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International Journal of Heat and Mass Transfer xxx (2010) xxx–xxx

HMT 7628 No. of Pages 8, Model 5G

17 April 2010ARTICLE IN PRESS

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

OO

FThe effect of local thermal non-equilibrium on forced convection boundarylayer flow from a heated surface in porous media

M. Celli a, D.A.S. Rees b,*, A. Barletta a

a Dipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale (DIENCA), Università di Bologna, Via dei Colli 16, I–40136 Bologna, Italyb Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK

a r t i c l e i n f o

E232425262728293031

Article history:Received 24 March 2010Available online xxxx

Keywords:Laminar flowForced convectionDarcy modelParabolic equationPorous mediumLocal thermal non-equilibriumThermal boundary layer

0017-9310/$ - see front matter � 2010 Published bydoi:10.1016/j.ijheatmasstransfer.2010.04.014

* Corresponding author.E-mail addresses: michele.celli3@unibo.it (M.

(D.A.S. Rees), antonio.barletta@unibo.it (A. Barletta).

Please cite this article in press as: M. Celli et al.,face in porous media, Int. J. Heat Mass Transfer

DPRa b s t r a c t

A steady two-dimensional forced convective thermal boundary layer flow in a porous medium is studied.It is assumed that the solid matrix and fluid phase which comprise the porous medium are subject to localthermal non-equilibrium conditions, and therefore two heat transport equations are adopted, one for eachphase. When the basic flow velocity is sufficiently high, the thermal fields may be described accuratelyusing the boundary layer approximation, and the resulting parabolic system is analysed both analyticallyand numerically. Local thermal non-equilibrium effects are found to be at their strongest near the leadingedge, but these decrease with distance from the leading edge and local thermal equilibrium is attained atlarge distances.

� 2010 Published by Elsevier Ltd.

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EC1. Introduction

Heat transfer in porous media arises in a very large number ofimportant applications and it is usually assumed that the temper-ature fields of the solid and fluid phases are identical locally; such asituation is generally known as local thermal equilibrium (LTE).The opposite situation is known as local thermal non-equilibrium(LTNE), and in these cases the solid matrix may have a differenttemperature from that of the saturating fluid, this being meant interms of averages over representative elementary volumes. Thus,hot fluid may flow into a cold, relatively insulating, porous matrixand there will exist a difference in the average local temperature ofthe two phases. This difference will take time to reduce to valuessuch that the phases are in LTE (see [1], for example), althoughthere are configurations for which LTNE persists even in the steadystate (e.g. [2,3]). When the phases are not in LTE the usual singleheat transport equation is replaced by a pair of equations, onefor each phase. First introduced by Anzelius [4] and Schumann[5], these equations use simple linear source/sink terms to modelthe local (i.e. microscopic) heat transfer between the phases atthe pore level. These early equations have been used in severalsubsequent studies, especially for the modeling of heat exchangers.The standard equations that are now used routinely are thosequoted by Nield and Bejan [6]. The review by Kuznetsov [7] con-

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Celli), D.A.S.Rees@bath.ac.uk

The effect of local thermal non(2010), doi:10.1016/j.ijheatma

centrates on forced convection phenomena in the presence ofLTNE, while the slightly more recent review by Rees and Pop [8]summarises much of the present knowledge of free and mixed con-vection and stability analyses.

In the present work a steady forced convection thermal bound-ary layer flow in a porous medium is studied. This configurationbears a great deal of similarity to the classical Pohlhausen problemin that a thermal boundary layer is induced by a step change in thetemperature of a semi-infinite flat plate [9]. The only difference isthat here we are dealing with saturated porous media and not withclear fluids. The study of thermally developing forced convection inporous media is also treated in recent papers by Nield andco-workers [10–12]. Here it is assumed that the two phases, solidand fluid, are not in local thermal equilibrium and therefore twoheat transport equations are adopted. Conditions are determinedwithin which the boundary layer approximation may be madeand we undertake an analytical and numerical study of the result-ing temperature fields. After a suitable rescaling, the resultingequations are found to depend on just one parameter, c, which,in turn, is defined in terms of the porosity of the medium andthe thermal conductivities of the two phases. The resulting thermalboundary layer equations are then solved in three different ways:(i) analytically by means of a small-x asymptotic analysis, (ii) ana-lytically using a large-x series solution and (iii) numerically bymeans of a Keller box code. The small-x analysis requires the useof two asymptotic regions, one of which is narrow and is such thatthe temperature of the fluid phase drops to the ambient value atleading order, and a second much thicker region within which

