57

HEAT CONDUCTION

PING CHENG and CHIN-TSAU HSU

Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

INTRODUCTION

In this chapter, we discuss conduction heat transfer in a porous medium. For steady heat conduction under local thermal equilibrium conditions it will be shown that the governing equation for heat conduction in a porous medium can be rewritten in a form similar to that of the classical heat conduction equation with an effective stagnant thermal conductivity consisting of two components: the first component represents the volumetric averaging of the thermal conductivities of the solid and the fluid phases while the second component represents the tortuosity effect due to the undulating thermal path across the fluid-solid interface. Exact numerical evaluations of the thermal conductivities of some idealized two- dimensional, spatially periodic media will be obtained based on closure modeling, or solving the two-dimensional conjugate heat conduction problem in the fluid and solid phase of a unit cell. Various analytical models for the calculation of the stagnant thermal conductivity will be reviewed.

GOVERNING EQUATIONS

The microscopic energy equations for the fluid and the solid phases under the no flow condition are

~ = V" (kfVTf) pfCpf ~t (la)

Ps C r = V.(k~VTs ) (lb) P~ o~t

with the boundary conditions at the fluid-solid interface being given by

Tf=T, on Afs (2)

58

nfs. k f V W e = nfs. k svT~ on Afs (3)

If the volumetric averaging process is applied to equation (1), the phase averaged equations are, see Nozad et al. [ 1]

(kf ) pfC pf a(r -V. [k- f V(r -I- V �9 7IIWfAfs as -[-Q fs (4)

a [(1-~)T~ ] : ~ . {ks ~[(1_ ~ ) ~ ]}_ ~ . / _ ~ !! Tsds) _ Qfs (5) PsCps 0t

m where T f and Ts are the volumetric temperatures, r is the porosity, V is the representative elementary volume, ds is the outward normal vector of the surface element from the fluid into the solid, i.e. ds = nfsdS where s is the interphase surface area, and Qf~ is the

interphase heat transfer per unit volume which is a source term in equation (4) but becomes a sink term in equation (5).

LOCAL THERMAL EQUILIBRIUM

We now assume that the local thermal equilibrium condition prevails such that

Tf -- Ts = T (6)

With this assumption, and assuming that the porosity is constant, equations (4) and (5) can be added to give

m T d s

s (7)

where

(pEp)m -- CpfC pf -~- (1- ~)ps C ps (8)

and

k m = ~)kf + (1 - r (9)

are the volumetric averaged thermal capacity and conductivity, respectively. At steady state condition, equation (7) reduces to

59

V" (km'V--'Y)+ V" [ '(kf V -ks) (10)

The area integrals in the above equations represent the tortuosity effect due to the undulating thermal path across the fluid-solid interface.

CLOSURE MODELING FOR STEADY HEAT CONDUCTION

In this section, we shall discuss the closures for equation (10). The microscopic energy equations for steady heat conduction in the fluid and solid phases are

V" (kf VTf ) = 0 in the fluid (11)

and

V. (ksVT S ) - 0 in the solid (12)

with the boundary conditions also given by equations (2) and (3). We now decompose T r and T s into

t

Z - Tf + Tf (13a)

= T~ + T' (13b)

t where T r and T" are the temperature deviations from the volumetric temperatures of the

fluid and the solid phases. Substituting equation (13) into equations (11)-(12) and boundary conditions (2)-(3) yields

V. (kfVTf') = ~ ., - --~--

, 1 ~ . (_~ i iT ,d s )_ ~ V" (ksVTs)=- 1---~ A~,

(14a)

Qfs (14b) 1 - ~

subject to the boundary conditions

T f t - y' ., on Afs (15)

-fs. k f V T / - . f . g s V T " +. fs . (ks - k f)VT on Afs (16)

60

In the above derivation, we have invoked the local thermal equilibrium condition and the constant porosity assumptions. The boundary condition (16) suggests that the deviation of the temperature from the phase averaged value is caused by the macroscopic thermal gradient incorporated with the difference in the thermal conductivities of the solid and the fluid phases. Therefore, it is plausible to assume that

