THERMAL CONDUCTIVITIES OF GAS-FILLED POROUS SOLIDS J O H N C.
H A R P E R A N D A H M E D
F.
E L S A H R I G I
Cniuersity of California, Dauis, Calif.
Thermal conductivity measurements were made on samples of highly porous freeze-dried apple, pear, and beef, and a polyurethane foam in the presence of Freon-1 2, COZ, NZ, Ne, He, and Hz over a pressure range of 0.005 mm. of Hg to atmospheric. Electrical analog measurements were made on a simplified model system. In all cases, the effective conductivity varied from a constant value in the low pressure region to a higher constant value at pressures near atmospheric. The difference between the high and low pressure values was equal to the pressure-independent conductivity for Nz, COz, and F-12, and became less than the pure gas conductivity for the lighter gases. A semitheoretical relationship was developed which expresses the effective conductivity as a series-parallel combination of the individual solid and gas conductivities. The gas conductivity was found to vary with pressure inversely as the quantity 1 f C/P, where C i s a constant for a particular gas and porous solid.
HE
effective thermal conductivity of a fluid-filled porous
T solid. a subject of interest in many areas of application, has been given considerable attention. The greatest attention has perhaps been given to solids of moderate or low porosity, such as catalyst pellets or reservoir rocks. Gas-filled solids with porosities ranging up to 90% or higher are involved in studies of insulating materials; porous solids of this type are also encountered in freeze-drying of food products. It is to be expected that these different types of materials will be different in heat conduction behavior and that general conclusions about one type might not apply to another. The results reported here ( 7 ) are concerned primarily with freeze-dried food materials. These were consolidated porous solids with porosities ranging from about 75 to 85%. The true solid-phase thermal conductivities were somewhat above 0.1 B.t.u./’(hr.) (ft.) (” F.), higher than those of the gases filling the pores but relatively low compared to those of rocks or refractory materials. The general physical characteristics of freeze-dried food products are discussed in detail elsewhere (3). T o demonstrate that the results are generally applicable, measurements were also made with a plastic foam and with an electrical analog. Permeability studies with these materials will be reported in a future paper. Two important questions regarding thermal conductivities of gas-filled porous solids are the relationship of the effective conductivity to the individual conductivities of the two phases and the effect of pressure on the thermal conductivity of the gas in the pore space. T o answer these questions, thermal conductivity was measured in four solids, with hydrogen, helium, neon, nitrogen, carbon dioxide, and Freon-12 in the pore spaces and the pressures ranging from 0.005mm. of Hg absolute up to atmospheric. A few measurements were also made with water vapor, to demonstrate that it behaved in a regular manner, so that the results can be applied with confidence to freeze-drying operations. Theory
Effect of Pressure on Gaseous Conduction. It is well known from kinetic theory that the thermal conductivity of an I Present address, Department of Food Science and Technology, University of Ein Shams, Shoubra. Cairo. Egypt.
318
ILEC FUNDAMENTALS
ideal gas is independent of pressure if the mean free path is small compared to the dimensions of the confining space. As the pressure of a gas in a porous solid is reduced, the mean free path increases until it becomes significant compared to the pore diameter, and the so-called “temperature jump” phenomenon is manifested. At still lower pressures, collisions among molecules become insignificant compared to collisions with solid surfaces, and free-molecule conduction prevails. The classical kinetic theory of heat conduction in rarefied gases is discussed in many texts--e.g., by Kennard ( 4 ) . Harper (2) compared the following three approaches to obtain the dependency on pressure of the gaseous thermal conductivity in a porous solid : 1. The standard kinetic theory expression relating thermal conductivity to density and mean free path is used, but an effective mean free path is defined that is based on collisions of molecules with the solid walls as well as with other molecules (6,9 ) . 2. .4n average pore diameter is used as the path length in the rarefied gas conduction relationships as presented by Kennard ( 4 ) . 3. Thermal conductivities are obtained by simple kinetic theory relationships from the results of Pollard and Present (7) on gaseous self-diffusion in capillary tubes.