-equilibrium on forced convection boundary layer flow from a heated sur-sstransfer.2010.04.014

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Nomenclature

c specific heat at constant pressureC1, C2 constantsh inter-phase heat transfer coefficientH scaled value of hk thermal conductivityK permeabilityp pressureT temperatureTw surface temperatureT1 temperature outside the boundary layerU1 base flow velocityu,v velocity componentsx,y Cartesian coordinates

Greek symbolsa effective thermal diffusivity/,U nondimensional solid temperature, Eq. (7)

u porosityc nondimensional parameter, Eq. (10)g boundary layer variable, Eq. (17)K threshold parameter, Eq. (56)l dynamic viscosityh,H nondimensional fluid temperature, Eq. (7)q mass densityn streamwise boundary layer variable, Eq. (17)

Superscript, subscriptsdimensional quantity

~ nondimensional quantityf fluid phases solid phase1 ambient conditions

2 M. Celli et al. / International Journal of Heat and Mass Transfer xxx (2010) xxx–xxx

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Cthe temperature of the solid phase drops to the ambient. Asymp-totic matching between these two regions is performed and de-scribed briefly. This analysis is guided by the earlier study ofRees and Pop [2] who considered the effect of LTNE on a verticalfree convective boundary layer flow. The two-layer structure isthen modeled numerically by using the results of the asymptoticanalysis to devise a modified outer boundary condition for the so-lid temperature. The numerical solution is further facilitated by theuse of properly rescaled variables.

Local thermal non-equilibrium between the solid matrix andthe fluid phase is found to be at its strongest near the leading edge,but the maximum difference between the temperatures of thephases decreases with distance from the leading edge, and localthermal equilibrium is attained at large distances.

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E2. Mathematical model

A steady flow in forced convection regime over a horizontal flatplate is assumed. A sketch of the plate and of the coordinate sys-tem is shown in Fig. 1. The Darcy model for the porous mediumis assumed to hold, and therefore the externally-generated fluidmotion is both uniform in space and constant in time. The imper-meable bounding surface, which is placed at �y ¼ 0, is held at thetemperature, Tw, in the range, �x > 0, but is held at the temperatureof the free stream, T1, in the range, �x < 0. The point, ð�x; �yÞ ¼ ð0;0Þ,at which the boundary condition for the temperature changes, willbe regarded the as leading edge of the surface.

Given that the solid and fluid phases are not in LTE, separateheat transport equations for the solid and fluid phases are em-ployed. Thus the governing mass, momentum and thermal energybalance equations in the steady state may be expressed in theform,

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140140

141142

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g

0

y

x

Bu

Fig. 1. A geometrical sketch of the problem.

Please cite this article in press as: M. Celli et al., The effect of local thermal nonface in porous media, Int. J. Heat Mass Transfer (2010), doi:10.1016/j.ijheatma

ED

P@�u@�xþ @

�v@�y¼ 0; ð1Þ

�u ¼ �Kl@�p@�x; �v ¼ �K

l@�p@�y; ð2Þ

uaf@2Tf

@�x2 þ@2Tf

@�y2

!þ hðqcÞf

ðTs � Tf Þ ¼ u@Tf

@�xþ v @Tf

@�y; ð3Þ

ð1�uÞas@2Ts

@�x2 þ@2Ts

@�y2

!þ hðqcÞs

ðTf � TsÞ ¼ 0: ð4Þ

The thermal boundary conditions are given by

�y ¼ 0 ð�x < 0Þ : Ts ¼ Tf ¼ T1;

�y ¼ 0 ð�x > 0Þ : Ts ¼ Tf ¼ Tw;

�y!1 : Ts; Tf ! T1: ð5Þ

The quantity, h, which appears in Eqs. (3) and (4), is a function ofthe microscopic geometry of the medium, the conductivities ofthe phases and the strength of the flow. Some values of h for stag-nant two- and three-dimensional media of various types are givenin some very recent papers by Rees [13,14].

