METHODS FOR CALCULATING THE THERMAL CONDUCTIVITY COEFFICIENT OF DRY BULK GRANULAR MATERIALS

A. S. Lyalykov

(Presented by Prof. G. I. Fuk, Doctor of Technical Sciences)

The study of the problem of heat transfer in disperse materials advanced considerably with the appearance of works [1, 2]. Nevertheless, the question of the theoretical calculation of the thermal-conductivity coefficient of dry bulk granular materials has not been sufficiently developed. The existing calculation methods have a number of substantial shortcomings. On the basis of an analysis of these methods, attempts are being made to create more reliable methods of calculation. Such a task was addressed, in particular, in our study [3]. In the review [4] it was intended “...to indicate which of these calculations are defective and why, and which may, and under what conditions, be taken as the basis for calculations,” but the author did not succeed in resolving the problem posed, since the question of the agreement with experiment of the formula of V. Z. Bogomolov [5] is considered more thoroughly in [1, 5], while the reasons for the defectiveness of other calculation methods are not shown.

The task of the present work is a critical analysis of the existing methods for calculating the thermal conductivity of dry bulk granular materials.

Studies in the field of the thermal conductivity of granular materials may be represented in the form of three main groups, each of which is based on a definite conception of the mechanism of thermal conductivity and of the structure of the material.

Some investigators—Smoluchowski [6], Hengst [12] (Hengst’s calculation formula is given in [13])—proceed from an in-depth study of the physics of phenomena at the interface between the solid and gas phases, which is very important for the case of granular materials because of the highly developed surfaces in the latter. The calculation relations obtained by these investigators for determining thermal conductivity take into account the temperature jump at the surface of the solid body, a question that was theoretically substantiated by Smoluchowski [7, 8, 9] and experimentally investigated by Smoluchowski, Gehrcke [10], Lazarev [11], and others.

Smoluchowski’s calculation relation for powder-like materials agrees with experiment [6], despite the fact that it does not take into account the thermal resistance of the grains, and porosity is taken into account very conditionally—by a constant numerical coefficient. This co-

agreement is, in our view, natural because the relative influence of the resistance of the grains themselves and of the pores on the thermal conductivity in powders is weak, while the main role here is played by the thermal resistances at the interface between the solid and gaseous phases.

With increasing grain size one should expect a decrease in the role of the thermal resistance over the interfacial surfaces and a determining influence of the thermal resistance of the bed, governed by porosity and by the properties of the grain substance.

Hengst [12] improved Smoluchowski’s method by taking into account the thermal conductivity of the grains themselves, as well as the deviation of the real structure from cubic packing, by means of a constant numerical coefficient \(B\).

Kling [13] noted that, for coarse-grained materials, calculations by Hengst’s formula agree better with experiment than calculations by Smoluchowski’s formulas.

To judge the agreement of calculations by the formulas of these authors with experimental data, we performed calculations presented in columns 8 and 9 of Table 1. In the calculations it was assumed: the thermal-conductivity coefficient in a monolith for quartz \(\lambda_M = 5.2\) kcal/m·h·deg [14], steel \(\lambda_M = 33\) [13], lead \(\lambda_M = 30\) [16]; the mean free path of an air molecule at atmospheric pressure \(L = 0.95 \cdot 10^{-7}\) m [11]; the temperature-jump coefficient \(\delta = C \cdot L = 1.65 \cdot 10^{-7}\) m, where \(C = 1.73\) is an empirical coefficient, averaged from the data of Smoluchowski, Gerke, and Lazarev; the thermal-conductivity coefficient of air \(\lambda_g = 0.022\) kcal/m·h·deg.

Comparison of the coefficients \(\lambda\) calculated by Smoluchowski’s formula with the experimental values shows that: 1) the ratio of the calculated values of \(\lambda\) to the experimental ones for the first four specimens is \(0.59\)—\(0.73\), while for the others it is considerably smaller, especially for granular materials whose grains are made of a substance that conducts heat well; 2) in all cases the calculated values of \(\lambda\) are smaller than those obtained experimentally.

