Chapter 15 - Porous Media - Chapter Notes, Chemical Engineering, Semester Chemical Engineering Notes | EduRev

Chemical Engineering : Chapter 15 - Porous Media - Chapter Notes, Chemical Engineering, Semester Chemical Engineering Notes | EduRev

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BOOKCOMP, Inc. — John Wiley & Sons / Page 1131 / 2nd Proofs /HeatTransferHandbook / Bejan
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CHAPTER 15
Porous Media
ADRIAN BEJAN
Department of Mechanical Engineering and Materials Science
Duke University
Durham,NorthCarolina
15.1 Introduction
15.2 Basic principles
15.2.1 Mass conservation
15.2.2 Flow models
15.2.3 Energy conservation
15.3 Conduction
15.4 Forced convection
15.4.1 Plane wall with constant temperature
15.4.2 Sphere and cylinder
15.4.3 Concentrated heat sources
15.4.4 Channels ?lled with porous media
15.4.5 Compact heat exchangers as porous media
15.5 External natural convection
15.5.1 Vertical walls
15.5.2 Horizontal walls
15.5.3 Sphere and horizontal cylinder
15.5.4 Concentrated heat sources
15.6 Internal natural convection
15.6.1 Enclosures heated from the side
15.6.2 Cylindrical and spherical enclosures
15.6.3 Enclosures heated from below
15.6.4 Penetrative convection
15.7 Other con?gurations
Nomenclature
References
15.1 INTRODUCTION
Heat and mass transfer through saturated porous media is an important development
andanareaofveryrapidgrowthincontemporaryheattransferresearch.Although
the mechanics of?uid ?ow through porous media has preoccupied engineers and
physicists for more than a century, the study of heat transfer has reached the status
1131
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CHAPTER 15
Porous Media
ADRIAN BEJAN
Department of Mechanical Engineering and Materials Science
Duke University
Durham,NorthCarolina
15.1 Introduction
15.2 Basic principles
15.2.1 Mass conservation
15.2.2 Flow models
15.2.3 Energy conservation
15.3 Conduction
15.4 Forced convection
15.4.1 Plane wall with constant temperature
15.4.2 Sphere and cylinder
15.4.3 Concentrated heat sources
15.4.4 Channels ?lled with porous media
15.4.5 Compact heat exchangers as porous media
15.5 External natural convection
15.5.1 Vertical walls
15.5.2 Horizontal walls
15.5.3 Sphere and horizontal cylinder
15.5.4 Concentrated heat sources
15.6 Internal natural convection
15.6.1 Enclosures heated from the side
15.6.2 Cylindrical and spherical enclosures
15.6.3 Enclosures heated from below
15.6.4 Penetrative convection
15.7 Other con?gurations
Nomenclature
References
15.1 INTRODUCTION
Heat and mass transfer through saturated porous media is an important development
andanareaofveryrapidgrowthincontemporaryheattransferresearch.Although
the mechanics of?uid ?ow through porous media has preoccupied engineers and
physicists for more than a century, the study of heat transfer has reached the status
1131
BOOKCOMP, Inc. — John Wiley & Sons / Page 1132 / 2nd Proofs /HeatTransferHandbook / Bejan
1132 POROUS MEDIA
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ofaseparate?eldofresearchduringthelastthreedecades(NieldandBejan,1999).
It has also become an established topic in heat transfer education, where it became a
partoftheconvectioncoursein1984(Bejan,1984,1995).Thereaderisdirectedto
agrowingnumberofmonographsthatoutlinethefundamentals(Scheiddeger,1957;
Bear, 1972; Bejan, 1987; Kaviany, 1995; Ingham and Pop, 1998, 2002; Vafai, 2000;
Pop and Ingham, 2001; Bejan et al., 2004).
Porous media and transport are becoming increasingly important in heat exchanger
analysis and design. It was pointed out in Bejan (1995) that the gradual miniatur-
izationofheattransferdevicesleadsthe?owtowardlowerReynoldsnumbersand
brings the designer into a domain where dimensions are considerably smaller and
structures considerably more complex than those covered by the single-con?guration
correlations developed historically for large-scale heat exchangers. The race toward
smallscalesandlargeheat?uxesinthecoolingofelectronicdevicesisthestrongest
manifestation of this trend. It is fair to say that the reformulation of heat exchanger
analysisanddesignasthebasisofporousmedium?owprinciplesisthenextareaof
growth in heat exchanger theory for small-scale applications.
Theobjectiveofthischapteristoprovideaconciseandeffectivereviewofsome
ofthe most basic results on heat transf er through porous media. This coverage is
anupdatedandcondensedversionofareviewpresented?rstinBejan(1987).More
detailed and tutorial alternatives were developed subsequently in Bejan (1995, 1999)
and Nield and Bejan (1999), to which the interested reader is directed.
15.2 BASIC PRINCIPLES
Thedescriptionofheatand?uid?owthroughaporousmediumsaturatedwith?uid
(liquidorgas)isbasedonaseriesofspecialconceptsthatarenotfoundinthepure-
?uid heat transfer. Examples are the porosity and the permeability of the porous
medium,andthevolume-averagedpropertiesofthe?uid?owingthroughtheporous
medium.Theporosityoftheporousmediumisde?nedas
f=
void volume contained in porous medium sample
totalvolumeofporousmediumsample
(15.1)
The engineering heat transfer results assembled in this chapter refer primarily to ?uid-
saturated porous media that can be modeled as nondeformable, homogeneous, and
isotropic. In such media, the volumetric porosityf is the same as the area ratio (void
areacontainedinthesamplecrosssection)/(totalareaofthesamplecrosssection).
Representative values are shown in Table 15.1.
Thephenomenonofconvectionthroughtheporousmediumisdescribedinterms
ofvolume-averaged quantities such as temperature, pressure, concentration, and
velocity components. Each volume-averaged quantity (?) is de?ned through the
operation
?=
1
V

