DISTRIBUTION OVER THE CROSS SECTION OF HEAT TRANSFERRED BETWEEN GRAINS IN A DRY BULK GRANULAR MATERIAL

A. S. Lyalykov

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

The calculation of the thermal-conductivity coefficient \(\lambda\) of a bulk granular material is usually based on consideration of heat transfer in an elementary typical cell of the granular material. As the most general form, let us represent the structure of a granular material as shown in Fig. 1. The grains of the material have an irregular shape. Each grain occupies a position of stable equilibrium, resting on three grains lying below it. The places where they touch—contacts—may be in the form of points and areas; moreover, within the areas there may be gaps caused by the microroughness of the grain surfaces. The spaces between the grains are filled with a gaseous medium. An elementary typical cell of such a material may be represented in the form of a cube \(abcd\), whose side is equal to the mean grain size. It includes parts of the contacting grains and the corresponding spaces filled with gas. The mean porosity of the elementary cell is equal to the overall porosity of the material.

Fig. 1.

One of the features of a granular material is the presence of contacts between grains, the influence of which on the thermal-conductivity coefficient must be taken into account in the calculation. Therefore, in developing a method for the theoretical calculation of \(\lambda\), it is important to determine the typical character of the contacts between grains and to establish what fraction of the total heat transferred by the cell falls directly on the contact, on the nearest zones, and on the remaining portions of the cross section; that is, to establish the distribution of the transferred heat over the cross section and the influence of the character of contacting upon it.

Here we shall not attempt to establish the typical character of contacting, but shall set ourselves the task of estimating the role of the regions of contact in the thermal process of a granular material for various cases of contact.

In order to obtain a qualitative picture of the phenomenon and to facilitate the use of mathematical apparatus, let us consider the distribution of the transferred heat over the cross section of a cell, using the notion of a “fictitious” layer whose particles are identical in size, have an ordered packing, and have a regular, for example spherical, shape. We shall confine ourselves to considering one of the contacts included in an elementary cell, since the phenomena in the remaining ones will have the same character.

A. Case in which there is a flat contact between the grains

Two spherical grains touch over a circular area whose radius is \(r_0\) (Fig. 2). This area may be the result of

Fig. 2.

deformation of the spheres due to external loads on the layer or due to the weight of the overlying layers of material. Let the horizontal planes passing through the points \(O'\) and \(O''\) correspond to the temperatures \(t_1\) and \(t_2\). The space between the grains is filled with a gaseous medium with thermal-conductivity coefficient \(\lambda_g\); for the material of the grains it is equal to \(\lambda_m\).

Heat fluxes over the cross section

With the existing temperature difference \((t_1 - t_2) = \Delta t\), the heat flux \(q_x\) in the section \(dx\,dy\) depends on the thermal resistance \(R_x\), which varies with the change of the coordinate \(x\). We assume that the isotherm-

ical planes are parallel to the \(x\)-axis. Then the heat fluxes in the region from \(x=0\) to \(x=r_o\) are:

\[ q'_x=\frac{\Delta t}{R_x}=\frac{\Delta t}{\dfrac{2r}{\lambda_m}}, \qquad q'_x=\frac{\lambda_m\Delta t}{2r}; \tag{a} \]

in the region from \(x=r_o\) to \(x=r\):

\[ q''_x=\frac{\Delta t}{R_x} =\frac{\Delta t}{\dfrac{2a}{\lambda_m}+\dfrac{\delta}{\lambda_g}} =\frac{\Delta t}{ \dfrac{2\sqrt{r^2-x^2}}{\lambda_m} + \dfrac{2\left(r-\sqrt{r^2-x^2}\right)}{\lambda_g} } \]

or

\[ q''_x= \frac{\Delta t}{ 2r\left[ \dfrac{1}{\lambda_g} + \left(\dfrac{1}{\lambda_m}-\dfrac{1}{\lambda_g}\right) \sqrt{1-\left(\dfrac{x}{r}\right)^2} \right] }. \tag{б} \]

In making the substitution \(\delta=2\left(r-\sqrt{r^2-x^2}\right)\), we do not take into account a certain displacement of the contours of the hemispheres due to deformation at the contact. In determining quantitative characteristics, the admissibility of such a substitution requires proof. In the present case, as will become clear below, the substitution \(\delta=2\left(r-\sqrt{r^2-x^2}\right)\), corresponding to the assumption of point contact, makes it possible to obtain a qualitatively correct representation, which is our immediate task.

