Here the heat transfer rate scales as q ≤ k m T h− T c Ra1/2 Considerable analytical, numerical, and experimental work has been done to esti-mate more accurately the overall heat transfe
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T c
T c
T c
T c
T c
T c
T h
T h
T h
T h
T h
T h
r o
r o
r o
r i
r i
r i
d i
d i
d1
d1
d N
d N
r o
g
H
H
H
g
g g
g g
( )g
( )d
( )a
( )h
( )e
( )b
( )i
( )f
( )c
l
g H
0 0
y
0
0 0
q⬙
Partition
Insulated Insulated
Insulated
Figure 15.5 Natural convection heat transfer in confined porous media heated from the
side: (a) rectangular enclosure; (b) rectangular enclosure with a horizontal partial partition;
(c) rectangular enclosure with a vertical full partition midway; (d) rectangular enclosure made
up ofN vertical sublayers ofdifferent K and α; (e) rectangular enclosure made up of N
horizontal sublayers ofdifferentK and α; ( f ) horizontal cylindrical enclosure; (g) horizontal
cylindrical annulus with axial heat flow; (h) horizontal cylindrical or spherical annulus with radial heat flow; (i) vertical cylindrical annulus with radial heat flow.
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Vertical boundary layers
Horizontal boundary layers
T
0 Tall system
II
III High Ra regimeH
RaH
IV Shallow system
T
y H
0
I Conduction
Conduction
Conduction
10⫺2
10⫺1
1
10
H/L
102
Figure 15.6 Four heat transfer regimes for natural convection in an enclosed porous layer heated from the side (From Bejan, 1984.)
(Bejan, 1984), that is, four ways to calculate the overall heat transfer rate q =
H
0 qdy These are summarized in Fig 15.6.
• Regime I: the pure conduction regime, defined by Ra H 1 In this regime, qis
approximately equal to the pure conduction estimatek m H (T h − T c )/L.
• Regime II: the conduction dominated regime in tall layers, defined by H/L 1
and (L/H )Ra1/2
H 1 In this regime, the heat transfer rate scales as q ≥
k m H(T h − T c )/L.
• Regime III: the convection-dominated regime (or high-Rayleigh-number regime),
defined by Ra−1/2 H < H/L < Ra1/2
H In this regime,qscales ask m (T h −T c )Ra1/2
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• Regime IV: the convection-dominated regime in shallow layers, defined by H/L
1 and (H/L)Ra1/2
H 1 Here the heat transfer rate scales as q ≤ k m (T h−
T c )Ra1/2
Considerable analytical, numerical, and experimental work has been done to esti-mate more accurately the overall heat transfer rateqor the overall Nusselt number:
kH (T h − T c )/L (15.98)
Note that unlike the single-wall configurations ofSection 15.5.1, in confined layers ofthicknessL the Nusselt number is defined as the ratio ofthe actual heat transfer
rate to the pure conduction heat transfer rate An analytical solution that covers the four heat transfer regimes smoothly is (Bejan and Tien, 1978)
Nu= K1+ 1
120K3 1
RaH H L
2
(15.99) whereK1(H/L, Ra H ) is obtained by solving the system
1
120δe· Ra2
H · K3 1
H
L
3
= 1 − K1= 1
2K1
H L
1
δe − δe
(15.100)
This result is displayed in chart form in Fig 15.7, along with numerical results from Hickox and Gartling (1981) The asymptotic values ofthis solution are
Nu∼
0.508 H LRa1H /2 as RaH → ∞ (15.101)
1+ 1
120
RaH H L
2
as H
The heat transfer in the convection-dominated regime III is represented well by
eq (15.101) or by alternative solutions developed solely for regime III: for example (Weber, 1975),
Equation (15.103) overestimates experimental and numerical data from three inde-pendent sources (Bejan, 1979) by only 14% More refined estimates for regime III were developed in Bejan (1979) and Simpkins and Blythe (1980), where the propor-tionality factor between Nu and(L/H )Ra1/2
H is replaced by a function of bothH/L
and RaH This alternative is illustrated in Fig 15.8 For expedient engineering cal-culations of heat transfer dominated by convection, Fig 15.7 is recommended for shallow layers (H/L < 1), and Fig 15.8 for square and tall layers (H/L 1) in the boundary layer regime, Ra−1/2 H < H/L < Ra1/2
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1 10
10
H
— L Ra H
Bejan and Tien (1978):
/ = 0
H L
Hickox and Gartling (1981)
/ = 0.5 0.2 0.1
H L
Nu
0.1 0.2
0.5
1
Figure 15.7 Total heat transfer rate through an enclosed porous layer heated from the side
(From Bejan and Tien, 1978.)