v
?dV (15.2)
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CHAPTER 15
Porous Media
ADRIAN BEJAN
Department of Mechanical Engineering and Materials Science
Duke University
Durham,NorthCarolina
15.1 Introduction
15.2 Basic principles
15.2.1 Mass conservation
15.2.2 Flow models
15.2.3 Energy conservation
15.3 Conduction
15.4 Forced convection
15.4.1 Plane wall with constant temperature
15.4.2 Sphere and cylinder
15.4.3 Concentrated heat sources
15.4.4 Channels ?lled with porous media
15.4.5 Compact heat exchangers as porous media
15.5 External natural convection
15.5.1 Vertical walls
15.5.2 Horizontal walls
15.5.3 Sphere and horizontal cylinder
15.5.4 Concentrated heat sources
15.6 Internal natural convection
15.6.1 Enclosures heated from the side
15.6.2 Cylindrical and spherical enclosures
15.6.3 Enclosures heated from below
15.6.4 Penetrative convection
15.7 Other con?gurations
Nomenclature
References
15.1 INTRODUCTION
Heat and mass transfer through saturated porous media is an important development
andanareaofveryrapidgrowthincontemporaryheattransferresearch.Although
the mechanics of?uid ?ow through porous media has preoccupied engineers and
physicists for more than a century, the study of heat transfer has reached the status
1131
BOOKCOMP, Inc. — John Wiley & Sons / Page 1132 / 2nd Proofs /HeatTransferHandbook / Bejan
1132 POROUS MEDIA
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[1132], (2)
ofaseparate?eldofresearchduringthelastthreedecades(NieldandBejan,1999).
It has also become an established topic in heat transfer education, where it became a
partoftheconvectioncoursein1984(Bejan,1984,1995).Thereaderisdirectedto
agrowingnumberofmonographsthatoutlinethefundamentals(Scheiddeger,1957;
Bear, 1972; Bejan, 1987; Kaviany, 1995; Ingham and Pop, 1998, 2002; Vafai, 2000;
Pop and Ingham, 2001; Bejan et al., 2004).
Porous media and transport are becoming increasingly important in heat exchanger
analysis and design. It was pointed out in Bejan (1995) that the gradual miniatur-
izationofheattransferdevicesleadsthe?owtowardlowerReynoldsnumbersand
brings the designer into a domain where dimensions are considerably smaller and
structures considerably more complex than those covered by the single-con?guration
correlations developed historically for large-scale heat exchangers. The race toward
smallscalesandlargeheat?uxesinthecoolingofelectronicdevicesisthestrongest
manifestation of this trend. It is fair to say that the reformulation of heat exchanger
analysisanddesignasthebasisofporousmedium?owprinciplesisthenextareaof
growth in heat exchanger theory for small-scale applications.
Theobjectiveofthischapteristoprovideaconciseandeffectivereviewofsome
ofthe most basic results on heat transf er through porous media. This coverage is
anupdatedandcondensedversionofareviewpresented?rstinBejan(1987).More
detailed and tutorial alternatives were developed subsequently in Bejan (1995, 1999)
and Nield and Bejan (1999), to which the interested reader is directed.
15.2 BASIC PRINCIPLES
Thedescriptionofheatand?uid?owthroughaporousmediumsaturatedwith?uid
(liquidorgas)isbasedonaseriesofspecialconceptsthatarenotfoundinthepure-
?uid heat transfer. Examples are the porosity and the permeability of the porous
medium,andthevolume-averagedpropertiesofthe?uid?owingthroughtheporous
medium.Theporosityoftheporousmediumisde?nedas
f=
void volume contained in porous medium sample
totalvolumeofporousmediumsample
(15.1)
The engineering heat transfer results assembled in this chapter refer primarily to ?uid-
saturated porous media that can be modeled as nondeformable, homogeneous, and
isotropic. In such media, the volumetric porosityf is the same as the area ratio (void
areacontainedinthesamplecrosssection)/(totalareaofthesamplecrosssection).
Representative values are shown in Table 15.1.
Thephenomenonofconvectionthroughtheporousmediumisdescribedinterms
ofvolume-averaged quantities such as temperature, pressure, concentration, and
velocity components. Each volume-averaged quantity (?) is de?ned through the
operation
?=
1
V

