The Numerical Modeling of Flows in the Complex Pipe Networks

 

Àrkady A. Atavin ¹), Vladimir V. Tarasevich ²)

 

¹) Institute for Water and Environmental Problems

of Siberian Branch of the Russian Academy of Sciences,

Papaninzev str., 105, Barnaul, 659099, Russia.

E-mail: atavin@iwep.altai.su.

²) Novosibirsk University of Architecture and Civil Engineering (NGASU)

NGASU, Leningradskaya, 113, Novosibirsk, 630008, Russia

tel.: (383-2) 66-47-87; fax: (383-2) 16-11-07; e-mail: tvv@iis.nsk.su

 

 

Abstract

The complex pipe systems are considered. The computation of fluid dynamics in such systems is the widespread and important problem in practice. The considered problems belong to the category of graph-defined problems. The flows in such system may be considered as intermediate kind of flows between one-dimensional and spatial flows.

 

The nonisothermal flows in quality of the general case are considered. The model of such flows is offered which represents in essence the further generalization of model of water hammer. The boundary conditions and initial data problem are discussed. The simplified model of a flow (in neglect by thermal processes) is offered regarding the considered network as system with the lumped parameters.

 

The methods of calculation are considered. Some specific difficulties are discussed which are caused by the scale heterogeneity. The combined implicit-explicit method of calculation for unsteady isothermal flows is described.

 

The suggested technique was used for computations of various real pipe systems. The examples of calculations are demonstrated: the flows in heat supply system, the pressure oscillation in hydrodriver of pressure die-casting machine, the transients in main pipeline of liquefied natural gas. The comparison to experimental data confirming good adequacy of models, is resulted.

 

Keywords: flow, liquid, pipe, system, model, calculation

 

Introduction

The notion "a complex pipe network" means some set of pressure pipelines, connected between themselves arbitrarily. Some equipment (pumps, valves and another devices) may be located in the sites of connection named nodes. The heating supply systems, the systems of industrial pipelines of nuclear and heat power plants, water-supply networks, gas and oil main pipelines, etc. may be the examples. Such system may be described by directed graph, at which arcs correspond to pipes of the system, and vertexes correspond to nodes of the system. Thus the orientation of arcs defines a direction of x axis. The considered problems belong to the category of graph-defined problems [1].

 

MATHEMATICAL FORMULATION OF PROBLEM

 

THE EQUATION DESCRIBING THE LIQUID FLOW IN PIPE

Nonisothermal Flow

Let's consider at first the general case of nonisotermal unsteady one-dimensional flow. Such parameters of flow as pressure p, absolute temperature T and velocity V are chosen in the capacity of unknown variables. Then the flows in each pipe of the network can be described by the following system of equations:

 

(1)

 

(2)

 

(3)

 

where t is time, x is longitudinal coordinate, r is density, w is cross - section area and c is wetted perimeter of pipe, t is tangential stress of friction determined according to Darcy - Weisbach formula [2].

 

Here , , where s_p is isothermal compressibility factor; s_T is coefficient of volumetric thermal expansion; C_p is thermal capacity under constant pressure, a_p is so-called coefficient of elastic repulse of pipe's wall [3]; and a_T is coefficient of linear expansion of material of pipe's wall.

 

In the equations (1) - (3) is nonisothermal velocity of perturbation's propagation, where a₀ is the velocity of water hammer wave [4] ;. q_w is rate of heat flux from environment to the flow:

 

(4)

 

where T_env is environmental temperature, K is heat transfer coefficient.

 

These equations were obtained taking into account the assumption that next estimations are valid for the actual range of pressure and temperature:

, .

The system of equations (1) - (3) belongs to hyperbolic type [5] and has three real characteristics: , and .

 

Isothermal Flow

The unsteady one-phase isothermal flows in the pipe systems are the important and widespread kind of considered flows. This case can be obtained as a result of passage to the limit under in formula (4). Then the energy equation degenerate into equation T = T_env = const and equations (1) - (3) transform into well-known water hammer equations [4, 2].

 

Boundary conditions

The functioning of nodes of system is described by the mixed system of algebraic and order differential equations in common case. This system of equations is used as the boundary conditions for equations (1) - (3). Thus one boundary condition is necessary for the extremity of pipe, where the liquid flows into the node, and two boundary conditions are necessary for the extremity of pipe, where the liquid outflows from the node.

 

The Problem of Initial Data

The parameters of steady flows are used as the initial data for the unsteady process as a rule. The computation of the steady solutions is quite difficult problem too [6, 7]. Some ways of the solving of this problem were discussed in the paper, among them new approach based on application of the artificial intelligence technique[6].

 

The simplified mathematical model

Plenty of pipes and nodes, complexity of system require significant computing resources. However for many kinds of problems we handle with rather smooth variations of parameters of a flow. The contribution of the distributed account of wave properties is negligible in this case. Therefore for smoothly varied currents (and processes) it is possible to take into account properties of elasticity and compressibility integrally, proceeding to model of system with the lumped parameters.