Each of these approaches leads to an expression of the following form for the dependencyof thermal conductivity on pressure :
The different approaches give somewhat different expressions for the constant, C, but they all agree that it is proportional to the ratio of the standard pressure mean free path to the pore diameter. In addition, it may be expected to depend on the accommodation coefficient, the heat capacity ratio, and possibly the Prandtl number. Furthermore, the second and third methods listed above lead to a value of C about 20% higher in the free-molecule region than in the continuum region, and the third method provides a basis for calculation of the continuous variation of C. Because information is lacking concerning accommodation coefficients and the exact con tribution of the gas to the effective conductivity of a porous solid, a reliable experimental verification of any of these relationships has not been possible. In view of these un-
certainties, the present recommendation is to use Equation 1 with C as an empirically determined constant for a paiticular material and gas over the entire pressure range. The experimental results included here make possible a reasonable estimate of the value of C for materials of the same general type as those studied. Combined Thermal Conductivity, Most previous work on the effective thermal conductivities of porous solids was done with refractory materials, porous rocks, or packed beds of particles, and hence cannot be considered representative of the type of materials studied here. Strong, Bundy, and Bovenkerk (8) investigated the conductivity of layers of glass fiber insulation. Their problem was complicated by the necessity of considering contact resistances, which are not present in the materials discussed here. For the combined conductivity. they merely added the separate conductivities on a volume fractiori basis, a procedure for which they gave no justification. Kessler (.i)investigated thermal conductivities in connection with fxeeze-dried substances. His major conclusions. however, were based on measurements with layers of glass spheres, which cannot be considered to be representative of freeze-dried foods. Furthermore, he was probably incorrect in attributing certain results to the effects of radiation. It has generally been found, by both observation and analysis, that radiaht heat transmission through porous materials a t moderate temperatures is insignificant. A more probable explanation of his results lies in the difference in the contact resistance contqibution with different sphere diameters. -4consolidated, open-pored, porous solid has continuous passages of both solid and gas for conduction of heat. Because the pores are arranged randomly rather than as parallel capillaries, there is also conduction through solid and gas in series. This over-all arrangement can be represented schematically (Figure 1). Experimental measurements in which temperature differences ahd the direction of the temperature gradient were varied showed that heat transfer by convection within the pores is insignificant. With reference to Figure 1, the combined conductivity may be expressed as follows :
series conduction, respectively. Because these three paths are not insulated from each other, there is mutual interaction. For example: the second part of the first term accounts for the contribution to continuous solid conduction of that portion of the solid that actually lies in the series path. This term is proportional to the solid path length, l,, and to an “effectiveness” factor, f 1 . Factor f i will depend on both the ratios of I , to I , and of k , to k,. Similar considerations apply to the gas conduction term of Equation 2 . The series conduction term is multiplied by a factor /a to account for the portion that has been utilized in the continuous solid and gas conduction paths. Equation 2 may be rewritten as:
in cthich y = X, k,, x = k,, k,, and the various coefficients are lumped into four constants. Four relationships are required in order to determine the four constants of Equation 3. One such relationship is given by the fact that the effective conductivity must lie between the two limiting cases of series and parallel conduction, as given in the following expressions. Series : X
Y =
(1
-
y = (1
-
In this expression, the three terms on the right side represent continuous solid conduction, continuous gas conduction, and
+e
(4)
Parallel :
(5)
e) + e x
For both of these equations, the slope a t x = 1 is equal to the porosity, e, and the same restriction must hold for the actual porous material. Another relationship is provided by the obvious requirement that, when k , is equal to k,? y = x = 1. A third condition is obtained from the limiting value of k , when k , is zero. This condition corresponds to continuous solid conduction in the absence of a gas phase and may be determined experimentally by measurements a t a sufficiently high vacuum. Denoting the zero pressure value by k,*, cy
-ks + Lkg
e)x
= k,*/k,
= yo
T h e fourth relationship needed to determine the four constants must be established empirically. Woodside and Messmer (70) used a treatment similar to the above for lowporosity unconsolidated materials, but they were unable to establish such a fourth condition. As discussed below, measurements on all the materials studied herein, including the electrical analog, showed an initial slope of 45’ on a plot of effective conductivity against gas conductivity. In terms of Equation 3, this result may be expressed as (dy/d.r)z=d = 1
There is no obvious theoretical explanation for this result, and for the present it must be taken as an empirical relationship. These four relationships, together with Equation 3, yield the following expressions for the four constants: a = Yo
Figure 1 . Schematic model for thermal conductivity of porous solid VOL. 3
NO. 4
NOVEMBER
1964
319
Model System. T h e freeze-dried apple and pear and the plastic foam all had structures that resembled a network of interconnected spherical bubbles. A simplified model of such a structure is a close-packed arrangement of spheres, with the spheres representing the gas and the interstitial space the solid. This arrangement has a porosity of 0.'4! compared to a n average of about 0.86 for the actual solids. This difference is to be expected. both because of the range of sizes of bubbles in the solids and because the bubbles are fused together, thus reducing the interstitial volume. Electrical analog measurements on a close-packed arrangement of uniform-diameter spheres xvere made as described below
Porosity. Porosities of the materials were determined by a simple gas displacement procedure. Dry nitrogen was introduced into a small chamber to a measured pressure, both with and without the sample. and the volume of nitrogen discharged upon release of pressure was measured. A simple calculation then gave the value of the porosity.
