Material and Energy Balances
An EXCEL Spreadsheet for Solving the Peng-Robinson Equation of State for Real Gas Mixtures
Brice Carnahan – University of Michigan
Introduction
The Peng-Robinson Equation of State for Pure Substances
The Peng-Robinson (P-R) equation of state [D.Y. Peng and D.B. Robinson, Industrial and Engineering
Chemistry Fundamentals, 15, p 59 (1976)] is one of the most popular equations for describing the PVT
(Pressure-Volume-Temperature) behavior of real, i.e., nonideal, pure substances, in particular of liquid and
gaseous hydrocarbons and common inorganic gases such as oxygen, nitrogen, and hydrogen sulfide. The PR equation is similar to the SRK equation of state shown in Section 5.3b of Felder, and has the following form:
P
=
R T
----------v -b
-
a(T)
------------------------------v (v + b) + b (v - b)
(1)
Here, R is the gas constant, P is the absolute pressure, T is the absolute temperature, v is the molar volume,
and b and a(T) are given by:
b
=
R Tc
0.07780 ---------Pc
(2)
R 2 Tc 2
a(T)
=
0.45724 ------------- (T)
Pc
(3)
T)
=
[1 + k (1 - ( T/Tc ) 0.5) ]2
(4)
k
=
0.37464 + 1.54226  - 0.26992 2
(5)
Here, Tc is the critical (absolute) temperature, Pc is the critical (absolute) pressure, and is the Pitzer acentric
factor shown for several gases in Table 5.3b of Felder. Thus the P-R equation has in effect just three
compound-specific experimentally-determined physical properties: Tc , Pc , and .
Note that if the compressibility factor z =Pv /(RT) [see equation (5.3-11) in Felder], then (1) can be written
as a cubic equation in z :
z3 -
(1 - B) z 2 + (A - 3 B 2 - 2 B) z - (A B - B 2 - B 3)
=
0
(6)
Here, B = Pb /(RT) and A = aP/(RT)2. Hence, using experimental values for , Tc, and Pc (see Tables
5.3b and B.1 in Appendix B of Felder) and given values for temperature T and pressure P, you can find v from
equation (1) or z from equation (6). Either of the equations can be solved numerically by using a root-finding
algorithm such as Newton's method described in Section A.2e (Appendix A) of Felder.
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Also note that k and (T) are dimensionless, b has units of vol/mol, a(T) has units of Pressure*(vol/mol)2
and R must be in units appropriate for those chosen for P, T, and v. For this problem, the parameters in the
equations should be in units of oK forT, atm for P, and m3/mole for v.
The Peng-Robinson Equation of State for Mixtures
As written, the P-R equation is intended for description of the PVT behavior of pure compounds.
However, it can also be used for mixtures of compounds by using "mixture-averaged" values for the equation
parameters. Let the values of parameters aii(T) and bi be the pure-component values of a(T) and b,
respectiverly, for the i(th) compound in a mixture. Also, let yi be the mole fraction of component i in the
mixture. Then "mixing" rules are applied to compute the mixture-averaged values of a(T) and b for a mixture
of n different compounds as follows:
b
=
a (T) =
where
aij
=
n
 yi bi
i =1
(7)
n
 yi
i =1
(8)
aji
n
 yj aij
j =1
=
( aii ajj ) 0.5
(9)
Problem Statement
Prepare a general-purpose EXCEL spreadsheet that:
(1) Contains bordered cells where the user will enter values for:
a) The mixture temperature, T (deg C)
b) The mixture pressure, P (atm)
c) The moles of each compound in a mixture containing any of the 14 different
(2) Computes the parameters for the Peng-Robinson (P-R) equation of state for all 14 of the pure
(3) Computes the mole fractions yi of components i
compound
(i =1,2,...n ) in the mixture,
(4) Determines "mixture-average" values of the P-R parameters b and a(T) for the specified mixture
using the mixing rules of equations (7) through (9),
(5) Solves equation (1) for the molar volume v (m3/mol) or equation (6) for the compressibility factor z of the mixture
(6) Calculates z or v as appropriate (depending on whether equation (1) or equation (6) is solved in part (5)).