These results, in our opinion, are explained by the following: 1) the method proceeds from cubic packing, which does not sufficiently fully reflect the actual character of grain arrangement; 2) the calculation formula does not make it possible to take into account the porosity of real materials; 3) the thermal resistance of the grains themselves is not taken into account.

As is known, these factors substantially affect the thermal conductivity of a material; therefore, in this form the calculations cannot be considered sufficiently reliable. Although the author showed agreement of his calculations with experiment, in particular for powders under deep rarefaction of the gas, it must be borne in mind that the role of the temperature jump increases with decreasing grain size and rarefaction of the gas, while the role of the resistances of the grains themselves decreases; therefore the latter could remain unnoticed, as could the role of porosity.

Comparison of the calculated values of \(\lambda\), computed by Hengst’s formula, with experimental data (column 9, Table 1) makes it possible to conclude that they agree better with one another; this should be regarded mainly as the result of introducing the correction coefficient \(B\), which eliminates the systematic error of the calculation. A certain role was played by allowance for the thermal conductivity of the grain material. However, here too, in some cases the ratios

\[ \frac{\lambda_{\text{calc.}}}{\lambda_{\text{exp.}}} \]

are \(0.78\) and \(1.33\)—\(1.39\).

This method does not take into account the influence of porosity on the thermal conductivity of the material and therefore, as can be seen from the table, with a change in porosity

Table 1

¹ For crushed dolomite, $\gamma = 2620\ \text{kg/m}^3$, thermal conductivity $\lambda_m = 5.2\ \text{kcal}/(\text{m}\cdot\text{h}\cdot\text{deg})$.

² For steel shot, $\gamma = 7800$, $\lambda_{\text{shot}} = 11800\ \text{kcal}/(\text{m}\cdot\text{h}\cdot\text{deg})$.

³ Result obtained with air motion in the instrument at $p = 650\ \text{mm Hg}$.

⁴ Value adopted on the basis of experiments of other authors and

\[ \frac{1}{c \cdot 388} \]

, [27].

⁵ Coefficients $\lambda$ calculated by us from the temperature-conductivity data, taken at $c = 0.11\ \text{kcal}/(\text{kg}\cdot\text{deg})$.

substantially changes the discrepancy between the calculated values of \(\lambda\) and the experimental ones. The presence in the calculation formula of a numerical empirical coefficient \(B\) reduces its theoretical value.

Another group of researchers—Krischer [14], Bernstein [15]—proceeds from an artificial representation of the layer as a set of plates of material (or small plates) and a gaseous medium. Two limiting cases of the greatest and least insulating effect are considered. It is assumed that the thermal conductivity of a granular material lies within the interval formed by the limiting values of \(\lambda\), which, according to Krischer, are calculated from the formulas:

\[ \lambda_{\min}=\lambda_2\frac{100}{\dfrac{\lambda_2}{\lambda_M}(100-m)+m}, \tag{1} \]

\[ \lambda_{\max}=\lambda_M\frac{100-m}{100}+\lambda_2\frac{m}{100}, \tag{2} \]

where \(m\) is the porosity in percent.

Such an approach, owing to the exceptional nature of the limiting cases on which it is based, and also to the neglect of the character of the contacts and of the structure, gives too wide a range of possible values of the coefficient of thermal conductivity of the granular bed (column 10 of Table 1) and therefore cannot serve as a calculation scheme. The thermal conductivities of steel, lead, quartzite, and air, as before, are taken to be respectively 33, 30, 5.2, and 0.022 \(\text{kcal}/\text{m}\cdot\text{h}\cdot^\circ\).

