A numerical method is proposed for solving the problem of pipeline transportation of natural gas through a non-horizontal gas pipeline, when the law of change in the hydrostatic pressure in time is specified at the inlet of the section, and the law of change in the mass flow of gas is set at the output.
The nonlinear, quasi-one-dimensional equations of pipeline gas transport in the isothermal approach are transformed using the natural logarithm of dimensionless gas density and the introduction of forward and backward traveling waves. In the newly constructed system of equations, convective and resistance forces are nonlinear, and the remaining components of the equations of traveling waves are linearized.
The equations of traveling waves involving scaled values of the speed of sound, length of a section, and gas density under normal conditions are presented in dimensionless variables and are approximated using a two-layer implicit scheme and an “upstream” scheme by A.A.Samarsky. The calculation of the forward wave, taking into account the input pressure value, was carried out according to the increasing discrete coordinate, and the backward wave, according to the decreasing discrete coordinate, where the flow velocity at the end of the section was first determined. According to the introduced substitutions, the condition at the exit from the site was transformed to a transcendental equation for the flow velocity, which was solved by multi-step using the Newton tangent method.
Due to the nonlinearity of the equations being solved in the computational domain, an iterative process was organized for each time step.
As an initial condition, we used the results of numerical integration of the dimensionless momentum conservation equation for a given constant gas mass flow rate, which significantly reduced the calculation time than in the case of the flow establishment method.
A computational experiment proved the applicability of the method for various (sinusoidal and discontinuous) options for setting the boundary functions for the inlet pressure and output mass gas flow rate, as well as a curved path.
It has been revealed that in long curved gas pipelines, intensive changes in gas-dynamic indicators occur near the border, where the indicators change in time. If the average boundary indicators tend to increase and decrease, then the ubiquitous transition process proceeds slowly, according to the average value of the hydrodynamic flow velocity.
At small lengths of the section, transients proceed intensively, where the role of the resistance force is negligible.
Calculations carried out for long, medium, and short distances showed that the proposed numerical method and calculation algorithm can be used in other boundary conditions too. The list of such conditions includes the tasks of taking into account the characteristics of the compressor and receiver, the outflow of gas into unlimited space, which were not solved due to their non-linearity.
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Mahkamov, Madaminjon K.; Bozorov, Orifjon Sh.; and Khujaev, Ismatulla K.
"FINITE-DIFFERENCE METHOD FOR CALCULATING A GAS PIPELINE LAID OVER ROUGH TERRAIN,"
Scientific Bulletin. Physical and Mathematical Research: Vol. 2
, Article 10.
Available at: https://uzjournals.edu.uz/adu/vol2/iss1/10