The common factor \(\dfrac{\Delta t}{2r}\) in equations \((a)\) and \((б)\) is a quantity proportional to the heat flux. Under conditions of steady flux and for a given granular material, \(\dfrac{\Delta t}{2r}\) is a constant quantity, which may be denoted by \(C_1\). Then we rewrite \((a)\) and \((б)\) in the form

\[ q'_x=C_1\lambda_m=\mathrm{const}, \tag{1} \]

\[ q''_x = C_1 \frac{1}{ \dfrac{1}{\lambda_g} + \left(\dfrac{1}{\lambda_m}-\dfrac{1}{\lambda_g}\right) \sqrt{1-\left(\dfrac{x}{r}\right)^2} }. \tag{2} \]

Expressions (1) and (2) characterize the variation of the heat fluxes with respect to the relative coordinate.

Heat transferred through the section between two grains, and its distribution over the section

In the \(xy\) plane, the total area through which heat passes from one grain to another is equal to the area of a circle of radius \(r_o\). The heat \(dQ_x\) passing per unit time through an annular element of section \(dS_x\) may be written as:

\[ dQ_x=\frac{\Delta t\, dS_x}{R_x}. \tag{в} \]

On the interval from \(x=0\) to \(x=r_0\)

\[ dQ_x=\frac{\Delta t\cdot 2\pi\cdot x\cdot dx}{\dfrac{2r}{\lambda_M}} =\frac{\lambda_M\Delta t\cdot 2\pi\cdot x\,dx}{2r}. \tag{г} \]

Similarly, for the interval from \(x=r_0\) to \(x=r\),

\[ dQ_x''=\frac{\Delta t\cdot 2\pi\cdot x\cdot dx}{\dfrac{2a}{\lambda_M}+\dfrac{\delta}{\lambda_2}} = \frac{\Delta t\cdot \pi\cdot x\cdot dx} {r\left[\dfrac{1}{\lambda_2}+\left(\dfrac{1}{\lambda_M}-\dfrac{1}{\lambda_2}\right)\sqrt{1-\left(\dfrac{x}{r}\right)^2}\right]} . \tag{д} \]

Equations (г) and (д) may be rewritten in the form:

\[ \frac{dQ_x'}{d\left(\dfrac{x}{r}\right)} = \Delta t\,\pi r\,\lambda_M \left(\frac{x}{r}\right) \bigg|_{\frac{x}{r}=0}^{\frac{x}{r}=\frac{r_0}{r}}, \tag{е} \]

\[ \frac{dQ_x''}{d\left(\dfrac{x}{r}\right)} = \Delta t\,\pi r\, \frac{\left(\dfrac{x}{r}\right)} {\dfrac{1}{\lambda_2}+\left(\dfrac{1}{\lambda_M}-\dfrac{1}{\lambda_2}\right)\sqrt{1-\left(\dfrac{x}{r}\right)^2}} . \tag{ж} \]

The derivatives obtained represent the equations for the distribution of heat with respect to the relative coordinate for the case under consideration.

The product \(\Delta t\pi r\), which is a factor in equations (е) and (ж), may be regarded as

\[ \Delta t\pi r=\frac{\Delta t\cdot 2\pi\cdot r^2}{2r}=2C_1S_0, \]

where \(S_0=\pi r^2\) is the total cross-sectional area of the heat flow between the grains. The quantity \(2C_1S_0\), like \(C_1\), is a quantity proportional to the heat flow. Let us denote \(2C_1S_0=C_2\).