v
?dV (15.2)
BOOKCOMP, Inc. — John Wiley & Sons / Page 1133 / 2nd Proofs /HeatTransferHandbook / Bejan
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TABLE 15.1 PropertiesofCommonPorous Materials
Porosity, Permeability, Surface per unit
Material f K(cm
2
) Volume (cm
-1
)
Agar–agar — 2× 10
-10
–4.4× 10
-9
Black slate powder 0.57–0.66 4.9× 10
-10
–1.2× 10
-9
7× 10
3
–8.9× 10
3
Brick 0.12–0.34 4.8× 10
-11
–2.2× 10
-9
Catalyst (Fischer–Tropsch,
granules only)
0.45 — 5.6× 10
5
Cigarette — 1.1× 10
-5
Cigarette ?lters 0.17–0.49
Coal 0.02–0.12
Concrete (bituminous) — 1× 10
-9
–2.3× 10
-7
Concrete (ordinary mixes) 0.02–0.07
Copper powder
(hot-compacted)
0.09–0.34 3.3× 10
-6
–1.5× 10
-5
Corkboard — 2.4× 10
-7
–5.1× 10
-7
Fiberglass 0.88–0.93 — 560–770
Granular crushed rock 0.45
Hair (on mammals) 0.95–0.99
Hair felt — 8.3× 10
-6
–1.2× 10
-5
Leather 0.56–0.59 9.5× 10
-10
–1.2× 10
-9
1.2× 10
4
–1.6× 10
4
Limestone (dolomite) 0.04–0.10 2× 10
-11
–4.5× 10
-10
Sand 0.37–0.50 2× 10
-7
–1.8× 10
-6
150–220
Sandstone (“oil sand”) 0.08–0.38 5× 10
-12
–3× 10
-8
Silica grains 0.65
Silica powder 0.37–0.49 1.3× 10
-10
–5.1× 10
-10
6.8× 10
3
–8.9× 10
3
Soil 0.43–0.54 2.9× 10
-9
–1.4× 10
-7
Spherical packings
(well shaken)
0.36–0.43
Wire crimps 0.68–0.76 3.8× 10
-5
–1× 10
-4
29–40
Source: Data from Nield and Bejan (1999), Scheidegger (1957), and Bejan and Lage (1991).
where ?istheactualvalueofthequantityatapointinsidethesamplevolume V .
Alternatively,thevolume-averagedquantityequalsthevalueofthatquantityaveraged
over the total volume occupied by the porous medium. The volume sample is called
representativeelementaryvolume(REV).ThelengthscaleoftheREVismuchlarger
thantheporescalebutconsiderablysmallerthanthelengthscaleofthemacroscopic
?ow domain.
15.2.1 Mass Conservation
The principle ofmass conservation or mass continuity applied locally in a small
regionofthe?uid-saturatedporousmediumis
D?
Dt
+??·v= 0 (15.3)
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CHAPTER 15
Porous Media
ADRIAN BEJAN
Department of Mechanical Engineering and Materials Science
Duke University
Durham,NorthCarolina
15.1 Introduction
15.2 Basic principles
15.2.1 Mass conservation
15.2.2 Flow models
15.2.3 Energy conservation
15.3 Conduction
15.4 Forced convection
15.4.1 Plane wall with constant temperature
15.4.2 Sphere and cylinder
15.4.3 Concentrated heat sources
15.4.4 Channels ?lled with porous media
15.4.5 Compact heat exchangers as porous media
15.5 External natural convection
15.5.1 Vertical walls
15.5.2 Horizontal walls
15.5.3 Sphere and horizontal cylinder
15.5.4 Concentrated heat sources
15.6 Internal natural convection
15.6.1 Enclosures heated from the side
15.6.2 Cylindrical and spherical enclosures
15.6.3 Enclosures heated from below
15.6.4 Penetrative convection
15.7 Other con?gurations
Nomenclature
References
15.1 INTRODUCTION
Heat and mass transfer through saturated porous media is an important development
andanareaofveryrapidgrowthincontemporaryheattransferresearch.Although
the mechanics of?uid ?ow through porous media has preoccupied engineers and
