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## Solving for f(omega) with pitzer’s acentric factor

The tables below highlights the Z factors for pressure from 50 to 6500 psia at temperature of 100 F.
 P (psia) Z (100 F) P (psia) Z (100 F) 50 0.97740 1500 0.48660 100 0.95450 1600 0.49510 200 0.90790 1700 0.50540 300 0.86000 1800 0.51690 400 0.81070 1900 0.52940 500 0.76010 2000 0.54240 600 0.70820 2500 0.61280 700 0.65580 3000 0.68660 800 0.60450 3500 0.76090 900 0.55770 4000 0.83510 1000 0.52020 4500 0.90860 1100 0.49540 5000 0.98160 1200 0.48260 5500 1.05390 1300 0.47860 6000 1.12560 1400 0.48060 6500 1.19670
Table 1: Z factors for pressures from 50 to 6500 psia at temperature of 100 F.
 P (psia) Z (200 F) P (psia) Z (200 F) 50 0.98650 1500 0.70230 100 0.97300 1600 0.69730 200 0.94640 1700 0.69430 300 0.92020 1800 0.69330 400 0.89450 1900 0.69390 500 0.86950 2000 0.69610 600 0.84540 2500 0.72300 700 0.82240 3000 0.76610 800 0.80060 3500 0.81690 900 0.78030 4000 0.87150 1000 0.76190 4500 0.92810 1100 0.74540 5000 0.98570 1200 0.73110 5500 1.04380 1300 0.71920 6000 1.10210 1400 0.70960 6500 1.16050
Table 2: Z factors for pressures from 50 to 6500 psia at temperature of 200 F.

## MATLAB Script

% solving for f(omega) with the Pitzer’s acentric factor.
pitzer = [0.01330 0.11304 0.17244 0.23561 0.34585 0.55335 0.84182]
for i=1:7
if pitzer(i) <= 0.49
fomega(i)=0.374640+(1.54226*pitzer(i))-(0.26992*pitzer(i)^2)
else
fomega(i)=0.379642+(1.48503*pitzer(i))-(0.164423*pitzer(i)^2)+(0.016666*pitzer(i)^3)
end
end
% Attraction parameter constant,Co-volume parameter constant,binary interaction coefficients, mole fractions
omgai = [0.42312848 0.45192604 0.45984739 0.45811880 0.39778691 0.39778691 0.39778691]
omgbi = [0.08046461 0.07926051 0.07843675 0.07791799 0.07510754 0.07510754 0.07510754]
deltam=[0 0.000986 0.007843 0.023942 0.037841 0.047445 0.26562214;
0.000986 0 0.003695 0.010541 0.010541 0.010541 0.010541;
0.007843 0.003695 0 0.002281 0.002281 0.002281 0.002281;
0.023942 0.010541 0.002281 0 0.000 0.000 0.000;
0.037841 0.010541 0.002281 0.000 0 0.000 0.000;
0.047445 0.010541 0.002281 0.000 0.000 0 0.000;
0.26562214 0.010541 0.002281 0.000 0.000 0.000 0]
molef=[0.679300 0.099000 0.110800 0.045000 0.052966 0.011941 0.000993]
% getting critical temperature and solving for reduced temperature
TciF=[-120.01 89.83 245.87 410.94 600.51 823.88 1060.94]
TciR=TciF + 460
pci=[662.81 752.19 581.03 481.06 385.00 253.07 174.67]
% setting gas constant to units of psia and R
R = 10.7316
T1=100+460
T2=200+460
TrR=TciR.\T1
% dimensional attraction
for i=1:7
aalpha(i)= ((omgai(i)*(R^2)*(TciR(i)^2))/pci(i))*((1+((fomega(i)*(1-(TrR(i))^0.5))))^2);
end
% dimensional co-volume
for i=1:7
beta(i)= (omgbi(i)*R*TciR(i))/pci(i);
end
% enitre mixture parameters
aam = 0.0;
bm = 0.0;
for i=1:7
bm = bm + molef(i)*beta(i);
for j=1:7
aam=aam+molef(i)*molef(j)*sqrt( aalpha(i)*aalpha(j) )*(1.0-deltam(i,j));
end
end
% setting pressure
P=[50 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500]
%solving for A, B, a1, b1, and c1 at temperature 560 R.
for i=1:30
A=(aam*P(i))/(R^2*T1^2);
B=(bm*P(i))/(R*T1);
a1=-(1-B);
b1=A-(3*B.^2)-(2*B);
c1=-((A.*B)-(B.^2)-(B.^3));
%solve for the roots
root1=roots([1 a1 b1 c1]);
RR(i) = root1(real(root1) >= 0 & imag(root1) == 0); %selecting positive real roots
end
%Repeat for temperature of 660 R.
TrR2=TciR.\T2
% dimensional attraction parameter
for i=1:7
aalpha2(i)= ((omgai(i)*(R^2)*(TciR(i)^2))/pci(i))*((1+((fomega(i)*(1-(TrR2(i))^0.5))))^2);
end
% dimensional co-volume parameters
for i=1:7
beta2(i)= (omgbi(i)*R*TciR(i))/pci(i);
end
% parameters for the entire mixture
aam2 = 0.0;
bm2 = 0.0;
for i=1:7
bm2 = bm2 + molef(i)*beta2(i);
for j=1:7
aam2=aam2+molef(i)*molef(j)*sqrt( aalpha2(i)*aalpha2(j) )*(1.0-deltam(i,j));
end
end
% setting pressure
P=[50 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500]
%find A2, B2, a12, b12, and c12 at temperature of 660 R.
for i=1:30
A2=(aam2*P(i))/(R^2*T2^2);
B2=(bm2*P(i))/(R*T2);
a12=-(1-B2);
b12=A2-(3*B2.^2)-(2*B2);
c12=-((A2.*B2)-(B2.^2)-(B2.^3));
%solving for the roots
root2=roots([1 a12 b12 c12]);
RR2(i) = root2(real(root2) >= 0 & imag(root2) == 0); %select real positive values only
end
disp('Z factors for T=560 R ')
disp(RR)
disp('Z factors for T=660 R ')
disp(RR2)
%creating two plots
for i=1:30
x1(i)= P(i);
y1(i)= RR(i);
x2(i)= P(i);
y2(i)= RR2(i);
end
plot(x1, y1),xlabel('P(psia)'), ylabel('Z factor')
hold on
plot(x2, y2),xlabel('P(psia)'), ylabel('Z factor')
Graph
Figure 1: Z factor vs. P(psia) plot obtained from MATLAB for two temperatures plots, blue plot (100 F), orange (200 F).