Then the equations for the distribution of heat with respect to the relative coordinate for the case of contact of grains over an area will finally be written as:

\[ \frac{dQ_x'}{d\left(\dfrac{x}{r}\right)} = C_2\lambda_M\cdot \left(\frac{x}{r}\right) \bigg|_{\frac{x}{r}=0}^{\frac{x}{r}=\frac{r_0}{r}}, \tag{3} \]

\[ \frac{dQ_x''}{d\left(\dfrac{x}{r}\right)} = C_2\, \frac{\left(\dfrac{x}{r}\right)} {\dfrac{1}{\lambda_2}+\left(\dfrac{1}{\lambda_M}-\dfrac{1}{\lambda_2}\right)\sqrt{1-\left(\dfrac{x}{r}\right)^2}} . \tag{4} \]

B. The case in which there is point contact between the grains

In contrast to the preceding case, with point contact there is no interval from \(x=0\) to \(x=r_0\) with a linear law of varia-

... of the distribution of heat fluxes and of the heat transferred. Consequently, the heat fluxes are determined by the relation

\[ q_x = C_1 \frac{1}{ \dfrac{1}{\lambda_2} + \left( \dfrac{1}{\lambda_M} - \dfrac{1}{\lambda_2} \right) \sqrt{1-\left(\dfrac{x}{r}\right)^2} }. \tag{5} \]

The distribution of the heat transferred is likewise described by a single equation, valid from \(x=0\) to \(x=r\):

\[ \frac{dQ_x}{d\left(\dfrac{x}{r}\right)} = C_2 \frac{ \left(\dfrac{x}{r}\right) }{ \dfrac{1}{\lambda_2} + \left( \dfrac{1}{\lambda_M} - \dfrac{1}{\lambda_2} \right) \sqrt{1-\left(\dfrac{x}{r}\right)^2} }. \tag{6} \]

B. The case in which there is a gap between the grains

(no contact)

Let the grains be separated by a gap whose magnitude along the axis \(o'o''\) is \(\delta_0\) (Fig. 3).

In this case the heat fluxes over the cross section are expressed as:

\[ q_x = \frac{\Delta t}{R_x} = \frac{\Delta t}{ \dfrac{2a}{\lambda_M} + \dfrac{\delta}{\lambda_2} + \dfrac{\delta_0}{\lambda_2} } \]

or

\[ q_x = C_1 \frac{1}{ \dfrac{1}{\lambda_2} + \dfrac{\delta_0\cdot 1}{d\cdot \lambda_2} + \left( \dfrac{1}{\lambda_M} - \dfrac{1}{\lambda_2} \right) \sqrt{1-\left(\dfrac{x}{r}\right)^2} }, \tag{7} \]

where \(d=2r\).

The same reasoning as in cases A and B leads to the following equation for the distribution of the heat transferred:

\[ \frac{dQ_x}{d\left(\dfrac{x}{r}\right)} = C_2 \frac{ \left(\dfrac{x}{r}\right) }{ \dfrac{1}{\lambda_2} + \dfrac{\delta_0\cdot 1}{d\cdot \lambda_2} + \left( \dfrac{1}{\lambda_M} - \dfrac{1}{\lambda_2} \right) \sqrt{1-\left(\dfrac{x}{r}\right)^2} }. \tag{8} \]

From (7) and (8) it is evident that, when there is a gap \(\delta_0\) between the grains, the heat fluxes \(q_x\) and the heat transferred \(Q_x\) depend, among other factors, on the relative magnitude of the gap \(\left(\dfrac{\delta_0}{d}\right)\).

The equations obtained, (1)—(8), make it possible to analyze the nature of the variation of the heat fluxes and the distribution of the heat transferred with respect to the relative coordinate, using specific materials as examples and for different types of contact \(A\), \(Б\), and \(B\).