In the field ofthermal insulation engineering, a more appropriate model for heat
transfer in the configuration of Fig 15.5a is the case where the heat flux q is
distributed uniformly along the two vertical sides of the porous layer In the high-Rayleigh-number regime (regime III), the overall heat transfer rate is given by (Bejan, 1983a)
Nu= 1
2
L H
4/5
where Ra∗H = KgβH2q/α m νk m The overall Nusselt number is defined as in eq
(15.98), whereT h −T cis the height-averaged temperature difference between the two sides ofthe rectangular cross section Equation (15.104) holds in the high-Rayleigh-number regime Ra∗−1/3 H < H/L < Ra ∗1/3 H
Impermeable partitions (flow obstructions) inserted in the confined porous me-dium can have a dramatic effect on the overall heat transfer rate across the enclosure
(Bejan, 1983b) With reference to the two-dimensional geometry of Fig 15.5b, in
the convection-dominated regimes III and IV the overall heat transfer rate decreases
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Figure 15.8 Heat transfer rate in regime III through a porous layer heated from the side
(From Bejan, 1979.)
steadily as the lengthl ofthe horizontal partition approaches L, that is, as the partition
divides the porous layer into two shorter layers The horizontal partition has practi-cally no effect in regimes I and II, where the overall heat transfer rate is dominated
by conduction Ifthe partition is oriented vertically (Fig 15.5c), in the
convection-dominated regime the overall heat transfer rate is approximately 40% of what it would have been in the same porous medium without the internal partition
The nonuniformity of permeability and thermal diffusivity can have a dominating effect on the overall heat transfer rate (Poulikakos and Bejan, 1983b) In cases where the properties vary so that the porous layer can be modeled as a sandwich ofvertical
sublayers of different permeability and diffusivity (Fig 15.5d), an important
parame-ter is the ratio ofthe peripheral sublayer thickness (d1) to the thermal boundary layer thickness (δT ,1) based on the properties ofthed1 sublayer (note thatδT ,1scales as
H · Ra −1/2 H,1 , where the Rayleigh number RaH,1 = K1gβH (T h − T c )/α1ν and where
the subscript 1 represents the properties ofthed1 sublayer) Ifd1 > δ T ,1, the heat
transfer through the left side of the porous system of Fig 15.5d is impeded by a
thermal resistance oforderδT ,1 /k1H Ifthe sublayer situated next to the right wall
(d N) has exactly the same properties as thed1sublayer, and ifδT ,1 < (d1, d N ), the
overall heat transfer rate in the convection-dominated regime can be estimated using
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eq (15.101), in which both Nu and RaH are based on the properties ofthe peripheral layers
When the porous-medium inhomogeneity may be modeled as a sandwich ofN horizontal sublayers (Fig 15.5e), the scale ofthe overall Nusselt number in the
convection-dominated regime can be evaluated as (Poulikakos and Bejan, 1983a)
Nu∼ 2−3/2Ra1/2
H
N
i=1
k i
k1
K i d iα1
K1d1αi
1/2
(15.105)
where both Nu and RaH,1are based on the properties ofthed1sublayer (Fig 15.5e).
The correlation ofeq (15.105) was tested via numerical experiments in two-layer systems
The heat transfer results reviewed in this section are based on the idealization that the surface that surrounds the porous medium is impermeable With reference to the
two-dimensional geometry ofFig 15.5a, the heat transfer through a shallow porous
layer with one or both end surfaces permeable is anticipated analytically in Bejan and Tien (1978) Subsequent laboratory measurements and numerical solutions for RaH values up to 120 validate the theory (Haajizadeh and Tien, 1983)
Natural convection in cold water saturating the porous-medium configuration of
Fig 15.5a was considered in Poulikakos (1984) Instead ofthe linear approximation
ofeq (15.60), this study used the parabolic model
whereγ ≈ 8.0 × 10−6K−2andT m = 3.98°C for pure water at atmospheric pressure.
