Simple methods for convection in porous media: scale analysis and the intersection of asymptotes

INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2003; 27:859–874 (DOI: 10.1002/er.922)

Simple methods for convection in porous media:scale analysis and the intersection of asymptotes

Adrian Bejann

Department of Mechanical Engineering and Materials Science, Duke University, Box 90300, Durham,

NC 27708-0300, U.S.A.

SUMMARY

This article outlines the basic rules and promise of two of the simplest methods for solving problems ofconvection in porous media. First, scale analysis is the method that produces order-of-magnitude resultsand trends (scaling laws) for concrete and applicable results such as heat transfer rates, flow rates, and timeintervals. Scale analysis also reveals the correct dimensionless form in which to present more exact resultsproduced by more complicated methods. Second, the intersection of asymptotes method identifies thecorrect flow configuration (e.g. B!eenard convection in a porous medium) by intersecting the two extremes inwhich the flow may exist: the many cells limit, and the few plumes limit. Every important feature of theflow and its transport characteristics is found at the intersection, i.e. at the point where the two extremescompete and find themselves in balance. The intersection is also the flow configuration that minimizes theglobal resistance to heat transfer through the system. This is an example of the constructal principle ofdeducing flow patterns by optimizing the flow geometry for minimal global resistance. The article stressesthe importance of trying the simplest method first, and the researcher’s freedom to choose the appropriateproblem solving method. Copyright # 2003 John Wiley & Sons, Ltd.

KEY WORDS: porous media; convection; scale analysis; constructal theory; intersection of asymptotes;self-organization in nature

1. OBJECTIVE: TRY THE SIMPLEST FIRST

The objective of this paper is to emphasize the freedom that educators and researchers have inchoosing methods to solve problems, present the results, and put them in practice. The field ofconvection in porous media is an excellent candidate for stressing this important message. It ismature enough, and at the same time it is rich: its results cover a wide spectrum of problems andapplications in thermal engineering, physics, geophysics, bioengineering, civil engineering, andmany other fields. These fields are united by several key phenomena, some of which are selectedfor analysis in this paper.

The opportunity that the maturity of our field offers is this: after a few decades ofdevelopment, we find that more than one method is available for attacking a certain type ofproblem. The papers collected in the present volume illustrate this aspect very well. Older

Received 31 January 2002Accepted 6 January 2003Copyright # 2003 John Wiley & Sons, Ltd.

nCorrespondence to: A. Bejan, Department of Mechanical Engineering and Materials Science, Duke University,Box 90300, Durham, NC 27708-0300, U.S.A.yE-mail: [email protected]

methods, such as analysis and experiments (direct laboratory measurements), are as eligible tobe used as the newer methods based on computational analysis. The point is that the availablemethods compete for the researcher’s attention. The researcher has the freedom to choose themethod that suits him or her.

Methods are literally ‘competitive’ because each is an example of a tradeoff between cost (effort)and accuracy. The simpler methods require less effort and produce less accurate results than themore complicated methods. The researcher enters the market place of methodology with apersonal profile: talent, time to work, interest in details, and user(s) of the results of thecontemplated work (e.g. customers, students). The match between researcher, problem andmethod is never the result of chance. It is an optimization decision (structure, configuration) in thesense of constructal theory and design (Bejan, 2000). This configuration maximizes theperformance (benefit) for all parties concerned. The intellectual work that goes on all over themap of research and education is a conglomerate of mind-problem-method matches of the kindillustrated in this paper. The so-called ‘knowledge industry’ thrives on the optimization of matchesand connections}it thrives on the development (growth) and optimization of structure.

Among the methods that have emerged in fluid mechanics and heat transfer, scale analysis(scaling) is one of the simplest and most cost effective. I have not researched the history of themethod. My impression is that the method developed independently on several work tables, aspersonal expressions of art or talent in research, and by appearing in isolated papers and booksover the years. I was attracted to this method by the work of Patterson and Imberger (1980) innatural convection. From the beginning, I felt that scale analysis deserves a self-standing rolenot only in research but also among the methods that we teach as ‘skeleton’ for the discipline(Bejan, 1984, 1995).

