ON THE THERMAL CONDUCTIVITY OF POROUS MATERIALS

G. M. SERYKH

Presented by Professor-Doctor G. I. FUKS

One of the important criteria of the suitability of building materials is their thermophysical characteristics in general, and thermal conductivity in particular.

The predominant majority of building and thermal-insulation materials have a porous structure. The shape of the pores is most often spherical or close to it. The maximum pore size may vary from fractions of a millimeter to several centimeters.

A considerable number of works has been devoted to the study of the thermophysical characteristics of porous materials: B. N. Kaufman [1], A. F. Chudnovsky [2], A. U. Franchuk [3], D. L. Timrot [4], Kamerer [5], Kingery [6], and a number of others. Another group of works—Maxwell [7], Russell [8], Eucken [9], Starostin [10], Ivantsov [11], Davidson [12], Loeb [13], Ribot [14], and others—is devoted to the analytical solution of the problem of heat propagation in a porous body.

The authors of the latter works proceed from Maxwell’s concept of a porous body as an aggregate of conductances and resistances. For example, Russell and Eucken consider a porous body as a special type of structure based on cubic packing. The body under consideration is divided into elementary cubes containing spherical pores filled with air (Fig. 1). As a result of considering this scheme and the corresponding mathematical calculations, Russell gives the dependence:

Fig. 1

\[ \frac{\lambda_{\mathrm{eff}}}{\lambda_{0}} = \frac{ m^{2/3} + \dfrac{\lambda_{0}}{\lambda_{v}}\left(1 - m^{2/3}\right) }{ m^{2/3} - m + \dfrac{\lambda_{0}}{\lambda_{v}}\left(1 - m^{2/3} + m\right) } \tag{1} \]

where \(m\) is the porosity, \(\lambda_{v}\) is the thermal conductivity of air, \(\lambda_{0}\) is the thermal conductivity of the skeleton, and \(\lambda_{\mathrm{eff}}\) is the effective thermal conductivity of the porous material.

Eucken gives the dependence:

\[ \frac{\lambda_{\mathrm{eff}}}{\lambda_{0}} = \frac{ 1+2m\,\dfrac{1-\dfrac{\lambda_{0}}{\lambda_{v}}}{2\dfrac{\lambda_{0}}{\lambda_{v}}+1} }{ 1-m\,\dfrac{1-\dfrac{\lambda_{0}}{\lambda_{v}}}{2\dfrac{\lambda_{0}}{\lambda_{v}}+1} } \tag{2} \]

F. D. Starostin replaces the mass of pores in a unit volume by a single pore placed within the unit volume (Fig. 2). Then, calculating the conductance of the sections \(c_{1}\), \(c_{2}\), \(c_{3}\), he finds the effective coefficient of thermal conductivity from the formula:

\[ \lambda_{\mathrm{eff}} = \frac{ \lambda_{0}^{2} m^{2/3}+(\lambda_{v}-\lambda_{0})\lambda_{0} }{ \lambda_{0}+(m^{2/3}-m)(\lambda_{v}-\lambda_{0}) } \tag{3} \]

This expression can be reduced to Russell’s formula by dividing the numerator and denominator of the right-hand side by \(\lambda_{v}\).

The drawback of this type of scheme is that it does not take into account convection and radiant heat transfer (Eucken and Starostin). The replacement of small pores by one large pore (Starostin) is also inadmissible, since this distorts the physical picture of heat propagation in a porous body.