## MATLAB Script Output

>> zfactor_final
pitzer =
0.0133 0.1130 0.1724 0.2356 0.3458 0.5534 0.8418
fomega =
0.3951
fomega =
0.3951 0.5455
fomega =
0.3951 0.5455 0.6326
fomega =
0.3951 0.5455 0.6326 0.7230
fomega =
0.3951 0.5455 0.6326 0.7230 0.8757
fomega =
0.3951 0.5455 0.6326 0.7230 0.8757 1.1539
fomega =
0.3951 0.5455 0.6326 0.7230 0.8757 1.1539 1.5232
omgai =
0.4231 0.4519 0.4598 0.4581 0.3978 0.3978 0.3978
omgbi =
0.0805 0.0793 0.0784 0.0779 0.0751 0.0751 0.0751
deltam =
0 0.0010 0.0078 0.0239 0.0378 0.0474 0.2656
0.0010 0 0.0037 0.0105 0.0105 0.0105 0.0105
0.0078 0.0037 0 0.0023 0.0023 0.0023 0.0023
0.0239 0.0105 0.0023 0 0 0 0
0.0378 0.0105 0.0023 0 0 0 0
0.0474 0.0105 0.0023 0 0 0 0
0.2656 0.0105 0.0023 0 0 0 0
molef =
0.6793 0.0990 0.1108 0.0450 0.0530 0.0119 0.0010
TciF =
1.0e+03 *
-0.1200 0.0898 0.2459 0.4109 0.6005 0.8239 1.0609
TciR =
1.0e+03 *
0.3400 0.5498 0.7059 0.8709 1.0605 1.2839 1.5209
pci =
662.8100 752.1900 581.0300 481.0600 385.0000 253.0700 174.6700
R =
10.7316
T1 =
560
T2 =
660
TrR =
1.6471 1.0185 0.7933 0.6430 0.5280 0.4362 0.3682
P =
Columns 1 through 8
50 100 200 300 400 500 600 700
Columns 9 through 16
800 900 1000 1100 1200 1300 1400 1500
Columns 17 through 24
1600 1700 1800 1900 2000 2500 3000 3500
Columns 25 through 30
4000 4500 5000 5500 6000 6500
TrR2 =
1.9412 1.2004 0.9350 0.7578 0.6223 0.5141 0.4339
P =
Columns 1 through 8
50 100 200 300 400 500 600 700
Columns 9 through 16
800 900 1000 1100 1200 1300 1400 1500
Columns 17 through 24
1600 1700 1800 1900 2000 2500 3000 3500
Columns 25 through 30
4000 4500 5000 5500 6000 6500
Z factors for T=560 R
Columns 1 through 10
0.9774 0.9545 0.9079 0.8600 0.8107 0.7601 0.7082 0.6558 0.6045 0.5577
Columns 11 through 20
0.5202 0.4954 0.4826 0.4786 0.4806 0.4866 0.4951 0.5054 0.5169 0.5294
Columns 21 through 30
0.5424 0.6128 0.6866 0.7609 0.8351 0.9086 0.9816 1.0539 1.1256 1.1967
Z factors for T=660 R
Columns 1 through 10
0.9865 0.9730 0.9464 0.9202 0.8945 0.8695 0.8454 0.8224 0.8006 0.7803
Columns 11 through 20
0.7619 0.7454 0.7311 0.7192 0.7096 0.7023 0.6973 0.6943 0.6933 0.6939
Columns 21 through 30
0.6961 0.7230 0.7661 0.8169 0.8715 0.9281 0.9857 1.0438 1.1021 1.1605
>>