Heat Transfer Handbook part 115 pps

15.3 CONDUCTION Pure thermal diffusion is the mechanism for heat transfer when there is no motion through the pores ofthe solid structure.. The overall thermal conductivity ofa porous me

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K = d2φ3

b = 1.75(1 − φ)

Brinkman’s (1947) modification ofthe Darcy flow model accounts for the transi-tion from Darcy flow to highly viscous flow (without porous matrix), in the limit of extremely high permeability:

The more appropriate way to write Brinkman’s equation is (Nield and Bejan, 1999)

∇P = −µ

Kv+ ˜µ∇2v (15.18)

which is similar to eq (15.17) without the body force term and multiplied byK/µ.

In eqs (15.17) and (15.18), two viscous terms are evident The first is the usual Darcy term, and the second is analogous to the Laplacian term that appears in the Navier–

Stokes equation The coefficient ˜µ is an effective viscosity Brinkman set µ and ˜µ

equal to each other, but in general that is not true The reader is referred to Nield and Bejan (1999) for a critical discussion ofthe applicability ofeq (15.18)

There are situations in which it is convenient to use the Brinkman equation One such situation is when flows in porous media are compared with those in clear fluids

The Brinkman equation has a parameterK (the permeability) such that the equation

reduces to a form of the Navier–Stokes equation asK/L2 → ∞ and to the Darcy

equation asK/L2→ 0 Another situation is when it is desired to match solutions in

a porous medium and in an adjacent viscous fluid

The two modifications ofthe Darcy flow model discussed above, the Forchheimer model ofeq (15.13) and the Brinkman model ofeq (15.17), were used simultane-ously by Vafai and Tien (1981) in a study of forced-convection boundary layer heat transfer In the presence of gravitational acceleration, Vafai and Tien’s momentum equations would read

µ |v|v =

K

None of the foregoing models account adequately for the transition from porous medium flow to pure fluid flow as the permeabilityK increases Note that in the

high-K limit, the terms that survive in eq (15.17) or (15.19) account for

momen-tum conservation only in highly viscous flows in which the effect of fluid inertia is negligible relative to pressure and friction forces A model that bridges the entire gap between the Darcy–Forchheimer model and the Navier–Stokes equations was proposed by Vafai and Tien (1981):

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ν

Kv+ b|v|v = −

Dv

Dt −

1

ρ∇P + ν ∇2v + g (15.20)

As the permeabilityK increases, the left-hand side vanishes and gives way to the

complete vectorial Navier–Stokes equation for Newtonian constant-property flow

The state ofthe art in the development offlow models and new questions about the older models are discussed in Nield and Bejan (1999)

15.2.3 Energy Conservation

Consider now the first law ofthermodynamics for flow through a porous medium

For simplicity, assume that the medium is isotropic and that radiative effects, viscous dissipation, and the work done by pressure changes are negligible For most cases, it

is acceptable to assume that there is local thermal equilibrium so thatT s = T f = T ,

whereT sandT fare the temperatures ofthe solid and fluid phases, respectively Also assume that heat conduction in the solid and fluid phases takes place in parallel so that there is no net heat transfer from one phase to the other Taking averages over an elemental volume ofthe medium give, for the solid phase (Nield and Bejan, 1999),

(1 − φ)(ρc) s ∂Ts

∂t = (1 − φ)∇ · (k s ∇T s ) + (1 − φ)q s (15.21)

and for the fluid phase,

φ(ρc p ) f ∂T f

∂t + (ρc p)fv· ∇T f = φ∇ · (k f ∇T f ) + φq f (15.22)

Here the subscriptss and f refer to the solid and fluid phases, respectively, c is the

specific heat ofthe solid,c pis the specific heat at constant pressure ofthe fluid,k is

the thermal conductivity, andq(W/m3) is the heat generation rate per unit volume.

In writing eqs (15.21) and (15.22) it has been assumed that the surface porosity

is equal to the porosity For example,−k s ∇T sis the conductive heat flux through the solid, and thus∇ ·(k s ∇T s ) is the net rate ofheat conduction into a unit volume ofthe

solid In eq (15.21), this appears multiplied by the factor (1− φ), which is the ratio

ofthe cross-sectional area occupied by solid to the total cross-sectional area ofthe medium The other two terms in eq (15.21) also contain the factor (1− φ), because

this is the ratio ofvolume occupied by solid to the total volume ofthe element In

eq (15.22) there also appears a convective term, due to the seepage velocity We

recognize that v· ∇T f is the rate ofchange oftemperature in the elemental volume due to the convection offluid into it, so this, multiplied by(ρcp ) f, must be the rate ofchange ofthermal energy, per unit volume offluid, due to the convection Note further that in writing eq (15.22), use has been made of the Dupuit–Forchheimer

relationship (Nield and Bejan, 1999), v = φV.