Fig. 2

A step forward in this process is Loeb’s theory, according to which a porous body with uniformly distributed porosity is divided into elementary tubes in the direction parallel to the direction of the heat flow (Fig. 3). Some of the tubes include pores, while some pass between the pores. The tubes that include pores contain a certain number of alternating regions with different conductivities. Neglecting the role of convection and assuming that the effective thermal conductivity is determined only by thermal conduction and radiation, Loeb calculates the effective thermal conductivity of a tube and then, passing to the whole porous material, obtains the expression:

\[ \frac{\lambda_{\mathrm{eff}}}{\lambda_{0}} = (1-m_{\mathrm{pl}}) + \frac{ m_{\mathrm{pl}} }{ \dfrac{1}{\lambda_{0}}(1-\Sigma l_{n}) + \dfrac{\Sigma l_{n}\lambda_{0}}{4\gamma_{2}\varepsilon\sigma_{0}dT_{0}^{3}} } , \tag{4} \]

where: \(m_{\mathrm{pl}}\) is the planar porosity, \(\Sigma l_{n}\) is the part of the tube length occupied by pores, \(\lambda_{0}\) is the coefficient of thermal conductivity of the principal material of the sample, \(\gamma_{2}\) is a geometric factor; for slit-shaped and cylindrical pores with

slots parallel to the heat flow, \(\gamma_2=1\); perpendicular to the heat flow, \(\gamma_e=\frac{\pi}{4}\) for spherical pores; \(\gamma_2=1\); \(d\) is the greatest dimension of the slit in the direction of the heat flow; \(e\) is a correction for the grayness of the radiating surface; \(T_0\) is the mean pore temperature in degrees; \(\sigma_0\) is the constant for an absolute black body.

Fig. 3

Loeb’s formula gives satisfactory agreement with experimental data for bodies with pores of diameter not greater than 3 mm. This is explained by the fact that it does not take into account convection in the pores and the curvature of the heat flow when it passes around the pores. Greater difficulty is caused by the calculation of \(\sum l_n\).

It is also necessary to point out an interesting relation obtained by Ribot. Ribot considers a porous body as a certain material with uniformly included pores of cubic shape. Having separated out in such a material a cube of size \((S_1+S_2)^3\) (see Fig. 4), and considering the heat flow to be rectilinear, Ribot writes the equation for a section with cross-sectional area \((S_1+S_2)^2\):

\[ Q=\frac{S_1}{S_1+S_2}\lambda_v\frac{T_1-T_2}{e} +\frac{S_2}{S_1+S_2}\frac{\lambda_0(T_1-T_2)}{e_1+e'}. \tag{5} \]

From it he derives the relation:

\[ \lambda_{\mathrm{eff}}=\lambda_v m^{1/3}+\lambda_0(1-m^{1/3}) \tag{6} \]

Besides shortcomings analogous to those of the preceding formulas, Ribot’s equation, as is evident from the initial expression, does not take into account the thermal resistance of the section of area \(S_1\) and length \(e'\).

As the investigations carried out show, the propagation of heat in a porous body is a complex process. Part of the heat is transferred through the skeleton of the basic material, and part through the pores. With a small temperature difference and pores of size less than 5 mm [15], [16], [17], the main part of the heat will be transferred through the skeleton material. The heat flow will for the most part pass around the pores. In the pores themselves, heat will be transferred by conduction and radiation. As the pore size increases, the share of heat transfer in them will increase because of the increase in radiation and the emergence of convection.

Fig. 4

If the thermal conductivity of the framework material is small, one can imagine a case in which the effective coefficient of thermal conductivity of a pore will be approximately equal to the coefficient of thermal conductivity of the base material. In this case the temperature field as a whole may be regarded as one-dimensional. Let us consider this simplest case.

Statement of the problem. We have a porous body in the form of an unlimited plate of thickness \(\delta\). The size of the average pore is more than \(5\) mm \([15--18]\). The temperature of the left wall is \(T_1\), of the right wall, \(T_2\). The temperature field is one-dimensional. The lines of heat flow are parallel straight lines. (See Fig. 5.) The material is absolutely dry.

Fig. 5.
(In the figure: “Heat flow.”)