- - N

Tt: = f . VT (17a)

m _ _

T" = g- VT (17b)

where f and g are vector functions in the microscopic coordinates. It can be shown that the right hand side of equation (14) is smaller than the left hand side. Thus, equation (14) reduces to

V- (kfVTe') = 0 (18a)

T p V. (k~V ~)=0 (18b)

Substituting equation (17) into equation (18) and equations (15)-(16) leads to

V2f = 0 (19a)

V 2g = 0 (19b)

subject to the boundary conditions

f = g on Afs (20)

nf.,. V f = Ant. ~ �9 Vg + ( A - 1)nfs . I on Afs (21)

where A - k / k f and I is a unit tensor. With the aid of equation (17a), the tortuosity

integral can be evaluated as

-~-~ T ' d s - CJ �9 VT (22a) At~

where

G = __1 ~ f nf, ds (22b) - - - V At.,

61

Thus, equation (10) can be written as

where the effective stagnant thermal conductivity S e is a tensor given by

K k, = [~) + ( 1 - (~)A]I + ( 1 - A)G (24)

which is also the expression obtained by Nozad et al.[ 1]. For isotropic media, G - G I

such that

K = k I (25a) ~ e e . . _

where

k~ = [(I) + (1-(I))A] + ( 1 - A ) 6 (25b)

is the scalar effective stagnant thermal conductivity.

THE EFFECTIVE STAGNANT THERMAL CONDUCTIVITY

It should be noted that equation (23) with the effective conductivity K e given by equation

(25) is the standard steady heat conduction equation for isotropic media, provided that the stagnant thermal conductivity ke can be measured or evaluated. It will be shown that the

magnitude of the effective stagnant thermal conductivity of a porous medium depends not only the porosity, the thermal conductivities of the solid and the fluid phases, but also on whether the particles are in good contact, point contact or no contact.

Experimental Investigations

A large number of experiments have been carried out to measure stagnant thermal conductivities of porous media saturated with fluids. A review of the literature on the methods used for the determination of the stagnant thermal conductivities is given by Tsotsas and Martin [2]. A compilation of early experimental data by a number of investigators [3-11] was given by Kunii and Smith [12], Krupiczka [13], Zehner and Schlunder [14], as well as by Crane and Vachon [15]. Most of these measurements were carried out for materials with solid/fluid thermal conductivity ratio in the range of 1 < k S / kf < 103 under atmospheric conditions. Thermal conductivities of porous materials

62

with higher values of k s/kf were obtained by Swift [16] and Nozad et al. [1] while those

with lower values of k ] kf were obtained by Prasad et al. [ 17].

Numerical Solutions

Nozad et al. [1] considered the stagnant thermal conductivity of arrays of touching square cylinders shown in Figure 1, where c/a is the touching parameter and the non-touching square cylinders is obtained by setting this parameter to zero. After numerical solutions of equations (19), subject to the boundary conditions (20) and (21), for the touching and non- touching square cylinders were obtained, Nozad et al. [1] determined the thermal conductivities of these geometries from equations (24) and (22b). Nozad et al. [1 ] found that their model with a touching parameter of 0.01 matched with existing experimental data for packed-sphere beds.

~ m

a

Figure 1 Arrays of touching square cylinders

Another approach to obtain the stagnant thermal conductivity is to solve numerically the multi-dimensional conjugate heat conduction problem in the fluid phase and in the solid phase in a unit cell. After the temperature distribution and the total heat flux have been computed, the effective thermal conductivity is determined based on the total heat flux through the unit cell. Deissler and Boegli [ 11] obtained a finite-difference solution for a cubic packing of spheres (q~ = 0.476). Similar numerical solutions were obtained for an orthorhombic structure (~) = 0.259) by Wakao and Kato [ 18], as well as by Wakao and Vortmeyer [19], assuming that both point contacts and finite area contacts have a contact radius r,. Wakao and Vortmeyer [19] found that the dimensionless contact

63

( )2 area, defmed as D = rs/d p where dp denotes the diameter of the solid particles, has an

important effect on the stagnant thermal conductivity when the solid/fluid conductivity is high.

Analytical Models

We shall now discuss the analytical models of the stagnant thermal conductivities of porous media consisting of particles with no contact, point contact and good contact.