 

For smooth-variation flow the assumed mathematical model can be simplified up to next form:

 

(5)

 

(6)

 

(7)

 

where L and d is the length and diameter of pipe; indexes «l» and «r» denote the beginning (x=0) and end (x=L) of pipe correspondingly; l is Darcy-Weisbach friction factor; p_* is supplementary parameter («integral pressure»), depended on elastic characteristics of liquid and pipe.

 

The weight coefficients q_l > 0 and q_r > 0 , q_l + q_r = 1, are chosen based on the requirement of the best nearness for frequency characteristics of both original and simplified models. One can put q_l = q_r = 1/2 under absence of the specialized information.

 

The method of computation

The application of a method of the characteristics for the calculations of nonisothermal flows (system (1) - (3)) seems to most expedient. This method is often applied for calculation of isothermal flows also (i.g. for water hammer equations) [2]. However the application of a method of characteristics in the classical form requires that Courant numbers C_r for each pipe must be equal 1 strictly. The various techniques are applied for overcoming of this restriction, which conduct to the irregular calculation, or are connected with correction of L or a₀ for some pipes, that introduces the additional errors (sometimes rather significant). The application of the schemes of the running calculation [5] permits to remove this restriction, since instead of strict equality the fulfilment of a more soft inequality C_r £ 1 is required.

 

The system can include both lengthy pipes and short pipes. In this case the length of short pipe will be the limiting factor restricting to mesh width. That leads to the calculation with unjustified small-sized step and large costs computing resources.

 

The explicit - implicit scheme of the running calculation with a variable time step is offered, which permits the calculation to be flexibly adapted to various scales of a system and to the change of character of process. For short pipes the calculation is executed under the implicit scheme, for long pipes under the explicit scheme. For smooth transients the time step can be increased (and the transition to the implicit scheme completely for all system is possible), for currents with large gradients the time step decreases (Courant numbers are to be near but at most 1 ).

 

If on all pipes, contiguous to some node, the calculation occurs under the explicit scheme then the boundary conditions can be easily solved separately for each node. The boundary conditions together with finite-difference equations constitute unified system of the equations in case of the implicit calculation. This system can be solved by direct methods (for example, Gauss elimination), by iterations or sweep method, similarly [1,6].

 

Results of calculations

The above-mentioned technique was used for computations of various actual pipe systems (hydrodrivers, pipe system of nuclear power plant, water supply networks, etc.). The results of research of transient in heat supply system are shown in Fig.1. This system represent the high-branched complex pipe network. The transient was caused by sudden failure at one of pumping plants disposed on the return pipeline. There curve 1 is result of calculation and curve 3 is experimental data for pressure in suction pipe of pump; curve 2 is the result of computation and curve 4 is experimental data for the pressure in the delivery pipe.

 

Fig. 1. Transient in heat supply system.

 

The results of research of unsteady flow in the of pressure die-casting machine is represented in Fig.2. The violent transient caused by impact of press-piston upon molten metal.

 

Fig. 2. Pressure oscillation in squeeze casting machine.

 

The problems arising under designing of the main pipeline of liquefied natural gas were considered. The failure on this pipeline as sudden break of a pipe was simulated. The results of calculations of pressure p, outflow velocity V and temperature T in a site of break are submitted in Fig. 3. The computations were performed by model (1) - (3). One can see from Fig. 3, that the wave processes represent high-frequency fluctuations with not great amplitude superimposed on the basic rather smooth transient. The simplified model (5) - (7) can be quite used for calculation of hydraulic parameters p and V of this basic transient.

 

 

Fig. 3. Transient in pipeline of liquefied natural gas.

 

Conclusion

Numerous computations of unsteady flows in various systems were performed by specified methods. Comparison with experimental data and analytical solutions (where it is possible) has confirmed a good degree of adequacy of used models. The durational computing practice has confirmed high efficiency of used technique for calculations of a wide range of flows in the complex pipe systems.

 

References

[1]     A.F.Voevodin, S.M.Shugrin, The numerical methods of calculating of one-dimensional systems.- Novosibirsk, Nauka, 1981 (in Russian).

[2]     V.L.Streeter, E.B.Wylie, Hydraulic transients. Mc Graw-Hill, 1968.

[3]      N.A. Kartvelishvili. Dynamics of pressure pipelines. Energia, Moscow, 1979. (in Russian)

[4]      N.E.Zukowsky, About the water hammer in water-supply pipelines // Proc. 4th Russian water-supply congress. - Moscow, Russia, 1899 (in Russian).

[5]      B.L.Rozhdestvensky, N.N.Yanenko, The systems of quasilinear equations and their applications to gas dynamics. Moscow, Nauka, 1978. (in Russian)

[6]     V.V.Tarasevich. The Simulation and Mathematical Modelling of the Complex Pipe Systems. 5-th (IMACS) World Congress on Scientific Computation, Modelling and Applied Mathematics. Berlin, August 1997. Proceedings, vol. 3., Computational Physics, Chemistry and Biology. p. 115-120. Editor by Achim Sydow.

[7]      A.P.Merenkov, V.Y.Hasilev, The theory of the hydraulic circuits. - Moscow, Nauka, 1985. (in Russian)