CALCULATION OF THE THERMAL CONDUCTIVITY COEFFICIENT OF A DRY BULK GRANULAR MATERIAL

(Experimental Verification)

A. S. LYALIKOV

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

In work [1] a relation was proposed for calculating the thermal conductivity coefficient of a dry bulk granular material, having the form

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

The present paper sets forth the results of experiments carried out to verify the agreement of calculations by (1) with experiment.

The thermal conductivity coefficient \(\lambda\) of samples of granular material was determined indirectly,

\[ \lambda = a c \gamma . \tag{2} \]

The thermal diffusivity coefficient \(a\) and the bulk density \(\gamma\) were found experimentally; the heat capacity \(c\), identical in the monolithic and crushed states of the substance, was taken from literature data.

To determine the coefficient \(a\), a one-dimensional heat-flow acalorimeter was designed and manufactured, consisting of a Dewar vessel with thermocouples placed in it along the axis of the vessel. The theoretical basis of the instrument is the solution of the problem of cooling of a uniformly heated semi-infinite rod in a medium of constant temperature. Calculations established that, in the acalorimeter used, the conditions of insulation of the lateral surfaces and of infinite length, which constitute the theoretical basis of the instrument, are reliably satisfied, and the error in determining the coefficient \(a\) does not exceed 6.6%.

According to the general scheme of the experimental setup (Fig. 1), its elements may be represented.

In Fig. 1 the numerals denote: 1 — one-dimensional heat-flow acalorimeter, 2 — refrigerator thermostat, 3 — GES-47 mirror galvanometer, 4 — galvanometer reading device, 5 — resistance box, 6 — laboratory autotransformer, 7 — drying cabinet, 8 — stopwatch or clock with a seconds hand.

Cooling of the calorimeter was carried out at small temperature gradients. The grain diameter of the materials tested was less than 5.6 mm. All this makes it possible to consider that convection and radiation were so small that they could be neglected.

Granular materials were tested in the form of steel, lead, copper, and aluminum balls, since the spherical shape of the grain was assumed in the theoretical calculation of \(\lambda\). A total of 10 specimens were tested, each specimen being tested 3 times, more rarely 2 or 4 times.

Fig. 1. Diagram of the experimental setup.

The values of \(\lambda\) for materials with spherical grains, determined from (2) on the basis of the experimental values of \(d\) and \(\gamma\), are given in Table 1. The values of \(\lambda\) calculated from (1), and the ratios of the calculated—

Table 1

the calculated and experimental values \(\dfrac{\lambda_{\text{calc.}}}{\lambda_{\text{exp.}}}\). The values of the heat capacities and coefficients of thermal conductivity of the grain material and of air were taken from the data of M. A. Mikheev [2]. The thermal resistance at the surface of separation of the solid body and the gas, taken into account in (1) by the quantity \(\delta_0\), for air at atmospheric pressure was found from the formula

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

If, according to the data of Smolukhovsky, Herke, and Lazarev, the value \(C=1.65\ \text{m}\cdot\text{ata}\) is taken as the mean value of the coefficient \(C\), then we obtain \(\delta_0=0.65\cdot 10^{-7}\ \text{m}\).

Examination of the table shows:

a) The experimental values of \(\lambda\) for a layer of spherical grains, regardless of \(\lambda_m\) of the grain substance, are grouped in a relatively narrow range of values from \(\lambda=0.417\) to \(\lambda=0.520\ \text{kcal}/\text{m}\cdot\text{h}^\circ\text{C}\).

b) The calculated values of \(\lambda\) are of the same order as the experimental ones.

c) The ratio \(\dfrac{\lambda_{\text{calc.}}}{\lambda_{\text{exp.}}}\), characterizing the agreement of the calculations with the experiment, is: for steel 1.08–1.15, for lead—0.923–1.01, aluminum—1.25–1.38, copper—1.18.

Figure 2 gives a graphical comparison of calculations by (1) with the results of the experiments performed. Also plotted here are the coefficients of thermal conductivity of steel, lead, and quartz obtained by other investigators [3, 4, 5, 6].

The data of Smolukhovsky are marked in Fig. 2 with an asterisk.

The data of our experiments and the results of experiments by other investigators, plotted in Fig. 2, show that:

a) in the experiment the same character of the dependence of \(\lambda\) on \(d\) is observed as in the calculation by (1); the existing scatter of points is explained by a certain nonuniformity in the state of the surfaces of the sample grains, by the presence of oxide films and work-hardened layers, which were noted in enlarged photographs;

Fig. 2. Comparison of calculations by formula (1) with experimental data.

b) our experimental data on the thermal conductivity of steel and lead coincide with the data of Kling, whose experimental method raises no doubts, and show some discrepancy with the results of Schumann and Foss, whose method is insufficiently reliable;

c) in the region of small sizes, satisfactory agreement between calculations by (1) and Smolukhovsky’s experiments for powdered iron is characteristic.

Thus, there is reason to conclude that the proposed calculation method makes it possible to determine, with sufficient reliability, the order of magnitude of the thermal conductivity of a granular material with rounded grains, and therefore it may be recommended for practical calculations.

REFERENCES

1. Lyalikov A. S. Calculation of the thermal conductivity coefficient of a dry bulk granular material [calculation method], in the present collection.

2. Mikheev M. A. Fundamentals of Heat Transfer. Gosenergoizdat, Moscow–Leningrad, 1949.

3. Schumann T. E. W. and Voss V., Heat Flow through granulated Material. Fuel in Science and Practice, Vol. XIII, No. 8, 1934.

4. King G. Forschung auf Gebiete des Ingenieurwesens, Bd. 9, No. 1, 1938.

5. Muldokandov G. N. Contact thermal conductivity of a layer of spherical particles, Zh. T. F., vol. XVII, issue 10, 1947.

6. Krischer O. Einfluss der Feuchtigkeit, Koernung und Temperature auf die Wärmeleitsfähigkeit körnigen Körper. Beihefte zum Gesundheits Ingenieur, H. 33, R. 1, 1934.