Let us cut out in the porous body a tube with cross-sectional area \(\Delta F\). Along the tube, through all sections perpendicular to its axis, an amount of heat \(\Delta Q_n\) will be transferred. The part of the heat passing through the area \(\Delta F_n\) will be transferred by thermal conduction, convection, and radiation. The part of the heat passing through the area \(\Delta F_0\) will be transferred only by thermal conduction. Therefore:

\[ \Delta Q_n = \left\{ \frac{\lambda_{\mathrm{usl}}^{k}}{l_{n_1}} + \frac{ C\left[\left(\frac{T_1}{100}\right)^4-\left(\frac{T_1'}{100}\right)^4\right] }{ T_1-T_1' } \right\} (T_1-T_1')\Delta F_n' + \frac{\lambda_0}{l_n}(T_1-T_1') \]

\[ \tag{7} \Delta Q_n = \left\{ \frac{\lambda_{\mathrm{usl}}^{(n)k}}{l_{nn}} + \frac{ C\left[\left(\frac{T_n'}{100}\right)^4-\left(\frac{T_2}{100}\right)^4\right] }{ T_n'-T_2 } \right\} (T_n'-T_2)\Delta F_n^n + \frac{\lambda_0}{l_{nn}}\Delta F_0^n(T_n'-T_2). \]

Denoting the second terms of the braces by \(\dfrac{\lambda_{\mathrm{usl}}^{il}}{l_{ni}}\), we may write:

\[ \Delta Q_n = \left( \frac{\lambda_{\mathrm{usl}}^{1k}}{l_{n_1}} + \frac{\lambda_{\mathrm{usl}}^{1l}}{l_{n_1}} \right) (T_1-T_1')\Delta F_n' + \frac{\lambda_0}{l_{n_1}}\Delta F_0'(T_1-T_1') \]

\[ \tag{8} \Delta Q_n = \left( \frac{\lambda_{\mathrm{usl}}^{(n)k}}{l_{nn}} + \frac{\lambda_{\mathrm{usl}}^{(n)l}}{l_{nn}} \right) (T_n'-T_2)\Delta F_n^n + \frac{\lambda_0}{l_{nn}}\Delta F_0^n(T_n'-T_2) \]

Here: \(\lambda_{\mathrm{usl}}^{k}\) is the conditional coefficient of thermal conductivity, taking into account heat transfer in the pore by thermal conduction and convection; \(\lambda_{\mathrm{usl}}^{l}\) is the conditional coefficient of thermal conductivity, taking into account heat transfer in the pore by radiation; \(\lambda_0\) is the coefficient of thermal conductivity of the framework; \(\Delta F\) is the part of the cross-sectional area of the elementary tube passing through the pore; \(\Delta F_0\) is the part of the cross-sectional area of the elementary

tube passing through the material of the skeleton; \(l_n\) is the length of the individual sections of the tube; \(T\) are absolute temperatures.

Solving equalities (8) with respect to the temperature difference, we have:

\[ T_1 - T'_1 = \frac{\Delta Q_n l_n} {\left(\lambda_{\mathrm{usl}}^{k}+\lambda_{\mathrm{usl}}^{l}\right)\Delta F_n^1+\lambda_o\Delta F_o^1} \tag{9} \]

\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]

\[ T'_n - T_2 = \frac{\Delta Q_n l_n} {\left(\lambda_{\mathrm{usl}}^{(n)k}+\lambda_{\mathrm{usl}}^{(n)l}\right)\Delta F_n^n+\lambda_o\Delta F_o^n} \]

Adding equations (9) term by term, we obtain:

\[ T_1 - T_2 = \Delta Q_n \left[ \frac{\delta} {\left(\lambda_{\mathrm{usl.av}}^{k}+\lambda_{\mathrm{usl.av}}^{l}\right)\Delta F_n^{\mathrm{av}}+\lambda_o\Delta F_o^{\mathrm{av}}} \right], \tag{10} \]

where

\[ \delta=\sum_{i=1}^{n} l_n, \]

\(\Delta F_n^{\mathrm{av}}\) and \(\Delta F_o^{\mathrm{av}}\) are the mean values of the areas \(\Delta F_n\) and \(\Delta F_o\), and \(\lambda_{\mathrm{usl.av}}^{k}\) and \(\lambda_{\mathrm{usl.av}}^{l}\) are the mean values of \(\lambda_{\mathrm{usl}}^{k}\) and \(\lambda_{\mathrm{usl}}^{l}\).

