Equations of State
Equations of state are mathematical relationships that describe the behavior of a substance, such as a gas or liquid, under different conditions of temperature, pressure, and volume. These equations help engineers and scientists understand and predict the properties and behavior of materials, making them essential for designing and optimizing technological processes and systems.
5 Key excerpts on "Equations of State"
eBook - ePub- Noah D. Manring, Roger C. Fales(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
...For many substances, the equation of state is extremely complex and difficult to describe exactly. An example of a relatively simple equation of state is that of an ideal gas. In this case, the ideal gas law is used to relate the pressure and temperature of the gas to the mass density using a determined gas constant. This result is given by (1.1) where is the mass density of the gas, is the absolute temperature of the gas (expressed in Kelvin or Rankine), is the absolute gas pressure, and is the universal gas constant that has been determined for the specific gas in question. Unfortunately, in the case of liquids, the equation of state is not so simple. However, since liquids are fairly incompressible it may be assumed that the mass density of a liquid will not change significantly with the exposed conditions of temperature and pressure. In this case, a first-order Taylor series approximation may be written to describe the small variations in density that occur due to changes in pressure and temperature. 1 This result is given by (1.2) where,, and represent a reference density, pressure, and temperature, respectively. The equation of state may be more meaningfully expressed by making the definitions, (1.3) where is the isothermal fluid bulk modulus and is the isobar fluid coefficient of thermal expansion. Using Equation (1.3) with Equation (1.2) yields the following equation of state for a liquid: (1.4) If the nearly incompressible assumption of this equation is correct, one may infer that the fluid bulk modulus is large and the thermal coefficient of expansion is small. Indeed, this is the case for hydraulic fluids, as shown in subsequent sections of this chapter. 1.2.2 Density-Volume Relationship To evaluate the relationship between fluid mass density and fluid volume, consider a fluid element of mass. This mass may be described as (1.5) where is the fluid mass density and is the volume of the fluid element...
- Ali Danesh(Author)
- 1998(Publication Date)
- Elsevier Science(Publisher)
...Developments in Petroleum Science, Vol. 47, Suppl. (C), 1998 ISSN: 0376-7361 doi: 10.1016/S0376-7361(98)80026-5 4 Equations of State The equality of fugacity of each component throughout all phases was proved, in Chapter 3, to be the requirement for chemical equilibrium in multicomponent systems. The fugacity coefficient, φ i, defined as the ratio of fugacity to pressure, of each component in any phase is related to pressure, temperature and volume by Eq.(3.31), (3.31) The fugacity coefficient can, therefore, be determined from the above with the aid of an equation relating pressure, temperature, volume and compositions, that is, an equation of state (EOS). In general, any equation of state which provides reliable volumetric data over the full range of the integral in Eq.(3.31) can be used to describe the fluid phase behaviour. Several types of EOS have been successfully applied to hydrocarbon reservoir fluids. The simplest, and highly successful equation, is the semi-empirical van der Waals type EOS with two or three parameters. Since 1873, when van der Waals improved the ideal gas equation by including parameters that represented the attractive and repulsive intermolecular forces, the equation has been revised and modified by numerous investigators. Other equations with many parameters have also been used to describe the phase behaviour, some with reasonable success. Amongst these equations, the Benedict-Webb-Rubin (BWR) type[l], which is an empirical extension to the virial EOS, can be applied to both liquid and vapour phases of reservoir fluids. These equations provide no additional reliability in phase behaviour studies, in spite of their complexity, in comparison with the van der Waals type EOS...