physicists for more than a century, the study of heat transfer has reached the status
1131
BOOKCOMP, Inc. — John Wiley & Sons / Page 1132 / 2nd Proofs /HeatTransferHandbook / Bejan
1132 POROUS MEDIA
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[1132], (2)
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———
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———
Long Page
* PgEnds: Eject
[1132], (2)
ofaseparate?eldofresearchduringthelastthreedecades(NieldandBejan,1999).
It has also become an established topic in heat transfer education, where it became a
partoftheconvectioncoursein1984(Bejan,1984,1995).Thereaderisdirectedto
agrowingnumberofmonographsthatoutlinethefundamentals(Scheiddeger,1957;
Bear, 1972; Bejan, 1987; Kaviany, 1995; Ingham and Pop, 1998, 2002; Vafai, 2000;
Pop and Ingham, 2001; Bejan et al., 2004).
Porous media and transport are becoming increasingly important in heat exchanger
analysis and design. It was pointed out in Bejan (1995) that the gradual miniatur-
izationofheattransferdevicesleadsthe?owtowardlowerReynoldsnumbersand
brings the designer into a domain where dimensions are considerably smaller and
structures considerably more complex than those covered by the single-con?guration
correlations developed historically for large-scale heat exchangers. The race toward
smallscalesandlargeheat?uxesinthecoolingofelectronicdevicesisthestrongest
manifestation of this trend. It is fair to say that the reformulation of heat exchanger
analysisanddesignasthebasisofporousmedium?owprinciplesisthenextareaof
growth in heat exchanger theory for small-scale applications.
Theobjectiveofthischapteristoprovideaconciseandeffectivereviewofsome
ofthe most basic results on heat transf er through porous media. This coverage is
anupdatedandcondensedversionofareviewpresented?rstinBejan(1987).More
detailed and tutorial alternatives were developed subsequently in Bejan (1995, 1999)
and Nield and Bejan (1999), to which the interested reader is directed.
15.2 BASIC PRINCIPLES
Thedescriptionofheatand?uid?owthroughaporousmediumsaturatedwith?uid
(liquidorgas)isbasedonaseriesofspecialconceptsthatarenotfoundinthepure-
?uid heat transfer. Examples are the porosity and the permeability of the porous
medium,andthevolume-averagedpropertiesofthe?uid?owingthroughtheporous
medium.Theporosityoftheporousmediumisde?nedas
f=
void volume contained in porous medium sample
totalvolumeofporousmediumsample
(15.1)
The engineering heat transfer results assembled in this chapter refer primarily to ?uid-
saturated porous media that can be modeled as nondeformable, homogeneous, and
isotropic. In such media, the volumetric porosityf is the same as the area ratio (void
areacontainedinthesamplecrosssection)/(totalareaofthesamplecrosssection).
Representative values are shown in Table 15.1.
Thephenomenonofconvectionthroughtheporousmediumisdescribedinterms
ofvolume-averaged quantities such as temperature, pressure, concentration, and
velocity components. Each volume-averaged quantity (?) is de?ned through the
operation
?=
1
V