From (10) we have:

\[ \Delta Q_n = \frac{(T_1-T_2)\left[\left(\lambda_{\mathrm{usl.av}}^{k}+\lambda_{\mathrm{usl.av}}^{l}\right)\Delta F_n^{\mathrm{av}}+\lambda_o\Delta F_o^{\mathrm{av}}\right]} {\delta} \tag{11} \]

Adding the elementary fluxes passing through all the areas \(\Delta F\), we shall have:

\[ Q_n = \frac{(T_1-T_2)\left[\left(\lambda_{\mathrm{usl.av}}^{k}+\lambda_{\mathrm{usl.av}}^{l}\right)F_n^{\mathrm{av}}+\lambda_o F_o^{\mathrm{av}}\right]} {\delta} \tag{12} \]

The expression for \(Q_n\) may also be written in the form:

\[ Q_n = \frac{\lambda_{\mathrm{eff}}(T_1-T_2)F}{\delta}. \tag{13} \]

If the specimen had no pores, the heat transmitted would be

\[ Q_o = \frac{\lambda_o(T_1-T_2)F}{\delta}. \tag{14} \]

From (12), (13), and (14) we obtain:

\[ \Lambda = \frac{\lambda_{\mathrm{eff}}}{\lambda_o} = \frac{\left(\lambda_{\mathrm{usl.av}}^{k}+\lambda_{\mathrm{usl.av}}^{l}\right)F_n^{\mathrm{av}}+\lambda_o F_o^{\mathrm{av}}} {\lambda_o F} \tag{15} \]

or

\[ \Lambda = \frac{\left(\lambda_{\mathrm{usl.av}}^{k}+\lambda_{\mathrm{usl.av}}^{l}\right)m_{n l}} {\lambda_o} - (1-m_{n l}) \tag{16} \]

where \(m_{n l}\) is the plane porosity, equal to

\[ \frac{F_n^{\mathrm{av}}}{F_o}. \]

Equation (16) may be rewritten as:

\[ \lambda_{\mathrm{eff}} = \left(\lambda_{\mathrm{usl.av}}^{k}+\lambda_{\mathrm{usl.av}}^{l}\right)m_{n l} - \lambda_o(1-m_{n l}). \tag{17} \]

Introduce the notation

\[ \frac{\lambda_{\mathrm{usl.av}}^{k}}{\lambda_v}=E_k, \]

where \(E_k\) is the convection coefficient.

It is known that

\[ E_k=f(\mathrm{Gr}\cdot \mathrm{Pr}), \tag{18} \]

therefore

\[ \lambda'_{\mathrm{cond.,avg.}}=f(Cr\cdot Pr)_{\lambda_v}. \tag{19} \]

As the experiments of Boyarintsev, Krausold, and others [15, 18] have shown, for small values of the argument \((Cr\cdot Pr)<1000\), \(E_k=1\), i.e., heat transfer in the pores occurs only by thermal conduction and radiation. In general,

\[ E_k=K(Cr\cdot Pr)_f^n . \tag{20} \]

Let us introduce the notation:

\[ \frac{ C\left[\left(\frac{T'_{i\,\mathrm{avg}}}{100}\right)^4- \left(\frac{T'_{(i-1)\,\mathrm{avg}}}{100}\right)^4\right] }{ (T_{i\,\mathrm{avg}}-T_{(i-1)\,\mathrm{avg}})\lambda_o } =Ki, \tag{21} \]

where \(K_i\) is the Kirchhoff criterion.

