the profiles of the scaled temperature win this case for the selected parameters m and X. Because ofX = 1, the quantity, w, coincides with the temperature, h. Inall of the cases shown in the figure, the fluid is heated orcooled by both cylindrical surfaces near the start of the ther-mal action on the fluid (small n = Z/Pe), depending onwhether TW > TIN giving X > 0 or TW < TIN resulting inX < 0, but the situation reverses further downstream thefluid flow. The temperature minimum evolves to a tempera-ture maximum, and the temperature slope on the wall sur-faces goes through zero (∂w/∂R = 0) resulting in a circularadiabatic line (AP in the figure), the position of which de-pends on the parameter m. Increasing m, shifts this position


Figure 2. Velocity distribution in cross section of annulus for selected valuesof parameters m and R1.

Figure 3. Pressure drop characteristic, U, as function of the Darcy number Dafor selected values of R1.

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towards the thermal start. The physical meaningof the negative temperature gradient on the sur-faces immediately follows from the temperaturedistributions.

From a practical point of view, it is importantto obtain the axial position of the adiabatic line(∂w/∂R = 0) on the circumference of the cylin-drical surfaces. As shown in Fig. 5, this positiondepends on the parameters m and R1, and theadiabatic line moves upstream with increasingm, but downstream with increasing R1, the latterbeing not illustrated graphically. This figure alsodisplays the axial position where the averagefluid temperature reaches the wall temperature,hW � �h. At this position, the driving tempera-ture difference for heat transfer, introducedwith the definition of the heat transfer coeffi-cient in Eq. (28), becomes zero, resulting in adiscontinuity of the Nusselt number.

5.2.2 Thermally Developing Flow with ThermalAsymmetry

With thermal asymmetry, basically three par-ticular cases may be distinguished: a) The fluidinlet temperature is below, b) between, and c)above the temperatures of the channel walls. Inthe following, these cases will be discussed sepa-rately.

Fluid inlet temperature below wall temperaturesFor TIN � TW1 and TIN � TW2, the fluid is

first heated by both cylindrical surfaces and itstemperature rises along the channel length (seeFig. 6). Upstream the line w = w1 in the 3D dia-grams, obtained by cutting the temperature field

w(R,n) by the plane w = w1, the fluid temperature is belowthe cold wall temperature. The temperature minimummoves toward the cold wall as the axial coordinate n = Z/Peincreases. When the temperature minimum reaches this wall,its surface becomes adiabatic in the corresponding flow crosssection, AP in the figure. At this position, which coincideswith the last temperature minimum, the heat flux changes itsdirection and, downstream this line, the fluid is cooled acrossthe cold wall. The adiabatic line is shown in the figure toslide upstream as the parameter m increases. Due to the heatgeneration within the porous medium, the temperature risesin a manner that, further downstream, also the hot cylindri-cal wall becomes adiabatic. This is the position of the firsttemperature maximum that necessarily appears on the hotwall. Between the last minimum (at the cold wall) and thefirst maximum (at the hot wall) of the temperature profiles,the fluid is heated by the hot, but cooled by the cold wall.Downstream of this maximum, the fluid is cooled acrossboth cylindrical walls.


Figure 4. Temperature fields, wall heat fluxes and Nusselt numbers at thermal symmetry(hW1 = hW2) for selected parameters m and X. Along the line w = 1 in 3D diagrams, fluid tem-perature coincides with wall temperature.

Figure 5. Positions of adiabatic lines on cylindrical wall surfaces and positionsof mean fluid temperature equal to wall temperature at thermal symmetry forR1 = 0.4 and X = 1.

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The Nusselt number of the cold (inner) surface experi-ences a discontinuity at a certain axial distances from thestart of the thermal action on the fluid (see Fig. 7). An in-crease of the parameter m shifts both the adiabatic lines andthe vertical Nusselt asymptotes upstream. In the range ofthe parameters shown in Fig. 6, no heat flux reversal occurson the hot (outer) wall.

