## Real Gases

At low temperatures and high pressures, the ideal gas equation ceases to apply. Under these conditions, you must deal with "real" gases.

A quick way of checking the validity of an ideal gas assumption is to look at the specific molar volume of the gas. The ideal gas equation is good to within about 1% if: Real gases require more complex equations of state than do ideal gases. There are many options, but we will focus on:

• Virial Equations of State
• Cubic Equations of State
• Compressibility Factor Equation
This topic will be an important part of Thermo II.

### Virial Equations of State

Virial equations are a family of equations of state of the general form: The parameters in the equation (B,C,D = ci) are called "virial coefficients". If ci=0 for i>0, the virial equation reduces to the ideal gas equation. Just as with the ideal gas equation, the temperatures and pressures used must be absolute.

The accuracy required determines the number of terms that are kept -- more terms makes the equation more accurate, but also more complicated to work with. Virial coefficients are different for each gas, but other than that are functions of temperature only.

Coefficients are normally obtained by making measurements of P, V, and T, and fitting the equation. These values are then published so that others may use them.

Many forms of the virial equation exist. Often, we will truncate the virial equation to A number of methods (correlations, etc.) are available to determine B. In order to improve accuracy and capture more behaviors, additional parameters are sometimes added. One example is the Benedict-Webb-Rubin (BWR) equation of state. All the constants must be supplied if you are to use this equation for a particular gas. It isn't always easy to find BWR coefficients for the gas you are interested in.

### Cubic Equations of State

Virial equations cannot represent thermodynamic systems where both liquid and vapor are present. A "cubic" EoS is need to do this. One such is the Soave-Redlich-Kwong (SRK) equation. where the constants are given by In this equation, the b term is a volume correction, while the a is a molecular interaction parameter. The constants all depend on the critical temperature and pressure of the gas. These can be looked up easily in a data table.

The "acentric factor", omega, is also easily looked up. It is related to the geometry of the gas molecule.

To use the SRK equation:

1. look up Tc, Pc, and the acentric factor
2. plug in and find a, b, and alpha
3. plug these into the SRK equation; the result will be a cubic equation in P, T, and V
4. solve for the unknown you seek
Solving the cubic equation typically requires an iterative ("trial-and- error") solution; most of the time you probably want to use a computer or calculator routine for this task.

EXAMPLE:
Carbon dioxide at 300 K and 6.8 atm flows at 100 kmol/h. Use the SRK equation of state to determine the volumetric flow. (Felder & Rousseau, 1986, Example 5.3-3)

Strategy: The pressure and temperature are known, so look up the critical properties and acentric factor, find the SRK equation constants, and solve the SRK equation for the specific molar volume.

The critical properties (from the back of the book) are so the constants become Inserting these into the SRK equation gives All of the terms have units of atm. The equation can be rearranged to make solution more convenient Solution of the Equation: This equation is cubic in V, so trying to solve it algebraically is a bit complicated. Consequently, we will use an iterative approach.

We can implement the iterative strategy in a number of ways -- and using a computer or a calculator solver is strongly recommended. All the approaches use the same basic approach. A value of V is guessed and the right hand side (RHS) of the equation calculated. The result will be compared to zero and a new value of V chosen and tried.

The ideal gas law is always a good place to get an initial value, because it will usually be "close" to the real solution: You also need to determine how close to the "exact" answer you want to be. This error tolerance is usually built-in to the solver routine, although you usually can adjust it (in Mathcad, look at the value of TOL). If you've got a particularly difficult system, sometimes it is useful to relax the tolerance to get a rough solution, then update your estimate, tighten the tolerance, and repeat the solution for a final value.

This is the type of problem that Mathcad handles very well: You can also use the "Given -- Find" block for these problems just like you would for solving simultaneous equations.

### Compressibility Factor Equation

Engineering calculations often require a tradeoff between ease of use and accuracy. The ideal gas equation is very easy to use, but of questionable accuracy for many cases. Virial and cubic equations of state are accurate, but not particularly convenient. A good compromise is a generalized compressibility factor equation.

The "compressibility factor", z, is defined so that Consequently, z=1 for an ideal gas. There are a wide variety of ways to obtain z -- it is a function of T and P and can be determined from any of the equations of state we have already discussed.

Most gas properties depend on composition, but according to the "Law of Corresponding States" a few properties are the same for all gases when expressed in terms of deviation from the critical point. We do this by using the "reduced temperature" and "reduced pressure" . Correlations and charts that take advantage of corresponding states to provide data for many gases are called "generalized". Thus, a "generalized compressibility factor chart" can be used to get z once we have the reduced temperature and pressure. The are typically several different views of the chart, depending on the pressure range.

Warning: Hydrogen and Helium are special cases. They require a correction when calculating reduced properties.

EXAMPLE:
It is necessary to store 1 lbmole of methane at a temperature of 122 F and a pressure of 600 atm. What is the volume of the vessel that must be provided? Compare results using the ideal gas law and the compressibility factor equation.

Using the ideal gas law: Using the compressibility factor equation  From the generalized compressibility factor chart, z=1.3 so the answer is 0.921 cubic feet.

You can also enter the compressibility factor charts using the reduced volume. Critical volume isn't tabulated -- it must be calculated from the critical temperatures and pressures.

### Real Gas Mixtures

Mixtures of real gases can be treated using any equation of state; however, finding virial equation coefficients for gas mixtures is a subject for Thermo II. For this course, we will use an approximate approach using the compressibility factor equation and "Kay's Rule", which states that PV=zmRT. The mixture compressibility factor, zm, is found from the generalized compressibility factor chart using the system temperature and pressure reduced using "pseudocritical constants" given by: EXAMPLE:
300 lbs of a mixture of 10 mol% propane, 20% n-butane, and 70% n-pentane is completely vaporized in a pipestill in one hour. At the outlet, the temperature and pressure are 515 F and 600 psia. What is the volumetric flow rate at the outlet in cfm at outlet conditions..

Basis: 100 moles mixture
10 mol propane * 44 lb/mole = 440 lbs
20 mol butane * 58 = 1160
70 mol pentane * 72 = 5040
Total 6640 lbs, AMW 66.4

Mass fractions: 0.0663 propane, 0.175 butane, 0.759 pentane.

Basis: 300 lbs mixture
300 * 0.0663 / 44 = 0.452 lbmoles propane fed
300 * 0.175 / 56 = 0.905 lbmoles butane
300 * 0.759 / 72 = 3.16 lbmoles pentane
4.53 lbmoles total

Check the ideality of the gas Since 17.4 < 320 the gas is not ideal.

The physical properties are:

Propane Butane Pentane Tc 369.9 K 425.17 K 469.8 K 665.8 R 765.31 R 845.64 R 42.0 atm 37.47 atm 33.3 atm
Applying Kay's Rule: From the compressibility factor chart, z=0.75
Basis: 1 hour flow References:

1. Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2nd Edition, John Wiley, 1986, pp. 193-208.
2. Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2005 3rd Edition, 2005, p. 201-212.
3. Hougen, Watson, Ragatz, Chemical Process Principles: Part I 2nd Edition, John Wiley, 1954.

R.M. Price
Original: 6/20/94
Modified: 10/6/95, 10/17/96, 10/22/98; 12/25/2004, 3/2/2005