Scaling is now used by many, yet the need to explain its rules and promise remains. Toaccomplish this in a compact and effective format is the first objective of this article. Scaleanalysis is so cost-effective that it is beneficial as a first step (preliminary, prerequisite) even insituations where the appropriate method is more laborious and the sought results are moreaccurate. The results of scale analysis serve as guide. They tell the researcher what to expectbefore the laborious method, what the ultimate (accurate) results mean, and how to report themin dimensionless form. The engineering advice to ‘try the simplest first’ fits perfectly in theoptimization of research.

The second objective of this paper is to explain the rules and the promise of another simplemethod: the intersection of asymptotes. This method was born by accident, in the search for aquick solution to a homework problem, namely the optimal spacing between parallel plates withnatural convection (Bejan, 1984; problem 11, p. 157). What led to the quick solution is athought of more permanent and general value. It is the thought that when I am challenged todescribe a complicated system or phenomenon (e.g. a flow structure), I help my understandingconsiderably if I describe the phenomenon in the simpler extremes (asymptotes) in which it maymanifest itself. The complicated phenomenon lies somewhere between the extremes, and itsbehaviour may be viewed as the result of the competition (clash, collision) between the extremes.This thought has helped us in many areas. For example, the highly complicated relationsbetween the thermodynamic properties of a real substance (e.g. stream tables) make more sensewhen viewed as the intersection of two extremes of thermodynamic behaviour: theincompressible substance model and the ideal gas model.

Although newer than scaling, the intersection of asymptotes method is now used frequently inthermal design and optimization (Sadeghipour and Razi, 2001). Optimization of global

Copyright # 2003 John Wiley & Sons, Ltd. Int. J. Energy Res. 2003; 27:859–874

A. BEJAN860

performance is an integral part of the physics of flow structures: flows choose certain patterns(shapes, structures, regimes) as compromises between the available extremes. Flows design forthemselves the paths for maximum (easiest) access (Bejan, 2000). To illustrate this ‘constructal’characteristic of natural flows, in this paper I focus on the classical phenomenon of naturalconvection in a porous layer saturated with fluid and heated from below (Horton and Rogers,1945; Lapwood, 1948). This phenomenon has been investigated with increasingly accurateanalytical, numerical and experimental tools, as shown by recent reviews (e.g. Nield and Bejan,1999) and several of the papers presented in this special issue. Just like scaling, the intersectionof asymptotes method provides a surprisingly direct alternative, a short cut to the mostimportant characteristics of the flow.

2. SCALE ANALYSIS

Scaling is a method for determining answers to concrete problems, such as the heat transfer ratein a configuration that is described completely. The results are accurate in an order ofmagnitude sense: the following examples exhibit an accuracy better than within a factor of 2 or1/2. The analysis is based on the complete problem statement, i.e. the conservation equationsand all the initial and boundary conditions. Partial differential equations are replaced by globalalgebraic statements, which are approximate. Scale analysis is a problem solving method}amethod of solution that should not be confused with dimensional analysis.

2.1. Forced convection boundary layers

A simple class of flows that can be described based on scale analysis are boundary layers.Figure 1 shows the thermal boundary layer in the vicinity of an isothermal solid wall (T0)embedded in a saturated porous medium with uniform flow parallel to the wall (U1, T1). Forsimplicity, we assume temperature-independent properties, so that the temperature field isindependent of the flow field. The flow field is known: it is the uniform flow U1, which in Darcyflow is driven by a constant pressure gradient (dP/dx). Unknown are the temperaturedistribution in the vicinity of the wall, and the heat transfer between the wall and the porousmedium.