Maxwell's Model and Its Extensions Assuming a sufficiently dilute dispersion of spheres, Maxwell [20] obtained the following expression for the stagnant thermal conductivity of a packed-sphere bed �9

k 2r + A ( 3 - 2r = (26)

kf 3 - ~ + A~

Equation (26), valid in the limit of the porosity approaches unity (~--~ 1), provides a

lower bound for the stagnant thermal conductivity of a packed-sphere bed. Subsequently, numerous attempts have been made to extend Maxwell's solution to higher order in (1-t~). A review of these early non-touching models has been given by Churchill [21 ].

Composite-Layer Models The simplest model for the calculation of the effective stagnant thermal conductivity is to assume that the porous medium consists of a fluid layer and a solid layer, either in parallel or in series with respect to the temperature gradient. As pointed out by Deisser and Boegli [11], these two arrangements represent the maximum and the minimum values of the effective stagnant thermal conductivity of a porous medium.

If the fluid and the solid layer are in parallel with respect to the thermal gradient, it can be shown that the effective stagnant thermal conductivity is

o r

k~ = kfr + k ~ ( 1 - r (27a)

ke/kf = r + ( 1 - (l))ks/kf (27b)

It should be noted that the effective stagnant thermal conductivity, given by equation (27), is identical to the first term of equation (24). Thus, under the parallel-layer model, the tortuosity effect is presumably neglected. The fluid and the solid layers are perpendicular to the temperature gradient, the effective stagnant thermal conductivity is

64

1 ~ (1 -~ ) = I (28a)

k k e k

or

k,/kf (28b) k , /k f (I- O) + (~k,/kf

Equations (27) and (28) are presented in Figure 2 as a function of the thermal conductivity ratio. It is seen that the values computed from equations (27) and (28) differ slightly if k s

is almost equal to kf and they differ substantially if k s is considerably different from kf.

105

10 4

10 3

ke/k f lO 2

10

1

layers in parallel layers in series

10-1 10 .2 10 "1 1 10 10 2 10 3 104 105

~=0 0.2

I 0.4 0.6 0.8

~= 0.2 0.4 0.6 0.8 1.0

ks/kf

Figure 2 Maximum and minimum values of the effective stagnant thermal conductivity

Kunii and Smith's Model Kunii and Smith [12] obtained the following approximate expression for the stagnant thermal conductivity of a packed-sphere bed:

k (1 -~ )

kf ~ + _ ~ (29)

65

where )~ = kf/ks is the fluid/solid conductivity ratio and r = r + (~) - 0.259)(r r is a linear interpolation between e 1 and ~2, which are the values of r corresponding to a

cubic packing of uniform spheres (# = 0.476) and to a tetrahedral packing of uniform

spheres (q~ = 0.259), respectively. The values of r and, r as a function of the solid/fluid thermal conductivity ratio is presented graphically by Kunii and Smith [ 12].

Zehner and Schlunder's Model An analytical expression for the effective stagnant thermal conductivity of a porous medium has been obtained by Zehner and Schlunder [14] who considered 1/8 of the cylinder, see Figure 3, instead of a cube as a unit cell. The unit cell consists of an outer cylinder with radius R and an inner cylinder with unit radius. Fluid is filled between the inner and outer cylinder while the inner cylinder consisting of both the solid and fluid phase with its interface Afs described by

2 z r 2 + = 1 (30)

[B - ( B - 1)z] z

where B is the shape factor to parametrize the geometrical effect of the solid particles. As shown in Figure 3, equation (30) assumes that the particles are in point contact.

!

1 - ( 1 z=0

1

i 'i I I I I

I I ~ z = i

~ k. t

z

Figure 3 Unit cell used by Zehner and Schlunder [14]

66

Based on this model, Zehner and Schlunder [ 14] obtained the following expression for the effective thermal conductivity of a packed bed:

2~/1-(DI(1-~,)B 1 B+I B-1 1 kc = 1 - ~ / 1 - r + ~ I n - - ~ (31) k e 1 - ZB ( 1 - ~B) 2 ZB 2 1 - kB

where the value of B is approximated by

B=c/X 0 (32)

with m = 10/9. The value of C depends on the shape of the particles and Zehner and Schlunder [14] suggested that C = 1.25, 1.4 and 2.5 for spheres, broken particles and cylinders, respectively. The stagnant thermal conductivity given by equations (31) and (32), as a function of ks/kf, for different values of porosity are presented in Figure 4. For comparison, the experimental data obtained by Nozad et al. [ 1 ] are also plotted in the same graph. It is noted from this graph that the Zehner-Schlunder model [14] with r = 0.4 is in good agreement with the experimental data for kJkf < 103. For higher values of the thermal conductivity ratio (k s/kf > 103), the Zehner-Schlunder model underpredicts the

stagnant thermal conductivity, see Kaviany [22].

