TransHyd: Computer Program for Transient Flow Analysis in Pipeline Networks

 

V. Anisimov ¹, I. Smogalev ¹ and V. Kriventsev ²

 

¹ Thermal Physics Department

Institute of Physics and Power Engineering

Bondarenko 1, Obninsk 249020, RUSSIA

e-mail: anisimov@hematic.obninsk.ru

² Research Laboratory for Nuclear Reactors

Tokyo Institute of Technology

2-12-1 O-okayama, Meguro-ku, Tokyo 152, JAPAN

Phone: +81-3-5734-3062, Fax: +81-3-5734-2959, e-mail: krivents@nr.titech.ac.jp

 

 

Abstract

A new computer program TransHyd is proposed for the transient flow analysis of complex pipeline networks. Areas of application include nuclear power plant equipment, heating systems, water-supply systems, oil and gas pipelines. TransHyd is the advanced version of the HydraNet program developed for a steady-state hydraulic network simulation [1].

Main features of the TransHyd software are summarized as follows:

·        Friendly user interface including the network editor that makes it possible to arrange very complicated hydraulic pipeline systems used in many fields of engineering. The network editor allows inserting pipes, expansions, contractions, wyes, pumps, pressure tanks etc. into the network map;

·        Library of numerous hydraulic elements (local resistances and friction factors for both laminar and turbulent flows) based on the wide-known handbook of I.E. Ideltchik [2];

·        Efficient, accurate and reliable computational techniques of solving the system of non-linear equations using the sparse matrix technology. It allows to analyze the very complex networks in a short time using different computational platforms including IBM-compatible PC.

In this paper, governing equation and main assumptions are discussed. Numerical method is briefly described also.

To verify the software, an application of TransHyd to the sample transient flow in a single pipe connecting two gas-liquid tanks is demonstrated. The exact analytical solution can be easy obtained for simplified case and this solution is compared with numerical on simulated by TransHyd. Numerical solution has shown a good agreement with analytical one in this case.

 

Keywords: pipeline networks; transient hydraulics; numerical analysis; Ideltchik's handbook

 

Governing Equations

Let us consider the physical and mathematical formulation of a problem. Considered here are the plane networks with an arbitrary topology. Fluid flows through the branches connecting network nodes (see Fig. 1). Some nodes can be given with supply or demand of mass flowrate. Some others can be supplied with pressure given by either constant or complex law value. The governing equations describing the fluid flow in such a system can be written as following:

- the mass conservation equation for every node

,

(1)

- and the pressure drops equation on the every branch of the network

,

(2)

where Gij is the mass flowrate on the branch connecting i- and j- nodes; Pi is the i-node pressure; N is the number of neighboring nodes for node i; Qi is the flowrate source in a node i; Lij and Fij are the length and cross-section of the branch; t is the time variable; r is the fluid density and g is the gravity constant. Also, in Eq.

(2), is the integral coefficient of total hydraulic resistance of the branch which includes local resistances and integral friction on the branch as well. The last means that xij is defined by following:

,

(3)

where xm and M are values and number of local resistances while lk is value of friction factor and K is number of sub-branches with different friction law and/or hydraulic diameters DH.

 

Equations

(1) and

(2) have been written under the following assumptions:

·        any branch of network can be replaced by the sequence of elements with defined cross-section (channels with friction factor) and elements with local resistances (expansion, contraction, etc.);

·        any branch with local resistance is assumed to be as short as its dynamic behavior can be neglected;

·        fluid is assumed to be incompressible in every separate branch;

·        flow interference between the sequential channels is negligible;

·        experimental data for local resistances and friction factors obtained for steady-state case can be applied to a transient analysis of networks.

 

Discretization

The full implicit scheme is used for finite-difference discretization of Eq.

