Finite-amplitude heat and fluid flow results for Rayleigh numbersKgγT h − T c 2H/α mν ofup to 104i.e., about 50 times greater than critical are also reported in Blake et al.. Research on h
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Figure 15.10 Asymptotes ofthe function Nu (RaH , Pr p) for convection in a porous layer heated from below (From Bejan, 1995.)
In this formulation Nu is a function of two groups, RaH and Prp, in which Prp accounts for the transition from Darcy to Forchheimer flow (Fig 15.10) In this formulation the Darcy flow result of eq (15.115) becomes
Nu
Prp ∼ 1 40
RaH
Prp
40< Ra H < Pr p (15.118)
The experimental data for convection in the entire regime spanned by the asymptotes given by eqs (15.116) and (15.118) are correlated by
Nu=
RaH 40
n +cRaH· Prp1/2n1/n
(15.119)
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wheren = −1.65 and c = 1896 are determined empirically based on measurements
reported by many independent sources The effects of fluid inertia and other depar-tures from Darcy flow are discussed in detail in Nield and Bejan (1999)
The correlations ofeqs (15.115)–(15.119) refer to layers with length/height ra-tios considerably greater than 1 They apply when the length (lateral dimensionL
perpendicular to gravity) ofthe confined system is greater than the horizontal length scale ofa single convective cell (i.e., greater thanH · Ra −1/2 H ) according to the scale analysis ofBejan (1984)
Natural convection studies have also been reported for porous layers confined
in rectangular parallelpipeds heated from below, horizontal circular cylinders, and horizontal annular cylinders The general conclusion is that the lateral walls have a convection-suppression effect For example, in a circular cylinder of diameterD and
height H (Fig 15.9b), in the limit D H the critical condition for the onset of
convection is (Bau and Torrance, 1982)
RaH = 13.56
H D
2
(15.120)
In inclined porous layers that deviate from the horizontal position through an angleφ (Fig 15.9c), convection sets in at Rayleigh numbers that satisfy the criterion
(Combarnous and Bories, 1975)
RaH >39.48
where it is assumed that the boundaries are isothermal and impermeable The average heat transfer rate at high Rayleigh numbers can be estimated by
Nu= 1 +∞
s=1
k s
1− 4π2s2
RaHcosφ
(15.122)
wherek s = 0 if RaHcosφ < 4π2s2andk s= 2 if RaHcosφ ≥ 4π2s2
In a porous medium confined in a wedge-shaped (or attic-shaped) space cooled
from above (Fig 15.9d), convection consisting ofa single counterclockwise cell
ex-ists even in the limit RaH → 0, because in this direction the imposed heating is not
purely vertical The same observation holds for Fig 15.9c Numerical solutions of
transient high-Rayleigh-number convection in wedge-shaped layers show the pres-ence ofa B´enard-type instability at high enough Rayleigh numbers (Poulikakos and Bejan, 1983b) WhenH/L = 0.2, the instability occurs above Ra H 620 It was found that this critical Rayleigh number increases asH/L increases.
The onset ofconvection in the layer ofFig 15.9a saturated with water near the state
ofmaximum density has been studied using linear stability analysis (Sun et al., 1970) and time-dependent numerical solutions ofthe complete governing equations (Blake
et al., 1984) In both studies, the condition for the onset of convection is reported graphically or numerically for a discrete series of cases The numerical results (Blake
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et al., 1984) for layers withT c = 0°C and 5°C ≤ T h≤ 8°C suggested the following criterion for the onset of convection:
KgH
αmν > 1.25 × 105exp
exp(3.8 − 0.446T h ) (15.123)
In this expressionT hmust be expressed in °C Finite-amplitude heat and fluid flow
results for Rayleigh numbersKgγ(T h − T c )2H/α mν ofup to 104(i.e., about 50 times greater than critical) are also reported in Blake et al (1984)
Nuclear safety considerations have led to the study of natural convection in
hori-zontal saturated porous layers (Fig 15.9a) heated volumetrically at a rate q Bound-ary conditions and observations regarding the onset ofconvection and overall Nusselt numbers are presented in Nield and Bejan (1999) It is found that convection sets in
at internal Rayleigh numbers (Kulacki and Freeman, 1979)
RaI =
k mβ
αmν
f
KgH3q
in the range 33 to 46, where the subscriptf indicates properties ofthe fluid alone Top
and bottom surface temperature measurements in the convection-dominated regime are adequately represented by (Buretta and Berman, 1976)
qH2
2k m (T h − T c ) ≈ 0.116Ra0I .573 (15.125)
whereT handT care the resulting bottom and top temperatures ifq is distributed
throughout the layer ofFig 15.9a The empirical correlation (11.25) is based on
experiments that reach into the high RaI range of103to 104
In this section attention is focused on buoyancy-driven flows that penetrate the en-closed porous medium only partially This class ofconvection phenomena have been
categorized as penetrative flows (Bejan, 1984) With reference to the two vertical cylindrical configurations sketched in Fig 15.9e, ifthe cylindrical space is slender
enough, the flow penetrates vertically to the distance
L y = 0.085r o· Rar o (15.126) where Rar o = Kgβr o (T h −T c )/α m ν and L y < H The overall convection heat transfer
rate through the permeable horizontal end is
q = 0.255r o k m (T h − T c )Ra r o (15.127)
A similar partial penetration mechanism is encountered in the two horizontal
geometries ofFig 15.9f The length oflateral penetration, L x, and the convection
heat transfer rate in the two-dimensional geometry,q(W/m), are (Bejan, 1984, 1995)
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L x = 0.16H · Ra1/2
q= 0.32k m (T h − T c )Ra1/2
where Rah = Kgβ(T h − T c )H/α m ν and L x < L.