~=o.o lO 4

lO 3

102

ke/kf

lO

experiments

Zehner & Schlunder's model

10.1 /

10 .2 10 1 1 10 102 103 104 105

ks/kf

0.1

0.38 0.40 0.42

0.8

1.0

Figure 4 Comparison of Zehner and Schlunder' s model with experimental data of Nozad et al. [ 1]

67

Area-Contact Model Since finite area contacts between spheres are likely to occur in a packed-sphere bed, Hsu et al. [23] postulated that the reason for the Zehner- Schlunder model underpredicting the stagnant thermal conductivity of a packed-sphere bed at high solid/fluid thermal conductivity ratios is due to the fact that their model assumes point contacts between spheres. To take into consideration the finite area contacts between spheres, Hsu et al. [23] modified the solid/fluid interface A fs by the following equation

2 Z

r z + = 1 (33) [(1 + a)B - ( B - 1)z] 2

where It is the deformation factor. The truncated sphere is indicated as the shaded area in Figure 5 where r s is the radius of the contact area which can be expressed in terms of o~

and B by imposing the condition: r = r s, z = 1. It follows from equation (33) that

1 r~ = 1 - (34)

!

,

1

I i glllllllllllllll, I i gllllllll//lll/i 1

,,

/ z = 0

R

/ z = O

Figure 5 Unit cell for the area-contact model

Based on the modification of Zehner and Schlunder's approach, Hsu et al. [23] obtained the following expression for the effective stagnant thermal conductivity of a packed sphere bed with finite contact areas:

68

ke = (1_ 1/1_ ~) + kf

"[ ] 41 ~ 1_ 1 (I+ o~B) 2

2 4 1 - r { (1 - xXI+ (z)B

[1- XB + (1- X)r [1- XB + (1 - X)aB] 2 In

B + 1 + 2~B _ (B- 1) } 2(1+ aB) 2 [1- XB + (1- X)aB](I+ aB)

l + a B (1+ a)BX

(35)

It should be noted that equation (35), with O~ = 0, reduces to equation (31) given by Zehner and Schlunder [14]. The value of B in equation (35) is also approximated by equation (32) with C and m being a function of 0~ as shown in Figure 6.

1.6

1.4

1.2

C 1.o

m.8 .6

.4

.2

m c

! i ! !

0 .2 .4 .6 .8 1.0

(Z

Figure 6 Variations of C and m with the deformation factor 0~

Computations were carried out for equation (35) with various values of o~ and (>

and it was found that equation (35), with (z = 0.002 and (~ = 0.42, matched well with the experimental data of Nozad et al. [ 1 ]. Results based on equation (35), with o~ = 0.002 and (~ = 0.42, are compared with the experimental data of Nozad et al. [1] in Figure 7, and the

69

computed results based on the two-dimensional model of Nozad et al. [1] are also presented in the same plot for comparison. It is shown that results based on equation (35)

are slightly lower than Nozad et al.'s model for ks/kf > 103 .

k e / k f

104

�9 experiments area-contact model (I)= 0.42 & (z =0.002

103 Hsu et al.'s 3D model c/a=0.13 ./. . . . . Zehner & Schlunder's model , , F / . . . . . . . Nozad's et al.'s numerical solution with c/a=0.01 .,F"~//

102 oO~

. . ~

10

I

10-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 -2 10 -1 1 10 102 10 3 104 105

k s / k f

Figure 7 Comparison of the area-contact model with other theoretical models and experimental data, see Hsu et al. [24]