Experimental Procedure
Electrically conducting spherical octants of 2-inch radius were made by molding a mixture of graphite and paraffin, with a small proportion of polybutene as a binder. To represent the condition of complete absence of gas, octants were machined from solid polystyrene. These octants were placed in a cubical box of proper dimensions in which two opposite faces were brass electrodes. T h e residual volume in the box w-as then filled \vith sodium chloride solutions over a range of concentrations, and the over-all resistance was measured with an a.c. bridge at a frequency of 1000 cycles per second. T h e resistivities of the salt solutions were measured in rhe same cell without the octants. and the resistivity of the graphite mixture was taken as that value at \vhich the effective combined resistance was equal to the resistance of the solution alone.
Materials. The samples of freeze-dried beef, apples, and pears Lvere prepared in a conventional tral- freeze-dryer. These materials liere selected as being typical of freezedried food products. T h e beef was cut so that heat flow \vas parallel to the grain. The fruit samples appeared to be isotropic. so there was no preferred direction of cutting. The hygroscopic nature of the dried fruits made it necessary to handle them with rubber gloves in a dry atmosphere. Experience showed that oxidation or other chemical changes that may occur over a period of rime with these materials did not affect thermal conductivities. T h e plastic foam \vas an open-pored polyurethane foam purchased at a local store. Thermal Conductivity, In the thermal conductivity measurements described by Harper ( 2 ) heat f l o ~ vrates were measured \\.ith heat flow transducers. These instruments gave someivhat erratic results. and in the present work, thermal conductivities were obtained with reference to a standard of Lucite plastic. In this arrangement. a sample either ' 2 or 3 , 4 inch thick and 2 inches square \vas placed in contact with a 1-inch-thick Lucite reference standard. .4 thinner piece of Lucite was placed on the other side of the sample, and this sandwich \vas clamped between a warm and a cold plate maintained at constant temperatures by; circulating water streams at approximately 110" and 80' F. T h e constant-temperature plates extended \\-ell beyond the edges of the sample. and the space in this extended portion was filled with plastic foam insulation. Copper-constantan thermocouples (36-gage) were cemented a t the center and near the edge of both Lucite surfaces in contact with the sample and on the opposite face of the reference Lucite. This entire arrangement was placed on a plate under a metal bell cover connected to a vacuum pump. T h e vacuum pump operated continuously. and the pressure was controlled b>- bleeding the desired gas into the system. T h e pressure could be controlled at a constant value from a few microns up to atmospheric. The thermal conductivity of the Lucite standard was measured in a similar arrangement Fvith reference to water and found to have a value of 0.116 B.t.u./'(hr.)(ft.)(' F . ) . T h e conductivities of the samples were calculated using this value together \vith the known thicknesses and the measured temperature differences. T h e temperature readings near the edge of the sample agreed closely with those in the center, indicating negligible error from radial heat flow. .4greement betlieen results \\.hen heating from either the top or bottom sho\\.ed that convection could be neglected. Radiant heat exchange through the Lucite was found to be negligible by measurements of the spectral transmittance with an infrared spectrophotometer over the wave length range of 2 to 16 micmns. I t \vas not possible to cut a single piece of fruit 2 inches square. so the fruit samples Lvere made up of sections of 'I2-inch thickness placed in a balsa wood holder. T h e beef and plastic inch thick. foam sample5 \?ere single pieces 320
l&EC FUNDAMENTALS
Electrical Analog. A repeating unit cell of a close-packed spherical arrangement can be represented by four spherical octants placed at alternate corners of a cube, in which the diagonal of a face of the cube is equal to the diameter of the spheres. As pointed out previously, the spherical portions in this model represent the pore space, and the interstitial volume represents the solid portion of the porous solid.