An attempt to narrow the interval of values of \(\lambda\) following from Krischer’s method belongs to Bernstein and consists in proposing a calculation scheme in the form of a packing of small plates. In this case the calculation formulas for the minimum insulating effect have the form:

\[ \frac{\lambda_{\max}}{\lambda_2} = \frac{4m}{1+\dfrac{\lambda_2}{\lambda_M}} + \frac{\lambda_M}{\lambda_2}(1-2m) \quad \text{— for } m \le 0.5 \tag{3} \]

and

\[ \frac{\lambda_{\max}}{\lambda_2} = \frac{4(1-m)}{1+\dfrac{\lambda_2}{\lambda_M}} + 2(m-1) \quad \text{— for } m>0.5. \tag{4} \]

Here \(m\) is the porosity as a fraction of unity.

The maximum insulating effect is still estimated by formula (1).

With the aid of experiment, the author himself established that the formulas he proposed give too wide a range of values of \(\lambda\). Therefore he came to the conclusion that it is necessary to take into account the air interlayers between the plates, which are the result of the roughness of their surfaces. Measurements showed the possibility of gaps between the plates of 0.05 mm. Taking the latter into account made it possible to carry out a calculation whose data agree better with experiment, but the author did not make any generalizations from this and did not give the form of a calculation formula that would take account of the presence of air interlayers.

In our opinion, attempts to improve Krischer’s method could not lead to serious success because its scheme does not take into account the nature of the packing of the grains, the mode of contact…

them; moreover, the method itself does not proceed from the fact that in a granular material, as compared with a continuous body, there are such features as a developed interface surface and numerous contacts between grains, which can affect the magnitude of the thermal conductivity of granular materials.

The third group of investigators—Bogomolov [5], Schumann and Voss [17]—is based on the premise that there is no contact between the grains, but that an air gap exists. The determining factor in the calculation is porosity.

Schumann and Voss’s approach to calculating the thermal conductivity of granular materials is distinctive. If it is assumed that the thermal-conductivity coefficient of a granular material does not depend on the grain size or on the method of packing, then it is natural to expect its dependence on the thermal-conductivity coefficient of the material and of the gas, and also on the fraction of the latter in the entire volume (i.e., the porosity \(m\)):

\[ \lambda = f(\lambda_m,\lambda_g,m). \tag{5} \]

The form of the function must satisfy a number of conditions that follow from the possibility of various variations of the granular material with respect to the coefficients \(\lambda_m\), \(\lambda_g\), and the value \(m\). These conditions are as follows:

\[ \begin{array}{llll} \text{If} & m=0, & \text{then} & \lambda=\lambda_m, \qquad (a)\\[6pt] \text{“} & m=1, & \text{“} & \lambda=\lambda_g, \qquad (b)\\[6pt] \text{“} & \lambda_g=0, & \text{“} & \lambda=0, \qquad (c)\\[6pt] \text{“} & \lambda_g=\lambda_m, & \text{“} & \lambda=\lambda_g, \qquad (d)\\[6pt] \text{“} & \lambda_g=\infty, & \text{“} & \lambda=\infty, \qquad (e)\\[6pt] \text{“} & \lambda_m=\infty, & \text{“} & \lambda=\infty, \qquad (f)\\[6pt] \text{“} & \lambda_m=0, & \text{“} & \lambda=\lambda_g m^3. \qquad (g) \end{array} \]

Conditions \((a)\), \((b)\), \((d)\), and \((e)\) are considered obvious. Condition \((c)\) is valid for point contact; \((f)\), only for the case when the contact is over an area; condition \((g)\) is accepted in the form of a hypothesis requiring experimental confirmation.

As a result of considering heat transfer in a cell of granular material, Schumann and Voss obtained equations for calculating the thermal-conductivity coefficient of a granular material from its porosity \(m\), \(\lambda_m\), and \(\lambda_g\), which do not satisfy conditions \((a)\) and \((g)\). An additional condition of the form is adopted:

\[ \lambda=\lambda_g \cdot m^3 + (1-m^3)\lambda', \tag{6} \]

by means of which this difficulty is eliminated. Here \(\lambda'\) is the thermal conductivity determined from the equation obtained by them, which does not satisfy conditions \((a)\) and \((g)\).

As a result, the thermal conductivity of a granular material is determined by the relationship of three equations.