Lumped Parameter Models Hsu et al. [24] proposed a lumped parameter model to obtain simple expressions for the effective stagnant thermal conductivities spatially periodic media. The essence of the method is to select a unit cell and assume one-dimensional heat conduction in the cell. Next divide the unit cell into layers in parallel to the heat flow direction and these layers may consist of fluid phase, solid phase or composite phase. The equivalent thermal resistance in the composite layer is obtained by assuming that the thermal resistances in the solid and the fluid phases are in series. A number of geometries, which had been considered previously, were obtained based on the lumped parameter method, and it is found that the results based on the lumped parameter models are in excellent agreement with existing numerical solutions. A review of the lumped parameter model has been given recently by Cheng and Hsu [25].

Square Cylinders Consider the stagnant thermal conductivity of an array of touching square cylinders as shown in Figure 1. Based on the lumped parameter method,

70

Hsu et al. [24] obtained the following expression for the effective stagnant thermal conductivity in the direction perpendicular to the cylinders"

k..._L.e = ~ta~tc ~(a(1-- ~tc) (1-- ~ta) + + (36)

kf ~, 1+(~ , - 1)]t a I + ( X - 1)7a7 c

where ~a = a / ~ and Yc = c/a . Equation (36) is presented in Figure 8 for ~= 0.36 and

for different values of the touching parameter c/a. The results show that the effect of the touching parameter c/a is important when the solid/fluid thermal conductivity ratio is high. For comparison, the experimental data of Nozad et al. [ 1 ], as well as Prasad et al. [ 17] and the numerical results of Nozad et al. [ 1 ] for c/a = 0.01 are also presented in Figure 8. It is shown that results computed based on the lumped parameter model for c/a = 0.01 are in excellent agreement with the experimental data.

ke/kf

10 4

10 3

10 2

10

0.4

experiments Nozad et al.'s touching model c/a =0.01

10 "1 " , < , ,K,K, , , . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . .

10 .2 1 O 1 1 10 102 103 104 105

0.1

0.02

0.01

0.001

c/a = 0.00

ks/kf

Figure 8 Comparison of stagnant thermal conductivity of touching square cylinders based on the lumped-parameter model with experimental data, Hsu et al. [24]

Since non-touching square cylinders is a special case of touching squares, the thermal conductivity of the former case can be obtained by letting c = 0 and consequently, Yc = 0 and ~fa = 1 -~. Under this condition, equation (36) becomes

71

x / 1 - ~ ke = ( 1 - x / l - ~ ) + (37)

1+ (~ - 1 ) 4 1 -

Equation (37) for r = 0.2, 0.4, and 0.7 are presented in Figure 9. For comparison, the numerical results of Nozad et al. [1] for t~ = 0.1, 0.36, 0.437 and 0.64 are also presented in the same graph. It is shown that they are in excellent agreement.

102

lO

ke/kf

10-~

2D lumped parameter model ........ Nozad's nontouching model

0.1

. . . . . . . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . o12" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

o - ~ 1 7 6 _ . . . . . . . . . . . . .

07

1 , , , i i i l l i i I t | l l l l i i i i i i i l l , i i i , i , 11 i i , i , i i i i i i i t l t t l l t i i i i i i

10 .2 101 1 10 102 103 104 10 S

.._~ 0.36 =0.4

""~0.437

ks/kf

Figure 9 Comparison of stagnant thermal conductivity of non-touching square cylinders based on the lumped-parameter model with numerical results of Nozad et al. [ 1]

Circular Cylinders Based on the lumped parameter method, Hsu et al. [24] obtained simple expressions for the stagnant thermal conductivity of in-line circular cylinders of diameter a with connecting width c, see Figure 10. Figure 11 is a comparison of the numerical results of Nozad et al. [ 1] for touching square cylinders with c/a = 0.01 and those based on the lumped parameter method for touching square and circular cylinders as well as the experimental data of a packed-sphere bed. It is shown that the stagnant thermal conductivity of the circular cylinders are slightly higher than that of square cylinders for k s ] kf > 1 and the behavior reverses itself when k s ] kf < 1. As shown

in Figure 12, similar characteristics are observed for non-touching circular and non- touching square cylinders with ~ = 0.4, 0.5, and 0.6. From Figure 12, we also see that the results from the present model agree excellently with the numerical results of Sabraoui and Kaviany[26].

