DIGITAL MODELING OF FILTERS OF UNCONSOLIDATED POROUS MEDIA

V. I. SES, V. S. KARIKH

(Submitted by the scientific seminar of the Scientific Research Institute of Electrical Engineering)

The use of a formula for the filtration coefficient of the form [1]

\[ k=\frac{\varepsilon^{2}}{5S_0^{2}(1-\varepsilon)^{2}}, \]

whose derivation is based on the concept of the hydraulic radius, leads to large errors (up to 100%); here \(\varepsilon\) is porosity, \(S_0\) is specific surface area.

The magnitude of the error is explained by the fact that the mathematical model of the filter reflects the properties of a random packing of spheres of one diameter. The density of such a model, the ratio of the volume of the solid phase \(V_t\) to the total packing volume, cannot be more than 0.74... for systematic packing and, for random packings, not more than 0.63 [3]. For real structures with an arbitrary distribution of granules by size, the density can approach unity arbitrarily closely.

To study the filtration coefficient, a digital model may be used in which the internal structure of the filter is modeled by a random packing of spheres whose sizes have the same distribution function as the granules of the filter.

Such a model is constructed on a digital computer, and then the necessary packing parameters are calculated on it. The main drawback of the digital model is the limited memory of computing machines and the significant amount of machine time required for constructing the model.

In [4] it is proposed, in order to accelerate the calculation, to replace the original packing, specified by the sizes and frequencies of the fractions, by a system of tetrahedra. Such a system is obtained by joining with segments the centers of the mutually contacting spheres in the original packing.

The present work describes an experiment to determine the dependence of the velocity of air filtration through a layer of ash on the granulometric composition of the ash and carries out a comparative analysis of the experimental results with calculated data obtained by the method described in [5].

Experiment

Air filtration was carried out through a 100 g charge of ash of a specified granulometric composition. The charge was placed in a cy-

a cylindrical container with a constant diameter and was compacted with a piston until a layer of the same height was obtained in all observations. Then air from a reservoir of limited volume with an initial pressure \(p_0\) was released into the atmosphere through the described filter. During the observations, the time was measured over which the pressure in the reservoir decreased to \(p_t\). In all observations \(p_0\) and \(p_t\) were constant. The pressure in the reservoir varies with time approximately according to an exponential law [5]:

\[ p_t = p_0 e^{-kt}, \tag{2} \]

where \(k\) is the filtration coefficient, \(t\) is time; or \(t = ck^{-1}\); here \(c = \lg p_t / \lg p_0\) is constant for all observations. Consequently, the experimentally estimated quantity was the time, whose magnitude is inversely proportional to the filtration coefficient.

The observation conditions are given in Table 1, where \(r_i\) is the mean size of the granules of the \(i\)-th powder fraction in \(\mu\), and \(f_i\) is the relative frequency of these granules in pieces.

Table 1

The experiment was planned for constructing a calibration curve, and not on the basis of optimality conditions for carrying it out; therefore, the corresponding compositions were selected.

The measurement results and the calculated values are given in Table 2. Column 1 of Table 2 gives the numbers of the experiments, whose conditions are specified in Table 1. Columns 2 and 3 give the calculated specific surface area and density; \(t\) is the observed filtration time.

Table 2

Analysis of variance of the experiment made it possible to neglect the second structural parameter, the density \(\rho\), for these conditions. It should be noted that the degree of compaction of the powder in all observations was the same; therefore, the change in density and specific surface area is explained by the granulometric composition. The packing density depends only on the relative sizes of the granules in this experiment and cannot serve as an unambiguous estimate of the filtration properties of the material.

On the basis of the experimental results, a regression equation was constructed for \(S_0\) as a function of \(T\), which has the following form:

\[ T = 0.238 S_0^{1.4}, \tag{4} \]

where \(T\) is the filtration time, and \(S_0\) is the specific surface area calculated on the digital model. The filtration time found from expression (4), and its deviation from the observed value \(\Delta\), are given in columns 6 and 7 of Table 2. The deviation of the observed values from the calculated ones does not exceed 6%; the results of the regression analysis and the magnitude of the deviations show that there is a close regression dependence between the filtration time and the specific surface area calculated on the digital model.

Discussion of the Experimental Results

The results of the experiment show that the tetrad method can be successfully used to estimate the magnitude of the specific surface area. An experimental justification for the application of this method to calculating the density of a structure is given in [6].

On this basis, one may conclude that it is correct to use the tetrad method [4] for estimating the geometric parameters of random structures, and especially for finding the optimum, with respect to some structural parameter (density, specific surface area, number of neighbors, etc.).

This method possesses all the advantages of digital models: the low labor intensity of the experiment, the short time required for its execution, the possibility of obtaining any information about the geometry of the structure, and, when modeling another object, the absence of any need for a fundamental restructuring of the model.

A drawback of the model is its abstraction from the shape of the grains of the mineralogical composition and from the state of the surface of the granules forming the structure, as well as the obtaining of biased estimates of the parameters. In some cases these shortcomings require the introduction of coefficients, the need for which disappears when relative values of the structural parameters are found.

LITERATURE

1. A. E. Scheidegger. The Physics of Flow Through Porous Media. Moscow, State Scientific-Technical Publishing House of Petroleum and Mining-Fuel Literature, 1960.

2. L. F. Toth. Arrangement on the Plane, on the Sphere, and in Space. Moscow, State Publishing House of Physical and Mathematical Literature, 1958.

3. Scott C. D., Nature, vol. 188, No. 4754, p. 908, 1960.

4. V. A. Vorob’ev, V. I. Ses’. Izvestiya TPI, vol. 251, Tomsk, Tomsk University Press, 1970.

5. L. S. Leibenzon. Collected Works. Vol. II, Moscow, Academy of Sciences of the USSR, 1953.

6. V. A. Vorob’ev, I. E. Naats, V. I. Ses’. Izvestiya vuzov, “Architecture and Construction,” Novosibirsk, No. 10, 1970.