The results of either electrical or thermal conductivity measurements in such heterogeneous systems can be expressed in terms of dimensionless ratios. so that the actual range of electrical conductivities used could be chosen to meet convenience. Since the range of interest lies in the region in which the resistivity of the octants is less than that of the solution, the question of contact resistance does not arise in these measuremen ts. Although this method is simple in principle, practical difficulties were encountered in that the resistivity of the graphite mixture was unstable, and repeated measurements were necessary for reliable results. .4 solid resistivity of about 200 ohmcm. is convenient for these measurements. and it is recommended that, for future work, other mixtures be sought that are more constant in electrical properties. Results and Discussion
Table I gives thermal conductivity measurements for six different gases in the four porous solids. A few measurements with water vapor showed its behavior to be consistent with that of the other gases. T h e plotted points in Figure 2 are results for the pear sample, which are typical of the others. T h e solid lines in this figure were calculated as described below. For all of the gases in each material, it may be seen that temperature-jump effects are not manifested until pressure is reduced to approximately 0.5 to 0.33 atm., and the thermal conductivity of the gas in the pore spaces a t atmospheric pressure is thus definitely known. .4t some point between 0.01 and 0.1 mm. of Hg, all of the measured conductivities for each sample come to a constant value, which represents conduction by the interconnected solid path in the absence of gas. This constant conductivity is the value of k,* for the sample in question. Figure 3 is a plot for three of the samples, with pressureindependent gas Conductivity as abscissa and effective conductivity as ordinate. T h e zero gas conductivity value corresponds to the low pressure limit as described above, and the remaining effective conductivities are the atmospheric pressure values. T h e points for the apple sample are omitted for clarity, since they lie very close to the pear points. All the
Table 1.
Measured Thermal Conductivities
P in CO 9
F-72 ke
P
k,
0.019 0.053 0.105 0.245 0,510 1.020 2.08 4.8 16.7 44.0 98.0 250 760
12.6 12.6 12.8 13.5 13.8 14.4 15.7 16 7 18.4 19.0 19.2 19.2 19.2
0.022 0.052 0.090 0.18 0.385 0.92 2.00 5.10 13.0 38.0 78.0 232 343
12.7 12.7 13.0 13.4 14.2 15.5 16.6 18.4 20.2 21.8 22.7 23.0 23.0
0.021 0.113 0.23 0.52 1.05 2.15 4.2 16.7 44.0 153 340 760
13.0 13.0 13.2 13.5 14.1 14.8 15.7 16.5 18.0 18.8 19.1 19.1 19.1
0.025 0.044 0.075 0.110 0.185 0.37 0.61 0 90 1.80 2.5 18.5 51 0 98 .O 220 760
13.0 13.1 13.2 13.3 13.7 14.2 15.: 15. I 17.7 18.7 22.5 23.2 23 2 23.2 23.2
0.019 0.055 0.128 0.240 0.570 1,100 2.25 4.80 16.60 38.0 125 30 1 760
21.6 22.1 22.5 23.2 24.2 25.3 26.1 27.0 28.3 28.6 28.9 28.9 28.9
0.021
21.8 22.1 22.5 23 4 24.7 26.3 27,s 28.9 30.0 31.2 32.4 32.4 32.4 32,4
0.050
0.020 0.056 0.120 0.252 0.57 1.21 2.44 4.70 17.0 46.0 120.0 280 760