The difference \((T_{i\,\mathrm{avg}}-T_{(i-1)\,\mathrm{avg}})\) can be determined by the following procedure. If one takes the average value \(d_n\), \(\lambda^k_{\mathrm{cond.,avg.}}\), \(\lambda^l_{\mathrm{cond.,avg.}}\) along the length of an elementary tube, then equation (9) can be rewritten as

\[ T_i-T_{(i-1)}=\Delta Q_n \frac{d_{n.\mathrm{avg}}} {(\lambda^k_{\mathrm{cond.,avg}}+\lambda^l_{\mathrm{cond.,avg}})\Delta F^{\mathrm{avg}}_n+\lambda_o\Delta F^{\mathrm{avg}}_o} . \tag{22} \]

According to (11):

\[ \Delta Q_n=\frac{T_1-T_2}{\delta} \left[(\lambda^k_{\mathrm{cond.,avg}}+\lambda^l_{\mathrm{cond.,avg}})\Delta F^{\mathrm{avg}}_n+\lambda_o\Delta F^{\mathrm{avg}}_o\right]. \tag{23} \]

From (22) and (23) it follows that

\[ T'_i-T'_{(i-1)}=\frac{T_1-T_2}{\delta}d_{\mathrm{avg}}, \tag{24} \]

where \(T'_i\) and \(T'_{(i-1)}\) are the temperatures at the pore boundaries in the direction of the heat flow.

Equation (16) includes the plane porosity, as the ratio \(\frac{F^{\mathrm{avg}}_n}{F}\). The dependence between the plane porosity \(m_{\mathrm{pl}}\) and the volumetric porosity \(m\) can be found from the following considerations. For spherical pores in a cubic packing:

\[ m=\frac{V_{\mathrm{por}}}{V_{\mathrm{tot}}} =\frac{\pi d^3 n^3}{6l^3}. \tag{25} \]

For round plane pores arranged on a square grid:

\[ m_{\mathrm{pl}}=\frac{F_n}{F} =\frac{\pi d^2 n^2}{12l^2}, \tag{26} \]

where: \(V_{\mathrm{por}}\) is the volume of all pores, \(V_{\mathrm{tot}}\) is the volume of the entire porous specimen, \(d\) is the pore diameter, \(n\) is the number of pores in a row, \(l\) is the specimen dimension, \(F_n\) is the area of all pores, and \(F\) is the area of the entire specimen.

Obviously, the area of one plane pore should not be taken by the diametral section, but as the mean integral cross-sectional area of the sphere:

\[ f_n=\frac{1}{d}\int_0^d \frac{\pi d^2}{4}\,dd \tag{27} \]

\[ f_n=\frac{\pi d^2}{12}. \tag{28} \]

From (25):

\[ d\cdot n = l\sqrt[3]{\frac{6m}{\pi}} \tag{29} \]

From (25), (26), and (28) we obtain:

\[ m_{pl}=0.4m^{2/3}. \tag{30} \]

Carrying out analogous calculations for volume pores of cubic form, and planar pores of square form, we obtain

\[ m_{pl}=m^{2/3}. \tag{31} \]

Fig. 6

For cylindrical pores with axes directed along the heat flow:

\[ m_{pl}=0.92m^{2/3} \tag{32} \]

In general form:

\[ m_{pl}=Lm^{2/3} \tag{33} \]

As experiments carried out on planar and three-dimensional electrical models have shown, the shape coefficient \(L\) depends on the porosity and varies from 0 to 1 (see Fig. 6).

Taking into account expressions (19), (20), (21), and (23), relation (16) may be rewritten in the form:

\[ \Lambda= \left[ \frac{\lambda_v}{\lambda_o}K(Cr\cdot Pr)_f^n+Ki \right]\cdot Lm^{2/3}+\left(1-Lm^{2/3}\right) \tag{34} \]

or, in dimensional form:

\[ \lambda_{\text{eff}}=(\lambda_{\text{cond.av}}^{k}+\lambda_{\text{cond.av}}^{l})Lm^{2/3}+\lambda_{0}(1-Lm^{2/3}). \tag{35} \]

The qualitative aspect of the problem posed can be studied on the basis of similarity theory. If it is assumed that the process of heat propagation in a porous body is steady, then it can be described by the following equations.

