CALCULATION OF THE THERMAL CONDUCTIVITY COEFFICIENT OF DRY GRANULAR BULK MATERIAL

(Calculation Method)

A. S. LYALIKOV

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

1

In [1] an analysis is given of the existing methods for calculating the thermal conductivity of dry granular bulk materials, on the basis of which it was noted that some of their shortcomings raise the question of creating more advanced calculation methods.

The aim of the present work is to set forth a calculation method that would reflect the unity of the conduction mechanism in fine-grained powdery and coarse-grained materials, taking into account all determining factors both for powdery materials and for coarse-grained ones: the influence of thermal resistance at the solid body–gas boundary, grain size, the porosity of the packing, the nature of the arrangement and contact of the grains, and the properties of the substances of the grains and of the gas filling the space between them. In another communication it is intended to show the results of an experimental verification of the proposed method for calculating the thermal conductivity coefficient of granular materials.

2

From consideration of heat fluxes and the propagation of heat transmitted through the cross-section between two grains [2], it follows that the region near the contact transmits a negligibly small amount of heat; therefore, the calculation of the thermal conductivity of a granular material must be carried out with allowance for the heat transmitted over the entire cross-section between the grains. It was also established there that the prerequisite for developing a reliable calculation method is a correct accounting of the nature of the contacting.

To clarify the nature of contacting and the manner in which grains are arranged in a layer, we carried out the simplest observations of the formation of the structure during the filling of a material, which made it possible to establish that:

1. The grains are not in a “suspended” state, but have point or flat contacts.

1. The most probable, from the stability conditions, is the presence of contact of each grain with three grains below it (and, consequently, with three grains above it).

Since, on the basis of [2], the heat transferred from grain to grain is practically the same for plane and point contact, we consider it possible to carry out the calculation on the basis of point contact between grains.

A real granular material consists of irregularly shaped grains arranged at random. But for a theoretical consideration it is convenient to use a “fictitious” granular layer—a layer with ordered identical spherical particles. Four schemes of ordered arrangement of spherical particles in a layer are known: cubic, square-pyramidal, plane-triangular, and tetrahedral. Of these schemes, only the last possesses the property that each grain rests on three underlying spheres; consequently, it corresponds more closely than the other schemes to the real packing of a granular material. For these reasons, the tetrahedral packing of spherical grains has been adopted by us as the model of the layer.

One of the most important properties of the structure of a granular material is the presence of developed separation surfaces between the solid body and the gas. Numerous investigations by M. Smoluchowski and P. Lazarev [3, 4, 5, 6, 7] led them to the conclusion that at the boundary between the solid and gaseous phases there may arise a temperature difference of finite magnitude (a “temperature jump”), associated with an additional thermal resistance at the phase-separation surface. This additional thermal resistance may be represented in the form of an additional gas layer, which must be taken into account if one calculates the temperature drop between the gas and the wall from the principal thermal resistance of the gas. M. Smoluchowski showed that the thickness of such an additional layer \(\delta_0\) can be calculated as

\[ \delta_0=\frac{C}{p}\cdot 10^{-7}\ \text{m}, \]

where \(p\) is the gas pressure, atm;

\(C\) is a constant quantity for each gas, determined from experiment, \(\text{m}\cdot\text{atm}\).

According to the experiments of Smoluchowski, Gerke [8], and Lazarev, for air the value of \(C\) is respectively equal to 1.53–1.70; 1.8; 1.34–1.96.

The additional thermal resistance at the phase-separation boundary can substantially affect the thermal conductivity of a loose material, especially as it is comminuted.

The method for calculating the thermal conductivity of powder materials [6], developed with allowance for the mechanism of heat transfer from a solid body to a gas, agrees with experiment.

An attempt to extend it to coarse-grained materials [9] is, in our opinion, expedient. However, the method [9] has shortcomings, which amount to the fact that the calculation is based on cubic packing, does not take into account the possible deviation of the porosity of real systems from the porosity of cubic packing, and therefore the possible errors of the theoretical calculation are compensated by a numerical coefficient determined from experiment for different systems of grain packing.