5.2 5.4 5.8 6.5 7.4 8.4 9.3 10.0 11.6 12.0 12.2 12.3 12.3
0.050 0.105 0.205 0.470 1 .oo 2.48 5,30 10.40 19.50 60.0 140.0 340.0 760
0.021 0.058 0.090 0.155 0,285 0.61 1.10
2.26 4.8 8.4 24.6 68.0 190 330 760
mm. Hg
k8 in B.t.u./(hr.) (ft.) ( ” F.) l o 3 -1‘2 iVe
5.4 5.6 5.7 6.2 7.1 8.4 9.9 11.5 13,l 14.2 15.4 15.6 15.6 15.6 l5,6
P 0.0053
0.0075 0.011 0.0194 0.031 0 057 0.104 0.195 0,380 0.80 1.82 4.8 10.0 31 .O 76.0 225 350 760
ke PEAR 12.6 12.6 12.6 12.6 12 6 12 9 13 3 14 0 l5,O 16.5 18.8 21.2 23.0 26 2 27.5 27.9 27.9 27.9
P
ka
P
k,
0.021 0.054 0.115 0.215 0 50 1 020 2 03 5 10 13.5 43.0 98.0 300 760
12.6 12.9 13.7 14.2 16.1 17.8 20.3 26.2 31.7 37.6 39.7 40.7 40,7
0.005 0.007 0.013 0.019 0.049 0.100 0.216 0.50 0,92 2.15 4.20 7.8 19.0 76.0 200 360 760
12.6 12.6 12.6 12.6 12.8 13.4 14.4 16.6 18.0 22.7 32.8 44.6 62.0 82.8 90.7 92.8 92.8
0,005 0.008 0.012 0.018 0.044 0.091 0.193 0,460 1.030 2.20 3.8 8.8 24.0 78.0 300 760
12.6 12.6 12.6 12.7 12.8 13.2 14.7 17.8 23.0 28.2 38.6 56.4 78.8 101 . o 108.0 108 .O
13.0 13.5 14.2 15.1 16.7 19.0 21.9 26.0 30.0 36.9 38.9 39.6 39.6
0.021 0 03 0.058 0.105 0.43 0.65 1.03 4.2 13.8 23.0 42.0 106 190 320 760
13.0 13.4 14.0
0.006 0.008 0.01 0.03 0.055 0.105 0,203 0.42 0.72 1.15 4.70 12.0 29.2 47.0 108 200 400 760
13.0 13.0 13.0 13.6 14.3 15.5 17.7 20.5 23.5 28.0 48.5 66.0 82.0 91 .O 105 112 115 115
0,007 0,010 0,025 0.042 0.064 0.103 0.160 0.33 0.73 1.03 5.10 11.5 23.5 55 0 110 250 760 0.005 0,007 0.013 0.023 0 051 0.104 0.220 0.410 0.840 1.670 3.10 5.30 11 . o o 20 00 57.0 112.0 250,O 760
BEEF 21.8 21 8 21 8 21.8 22.2 23 0 24 2 25 8 27 8 29 31 6 33 2 35 2 3’ 0 37 75 37.8 37 8 37 8
0.0065 0.0082 0.011 0.026 0.053 0.110 0.192
0,510 1.43 2.56 5.20 7.20 9.30 24 0 49.0 107 36G 760
’
H2
k,
APPLE 13.0 0.018 13.0 0.049 13.0 0.105 13.0 0.215 13.2 0.49 13.4 1.10 13 8 2.1 14 1 4.9 14.9 12.6 16 0 44.0 18 3 100 23 5 230 25 8 760 27,3 28 3 29 0 29.1 29.1
0.005
He
P
0 019 0 052 0 105 0.200 0,510 1 04 2 27 4 90 13 0 44 0 98 0 145 310 760
PLASTIC FOAM 5 4 o 027 5 4 0 072 5 4 0 135 5 4 0 2’5 5 6 0 50 6 2 0 95 6 9 2 28 8 9 4 9 12 1 5 2 13 9 9 5 15 8 15 5 16 7 28 0 17 2 84 0 19 3 200 20 3 380 20 4 760 20 4 20 4
21.7 22.3 23.2 24.7 27.7 31 1 34.0 36 4 41 . O 45.9 48.0 48,6 49.0 49.0
5.4 5.8 6.5 8.2 9.9 12.3 16.5 20.4 19.7 23.4 25.5 28.1 31.6 31.9 31.9 31.9
VOL.
0,005 0.007 0.011
15.0 18.5 21 . o 23.0 38.0 56.0 65.0 75.0 91 97 98 98
0.019 0.053 0.106 0.210 0.49 0.95 2.28 5.00 10.0 19.8 61 0 142.0 340 760
21.6 21.7 21.8 21.6 22.3 23.7 25.8 29.2 35.6 45.9 57.1 66.1 74.5 90.0 97.6 97.6 97.6
11.5 23.5 53.0 125.0 260 760
0.006 0.008 0.020 0.039 0.086 0.163 0.42 0.73 1.32 5.2 9.0 20.0 51 . O 105 143 239 400 760
5.4 5.4 5.4 5.5 6.2 7.2 11.3 16.5 24.5 44.9 53.3 63.8 74.5 77.4 78.3 79.5 79.5 79.5
0.006 0.0)8 0 013 0.020 0.050.094 0.140 0.380 0.68 1, 3 0 5.00 8.80 19,s 36 0 86.0 210 400 760
3
NO. 4
0,005 0 00’ 0.013 0.019 0 046 0.096 0.20; 0.49 0.97 2 10 5.20
21.6 21.8 21.7 21.6 22.4 24 0 27.1 33.1 42.1 53.2 65.1 75 . O 84.5 96.0 109.0 115.5 115,5
NOVEMBER 1 9 6 4
5.4 5.4 5.4 5.4 6.0 6.9 8.4 15 2 23.9 33.8 54,6 64.0 76.8 83 9 89.6 91 . 9 92.9 92.9
321
-
Table II.
