ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Industrial Design Data
...
Analysis of Porous Thermal Insulating Materials LEONARD TOPPER Deparfmenf of Chemical Engineering, The Johns Hopkins University, Baltimore, Md.
HE resistance to heat transfer of a plastic or other solid material in which gas bubbles are dispersed depends on the average bubble size, as well as on the total porosity of the solid. The approximate analysis presented here is a first step toward the development of means to predict the effective thermal conductivity of a porous material from the properties of its component materials. Experimental data beyond those now available in the literature will be needed to assess the practical usefulness of the proposed relations.
solid and radiation through the gas contribute to the transfer of of heat; convection is negligible in pores smaller than ‘/8 inch ( 1 ) . The thermal resistance is the sum of the resistances of the solid layer of area X 2 and thickness X(l - E1/3)/(N)l’3,and of the composite gas-solid layer of thickness X ( E / N ) 1 / 3 . The area for heat conduction through the gas is X2E213,and through the solid channels is X2(1 - E2’3). The conductance of the single complete solid layer is
X K S ( N ) ” ~1/ (- E ” 3 )
Voids are assumed to be cubes
Suppose the porous insulation to be a cube of side X , and that the fraction of the total volume occupied by pores is E (Figure 1). Assume the voids to be uniformly distributed as N identical cubes arranged in a simple cubic lattice, the face of the voids being parallel to the respective faces of the cube of insulation. Assume the voids 6lled with a gas such as air that is transparent to thermal radiation, and that the solid surface a t the void has an emissivity of unity (an opaque solid is implied). T h e heat flow is assumed to be normal to a face of the cube, as it would be in a slab having a small thickness compared with the length and width. The thermal resistance of the cube is equivalent to that of (N)’lSlayers in series, where each layer is like that represented in Figure 1. The thermal resistance of the layer of cross-sectional area X 2 and thickness X / ( N ) 1 / 3 is evaluated. Conduction through gas and
-
DIRECTION OF HEAT FLOW Figure 2.
Schematic diagram of expanded solid spherical voids
The conductance of the composite layer is
The hR arises in expressing the radiant heat transmission acrom a single void as qR =
hR
x2
(!f2
-
Tl)(E/N2’3)
(1)
When (T1- TI) is small, hs is a function of TIalone (2). The resistance of the single layer drawn in Figure 1 is then
Figure 1.
July 1955
Schematic diagram of expanded solid cubical voids
INDUSTRIAL AND ENGINEERING CHEMISTRY
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Equation 7 and Equation 4 are compared in Figure 3. I n this connection the maximum porosity attainable with uniform spheres in a simple cubic lattice is E = 0.524 (actual expanded materials have nonuniform voids and are not limited in porosity). Equation 8 yields thermal radiation for opaque solid
x‘;l ao
o
(2)
a
az
as
a4
a6
0.6
0.7
0,.
LO
0..
E Figure 3.
Comparison of thermal resistance of expanded solids (radiation neglected)
Ks = 0.1 1; Kc = 0.01 ( 1 ) Cubical voids (Equation 4) (2) Spherical voids (Equation 7) and that of the ( N ) ’ / 3layers that make up the entire cube is
Ks
+
E1/3
[(hg) + KaE2/8+ Ks(1 - EZ/3)It
The radiant heat transfer through low temperature insulation is of secondary magnitude compared with conduction (3)) so that the calculated effective thermal conductivity can adequately be estimated even though the contribution of radiation has been very crudely calculated. When the solid is opaque to thermal radiation, the radiant contribution may be estimated in terms of hE, defined according to Equation 1; another approach is preferable when the solid is essentially transparent. I n the former case, radiation is transmitted only through the gas spaces, so that
~ E Q U >T hR ~ ~> ~0.8 . esuTaav.
(8)
I n Equation 8, €8 is the emissivity of the solid a t the surface of the void and u is the Stefan-Boltzmann constant. The term to the left of h~ is based on the total radiation from one face of the cubical void whose average temperature is Tav.;the term to the right is based on the radiation that is transmitted directly from the hotter face to the parallel cooler face, in the direction of heat flow, without being intercepted by the connecting walls of the void. The temperature along the connecting (3) , the walls will determine the magnitude of h ~ within
I n these equations, K Q is the thermal conductivity of the gas and K s that of the solid. As N becomes very large, the contribution of radiation to the heat transfer drops out and
1
ap-
proaches the limit
The estimat,ion of hR is deferred. Voids are assumed to be uniform spheres
It is interesting to compare Equation 4 with the thermal resistance of a matrix of voids which are uniform spheres distributed in a simple cubic lattice (Figure 2). Disregarding thermal radiation, the thermal conductivity of the gas-solid composite layer of thickness dy in the direction of heat flow is dv
Figure 4. Effect of number of voids per unit volume on thermal resistance of expanded solids (Equation 3)
Note that r 2 = R2 - yz. Then the resistance of the lowest half layer of composite gas-solid of Figure 2 is
KS = 0.1 1; Ka = 0.01; E = 0.80; X = 0.25 ft. (1) hR = 0.2 ( T = 40’ F.) (2) hR = 0.4 (T = 170’ F.)