(2) to guarantee stability. Then, the system of non-linear equations is linearized in a special way that results in the system of linear equations written for each node of the network. This system is solved by Newton-Raphson iterative procedure. The results are the pressure and flowrate distributions for the given time step. The use of direct Gauss's method based on sparse matrix technology helps to decrease needs for computer memory and CPU time.

 

Calculation of Sample Problem

For verification of algorithm and its realization in the TransHyd program, a sample problem has been chosen as following. The hydraulic system consisting of two gas-liquid tanks connected by pipe has been considered (see Fig. 2). The tanks were filled with liquid at the bottom and gas at the top. If the levels of liquid are different then flow redistributes between two tanks until the same levels are reached.

The governing equations describing the flow in such a system can be written as follows

,

(4)

where B is the integral hydraulic resistance of pipe, c is the integral coefficient reflecting dynamic properties of the connecting pipe. The difference in the levels of liquid between two tanks is

,

where is the initial difference in levels. Also, P0 is the gas pressure that is assumed to be a constant during the process (to simplify deriving of analytical solution).

The initial boundary condition is given by .

Using an additional assumption that the variation of the fluid level in the left tank to be insignificant (Dh << H-h0 and flow regime in the pipe to be laminar (b=B|G|=constant), the following analytical solutions are possible:

a)     In case of the friction forces in the pipe being much greater than the inertia forces, the monotone dumping transient process is realized:

,

(5)

 

,

 

where

b)     Contrariwise, when friction factor is low or, in other words, > 0, the oscillatory dumping process occurs in such a system:

, .

(6)

The realization of both cases is shown in

Fig. 3 and

Fig. 4. The solid lines on these maps show the exact analytical solutions defined by (5) and (6). The marks present the numerical solution obtained with TransHyd. The good agreement of the exact and numerical solutions proves the efficiency of the numerical methods briefly described here and realized in the TransHyd program.

 

Conclusion

In this paper, a new version of the computer program TransHyd for transient simulation of hydraulic (pipeline) networks has been presented. The method of calculation is based on the simple pressure drop and mass conservation equations, which are normally used for calculation of such a system. Even for transient problem, friction factors and local resistance coefficients were used as the same as they obtained from experimental data for steady-state case. This wide-accepted common assumption is used because the real data on transient flow are limited by a few very simplified cases like a friction factor for straight tube. At once, a variety of experimental relations for steady-state flow was collected at the Ideltchik's handbook of hydraulic resistances. TransHyd allows to use these data in calculation via the library of hydraulic resistances and friction factors integrated into software.

The TransHyd program has been verified on a simplified problem that allows to obtain an exact analytical solution to compare with. This comparison has shown a good agreement with numerical simulation performed by TransHyd. It should be noted here, that some further verification should be done for more complex network systems. Nevertheless, the sample problem described in this paper, which has an easy-to-obtain exact analytical solution, allows us to recommend TransHyd for transient analysis of more complex pipeline systems.

 

REFERENCES

1.      I.P. Smogalev, V.V. Anisimov and V.I. Kriventsev, PC Software of Hydraulic Analysis of Pressure Drops. Nuclear Energy, 1991, Vol.70, No.6, pp.402-403

2.      I.E. Ideltchik, Handbook of Hydraulic Resistance, Second Edition, Hemisphere, New York, 1986.

3.      V.V. Anisimov, A.I. Groshev and V.I. Kriventsev, The development of methods and software for complex hydraulic network analysis. Simulation of Heat Transfer and Fluid Flow for Nuclear Power Plant Equipment. Proc. of Thermal Physics Dept., INPE, Obninsk, 1993, pp.64-76

 

 

Fig. 1 Schematic Map of a Plain Hydraulic Network

 

 

 

Fig. 2 Sample of the "Tank-to-Tank" pipeline connected system

 

Fig. 3 The monotone dumping transient process into the "tank-to-tank" sample hydraulic system (case a).

 

Fig. 4 The oscillatory dumping process into the sample hydraulic system (case b).