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Steady Flow of Natural Gas in Pipeline Networks via Solution of a Nonlinear Differential-Algebraic System of Equations

Shriram Srinivasan, Kaarthik Sundar

Abstract

In the consideration of steady-state flow of gas in pipeline networks, the exclusion of gravity and nonlinear inertial effects (convective acceleration) leads to a fortuitous simplification in the governing equations to yield a system of nonlinear algebraic equations. Consequently, there are no studies that quantify the effect of gravity and inertial effects on the flow of gas in pipeline networks or delineate regimes of flow conditions wherein the effects are significant or negligible. In addressing this need, we consider the steady-state flow equations in pipeline networks without neglecting the gravitational and inertial terms and in place of a system of algebraic equations (one for each pipe), this approach results in a nonlinear system of first-order ordinary differential equations (ODEs) which are coupled through algebraic equations that appear in the form of boundary conditions on the pressure and balance of mass flows at either end. One of our main contributions in this article is to demonstrate how the Newton-Raphson algorithm can still be used to solve the coupled nonlinear differential-algebraic system by utilizing the appropriate forward sensitivity ODEs to evaluate the Jacobian terms arising in the iterative scheme. We also propose a variable transformation that alleviates the poor scaling of the ODE, and we introduce a two-point collocation scheme as a coarse approximation of the system from which to find initial guesses for the Newton iterations. Simulation studies were conducted for a single pipe as well as a large-scale pipeline network with real data. From these studies, we concluded that while the effect of gravity is important, the inertial effect was negligible in all cases. The proposed methodology is applicable to a wide class of pipeline and thermal-fluid networks beyond natural gas, including liquid pipelines and hydrogen transport.

Steady Flow of Natural Gas in Pipeline Networks via Solution of a Nonlinear Differential-Algebraic System of Equations

Abstract

In the consideration of steady-state flow of gas in pipeline networks, the exclusion of gravity and nonlinear inertial effects (convective acceleration) leads to a fortuitous simplification in the governing equations to yield a system of nonlinear algebraic equations. Consequently, there are no studies that quantify the effect of gravity and inertial effects on the flow of gas in pipeline networks or delineate regimes of flow conditions wherein the effects are significant or negligible. In addressing this need, we consider the steady-state flow equations in pipeline networks without neglecting the gravitational and inertial terms and in place of a system of algebraic equations (one for each pipe), this approach results in a nonlinear system of first-order ordinary differential equations (ODEs) which are coupled through algebraic equations that appear in the form of boundary conditions on the pressure and balance of mass flows at either end. One of our main contributions in this article is to demonstrate how the Newton-Raphson algorithm can still be used to solve the coupled nonlinear differential-algebraic system by utilizing the appropriate forward sensitivity ODEs to evaluate the Jacobian terms arising in the iterative scheme. We also propose a variable transformation that alleviates the poor scaling of the ODE, and we introduce a two-point collocation scheme as a coarse approximation of the system from which to find initial guesses for the Newton iterations. Simulation studies were conducted for a single pipe as well as a large-scale pipeline network with real data. From these studies, we concluded that while the effect of gravity is important, the inertial effect was negligible in all cases. The proposed methodology is applicable to a wide class of pipeline and thermal-fluid networks beyond natural gas, including liquid pipelines and hydrogen transport.
Paper Structure (13 sections, 34 equations, 4 figures, 2 tables)

This paper contains 13 sections, 34 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Steady flow in a pipe
  3. Nondimensional equations
  4. Closed-form first integrals for ideal gas EoS
  5. Steady flow in a network of pipelines
  6. Solution approach
  7. Reformulation for networks
  8. Two-point collocation
  9. Results
  10. Case study 1
  11. Case study 2
  12. Case study 3
  13. Conclusion

Figures (4)

  • Figure 1: The figure shows the relative pressure change at the withdrawal end with respect to the pressure computed for a horizontal pipe. For both the ideal and non-ideal EoS, the gravitational effects are significant and the relative change in pressure is a nonlinear function of the pipe inclination. We can also confirm that the choice of the EoS has increasing significance on the computational results as the (positive) angle of inclination increases
  • Figure 2: Northwest pipeline network. The pipeline is shown in orange, the black squares are the compressors, the red/green circles are the withdrawals/injections. The radius of the circle corresponds to the magnitude of injection/withdrawal. The network has a total length of 9228 km, comprising 977 nodes (of which 183 are injection/withdrawal nodes), 1033 edges, and 41 compressors.
  • Figure 3: The abscissa shows the relative difference in predicted nodal pressures due to gravity in the form of a (normalized) histogram as well as the empirical cumulative density function (CDF). The ideal gas EoS is assumed.
  • Figure 4: The abscissa shows the relative difference in predicted nodal pressures due to gravity in the form of a (normalized) histogram as well as the empirical cumulative density function (CDF). The non-ideal EoS is assumed.

Theorems & Definitions (2)

  • Definition 1
  • Remark 1