The parabolic density model is valid in the temperature range 0 to 10°C In the convection-dominated regime Nu 1, the scale analysis (Bejan, 1995) leads to the
Nusselt number correlation (Bejan, 1987)
Nu= c3
L/H
Ra−1/2 γh + c4Ra−1/2 γc (15.107) where Raγh = KgγH(T h − T m )2/α m ν, Ra γc = KgγH (T m − T c )2/α mν, and where
the Nusselt number is defined in eq (15.98) For the convection-dominated regime, the numerical study (Poulikakos, 1984) tabulated results primarily for the caseT c=
0°C, Th = 7.96°C; using these data for cases in which T candT hare symmetrically positioned aroundT m(i.e., when Raγh= Raγc), the scaling-correct correlation in eq
(15.107) takes the form (Bejan, 1987)
Nu≈ 0.26 L
HRa
1/2
In other words, the two constants that appear in eq (15.107) satisfy the relationship
c3≈ 0.26(1 + c4) More experimental data for the high-Rayleigh-number regime in
vertical layers with Raγh= Raγcare needed to determinec3andc4uniquely
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15.6.2 Cylindrical and Spherical Enclosures
The convection occurring in a porous medium confined in a horizontal cylinder with
disk-shaped ends at different temperatures (Fig 15.5f ) has features similar to the problem ofFig 15.5a A parametric solution for the horizontal cylinder problem
is reported in Bejan and Tien (1978) The corresponding phenomenon in a porous
medium in the shape ofa horizontal cylinder with annular cross section (Fig 15.5g)
is documented in Bejan and Tien (1979)
A pivotal important geometric configuration in thermal insulation engineering is a
horizontal annular space filled with fibrous or granular insulation (Fig 15.5h) In this
configuration the heat transfer is radial between the concentric cylindrical surfaces ofradiir i and r o , unlike in the earlier sketch (Fig 15.5g), where the cylindrical
surfaces were insulated and the heat transfer was axial Experimental measurements
and numerical solutions for the overall heat transfer in the configuration of Fig 15.5h
have been reported in Caltagirone (1976) and Burns and Tien (1979) These results were correlated based on scale analysis (Bejan, 1987) in the range 1.19 ≤ ro /r i ≤ 4
and the results are correlated by
Nu= qactual
q
conduction
≈ 0.44Ra1/2
r i
ln(ro /r i )
1+ 0.916(r i /r o )1/2 (15.109)
where Rar i = Kgβr i (T h − T c )/α m ν and q
conduction = 2πk m (T h − T c )/ ln(r o /r i ) This
correlation is valid in the convection-dominated limit, Nu1
Porous media confined to the space formed between two concentric spheres are
also an important component in thermal insulation engineering Figure 15.5h can be
interpreted as a vertical cross section through the concentric-sphere arrangement Nu-merical heat transfer solutions for discrete values of Rayleigh number and radius ratio are reported graphically in Burns and Tien (1979) Using the scale analysis method (Bejan, 1984, 1995) the data that correspond to the convection-dominated regime (Nu 1.5) are correlated within 2% by the scaling-correct expression (Bejan, 1987)
Nu= qactual
qconduction
= 0.756Ra1/2
r i
1− r i /r o
1+ 1.422(r i /r o )3/2 (15.110)
where Rar i = Kgβr i (T h − T c )/α m ν and qconduction = 4πk m (T h − T c )/(r−1
o ) In
terms ofthe Rayleigh number based on the insulation thickness Rar o −r i = Kgβ(r o−
r i )(T h − T c )/α mν, the correlation (15.110) becomes
Nu= 0.756Ra1/2
r o −r i
r i /r o − (r i /r o )21/2
1+ 1.422(r i /r o )3/2 (15.111)
In this form, the Nusselt number expression has a maximum inr i /r o (at r i /r o = 0.3).
Heat transfer by natural convection through an annular porous insulation oriented
vertically (Fig 15.5i) was investigated numerically (Havstad and Burns, 1982) and
experimentally (Prasad et al., 1985) For systems where both vertical cylindrical
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surfaces may be modeled as isothermal (ThandT c), results were correlated with the five-constant empirical formula (Havstad and Burns, 1982)
Nu= 1 + a1
r i
r o
1− r i
r o
a2
· Raa4
r o
H
r o
a5
exp
−a3r i
r o
(15.112)
wherea1 = 0.2196, a2 = 1.334, a3 = 3.702, a4 = 0.9296, and a5 = 1.168
and where Rar o = Kgβr o (T h − T c )/α mν The Nusselt number is defined as Nu =
qactual/qconduction, whereqconduction = 2πk m H (T h − T c )/ ln(r o /r i ) The correlation
ofeq (15.112) fits the numerical data in the range 1 ≤ H/r o ≤ 20, 0 < Ra r o <
150, 0 < ri /r o ≤ 1, and 1 < Nu < 3 In the boundary layer convection regime (at
high Rayleigh and Nussselt numbers), the scale analysis ofthis two-boundary-layer problem suggests the following scaling law (Bejan, 1987):
Nu= c1
ln(r o /r i )
c2+ r o /r i
r o
HRa
1/2
where RaH = KgβH (T h − T c )/α mν Experimental data in the convection-dominated
regime Nu 1 are needed to determine the constants c1 andc2(note that Havstad and Burns’s data are for moderate Nusselt numbers 1< Nu < 3, i.e., for cases where
pure conduction plays an important role) There is a large and still-growing volume of additional results for enclosed porous media heated from the side, for example, with application to cavernous bricks and walls for buildings (Lorente et al., 1996, 1998;
Lorente 2002; Lorente and Bejan, 2002; Vasile et al., 1998)
The most basic configuration ofa confined porous layer heated in the vertical
direc-tion is shown in Fig 15.9a An important difference between heat transfer in this
configuration and heat transfer in confined layers heated from the side is that in Fig
15.9a convection occurs only when the imposed temperature difference or heating rate exceeds a certain, finite value Recall that in configurations such as Fig 15.5a,
convection is present even in the limit ofvanishingly small temperature differences (Fig 15.6)
Assume that the fluid saturating the porous medium ofFig 15.9a expands upon
heating (β > 0) By analogy with the phenomenon ofB´enard convection in a pure
fluid, in the convection regime the flow consists offinite-sized cells that become more slender and multiply discretely as the destabilizing temperature differenceT h − T c
increases IfT h − T c does not exceed the critical value necessary for the onset of convection, the heat transfer mechanism through the layer of thickness H is that of pure thermal conduction Ifβ > 0 and the porous layer is heated from above (i.e., if
T handT c change places in Fig 15.9a), the fluid remains stably stratified and the heat
transfer is again due to pure thermal conduction:q= k m L(T h − T c )/H.