Material Pear Apple Beef Foam Spheres a I d u d e s efect
I
k, *
e
ka 0.15 0.15 0.15 0.055
Constants for Porous Splids
P
o(
0.87 0.013 0.087 0.83 0.86 0.013 0.087 0.78 0.76 0.022 0.145 0.58 0,850 0,0054 0.097 0.76 0.74 0.174 0.656 of slight compression of sample in thermal conductivity apparatus.
I
I
I
6 0.50 0.62 0.65 0.59 0.494
r 0.089 0.138 0.27 0.142 0.170
$
0.65
0.62 0.61 0.67 0.67
Obsd. Pore Diam., .Microns 70-200 130-300 30-120 200-600
I i -- 1
0 IO
LL 0. I
? r
009
9 >-
008
t
007 V
3
4
006
V 0
2
ay
005 004
I-
W
003
IV -
k!
002
LL
w
001
GAS T H E R MAL CO ND UCT IV IT Y, B.t.u/ hr.x f t.x
OF:
I _ Figure 4. Logarithmic plot of effective thermal conductivity O0.01 0.I IO 10
of freeze-dried pear
PRESSURE, mm.Hg
Figure 2.
Thermal conductivity results for freeze-dried pear lines calculated from Equations 1 and 3
I
GAS TH E R MA L CON DUCT I VI T Y, BLu/ hr! f t.a O E
Figure 3. Relationship of effective thermal conductivity to gas thermal conductivity 322
l&EC FUNDAMENTAL
curves have an initial slope of 45'. When the gas and solid conductivities are equal, this common value is also the effective Conductivity. Consequently, the intersection of the curve for each material with a 45' line through the origin represents the point a t which the effective, gas, and solid conductivities are equal and yields the value of the solid conductivity. These points of intersection can be determined more accurately on a logarithmic plot, making use of the fact that the slope of the curve a t the intersection must be equal to e on logarithmic as well as linear coordinates. With a knowledge of k,, k,*, and e, the constants appearing in Equation 3 can be calculated from Equation 6. The values of all the constants relating to the different materials are given in Table 11. An attempt was made to check values of k , by measuring the conductivity of samples of pear and beef compressed into cakes by a pressure of 14,000 p.s.i. Atmospheric pressure conductivities with nitrogen and helium, were, respectively, 0.092 and 0.122 for the beef, and 0.087 and 0.108 for the pear. The fact that different values were obtained for the two gases shows that there was still appreciable pore volume. The data are not sufficient to check the values of k , quantitatively, but they show that k , is substantially above the conductivity of helium and is consistent with values given in Table 11. The data for the pear sample are plotted logarithmically in Figure 4. The dashed lines in this figure represent the limiting cases of parallel and series conduction for k, = 0.15, and the
Table 111.
Electrical Analog Results
Solution Resistivity Dioided by Solid Resistiuity
1.23 0.69 0.54 0.46 0.36 0.30 0.22 0.18 0.14 0.11 0.0 a
Solution Resistivity Divided by Effective Resisttvity 1.15 0.78 0 68 0 61 0.52 0 46 0 38 0.35 0.31 0.28 0.174~
Ai,erage of rereral miasurcments with plastic octunts in different
solutions.
solid line is calculated by Equation 3. The data lie much closer to the parallel than to the series limit. The electrical resistivity results are shown in Table 111. The resistivity of the graphite \vas determined from a logarithmic plot as the point a t tvhich the effective resistivity was equal to the solution resistivity. A plot of these data on the same basis as Figure 3 also shows a 45' slope at the left-hand end. The slope of the line at the point corresponding to equal resistivities of both phases has a value of about 0.7, whereas 6 for this case is 0.74. This small discrepancy may well be caused by the unstable nature of the graphite mixture, discussed above, and verification of these results with a more suitable solid conductor would be of interest. There is no obvious theoretical basis for this initial slope of 45': and the data are not precise enough to show whether this relationship is exact or merely an approximation. In view of the rather considerable differences in structure represented by the five sets of measurements, it would seem safe to conclude that the relationship is valid for consolidated porous structures hvith porosities above about 0.7. No other data found in the literature were suitable to use for further verification. The data of LVoodside and Messmer (70) for unconsolidated materials of low porosity do not show this constant slope. An attempt was made, but without success, to obtain an analytical expression for k e f for the spherical model system. The experimental results may be represented approximately by the expression k,*/k, =
(1 - e )
The values of i: in Table I1 all lie between 0.6 and 0.7. This variation is actually within the limits of experimental error, for the values of k , and (1 - e ) for the porous materials cannot be considered reliable to more than one place in the second decimal. On the other hand, the result for the model system should have an error of no more than 2y0. The porosity is kno\vn precisely from the geometry, and the value of LY is the average of a number of closely reproducible measurements with accurately machined plastic octants. In the absence of specific experimental measurements, it is recommended that the value of $ for the model sphere system be used to predict the zero-pressure thermal conductivity of porous materials of the t)pe investigated here. The thermal conductivity of the gases in the pore spaces for the different materials can be calculated over the entire pressure range of the experimental data by using Equation 3 with the constants listed in Table 11. This method is not reliable enough to check theoretical relationships for constant C in
Table IV.