The analysis refers to the boundary layer regime, which is based on the assumption that theregion in which the thermal effect of the wall is felt is slender,

dT � x ð1Þ

The boundary layer region has the length x and thickness dT. The latter is defined as thedistance y where the temperature is practically the same as the approaching temperature (T1),or where qT/qyffi0. The thermal boundary layer thickness dT is the unknown geometric featurethat is also the solution to the heat transfer problem. The wall heat flux is given by

q00 ¼ k �@T@y

� �y¼0

ð2Þ

where k is the effective thermal conductivity of the porous medium saturated with fluid. Thescale of the temperature gradient appearing in Equation (2) is given by

�@T@y

� �y¼0

� �T1 � T0dT � 0

¼DTdT

ð3Þ


SIMPLE METHODS FOR CONVECTION IN POROUS MEDIA 861

where DT is the overall temperature difference that drives q00: In conclusion, the heat flux scale isgiven by

q00 � kDTdT

ð4Þ

which means that in order to estimate q00 we must first estimate dT.The required equation for dT is provided by the energy equation (e.g. Nield and Bejan, 1999),

u@T@x

þ v@T@y

¼ a@2T@y2

ð5Þ

where /=k/(rcp)fluid is the thermal diffusivity of the saturated porous medium. The volume-averaged velocity components of the uniform flow are u=U1 and v=0, such that Equation (5)becomes

U1@T@x

¼ a@2T@y2

ð6Þ

To determine the order of magnitude of @2T/@y2, we use the same technique as in Equation (3):we look at the change in @T/@y from y=0 to �dT,

@2T@y2

¼@

@y@T@y

� ��

@T=@y� �

dT� @T =@y� �

0

dT � 0¼

0þ DT=dTdT

¼DT

d2Tð7Þ

Figure 1. Thermal boundary layer near an isothermal wall with parallel flow.


A. BEJAN862

Similarly, for @T/@x we look at the change in T along the system, from x=0–x, at a constant ysufficiently close to the wall:

@T@x

�ðT Þx � ðT Þx¼0

x� 0¼

T0 � T1x

¼DTx

ð8Þ

Together, Equations (6)–(8) produce

U1DTx

� aDT

d2Tð9Þ

which is the approximate algebraic statement that replaces the partial differential equation (6).The boundary layer thickness follows from Equation (9),

dT �axU1

� �1=2

ð10Þ

and so does the conclusion that dT increases as x1/2, as shown in Figure 1. The heat fluxdecreases as x–1/2, cf. Equation (4),

q00 � kDTU1

ax

� �1=2

ð11Þ

The dimensionless version of this heat transfer rate is given by

Nux � Pe1=2x ð12Þ

where the Nusselt and Peclet numbers are given by

Nux ¼q00xkDT

Pex ¼U1xa

ð13Þ

These estimates are valid when the slenderness assumption (1) is respected, and this translatesinto the requirement that Pex41, or that U1 and/or x must be sufficiently large.

How approximate is this heat transfer solution? The exact solution to the same thermalboundary layer problem is available in closed form, after solving Equation (6) in similarityformulation. The details of this analysis may be found in Bejan (1995). The similarity solutionfor the local heat flux is given by

Nux ¼ 0:564 Pe1=2x ð14Þ

This agrees with the scaling estimate (12), and Cheng (1977) obtained numerically the sameresult as Equation (14). The heat flux averaged over a wall of length L,

q00 ¼1

L

Z L

0

q00 dx ð15Þ

can be estimated based on Equation (14),

Nu ¼ 1:128 PeL ð16Þ

where corresponding Nusselt and Peclet number definitions are given by

Nu ¼q00LkDT

PeL ¼U1La

ð17Þ

Equation (16) shows that the scaling estimate (12) is again accurate within a factor of order 1.Furthermore, scale analysis makes no distinction between local flux and wall-averaged



heat flux}both have the same scale, and the correct scale is delivered by scale analysis,Equation (11).

The analysis that produced Equation (11) did not require the statement that the walltemperature is uniform, DT=constant. We made this assumption only later, when weformulated Nux and Nu: Equation (11) is the general and correct relation between the heat fluxscale ðq00Þ and wall excess temperature scale (DT) in the boundary layer configuration. Equation(11) can be used in situations other than the isothermal-wall case of Figure 1. For example,when the wall heat flux is uniform, Equation (11) delivers the scale and character of the walltemperature distribution, DT=T0(x)�T1. In local Nusselt number formulation, the scalingresult is

Nux ¼q00x

k½T0ðxÞ � T1�� Pe1=2x ð18Þ

The corresponding local and overall Nusselt numbers derived from the similarity solution to thesame problem are

