As the art of pipeline flow simulation has advanced, the rigor and, we trust, the accuracy of the many elements making up a flow model have increased. Steady-state models have given way to transient solutions, fluid properties are calculated and tracked along the pipeline, and configurations are represented in more detail. Isothermal models have been replaced with solutions of the energy flow equation supported by real-fluid thermodynamics and ground heat flow models. Accurate fluid properties and thermodynamics require accurate equations of state. There are many more equations of state than could be reasonably discussed in a single paper. This tutorial reviews current practice using equations of state in the simulation of fluid flow in pipelines, starting with fundamental considerations, following with a discussion of several ancient & modern equations of state, and concluding by discussing what's reasonable to use. Reasonable is a subjective word, and the decisions as to which equations to consider and which to use for what are based on the author's experience in simulating the flow of gases, liquids, supercritical fluids, two-phase systems, on preferences arising from that experience, and on externally imposed requirements.

What's an Equation of State?

An equation of state is a relationship between state variables, such that specification of two state variables permits the calculation of the other state variables. There are many state variables; usually in fluid dynamics we talk about pressure, temperature, and density because these variables appear in the equations of motion.

What Do We Want from an Equation of State?

In pipeline flow simulations we use equations of state for the following:

  • Determine the density from the temperature & pressure for:

    • Linepack calculations

    • Flowmeter calibration

    • Pressure drop calculations

    • Determine thermodynamic variables for;

    • Thermal modeling

    • Compressor calculations

    • Vapor-liquid equilibrium

  • These uses imply certain characteristics of an effective equation of state:

    • Accuracy (<0.1% for custody transfer flow meters)

    • Applicable over wide temperature and pressure ranges

    • Applicable over wide range of compositions

    • Rigorous (for thermodynamics; Not quite the same thing as accuracy)

    • Works for liquids too

    • Easy to use !!!

There is usually a contradiction between the last characteristic and the others. How does the Ideal Gas Law Stack Up? The ideal gas law was originally developed in the form of equation (6), rather than equation (1) because it is easier to measure gas volumes than gas densities, and because chemists tend to think in terms of moles rather than masses. Although the ideal gas law was originally derived from Boyle's law and Charles' law, it can also be obtained from the kinetic theory of gases. Getting the ideal gas law from the kinetic theory of gases requires a couple of assumptions which give us some insight into the physics:

  • The gas molecules occupy no volume, which was already implied by Boyle's law.

  • There are no forces between the molecules except at the instant of collision.

For a gas at atmospheric (standard) conditions, these two assumptions are nearly satisfied.

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