1
i
R
tan-‘ { ( K s X 2 ) / [ ( Ks K G ) T N ” ~] B2)1’2 - rNz’a(Ks - K a ) ( ( K s X z ) / [ ( K s R Q ) T A ’ ~ ‘-~ ]R2)’”
limits imposed by Equation 8. Figure 4 illustrates the importance of radiation; it is a plot of Equation 3, based on the ~ ~ that . assumption that hx = u ~ s T 3and The resistance o f the cube of insulation, neglecting radiation, is c8 is unity. The estimation of the ( R radiant contribution when the solid is ~. appreciably transparent is more difficult 1 -x 2~~113 tan-1 { ( K s X 2 ) / [ ( Ks K Q ) ~ N -~R /2 ~ } 1]/ 2 _ than when it is opaque and requires + rN1’3(Ks K c .) .{.( K s X 2 ) / I ( K s Ka)rN2/31- R21*/’ (6) UA KsXz additional information of other maSince R = X(3E/4rN)‘Ia terial properties: the reflectivity of the solid a t the surface of the void, (3E/4a)’/3 and the absorption coefficient of the [l tan-’ [KS/a(KS - Ka) - ( 3 E / (‘ 1 solid itself. The radiant contribuKs r ( K s - K a ) / K s / r ( K s - Ka) tion will be greater than that included
-
-
1
(3‘3] { +
1378
(5)
(
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 7
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT in Equation 3 ; the over-all heat transfer coefficient, U , will be greater than that from Equation 3 by an amount- less than 4uTaay.. The extent of this additional contribution from radiant heat transmission through the solid will be controlled b y the average distance the radiation must travel in penetrating the insulation and b y the number of internal reflections made b y the radiation. Thermal resistance i s independent of void shape
The value of insulation is usually expressed as an effective thermal conductivity, in the units B.t.u./(hr.)(sq. ft.)( F./inch). This measure is 12 times the reciprocal of our X / ( U A ) . Figures 3 and 4, predicated on opaque solids, predict effective thermal conductivities of about 0.2-0.3; these are common values for practical materials. Available data are not adequate for a more severe test of the theory. The idealization of cubical voids should be an adequate one for the analysis of highly porous materials. I n these, the thermal resistance of the gas-solid composite is essentially independent of the form of the individual voids. O
Acknowledgment
This problem was suggested for examination by R. K. N7itt of
The Johns Hopkins University. Valuable criticisms were given by F. E. Towsley of The Dow Chemical Co. Nomenclature
E
=
total fraction of voids thermal conductivity of gas and solid, respectively, B.t.u./(hr.)(Fq. ft.)(’ F./ft.) number of individual voids radius of spherical void, feet ”. = temperatures at opposite faces of a void; average tem erature, O R. over-ai heat flow per unit temperature difference, B.t.u./(hr.)( F.) length of cube of insulation feet equivalent heat transfer coefficient for radiation, B.t.u./(hr.)(sq. ft.)( F.) coordinates in spherical void, Figure 2 emissivity of solid at surface of void Stefan-Boltzmann constant, 0.173 X 10-8, B.t.u./ (hr.)(sq. ft.)(’ R.4) O
O
literature cited (1) King, W. H., Mech. Eng., 54, 347 (1932). (2) McAdams, W. H., “Heat Transmission,” 3rd ed., McGraw-Hill. New York, 1954. (3) McIntire, 0. R., and Kennedy, R. N., Chem. Eng. Progr., 44,727 (1 948). RECEIVED for review May 18, 1954.
ACCEPTED February 25, 1955
Fluid Flow through Porous Aggregates E f f e c t of Porosity and Particle Shape on Kozeny-Carman Constants M. R. J. WYLLIE
AND
A. R. GREGORY
Gulf Research & Development Co., Pittsburgh, Pa.
ITERATURE on the permeability of porous media to fluid single alternative method, proposed by Rapoport and Leas (I9), flow is voluminous. For a general review, reference may utilizes the concept of stagnant fluid within consolidated porous be made to papers by Carman ( 5 ) , Sullivan and Hertel (W), media in a manner analogous to that discussed by Carman ( 4 ) and Hawksley ( I S ) . for clay aggregates. The constant of 5.0 is retained. During the last two decades, considerable interest has centered It was shown by Wyllie and Spangler (29) that if the constant on the Kozeny ( 1 4 ) equation as modified by Carman ( 5 ) . This of the Kozeny-Carman equation were increased to values conequation relates the permeability of a porous medium to its siderably greater than 5.0, the equation, when modified for the specific surface area and porosity. The equation is applicable effects of pore-size distribution, appeared to apply not only t o only under conditions of viscous flow. As derived, i t is also single-phase fluid flow in consolidated porous media but also descriptive only of flow in unconsolidated porous media. to multiphase flow. The Kozeny-Carman constants used were T o the oil industry, the problems of single- and multiphase based on electrical resistivity. measurements. In spite of t h e fluid flow in porous sedimentary rocks are basic to all oil recovery apparent agreement noted, doubt as to the general applicability processes. The sedimentary rocks that constitute petroleum of the equation remains. This doubt stems, in part, from t h e reservoirs are generally consolidated; unconsolidated sands are fact that the independent measurement of the surface areas of relatively rare. It is thus natural that a number of attempts consolidated porous media is extremely difficult. Thus, measshould have been made to apply the Kozeny-Carman equation urements of surface areas by gas adsorption methods, although t o consolidated porous media. A summary of the principal easily made, undoubtedly reflect in many cases a component of methods suggested has been given by Wyllie and Spangler (99). surface area which is not involved in flow processes (15). Many A more recent attempt is that of Cornel1 and Katz (6). measurements of permeability in systems involving two or more Attempts to render the Kozeny-Carman equation applicable fluid phases are also of dubious accuracy. t o consolidated porous media have centered on methods of deThe primary assumption made in extensions to the Kozenytermining an appropriate constant in the equation. The value Carman equation that involve altering the constant of the equaof 5.0 is generally employed for the constant when the equation tion is that the shape factor incorporated in the constant has a is applied to flow through unconsolidated porous media. For value of about 2.0 to 3.0 for all porous media. This assumption consolidated porous media, as suggested originally b y Rose and appears to lack any serious experimental foundation. Bruce (Zl), much larger constants seem to be required, The I n this article the measurement of the Kozeng-Carman conJuly 1955
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