SettingTs = T f = T and adding eqs (15.21) and (15.22) yields

(ρc)m ∂T ∂t + (ρc) fv· ∇T = ∇ · (k m ∇T ) + q

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where

(ρc) m = (1 − φ)(ρc) s + φ(ρc p ) f (15.24)

km = (1 − φ)k s + φk f (15.25)

q

m = (1 − φ)q

s + φq

are, respectively, the overall heat capacity per unit volume, overall thermal conduc-tivity, and overall heat production per unit volume ofthe medium Equation (15.24) may also be written as

(ρc) m = (ρc p ) fσ (15.27) whereσ is the heat capacity ratio,

σ = φ + (1 + φ) (ρc)s

The energy balance ofeq (15.24) becomes

σ∂T

∂t + v · ∇T = ∇ · (α m ∇T ) +

q

m

whereαmis the thermal diffusivity of the fluid saturated porous medium,

αm= k m

Ifthe assumption oflocal thermal equilibrium is abandoned, account must be taken for the local heat transfer between solid and fluid Equations (15.21) and (15.22) are replaced by

(1 − φ)(ρc) s ∂T

∂t = (1 − φ)∇ · (k s ∇T s ) + (1 − φ)q s+ h(T f − T s )

(15.31)

φ(ρc p ) f ∂Tf

∂t + (ρc p ) f v· ∇T = φ∇ · (k f ∇T ) + φq f+ h(T s − T f ) (15.32) whereh is the heat transfer coefficient A critical aspect of using this approach lies in

the determination ofthe appropriate value ofh Experimental values of h are found

in an indirect manner and methods are reviewed by Nield and Bejan (1999)

In situations where the fluid that saturates the porous structure is a mixture oftwo

or more chemical species, the equation that expresses the conservation ofspecies is (Bejan, 1995)

φ∂C ∂t + v · ∇C = D ∇2C + ˙m

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In this equationC is the concentration of i, expressed in kilograms of i per unit

volume ofporous medium;D is the mass diffusivity of i through the porous medium

with the fluid mixture in it; and ˙m

i is the number ofkilograms ofi produced by a

chemical reaction per unit time and per unit volume ofporous medium

15.3 CONDUCTION

Pure thermal diffusion is the mechanism for heat transfer when there is no motion through the pores ofthe solid structure The conservation ofenergy is described by

eq (15.29), in which the convection term is absent:

σ∂T

∂t = ∇ · (α m ∇T ) +

q

Except for the heat capacity ratioσ that appears on the left side, eq (15.34) is the

same as the equation for time-dependent conduction through a solid (Bejan, 1993)

This means that the mathematical methods developed for conduction in solids (Bejan, 1993; Carslaw and Jaeger, 1959) apply to porous media saturated with stagnant fluid

For example, the thermal penetration depth due to time-dependent conduction into a semiinfinite porous medium without fluid motion is oforder(αmt/σ)1/2 The reader

is directed to Chapter 3 in this book for additional mathematical solutions for key configurations

The overall thermal conductivity ofa porous medium depends in a complex fash-ion on the geometry ofthe medium (Nield and Bejan, 1999; Nield, 1991) Ifthe heat conduction in the solid and fluid phases occurs in parallel, the overall conductivity

k Ais the weighted arithmetic mean ofthe conductivitiesk sandk f:

k A = (1 − φ)k s + φk f (15.35)

On the other hand, ifthe structure and orientation ofthe porous medium is such that the heat conduction takes place in series, with all the heat flux passing through both solid and fluid, the overall conductivitykH is the weighted harmonic mean ofks

andkf:

1

kH =

1− φ

φ

In general,k Aandk H will provide upper and lower bounds, respectively, on the actual overall conductivitykm It is always true thatkH ≤ k A, with equality ifand only

ifks = k f For practical purposes, a rough-and-steady estimate forkmis provided by

kG, the weighted geometric mean ofksandkf, defined by

kG = k1 −φ

This provides a good estimate as long asksandkfare not too different from each other (Nield, 1991) More complicated correlation formulas for the conductivity of packed beds have been proposed Experiments by Prasad et al (1989) showed that eqs