Nux ¼q00x

k½T0ðxÞ � T1�¼ 0:886 Pe1=2x ð19Þ

Nu ¼q00L

kð %TT � T1Þ¼ 1:329 Pe

1=2L ð20Þ

where %TT is T(x) averaged from x=0 to L. Once again, the scaling result (18) anticipates within12 and 33 per cent the similarity solution, Equations (19) and (20). The trends are identical, andcorrect. To appreciate how simple, direct and cost-effective scale analysis is, the reader shouldtry to solve the problems using other methods. Several other examples of forced convectionboundary layers over flat and curved walls, steady and time-dependent, are reviewed in Nieldand Bejan (1999).

2.2. Forced convection wakes

The generality of the scaling law (9, 10) is illustrated further by the solution for the temperaturefield in the wake of a concentrated heat source. Figure 2 shows two key configurations in asaturated porous medium, the thermal wake behind a point heat source, q[W], and the wakebehind a line source, q0[Wm�1]. The convection and diffusion of heat in the wake are governed

Figure 2. Thermal wakes behind (a) a point heat source, and (b) a line heat source, in a porous mediumwith forced convection.


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by Equation (5) or its equivalent in cylindrical co-ordinates. The transversal length scale of thewake is the dT scale given by Equation (10).

There are two key questions in the description of a thermal wake, the rate of lateral spreading,which is answered by Equation (10), and the temperature decay in the downstream direction.The latter is represented by the longitudinal variation of the wake temperature averaged overthe x=constant plane. It results from the conservation of energy in a control volume oftransversal dimension dT. The energy conservation statement for Figure 2(a), i.e. for a cylinderof radius dT and length x, is given by

q � rcpU1d2T Tc � T1ð Þ ð21Þ

where Tc(x) is the wake centreline temperature, and (Tc�T1) is the order of magnitude of thewake excess temperature relative to the free stream temperature. Combining Equations (21) and(10) we obtain the longitudinal distribution of the temperature in the wake,

TcðxÞ � T1 �qkx

ð22Þ

This result shows that the wake excess temperature decreases as x�1 in the downstreamdirection. The corresponding result for the wake behind the line source (Figure 2(b)) is given by

TcðxÞ � T1 �q0

ðrcpÞf ðU1axÞ1=2ð23Þ

The scaling laws (22) and (23) agree, in an order of magnitude sense with the formulas derivedfrom the similarity solutions to the same problems (e.g. Bejan, 1995).

2.3. Natural convection boundary layers

Consider next the natural convection boundary layer near a vertical impermeable wallembedded in a saturated porous medium at a different temperature. The boundary conditionsare indicated in Figure 3, where the gravitational acceleration points in the negative y direction.The equations for mass conservation and Darcy flow,

@u@x

þ@v@y

¼ 0 ð24Þ

u ¼ �Km@P@x

v ¼ �Km

@P@y

þ rg� �

ð25Þ

can be rewritten as a single equation

@2c@x2

þ@2c@y2

¼ �Kgbn

@T@x

ð26Þ

where c(x,y) is the streamfunction for volume averaged flow, u=@c/@y and v=�@c/@x. InEquation (26) we used the Boussinesq approximation r=r0=[1�b(T�T0)], which effects thecoupling of the velocity field to the temperature field. The energy conservation equation forboundary layer flow is given by, cf. Equation (5),

@c@y

@T@x

�@c@x

@T@y

¼ a@2T@x2

ð27Þ



The problem is to determine the heat transfer rate between the wall and the medium, q00�kDT/dT, where DT=T0�T1, and dT is the thickness of the boundary layer region (dT5y). Byinvoking the temperature boundary conditions sketched in Figure 3, and using the rules ofscaling analysis outlined in Sections 2.1, we find that the terms of Equations (26) and (27) arerepresented by the following scales:

c

d2T;cy2

�KgbDTndT

ð28Þ

cDTydT

;cDTdT y

� aDT

d2Tð29Þ

On the left side of Equation (28) we retain the first scale, because c=d2T > c=y2: On the left side ofEquation (29), the two scales are represented by the same order of magnitude, cDT/(ydT).Together, Equations (28) and (29) are sufficient for determining the two unknown scales, dT and c,

dTy

� Ra�1=2y c � a Ra1=2y ð30Þ

Figure 3. Natural convection boundary layer near a heated vertical impermeable wall.


