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# Equation of State of Fluids

December 12, 2013 By: Hsin-Hua Tsuei

The equation of state describes a fluid’s density variation at different conditions. According to thermodynamics, density is usually a function of either pressure or temperature, or both.  There are two ends of the spectrum that we are accustomed to – liquid and ideal gas. For liquid, in many analysis and modeling approaches, a fully incompressible flow assumption, which assumes the fluid density remains unchanged throughout the system, is often used to simplify the governing equations. Similarly, the ideal gas law is a widely used approach for modeling gaseous species.  However, many fluids exist in between the two extremes in engineering systems and applications. For example, since the industrial revolution, engineers have understood the importance of steam, and have been working with it for centuries. Yet steam can’t be described accurately by the ideal gas law.  Similar conditions exist for refrigerants, which are critical for providing cooling air on hot summer days.  This begs the question: How do you properly model the equation of state of complex fluids? It can’t be described by using a simplified approach.

For years, the ASME has invested significant resources towards the development steam tables, in an effort to provide the complex equation of state of steam for industrial applications, which extend to a wide range of industrial systems in power generation, processing, etc. The steam density is, in general, a function of both pressure and temperature.  By using the steam table, any analytical tool can use interpolation in the pressure and temperature range to find the appropriate steam density.  This is often referred to as the “table look-up” method.  A similar effort has been undertaken to generate refrigerant equations of state by companies working with different types of refrigerants, such as R123, R134, etc. What if these tables are not available? Several types of real gas modeling methods exist, using a modified ideal gas law, such as the well-known Ridlich-Kwong equation, or the wet Ridlich-Kwong equation for steam, refrigerants, hydrocarbons, etc.  This provides a convenient, first-order modeling option for complex equations of state without having to generate an extensive property table.

For liquids, the equation of state is often only related to pressure, or the so-called “bulk modulus”. This is essentially the speed of sound in a liquid, which determines how fast a signal (or pressure oscillation) would travel in liquid.  For a pure incompressible flow approach, assuming the density is constant, the speed of sound is infinite. For example, if you are fortunate enough to hitch a submarine ride on board the USS Dallas and want to send out a “ping” signal to your buddy on the USS Houston, located at the other end of the Pacific, the ping signal should be detected instantaneously without delay.  But we know this is not the case. The finite compressibility of liquid allows a small density change, which limits the signal propagating speed, to about 1,500 m/s in water. This is high, but not infinite. For many liquid processing systems, the finite compressibility is often not significant enough to alter the machine performance, and a fully incompressible flow assumption of liquid density is adequate.  However, when working with devices such as solenoids, piezo-electric transducers, actuation valves, etc., the finite compressibility should be taken into consideration to accurately capture pressure oscillation, or acoustics, in liquid.  If not, the pressure oscillation generated by these devices will be “felt” throughout the system immediately without the appropriate propagation speed or decay. The results are often misleading.

When working with fluids, if a straightforward incompressible flow or ideal gas assumption is not applicable, maybe it’s time to think about including a complex equation of state to account for fluid density variations.