Constants for Gases in Porous Materials
k""
Gus F-12
0 0 0 0 0 0
coz
NP
Ne
He
Hz
006 0099 0154 0279 083 106
Pear 2 2 3 6 3 8 5 5 113 118
C . .2lm. Apple 3 1 1 ' 2 3 5 6 106 108
Hg Beef 0 9 1 8 1 8 3 9 6 4 6 9
Foam 1 3 1 4 2 1 4 0 5 4 4 7
Equation 1. Table IV lists values of C selected to give agreement with the experimental data in the central portion of the pressure range. For each of the materials, C correlates reasonably \veil \vith the pressure-independent thermal conductivity of the free gas. Results for the apple, pear, and foam samples also vary inversely with the microscopically observed pore diameters listed in Table 11: as is expected by theory. The fact that the beef sample does not fall into line with the others on the basis of pore diameters may reflect either its different type of structure o r its different porosity. ;Ifore data are needed for a Xvider range of material properties in order to establish any general correlation for the constant C. The lines in Figure 2 are calculated from Equations 1 and 3 and the constants in Tables I1 and IV. Deviations are generally no greater than about 47,, and in most cases appear to be caused by experimental errors rather than a deficiency in the correlation. For example, any failure of the points in Figure 3 to lie on a smooth line is a result of experimental errors and will be reflected in the agreement between calculated and experimental values as shown in Figure 2. The calculated conductivities are still increasing at atmospheric pressure, whereas the experimental data all come to some constant value at lower pressures. The correlation could be improved in this respect if the value of C were allowed to decrease with increase in pressure, as indicated by theory (2). Even though no general correlations are available for predicting the various constants in Equations 1 and 3: effective thermal conductivities Lvith all but the very light gases can be calculated with reasonable engineering accuracy from a knowledge of the upper limit of the effective conductivity (usually the atmospheric pressure value). The data in Table I show that the difference between the high and low pressure limits is equal to the pressure-independent gas thermal conductivity within the limits of experimental accurac)., a t least for gases through nitrogen. The sum of k,* obtained from this fact and k, given by Equarion 1. with a reasonable value of C! gives an effective conductivity acceptable for most engineering purposes. For example, Table IV shoivs that C ranges from 1.8 to 3.8 for nitrogen in the different materials. If k , calculated for this range of C is added to k,* obtained as stated above, the resulting effective conductivity has a maximum spread of about 12'30. If an average value of C were used, the maximum error would thus be only- 670. Judicious selection of C lvould result in an even smaller error. Nomenclature a0
= fraction of cross-sectional area for continuous gas
a,
=
ago
=
C
= =
,+I,
j?.1 3
path fraction of cross-sectional area for continuous solid path fraction of cross-sectional area for gas and solid in series constant in Equation 1. mm. Hg constants in Equation 2 VOL. 3
NO. 4
NOVEMBER 1 9 6 4
323
effective thermal conductivity of p o r o u solid, B.t.u. ( h r J ( f t , ) ( ' F.) = thermal conductivity of porous solid a t zero gas pressure. B.t.u. '(hr.)(ft.)(O1 7 . ) = thermal conductivity of gas in pore space, B t.u.,' (hr.)(ft.)(' F.) = thermal conductivity of free gas at atmospheric pressure, B.t.u., (hr.)(ft.)(' F.) = thermal conductivity of solid material, B.t.u. I' (hr.)(ft.)(' F.) = fraction of series path length occupied by gas = fraction of series path length occupied by solid = gas pressure, m m . H g = k;k, = kJk, = k,*:k,.