As the calculation scheme we adopt the tetrahedral packing of grains (Fig. 1). It has a constant porosity \(m = 25.95\%\). Changes—

...the porosity for real structures will be taken into account at the end of the calculation. We shall regard the contacts between grains as point contacts.

Let there be a temperature difference \(t_1-t_2=\Delta t\) between neighboring rows of grains. If it is assumed that the isothermal surfaces are horizontal planes, then the temperature difference between neighboring grains in the direction \(oo\) will also be equal to \(\Delta t\). Consider the transfer of heat between two grains, isolating an element as shown in Fig. 2. We take into account the thermal resistance of the grain material, of the gaseous medium, and the additional resistance at the phase boundary, which we introduce into the calculation in the form of an additional gas layer \(\delta_0\). The latter makes it possible to regard the grains as separated by a gap \(2\delta_0\). Then, for the heat transferred between the grains through the annular element \(dx\), one may write:

Fig. 1.

Fig. 2.

\[ dQ_0=\frac{\Delta t}{R_x}\,dS = \frac{\Delta t}{\dfrac{2\delta_{x.m}}{\lambda_m}+ \dfrac{2(\delta_0+\delta_{x.g})}{\lambda_g}}\,dS . \tag{1} \]

Here \(R_x\) is the specific thermal resistance of the annular element \(dx\);

\(\lambda_m\) is the thermal conductivity coefficient of the grain substance;

\(\lambda_g\) is the thermal conductivity coefficient of the gas.

But

\[ \delta_{x.m}=\sqrt{r^2-x^2}, \]

\[ \delta_0+\delta_{x.g}=\delta_0+\left(r-\sqrt{r^2-x^2}\right), \]

\[ dS=2\pi x\,dx, \]

therefore

\[ dQ_0= \frac{\Delta t\,\pi x\,dx} {\dfrac{\delta_0+r}{\lambda_g} +\left(\dfrac{1}{\lambda_m}-\dfrac{1}{\lambda_g}\right)\sqrt{r^2-x^2}} . \]

Let us denote \(\delta_0+r=b\) and \(1-\dfrac{\lambda_g}{\lambda_m}=k\).

Then

\[ dQ_0=\Delta t\,\pi\lambda_2\frac{x\,dx}{b-k\sqrt{r^2-x^2}} . \tag{3} \]

The total amount of heat transferred from the given grain to the three grains below it will be

\[ Q=3\Delta t\,\pi\lambda_2\int_0^x \frac{x\,dx}{b-k\sqrt{r^2-x^2}} . \tag{4} \]

For a temperature gradient of \(1\,\dfrac{^\circ C}{m}\), the temperature drop over one row of grains in tetrahedral packing will be

\[ \Delta t=\frac{r}{3}\sqrt{24}. \tag{a} \]

The area of the layer attributable to one grain is

\[ S_T=2\sqrt{3}\cdot r^2 . \tag{b} \]

Since the coefficient of thermal conductivity is the heat transferred per unit time through unit area at a temperature gradient equal to unity, then, substituting (a) into (4) and dividing by (b), we obtain:

\[ \lambda=\frac{\sqrt{2}\pi\lambda_2}{r}\int_0^x \frac{x\,dx}{b-k\sqrt{r^2-x^2}} . \tag{5} \]

As a result of integration within the limits from \(x=0\) to \(x=r\), we obtain

\[ \lambda=\frac{\sqrt{2}\pi\lambda_2}{r}\cdot\frac{b}{k} \left[ \frac{1}{k}\ln \frac{1+\dfrac{\delta_0}{r}} {(1-k)+\dfrac{\delta_0}{r}} -1 \right]. \tag{6} \]

Expression (6) was obtained for a granular material having a tetrahedral packing with constant porosity \(m=25.95\%\). The change in porosity in systems differing in packing density from tetrahedral packing shall be taken into account from the following considerations. The area attributable to one grain, to which the heat transferred in determining \(\lambda\) is referred, is variable for the systems mentioned and depends on the porosity. It can be expressed through the porosity and the volume of the layer attributable to one grain, if the height of this volume is regarded as constant and equal to \(h=\dfrac{r}{3}\sqrt{24}\), as for tetrahedral packing.