The heat-conduction equation for the solid body:

\[ a\left(\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}}\right)=0. \tag{36} \]

The heat-conduction equation for the gas in the pores:

\[ \omega_x\frac{\partial T}{\partial x}+\omega_y\frac{\partial T}{\partial y}+\omega_z\frac{\partial T}{\partial z} = a\left(\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}}\right). \tag{37} \]

The heat-exchange equation in the pores:

\[ \alpha_k\Delta T+C_0E_n\left[\left(\frac{T_1}{100}\right)^4-\left(\frac{T'_1}{100}\right)\right] = \lambda_{\text{eff}}\frac{\partial T}{\partial x}. \tag{38} \]

The equation of motion in the pores:

\[ \rho\left(\omega_x\frac{\partial\omega_x}{\partial x} +\omega_y\frac{\partial\omega_x}{\partial y} +\omega_z\frac{\partial\omega_x}{\partial z}\right) = \rho g_x-\frac{\partial p}{\partial x} +\mu\left(\frac{\partial^2\omega_x}{\partial x} +\frac{\partial^2\omega_x}{\partial y} +\frac{\partial^2\omega_x}{\partial z}\right). \tag{39} \]

The equations for the \(y\) and \(z\) axes are written analogously.
Here: \(\alpha_k\) is the heat-transfer coefficient, \(E_n\cdot C_0=C\) is the radiation coefficient, \(a\) is the thermal diffusivity, \(\rho\) is the density, \(p\) is the pressure, \(\mu\) is the viscosity coefficient.

Introducing scale transformations:

\[ \begin{aligned} x&=x_*X, & \omega_x&=\omega_*W_x, & T&=T_*\Theta, & \lambda_{\text{eff}}&=\lambda_*\Lambda, & p&=p_*P,\\ y&=y_*Y, & \omega_y&=\omega_*W_y, & \rho&=\rho_*P, & \alpha_k&=\alpha_{k*}A_k, & \mu&=\mu_*M,\\ z&=z_*Z, & \omega_z&=\omega_*W_z, & a&=a_*A, & g&=g_*G, & c&=c_*C, \end{aligned} \tag{40} \]

and substituting them into equations (36)—(39), we obtain:

\[ \frac{\omega_*T_*}{x_*}=\frac{a_*T_*}{x_*^2}, \tag{41} \]

\[ \alpha_*T_*=C_*T_*^4=\lambda_*\frac{T_*}{x_*}, \tag{42} \]

\[ \frac{\rho_*\omega_*^2}{x_*} = \rho_*g_*= \frac{p_*}{x_*} = \frac{\mu_*\omega_*}{x_*^2}, \tag{43} \]

Assuming

\[ a_*=a_0,\qquad \lambda_*=\lambda_0,\qquad T_*=T_{\text{av}},\qquad x_*=d_0 \tag{44} \]

and taking expression (29) into account, we obtain:

\[ W_x=Pe\cdot m,\qquad X=\frac{x}{d_0},\qquad Y=\frac{y}{d_0},\qquad Z=\frac{z}{d_0}, \tag{45} \]

where \(d_0\) is the distance between the centers of two pores in a cubic packing.