2.1. Nondimensionalization

Given that this is a forced convection problem, the velocity fieldis given by,

ð�u; �vÞ ¼ ðU1;0Þ; ð6Þ

and this may be used immediately in Eq. (3).A semi-infinite domain such as the one described in Fig. 1 im-

plies that there is no natural physical lengthscale on which to basea Péclet number and to use for nondimensionalization. However,the quantity, af =U1, has the dimensions of length, and may be usedfor this purpose. It turns out to be a little more convenient to useuaf =U1 as the lengthscale, and therefore let us introduce nondi-mensional variables using the following scalings,

ð�x; �yÞ ¼ uaf

U1ð~x; ~yÞ; ðTf ; TsÞ ¼ ðTw � T1Þðh;/Þ þ T1: ð7Þ

On using Eqs. (3) and (7), Eqs. (3) and (4) become,

@2h@~x2 þ

@2h@~y2 þ Hð/� hÞ ¼ @h

@~x; ð8Þ

-equilibrium on forced convection boundary layer flow from a heated sur-sstransfer.2010.04.014

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M. Celli et al. / International Journal of Heat and Mass Transfer xxx (2010) xxx–xxx 3

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@2/@~x2 þ

@2/@~y2 þ Hcðh� /Þ ¼ 0; ð9Þ

where

H ¼ uhaf

U21ðqcÞf

; c ¼ ukf

ð1�uÞks¼ uaf ðqcÞsð1�uÞasðqcÞf

: ð10Þ

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2.2. Boundary layer approximation

The system formed by Eqs. (8) and (9) is elliptic, and therefore,for O(1) values of H, a fully elliptic numerical simulation should beundertaken to obtain the resulting temperature fields of the twophases. The forced convection regime assumed implies that a highvelocity basic flow is taken into account so that the limit U1 � 1 isconsidered. From the definition of H in Eq. (10), U1 � 1 impliesthat H � 1. Assuming now that H� 1, it is necessary to balancethe magnitudes of the last three terms in Eq. (8), and it may beshown easily that the appropriate orders of magnitude are~x ¼ OðH�1Þ and ~y ¼ OðH�1=2Þ. Therefore the scalings,

~x ¼ H�1x; ~y ¼ H�1=2y; ð11Þ

are introduced into Eqs. (8) and (9) to obtain,

H@2h@x2 þ

@2h@y2 þ ð/� hÞ ¼ @h

@x; ð12Þ

H@2/@x2 þ

@2/@y2 þ cðh� /Þ ¼ 0: ð13Þ

Formally allowing H? 0 yields the boundary layer approximationnaturally, and the governing equations are now,

@h@x¼ @2h@y2 þ ð/� hÞ; ð14Þ

@2/@y2 þ cðh� /Þ ¼ 0: ð15Þ

This system of equations is parabolic, and although it may be shownthat there does not exist a self-similar solution, there is neverthe-less a nonsimilar formulation which may be found; this analysis fol-lows below.

2.3. Boundary layer transformation

It is well-known that the equation formed by the first two termsof Eq. (14) admits the self-similar solution,

h ¼ erfcy

2ffiffiffixp

� �; ð16Þ

The solution shown in Eq. (16) can be found, for instance, in Özis�ik[15], and it motivates the following coordinate transformations,

g ¼ y2ffiffiffixp ; n ¼

ffiffiffixp

; h ¼ Hðn;gÞ; / ¼ Uðn;gÞ; ð17Þ

where the upper case characters, H and U, will henceforth denotethe respective temperatures of the fluid and solid phases in thisnew coordinate system, while the lower case characters, h and /,correspond to the solution written in Cartesian coordinates. Eqs.(14) and (15) become

@2H@g2 � 2n

@H@nþ 2g

@H@gþ 4n2ðU�HÞ ¼ 0; ð18Þ

@2U@g2 þ 4n2cðH�UÞ ¼ 0; ð19Þ

which are to be solved subject to the boundary conditions,

Please cite this article in press as: M. Celli et al., The effect of local thermal nonface in porous media, Int. J. Heat Mass Transfer (2010), doi:10.1016/j.ijheatma

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g ¼ 0 and n > 0 : H ¼ U ¼ 1;g!1 and n > 0 : H;U! 0: ð20Þ

It is important to note that the coefficient, n2, of the source/sinkterms in Eqs. (18) and (19) plays the same role as H does in Eqs.(8) and (9). Therefore it is possible to observe immediately thatlarge values of n will correspond to local thermal equilibrium, whilelocal thermal non-equilibrium effects will be at their strongest nearthe leading edge, n = 0. These equations have only one nondimen-sional parameter, namely c.

3. Asymptotic analyses

The numerical solution of Eqs. (18) and (19) will be preceded byasymptotic analyses close to and far from the leading edge at n = 0.The near-leading-edge analysis is performed first because it pro-vides the boundary conditions that are essential for the numericalsolution.