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(15.35)–(15.37) gave reasonably good results provided thatk f was not significantly greater thank s Additional thermal conductivity models are discussed in Nield and Bejan (1999)

15.4 FORCED CONVECTION 15.4.1 Plane Wall with Constant Temperature

The newer results developed for heat transfer through porous media refer to forced and natural convection The heat transfer results listed next refer to a uniform unidi-rectional seepage flow through a homogeneous and isotropic porous medium They are based on the idealization that the solid and fluid phases are locally in thermal equilibrium

Consider the uniform flow (u, T∞) parallel to a solid wall heated to a constant temperatureTw , as shown in Fig 15.1b The boundary layer solution for the local

Nusselt number is available analytically (Bejan, 1995):

Nux = q

Tw − T∞

x

km = 0.564Pe1x /2 (15.38)

where Pex is the P´eclet number based on the local longitudinal position, Pex =

ux/αm The heat fluxqand the heat transfer coefficientq/(Tw − T∞) decrease

asx −1/2 The overall Nusselt number based on the heat flux,qaveraged fromx = 0

to a given wall lengthx = L is

NuL = q

Tw − T∞

L

km = 1.128Pe

1/2

where the overall P´eclet number is PeL = uL/α m The total heat transfer rate through

the wall isq = qL Related boundary layer solutions are reviewed in Nield and

Bejan (1999)

The local Nusselt number for boundary layer heat transfer near a wall with constant heat flux is also available in closed form (Bejan, 1995):

Nux = T q

w (x) − T∞

x

k m = 0.886Pe1x /2 (15.40)

where Pex = ux/α m The temperature differenceT w (x) − T∞increases asx1/2 The

overall Nusselt number that is based on the average wall temperatureT w(specifically, the temperature averaged fromx = 0 to x = L is

NuL= q

Tw − T∞

L

k m = 1.329Pe

1/2

where PeL = uL/α m Equations (15.38)–(15.41) are valid when the respective P´eclet

numbers are greater than 1 in an order-of-magnitude sense Mass transfer counter-parts to these heat transfer formulas are obtained through the notation transformation

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Nu → Sh, q → j, T → C, k m → D, and α m → D, where Sh, j, C, and

D are the Sherwood number, mass flux, species concentration, and mass diffusivity

constant

From a fluid mechanics standpoint, these results are valid ifthe flow is parallel, that is, in the Darcy regime or Darcy–Forchheimer regime through a homogeneous and isotropic porous medium The special effect of the flow resistance exerted by the solid wall was documented numerically in Vafai and Tien (1981)

15.4.2 Sphere and Cylinder

Consider the thermal boundary layer region around a sphere or around a circular cylinder that is perpendicular to the uniform flow with volume-averaged velocityu.

As indicated in Fig 15.1c, the sphere or cylinder radius is r0and the surface temper-ature isTw The distributions ofheat flux around the sphere and cylinder in Darcy flow were determined in Cheng (1982) With reference to the angular coordinateθ

defined in Fig 15.1c, the local peripheral Nusselt numbers are, for the sphere,

Nuθ= 0.564

ur

0θ

αm

1/2

3

2θ

1/2

sin2θ



1

3cos

3θ − cos θ +2

3

−1/2

(15.42) and for the cylinder,

Nuθ= 0.564

ur

0θ

αm

1/2 (2θ)1/2sinθ(1 − cos θ) −1/2 (15.43)

The P´eclet number is based on the swept arcr0θ: namely, Peθ= ur0θ/α m The local Nusselt number is defined as

Nuθ= T q

w − T∞

r0θ

Equations (15.42) and (15.43) are valid when the boundary layers are distinct (thin), that is, when the bondary layer thicknessr0·Pe−1/2θ is smaller than the radiusr0 This requirement can also be written as Pe1θ/2 1 or Nuθ 1

The conceptual similarity between the thermal boundary layers ofthe cylinder

and the sphere (Fig 15.1c) and that ofthe flat wall (Fig 15.1b) is illustrated further

by Nield and Bejan’s (1999) correlation of the heat transfer results for these three configurations The heat flux averaged over the area ofthe cylinder and sphere,q, can

be calculated by averaging the local heat fluxqexpressed by eqs (15.42)–(15.44).