In a semi-infinite porous medium bounded from below or from above by a
horizon-tal surface with alternating zones of heating and cooling (Fig 15.9g), the
buoyancy-driven flow penetrates vertically to a height or depth approximately equal toλ·Ra1/2
λ , where Raλ = Kgβλ(T h − T c )/α mν and λ is the distance between a heated zone and
an adjacent cooled zone Numerical and graphic results are reported in Poulikakos and Bejan (1984a) for the Raλrange 1 to 100
There are many other circumstances in which penetrative flows can occur, steady and transient (Nield and Bejan, 1999) For example, in a porous medium heated and cooled along the same vertical wall ofheightH , the incomplete penetration can be either horizontal (Fig 15.9h) or vertical (Fig 15.9i) (Poulikakos and Bejan, 1984b).
In the case ofincomplete horizontal penetration, the penetration length and convective heat transfer rate scale as
L x ∼ H · Ra1/2
q∼ k m (T h − T c )Ra1/2
The derivation of these order-of-magnitude results can also be found in Bejan (1995)
They are valid ifRa1H /2 < L/H and Ra H 1 The corresponding scales ofincomplete vertical penetration (Fig 15.9i) are
L y ∼ H
L H
2/3
· Ra−1/3 H (15.132)
q∼ k m (T h − T c )
L
H · RaH
1/3
(15.133)
and are valid ifRa1H /2 > L/H and Ra1/2
H > H/L The penetrative flows ofFig 15.9h and i occur when the heated section T his situated above the cooled sectionT c When
the positions ofT handT care reversed, the buoyancy-driven flow fills the entire space
H × L.
Research on heat and mass transfer in porous media has grown impressively during the past two decades, beyond the fundamental results highlighted in Sections 15.1 and 15.6 The most up-to-date review ofthe current state ofthe literature on heat transfer in porous media is provided in the latest edition of Nield and Bejan’s (1999)
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book, which contains over 1600 references and in a new book (Bejan et al., 2004)
This closing section is a briefreview ofsome ofthe subfields that have emerged
When the flow is due to a combination ofdriving forces, buoyancy and imposed pressure differences, the heat transfer characteristics depend greatly on which force dominates Two papers (Lai et al., 1991; Vargas et al., 1995) review most ofwhat is known on mixed convection in two-dimensional external flow, along walls (vertical, inclined, horizontal) and in wedge-shaped domains The free and forced-convection effects are governed by the Rayleigh and P´eclet numbers, respectively,
Rax= g x βKx(Tα w − T∞)
wherex is the position from the leading edge measured along the wall and g x is the acceleration component aligned withx The graphic presentation ofthe heat transfer
results suggests that, in broad terms, free convection is the dominating effect when
Rax /Pe x > O(1).
