1.0 Introduction

Dynamic models for the flow in pipelines have reached a degree of maturity, indicated by the commercial availability of a variety of off-line and real-time software products from several vendors. The models in current use are based on three different numerical techniques and a number of physical approximations. Our purpose in this paper is to discuss these different approaches, and to present results for simulations of a simple transient scenario, showing how the different numerical approaches compare for a fixed set of physical assumptions, and also showing the effects of various physical approximations with a fixed numerical scheme.

2.0 Equation of Motion

The equations of motion presented below are the generally accepted one-dimensional equations for pipe flow. The momentum and mass equations derive from the Navier-Stokes equations for fluid flow, with the viscous stress terms replaced by the Fanning/Darcy/Weisbach friction force term.

2.3 Energy (Temperature) Equation

The energy equation takes into account the balance between work on the fluid by pressure, friction, and gravitational forces and the changes in the internal energy, kinetic energy of bulk motion, and gravitional potential energy of the fluid. Many formulations of the energy equation are available, differing primarily in the choice of thermodynamic functions used to express the internal energy and its conversion to and from mechanical work. The particular form presented here was developed by Dr. Glenn Bernard.

2.4 Equation of State

The equation of state provides a relation between the pressure, density, and temperature of the fluid.

3.0 Numerical Methods

Differential equations describing real systems, as opposed to ideal systems usually dealt with in mathematics courses, are usually not solvable by analytic methods. The difficulty lies both in the form of the fundamental equations and in the dependance of various properties on the physical variables. Numerical methods are based on approximating the derivatives appearing in the differential equations by averages calculated over some interval in space or time. In some formulations, it is also necessary to assume that certain quantities appearing in the equations do not change over the intervals. The numerical techniques commonly used for pipeline flow simulations may be divided into explicit methods, implicit methods, and the method of characteristics. Typical numerical procedures include the following sequence:

1. Initial profiles are established with a steady-state solution

2. Changes in the dependent variables are calculated for a time interval by numerical approximations to the equations of motion

3. The changes are added to the previous vs1ues to establish the new profile at the end of the time interval

3.1 Explicit Integration

In explicit methods, the numerical approximation of the equations of motion are expressed in terms of the values at the beginning of the time interval, as indicated in Figure 1a. These values are known from the previous time step or the initial profiles, so the calculation of the changes in the flow variables is explicit. Our explicit method uses the equations of motion in terms of the physical variables velocity, density, pressure, and temperature.

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