Fluid inlet temperature between wall temperaturesIn this case, TC ≤ TIN ≤ TH and, except for the limits

TIN = TC or TIN = TH, the fluid is first cooled by the coldand heated by the hot wall. For TIN = TC, the cold wall sur-

face is at small values n = Z/Pe thermally inactive, that is tosay, there is an adiabatic surface portion near the thermalstart (see Fig. 8). In this case, the temperature field does notexhibit a minimum, but may reach a maximum that firstwould appear on the hot wall (R = 1). However, in the rangeof the parameters shown in Fig. 8, the channel length n = Z/Pe = 0.2 is too short for the temperature maximum to estab-lish, and the porous matrix is heated across the hot, andcooled across the cold wall.

Fluid inlet temperature above wall temperaturesFor TIN � TH � TC, the dimensionless temperature hW1 is

greater than unity, hW1 � 1, and the Eckert number Ec de-fined in Eq. (6) becomes negative. This results in a negativeparameter X, introduced in Eq. (23). The temperature pro-files visualized in Fig. 9 show a minimum that moves towardthe hot wall along the fluid flow direction. Note that – for il-lustration reason – the diagrams show the negative values ofthe scaled temperature w, the temperature profile actuallyexhibits a maximum. The fluid is cooled down as it flowsalong the annulus, and the temperature maximum reachesthe hot wall, which, at that axial position becomes adiabatic,


Figure 6. Temperature fields, wall heat fluxes and Nusselt numbers at thermalasymmetry for fluid inlet temperature below wall temperatures at selected pa-rameters m and X.

Figure 7. Axial positions of adiabatic line, ∂w/∂R = 0, and of vertical Nusseltnumber asymptote, hW � �h, on inner surface for R1 = 0.4.

Figure 8. Temperature fields, wall heat fluxes and Nusselt numbers at thermalasymmetry for fluid inlet temperature equal to cold wall temperature,hW1 = 0.

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∂w/∂R = 0 (see Fig. 10). The hot wall heat flux reverses thedirection and the corresponding Nusselt number experiencesa discontinuity at hW1 � �h, the position of which movesslightly downstream, as the parameter m increases.

5.2.3 Thermally Developed Flow

In the hydrodynamically and thermally developed flow re-gion, the temperature profile does not depend on the axialcoordinate, ∂w/∂n = 0, and the energy equation (25) reducesto:

1R

∂∂R

R∂w∂R

� �� f 2 m�R1�R� � � 0 (41)

which basically describes the heat conduction in a cylindricalwall with a heat source the strength of which depends on theradial position. An integral of Eq. (41) can be obtained interms of the Bessel functions, but the somewhat clumsy solu-tion, being less appropriate for practical purposes, will notbe reported here.

For m = 0, that is, for an annulus without a porous insert,the function f becomes zero, f = 0, and the solution to Eq.(41) takes the form:

h � 1 � �hW1 � 1� lnRlnR1

(42)

The corresponding exact expression for the Nusselt num-bers have been reported in earlier papers [26, 29] and ap-proximated by:

Nu1∞ � 2�3342 � 1�6658 R�3�41 (43)

Nu2∞ � 1�786 � 2�214 R0�581 (44)

at thermal symmetry, and by:

Nu1∞ � 5�104 � 2�437 R�3�41 (45)

Nu2∞ � 3�902 � 3�639 R0�581 (46)

at thermal asymmetry.For a porous channel (m ≠ 0), the Nusselt numbers Nu1

and Nu2 are illustrated in Fig. 11, for both thermal symmetryand thermal asymmetry. As follows from the figure, a largerparameter m corresponds to a larger Nusselt number. Forthe thermal symmetry, hW1 = hW2, the Nusselt numbers arealways positive and are independent of the parameter X.

For the thermal asymmetry at m ≠ 0 (see Fig. 11), theNusselt number Nu2 of the outer cylindrical surface may be-come negative, which, by Eq. (36), will happen when:

∂w�∂R� �W2

�w2 � �w� � � 0

that is, when both the nominator and denominator have thesame sign.

