Ideal compressibility factor

The above equation is valid at low pressures where the assumptions hold. However, at typical reservoir temperatures and pressures, the assumptions are no longer valid, and the behaviour of hydrocarbon reservoir gases deviate from the ideal gas law. In practice, it is convenient to represent the behaviour of these real gases by introducing a correction factor known as the gas deviation factor, (also called the dimensionless compressibility factor, or z-factor) into the ideal gas law ... 

Compressibility of Natural Gas All gases deviate from the perfect gas law at some combinations of temperature and pressure, the extent depending on the gas. This behavior is described by a dimensionless compressibility factor Z that corrects the perfect gas law for real-gas behavior, FV = ZRT. Any consistent units may be used. Z is unity for an ideal gas, but for a real gas, Z has values ranging from less than 1 to greater than 1, depending on temperature and pressure. The compressibihty faclor is described further in Secs. 2 and 4 of this handbook. 

Ideal gas obeys the equation of state PV = MRT or P/p = MRT, where P denotes the pressure, V the volume, p the density, M the mass, T the temperature of the gas, and R the gas constant per unit mass independent of pressure and temperature. In most cases the ideal gas laws are sufficient to describe the flow within 5% of actual conditions. When the perfect gas laws do not apply, the gas compressibility factor Z can be introduced ... 

Since non-ideal gases do not obey the ideal gas law (i.e., PV = nRT), corrections for nonideality must be made using an equation of state such as the Van der Waals or Redlich-Kwong equations. This process involves complex analytical expressions. Another method for a nonideal gas situation is the use of the compressibility factor Z, where Z equals PV/nRT. Of the analytical methods available for calculation of Z, the most compact one is obtained from the Redlich-Kwong equation of state. The working equations are listed below ... 

Compressibility is experimentally derived from data about the actual behavior of a particular gas under pVT changes. The compressibility factor, Z, is a multiplier in the basic formula. It is the ratio of the actual volume at a given pT condition to ideal volume at the same pT condition. The ideal gas equation is therefore modified to ... 

For a non-ideal gas, equation 2.15 is modified by including a compressibility factor Z which is a function of both temperature and pressure ... 

At very low pressures, deviations from the ideal gas law are caused mainly by the attractive forces between the molecules and the compressibility factor has a value less than unity. At higher pressures, deviations are caused mainly by the fact that the volume of the molecules themselves, which can be regarded as incompressible, becomes significant compared with the total volume of the gas. 

Equations 4.55 and 4.57 are the most convenient for the calculation of gas flowrate as a function of Pi and Pi under isothermal conditions. Some additional refinement can be added if a compressibility factor is introduced as defined by the relation Pv -= ZRT/M, for conditions where there are significant deviations from the ideal gas law (equation 2.15). 

FIGURE 4.28 A plot of the compression factor, Z, as a function of pressure for a variety of gases. An ideal gas has Z = 1 for all pressures. For a few real gases with very weak intermolecular attractions, such as H2, Z is always greater than 1. For most gases, at low pressures the attractive forces are dominant and Z 1 (see inset). At high pressures, repulsive forces become dominant and Z 1 for all gases. 

We can assess the effect of intermolecular forces quantitatively by comparing the behavior of real gases with that expected of an ideal gas. One of the best ways of exhibiting these deviations is to measure the compression factor, Z, the ratio of the actual molar volume of the gas to the molar volume of an ideal gas under the same conditions ... 

The compression factor of an ideal gas is 1, and so deviations from Z = I are a sign of nonideality. Figure 4.28 shows the experimental variation of Z for a number of gases. We see that all gases deviate from Z = 1 as the pressure is raised. Our model of gases must account for these deviations. 

The presence of intermolecular forces also accounts for the variation in the compression factor. Thus, for gases under conditions of pressure and temperature such that Z > 1, the repulsions are more important than the attractions. Their molar volumes are greater than expected for an ideal gas because repulsions tend to drive the molecules apart. For example, a hydrogen molecule has so few electrons that the its molecules are only very weakly attracted to one another. For gases under conditions of pressure and temperature such that Z < 1, the attractions are more important than the repulsions, and the molar volume is smaller than for an ideal gas because attractions tend to draw molecules together. To improve our model of a gas, we need to add to it that the molecules of a real gas exert attractive and repulsive forces on one another. 

The theoretical equation of state for an ideal rubber in tension, Eq. (44) or (45), equates the tension r to the product of three factors RT, a structure factor (or re/Eo, the volume of the rubber being assumed constant), and a deformation factor a—l/a ) analogous to the bulk compression factor Eo/E for the gas. The equation of state for an ideal gas, which for the purpose of emphasizing the analogy may be written P = RT v/Vq) Vq/V), consists of three corresponding factors. Proportionality between r and T follows necessarily from the condition dE/dL)Ty=0 for an ideal rubber. Results already cited for real rubbers indicate this condition usually is fulfilled almost within experimental error. Hence the propriety of the temperature factor... 

The compressibility factor z of methane is always less than 1.0 in normal temperature ranges (i.e., between —40° and 50° C). Furthermore, the compressibility factor decreases as the pressure rises or the temperature falls. Therefore, less energy is needed to pump a given volume of methane (measured at standard volume) at any given normal temperature than would be expected at that temperature if the methane were an ideal gas. This effect is more marked at higher pressures. Similarly, as the pressure is increased at a constant temperature, more methane (measured at standard volume) can be stored in a given volume than would be predicted from the ideal gas equation. 

If no Mollier diagram is available, it is more difficult to estimate the ideal work in compression or expansion processes. Schultz (1962) gives a method for the calculation of the polytropic work, based on two generalised compressibility functions, X and Y which supplement the familiar compressibility factor Z. 

