Calculation of the Operating Static Characteristic of the ANV Extrusion Press

A. A. Tatarnikov

(Presented by the scientific seminar of the Department of ATPPP)

The calculation of the operating static throughput characteristic of the extrusion press of a continuous vulcanization unit (ANV) is necessary in developing an automatic control system (ACS) for the diameter of the rubber sheath of a cable product, in order to determine the transfer coefficients for the various channels of external influences affecting the value of the sheath diameter, and in developing technological regimes for extruding the sheath. By the static characteristic we shall understand the dependence \(n_{\text{ш}} = f(V_{\text{т}})\) for a given design of the cable product and of the die (forming tool), where \(n_{\text{ш}}\) is the number of revolutions of the extrusion-press screw, and \(V_{\text{т}}\) is the speed of motion of the cable-product blank.

The fundamentals of the theory of calculation and the calculation of the productivity of extrusion presses are considered in detail, for example, in [1, 2, 3]. However, in these and other literary sources the questions considered are mainly those associated with the production of non-cable products from plastic compounds. One of the differences between the production of cable and non-cable products using presses is as follows. In the process of applying a sheath to the blank of a cable product, the plastic or rubber compound is pressed onto the moving cable blank, which passes through the die at a certain speed \(V_{\text{т}}\), i.e., the inner surface of the “annular” die is movable. In the manufacture of products from plastic, however, the inner surface of the “annular” die remains stationary. In [3] it is noted that the motion of the cable blank through the die affects the amount of compound extruded by the extrusion press at the same screw speed.

In the proposed article an equation is derived for the statistical characteristic of the operation of an extrusion press, taking into account the speed of motion of the cable-product blank \(V_{\text{т}}\).

The productivity of a screw press may be expressed by the equation [1]

\[ Q_{\text{м}} = Q_{\text{д}} - Q_{\text{р}} - Q_L, \tag{1} \]

where
\(Q_{\text{м}}\) is the volumetric productivity of the screw press;
\(Q_{\text{д}}\) is the volume of the direct flow;
\(Q_{\text{р}}\) is the magnitude of the reverse flow in volumetric units;
\(Q_L\) is the magnitude of leakage through the clearance between the screw and the cylinder.

Determining the quantities \(Q_d\), \(Q_p\), and \(Q_L\) by integrating the equation of motion of a viscous fluid and substituting the results of the calculations into equation (1), we obtain the complete flow equation in the form [1]

\[ Q_{\mathrm{m}}=\alpha n_{\mathrm{sh}}-\beta\left(\frac{\Delta P}{\mu}\right)-\gamma\left(\frac{\Delta P}{\mu}\right), \]

where \(n_{\mathrm{sh}}\) is the number of revolutions of the screw worm;
\(\alpha,\ \beta,\ \gamma\) are constants determined by the design of the worm. During operation the values of these coefficients change somewhat [1, 2].
\(\Delta P\) is the pressure drop created by the worm;
\(\mu\) is the viscosity of the rubber compound.

The pressure drop \(\Delta P\) created by the worm is used to form the flow of the rubber compound through the die, \(Q_{\mathrm{m}}\). For steady flow in section I (the channels feeding the die) and section II (the die), Fig. 1, the equations may be written as

\[ \left. \begin{aligned} Q'_{\mathrm{m}} &= k_1\,\frac{\Delta P_1}{\mu_1},\\ Q''_{\mathrm{m}} &= k_{\mathrm{m}}\,\frac{\Delta P_2}{\mu_2}+k_{\mathrm{t}}V_{\mathrm{t}} \end{aligned} \right\}. \tag{3} \]

where \(\Delta P_1\) and \(\Delta P_2\) are, respectively, the pressure losses in sections I and II;
\(\mu_1\) and \(\mu_2\) are the viscosity of the rubber compound in sections I and II;
\(k_1\) and \(k_{\mathrm{m}}\) are, respectively, the constants of the feeding channel and of the die;
\(k_{\mathrm{t}}\) is the constant determining the increase in die output due to the motion through it of the cable-product blank.

Fig. 1. Working channel of the extrusion press

Fig. 2. Graph for determining the coefficient \(\varphi_1\) in equation (6) for an annular die.
\(d_{\mathrm{m}}\) is the outside diameter of the die; \(d_{\mathrm{z}}\) is the diameter of the cable blank.