We shall represent the Peclet criterion in the form:

\[ m\cdot Pe=Re\cdot Pr\cdot m. \tag{46} \]

From (42), the first and second terms give:

\[ z_k = Bi \cdot m; \qquad \Lambda = \frac{\lambda_{\mathrm{eff}}}{\lambda_o}; \qquad \Theta = \frac{T}{T_{\mathrm{cp}}}. \tag{47} \]

The second and third terms give, under condition (44):

\[ C=\frac{g_{cp}\cdot d \sqrt[3]{\frac{\pi}{6m}}}{\lambda_o T_{\mathrm{cp}}}, \tag{48} \]

where \(d\) is the pore diameter,
or

\[ C = Ki \cdot m. \]

From (43)

\[ G=Fr \tag{49} \]

Applying Galileo’s criterion and multiplying it by the simplex \(\frac{\rho-\rho_0}{\rho}\), we obtain Archimedes’ criterion

\[ Ca = Fr \cdot Re^2, \tag{50} \]

\[ Ar = Ca \cdot \frac{\rho-\rho_0}{\rho}, \tag{51} \]

where \(\rho\) and \(\rho_0\) are the densities of air at two points of the pore. Since the difference in fluid densities is determined by the temperature difference \(\Delta T\), then

\[ \frac{\rho-\rho_0}{\rho}=\beta \cdot \Delta T_{\mathrm{cp}}, \tag{52} \]

where \(\beta\) is the coefficient of volumetric expansion.

Substituting (52) into (51), we obtain the Grashof criterion.
Taking into account (45–52), we obtain a criterion relation of the form:

\[ \lambda=f(Re;\ Pr;\ Cr;\ X;\ Y;\ Z;\ \Theta;\ Bi;\ Ki;\ M) \tag{53} \]

Relation (53) is consistent with the \(\pi\)-theorem, according to which the number of criteria-complexes is equal to

\[ n-k=13-10=3. \]

The number of criteria-complexes is

\[ N-n=21-13=8. \]

Here \(N\) is the total number of quantities entering into expressions (36), (37), (38), (39);
\(n\) is the total number of scales with different dimensions;
\(k\) is the number of scales with independent dimensions.

1. Since the motion of air in the pores is not forced, \(Re\) drops out [18].

2. Since \(\lambda_{\mathrm{eff}}\) does not depend on the coordinates, but is a parameter of the entire mass of the material, \(X\), \(Y\), and \(Z\) drop out.

3. Since we assume that the temperature interval is small and that the coefficient of thermal conductivity, consequently, does not depend on \(T\), \(\Theta\) drops out.

4. Since \(Bi=f(\alpha_k)\), and \(\alpha_k=f(Cr \cdot Pr)\), consequently \(Bi=f(Cr \cdot Pr)\); that is, \(Bi\) also drops out.

5. Since the pores are filled with air, and for air \(Pr\) has quite definite values, the criterion \(Pr\) also drops out, and then expression (53) takes the form:

\[ \lambda=f(Cr;\ Ki;\ m) \tag{54} \]

The criterion dependence obtained in this way has the same form as dependence (34).

Both relation (34) and the relations given by other authors show certain deviations from the experimental data. See Fig. 7.

1—the experimental curve of Kaufman; 2—the curve calculated by Russell’s formula; 3—the curve calculated from equation (34); 4—the curve calculated by Eucken’s formula; 5—the curve calculated by Ribaud’s formula.

As calculations have shown, the term \((\lambda_{\text{cond.skel.}} + \lambda'_{\text{cond.skel.}}) L \cdot m_i^{2/3}\) of equation (34) is small in comparison with the term \(\lambda_0(1 - Lm_i^{2/3})\). The equidistant displacement of curve 3 from curve 1 is due to the quantity \(\lambda_0(1 - Lm_i^{2/3})\), that is, to the fraction of heat passing through the skeleton of the porous material. To determine the magnitude of this displacement, experiments were carried out on an electrical model [19], [20].

Fig. 7

Having found on the electrical model the surfaces or lines of equal potential, we were able to determine the temperature field of the specimen and to clarify the nature of the motion of the heat flux. Taking into account relation (33) between volume and plane porosities, we carried out the first series of experiments on a plane electrical model [21], [22].