v
?dV (15.2)
BOOKCOMP, Inc. — John Wiley & Sons / Page 1133 / 2nd Proofs /HeatTransferHandbook / Bejan
BASIC PRINCIPLES 1133
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[1133], (3)
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———
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Long Page
* PgEnds: Eject
[1133], (3)
TABLE 15.1 PropertiesofCommonPorous Materials
Porosity, Permeability, Surface per unit
Material f K(cm
2
) Volume (cm
-1
)
Agar–agar — 2× 10
-10
–4.4× 10
-9
Black slate powder 0.57–0.66 4.9× 10
-10
–1.2× 10
-9
7× 10
3
–8.9× 10
3
Brick 0.12–0.34 4.8× 10
-11
–2.2× 10
-9
Catalyst (Fischer–Tropsch,
granules only)
0.45 — 5.6× 10
5
Cigarette — 1.1× 10
-5
Cigarette ?lters 0.17–0.49
Coal 0.02–0.12
Concrete (bituminous) — 1× 10
-9
–2.3× 10
-7
Concrete (ordinary mixes) 0.02–0.07
Copper powder
(hot-compacted)
0.09–0.34 3.3× 10
-6
–1.5× 10
-5
Corkboard — 2.4× 10
-7
–5.1× 10
-7
Fiberglass 0.88–0.93 — 560–770
Granular crushed rock 0.45
Hair (on mammals) 0.95–0.99
Hair felt — 8.3× 10
-6
–1.2× 10
-5
Leather 0.56–0.59 9.5× 10
-10
–1.2× 10
-9
1.2× 10
4
–1.6× 10
4
Limestone (dolomite) 0.04–0.10 2× 10
-11
–4.5× 10
-10
Sand 0.37–0.50 2× 10
-7
–1.8× 10
-6
150–220
Sandstone (“oil sand”) 0.08–0.38 5× 10
-12
–3× 10
-8
Silica grains 0.65
Silica powder 0.37–0.49 1.3× 10
-10
–5.1× 10
-10
6.8× 10
3
–8.9× 10
3
Soil 0.43–0.54 2.9× 10
-9
–1.4× 10
-7
Spherical packings
(well shaken)
0.36–0.43
Wire crimps 0.68–0.76 3.8× 10
-5
–1× 10
-4
29–40
Source: Data from Nield and Bejan (1999), Scheidegger (1957), and Bejan and Lage (1991).
where ?istheactualvalueofthequantityatapointinsidethesamplevolume V .
Alternatively,thevolume-averagedquantityequalsthevalueofthatquantityaveraged
over the total volume occupied by the porous medium. The volume sample is called
representativeelementaryvolume(REV).ThelengthscaleoftheREVismuchlarger
thantheporescalebutconsiderablysmallerthanthelengthscaleofthemacroscopic
?ow domain.
15.2.1 Mass Conservation
The principle ofmass conservation or mass continuity applied locally in a small
regionofthe?uid-saturatedporousmediumis
D?
Dt
+??·v= 0 (15.3)
BOOKCOMP, Inc. — John Wiley & Sons / Page 1134 / 2nd Proofs /HeatTransferHandbook / Bejan
1134 POROUS MEDIA
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T
w
T
w
T
w
uT ,

