NUMERICAL SOLUTIONS OF THE PROBLEM FOR DETERMINING THE LOCATIONS AND VOLUME OF LEAKAGE IN CASE OF UNSTEADY FLUID MOVEMENT

This paper considers the numerical solution of the inverse problems on the pipeline network. The task is to determine the locations and volumes of leaks if there are any points in the pipeline with unsteady fluid movement. Methods for solving a linear differential equation are considered.

Consider the process of unsteady laminar fluid flow in a pipeline network containing 3 peaks and 2 nodes. N = 3, M = 2. N is a set of vertices, M is the number of nodes.

Then the process of unsteady isothermal laminar fluid flow with constant density on the linear section (1, 2) and (2, 3) of the pipeline network can be described by the following two linear differential equations [1] - [3]:

In this work, the following values of the parameters involved in the problem [4] are taken:

     §ґ§а§Ф§Х§С  

Let there be a leak in the section (1,2) at the point  = 30, and the leakage volume is determined by the function

Then system (1) - (2) is written in one grid, without superscripts, in the form:

At the initial point , boundary conditions of the fourth kind are set:

Here  and  are, respectively, the pressure and flow rate of the fluid, which determine the state of the process at time  at point

where  We solve equation (5) with respect to

At the point  the last formula is written as

From the last two relations we derive the equality:

We substitute the found value of the left difference derivative into (6). Then

The last equality is a three-point difference scheme. To bring it to its standard form, we introduce the notation:

Remark. Using  we will find the location of the leak  and fluid flow

That is, the inverse problem is being solved. Since we have no experimental measured data for unknowns  and , calculated by   serves as measured data.

The paper proposes a numerical approach for determining the locations and volumes of leaks in case of unsteady fluid movement in pipeline networks of complex structure.  A feature that causes particular difficulties in the numerical solution is the presence of nonseparated boundary conditions at the nodes of the network, arising due to the fulfillment of an analogue of Kirchhoff's law on the preservation of material balance and the continuity of the pressure value.

References:

1. Adamkowski A. Analysis of transient flow in pipes with expanding or contracting sections //J. Fluids Engng 2003; V125, №4, 716-722.
2. Aida-zade K.R., Asadova J.A. Study of transients in oil pipelines // Aut. and Remote Control. 2011. V. 72. № 12. P. 2563–2577.
3. Aida-zade K.R., Ashrafova E.R. Localization of the points of leakage in an oil main pipeline under non-stationary conditions // J. of Eng. Phys. and Therm. 2012. V. 85. № 5. P. 1148–1156.
4. Smirnov M.E., Verigin A.N., Nezamaev N.A. Computation of unsteady regimes of pipeline systems // Russian J. of Applied Chemistry. 2010. V. 83. № 3. P. 572–578.