The corresponding class ofmixed convection problems concerning the embedded sphere and horizontal cylinder in a uniform vertical flow was treated in Cheng (1982)
Inertial effects were introduced in the modeling of these problems by several authors
The most comprehensive treatment is the unifying analysis (Nakayama and Pop, 1991) for mixed convection based on the Darcy–Forchheimer model, which is valid for plane walls and axisymmetric bodies of arbitrary shape This unifying treatment and other configurations (internal flow) can also be found in Nield and Bejan (1999)
Another combination or competition ofdriving forces occurs when the local den-sity variations that cause buoyancy are due not only to temperature gradients but also
to concentration gradients This class ofphenomena is also known as double-diffusive
or thermohaline convection As an example, consider the onset ofconvection in a
hor-izontal porous layer subjected to heat and mass transfer between the confining bottom
and top walls (Fig 15.9a) An additional buoyancy effect is due to the concentration ofconstituent i maintained along the bottom wall ( C b) and the top wall (C t) The linearized density–temperature relation is
ρ ≈ ρb1− β(T − T h ) − β C (C − C b ) (15.136) whereβCis the concentration expansion coefficient,
βC= −1 ρ
∂ρ
∂C
P
(15.137)
For saturated porous layers confined between impermeable walls with uniformT and
C distributions, convection is possible if(Nield, 1968)
RaH + RaD,H > 39.48 (15.138)
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where
RaH =KgβH (T h − T c )
RaD,H =Kgβ C H (C b − C t )
withD as the mass diffusivity of constituent i through the solution-saturated porous
medium Therefore, becauseβCcan be positive or negative, the effect of mass transfer from below can be, respectively, either to decrease or increase the critical RaH for the onset ofconvection Alternatives to eq (15.138) for horizontal porous layers subjected to other boundary conditions are presented in Nield and Bejan (1999) and Nield (1968)
In the single-wall vertical flow configuration ofFig 15.3a, the additional buoyancy
effect caused by the imposed concentration differenceC w − C∞ can either aid or oppose the familiar flow due toT w − T∞(Bejan, 1984, 1995) An important role is played by the buoyancy ratio
N =βC (C w − C∞)
β(T w − T∞) (15.141)
In heat-transfer-driven flows (|N| 1), the heat transfer rate is given by eq (15.66).
The overall mass transfer rate can be estimated based on the scaling laws
j
D(C w − C∞) ∼
Ra1H /2· Le1/2 for Le 1
Ra1H /2· Le for Le 1 (15.142) wherej/(kg/s · m) is the overall mass transfer rate per unit length and Le is the
Lewis number ofthe solution-saturated porous medium, αm /D In mass transfer–
driven situations(|N| 1), the overall mass transfer rate is
j
D(C w − C∞) = 0.888(Ra H · Le|N|)1/2 (15.143)
for all Lewis numbers The corresponding overall Nusselt number obeys the scaling laws
q
k(T w − T∞)∼
(Ra H |N|)1/2 for Le 1
Le−1/2 (Ra H |N|)1/2 for Le 1 (15.144) The order-of-magnitude results of eqs (15.142) and (15.144) agree within 15%
with overall heat and mass transfer calculations based on similarity solutions to the same problem (Bejan and Khair, 1985) The corresponding enclosure problem, where the vertical walls are maintained at different temperatures and concentrations, the
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heat and mass transfer due to convection driven by combined buoyancy effects was documented in terms ofnumerical experiments in Trevisan and Bejan (1985, 1986)
Another expanding area is the study ofconduction and convection in the presence ofphase change: melting, solidification, boiling, and condensation These problems are recommended by specialized applications in diverse fields such as geophysics, manufacturing, small-scale heat exchangers, and spaces filled with fibers coated with energy storage (phase-change) material These and other applications are treated in detail in the most recent edition ofNield and Bejan (1999)
NOMENCLATURE
Roman Letter Symbols
A cross-sectional area, m2
a fissure spacing, m
B width ofstack, m
combination ofterms, dimensionless
Be Bejan number, dimensionless
b fissure spacing, m
coefficient in Forchheimer’s modification of Darcy’s law, m−1 stratification parameter, dimensionless
C constituent concentration, kg/m3
c specific heat, J/kg · K
c p specific heat at constant pressure, J/kg · K
D mass diffusivity, m2/s
diameter ofround tube, m distance between parallel plates, m
D p fiber diameter, m
D/Dt material derivative operator, s−1
d diameter, m
peripheral sublayer thickness, m
g gravitational acceleration, m/s2
h heat transfer coefficient, W/m2· K
h m local mass transfer coefficient, m/s
j constituent mass flow per unit length, kg/m · s
j constituent mass flow per unit area, kg/m2· s
K permeability, m2
k thermal conductivity, W/m · K
k A overall average thermal conductivity, W/m · K
k G weighted geometric mean thermal conductivity, W/m · K
k H weighted harmonic mean thermal conductivity, W/m · K
k m porous medium thermal conductivity, W/m · K