uT ,

uT ,

u
uT ,

ug ,
x
wg ,
z
v, g
y
r
0
() e
() c
() a
() f
() d
() b
q
q
q
y
y

yr or
r
y
x
x
x
x
x
z
Heated Insulated
Tx,y ()
Tx,r ()
D

q
Figure 15.1 Con?gurations for forced-convection heat transfer: (a) Cartesian coordinate
system; (b) boundary layer development over a ?at surface in a porous medium; (c) boundary
layer development around a cylinder or sphere embedded in a porous medium; (d) point heat
source in a porous medium; (e) horizontal line source in a porous medium; (f ) duct ?lled with
porous medium.
where D/Dt is the material derivative operator:
D
Dt
=
?
?t
+ u
?
?x
+ v
?
?y
+ w
?
?z
(15.4)
and where v (u, v, w) is the volume-averaged velocity vector (Fig. 15.1a). For ex-
ample, the volume-averaged velocity component u in the x direction is equal to
Page 5


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CHAPTER 15
Porous Media
ADRIAN BEJAN
Department of Mechanical Engineering and Materials Science
Duke University
Durham,NorthCarolina
15.1 Introduction
15.2 Basic principles
15.2.1 Mass conservation
15.2.2 Flow models
15.2.3 Energy conservation
15.3 Conduction
15.4 Forced convection
15.4.1 Plane wall with constant temperature
15.4.2 Sphere and cylinder
15.4.3 Concentrated heat sources
15.4.4 Channels ?lled with porous media
15.4.5 Compact heat exchangers as porous media
15.5 External natural convection
15.5.1 Vertical walls
15.5.2 Horizontal walls
15.5.3 Sphere and horizontal cylinder
15.5.4 Concentrated heat sources
15.6 Internal natural convection
15.6.1 Enclosures heated from the side
15.6.2 Cylindrical and spherical enclosures
15.6.3 Enclosures heated from below
15.6.4 Penetrative convection
15.7 Other con?gurations
Nomenclature
References
15.1 INTRODUCTION
Heat and mass transfer through saturated porous media is an important development
andanareaofveryrapidgrowthincontemporaryheattransferresearch.Although
the mechanics of?uid ?ow through porous media has preoccupied engineers and
physicists for more than a century, the study of heat transfer has reached the status
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ofaseparate?eldofresearchduringthelastthreedecades(NieldandBejan,1999).
It has also become an established topic in heat transfer education, where it became a
partoftheconvectioncoursein1984(Bejan,1984,1995).Thereaderisdirectedto
agrowingnumberofmonographsthatoutlinethefundamentals(Scheiddeger,1957;
Bear, 1972; Bejan, 1987; Kaviany, 1995; Ingham and Pop, 1998, 2002; Vafai, 2000;
Pop and Ingham, 2001; Bejan et al., 2004).
Porous media and transport are becoming increasingly important in heat exchanger
analysis and design. It was pointed out in Bejan (1995) that the gradual miniatur-
izationofheattransferdevicesleadsthe?owtowardlowerReynoldsnumbersand
brings the designer into a domain where dimensions are considerably smaller and
structures considerably more complex than those covered by the single-con?guration
correlations developed historically for large-scale heat exchangers. The race toward
smallscalesandlargeheat?uxesinthecoolingofelectronicdevicesisthestrongest
manifestation of this trend. It is fair to say that the reformulation of heat exchanger
analysisanddesignasthebasisofporousmedium?owprinciplesisthenextareaof
growth in heat exchanger theory for small-scale applications.
Theobjectiveofthischapteristoprovideaconciseandeffectivereviewofsome
ofthe most basic results on heat transf er through porous media. This coverage is
anupdatedandcondensedversionofareviewpresented?rstinBejan(1987).More
detailed and tutorial alternatives were developed subsequently in Bejan (1995, 1999)
and Nield and Bejan (1999), to which the interested reader is directed.
15.2 BASIC PRINCIPLES
Thedescriptionofheatand?uid?owthroughaporousmediumsaturatedwith?uid
(liquidorgas)isbasedonaseriesofspecialconceptsthatarenotfoundinthepure-
?uid heat transfer. Examples are the porosity and the permeability of the porous
medium,andthevolume-averagedpropertiesofthe?uid?owingthroughtheporous
medium.Theporosityoftheporousmediumisde?nedas
f=
void volume contained in porous medium sample
totalvolumeofporousmediumsample
(15.1)
The engineering heat transfer results assembled in this chapter refer primarily to ?uid-
saturated porous media that can be modeled as nondeformable, homogeneous, and
isotropic. In such media, the volumetric porosityf is the same as the area ratio (void
areacontainedinthesamplecrosssection)/(totalareaofthesamplecrosssection).
Representative values are shown in Table 15.1.
Thephenomenonofconvectionthroughtheporousmediumisdescribedinterms
ofvolume-averaged quantities such as temperature, pressure, concentration, and
velocity components. Each volume-averaged quantity (?) is de?ned through the
operation
?=
1
V