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where Ray is the Darcy-modified Rayleigh number, Ray=KgbyDT/(an). From the c scale weconclude that the vertical velocity scale is v�c/dT, or v�(a/y) Ray�KgbDT/n. From the dTscale we deduce the heat flux, or the Nusselt number,

Nuy ¼q00ykDT

� Ra1=2y ð31Þ

This scale agrees with the similarity solution to the problem of the boundary layer along anisothermal wall, T0 (Cheng and Minkowycz, 1977),

Nuy ¼q00y

T0 � T1ð Þk¼ 0:444 Ra1=2y ð32Þ

Nu ¼q00H

T0 � T1ð Þk¼ 0:888 Ra

1=2H ð33Þ

If the wall is with uniform heat flux, then the local wall temperature and the boundary layerthickness must vary such that

q00 � kT0ðyÞ � T1

dT¼ constant ð34Þ

Combining this with the dT scale (30), we conclude that

dTy

� Ra�1=3ny ð35Þ

where Rany is the Darcy-modified Rayleigh number based on heat flux, Rany=Kgby2q00/(ank).The local heat transfer rate must therefore scale as

Nuy ¼q00

T0ðyÞ � T1

yk� Ra

1=3ny ð36Þ

The numerical solution to the similarity for formulation of this problem is given by (Bejan,1995)

Nuy ¼q00

T0ðyÞ � T1

yk¼ 0:772 Ra

1=3ny ð37Þ

Nu ¼q00

%TT0 � T1

Hk¼ 1:03 Ra

1=3nH ð38Þ

where H is the wall height. In conclusion, Equations (36)–(38) show that the exact results areanticipated within 23 per cent by the results of scale analysis. Many more examples of scaleanalysis of convection in porous media can be found in the current literature. Importantexamples are the plumes rising above point and line sources, natural convection in cavities filled



with porous media and heated from the side, and porous layers heated from below. The latterforms the subject of the alternative method outlined in the next section.

3. THE METHOD OF INTERSECTING THE ASYMPTOTES

3.1. Porous layer saturated with fluid and heated from below

Assume that the system of Figure 4 is a porous layer saturated with fluid and that, if present,the flow is two-dimensional and in the Darcy regime. The height H is fixed, and the horizontaldimensions of the layer are infinite in both directions. The fluid has nearly constant propertiessuch that its density–temperature relation is described well by the Boussinesq linearization. Thevolume averaged equations that govern the conservation of mass, momentum and energy areEquation (24) and

@u@y

2@v@x

¼ 2Kgbn

@T@x

ð39Þ

u@T@x

þ v@T@y

¼ a@2T@x2

þ@2T@y2

� �ð40Þ

The horizontal length scale of the flow pattern (2Lr), or the geometric aspect ratio of one roll, isunknown (Nelson and Bejan, 1998).

3.2. The many counterflows regime

In the limit Lr!0 each roll is a very slender vertical counterflow (Figure 5). Because ofsymmetry, the outer planes of this structure (x=�Lr) are adiabatic: they represent the centreplanes of the streams that travel over the distance H. The scale analysis of the H� (2Lr) regionindicates that in the Lr/H! 0 limit the horizontal velocity component u vanishes. This scaleanalysis is not shown because it is well known as the defining statement of fully developed flow

Figure 4. Horizontal layer saturated with fluid and heated from below.