3.1. Close to the leading edge

The system formed by Eqs. (18) and (19) may be solved in theregion relatively close to the leading edge by searching for a solu-tion in the form of a power series in the variable n,

Hðn;gÞ ¼ H0ðgÞ þ nH1ðgÞ þ n2H2ðgÞ þ � � � ;Uðn;gÞÞ ¼ U0ðgÞ þ nU1ðgÞ þ n2U2ðgÞ þ � � � :

ð21Þ

At leading order in the expansion, the equations for H0 and U0 are,

H000 þ 2gH00 ¼ 0; U000 ¼ 0; ð22Þ

and their solutions are

H0 ¼ erfcðgÞ; U0 ¼ 1: ð23Þ

In writing down the above expression for U0, it is important tonote that it is impossible to solve U000 ¼ 0 subject to both U0(0) = 1and U0 ? 0 as g ?1. The boundary condition at the heated sur-face was employed, the implication being that the temperatureof the solid phase near the leading edge must then vary over alengthscale which is much greater than that represented by O(1)values of g. This is consistent with a previous analysis in [2]. It isalso clear that Eq. (15) provides a means of satisfying the far fieldboundary condition; in this regime where y = O(1) the leading or-der temperature of the fluid phase, H0, has already decayed tozero, leaving a system which admits an exponentially decayingsolution. Therefore it is natural to attempt a corresponding solu-tion in this ‘outer’ region where y = O(1) (noting that the ‘inner’ re-gion where g = O(1) corresponds to y = O(x1/2) = O(n)). Thus anouter solution of the form,

hðx; yÞ � h0ðyÞ þffiffiffixp

h1ðyÞ þ xh2ðyÞ þ � � � ;/ðx; yÞ � /0ðyÞ þ

ffiffiffixp

/1ðyÞ þ x/2ðyÞ þ � � �ð24Þ

is sought.On taking h0 = 0 because of the superexponential decay of H0 in

the inner region, the leading order equations in the outer regionare,

h2 ¼ /0; /000 � c/0 ¼ 0: ð25Þ

Given that U0 = 1, the appropriate matching condition for /0 is that/0(0) = 1. Hence /0 and h2 are given by,

h2 ¼ /0 ¼ e�ffifficp

y; ð26Þ

which shows that the leading order terms of both temperaturefields decay exponentially; this has important implications laterfor the numerical solutions.

-equilibrium on forced convection boundary layer flow from a heated sur-sstransfer.2010.04.014

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These solutions now provide us with the appropriate match-ing conditions for the inner solutions at the next order in n.On expanding the solutions given in Eq. (26) about y = 0 oneobtains,

h � 1� ffiffifficp

yþ cy2

2þ � � �

� �x; ð27Þ

/ � 1� ffiffifficp

yþ cy2

2þ � � � ; ð28Þ

and if y is replaced by 2gn then the large-g matching conditions forsmall values of n are found to be,

H � n2 � 2ffiffifficp

gn3 þ � � � ; U � 1� 2ffiffifficp

gnþ 2cg2n2 þ � � � : ð29Þ

Therefore the following conditions as g ?1 may be written,

U1 � �2ffiffifficp

g; U2 � 2cg2; H2 � 1: ð30Þ

The next terms in the inner region arise at O(n). From the expansionat this order one obtains the system

H001 þ 2gH01 � 2H1 ¼ 0; U001 ¼ 0: ð31Þ

which is to be solved subject to the conditions,

g ¼ 0 : H1 ¼ 0; U1 ¼ 0;g!1 : H1 ! 0; U1 � �2

ffiffifficp

g: ð32Þ

The solutions are

H1 ¼ 0; U1 ¼ �2ffiffifficp

g: ð33Þ

The solution for /1 is clearly passive, and it is caused by the leadingorder exponential properties of /0.

At OðffiffiffixpÞ in the outer region, the governing equations are

h001 þ /1 � h1 � 2h3 ¼ 0; /001 þ cðh1 � /1Þ ¼ 0; ð34Þ

which are to be solved subject to

y ¼ 0 : h1 ¼ /1 ¼ 0; y!1 : h1;/1 ! 0: ð35Þ

Given that these equations are homogeneous, the solutions are

h1 ¼ 0; h3 ¼ 0; /1 ¼ 0: ð36Þ

At O(n2) in the inner region, the governing equations are,

H002 � 4H2 þ 2gH02 þ 4erfðgÞ ¼ 0; U002 � 4cerfðgÞ ¼ 0; ð37Þg ¼ 0 : H2 ¼ 0; U2 ¼ 0;

g!1 : H2 ! 1; U2 � 2cg2; ð38Þ

and the solutions are

H2 ¼ erfðgÞ; U2 ¼ g2e�g2cffiffiffiffi

pp þ cð1þ 2g2ÞerfðgÞ: ð39Þ

This is a reasonable point at which to stop the expansion for the in-ner region because three terms have now been obtained for U. AtO(x) for the outer region one has

h002 þ /2 � h2 � 2h4 ¼ 0; /002 þ cðh2 � /2Þ ¼ 0: ð40Þ

which are to be solved using Eq. (26) and subject to

y ¼ 0 : /2 ¼ 0; y!1 : /2 ! 0; ð41Þ

the solution for Eq. (40) is

h4 ¼e�y

ffifficp

42c� 2þ y

ffiffifficpð Þ; /2 ¼ e�y

ffifficp y

ffiffifficp2

� �: ð42Þ

To summarise, the solutions found by the power series expansionfor both the inner and the outer regions are:

Please cite this article in press as: M. Celli et al., The effect of local thermal nonface in porous media, Int. J. Heat Mass Transfer (2010), doi:10.1016/j.ijheatma

H � erfcðgÞ þ n2erfðgÞ;

U � 1� 2ffiffifficp

ngþ g2ce�g2ffiffiffiffi

pp þ cð1þ 2g2ÞerfðgÞ

" #n2; ð43Þ

h � xe�yffifficp; / � e�y

ffifficpþ x

yffiffifficp

2

� �e�y

ffifficp:

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F

3.2. Far from the leading edge

The solutions of Eqs. (18) and (19) at large distances from theleading edge are considered now. When n� 1 there are two possi-ble order-of-magnitude balances that may be obtained from Eq.(19). The first suggests that there is an inner layer of widthg = O(n�1) within the main boundary layer, for which we assumethat g = O(1). The second, and more reasonable one on physicalgrounds, is that h � / = O(n�2), and that there is only one asymp-totic region. This second balance corresponds to the approach toLTE. Therefore one may investigate the region far from the leadingedge by searching for a solution expressed as a power series expan-sion of the form

Hðn;gÞ � H0ðgÞ þ n�1H1ðgÞ þ n�2H2ðgÞ þ � � � ;Uðn;gÞ � H0ðgÞ þ n�1H1ðgÞ þ n�2H2ðgÞ þ � � � :

ð44Þ

The leading order, i.e. O(1), equations are

H000 þ 2gH00 þ 4ðU2 �H2Þ ¼ 0; ð45ÞU000 þ 4cðH2 �U2Þ ¼ 0; ð46Þ

while the boundary conditions are

g ¼ 0 : H ¼ U ¼ 1;g!1 : H;U! 0: ð47Þ

Given the above argument, one may reasonably assume thatU0 = H0. On multiplying Eq. (45) by c and adding it to Eq. (46),one obtains

ð1þ cÞH000 þ 2gcH00 ¼ 0; ð48Þ

the solution for which is,

U0 ¼ H0 ¼ erfc gffiffiffiffiffiffiffiffiffiffiffiffi

c1þ c

r� �: ð49Þ

At O(n�1) of the expansion the governing equations are,

H001 þ 2H1 þ 2gH01 þ 4ðU3 �H3Þ ¼ 0; ð50ÞU001 þ 4cðH3 �U3Þ ¼ 0: ð51Þ

The terms in H3 and U3 may be eliminated by means of the sameprocess of combination of the equations, and this results in the fol-lowing equation,

ð1þ cÞH001 þ 2gcH01 þ 2cH1 ¼ 0: ð52Þ

Given the above scaling argument that H �U = O(n�2), it has beenassumed that H1 = U1. Eq. (52) has the general solution,

U1 ¼ H1 ¼ C1e�c

1þcg2erfi g

ffiffiffiffiffiffiffiffiffiffiffiffic

1þ c

r� �þ C2e�

c1þcg

2; ð53Þ

where C1 and C2 are constants. This solution should satisfy zeroboundary conditions for H1 at both g = 0 and as g ?1. This impliesthat C2 = 0, but C1 remains indeterminate. Clearly, then, the functionmultiplying C1 in Eq. (53) is an eigensolution of the equation.Although further terms in this expansion will only be expressedin terms of C1 and any further eigensolutions which might arise, itis possible, nevertheless, to determine precisely the leading order

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departure from LTNE. If Eq. (49) is substituted into Eqs. (45) and(46), then it is straightforward to show that

H2 �U2 ¼U0004c¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

pð1þ cÞ3

sge�

c1þcg

2: ð54Þ

Eq. (54) corresponds to an O(n�2) difference between the tempera-tures of the phases. It is also worthy of note that the magnitude ofthis difference decreases as c increases; the mathematical reasonlies in the fact that a large value of c forces the difference betweenthe temperatures to be small, as seen in Eq. (19).