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(7) Computes the following associated information once z and v are known:
a) total actual volume of the mixture, V in m3
b) the Pseudocritical Temperature T 'c in oK and Pseudocritical Pressure P'c in atm using
c) the Pseudoreduced Temperature T 'r = T/T 'c and the Pseudoreduced Pressure,
P'r = P/P'c, as described in Section 5.4c of Felder.
Organize your spreadsheet as you like, clearly labeling and outlining the cells to be filled in with data (use
the Bordering feature from the Format menu). Gridlines and the row and column labels can be removed from
the printed spreadsheet by checking appropriate entries in the dialog box for the Page Setup entry of the "File"
Menu (e.g., by printing in horizontal (landscape) mode). The reduce/enlarge feature for laser printing can be
used to fit the entire sheet onto one or two pages if it is too big otherwise.
Test Cases
You should test your spreadsheet carefully to insure that it can handle general mixtures of the given
compounds. Once you have done that, use your spreadsheet to solve the following problems:
1. 10 moles of pure N2 at -100 oC and 100 atm.
2. 100 moles of pure CO2 at 27 oC and 6.8 atm (see example 5.3-3 in Felder).
3. a mixture containing 30 moles of ethane, 15 moles of hydrogen sulfide, 20 moles of methane, and 50
4. a mixture containing 1 mole of each of the 14 substances at 50 oC and 1000 atm.
In each case, use the values computed in your spreadsheet for the pseudoreduced temperature and
pressure to determine the compressibility factor z predicted by the Generalized Compressibility Charts in
Section 5.3 of Felder. Write the values from the charts by hand on the corresponding spreadsheets, and note
the degree of agreement (or disagreement) with the compressibility factors computed from the P-R equation
of state.
What to turn in
You have one week to complete the problem. Please hand in:
1. One printed spreadsheet showing the cell formulas (use the View feature of the Options menu to see
the formulas). If a cell formula cannot be seen completely (because of limited cell space), widen the columns
before printing so that the complete formula in every cell on the spreadsheet will show up.
2. A printed spreadsheet for each of the four problems assigned above.
Hints
The easiest way to implement the Newton's method solution of an equation f(x) = 0 is to set up several
cells as follows:
1) A "trigger" cell that normally contains a 0; when the content is set to something else, the iterative
calculation is triggered, and proceeds until no cell in the sheet changes by more than a specified tolerance
value during one complete iteration.
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moles of n
2) A cell for holding xold, the "old" value of the solution variable x.
It should be conditionally assigned
3. A cell that evaluates f(xold).
4. A cell that evaluates f'(xold).
5. A cell that calculates xnew, the new value of x using Newton's formula (A.2-2):
x new = xold - f(xold) /f'(xold).
Once the sheet is set up, choose the Calculations tab from the Preferences entry of the File menu, click
on the Iteration box. You can set the maximum number of iterations (default value = 100) and the relative
tolerance to be used in testing for convergence (default value = 0.001) if you like. Then click on the OK
button. To begin the iterative solution, set the trigger cell in the spreadsheet to a nonzero value.
Note that Stanley Sandler, in his thermodynamics text Chemical and Engineering Thermodynamics,
suggests that Newton's Method usually converges faster and is somewhat more reliable for solving equation
(6) than equation (1). However, you may choose to solve either of the equations in your spreadsheet.
One easy way to implement the mixing rule calculations of equations (7) through (9) is to use the array
functions MMULT and TRANSPOSE in appropriate cell formulas. For example, if the 14 yi values are in one
column (say C21:C34) and the 14 bi values are in another column (say G21:G34), then the parameter b given
by equation (7) could be computed in a cell with the array formula:
= MMULT(TRANSPOSE(C21:C34),G21:G34))
i.e., b is given by the matrix product of the one row matrix (row vector) TRANSPOSE(C21:C43) and the one
column matrix (column vector) G21:G34.
Once the formula is typed into the formula bar, you must hold down the command key and then press the
return key (on a Mac) to enter the array formula into the cell. On the PC, the key combination for entering the
formula is <Ctrl><Enter>.
BC
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the initial v