A. BEJAN868

(e.g. Bejan, 1995, p. 97). Equations (39)–(40) reduce to

@v@x

¼Kgbn

@T@x

ð41Þ

v@T@y

¼ a@2T@x2

ð42Þ

which can be solved exactly for v and T. The boundary conditions are qT/qx=0 at x=�Lr,and the requirement that the extreme (corner) temperatures of the counterflow regionare dictated by the top and bottom walls, T(�Lr,H)=Tc and T(Lr, 0)=Th. The solution isgiven by

vðxÞ ¼a2H

Rap2pH2Lr

� �2" #

sinpx2Lr

� �ð43Þ

T ðx; yÞ ¼n

KgbvðxÞ þ

nKgb

2yH21

� � a2H

Rap2pH2Lr

� �2" #

þ Th2Tcð Þ 12yH

� �ð44Þ

where the porous-medium Rayleigh number Rap=KgbH(Th–Tc)/(an) is a specifiedconstant. The right side of Figure 5 shows the temperature distribution along the verticalboundaries of the flow region (x=�Lr): the vertical temperature gradient qT/qy is inde-pendent of altitude. The transversal (horizontal) temperature difference (DTt) is also aconstant,

DTt ¼ T x ¼ Lrð Þ2T x ¼ 2Lrð Þ ¼n

KgbaH

Rap2pH2Lr

� �2" #

ð45Þ

Figure 5. The extreme in which the flow consists of many vertical and slender counterflows.



The counterflow convects heat upward at the rate q0, which can be calculated using Equations(43) and (44):

q0 ¼Z L

�LðrcpÞfvT dx ð46Þ

The average heat flux convected in the vertical direction is q00=q0/2Lr, hence the thermalconductance expression

q00

DT¼

k8H Rap

Rap2pH2Lr

� �2" #2

ð47Þ

This result is valid provided the vertical temperature gradient does not exceed the externallyimposed gradient, (–qT/qy)5DT/H. This condition translates into

LrH

>p2Ra21=2

p ð48Þ

which in combination with the assumed limit Lr/H! 0 means that the domain of validity ofEquation (47) is wide when Rap is large. In this domain the thermal conductance q00/DTdecreases monotonically as Lr decreases, cf. Figure 6.

3.3. Few plumes regime

As Lr increases, the number of rolls decreases and the vertical counterflow is replaced by ahorizontal counterflow in which the thermal resistance between Th and Tc is dominated by twohorizontal boundary layers, as in Figure 7. Let d be the scale of the thickness of the horizontalboundary layer. The thermal conductance q00/DT can be deduced from the heat transfer solutionfor natural convection boundary layer flow over a hot isothermal horizontal surface facingupward, or a cold surface facing downward. The similarity solution for the horizontal surfacewith power-law temperature variation (Cheng and Chang, 1976) can be used to develop ananalytical result, as we show at the end of this section.

A simpler analytical solution can be developed in a few steps using the integral method.Consider the slender flow region d� (2Lr), where d52Lr, and integrate Equations (24), (39) and(40) from y=0 to y!1, i.e. into the region just above the boundary layer. The surfacetemperature is Th, and the temperature outside the boundary layer is T1 (constant). The originx=0 is set at the tip of the wall section of length 2Lr. The integrals of Equations (24) and (40)

Figure 6. The geometric maximization of the thermal conductance of a fluid-saturated porous layer heatedfrom below.


A. BEJAN870

yield

d

dx

Z 1

0

u T2T1ð Þ dy ¼ 2a@T@y

� �y¼0

ð49Þ

The integral of Equation (39), in which we neglect qv/qx (cf. boundary layer theory), leads to

u0ðxÞ ¼Kgbn

d

dx

Z 1

0

T dy ð50Þ

where u0 is the velocity along the surface, u0=u(x,0). Reasonable shapes for the u and T profilesare the exponentials

uðx; yÞu0ðxÞ

¼ exp 2y

dðxÞ

� �¼

T ðx; yÞ2T1Th2T1

ð51Þ

which transform Equations (49) and (50) into

d

dxu0dð Þ ¼

2ad

ð52Þ

u0 ¼Kgbn

ðTh2T1Þdddx

ð53Þ

These equations can be solved for u0(x) and d(x),

dðxÞ ¼9an

Kgb Th2T1ð Þ

� �1=3x2=3 ð54Þ

The corresponding solution for u0(x) is of the type u0�x–1/3, which means that the horizontalvelocities are large at the start of the boundary layer, and decrease as x increases. This isconsistent with the geometry of the H � 2Lr roll sketched in Figure 7, where the flow generatedby one horizontal boundary layer turns the corner and flows vertically as a relatively narrowplume (narrow relative to 2Lr), to start with high velocity (u0) a new boundary layer along theopposite horizontal wall.