In order to cover as fully as possible the range of materials with different grain thermal conductivities, we calculated the quantities \(q_x\) and \(\dfrac{dQ_x}{d\left(\dfrac{x}{r}\right)}\) for a granular material with grains made of well-

...of a heat-conducting material—steel \((\lambda_m = 30\ \text{kcal}/\text{m}\cdot\text{h}^\circ)\) and of a poorly conducting material—quartz \((\lambda_m = 5.2\ \text{kcal}/\text{m}\cdot\text{h}^\circ)\). The thermal conductivity of air was taken as \(\lambda_g = 0.022\). The numerical values of the constants \(C_1\) and \(C_2\) were of no interest to us, since in plotting the relationships under consideration they affect only the scale of the graph and do not change the character of the functions. Therefore, for simplicity, one may take \(C_1=1\) and \(C_2=1\). The results of the calculations for cases \(A\), \(Б\), and \(B\) (the last for two values \(\delta_0/d = 0.01\) and \(\delta_0/d = 0.1\)) are shown in Figs. 2, 3, and 4.

Fig. 3.

As was found, in case \(Б\) the heat fluxes and heat distribution differ from \(A\) in that over a very limited segment from \(x=0\) to \(x=r_0\) the linear character of the function is replaced by a curvilinear one with a large radius of curvature; therefore the calculations for case \(Б\) were made only for a granular material of steel. In plotting the curves characterizing the variation of the heat fluxes and the distribution of the heat transferred, it turned out that the curves for cases \(A\) and \(Б\) practically coincide; therefore the curves for case \(Б\) are not presented.

In calculating the ratio \(r_0/r\) entering into (3) and (4), the relation between force, deformation, and the mechanical properties...

material and the size of the sphere \([2,\ \text{p. }353]\), which, for the case of spheres of identical diameter and properties, can be transformed into the form

\[ r_0 = 0.712 \sqrt[3]{\frac{p_0 \cdot d(1-\sigma^2)}{E}}, \tag{9} \]

where \(p_0\) is the force at the contact;
\(d = 2r\) is the grain diameter;
\(E\) is the modulus of elasticity of the grain material;
\(r_0\) is the radius of the area formed as a result of deformation;
\(\sigma\) is Poisson’s ratio.

Equation (9) includes the force \(p_0\) deforming the sphere. Considering tetrahedral packing (three contact points for each sphere), we found it as a function of the grain size and the total pressure in the layer \(P_z\), and it is expressed as

\[ P_0 = \frac{3\sqrt{3}}{2(\sqrt{24}+\sqrt{6})}\cdot d^2 \cdot P_z = 0.353\, d^2 \cdot P_z . \]

Substituting this into (9), we obtain:

\[ \frac{r_0}{r} = 2 \cdot 0.712 \sqrt[3]{\frac{0.353\, P_z(1-\sigma^2)}{E}} . \tag{10} \]

Taking for steel

\[ E = 2.1 \cdot 10^6 \frac{\text{kg}}{\text{cm}^2} = 2.1 \cdot 10^{10} \frac{\text{kg}}{\text{m}^2}, \]

\[ \sigma = 0.3,\qquad \gamma_m = 7900 \frac{\text{kg}}{\text{m}^3}, \]

and the thickness of the overlying layer \(z = 0.202\ \text{m}\), with tetrahedral packing of porosity \(m = 0.259\), we shall have:

\[ \gamma = (1-m)\gamma_m = 5850 \frac{\text{kg}}{\text{m}^3}, \]

\[ P_z = \gamma \cdot z = 1727 \frac{\text{kg}}{\text{m}^2}. \]

With the aid of (10) we obtain:

\[ \frac{r_0}{r} = 6.23 \cdot 10^{-3}. \]

The calculations of \(\frac{r_0}{r}\) for quartz are analogous.

The values of \(\frac{x}{r}\) corresponding to the maximum of the curve of distribution of the heat transferred (when the latter is difficult to identify in the graphical construction of this function) were determined by the usual rules of mathematics, with subsequent determination of the maximum itself.

In the figures, the corresponding quantities with the subscript “с” refer to steel, and with “к” to quartz. The index \(A\) refers to case \(A\), the index 1 to case \(B\) for \(\delta_0/d = 0.01\), and the index 2 for \(\delta_0/d = 0.1\).