v
?dV (15.2)
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BASIC PRINCIPLES 1133
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TABLE 15.1 PropertiesofCommonPorous Materials
Porosity, Permeability, Surface per unit
Material f K(cm
2
) Volume (cm
-1
)
Agar–agar — 2× 10
-10
–4.4× 10
-9
Black slate powder 0.57–0.66 4.9× 10
-10
–1.2× 10
-9
7× 10
3
–8.9× 10
3
Brick 0.12–0.34 4.8× 10
-11
–2.2× 10
-9
Catalyst (Fischer–Tropsch,
granules only)
0.45 — 5.6× 10
5
Cigarette — 1.1× 10
-5
Cigarette ?lters 0.17–0.49
Coal 0.02–0.12
Concrete (bituminous) — 1× 10
-9
–2.3× 10
-7
Concrete (ordinary mixes) 0.02–0.07
Copper powder
(hot-compacted)
0.09–0.34 3.3× 10
-6
–1.5× 10
-5
Corkboard — 2.4× 10
-7
–5.1× 10
-7
Fiberglass 0.88–0.93 — 560–770
Granular crushed rock 0.45
Hair (on mammals) 0.95–0.99
Hair felt — 8.3× 10
-6
–1.2× 10
-5
Leather 0.56–0.59 9.5× 10
-10
–1.2× 10
-9
1.2× 10
4
–1.6× 10
4
Limestone (dolomite) 0.04–0.10 2× 10
-11
–4.5× 10
-10
Sand 0.37–0.50 2× 10
-7
–1.8× 10
-6
150–220
Sandstone (“oil sand”) 0.08–0.38 5× 10
-12
–3× 10
-8
Silica grains 0.65
Silica powder 0.37–0.49 1.3× 10
-10
–5.1× 10
-10
6.8× 10
3
–8.9× 10
3
Soil 0.43–0.54 2.9× 10
-9
–1.4× 10
-7
Spherical packings
(well shaken)
0.36–0.43
Wire crimps 0.68–0.76 3.8× 10
-5
–1× 10
-4
29–40
Source: Data from Nield and Bejan (1999), Scheidegger (1957), and Bejan and Lage (1991).
where ?istheactualvalueofthequantityatapointinsidethesamplevolume V .
Alternatively,thevolume-averagedquantityequalsthevalueofthatquantityaveraged
over the total volume occupied by the porous medium. The volume sample is called
representativeelementaryvolume(REV).ThelengthscaleoftheREVismuchlarger
thantheporescalebutconsiderablysmallerthanthelengthscaleofthemacroscopic
?ow domain.
15.2.1 Mass Conservation
The principle ofmass conservation or mass continuity applied locally in a small
regionofthe?uid-saturatedporousmediumis
D?
Dt
+??·v= 0 (15.3)
BOOKCOMP, Inc. — John Wiley & Sons / Page 1134 / 2nd Proofs /HeatTransferHandbook / Bejan
1134 POROUS MEDIA
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T
w
T
w
T
w
uT ,