The thermal resistance of the geometry of Figure 7 is determined by estimating the local heatflux k(Th�T1)/d(x) and averaging it over the total length 2Lr:

q00 ¼3

4

� �1=3kDTH

Th2T1DT

� �4=3

Ra1=3pHLr

� �2=3

ð55Þ

Figure 7. The extreme in which the flow consists of a few isolated plumes.



The symmetry of the sandwich of boundary layers requires Th � T1 ¼ 12DT ; such that

q00

DT¼

31=3k4H

Ra1=3pHLr

� �2=3

ð56Þ

The goodness of this result can be tested against the similarity solution for a hot horizontalsurface that faces upward in a porous medium and has an excess temperature that increases asxl. The only difference is that the role that was played by (Th–T1) in the preceding analysis isnow played by the excess temperature averaged over the surface length 2Lr. If we use l=1/2,which corresponds to uniform heat flux, then it can be shown that the solution of Cheng andChang (1976) leads to the same formula as Equation (56), except that the factor 31/3=1.442 isreplaced by 0.816(3/2)4/3=1.401.

Equation (56) is valid when the specified Rap is such that the horizontal boundary layers donot touch. We write this geometric condition as d(x=2Lr)5H/2 and, using Equation (54), weobtain

LrH5

1

24Ra1=2p ð57Þ

Since in this analysis Lr/H was assumed to be very large, we conclude that the Lr/H domain inwhich Equation (56) is valid becomes wider as the specified RaH increases. The importantfeature of Equation (56) is that in the ‘few rolls’ limit the thermal conductance decreases as thehorizontal dimension Lr increases. This second asymptotic trend has been added to Figure 6.

3.4. Intersection of asymptotes

Figure 6 presents a bird’s-eye view of the effect of flow shape on thermal conductance. Eventhough we did not draw completely q00/DT as a function of Lr, the two asymptotes tell us that thethermal conductance is maximum at an optimal Lr value that is close to their intersection. Thereis a family of such curves, one curve for each Rap. The q

00/DT peak of the curve rises, and the Lr

domain of validity around the peak becomes wider as Rap increases. Looking in the direction ofsmall Rap values we see that the domain vanishes (and the cellular flow disappears) when thefollowing requirement is violated

1

24H Ra1=2p �

p2H Ra�1=2

p � 0 ð58Þ

This inequality means that the flow exists when Rap512p=37.7. This conclusion isextraordinary: it agrees with the stability criterion for the onset of two-dimensional convection(Horton and Rogers, 1945; Lapwood, 1948), namely Rap>4p2=39.5, which is derived based ona lengthier analysis and the assumption of initial disturbances.

We obtain the optimal shape of the flow, 2Lr,opt/H, by intersecting the asymptotes (47) and(56):

p2H

2Lr;optRa21=2

p

� �2

þ25=631=6H

2Lr;optRa21

p

� �1=3

¼ 1 ð59Þ

Over most of the Rap domain where Equation (58) is valid, Equation (59) is approximated wellby its high Rap asymptote:

2Lr;optH

ffi pRa21=2p ð60Þ


A. BEJAN872

The maximum thermal conductance is obtained by substituting the Lr,opt value of Equation (59)in either Equation (56) or Equation (47). This estimate is an upper bound, because theintersection is above the peak of the curve. In the high-Rap limit (60) this upper-bound assumesthe analytical form

q00

DT

� �max

Hk

31=3

24=3p2=3Ra2=3p ð61Þ

Towards lower Rap values the slope of the (q00/DT)max curve increases such that the exponent of

Rap approaches 1. This behaviour is in excellent agreement with the large volume ofexperimental data collected for B!eenard convection in saturated porous media (Cheng, 1978).The ‘less than 1’ exponent of Rap in the empirical Nu(Rap) curve, and the fact that this exponentdecreases as Rap increases, has attracted considerable attention from theoreticians during thelast two decades (Nield and Bejan, 1999).