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L x penetration length, m
L y penetration height, m
Le Lewis number, dimensionless
m mass produced in chemical reaction, kg/m3
N buoyancy ratio, dimensionless
number ofhorizontal sublayers, dimensionless
Nu Nusselt number, dimensionless
NuH Nusselt number based on height, dimensionless
NuL Nusselt number based on wall length, dimensionless
Nuθ peripheral Nusselt number, dimensionless
Nux Nusselt number based on local longitudinal position,
dimensionless
Nuy Nusselt number based on heat flux, dimensionless
P pressure, Pa
Pe P´eclet number, dimensionless
PeL overall P´eclet number, dimensionless
Pex P´eclet number based on local longitudinal position,
dimensionless
Prp porous medium Prandtl number, dimensionless
q heat transfer rate, W
q heat transfer rate per unit length, W/m
q heat transfer rate per unit area, W/m2
q volumetric heat generation rate, W/m3
R parameter defined in eq (15.91), dimensionless
r radial coordinate, m
spherical coordinate, m
r i inner radius, m
r o outer radius, m
RaH Rayleigh number based on height and temperature difference,
dimensionless
Ra∗H Rayleigh number based on heat flux, dimensionless
RaI internal Rayleigh number based on volumetric heat generation
rate, dimensionless
Ray Darcy-modified Rayleigh number, dimensionless
Rayleigh number based on heat flux, dimensionless
Ra∗∞,y Rayleigh number for inertial flow based on heat flux,
dimensionless
Raγc Rayleigh number for the cold side of a porous medium
saturated with fluid near the density maximum, dimensionless
Raγh Rayleigh number for the hot side of a porous medium
saturated with fluid near the density maximum, dimensionless
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Sh Sherwood number, dimensionless
T temperature, K
u velocity component in thex direction, m/s
V volume, m3
V volume averaged velocity vector, m/s
w velocity component in thez direction, m/s
x Cartesian coordinate, m
y Cartesian coordinate, m
z Cartesian coordinate, m
Greek Letter Symbols
αm porous medium thermal diffusivity, m2/s
αm empirical factor in the density–temperature relation for water,
dimensionless
β coefficient of thermal expansion, K−1
βC coefficient of concentration expansion, m3/kg
γ vertical temperature gradient, K/m
δT thermal boundary layer thickness, m
η similarity variable, dimensionless
λ distance, m
µ dynamic viscosity, Pa· s
ν kinematic viscosity, m2/s
ρ density, kg/m3
σ capacity ratio, dimensionless
τ time, dimensionless
φ porosity, dimensionless
spherical coordinate, rad angle, rad
ψ stream function, m2/s
spherical coordinate, dimensionless
ω wall thickness parameter, dimensionless
Subscripts
b bottom or bulk
f fluid (liquid or gas) phase
G geometric mean
H harmonic mean
L property based on plate length
m bulk property
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property ofthe state ofmaximum density porous medium
p constant pressure condition
s solid phase
x local property
∞ free stream condition
REFERENCES
Bau, H H., and Torrance, K E (1982) Low Rayleigh Number Thermal Convection in a
Vertical Cylinder Filled with Porous Materials and Heated from Below, J Heat Transfer,
104, 166–172
Bear, J (1972) Dynamics of Fluids in Porous Media, American Elsevier, New York.
Bejan, A (1978) Natural Convection in an Infinite Porous Medium with a Concentrated Heat
Source, J Fluid Mech., 89, 97–107.
Bejan, A (1979) On the Boundary Layer Regime in a Vertical Enclosure Filled with a Porous
Medium, Lett Heat Mass Transfer, 6, 93–102.
Bejan, A (1983a) The Boundary Layer Regime in a Porous Layer with Uniform Heat Flux
from the Side, Int J Heat Mass Transfer, 26, 1339–1346.
Bejan, A (1983b) Natural Convection Heat Transfer in a Porous Layer with Internal Flow
Obstructions, Int J Heat Mass Transfer, 26, 815–822.
Bejan, A (1984) Convection Heat Transfer, Wiley, New York.
Bejan, A (1987) Convective Heat Transfer in Porous Media, in Handbook of Single-Phase
Convective Heat Transfer, S Kakac¸, R K Shah, and W Aung, eds., Wiley, New York.
Bejan, A (1993) Heat Transfer, Wiley, New York.
Bejan, A (1995) Convection Heat Transfer, 2nd ed., Wiley, New York.
Bejan, A (1997) Advanced Engineering Thermodynamics, 2nd ed., Wiley, New York.
Bejan, A (1999) Heat Transfer in Porous Media, in Heat Exchanger Design Update, G F.
Hewitt, ed., Begell House, New York, Vol 6, Sec 2.11
Bejan, A (2000) Shape and Structure, from Engineering to Nature, Cambridge University
Press, Cambridge, UK
Bejan, A., and Anderson, R (1981) Heat Transfer across a Vertical Impermeable Partition
Imbedded in a Porous Medium, Int J Heat Mass Transfer, 24, 1237–1245.
Bejan, A., and Anderson, R (1983) Natural Convection at the Interface between a Vertical
Porous Layer and an Open Space, J Heat Transfer, 105, 124–129.
Bejan, A., and Khair, K R (1985) Heat and Mass Transfer by Natural Convection in a Porous
Medium, Int J Heat Mass Transfer, 28, 909–918.
Bejan, A., and Lage, J L (1991) Heat Transfer from a Surface Covered with Hair, in
Con-vective Heat and Mass Transfer in Porous Media, S Kakac¸, B Kilkis, F A Kulacki, and
F Arinc, eds., Kluwer Academic, Dordrecht, The Netherlands, pp 823–845