Besides of the limiting situation m = 0, governed by theproperties of the porous insert, there are further limitingcases generated by variation of the channel width, that is tosay, by the variation of the parameter R1. For R1 = 0, the an-nular channel becomes a circular tube, and the Nusselt num-ber Nu1 looses its significance. The Nusselt number Nu2 ofthe other surface would, intuitively, be expected to becomeidentical with the one of a circular tube. This, however, isnot the case because of the boundary conditions at R = R1

that remain unchanged even for R1 = 0. For a circular tube,both the fluid velocity and the temperature distribution haveextremal values at R1 = 0.

For R1 → 1, formally, a flat channel is constructed. In thiscase the Nusselt numbers Nu1 and Nu2 are expected to de-


Figure 9. Temperature fields, wall heat fluxes and Nusselt numbers at thermalasymmetry for fluid inlet temperature above wall temperatures, hW1 = 3/2,hW2 = 1.

Figure 10. Axial positions of adiabatic line and vertical Nusselt asymptote onhot wall, hW1 = 3/2, hW2 = 1.

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pend on the parameter m and on the thermal asymmetryhW1. Fig. 11 does, however, not indicate such a dependency.Instead, the Nusselt numbers Nu1 and Nu2 tend to the samevalue, namely, Nu1 = Nu2 = Nu = 4.0, independently of m orX, which is the Nusselt number for an empty flat channel atthermal asymmetry [33]. This, at first sight unexpected be-havior, is rooted in the shape of the function f(m,R1,R) thatgoverns the strength of the heat source in the energy equa-tion. As shown in Fig. 12, the values of this function becomesmaller the larger R1. For R1 → 1, the function f(m,R1,R)tends to zero, independently of m. Physically, this means thatthe annular channel becomes very narrow, and, due to thenonslip boundary conditions on the wall, the velocity in thewhole cross section gradually reduces to zero.

5.2.4 Plug Flow

Setting in Eq. (5) W = 1, one formulates a plug flow withthe energy equation:

∂w∂n

� ∂2w∂R2 �

1R

∂w∂R

� 1 (47)

w � DaEcPr

H

n � ZPe

(48)

With the boundary conditions stated in Eq. (26), the defi-nitions of the Nusselt numbers remain the same, Eqs. (35)and (36), while Eq.(33) for the average temperature reducesto:

�w � 11 � R1

� 1

R1

wdR (49)

In the present considerations, Eq. (47) is simplified by re-quiring the plug flow to be thermally developed, ∂w/∂n = 0.Then, one obtains:

w � w2 �14

1 � R2� �� w1 � w2 �14

1 � R21

� �� lnRlnR1

(50)

�w � w2 � 1

12

2� 3�R21� �R1

1�R1

� w1 � w2 � 1

41 � R2

1

� �� 1 � R1 � R1 lnR1

1 � R1� � lnR1(51)

which lead to the following analytical expressions for theNusselt numbers Nu1 and Nu2:

Nu1 � (52)6 1 � R1� �2 1 � R2

1 � 4�w1 � w2� � 2 R21 lnR1

� �3 1 � R1� � R2

1 � 1 � 4 w1 � w2� �� R1 � 2R1 R3

1 � 1 � 6 w1 � w2� �� lnR1

Nu2 � (53)6 1 � R1� �2 1 � R2

1 � 4 w1 � w2� � � 2 lnR1� �

3 1 � R1� � R21 � 1 � 4 w1 � w2� �� 2 R3

1 � 1 � 6 R1 w1 � w2� �� lnR1


Figure 11. Nusselt numbers in thermally symmetric fully developed flowregime, hW1 = hW2, and thermally asymmetric fully developed flow regime forselected parameters m and X.

Figure 12. Distribution of function f(m,R1,R) in annular cross section for se-lected values of m and R1.