From the ideal gas equation, it is found that for 1 mole of gas, PV/KT = 1, which is known as the compressibility factor. For most real gases, there is a large deviation from the ideal value, especially at high pressure where the gas molecules are forced closer together. From the discussions in previous sections, it is apparent that the molecules of the gas do not exist independently from each other because of forces of attraction even between nonpolar molecules. Dipole-dipole, dipole-induced dipole, and London forces are sometimes collectively known as van der Waals forces because all of these types of forces result in deviations from ideal gas behavior. Because forces of attraction between molecules reduce the pressure that the gas exerts on the walls of the container, van der Waals included a correction to the pressure to compensate for the "lost" pressure. That term is written as w2a/V2, where n is the number of moles, a is a constant that depends on the nature of the gas, and V is the volume of the container. The resulting equation of state for a real gas, known as van der Waals equation, is written as... 

A typical use for this model would be to solve for the number of moles of a gas, given its identity, pressure, volume, and temperature. The iterative solver is used for this purpose. You must decide which variable to choose for iteration and what a reasonable initial guess is. Real gases approach ideal behavior at low pressure and moderate temperatures. Since the compressibility factor z is 1 for an ideal gas, and since knowing z along with P, V, and T allows a calculation of n, we choose z as the iteration variable and 1.0 as the initial guess. 

Numerous representations have been used to describe the isotherms in Figure 5.5. Some representations, such as the Van der Waals equation, are semi-empirical, with the form suggested by theoretical considerations, whereas others, like the virial equation, are simply empirical power series expansions. Whatever the description, a good measure of the deviation from ideality is given by the value of the compressibility factor, Z= PV /iRT), which equals 1 for an ideal gas. 

As we have already seen, the universal chart of gases assumes that all gaseous species exhibit the same sort of deviation from ideal behavior at the same values of 7, and V. This fact, known as the principle of corresponding states, is analytically expressed by the deviation parameter (or compressibility factor ) Z. For n =, ... 

Practically, in environmental applications, the temperature ranges from 20 to 400 °C and pressure from 1 to 40 atm. According to available data (Perry and Green, 1999) for air, nitrogen, oxygen, and hydrogen, the compressibility factor is practically unity and only in severe conditions of pressure and temperature varies from 0.98 to 1.02. Thus, the ideal gas law can be safely used in most environmental applications. 

Results from a series of Odyssey simulations of non-ideal gases are shown in Figure 6. The compression factor PV/nRT is plotted as a function of the pressure for two systems. The first system is a mixture of hydrogen and helium (T-120 K 90 and 10 molecules, respectively) as it might be encountered in the atmosphere of Jupiter. The second system is pure gaseous ammonia (7 298 K 50 molecules). 

In this example line 902 evaluates the function (2.6) and stores its value in the variable F. We print X and F to follow the iteration. The bracketing interval is chosen on the basis of a priori information. We know that in this example the compressibility factor PV/(RT) is close to one, and use the lower and upper limits xL = v°/2 and Xy = 2v°, respectively, where v° is the ideal molar volume... 

While an ideal-gas law serves very well under many circumstances, there are also circumstances in which non-ideal behavior can be significant. A compressibility factor Z is an often-used measure of the extent of nonideality,... 

For an ideal gas, Z = 1. In general, the law of corresponding states provides that the compressibility factor depend on the reduced temperature and pressure,... 

Figure 3.2 illustrates the relatively complex nature of the compressibility factor s dependence on temperature and pressure. It is evident that there can be very substantial departures from ideal-gas behavior. Whenever possible, it is useful to represent the equation of state as an algebraic relationship of pressure, temperature, and volume (density). Certainly, when applied in computational modeling, the benefits of a compact equation-of-state representation are evident. There are many ways that are used to accomplish this objective [332], most of which are beyond our scope here. 

Figure 2.5 Compressibility factor Z(P) for C02 at 40°C (cf. Fig. 2.2), comparing the Van der Waals approximation (solid line) with experimental values (circles, dotted line) and with the ideal gas approximation (dashed line).

In 1901, H. Kamerlingh Onnes introduced a fundamentally new and improved description of real gas PVT properties in terms of the virial equation of state. [The word virial, deriving from the Latin word viris ( force ) was introduced into physics by R. Clausius, whom we shall meet later.] This equation expresses the compressibility factor Z(Vm, T) in terms of a general power series expansion in inverse molar volume Vm. The starting point for the virial expansion is the ideal limiting behavior (2.12), which can also be expressed as... 

One of the limitations in the use of the compressibility equation of state to describe the behavior of gases is that the compressibility factor is not constant. Therefore, mathematical manipulations cannot be made directly but must be accomplished through graphical or numerical techniques. Most of the other commonly used equations of state were devised so that the coefficients which correct the ideal gas law for nonideality may be assumed constant. This permits the equations to be used in mathematical calculations involving differentiation or integration. 

Equation 6-17 can be used to calculate gross heating value or net heating value. In either case, the values must be converted from ideal gas to real gas at standard conditions. This is done by dividing the ideal value by compressibility factor of the gas at standard conditions. 

First, calculate gross heating value of ideal gas and compressibility factor of the gas at standard conditions. 

Although integration is possible in closed form, it may be more convenient to perform the integration numerically. With more accurate and necessarily more complicated equations of state, numerical integration will be mandatory. Example 6.13 employs the van der Waals equation of steam, although this is not a particularly suitable one the results show a substantial difference between the ideal and the nonideal pressure drops. At the inlet condition, the compressibility factor of steam is z = PV/RT = 0.88, a substantial deviation from ideality. 

---