eBook - ePubCompressors
Selection and Sizing
- Royce N. Brown(Author)
- 2011(Publication Date)
- Gulf Professional Publishing(Publisher)
...In the past, these equations required the use of a mainframe computer not only to solve the equations themselves, but to store the great number of constants required. This has been true particularly if the gas mixture contains numerous components. With the power and storage capacity of personal computers available, the equations have the potential of becoming more readily available for general use. The Equations of State calculations are covered in more detail later in Chapter 2 in the section entitled “Real Gas Tools.” Mollier Charts Another form in which gas properties are presented is found in plots of pressure, specific volume, temperature, entropy, and enthalpy. The most common form, the Mollier chart, plots enthalpy against entropy. A good example of this is the Mollier chart for steam. Gases are generally plotted as pressure against enthalpy (P-h charts). These are also sometimes referred to as Mollier charts. The charts are readily available for a wide range of pure gases, particularly hydrocarbons and refrigerants. Some of the more common charts are included in Appendix B. First Law of Thermodynamics The first law of thermodynamics states that energy cannot be created or destroyed, although it may be changed from one form to another...
eBook - ePub- Pasquale M. Sforza(Author)
- 2011(Publication Date)
- Butterworth-Heinemann(Publisher)
...Introduction Equations that describe flow through a duct within the constraints of a steady one-dimensional approximation are developed in this section. The equations will be applied to simpler flow situations in other propulsion subsystems, such as combustors, nozzles, inlets, and turbomachinery cascades. The equations are particularly simple to deal with, yet they are commonly acknowledged to be reasonably accurate, even for fairly complex problems. This attribute makes the equation set to be developed particularly useful for preliminary design purposes. The analysis accounts for the following effects: • Changes in combustor cross-sectional area • Variations in gas molecular weight and specific heat • Exchange of heat with surroundings • Drag caused by internal bodies or solid particles in the flow • Losses due to friction on the combustor walls These equations are developed in such a manner that close correspondence with the combustor problems at hand is maintained without significant loss in generality. The format due to Shapiro (1953) is followed, where the basic equations are treated sequentially and then a final set of equations are developed from them. 2.2. Equation of State The basic assumption is that fluid flowing through the system behaves like a perfect gas and therefore follows the perfect gas equation: (2.1) The perfect gas law will be useful here in logarithmic differential form: (2.2) In flows with chemical reactions, as in combustors, afterburners, and rocket nozzles, the molecular weight may change substantially and the effects may need to be taken into account. 2.3. Speed of Sound The speed of sound is the speed at which an infinitesimal pressure disturbance is propagated in a compressible gas. Because the pressure change is slight, the accompanying density change is similarly slight and the process is so rapid little heat can be transferred...
eBook - ePub- Nayef Ghasem(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
...1 Thermodynamics and Fluid-Phase Equilibria DOI: 10.1201/9781003167365-1 At the end of this chapter, students should be able to: Estimate the vapor pressure of pure components. Determine the boiling point and dew point of a mixture. Estimate the molar volume using the equation of state (EOS). Plot the effect of temperature versus density. Use UniSim/Hysys, Aspen Plus, PRO/II, SuperPro, and Aveva Process Simulation software packages to estimate physical properties. 1.1 INTRODUCTION Phase-equilibrium thermodynamics deals with the relationships that govern the distribution of a substance between gas and liquid phases. When a species is transferred from one phase to another, the transfer rate decreases with time until the second phase is saturated with the species, holding as much as it can hold at the prevailing process conditions. When concentrations of all species in each phase cease to change, the phases are at phase equilibrium. When two phases are in contact, a redistribution of the components of each phase occurs through evaporation, condensation, dissolution, and precipitation until a state of equilibrium is reached in which the temperatures and pressures of both phases are the same, and the compositions of each phase no longer change with time. A species’ volatility is the degree to which the species tends to be transferred from the liquid phase to the vapor phase. The vapor pressure of a species is a measure of its volatility. Estimation of vapor pressure can be carried out by empirical correlation. When a liquid is heated slowly at constant pressure, the temperature at which the first vapor bubble forms is called bubble point temperature. When the vapor is cooled slowly at constant pressure, the temperature at which the first liquid droplet forms is known as dew point temperature. 1.2 BOILING POINT CALCULATIONS When heating a liquid consisting of two or more components, the bubble point is where the first formed bubble of vapor...
