EXAMPLE $5-1$ We are given the pressure $P$ and the liquid composition $x.$ We

want to find the bubble$-$point temperature and the vapor composition as discussed

under "Phase equilibrium" in Sec. $2-2\mathrm{.}6.$ For simplicity let us assume a binary system

of components $1$ and $2.$ Component $1$ is the more volatile, and the mole fraction of

$1$ in the liquid is $x$ and in the vapor is $y.$ Let us assume also that the system is ideal:

Raoult's and Dalton's laws apply.

In liquid:

$P_1=x{P}_1^0$

$P_2=(1-x)P_2^0$

In vapor:

$P_1=yP$

$P_2=(1-y)P$

$124$ COMPUTER SIMULATION

$B_2=2\mathrm{.}585$

where $P_j=$ partial pressure of $j$ th component

$P_j^0=$ vapor pressure of $j$ th component which is a function of temperature $T,R$

$ln{P}_1^0=\frac{A_1}{T}+B_1$

$ln{P}_2^0=\frac{A_2}{T}+B_2$

Equating partial pressures in liquid and vapor gives

$P=x{P}_1^0+(1-x)P_2^0$

$y=\frac{x{P}_1^0}{P}$

Our convergence problem is to find the value of temperature $T$ that will satisfy

Eq$.(5-4).$ The procedure is as foilows:

$1$ Guess a temperature $T.$

$2$ Calculate the vapor pressures of components $1$ and $2$ from Eqs. $(5-3).$

$3$ Calculate a total pressure $P^{\text{c}\text{a}\text{l}\text{e}}$ from Eq$.(5-4).$

$P^{\text{c}\text{a}\text{l}\text{c}}=x{P}_1^0+(1-x)P_2^0$

$4$ Compare $P^{\text{c}\text{a}\text{l}\text{c}}$ with the actual total pressure given, $P.$

$5$ If $P^{\text{c}\text{a}\text{l}\text{c}}$ is greater than $P,$ the guessed temperature was too high and we must

make another guess that is lower.