Question

For the system methane and ethane for equimolar mixture at temperature 100°C and pressure 35bar, estimate the fugacity of gases

through application of the Generlized Correlations.

Answer 1

We will first find the pseudo critical properties of the mixture to find the reduced temperature and pressure.

Critical temperature of methane is, $T_{c}^{m}=190.6K$

Critical pressure of methane is

Pm 46bar

Critical temperature of ethane is, $T_{c}^{e}=305.3K$

Critical pressure of ethane is $P_{c}^{e}=49bar$

Now the pseudo critical temperature of the mxture will be, $T_{c}'=0.5*190.6+0.5*305.3=247.95K$

pseudo critical pressure of the mxture will be, $P_{c}'=0.5*46+0.5*49=47.5bar$

Now the given temperature is $T=100^{0}C=373K$ and pressure is $P=35bar$

Now the reduced temperature is calculated as, $T_{r}=\frac{T}{T_{c}'}=\frac{373}{247.95}=1.5$

The reduced pressure is calculated as, $P_{r}=\frac{P}{P_{c}'}=\frac{35}{47.5}=0.74$

Now we can calculate the fugacity using the formulae,

$ln\frac{f}{P}=\int_{0}^{P_{r}}\frac{Z-1}{P_{r}}dP_{r}$

We can calculate the integral analytically using simpson's 1/3 rd rule at constant temperature by taking compressibility factors from the isotherms on the generalised compressibilty chart.

TK 500 3.00 2009 1.00 -1.60 1 0.90 o -1.40 1.5 1.30 0.80 0.85 120 15 10.90 10 0.90 1.00 0.80 0.95 PV 0.70 Z= RT 0.60 17:28 1.40 1.20 070 WA 14:00 .00 0.95 10.90 Z 0.95 Reduced Temperature TY = T/T, 0.50 20.85 0.80 0.40 0.90 0.00 N0.75 0.10 0.05 P 0.30 0.0 0.1 0.2 0.8 0.9 1.0 Reduced Pressure, Pr = P/PC

From the generalised compressibility chart we will take points equidistant $P_{r}$ values from 0 to 0.74 and we will find the corresponding .

The Simpson's 1/3rd rule is given as,

b a f(x)dx = $[F(xo)+4{f(x, J+f(x3) +...+ f(x-1)}+...] ...+2{f(x2)+ f(xx)+...+ f(x,-2)}+ f(x,)}] f(x)+4 "Ef(x;)+2 Ef(x;)+F(x,) i=2 i=even $(10)+49 (*)+ 16) - 10) = * Efe) +2 & fic f(xo)+4 Ef(x;)+2 Žf(x)) + f(x)

Here $f(x)=\frac{Z-1}{P_{r}}$ and $dx=dP_{r}$

The limits are $a=0$ and $b=P_{r}=0.74$

We will choose an interval of $h=0.02$ , so there will be a total of $n=37$ points

We will do the rest of the calculation in excel,

0905 TL 5.00 -3.00 2.00 1 880 2009 1.60 0.9 1.46 2.20 3.00 60 -0.80 2.00 80 60 1.30 1.50 40 1.30 $120 1.10 0.8 0.85 1.20 15 120 0.90 1.00 0.90 0.7 0.80 0.95 1 0.6 30 1.60 1.40 1.20 1:10 T 05 1.00 10.95 40.90 - 209 10.231 10:60 R Tel TI 0.5 0.65 100 15 10.85 12 10) 10.80 0.70 0.4 10.pl 10.75 0.10 0.05 P 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 1 f(x) 이 1 0 0.995 | 1 | 0.02 0.995 | 2 10.04 0.995 | 3 10.06 0.995 15 | 0.1 | 0.99 | 0.12 | 0.99 | 7 8 10.14 10.16 | 0.99 | 0.99 | 9 | 0.18 0.985 | 101 10.2 0.98 | 11 12 131 | 0.22 | 0.24 0.26 | 0.98) | 0.98 0.98 | 14 0.28 0.975 | 16 0.32 0.975 | 17 10.34 0.975 0.975

24 33 35 18 0 36 0.97 19 2.38 0.97 20 0.4 0.965 21 0.42 0.965 22 0.44 0.965 23 0.46 0.965 0.48 0.96 26 0.52 0.96 27 0 54 0.96 28 0.56 0.955 29 0 55 0.955 30 0.6 0.95 31 0.62 0.95 32 0.64 0.945 34 0.68 0.94 0.7 0.94 0.96 0.72 0.94

We will get the value of integral as 0.686

So,

$ln\frac{f}{P}=0.686$

$\Rightarrow$$f=69.5bar$

So the fugacity of equimolar ethane methane mixture is 69.5 bar