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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
solving for f
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
>>