The results are for the sphere,

NuD = 1.128Pe1/2

and for the cylinder,

NuD = 1.015Pe1/2

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In these expressions, the Nusselt and P´eclet numbers are based on the diameter

D = 2r0:

NuD =T q

w − T∞

D

k m PeD =

uD

15.4.3 Concentrated Heat Sources

In the region downstream from the hot sphere or cylinder of Fig 15.1c, the heated

fluid forms a thermal wake whose thickness increases asx1/2 This behavior is

il-lustrated in Fig 15.1d and e, in which x measures the distance downstream from

the heat source Seen from the distant wake region, the embedded sphere appears

as a point source (Fig 15.1d), while the cylinder perpendicular to the uniform flow

(u, T∞) looks like a line source (Fig 15.1e).

Consider the two-dimensional frame attached to the line sourceqin Fig 15.1e.

The temperature distribution in the wake region,T (x, y), is

T (x, y) − T∞= 0.282 k q

m

αm

ux

1/2

e −uy2/4α m x (15.48)

This shows that the width ofthe wake increases asx1/2, while the temperature excess

on the centerline [T (x, 0) − T∞] decreases asx −1/2 The corresponding solution for

the temperature distributionT (x, r) in the round wake behind the point source q of

Fig 15.1d is

T (x, r) − T∞= q

4πk mx e −ur

In this case, the excess temperature on the wake centerline decreases asx−1, that is,

more rapidly than on the centerline ofthe two-dimensional wake Equations (15.48) and (15.49) are valid when the wake region is slender, in other words, when Pex  1

When this P´eclet number condition is not satisfied, the temperature field around the source is dominated by the effect of thermal diffusion, not convection In such cases, the effect of the heat source is felt in all directions, not only downstream

In the limit where the flow (u,T∞) is so slow that the convection effect can be neglected, the temperature distribution can be derived by the classical methods of pure conduction A steady-state temperature field can exist only around the point source (Bejan, 1993),

T (r) − T∞= q

The pure-conduction temperature distribution around the line source remains time-dependent When the timet is sufficiently long so that (x2+ y2)/(4α m t)  1, the

excess temperature around the line source is approximated by

T (x, y, t) − T∞ q

4πk m



ln 4αmt

σ(x2+ y2) − 0.5772



(15.51)

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15.4.4 Channels Filled with Porous Media

Consider now the forced-convection heat transfer in a channel or duct packed with a

porous material as in Fig 15.1f In the Darcy flow regime the longitudinal

volume-averaged velocityu is uniform over the channel cross section When the temperature

field is fully developed, the relationship between the wall heat fluxqand the local

temperature difference (T w − T b) is analogous to the relationship for fully developed heat transfer to slug flow through a channel without a porous matrix (Bejan, 1995)

The temperatureT bis the mean or bulk temperature ofthe stream that flows through the channel,

Tb= A1

in whichA is the area ofthe channel cross section In cases where the confining wall

is a tube with the internal diameterD, the relation for fully developed heat transfer

can be expressed as a constant Nusselt number (Rohsenow and Choi, 1961):

NuD =







q(x)

Tw − T b(x)

D

k = 5.78 (tube,T w= constant) (15.53)

q

Tw(x) − Tb(x)

D

km = 8 (parallel plates,q= constant) (15.54)

When the porous matrix is sandwiched between two parallel plates with the spacing

D, the corresponding Nusselt numbers are (Rohsenow and Hartnett, 1973)

NuD =







q(x)

Tw − T b(x)

D

km = 4.93 (parallel plates, Tw = constant) (15.55)

q

Tw(x) − Tb(x)

D

km = 6 (parallel plates, q= constant) (15.56)

The forced-convection results of eqs (15.53)–(15.56) are valid when the temper-ature profile across the channel is fully developed (sufficiently far from the entrance

x = 0) The entrance length, or length needed for the temperature profile to become

fully developed, can be estimated by noting that the thermal boundary layer thick-ness scales as (α m x/u)1/2 Setting(α m x/u)1/2 ∼ D, the thermal entrance length

x T ∼ D2u/α mis obtained Inside the entrance region 0< x < x T, the heat transfer

is impeded by the forced-convection thermal boundary layers that line the channel walls, and can be calculated approximately using eqs (15.38)–(15.41)

