SELECTION OF AN OPTIMALITY CRITERION FOR DETERMINING THE RATIONAL GEOMETRY OF CHANNELS IN EXTRUSION MACHINES

R. N. LYUBLINSKII, N. M. OSKORBIN

(Presented by the scientific and technical seminar of the Department of Automation and Telemechanics)

Extrusion screw machines (ESM) have found wide application for applying rubber coverings to conductors and cable blanks. However, many design parameters of ESM have been selected on the basis of accumulated experience, and their optimality lacks proper substantiation.

In solving an optimization problem, the most difficult and least formalized stage is the choice of a criterion for the operating efficiency of the object being optimized. The problem of choosing an optimality criterion should be posed as follows: to choose a criterion for the efficiency of the device being optimized on the basis of a concretization of the general efficiency criterion of the system in which the optimized device is included as an element. The formulation of precisely such a problem is dictated by the fact that, for the system, the efficiency criterion is, as a rule, known, whereas for an element of this system it is necessary to find a particular criterion that is more sensitive to the varied parameters of the given element and does not contradict the general efficiency criterion for the entire system. Concretization of the general efficiency criterion is carried out either at a qualitative level or, if the model for the general criterion is known, analytically.

In the present work, this approach is used to select the optimality criterion for the channels of a die holder, which is one of the main units of an ESM. At the same time, certain questions of the methodology of formalized detailing of the general criterion are shown.

A schematic representation of an extrusion screw machine for processing rubber compounds is shown in Fig. 1. The extruder consists of screw 1 and cylinder 2, which form the screw process, and of a forming head consisting of die holder 3, die 4, and matrix 5. The essence of the extrusion process is as follows: a rubber compound in the form of preheated strips of a certain width and thickness is fed into the loading hopper; the rotating screw captures the materi—

Fig. 1. Schematic representation of an extruder

homogenizes it, compacts it, and feeds it into the extrusion head; the rubber compound passes through the channels of the mandrel holder and, through the annular gap between the mandrel and the die, which form the shaping zone, is extruded in the form of a monolithic cylindrical tube tightly enveloping the conductor or cable blank passing through the internal channel of the mandrel.

The general efficiency criterion of an extrusion machine operating in a complex technological scheme must be oriented toward the efficiency criterion of the entire production process [1]. Detailing the production efficiency criterion down to the efficiency criteria of individual units of the extrusion-machine type can be carried out analogously to detailing the efficiency criterion of an extrusion machine down to the efficiency criteria of its individual parts, and is not considered in the present work.

As the efficiency criterion of an extrusion machine, let us choose the volumetric output with respect to the extruded material, subject to constraints on the quality indicators of the manufactured product as defined by the corresponding GOST standard [2].

The optimization problem for the structural dimensions of the mandrel-holder channels \(X\) is formulated as follows: determine the value of the vector \(X^*\) satisfying the following conditions:

\[ Q_2(X^*)=\max_X Q_2(X); \tag{1} \]

\[ X\in\Omega, \tag{2} \]

where \(\Omega\) is the region of admissible dimensions within the framework of the selected mandrel-holder design, and the constraint on product quality, by which is meant the value of the eccentricity, that is,

\[ \tau \le R(\varphi)<\tau+\varepsilon, \tag{3} \]

where \(\tau\) is the nominal radius;
\(\varepsilon\) is the permissible change in radius,
\(\varphi\in[0,2\pi]\).

For the volumetric output of the extrusion machine, we write the material-balance equation:

\[ Q_2=Q_{\mathrm{p}}-Q_{\mathrm{y}}, \tag{4} \]

where \(Q_{\mathrm{p}}\) is the output of the feed press;
\(Q_{\mathrm{y}}\) is the leakage flow of the rubber compound at the cylinder walls.

According to [3]:

\[ Q_{\mathrm{p}}=a\cdot\omega, \tag{5} \]

where \(a\) is a constant characterizing the geometric shape of the screw and the properties of the rubber compound.

If the flow equation of the rubber compound is considered in the form of a power law, then the expressions for \(Q_2\) and \(Q_{\mathrm{y}}\) [3] have the form:

\[ Q_{\mathrm{y}}=\frac{k_1}{\mu_1^{n_1}}\cdot P^{n_1}; \tag{6} \]

\[ Q_2=\frac{k_2}{\mu_2^{n_2}}\cdot P^{n_2}, \tag{7} \]

where \(k_1\), \(k_2\) are constants characterizing the structural dimensions of

1) the coupling between the cylinder and the screw;
2) the extrusion head.

\(\mu_1, n_1;\ \mu_2, n_2\) are quantities characterizing the rheological properties of the rubber compound:

1) of the leakage flow;
2) in the extrusion head;

$\mu_1, \mu_2$ — consistency coefficients;
$n_1, n_2$ — flow indices;
$P$ — pressure developed by the press.