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4. Numerical solutions

The numerical scheme which was used to solve numerically thesystem of Eqs. (18) and (19) is a standard Keller box method, whichis a generic method for time or space marching problems, [16].Most often, the system is reduced to first order form in g, theresulting equations discretised half way between the grid pointsin both the g and n directions, and it is therefore of second orderaccuracy. An initial profile is not required, because the system ofparabolic partial differential equations reduces to an ordinary dif-ferential system at n = 0, and a suitably written code will be able tosolve this system easily. Although the present system of equations

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0 5 10 15 200

20

40

60

80

y

0 5 10 15 200

20

40

60

80

y

0 5 10 15 200

20

40

60

80

y

x

Fig. 2. Isotherms for both the solid and fluid phases for c = 0.01 (upper), c = 0.1(middle), c = 1 (lower). Dashed lines refer to the solid phase and solid lines refer tofluid phase.

Please cite this article in press as: M. Celli et al., The effect of local thermal nonface in porous media, Int. J. Heat Mass Transfer (2010), doi:10.1016/j.ijheatma

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is linear, the general methodology uses a multidimensionalNewton–Raphson iteration scheme to solve the discretised equa-tions. In our case the corresponding iteration matrix is computednumerically, and a standard block-Thomas algorithm is used tosolve the iteration equations. The x variable is treated here as atime-like variable, and the solution is marched downstream.

In the present implementation we have adopted a backwarddifference method in x in order to maximise numerical stability,and good accuracy is ensured by employing the small steplength,d n = 10�2. In the g-direction, the steplength, dg = 10�2, is takenwith gmax = 20. However, in the above small-n analysis we deter-mined that the leading order temperature fields in the outer regiondecay as expð�y

ffiffifficp Þ, i.e. as expð�2gnffiffifficp Þ from the point of view of

the boundary layer. Therefore it is clear that this rate of decay can-not be contained within a finite computational domain when n issufficiently small. However, it is possible to model this rate of de-cay by adopting a different set of boundary conditions from thosegiven earlier in Eq. (20). We used the following,

O

H0 þ 2nffiffifficp

H ¼ 0; U0 þ 2nffiffifficp

U ¼ 0; at g ¼ gmax: ð55Þ

410

PRThese conditions were used from n = 0 until the first value of n at

which both H and U are less than 10�6, and thereafter the condi-tions given in Eq. (20) were adopted. In this way the full presence

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0 2 4 6 8 100

10

20

30

40

y

0 2 4 6 8 100

10

20

30

40

y

0 2 4 6 8 100

10

20

30

40

y

x

Fig. 3. Isotherms for both the solid and fluid phases for c = 0.01 (upper), c = 0.1(middle), c = 1 (lower). Dashed lines refer to the solid phase and solid lines refer tofluid phase. This figure has a different vertical resolution to that shown in Fig. 2.

-equilibrium on forced convection boundary layer flow from a heated sur-sstransfer.2010.04.014

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0.0 0.5 1.0 1.5 2.00

2

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y

0.0 0.5 1.0 1.5 2.00

2

4

6

8

y

x0.0 0.5 1.0 1.5 2.0

0

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6

8

y

Fig. 4. Isotherms for both the solid and fluid phases for c = 0.01 (upper), c = 0.1(middle), c = 1 (lower). Dashed lines refer to the solid phase and solid lines refer tofluid phase. This figure has a different vertical resolution to that shown in Fig. 2.

0 1 2 3 4 5

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0.01, 0.1,1,10

,

Fig. 5. Variation of the rates of heat transfer, @h/@g and @//@g, on the plate g = 0versus n, for different values of c. Dashed lines refer to the solid phase and solidlines to the fluid phase.

0.001 0.01 0.1 1 100

5

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20

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x

0.001,1,10,100,1000

Fig. 6. Behaviour of the parameter K as a function of x for different values of c.

6 M. Celli et al. / International Journal of Heat and Mass Transfer xxx (2010) xxx–xxx

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condition.Figs. 2–4 show the temperature profiles obtained by the

numerical simulations, and are plotted in the (x,y)-coordinates,as defined in Eq. (11). The three frames in Fig. 2 are drawn usingthe same range of values of x and y, but for different values of c:the upper frame refers to c = 0.01, the central frame to c = 0.1and the lower refers to c = 1. Dashed lines refer to the solidphase and solid lines refer to fluid phase. In this figure it is clearthat the thickness of the thermal boundary layer at all values of xis strongly dependent on the magnitude of c. This is consistentwith Eq. (49) which shows clearly that the large-x thickness ofthe boundary layer increases as c decreases. Likewise, near theleading edge, the thickness of the thermal boundary layer corre-sponding to the solid phase increases as c decreases; see Eq. (25).That this should be so is understood by referring to the physicalimplications of different values of c. When c is small, heat is con-ducted easily away from the heated surface, and only a smallamount is imparted to the fluid. On the other hand, when c islarge, heat is transferred easily to the fluid and is therefore ad-vected downstream, which decreases the thickness of the bound-ary layers. The ease of heat transfer between the phases may alsobe seen in Fig. 7 local thermal equilibrium is attained at muchsmaller values of x when c = 1 than when c = 0.01. Figs. 3 and4 show the same information as Fig. 2, but for a progressively