uT ,

uT ,

u
uT ,

ug ,
x
wg ,
z
v, g
y
r
0
() e
() c
() a
() f
() d
() b
q
q
q
y
y

yr or
r
y
x
x
x
x
x
z
Heated Insulated
Tx,y ()
Tx,r ()
D

q
Figure 15.1 Con?gurations for forced-convection heat transfer: (a) Cartesian coordinate
system; (b) boundary layer development over a ?at surface in a porous medium; (c) boundary
layer development around a cylinder or sphere embedded in a porous medium; (d) point heat
source in a porous medium; (e) horizontal line source in a porous medium; (f ) duct ?lled with
porous medium.
where D/Dt is the material derivative operator:
D
Dt
=
?
?t
+ u
?
?x
+ v
?
?y
+ w
?
?z
(15.4)
and where v (u, v, w) is the volume-averaged velocity vector (Fig. 15.1a). For ex-
ample, the volume-averaged velocity component u in the x direction is equal to
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BASIC PRINCIPLES 1135
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fu
p
, where u
p
is the average velocity through the pores. In many single-phase ?ows
through porous media, the density variations are small enough so that the D?/Dt
term may be neglected in eq. (15.3). The incompressible ?ow model has been in-
vokedinthedevelopmentofthemajorityoftheanalyticalandnumericalresultsre-
viewed in this section. The incompressible ?ow model should not be confused with
the incompressible-substance model encountered in thermodynamics (Bejan, 1997).
15.2.2 Flow Models
The most frequently used model for volume-averaged ?ow through a porous medium
is the Darcy ?ow model (Nield and Bejan, 1999; Bejan, 1995). According to this
model, the volume-averaged velocity in a certain direction is directly proportional to
the net pressure gradient in that direction,
u=
K
µ 
-
?P
?x

(15.5)
Inthreedimensionsandinthepresenceofabodyaccelerationvector g= (g
x
,g
y
,g
z
)
(Fig. 15.1a), the Darcy ?ow model is
v=
K
µ (-?P +?g) (15.6)
The proportionality factor K in Darcy’s model is the permeability ofthe porous
medium. The units of K are m
2
. In general, the permeability is an empirical constant
that can be determined by measuring the pressure drop and the ?ow rate through a
column-shapedsampleofporousmaterial,assuggestedbyeq.(15.5).Theperme-
ability can also be estimated from simpli?ed models of the labyrinth formed by the
interconnectedpores.Modelingtheporesasabundleofparallelcapillarytubesof
radius r
0
yields (Bejan, 1995)
K =
pr
4
0
8
N
A
(15.7)
where N isthenumberoftubescountedonacrosssectionofarea A. Modeling the
poresasastackofparallelcapillary?ssuresofwidth B and ?ssure-to-?ssure spacing
a+ b yields the permeability formula (Bejan, 1995)
K =
b
3
12(a+b)
(15.8)
Modelingtheporousmediumasacollectionofsolidspheresofdiameter d, Kozeny
obtained the relationship (Nield and Bejan, 1999)
K ~
d
2
f
3
(1-f)
2
(15.9)
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