It should be noted that in the present work it is assumed that

\[ n_1 = n_2, \]

since the flow index does not depend on the operating mode of the extruder [3]. Substituting expressions (5) and (6) into (4), we obtain

\[ Q_2 = a\omega - \frac{\kappa_1}{\mu_1^{n_1}} \cdot P^{n_1}, \tag{8} \]

but from expression (7) we have

\[ P^{n_1} = \frac{Q_2 \mu_2^{n_1}}{\kappa_2}. \tag{9} \]

Finally, expression (8), taking (9) into account, can be written in the following form:

\[ Q_2 = \frac{a\omega \kappa_2 \mu_2^{n_1}}{\kappa_2 \mu_1^{n_1} + \kappa_1 \mu_2^{n_1}}. \]

The design dimensions of the die-holder channels determine only the value $\kappa_2$. Since $Q_2$ is a monotonically increasing function of $\kappa_2$, the problem of optimizing the design dimensions of the die-holder channels is then formulated as follows: determine the value of the vector $X^*$ satisfying the following conditions:

\[ \kappa_2(X^*) = \max_X \kappa_2(X); \tag{10} \]

\[ X \in \Omega; \tag{11} \]

\[ \tau \leq R(\varphi) < \tau + \varepsilon. \tag{12} \]

The shape coefficient of the head $\kappa_2$ depends both on the design dimensions of the die-holder channels and on the dimensions of the forming zone. This dependence is written in the following form [3]:

\[ \frac{\mu_2}{\kappa_2^{1/n}} = \frac{\mu_k}{\kappa_k^{1/n}} + \frac{\mu_3}{\kappa_3^{1/n}}. \tag{13} \]

Taking into account that the shape coefficient $\kappa > 0$ and $n > 1$, one may write, in the domain of positive real values:

\[ \kappa(X^*) = \max_X \kappa(X) \to \kappa^{1/n}(X^*) = \max_X \kappa^{1/n}(X); \tag{14} \]

\[ \kappa^{1/n}(X^*) = \max_X \kappa^{1/n}(X) \to \kappa^{-1/n}(X^*) = \max_X \kappa^{-1/n}(X). \tag{15} \]

Then, according to expressions (14) and (15), from formula (13) the following expression follows, under the assumption that $\kappa_3 = \mathrm{const}$:

\[ \kappa_2(X^*) = \max_X \kappa_2(X) \to \kappa_k(X^*) = \max_X \kappa_k(X). \]

Let us note that the constraints on the vector $X$ are preserved. Let us now consider the behavior of the constraint on the thickness of the rubber shell. The uniformity of flow along the perimeter of the extruded shell is determined by the uniformity of the pressure distribution, which makes it possible to replace constraints (12) by constraints associated with the nonuniformity of pressure along the perimeter:

\[ S^2 = \sum_{i=1}^{N} (P_{1i} - P_{1\mathrm{cp}})^2 < S_0^2, \tag{16} \]

where \(P_{1i}\) is the measured value of the pressure around the perimeter at the entrance to the forming zone;

\[ P_{1\mathrm{av}}=\frac{1}{N}\sum_{i=1}^{N} P_{1i} \]

is the average pressure value;

\(N\) is the number of pressure-measurement points;

\(S_0^2\) is the value induced by the quantity \(\varepsilon\) from constraint (12).

The form of the functional relationship between \(\varepsilon\) and \(S_0^2\) is determined by the dimensions of the forming zone, which are taken to be the same as in the general formulation of the optimization problem. This relationship can be obtained statistically.

Finally, the optimization problem for the design parameters of the die-holder channels is posed as follows: determine the value of the vector \(X^*\) such that the following conditions are satisfied:

\[ \kappa_{\mathrm{k}}(X^*)=\max_X \kappa_{\mathrm{k}}(X); \tag{17} \]

\[ X\in\Omega; \tag{18} \]

\[ S^2<S_0^2. \tag{19} \]

The solution of the optimization problem (17–19) can be carried out analytically or by statistical methods of the theory of experimental design. In the latter case, the optimality criterion \(\kappa_{\mathrm{k}}\) is calculated on the basis of experimental data by the formula

\[ \kappa_{\mathrm{k}}(X)=\frac{Q_2\mu_2^{\,n}}{(P-P_{1\mathrm{av}})^n}, \]

and the constraints are measured according to formula (16).

REFERENCES

1. V. M. Dobkin. Selection of economic criteria for optimizing the operating and design parameters of reactors. Chemical Industry, 1968, No. 3.

2. Economic Evaluation of Machines for Plastics Processing. Edited by V. G. Chirkov, Moscow, Mashinostroenie, 1968.

3. R. V. Torner. Basic Processes of Polymer Processing. Moscow, Khimiya, 1972.