The simulation of functioning of the systems of hydro-automatics under the flow control.

 

SHERONOSOVA TATiANA

 

Postgraduate student

Novosibirsk State University of Architecture and Civil Engineering (NGASU), Leningradskaja 113, Novosibirsk, 630008, Russia

 

 

Abstract

The paper is devoted to elaboration of a technique of «direct» mathematical modeling of automatic control units in the systems of technological pipelines of thermal and atomic power stations, hydrodrivers and so on.

The problem of simulation of disturbances is considered. For the complete description of disturbances it is necessary to simulate external parts of system, i.e. all considered system wholly. However it is possible to isolate the unit of automatic regulation in a separate subsystem if to set the disturbances from external environment as incoming Riemann's invariants.

The offered approach is illustrated for the system of pressure automatic control in a return pipeline of a heat supply system.

For this purpose the mathematical model of the actuator valve was elaborated. It allows to watch the dynamics of its functioning evidently. The mathematical model represents a system of the differential and algebraic equations, which are solved numerically by finite difference method.

The results of accounts of system reaction on disturbances of various amplitude and duration are demonstrated. The contribution of the various factors (friction, inertness) to dynamics of process is specified.

The results may be applied in accounts of thermohydraulic parameters of complicated pipe systems in large heat and power objects.

 

Keywords: pipes, flows, hydraulics, automatic control, valve, pump, power plant, numerical simulation.

 

Introduction

Nowadays the main role in thermal and electrical energy production belongs to large heat and atomic power plants. Pipe systems are an important part of such plants as well as systems of heat supply and consumption. It is impossible to ensure the reliability and economic work of pipelines and its equipment without application of systems of automatic control.

The automatic control equipment has a broad classification depending on goals and control parameters. [1].

It is quit difficult to experiment at considered object as a whole. The mathematical modeling seems to be the most effective way which allows to predict system response to disturbances rather precisely. The numerical experiment is the most convenient and exact means of detection of unknown relationships. It allows to watch dynamics of functioning of regulation system directly.

The complexity of a problem is impossibility of assignment of disturbances of various flow parameters separately from each other, as they are parameters of indivisible hydrodynamic process. The most adequate results can be obtained by modeling of complete system, into which the systems of automatic control include as separate subsystems [2]. However, the modeling as a whole requires significant computing and mathematical resources. Therefore the problem of decomposition of system is important one. That gives the opportunities to research of subsystems (in particular, systems of automatic control) independently from each other, but in view of their interaction with overall system.

 

Formulation of problem

 

Decomposition of researched system (Mathematical test stand).

Let's demonstrate a technique of account on an example of system of automatic pressure control in a return pipeline of a heat supply system. The analysed system is located on inlet duct of pumps. (Fig.1).

 

 

Fig.1. Modelling scheme of pressure pipeline.

 

Assume that the regulating valve is set in system(top figure on Fig. 1.). Here S1, S2 are external parts of the system; A is the control point; P1, P2 are the pressures before and after the valve; Q is the valve discharge.

It is necessary to take into account the reaction of external parts S1 and S2 in response to functioning if CV though isolation of the control unit in a separate subsystem. To make up the exact model we replace this parts to incoming Riemann invariants [3] r₁, s₂ :

(1); (2) .

Where: ñ₁, c₂ is velocity of perturbation's propagation; w₁, w₂ are areas of cross-sections of inlet and outlet pipelines accordingly; P_A, P_B are pressures in points A and B; Q is discharge; r is liquid density.

 

THE CONSIDERED scheme of the CONTROL VALVE

 

 

Fig.2. Scheme of automatic pressure control system.

 

Scheme of automatic pressure control system is given on Fig. 2. On this figure: WP is water pump; CHW is check valve; AV is actuator valve; PV is pilot valve; M is the AV counterweight. The constant pressure P_reg should be provided on inlet duct of the pump (the point A). If the pressure in the point A begin to decrease, the balance of all system will be broken. The pressure in the pulse chamber will decrease too. The valve rod blocks top nozzle and transmitting of high pressure from a point B to a chamber 1 begins to move down rod of AV, reducing the charge through it and increasing its resistance. In result the pressure in a point A begins to increase. When the pressure increases in a point A the valve PV covers port. The high pressure can't transfer into chamber 2 and chamber 1. Therefore will take place overflow of a liquid from a chamber 1 in an atmosphere, and the rod of actuator valve moves up. The resistance of the valve decreases and the discharge through it increases. In the end the pressure in a point A falls.