Please cite this article in press as: M. Celli et al., The effect of local thermal nonface in porous media, Int. J. Heat Mass Transfer (2010), doi:10.1016/j.ijheatma

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smaller range of values of y and x. The variation with c of theboundary layer thickness of the solid phase near the leading edgemay be seen clearly, but it is also possible to see that the bound-ary layer thickness of the fluid phase is independent of c there.This independence is consistent with Eq. (23). At slightly in-creased distances from the leading edge the boundary layerthickness of the fluid phase increases rapidly at a rate which isdependent on c.

Although c is the sole parameter of the system being studied,Figs. 2–4 already indicate substantial variations of the resultingtemperature fields as the parameter varies. The surface rate of heattransfer is an important quantity, and the manner in which its evo-lution with n varies with c is shown in Fig. 5. All of our previousobservations are seen clearly in this figure, particularly the magni-tude of the rate of heat transfer far from the leading edge. Giventhat we have plotted the g-derivative of the temperatures, the solidphase now has a zero rate of heat transfer at the leading edge, butthis varies rapidly as n increases, and does so at a c-dependent rate.However, of most interest is the distance at which one might saythat local thermal equilibrium has been attained. Such a criterionis necessarily arbitrary; we can define xLTE to be the distance be-yond which the relative difference of the surface rates of heattransfer, K, is less than 1%, namely, that

K ¼ 2@h@y

����y¼0� @/@y

����y¼0

!@h@y

����y¼0þ @/@y

����y¼0

!6 0:01:

,ð56Þ

The behaviour of the parameter K as a function of x is shown inFig. 6. This figure confirms the results previously obtained in

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511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565

0.01 0.1 1 10 100 10000

2

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LTEx

Fig. 7. Variation with c of the distance, xLTE, at which local thermal equilibrium isattained.

Table 1Variation with c of the distance, xLTE, at which local thermal equilibrium is attained.Threshold for the achieving of LTE is chosen to be when the parameter K 6 0.01.

c xLTE

0.01 10.0110.02 9.83450.05 9.46180.1 9.00000.2 8.24840.5 6.78081 5.42892 4.02805 2.502710 1.677020 1.085850 0.6053100 0.3856200 0.2450500 0.13321000 0.0841

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EFig. 5 for which as c increases the heat transfer between the twophases decreases more quickly. Thus also the distance from theleading edge xLTE defined by Eq. (56) decreases as c increases. Thisbehaviour is displayed in Fig. 7 and the relative Table 1. The valueof xLTE, in fact, decreases monotonically as c increases.

5. Conclusion

A steady two-dimensional forced convection thermal boundarylayer in a porous medium has been studied both analytically andnumerically. The main focus has been on the effect of local thermalnon-equilibrium between the solid and fluid phases. Separateasymptotic analyses which are valid at small distances and at largedistances from the leading edge have been obtained. In particular,we have determined that the boundary layer in the near-leading-edge region splits into two well-defined asymptotic regions, onein which y is the natural variable, and the other in which g is thenatural variable. This analysis was used to obtain an effective outerboundary condition for the numerical study which would take intoaccount accurately the presence of the outer region. The large-nasymptotic analysis shows that local thermal equilibrium is at-tained in this limit, with local thermal non-equilibrium effectsbeing of O(x�1) in magnitude. A full numerical solution was alsoobtained and solutions presented.

We found that:

� the thickness of the boundary layers of each phase dependsstrongly on the value of c;

566Please cite this article in press as: M. Celli et al., The effect of local thermal nonface in porous media, Int. J. Heat Mass Transfer (2010), doi:10.1016/j.ijheatma

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� the thickness of fluid boundary layer near the leading edge isindependent of c;� the boundary layer corresponding to the solid phase is always

thicker than that of the fluid phase;� local thermal equilibrium is attained at decreasing distances

from the leading edge as c increases; this is related to theincreasing ease of heat transfer between the phases.

