1. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Stencil used in the finite difference method

2. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Local errors for p, m·, and T as a function of grid points N

3. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Outlet boundary condition for mass flow for the hydraulic model

4. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Results for hydraulic model. Left: Inlet mass flow, and right: outlet pressure.

5. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Difference between implicit cell centered method and backward Euler upwind method for the hydraulic model. Left: Difference in inlet mass flow, and right: difference in outlet pressure.

6. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Validation of hydraulic model using the implicit cell centered finite difference method. Left: Inlet mass flow, and right: outlet pressure. Modeled results show good agreement with measured values.

7. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Effect of grid refinement for results in Fig. 6. Data presented above is the absolute difference between the selected grid and the high resolution solution (Δx = 0.25 km).

8. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Boundary conditions used in full nonisothermal model. Left: Inlet mass flow f(t), and right: inlet temperature g(t).

9. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Modeled flow results. Top: Inlet pressure, middle: outlet mass flow, and bottom: outlet temperature.

10. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Outlet gas temperature computed using the implicit cell centered method for two different grid spacings, Δx = 0.5 km and Δx = 1 km. For a finer grid the oscillations in outlet temperature are reduced.

11. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Temperature profile along 200 km pipeline computed using the implicit cell centered method with Δx = 1 km. (a) t = 1 h. Before the inlet temperature is reduced the temperature profile is nice and smooth. (b) t = 1 h 40 min. A discontinuous change in inlet temperature introduces nonphysical oscillations in the temperature profile. (c) t = 2 h 10 min. Oscillations are still present but have been slightly damped. (d) t = 4 h 40 min. Oscillations still present and have moved along the pipeline with the gas velocity u.

12. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Left Unphysical oscillations which occur when a discontinuous change in inlet temperature is introduced using the implicit cell centered method and the backward Euler centered method. The oscillations are most dominant in the implicit cell centered method. Right: Temperature profile at the inlet of the pipe using the implicit cell centered method for different grid sizes.

13. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Temperature profile along 200 km pipeline using the two solution strategies. The difference between the two solution strategies is how the energy equation is solved. In the first case the implicit cell centered method is used (CC) and all governing equations are solved simultaneously, while in the second case the hydraulic and thermal model are solved separately (Split). The hydraulic model is solved using the implicit cell centered method while the thermal model is solved using the backward Euler upwind method. (a) t = 1 h 40 min. A discontinuous change in inlet temperature introduces nonphysical oscillations in the case of the implicit cell centered method. (b) t = 1 h 40 min. A close up view of the temperature profile shows that the when the hydraulic and thermal model are solved separately using different discretizations no such oscillations are introduced. (c) t = 2 h 10 min. Oscillations have been damped, but are still present in the implicit cell centered method. (d) t = 4 h 10 min. A difference between the two solution strategies can still be seen.

14. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Outlet temperature as a function of time using the two solution strategies. Solving all three governing equations simultaneously using the implicit cell centered method (CC) introduces small oscillations in the outlet temperature. When solving the hydraulic and thermal model separately using the backward Euler upwind method for the thermal model (Split) no such oscillations are present.

15. Date of download: 10/16/2017
Copyright © ASME. All rights reserved.
From: Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes
J. Offshore Mech. Arct. Eng. 2014;136(3): doi: /
Figure Legend:
Validation of nonisothermal model using operational data from 650 km offshore pipeline. Top: Inlet pressure, middle: outlet mass flow, and bottom: outlet temperature.