The experimental verification carried out by Schumann and Foss yielded results from which it follows:

It is evident that the calculation errors are systematic in nature, in the direction of smaller values of $\lambda$, and reach a considerable magnitude.

The main shortcoming of this method, in our opinion, is the formalistic approach, which consists in the fact that the influence of porosity is taken into account not by means of some equation reflecting the objective physical characteristics of the material structure, but by means of an equation that is a formal condition that the solution satisfy conditions $(a)$ and $(g)$. In this connection, when the porosity decreases, the calculated values of $\lambda$ decrease in such a way that (as is seen for quartz) the discrepancy between the calculated data and experiment increases. This same circumstance, apparently, is the reason for the systematic calculation errors toward smaller values.

Much attention was devoted by V. Z. Bogomolov [5] to calculations of the thermal conductivity of two-component dispersed systems. As the calculation scheme for a granular material he uses a tetrahedral packing of spherical grains $(m = 25.95)$. It is assumed that the temperature gradient exists only in one direction. It is accepted that, at the places corresponding to contacts, the grains are separated by an air interlayer, half of whose thickness is denoted by $a$. The calculation is performed for two cases: with and without allowance for the thermal resistance of the grains. The calculation expressions have the form:

$$ \lambda = \lambda_2 \cdot 3\pi \left[\ln \frac{0.184(r + a)}{a} - \frac{r}{r + a}\cdot 0.09 \right]; \tag{7} $$

$$ \lambda = \frac{3\pi \lambda_2}{\kappa} \left[ \frac{1}{\kappa}\ln \frac{1 - 0.82\,\dfrac{r}{b}\cdot \kappa}{1 - \kappa \cdot \dfrac{r}{b}} - 0.09\,\frac{r}{b} \right], \tag{8} $$

where

$$ \kappa = 1 - \frac{\lambda_2}{\lambda_m}, $$

$$ b = r + a. $$

The use of equations (7) and (8) for practical calculations is possible when reliable values of $a$—half the thickness of the ...

air interlayer. V. Z. Bogomolov considered it possible, from purely geometrical considerations, to relate the value \(a\) to the porosity \(m\) by the equation

\[ \frac{a}{r}=\frac{m-26}{3(100-m)} . \tag{9} \]

Here \(m\) is in percent.

Let us trace the course of the calculations that led to formulas (7) and (8). Having written the Fourier equation for the heat transmitted between two grains,

\[ Q_0=\lambda_2 \frac{\sqrt{3}(r+a)}{2}\int_0^{2\pi} d\varphi \int_0^x \frac{x\,dx}{y}, \]

and making the substitution \(y=(r+a)-\sqrt{r^2-x^2}\), the author gives the solution of the integral

\[ \int \frac{x\,dx}{y} = \int \frac{x\,dx}{(r+a)-\sqrt{r^2-x^2}} \tag{9} \]

in the form

\[ \int_0^x \frac{x\,dx}{(r+a)-\sqrt{r^2-x^2}} = \left[ \sqrt{r^2-x^2}+2(r+a)\ln\left(r+a-\sqrt{r^2-x^2}\right) \right]_0^x . \tag{10} \]

Hence

\[ Q_0=\lambda_2\sqrt{3}(r+a)\pi \left[ \sqrt{r^2-x^2}+2(r+a)\ln\left(r+a-\sqrt{r^2-x^2}\right) \right]_0^x . \]

Assuming that the radius of the area of the circle of heat flow from sphere to sphere must be taken as the upper limit, the author found it possible to use, as the radius of this circle, the distance from the point of contact to the center of triangle \(abc\), i.e., the quantity \(oa\) (Fig. 1):

\[ oa=\frac{\sqrt{3}\,r}{3}. \]

Taking into account that three heat flows pass through each sphere (corresponding to the number of contact points), the amount of heat transferred is found. Referring the latter to the area that falls on each sphere together with the voids,

\[ S=2\sqrt{3}(r+a)^2, \]

the author obtained the calculation relation (7).

Fig. 1.