 

Mathematical formulation of problem and outline of algorithm of calculations

The flows in a pipes are described as follows:

The pressure difference between points A, C, D and B accordingly:

, (3)

, (4)

, (5)

where P_a, P_b, P_c, P_d are pressures in the points A,B,C and D; f (Q) is bump characteristic.

(6)

Here a_wp, b_wp, c_wp are coefficients of bump characteristic; x_chw - resistance of the check valve;

(7), (8)

where z is dimensionless resistance of the valve; n is number of the main pipeline brunches; w is area of a pipe cross section; x_kr is variable resistance of the valve; x_krⁿ is resistance of the completely opened valve; x_krⁿ is the maximum valve stroke; x_kr is a current valve stroke; N - experimental exponent.

 

Equation describing the valve movement is:

⁽⁹⁾

^{where , mk is mass of mobile parts of the valve; mpk is mass of} mass of counterweight of the valve; k is the lever ratio of a valve counterweight; S_m is effective area of a membrane; P₁ is pressure in a chamber 1; S_z is valve area ;

, , F_str - force of "dry" friction in stuffing box seal; a is an inclination of rod to horizon; b is an inclination of the level to horizon.

 

The pilot valve:

(10), (11)

The pressure in a chamber 2 depends on pressure Ð_a and changes in limits from P_atm up to P_b.

(12)

Here P₂ is pressure in a managing chamber of the device; x_sl is drain resistance from the chamber 1 to the chamber 2; Q_sl is the discharge from a chamber 1; D₀ is membrane diameter; D is accuracy of regulation.

The implicit finite-difference scheme is used for solving (9) and (11), which are solved together with non-linear equations (3)-(8), (10), (12) and input disturbance functions (1)-(2).

 

Results of calculations

The numerical experiments have allowed to specify the contribution of the various factors (such as force of friction, system inertness etc.) in dynamics of process of automatic control. Also this experiments have allowed to watch their influence to a time of automatic's operation necessary to establish an predetermined stationary mode.

The diagram on Fig.3 shows the dynamics of regulating process: the 1^st curve is the pulse shape of given disturbances, the 2^ndcurve is the valve stroke, the 3^d curve is the valve discharge, the 4^th curve is regulated pressure, 4a is initial pressure.

The diagram on Fig.4 shows response time of system after smooth disturbance in the form shown on Fig.3. Such disturbance simulates transient from one stationary mode of operations to other with amplitude A_imp and pulse rise time t_imp. The dependence of time of system relaxation t_resp on amplitude A_imp is shown. Here the 1^st curve corresponds to disturbance of a pulse t_imp = 1c, 2^nd curve corresponds to t_imp = 25c, the 3^d curve corresponds to t_imp = 50 c, the 4^th curve corresponds to t_imp = 100c. These diagrams allow to estimate reaction of automatic control system under transients.

 

 

Fig.3. Reaction of the control system in response to given disturbance.

 

 

Fig.4. Response time of system after smooth disturbance in relation to amplitude of pulse.

 

Conclusion

The spectrum analysis, transfer functions and the like technique is used in classical calculation methods of control systems [1], that allow to estimate, basically, only indirect characteristics of process. The account of the nonlinear factors represents significant complexity for these methods. Besides this technique does not allow to take into account in full measure reaction of other parts of the considered object to work of system of regulation.

The offered approach enables directly to set the form of a signal on an input and to receive the form of a signal on an output, taking into account all hydraulic parameters and nonlinear effects.

The application of concept of Riemann invariants has allowed correctly to set of disturbances from external parts of overall system.

The offered technique can be successfully applied in accounts of thermohydraulic parameters of complex pipelines in large heat power plants, systems of pressure pipes in hydroelectric stations, hydrodrives and other machinery.

 

Acknowledgements

The special gratitude is expressed to my research advisor Prof., Dr. Ph.-Math. Vladimir V. Tarasevich for the invaluable help in preparation of this material.

 

References

[1]     A.A.Voronov, D.P Kim and others. The theory of automatic control. In two parts, under edition A.A.Voronov, second edition, Vyschaja Shkola, Moscow, 1986. (in Russian).

[2]     V.V.Okolnishnikov, V.V.Tarasevich. Mathematical Modelling and Simulation of Nonlinear Dynamics in Systems of Automatic Control and Governing of Heat Power Plants. Proc. Of Second Int. Scientific and Technical Conf. «Dynamics of System, Machinery and Engines» (November 1997, Omsk, Russia), vol.3, p. 39. (in Russian).

[3]     B.L.Rozhdestvensky, N.N.Yanenko. Quasylinear Equation System and their Applications to Gas Dynamics. Nauka, Moscow, 1978. (in Russian).