The experimental setup shown in Fig. 8 consisted of the following:

A sheet of aluminum foil 1, of thickness \(\delta = 0.0015\) mm, was glued with paraffin to a sheet of Whatman paper 2, previously impregnated in paraffin, of size \(300 \times 420\) mm; it was inserted between two massive busbars 3 and fastened together with the busbars by bolts to a special board. To obtain reliable contact between the foil and the busbars, a thin copper wire 5, stretched by two springs 4, was laid between them. The electrical circuit consisted of a battery 6, an ammeter 7, a single-pole knife switch 8, and a voltmeter 9. Each sheet of foil was first checked for uniformity by means of a null galvanometer 10 with constant \(C = 1.8 \cdot 10^{-9}\) and two probes 11. Then holes of various shapes and arrangements were cut in the sheet of foil, and the equipotential lines were again recorded.

The form of these lines for circular pores with a square arrangement is shown in Fig. 8a. As can be seen from it, the heat flux, going around the pores, travels a path greater than the length of the specimen, while the cross-sectional area through which the main mass of the heat flux passes is considerably smaller than the area of the specimen.

From expression (17), under the condition that the thermal conductivity of the pore may be neglected, we have:

\[ \frac{\lambda_{\text{eff}}}{\lambda_0} = (1 - m_{n,l}). \tag{55} \]

The expression on the right-hand side of equation (55) represents the fraction of the skeleton of the porous body theoretically participating in the transfer of heat.

Fig. 8

In the graph:
vertical axis: \(\dfrac{\lambda_{\mathrm{eff}}}{\lambda_0}\)
horizontal axis: \(m\)

Legend:

Fig. 9

However, as experiments have shown, not all of the framework fraction works with the same heat load. Therefore the term standing in the right-hand part of expression (55) will be less than \((1 - m_{p.l.})\). To determine its magnitude, we measured the resistance of the specimen as a whole and with pores by means of a bridge circuit. Using the electrothermal analogy, we found the ratio \(\dfrac{\lambda_{\mathrm{eff}}}{\lambda_0}\) and plotted it on the graph (Fig. 9), where: 1 is the straight line \((1 - m_{p.l.})\), 2 is the experimental curve found on the plane model.

One can also use the curve obtained on a volumetric model, or find it from expression (33). Obviously, the difference in ordinates \(\varphi\) will be the correction that takes into account the deviation of the flow from rectilinearity and its nonuniform distribution over the cross section.

Taking this into account, expressions (34) and (35) take the form:

\[ \Lambda = \left[ \frac{\lambda_v}{\lambda_0} Ki(Cr \cdot Pr)_f^n + Ki \right] Lm^{2/3} + (1 - Lm^{2/3} - \varphi) \tag{56} \]

\[ \lambda_{\mathrm{eff}} = \left(\lambda_{\mathrm{cond.avg}}^{\kappa} + \lambda_{\mathrm{cond.avg}}^{\lambda}\right)Lm^{2/3} + \lambda_0(1 - Lm^{2/3} - \varphi) \tag{57} \]

The curve calculated from expression (57) is shown in Fig. 7 under No. 6. As is seen from Fig. 7, this curve gives close agreement with the experimental data.

References

1. B. N. Kaufman — Thermal Conductivity of Building Materials, Moscow, GILSA, 1955.

2. A. F. Chudnovskii — Heat Transfer in Disperse Media, Moscow, GITSL, 1954.

3. A. U. Frank-Kamenetskii — Thermal Conductivity of Building Materials as a Function of Moisture, Stroiizdat, 1941.

4. D. L. Timrot — Determination of the Thermal Conductivity of Building and Insulating Materials, Energoizdat, 1932.

5. Kappeger
   1) Mitt. Forsch. fur Wärmeschutz, No. 4, 1925.
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22. N. I. Druzhinin — The Method of Electrodynamic Analogies and Its Application in the Study of Filtration, GEI, 1956.