It is necessary to note the following circumstances.

1) The solution of integral (9) does not correspond to expression (10). We have found it in the form

\[ \int_0^x \frac{x\,dx}{(r+a)-\sqrt{r^2-x^2}} = \left[ \sqrt{r^2-x^2}+(r+a)\ln\left(r+a-\sqrt{r^2-x^2}\right) \right] . \tag{11} \]

2) As the upper limit, \(x=\dfrac{\sqrt{3}\,r}{3}\simeq 0.578\cdot r\) is taken.

It is difficult to agree with such a substitution of the limit if no justification is given.

Similarly, for the case of calculating thermal conductivity with allowance for the thermal resistance of the grain material,

\[ Q_0=\int_0^{2\pi} d\varphi\,\frac{\sqrt{3}}{2}\cdot b\cdot \lambda_z \int_0^x \frac{x\,dx}{b-k\sqrt{r^2-x^2}} . \tag{12} \]

The solution of the integral

\[ \int_0^x \frac{x\,dx}{b-k\sqrt{r^2-x^2}} \]

was found by the author in the form

\[ \int_0^x \frac{x\,dx}{b-k\sqrt{r^2-x^2}} = -\frac{1}{k} \left[ 2\,\frac{b}{k}\ln\left(\frac{b}{k}-\sqrt{r^2-x^2}\right) +\sqrt{r^2-x^2} \right]_0^x . \tag{13} \]

The same operations as in the preceding case, and the use of the upper limit

\[ x=\frac{\sqrt{3}\cdot r}{3}, \]

led the author to calculation formula (8).

The two circumstances mentioned in the preceding case must also be noted here.

1) The solution of the integral given by equation (13) was obtained by us in another form:

\[ \int_0^x \frac{x\,dx}{b-k\sqrt{r^2-x^2}} = -\frac{1}{k} \left[ \frac{b}{k}\ln\left(\frac{b}{k}-\sqrt{r^2-x^2}\right) +\sqrt{r^2-x^2} \right] . \tag{14} \]

2) The admissibility of using, as the upper limit, the value

\[ x=\frac{\sqrt{3}\cdot r}{3} \]

without justification is doubtful.

The author obtained good agreement of the calculations by the formulas he proposed for 19 samples of various soils (deviation up to 15%). The results of calculations by formulas (8) and (9) for the granular materials entered in Table 1 are given in column 11 of that table. It is seen that the deviations are not so substantial, if one takes into account that the calculation is carried out by a purely theoretical method, without any empirical coefficients. However, attention is drawn to the systematic character of the calculation errors toward lower values of \(\lambda\) and to the change in the magnitude of the errors with porosity. The latter is especially evident in the example of quartz, for which the porosity range from \(m=51.5\) to \(m=31.2\%\) is covered. In this porosity interval, the ratios

\[ \frac{\lambda_{\text{calc}}}{\lambda_{\text{exp}}} \]

for the five samples considered are:

\[ \begin{aligned} m&=51.5, & \frac{\lambda_{\text{calc}}}{\lambda_{\text{exp}}}&=0.675,\\ m&=50.8, & \frac{\lambda_{\text{calc}}}{\lambda_{\text{exp}}}&=0.687,\\ m&=50.0, & \frac{\lambda_{\text{calc}}}{\lambda_{\text{exp}}}&=0.750,\\ m&=42.0, & \frac{\lambda_{\text{calc}}}{\lambda_{\text{exp}}}&=0.841,\\ m&=31.2, & \frac{\lambda_{\text{calc}}}{\lambda_{\text{exp}}}&=1.466. \end{aligned} \]

Figure 2 shows the calculated curve \((\lambda = f(m))\) for quartz and plots the experimental points that determined the behavior of the function \(\lambda = f(m)\) from experimental data. Consideration of the behavior of the calculated and experimental curves, their intersection, and their sharp divergence in the region of values of \(m\) close to the porosity of tetrahedral packing, \(m = 25.95\), permits the conclusion that equation (9), to which an essential role is assigned in the method under consideration, is still far from perfect.