4. CONCLUSION: THE RESEARCHER IS FREE TO CHOOSE THE METHOD

In this paper I have outlined the basic rule for two methods of solution for problems ofconvection in porous media: scale analysis, and the intersection of asymptotes. These are two ofthe simplest methods that are available. They yield concrete results for engineering items, suchas heat transfer rates, flow rates, velocities, temperature differences, and time intervals. Theydistinguish themselves from other methods because they offer a high return on investment:because they are so simple, they deserve to be tried first, as preliminaries, even in problemswhere more exact results are needed. Simple methods identify the proper dimensionlessformulation for presenting more exact (and more expensive) results developed based on morecomplicated methods (analytical, numerical, experimental).

Other simple methods are available, for example, the integral method (Karman-Pohlhausen),and similarity formulations. A word of caution goes with the use of all the methods that are‘simple’. More complicated problems with nonsimilar and singular solutions may require moreadvanced treatments from the start.

The intersection of asymptotes method relied on an additional principle that appliesthroughout the physics of the flow systems. That principle is the maximization of access forcurrents in systems far from equilibrium. In Section 3, we invoked this principle when weminimized the global thermal resistance encountered by the flow of heat across the horizontalporous layer. The intersection of the two asymptotes is an approximation of the flow geometrythat minimizes the global thermal resistance. The same principle has been used to predict flowgeometry and transitions between flow regimes in a great variety of configurations (Bejan, 2000).

The most important conclusion is that by learning simple methods, and using them correctly,the young researcher learns two important lessons. One is ‘try the simplest first’. These simplemethods are valuable throughout fluid mechanics and heat transfer, not only in porous media.For the other lesson, I paraphrase from my own course on convection (Bejan, 1995, p. 55): Inthis paper we have seen the open competition among methodologies, in the search forengineering answers to the basic questions of convective heat transfer. The laminar boundarylayer near an isothermal flat plate was the simplest and, historically, oldest setting in which towitness this competition. Despite what the pure scientists and pure engineers among us maywant us to believe, there can be no official winner in such a competition. An individual



researcher with a personal mathematics background and, most important, with a personalsupply of curiosity and time can and should judiciously evaluate the worthiness of any of thesemethodologies relative to his ability and taste. And he or she is free to choose.

REFERENCES

Bejan A. 1984. Convection Heat Transfer. Wiley: New York.Bejan A. 1995. Convection Heat Transfer (2nd edn). Wiley: New York.Bejan A. 2000. Shape and Structure, from Engineering to Nature. Cambridge University Press: Cambridge, UK.Cheng P. 1977. Combined free and forced convection flow about inclined surfaces in porous media. International Journal

of Heat and Mass Transfer 20:807–814.Cheng P. 1978. Heat transfer in geothermal systems. Advances in Heat Transfer 14:1–105.Cheng P, Chang JD. 1976. On buoyancy induced flows in a saturated porous medium adjacent to impermeable

horizontal surfaces. International Journal of Heat and Mass Transfer 19:1267–1272.Cheng P, Minkowycz WJ. 1977. Free convection about a vertical flat plate embedded in a saturated porous medium with

application to heat transfer from a dike. Journal of Geophysical Research 82:2040–2044.Horton CW, Rogers FT. 1945. Convection currents in a porous medium. Journal of Applied Physics 16:367–370.Nelson RA, Jr. Bejan A. 1998. Constructal optimization of internal flow geometry in convection. Journal of Heat

Transfer 120:357–364.Nield DA, Bejan A. 1999. Convection in Porous Media (2nd edn). Springer: New York.Lapwood ER. 1948. Convection of a fluid in a porous medium. Proceedings of the Cambridge Philosophical Society 44:

508–521.Patterson J, Imberger J. 1980. Unsteady natural convection in a rectangular cavity. Journal of Fluid Mechanics 100:

65–86.Sadeghipour MS, Razi YP. 2001. Natural convection from a confined horizontal cylinder: the optimum distance between

the confining walls. International Journal of Heat and Mass Transfer 44:367–374.


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Simple methods for convection in porous media: scale analysis and the intersection of asymptotes