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For thermal symmetry, w1 = w2, it follows:

Nu1 � 6 1 � R1� � R21 � 1 � 2 R2

1 lnR1� �

R1 3 � 3R21 � 2 1 � R1 � R2

1

� �lnR1

� � (54)

Nu2 � 6 1 � R1� � R21 � 1 � 2 lnR1

� �3 � 3R2

1 � 2 1 � R1 � R21

� �lnR1

(55)

which are independent of w.The Nusselt numbers Nu1 and Nu2 according to Eqs. (52)

and (53) become zero at:

w1 � w2 � 1

41 � R2

1 � 2R21 lnR1

� �(56)

w1 � w2 � 1

41 � R2

1 � 2 lnR1� �

(57)

respectively. For the parameters not satisfying these expres-sions, the porous matrix is only heated or only cooled acrossthe cylindrical surfaces.

6 Conclusions

The temperature distribution in laminar forced convectionwith the Darcy dissipation in a porous annular channel ex-posed to a thermal asymmetry results in asymmetric heatfluxes at the channel boundaries. The parameters governingthe heat transfer are the dimensionless quantities like theDarcy, the Eckert and the Péclet number, which is wellknown from the publications dealing with the symmetricheat transfer in circular tubes or parallel plate channels. Inaddition, the thermal asymmetry imposed as the boundarycondition substantially affects the heat transfer in the porousmedium. Although very limited in the present paper, theparameter variations clearly show the main effects of ther-mal asymmetry on the heat transfer.

The common figure of heat transfer at thermal symmetry,where, in the developing region, the Nusselt number conti-nually decreases along the flow direction, changes dramati-cally in the case of a thermal asymmetry. With thermal asym-metry, at least one of the Nusselt numbers may run througha discontinuity, thereby jumping from infinite positive toinfinite negative, or first become zero, at a certain distancefrom the start of the thermal action. This behavior is illus-trated in the paper for three characteristic thermal arrange-ments, namely, for the fluid inlet temperature below, be-tween, and above the temperatures of the channel walls. Inthe fully developed heat transfer region, the Nusselt num-bers can be positive or negative for the same inlet condi-tions, depending on the heat source strength. In the ther-mally developed region of a plug flow, analytical equationsare obtained for the Nusselt numbers.

Received: February 21, 2006

Symbols used

A cross selection of annulusAP adiabatic point (line)cp specific heat at constant pC1 constantC2 constantd hydraulic diameter, Eq. (34)Da Darcy number, Eq. (6)Ec Eckert number, Eq. (6)f function, Eq. (22)I0(f) zero-order modified Bessel function of the first

kindk thermal conductivityK permeability of porous mediumK0(f) zero-order modified Bessel function of the second

kindm parameter, Eq. (7)Nu Nusselt number, Eqs. (35) and (36)p pressureP dimensionless pressure, Eq. (6)Pe Péclet number, Eq. (6)Pr Prandtl numberq heat fluxr radial coordinateR dimensionless radial coordinateRe Reynolds number, Eq. (6)T temperature�T average of TU wetted perimeter of annulusw local velocity in z direction�w average fluid velocityW dimensionless velocityz axial coordinateZ dimensionless axial coordinate

Greek symbolsa heat transfer coefficient, Eqs. (28) to (30)d thickness of porous layere heat source strength, Eq. (3)U pressure drop function, Eq. (7)D determinate, Eq. (14)D1 determinate, Eq. (15)D2 determinate, Eq. (16)j thermal diffusivitym kinematic viscosityl dynamic viscosityh dimensionless temperature Eq. (6)�h average of hq fluid densityX parameter, Eq. (23)n dimensionless axial coordinate, Eq. (24)f dimensionless axial coordinate, Eq. (9)w scaled temperature difference Eq. (24)�w average of wN velocity function, Eq. (8)


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IndicesC coldeff effective propertyH hotIN inletW wall1 inside surface2 outside surface∞ fully developed

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effect of thermal asymmetry on laminar forced convection heat transfer in a porous annular channel

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