?2

c

Figure 10 Arrays of touching circular cylinders

ke/kf

104

103

102

10

10-1 10-2

�9 experiments 2d model with square cylinder c/a=0.01

....... 2d model with circular cylinder c/a=0.01 Nozad et al.'s touching model with c/a=0.01 /•../d

�9 ~ , Y ' ~ ' J . I (I:) = 0.36

i~....~. -'~- ~ oO"

1 . * . . . . . . . I . . . . . . . . I . . . . . . . I I i I . . . . . | l i i i i . . . . I i . . . . . . . I . . . . . . . .

10 "1 1 10 10 2 103 104 10 5

ks/kf

Figure 11 Comparison of stagnant thermal conductivity of touching square and touching circular cylinders, Hsu et al. [24]

73

10

ke/k f

T

1

2d model for square cylinder ~ Sabraoui & ....... 2d model for circular cylinder ,,~ Kaviany (1993)

§ in-line square cylinder �9 in-line circular cylinder 0.4

§ +

. . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "k t �9 oO.. . . . . . . . . . . . . . . .

o . . - "

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,~ ~""'~ 4 ~ ' 11. O--0.6

. . . . . . . . . . . . . . . . . . . . . . . i i i . . . . . . . . . . . . . . i i i i

10 102 103 104 105

ks/kf

Figure 12 Stagnant thermal conductivity of non-touching square and circular cylinders based on the lumped-parameter model, see Hsu et al. [24]

Cubes The stagnant thermal conductivity of an array of in-line cubes, with connecting bars shown in Figure 13, is also given by Hsu et al. [24] based on the lumped parameter method. They obtained the following expression for the stagnant thermal conductivity:

__-k _ 1 - y2a_ 2~/a~/c -t" 2~/c~ 2a "t- ,y~/2a + Y2( 1 - a ~'~) + 23',3'~ (1 - 7,) (38) kf )L 1+ ~r 1) 1+ ~gc~a(~-- 1)

For non-touching cubes, where 7c ---> 0 and 7a ---> (1 - r V3 , the above equation reduces

to

( 1 - r 2/3 ke = [ 1 - ( 1 - r 2/3] + (39) kf 1+ (~,- 1X1- , ) V3

Figure 14 is a comparison of experimental data of the thermal conductivity of packed- sphere beds with models based on touching cubes (with c/a = 0.13) and touching square cylinders with two values of the touching parameters (c/a = 0.01 and 0.02). It is shown that the results based on the touching cubes model agree better with experimental data than

74

that of touching square cylinders model. For comparison, the results for non-touching square cylinders and non-touching cubes are also plotted in Figure 14. It is shown that the stagnant thermal conductivity of the three-dimensional non-touching cubes model is higher than that of the two-dimensional model of non-touching square cylinders.

r- . . . . . . - '" i

1

s j j s SJ~"

"" "-- ...................

Figure 13 In-line touching cubes

10 4

103

102

ke/kf

10

10-1

10-2

�9 experiments 0.02

- - - - 3D model c/a = 0.00 / ~ /

- - - - 3D model c/a = 0.13 7 , ~ /

. . . . 2D model, c/a= 0.00, 0.01 & 0.02 . 1

�9 �9 j �9 ul � 9 �9 ~ . . . . . . . . . . . .

~ , ~ , ~ . ~ ~ ~ . . . . . . . .

,=0.36

| j | , , , , , i , i , , i t l , i , i J l , , , , n i , , , , , , , i , , , , , , , , I , j . . . . . . I , i , , , , ,

101 1 10 10 2 103 104 105

ks/kf

Figure 14 Comparison of the theoretical models with the experimental data

75

C O N C L U D I N G R E M A R K S

Conduction heat transfer in porous media has been reviewed in this chapter. Our attention has been confined to steady heat conduction under local thermal equilibrium conditions. Particular attention has been given to the effective stagnant thermal conductivity of a porous medium which is saturated with a single component fluid. Further work is required for transient heat conduction in porous media under non-local thermal equilibrium conditions, and for the thermal conductivity of a porous medium which is saturated with a multi-component fluid.

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