One important application ofthe results for a channel packed with a porous ma-terial is in the area of heat transfer augmentation The Nusselt numbers for fully de-veloped heat transfer in a channel without a porous matrix are given by expressions similar to eqs (15.53)–(15.56) except that the saturated porous medium conductivity

km is replaced by the thermal conductivity ofthe fluid alone,k f The relative heat

transfer augmentation effect is indicated approximately by the ratio

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hx (with porous matrix)

hx (without porous matrix) ∼

km

in whichhxis the local heat transfer coefficientq/(Tw −T b) In conclusion, a

signif-icant heat transfer augmentation effect can be achieved by using a high-conductivity matrix material, so thatkmis considerably greater thankf

Key results for forced convection in porous media have also been developed for tree networks ofcracks (constructal theory) (Bejan, 2000), time-dependent heating, annular channels, stepwise changes in wall temperature, local thermal nonequilib-rium, and other external flows (such as over a cone or wedge) These are also reviewed

in Nield and Bejan (1999), which also describes improvement in porous-medium modeling that account for fluid inertia, thermal dispersion, boundary friction, non-Newtonian fluids, and porosity variation

The concepts of heatfunctions and heatlines were introduced for the purpose of visualizing the true path ofthe flow ofenergy through a convective medium (Bejan, 1995) The heatfunction accounts simultaneously for the transfer of heat by conduc-tion and convecconduc-tion at every point in the medium The heatlines are a generalizaconduc-tion ofthe flux lines used routinely in the field ofconduction The concept ofheatfunction

is a spatial generalization ofthe concept ofthe Nusselt number, that is, a way ofindi-cating the magnitude ofthe heat transfer rate through any unit surface drawn through any point on the convective medium The heatline method was extended recently to several configurations ofconvection through fluid-saturated porous media (Morega and Bejan, 1994)

15.4.5 CompactHeatExchangers as Porous Media

An important application ofthe formalism offorced convection in porous media is

in the field ofheat exchanger simulation and design Heat exchangers are a century-old technology based on information and concepts stimulated by the development oflarge-scale devices The modern emphasis on heat transfer augmentation, and the more recent push toward miniaturization in the cooling ofelectronics, have led to the development ofcompact devices with much smaller features than in the past

These devices operate at lower Reynolds numbers, where their compactness and small dimensions (“pores”) make them candidates for modeling as saturated porous media

Such modeling promises to revolutionize the nomenclature and numerical simulation ofthe flow and heat transfer through heat exchangers (Bejan et al., 2004)

To illustrate this change, consider Zhukauskas’s (1987) classical chart for the pressure drop in crossflow through arrays ofstaggered cylinders (e.g., Fig 9.38 in Bejan, 1993) The four curves drawn on this chart for the transverse pitch/cylinder diameter ratios 1.25, 1.5, 2, and 2.5 can be made to collapse into a single curve (Bejan and Morega, 1993), as shown in Fig 15.2 The technique consists oftreating the bundle as a fluid-saturated porous medium and using the volume-averaged velocityU,

the pore Reynolds numberUK1/2 /ν on the abscissa, and the dimensionless pressure

gradient group(∆P /L)K1/2 /ρU2on the ordinate

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Figure 15.2 Porous medium representation ofthe classical pressure-drop data for flow through staggered cylinders and stacks ofparallel plates (From Bejan and Morega, 1993)

The method ofpresentation ofFig 15.2 deserves to be extended to other heat exchanger geometries Another reason for pursuing this direction is that the heat and fluid flow process can be simulated numerically much more easily ifthe heat ex-changer is replaced at every point by a porous medium with volume-averaged properties

Another important application ofporous media concepts in engineering is in the optimization ofthe internal spacings ofheat exchangers subjected to overall volume constraints Packages ofelectronics cooled by forced convection are examples ofheat exchangers that must function in fixed volumes The design objective is to install

as many components (i.e., heat generation rate) as possible, while the maximum temperature that occurs at a point (hot spot) inside the given volume does not exceed a specified limit A very basic trade-offexists with respect to the number ofcomponents installed (Bejan and Sciubba, 1992) regarding the size ofthe pores through which the coolant flows This trade-off is evident when the two extremes are imagined, these extremes being numerous components (small pores) and few components (large spacings)

When the components and pores are numerous and small, the package functions

as a heat-generating porous medium When the installed heat generation rate is fixed, the hot-spot temperature increases as the spacings become smaller, because in this limit the coolant flow is being shut off gradually In the opposite limit, the hot-spot temperature increases again because the heat transfer contact area decreases

as the component size and spacing become larger At the intersection ofthese two