ke
=
Le*
ko
kx
4 1,
P X
Y YO
GREEKLETTERS cy, p, 6, < = constants in Equation 3
4
= =
e
yo '(1 -
E)
porosiry (volume fraction of voids in porous solid)
literature Cited (1) El Sahrigi. A . F., Ph.D. dissertation, L n i v . of Calif., Davis; Calif., 1963. (2) Harper, J. C., A.1.Ch.E. J . 8, 298 (1962). (3) Harper, J. C., Tappel, A . I,..Adzman. Food Res. 7 , 171 (1957). (4) TKennard,E. H.. "Kinetic Theory of Gases," McGraw-Hill, h e w York: 1939. (5) Kessler, H. G., Chcm-Ingt. Techn. 34, 163 (1962). 16) Kistler. S.S..1. Phw. Chem. 39, 79 119'35'1. (7) Pollard, \V. G., Preknt, S.D.>khjs.'Rei,. 73, 762 (1948). (8) Strong, H. M., Rundy, F. P., Bovenkerk: H. P., J . -ippi. Phys. 31, 39 (1960). (9) Verschoor, J. D., Greebler, P., Trans. .4m. Suc. .\.lech. Engrs. 74, 961 (1952). (10) \Voodside, \IT., Messmer, J. I I . , J . Appl. Phys. 32, 1688 (1961).
RECEIVED for review October 3, 1963 ACCEPTED May 1 , 1964 Investigation supported in part by PHS Research Grant EF-151 from the Division of En\ironmental Engineering arid Food Protection, Public Health Service.
EFFECTS OF SOLID THERMAL PROPERTIES ON HEAT TRANSFER T O GAS FLUIDIZED BEDS E. N . Z I E G L E R , ' L . B . K O P P E L , * A N D W . T . B R A Z E L T O N a Chemical Engineering Daision, Argonne .Vational Laborator), Argonne, Ill.
Heat transfer coefficients were measured from a sphere and a cylinder to a bed of solids fluidized in an air stream. The solid particles varied only in their thermal properties. The heat transfer coefficient was found to increase with solid heat capacity and was independent of solid thermal conductivity. A model which predicts the observed relation of heat transfer coefficient to solid thermal properties i s based on particle heat absorption, and utilizes a statistical distribution of particle residence times. Particle residence times back-calculated from the model agree with residence times observed experimentally and calculated from simple theoretical models. A maximum Nusselt number of 7.2 i s predicted for any fluidized system satisfying the assumptions of the model.
HE mechanism of heat transport betlieen the particles and Theat exchange surfaces :s' generally believed to be one of particle heat absorption (or release)! with the gas serving as a hear transfer meeium and as a stirring agent for the solids. T-his type of mechanism has been suggested by various investigators (4. IO. 75,24, 28) and has recently been experimentally verified ( 2 9 ) . The particles leave the core of the bed. arrive at the surface. and absorb (or release) heat, then return TO the core of the bed. Since the particles are presumed to be responsible for the rapid exchange of heat ar rhe,surface. it is of interest to determine the effect of particle thermal properties on the heat transfer coefficient. L-nfortunately. most solids of similar density have approximately the same hear capacity, and vice versa. l'herefore, it is difficult to alter the heat capacity of the solids without correspondingly changing the density of the particles, and hence their fluidization charactaristics (22). €-lo\\-ever: three materials of Ihr same density, Lvhose thermal properries vary over a reasonable range: have been found and 1
Prrsent address. Esso Research and Engineering Co.. Linden,
N.J.
*
3
324
Present address. Purdue University, Lafayette, Ind. Present address, Sorthwestern University, Evanston. I11 I&EC
FUNDAMENTALS
are listed in Table 1: together u-ith their pertinent properties. A tenfold variation in thermal conductiviry and a tbvofold variation in specific heat are obtained Lvithout significant variarion in density. This discovery permitted a n investigation of the explicit effect of particle thermal properties on the heat transfer coefficient. Juxtaposed with the experimental investigation: a throretical model has been developed lvhich predicts the observed dependence of heat transfer coefficient on solid thermal properties.
Table 1.
Material Copper Nickel Solder ( 50Yc Pb50TC Sn)
Properties of Solid Particles
Thermal Conductiility Heat Capaczty ( 7 7 " F.), Denszty,. (77" F.)," B.t.u./(Hr.) Lb./Cu. Ft. B . t . u . / ( L b . ) ( " F . ) ( F t . ) j ° F . ) 559 0 092 223 555 0 108 53 554
0.053
27