Since

\[ m=1-\frac{V_{tv.}}{V_{poln.}}, \]

where \(V_{tv.}\) is the volume of the solid phase in the cell, equal to the volume of the sphere

\[ V_{tv.}=\frac{4}{3}\pi r^3, \]

\(V_{\text{full}}\) is the full volume of the cell,

\[ V_{\text{full}}=\frac{\frac{4}{3}\pi r^{3}}{(1-m)}, \]

then the area per one grain will be

\[ S=\frac{V_{\text{full}}}{h}=\frac{4\pi r^{2}}{\sqrt{24(1-m)}} . \tag{7} \]

Since expression (6) was obtained on the basis of the area corresponding to tetrahedral packing, for systems with a different porosity it is necessary to take into account the change in area, for which (6) should be multiplied by the ratio \(\frac{S_T}{S}\):

\[ \lambda= \frac{2\sqrt{3}\,r^{2}}{\dfrac{4\pi r^{2}}{\sqrt{24(1-m)}}}\cdot \frac{\sqrt{2}\pi\lambda_g}{r}\cdot \frac{b}{k} \left[ \frac{1}{k}\ln \frac{1+\dfrac{\delta_0}{r}} {(1-k)+\dfrac{\delta_0}{r}} -1 \right]. \tag{8} \]

Taking into account that \(b\) and \(r\) are numerically close and may, without loss of accuracy, be canceled, we finally obtain:

\[ \lambda=\frac{6(1-m)\lambda_g}{k} \left[ \frac{1}{k}\ln \frac{1+\dfrac{\delta_0}{r}} {1+\dfrac{\delta_0}{r}-k} -1 \right]. \tag{9} \]

Formula (9) takes into account all the principal factors affecting the thermal conductivity of a granular material. In calculations of the coefficient \(\lambda\) by (9), it is assumed that use is made of research results concerning the additional thermal resistance at the interface between the solid and gaseous phases, represented in (9) by the quantity \(\delta_0\).

It should be recalled that relation (9) was obtained for a granular material with spherical grains. Its use for materials with grains of non-spherical shape requires an additional study of the influence of shape on the thermal properties of the material.

REFERENCES

1. Lyashkov A. S. Methods for calculating the coefficient of thermal conductivity of bulk granular materials. “Izv. TPI,” vol. 109, 1960.

2. Lyashkov A. S. Distribution of heat transferred with dry bulk material between grains. “Izv. TPI,” vol. 109, 1960.

3. Smoluchowski M. Über wärmeleitung in verdünnten Gasen. Ann. d. Phys. u. Chem., Bd. 64, s. 101, 1898.

4. Smoluchowski M. Über den Temperatursprung bei Wärmeleitung in Gasen. Sitzungsberichte der Kaiserlichen Akad. der Wiss., Bd. CVII, s. 304, Wien, 1899.

5. Smoluchowski M. Weitere Studien über den Temperatursprung bei Wärmeleitung in Gasen. Sitzungsberichte der Kaiserlichen Akad. der Wiss., Bd. 108, Abr. II-a, s. 5, Wien, 1899.

6. Smoluchowski M. Conductibilité calorifique des corps pulvérisés. Bulletin international de L’Akademie des Sciences de Cracovie, No. 5, 1910.

7. Lazarev P. On the temperature jump in heat conduction at the boundary of a solid and a gas. Journal of the Russian Physico-Chemical Society, phys. part, XLIII, issue 3, 1911.

8. Gehrcke E. Über den Wärmeleitung Verdünnter Gase. Ann. Phys., 4. Folge, Bd. 2, Leipzig, 1900.

9. Hengst G. Die Wärmeleitfähigkeit pulverförmiger Wärmeisolierstoffe bei hohen Gasdruck. Diss., München, 1934. (Cited after Kling G., “Forschung auf Gebiete des Ingenieurwesens,” Bd. 9, No. 1, 1938).