Having neglected the membrane mass with the rod and the locking – regulatory body, at moving up, the shut – off and regulating body for the Δh motion, is described by the following equation :
where: ωСdh – the volume part of the over – membrane chamber with the dh height, having filled with the water during dt; μVL, ωVL – the flow rate coefficient and the inlet section of the calibrated hole; μSL(h), ωSL – the flow rate coefficient and the input section of the branch hole; НС – the head in the over – membrane chamber; Н = НIN – Нс – the head outflow through the calibrated hole; НIN – the head at the inlet of the calibrated hole.
Since the system dynamics is studied at the slight movement of the locking – regulatory body (Δh), then in order to be simplified, we linearize the non – linear equation (1). For the linearization, we’ll introduce the variables deviation from the non – initial values. So, we’ll denote:
h = ho + Δh; H = Ho + ΔH.
So, the non – linear function
we present in the following form:
where D (Δh; ΔH) – is the non – linear one, having contained the Δh and ΔH product and their degrees, which are over the first one. Because of the small deviation values Δh and ΔH, the non – linear part of the series can be neglected, and, thus, to be replaced the non – linear function by its linear approximation:
Having given, that Q (h0; Н0) is equal to and substituted (3) into (1), we’ll yield the following:
Having given, that
we’ll yield the following:
So, the resulting equation of the transient regime (e.g. the dynamics) in the coordinates’ increments, we’ll give the dimensionless form, by means of the h and H relative deviations introducing:
where hН and НН – are some constant baseline values of the water level and the moving (in our case, the head in the over – membrane chamber).
Having substituted the Δh and ΔH values, we’ll get the following:
For the dimensionless receiving of all the equation terms, we’ll divide it by the coefficient хIN and then, having taken the Laplace transforms (e.g. the entries in the pictures, or in the operator form), we’ll obtain the dynamics equation, in the following form:
(Т⋅р – 1) хINPUT = КхOUT, (8)
where – the time constant, having obtained from the equation (7), by means of dividing by the coefficient at хINPUT;
– is the transfer coefficient;
р – the symbol (e.g. the operator) of the differentiation.
The characteristic equation of the equation (8) will have the following form:
Т⋅р – l = 0, (9)
Thus, from the equation (10), it is clear, that the check valve operation will be stable, if р has the negative real part. So, the time constant Т, in this case, must be the negative one, i.e. the denominator will be less than zero, in other words, it is presented itself the decreasing function.
Indeed, at the h increase, the flow rate through the calibrated hole is decreased. Thus, here, there is always observed the inequality.
The work is submitted to the International Scientific Conference «Actual problems of science and education», France (Marseilles), June, 2-9, 2013, came to the editorial office оn 29.04.2013.