Finally, it is important to note that the flows we have consid-ered may be subject to thermoconvective instability. Papers byRees and Bassom [17] and Rees [18–20] are concerned with variousaspects of the instability of free convection boundary layers, whileRees [21] is concerned with linear instability and the nonlinearvortex development in a mixed convection flow where forced con-vection effects are strong compared with buoyancy. The situationdescribed in [21] could be applied to the flow considered here ifweak buoyancy forces are allowed to exist. As the flow developsdownstream, the local Rayleigh number, which is proportional tothe boundary layer thickness, also increases and eventually be-comes sufficiently large that thermoconvective instability will oc-cur. It is intended to report on this aspect elsewhere.

References

[1] D.A.S. Rees, A.P. Bassom, P.G. Siddheshwar, Local thermal non-equilibriumeffects arising from the injection of a hot fluid into a cold porous medium, J.Fluid Mech. 594 (2007) 379–398.

[2] D.A.S. Rees, I. Pop, Vertical free convective boundary-layer flow in a porousmedium using a thermal nonequilibrium model, J. Porous Media 3 (2000) 31–44.

[3] D.A.S. Rees, Vertical free convective boundary-layer flow in a porous mediumusing a thermal nonequilibrium model: elliptical effects, J. Appl. Math. Phys.(ZAMP) 54 (2003) 437–448.

[4] A. Anzelius, Über Erwärmung vermittels durchströmender, Medien. Zeit. Ang.Math. Mech. 6 (1926) 291–294.

[5] T.E.W. Schumann, Heat transfer: liquid flowing through a porous prism, J.Franklin Institute 208 (1929) 405–416.

[6] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer, New York,2006.

[7] A.V. Kuznetsov, Thermal nonequilibrium forced convection in porous media,in: D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media,Pergamon, 1998, pp. 103–129.

[8] D.A.S. Rees, I. Pop, Local thermal nonequilibrium in porous mediumconvection, in: D.B. Ingham, I. Pop (Eds.), Transport Phenomena in PorousMedia III, Pergamon, 2005, pp. 147–173.

[9] K. Pohlhausen, Der Wärmeaustausch zwischen festen Körpen undFlüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung, Z. Angew.Math. Mech. 1 (1921) 115–124.

[10] D.A. Nield, A.V. Kuznetsov, M. Xiong, Effect of local thermal non-equilibriumon thermally developing forced convection in a porous medium, Int. J. HeatMass Transfer 45 (2002) 4949–4955.

[11] A.V. Kuznetsov, D.A. Nield, Thermally developing forced convection in abidisperse porous medium, J. Porous Media 9 (2006) 393–402.

[12] A.V. Kuznetsov, D.A. Nield, Thermally developing forced convection in a porousmedium occupied by a rarefied gas: parallel plate channel or circular tube withwalls at constant heat flux, Transp. Porous Media 76 (2009) 345–362.

[13] D.A.S. Rees, Microscopic modelling of the two-temperature model forconduction in heterogeneous media, J. Porous Media 13 (2010) 125–143.

[14] D.A.S. Rees, Microscopic modelling of the two-temperature model forconduction in heterogeneous media: three-dimensional media, in:Proceedings of the Fourth International Conference on Applications ofPorous Media, Istanbul, Turkey, Paper 15, August 2009.

[15] M.N. Özis�ik, Heat conduction, second ed., Wiley, John and Sons, 1993.[16] H.B. Keller, T. Cebeci, Accurate numerical methods for boundary layer flows 1.

Two dimensional Flows, in: Proceedings of the International Conference onNumerical Methods in Fluid Dynamics, Lecture Notes in Physics, Springer, NewYork, 1971.

[17] D.A.S. Rees, A.P. Bassom, The nonlinear nonparallel wave instability of freeconvection induced by a horizontal heated surface in fluid-saturated porousmedia, J. Fluid Mech. 253 (1993) 267–296.

[18] D.A.S. Rees, Vortex instability from a near-vertical heated surface in a porousmedium. I Linear theory, Proc. Roy. Soc. A457 (2001) 1721–1734.

[19] D.A.S. Rees, Vortex instability from a near-vertical heated surface in a porousmedium. II Nonlinear evolution, Proc. Roy. Soc. A458 (2002) 1575–1592.

[20] D.A.S. Rees, Nonlinear vortex development in free convective boundary layersin porous media. in: Proceedings of NATO ASI, Neptun, Romania, June 9–202003, pp. 449–458.

[21] D.A.S. Rees, The onset and nonlinear development of vortex instabilities in ahorizontal forced convection boundary layer with uniform surface suction,Transp. Porous Media 77 (2009) 243–265.

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