If, using tetrahedral packing and the mechanism of heat conduction in a granular material adopted by the author, one maintains rigor in the use of the mathematical apparatus, then we arrive at the following expression for the coefficient of thermal conductivity:

\[ \lambda = \frac{\sqrt{2}\,\pi\,\lambda_2}{\kappa} \left[ \frac{1}{\kappa}\ln \frac{1}{1-\frac{r}{b}\cdot \kappa} - \frac{r}{b} \right]. \tag{15} \]

The latter is obtained from (12) by using the solution (14) and the limits \(x = r\) and \(x = 0\), also replacing the temperature gradient

\[ \Delta t = \frac{\sqrt{3}(r+a)}{2} \quad \text{by the more exact expression} \quad \Delta t = \frac{\sqrt{24}(r+a)}{3\cdot 2}. \]

Fig. 2.

Column 12 of Table 1 gives the values of \(\lambda\), calculated from (15) using (9) to determine the relative gap between

grains. It is obvious that the adopted mechanism and the corresponding relations (15) and (9) agree unsatisfactorily with experiment.

In addition to these three main directions in the theory of calculating the thermal conductivity of dry granular materials, a number of other proposals should be briefly considered.

A. A. Nekrasov [18], taking into account the deformation of grains due to pressure in a layer of spheres having a cubic packing and contact over areas, considers the heat flow through a spherical particle on the basis of Fourier’s law (Fig. 3):

\[ dQ=-\lambda_{\mathrm{m}}\cdot S\cdot \frac{dT}{dL}\cdot d\tau \tag{16} \]

or

\[ dQ=-\frac{dT}{\frac{1}{\lambda_{\mathrm{m}}}\cdot \frac{dL}{S}}\cdot d\tau . \tag{17} \]

Fig. 3.

By analogy with Ohm’s law it is assumed that the quantity

\[ \frac{1}{\lambda_{\mathrm{m}}}\cdot \frac{dL}{S}=dR_0' \tag{18} \]

is the thermal resistance of an elementary layer \(dL\), having area \(S\); \(T\) is temperature, \(\tau\) is time.

The integral of expression (18), taking into account the deformation of the sphere, was found by the author in the form

\[ R_0'=\frac{1}{\pi\cdot \lambda_{\mathrm{m}}\cdot b}\cdot \ln \frac{2a+b}{2a-b}, \tag{19} \]

where the following notation is used:

\[ a=r-r_0, \]

\[ b=\sqrt{r^2-r_0^2}. \]

Verification has shown that an error was made here, since integration of (18) gives:

\[ R_0'=\frac{1}{\pi\cdot \lambda_{\mathrm{m}}\cdot r}\cdot \ln\left(\frac{2d}{\delta}-1\right), \tag{20} \]

where \(\delta=2r_0\) is the linear deformation of the sphere,

\(d=2r\) is the diameter of the sphere.

Therefore the author’s subsequent solution and the result obtained by him,

\[ \lambda= \frac{2\pi\lambda_{\mathrm{m}}\sqrt{\,2r+k\Sigma^{2/3}\,}} {2\sqrt{\,2r-k\Sigma^{2/3}\,}} \cdot \frac{1}{ \ln \frac{ 2k\Sigma^{2/3}-4r-\sqrt{\,4r^2-k^2\Sigma^{2/3}\,} }{ 2k\Sigma^{2/3}-4r+\sqrt{\,4r^2-k^2\Sigma^{2/3}\,} } } \tag{21} \]

cannot be considered correct.

The form of equation (21) also gives grounds for doubting its suitability for practical calculations for the following reasons:

1. It does not take into account the thermal conductivity of the medium filling the cells formed by the grains. As is known \[17, 13, 19, 15, 5, etc.\], it is precisely the latter that is one of the determining factors in the thermal conductivity of a granular material.