The onset ofconvection in an infinitely long porous layer heated from below as examined on the basis oflinearized hydrodynamic stability analysis (Nield and Bejan, 1999; Horton and Rogers, 1945; Lapwood, 1948) For fluid layers confined between
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T c
T c
T c C t
T c
T c
T c
T c
T c
T h
T h
T h
T h
T h
T h C b
T h
T h
T h
T h
L y
L x
L y
L y
L y
L
r o
L x
T c T h T c
distr
g
L
H
H H
g
g g
g g
g g
( )g
( )d
( )a
( )h
( )e
( )b
( )i
( )f
( )c
Permeable end
L
D
L x
y
g H
Permeable end
Figure 15.9 Natural convection heat transfer in confined porous layers heated from below
(a–d), and due to penetrative flows (e–i): (a) rectangular enclosure; (b) vertical cylindrical en-closure; (c) inclined rectangular enen-closure; (d) wedge-shaped enen-closure; (e) vertical cylindrical enclosure; ( f ) horizontal rectangular enclosure; (g) semi-infinite porous medium bounded by
a horizontal surface with alternate zones of heating and cooling; (h) shallow rectangular en-closure heated and cooled from one vertical wall; (i) slender rectangular enen-closure heated and
cooled from one vertical wall
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impermeable and isothermal horizontal walls, it was found that convection is possible ifthe Rayleigh number based on height exceeds the critical value
RaH = KgβH (Tα h − T c )
A much simpler closed-form analysis based on constructal theory (Nelson and Bejan, 1998; Bejan, 2000) predicted the critical Rayleigh number 12π = 37.70, which
approaches within 5% the hydrodynamic stability result For a history ofthe early theoretical and experimental work on the onset ofB´enard convection in porous media, and for a rigorous generalization of the stability analysis to convection driven by combined buoyancy effects, the reader is directed to Nield (1968), where it is shown that the critical Rayleigh number for the onset of convection in infinitely shallow layers depends to a certain extent on the heat and fluid flow conditions imposed along the two horizontal boundaries
Of practical interest in heat transfer engineering is the heat transfer rate at Rayleigh numbers that are higher than critical There has been a considerable amount ofan-alytical, numerical, and experimental work devoted to this issue Reviews ofthese advances may be found in Nield and Bejan (1999) and Cheng (1978) Constructal theory anticipates the entire curve relating heat transfer to Rayleigh number (Nelson and Bejan, 1998; Bejan, 2000)
The scale analysis ofthe convection regime with Darcy flow (Bejan, 1984) con-cludes that the Nusselt number should increase linearly with the Rayleigh number, whence the relationship
Nu≈ 1
40RaH for RaH > 40 (15.115) This linear relationship is confirmed by numerical heat transfer calculations at large Rayleigh numbers in Darcy flow (Kimura et al., 1986) The experimental data com-piled in Cheng (1978) show that the scaling law ofeq (15.115) serves as an upper bound for some of the high-RaH experimental data available in the literature
Most ofthe data show that in the convection regime Nu increases as Ran H, wheren
becomes progressively smaller than 1 as RaH increases This behavior is anticipated
by the constructal-theory solution (Nelson and Bejan, 1998; Bejan, 2000) The expo-nentn ∼ 1
2 revealed by data at high Rayleigh numbers was anticipated based on a scale analysis ofconvection rolls in the Forchheimer regime (Bejan, 1995):
Nu
Prp ∼
RaH
Prp
1/2
RaH > Pr p
(15.116)
where Prp is the porous-medium Prandtl number for the Forchheimer regime (Bejan,
1995),
Prp = H ν