Fig. 4

The graphs in Figs. 2, 3, and 4 make it possible to note the following.

a) When there is an ideal area contact between grains (case \(A\), Fig. 2), the latter corresponds to a maximum heat flux, which has a constant value for values of \(x\) from \(x=0\) to \(x=r_0\). For \(x>r_0\) it decreases sharply. The magnitudes of the maximum value \(q_x\) and the steepness of the curve are determined by the thermal-conductivity coefficient of the grain material and are most pronounced for materials with high \(\lambda_m\).

The curve of the distribution of the heat transferred likewise consists of two sections: from 0 to \(r_0\),

\[ \frac{dQ_x}{d\left(\frac{x}{r}\right)} \]

increases linearly, and thereafter follows a curve with a maximum. Its magnitude and the position of its maximum are determined by the thermal-conductivity coefficient of the grain material \(\lambda_m\).

The area under the heat-distribution curve is the total quantity of heat transferred from grain to grain. As is seen from the graph, it is the greater, the greater the thermal conductivity of the solid material \(\lambda_m\). The fraction of heat transferred directly through the contact, despite the greater heat fluxes, is small. This is connected with the small

contact area. Moreover, through a zone wider than the contact (for example, up to $\dfrac{x}{r}=0.05$ or even $\dfrac{x}{r}=0.1$), a comparatively small fraction of heat is transferred, which decreases as the thermal conductivity of the material $\lambda_m$ decreases. At the same time, the role of heat transfer through the remaining regions increases.

We shall not analyze case B, since it differs insignificantly from A.

b) As can be seen from Figs. 3 and 4, the curves of heat fluxes sharply change their character in the interval of ratios under consideration, $\dfrac{\delta_0}{d}=0.01—0.1$. For $\dfrac{\delta_0}{d}=0.01$ this is a steeply falling curve; for $\dfrac{\delta_0}{d}=0.1$ it is a weakly decreasing line.

It is interesting to note that, for identical values of $\dfrac{\delta_0}{d}$ (especially at elevated ones), the magnitudes of the heat fluxes for quartz and steel are approximately the same.

It is also seen in Fig. 3 that the heat transferred depends mainly on the relative gap $\dfrac{\delta_0}{d}$ and depends little on the thermal conductivity of the grains $\lambda_m$. Through the region lying near $\dfrac{x}{r}=0$, a negligible amount of heat is transferred.

For relatively large $\dfrac{\delta_0}{d}$ (for example, 0.1), the distribution of the heat being transferred is such that the region $\dfrac{x_0}{r}>0.5$ becomes, in terms of heat transfer, approximately equivalent to the region $\dfrac{x}{r}<0.5$.

c) Figure 4 gives the dependences of the heat fluxes $q$ and of the transferred heat $\dfrac{dQ}{d\left(\dfrac{x}{r}\right)}$ for cases A, $B_1\left(\dfrac{\delta_0}{d}=0.01\right)$ and $B_2\left(\dfrac{\delta_0}{d}=0.1\right)$ for one and the same material (quartz). It is seen that the heat fluxes and the amount of heat transferred depend on the mode of contacting; moreover, it may be estimated approximately that the heat transferred in cases A, $B_1$, and $B_2$ is in the ratio $3:2:1$.

From this examination of Figs. 2, 3, and 4 we draw three conclusions that must be taken into account in developing a method for the theoretical calculation of the thermal conductivity of a granular material.

1. In any of the three contacting cases considered, through the contact and the region from $\dfrac{x}{r}=0$ to $\dfrac{x}{r}\simeq 0.02—0.03$, a negligibly small amount of heat is transferred because of the negligible area falling on the contact and this region.

1. The calculation of the thermal conductivity of a granular material must be carried out with allowance for the heat transferred over the entire cross section up to

\[ \frac{x}{r}=1, \]

especially for nonmetallic materials.

1. A prerequisite for developing a reliable calculation method is the correct choice of the nature of the contact.

REFERENCES

1. A. S. Lyatikov. On the question of the thermal conductivity of granular material, dissertation, Tomsk, 1956.
2. Mechanical Engineering. Encyclopedic Handbook, vol. 1, book 1, Moscow, 1947.