6.18c. Flash calculations: the Rachford-Rice method for ideal mixtures\ (a) Show that the problem of flash vaporization of a multicomponent ideal mixture can be reformulated as suggested by Rachford and Rice (Doherty and Malone, 2001):\

f(\\\\phi )=\\\\sum_(i=1)^C (z_(i)(m_(i)-1))/(1+\\\\phi (m_(i)-1))=0 where m_(i)=(P_(i))/(P)\ x_(i)=(z_(i))/(1+\\\\phi (m_(i)-1)),i=1,2,cdots,C\ y_(i)=m_(i)x_(i),i=1,2,cdots,C\ D=\\\\phi F,W=(1-\\\\phi )F

\ Equation (6-109) is solved iteratively for

\\\\phi

; all other variables are calculated explicitly from equations (6-110) to (6-112).\ (b) Solve Example 6.2 using the Rachford-Rice method.\ \ Example 6.2 Flash Vaporization of a Ternary Mixture\ A liquid containing

50mol%

benzene (A),

25mol%

toluene (B), and

25mol%

oxylene

(C)

is flash-vaporized at

1atm

and

373K

. Compute the amounts of liquid and vapor products and their composition. These components form ideal mixtures. The vapor pressures of the three components at

373K

are:

P_(A)=178.8kPa

,

P_(B)=73.6kPa

, and

P_(C)=26.3kPa

. Therefore, for a total pressure of

1atm

and a temperature of

373K,m_(A)=1.765,m_(B)=0.727,m_(C)=0.259

.\ Solution\ Equation (6-4) and Raoult's law apply for each of the three components in the mixture. Also, the mol fractions must add up to 1.0 in each of the two phases leaving the separator. The Mathcad program of Figure 6.3 is easily modified to include an additional component and solve for

D,W

, and the vapor and liquid compositions (see Problem 6.3). The results are:

D=0.297,y_(AD)=0.719,y_(BD)=

0.198,y_(CD)=0.083,x_(AW)=0.408,x_(BW)=0.272,x_(CW)=0.320

.

Example 6.2 Flash Vaporization of a Ternary Mixture A liquid containing 50mol% benzene (A), 25mol% toluene (B), and 25mol% oxylene (C) is flash-vaporized at 1atm and 373K. Compute the amounts of liquid and vapor products and their composition. These components form ideal mixtures. The vapor pressures of the three components at 373K are: PA=178.8kPa, PB=73.6kPa, and PC=26.3kPa. Therefore, for a total pressure of 1 atm and a temperature of 373K,mA=1.765,mB=0.727,mC=0.259. Solution Equation (6-4) and Raoult's law apply for each of the three components in the mixture. Also, the mol fractions must add up to 1.0 in each of the two phases leaving the separator. The Mathcad program of Figure 6.3 is easily modified to include an additional component and solve for D,W, and the vapor and liquid compositions (see Problem 6.3). The results are: D=0.297,yAD=0.719,yBD= 0.198,yCD=0.083,xAW=0.408,xBW=0.272,xCW=0.320