Determine the values \(\Delta P_1,\ \Delta P_2\) from (3) and substitute them into (2), taking into account that \(\Delta P=\Delta P_1+\Delta P_2\). After some transformations we obtain

\[ Q_{\mathrm{m}} = \frac{\alpha n_{\mathrm{sh}}} {1+(\beta+\gamma)\left(\frac{1}{k_1}\frac{\mu_1}{\mu}+\frac{1}{k_{\mathrm{m}}}\frac{\mu_2}{\mu}\right)} + \frac{ (\beta+\gamma)\frac{k_{\mathrm{t}}}{k_{\mathrm{m}}}\frac{\mu_2}{\mu}\,V_{\mathrm{t}} } { 1+(\beta+\gamma)\left(\frac{1}{k_1}\frac{\mu_1}{\mu}+\frac{1}{k_{\mathrm{m}}}\frac{\mu_2}{\mu}\right) }. \tag{4} \]

We assume, in the first approximation [1, 2, 3], that the flow of the rubber

the mass is isothermal and the quantity \(\mu_1=\mu_2=\mu\). Then (4) is transformed into the form

\[ Q_{\mathrm{m}}= \frac{\alpha n_{\mathrm{sh}}} {1+(\beta+\gamma)\left(\frac{1}{K_1}+\frac{1}{K_{\mathrm{m}}}\right)} + \frac{(\beta+\gamma)\dfrac{K_{\mathrm{t}}}{K_{\mathrm{m}}}\cdot V_{\mathrm{t}}} {1+(\beta+\gamma)\left(\frac{1}{K_1}+\frac{1}{K_{\mathrm{m}}}\right)} . \tag{5} \]

The second term in equation (5) determines the effect on the output of the extruder of the speed of motion of the billet \(V_{\mathrm{t}}\). The amount of plastic or rubber compound extruded \(Q_{\mathrm{t}}\) as a result of the motion of the billet can be determined from the equation

\[ Q_{\mathrm{t}}=K_{\mathrm{t}}V_{\mathrm{t}} =\frac{1}{2}\varphi_1\left(\frac{\pi d_{\mathrm{m}}^2}{4}-S_0\right)\cdot V_{\mathrm{t}} =\frac{1}{2}\varphi_1\cdot\varphi_2\left(\frac{\pi d_{\mathrm{k}}^2}{4}-S_0\right)\cdot V_{\mathrm{t}}, \tag{6} \]

where

\(d_{\mathrm{m}}\) — diameter of the die;

\(d_{\mathrm{k}}\) — diameter of the sheath of the cable product;

\(S_0\) — cross section of the cable-product billet;

\(\varphi_1\) — coefficient taking into account the nonlinearity of the velocity profile in the annular gap of the die. The values are determined from the graph in Fig. 2 [3];

\[ \frac{1}{2}\varphi_1 V_{\mathrm{t}}=V_{\mathrm{av}} \]

— average velocity of motion of the mass in the die due to the motion of the cable billet;

\[ \varphi_2= \frac{\dfrac{\pi d_{\mathrm{m}}^2}{4}-S_0} {\dfrac{\pi d_{\mathrm{k}}^2}{4}-S_0}. \]

Substituting the values of \(Q_{\mathrm{t}}\) from (6) and \(Q_{\mathrm{m}}=\left(\dfrac{\pi d_{\mathrm{k}}^2}{4}-S_0\right)V_{\mathrm{t}}\) into (5) and solving the resulting expression with respect to \(n_{\mathrm{sh}}\), we finally obtain

\[ n_{\mathrm{sh}}= \left(\frac{\pi d_{\mathrm{k}}^2}{4}-S_0\right) \left\{ \frac{ 1+(\beta+\gamma)\left[ \frac{1}{K_1}+\frac{1}{K_{\mathrm{m}}}\left(1-\frac{\varphi_1\cdot\varphi_2}{2}\right) \right] }{\alpha} \right\}\cdot V_{\mathrm{t}}. \tag{7} \]

The obtained equation (7) represents an analytical expression for the static operating characteristic of the extruder, taking into account the speed of motion of the cable billet.

Solving equation (7) with respect to \(d_{\mathrm{k}}\) and then taking partial derivatives with respect to various disturbing actions, for example, \(n_{\mathrm{sh}}\) and \(V_{\mathrm{t}}\), we obtain the values of the corresponding transfer coefficients.

LITERATURE

1. Extrusion of plastic compounds on worm presses (fundamentals of theory). Collection of translations of GNIINTI. Moscow, GNTIML Publishing House, 1960.

2. G. Schenkel. Screw Presses for Plastics. Leningrad, GKhI Publishing House, 1962.

3. E. Bernhardt. Processing of Thermoplastic Materials. Moscow, Khimiya Publishing House, 1965.