2. The quantity \(\lambda_m\) enters the equation as a multiplier. This makes it possible to assume that the \(\lambda\) of granular materials whose grains consist of substances close in Poisson’s ratio \(\sigma\) and compression modulus \(E\), and whose grain sizes are equal, will be proportional to the coefficients of thermal conductivity of the substances of which the grains consist. The latter contradicts experiment \[17, 13, 19, 5, etc.\].

In Nekrasov’s work there is clearly an erroneous initial premise. The calculation considered is based on the assumption that heat transfer between neighboring grains takes place only through contact, while the air gaps are considered non-heat-conducting. In reality, the role of contacts is insignificant, and the main share of heat is transferred outside the contact \[3\].

Smith’s study \[20\] does not provide an effective calculation method because of the empirical manner in which the structural factor, introduced into the calculation as a cornerstone, is determined. But it advances more general ideas about heat transfer in a dry granular material, on the basis of which one can explain and link the influence on thermal conductivity of all the principal factors: porosity, the thermal conductivity of the grain material and the gas, and the size and shape of the grains.

One cannot fail to note, as a major step forward in calculating the thermal conductivity of building materials, the work of B. N. Kaufman \[2\]. Being a generalization of a large amount of experimental data, it makes it possible to calculate reliably the \(\lambda\) of most widely used building materials. Unfortunately, questions of purely theoretical calculation are not given due attention in it.

The calculation formulas for the generalized conductivity of heterogeneous systems by V. I. Odelevskii \[23\], as Chudnovskii points out, have not yet been extended to thermal conductivity. This also follows from the work of A. A. Babanov \[4\].

The variants for calculating thermal conductivity proposed by G. I. Pokrovskii and V. G. Budychev \[24\], F. D. Starostin \[25\], Russell \[26\], and A. U. Franchuk \[27\] apply either to moist granular materials or to porous non-bulk materials. We have come to the conclusion \[3\] that they cannot be applied to calculations of the thermal conductivity of dry bulk granular materials.

As a result of the above analysis of existing methods for calculating the thermal conductivity of granular materials, the following conclusions may be drawn.

1. The calculation methods \[6, 12\] based on taking into account the temperature jump at the solid–gas interfaces have a number of shortcomings: \[12\] does not take into account porosity or the character of grain packing, while \[6\], in addition, does not take into account the thermal resistance of the grains. As a result, for coarse-grained materials the calculation either agrees insufficiently with experiment \[6\], or the existing systematic discrepancies between calculation and experiment are eliminated by means of a constant numerical coefficient \[12\].

2. Methods based on calculating the limiting values of \(\lambda\) of a granular material \[14, 15\] give too wide a range of \(\lambda\) and cannot be used for calculations of the thermal conductivity of granular materials.

1. In one of the methods [17], based on the assumption that there is no contact between grains but that an air gap is present, the porosity of the material is connected with the thermal process only formally; therefore, the calculated values of \(\lambda\) deviate substantially and systematically from the experimental ones.

The second theory [5] proceeds from tetrahedral packing and gives a more substantiated relationship between the size of the air gap and the porosity; however, this theory too is not without shortcomings. If one takes into account that the calculation relations were apparently obtained empirically, then its value is considerably reduced.

1. The empirical character of the structural factor introduced into the calculation method [20] does not at present allow this method to be regarded as effective, and raises the problem of finding ways for a theoretical determination of the structural factor.

2. The questions of heat transfer through contact points and their role in the process, as well as the influence of grain shape on the thermal conductivity of a granular material, have been studied quite insufficiently.

3. The noted shortcomings of the existing methods raise the question of creating more perfect methods for calculating the thermal conductivity of granular materials. Such a method must reflect the unity of the conduction mechanism for both powder-like and coarse-grained materials and, consequently, of the calculation relation for determining their thermal conductivity. It must take into account all the principal determining factors both for fine-grained (powder) materials and for coarse-grained ones: the possible influence of thermal resistances at the solid body—gas boundaries, grain size, porosity of the packing, the nature of the arrangement and contact of the grains, and the properties of the substance of the grains and of the gas filling the space between the grains.

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