Consider a vapor/liquid separation performed by flash vaporization in a drum:

The feed stream has flow rate F and compositions (in mole fraction) zi for n components, giving compositions z1, z2, ..., zn. In the drum, this feed is separated into a liquid phase that leaves the system with flow rate L and compositions xi and a vapor phase that leaves the system with flow rate V and compositions yi.

The thermodynamic principles governing this process are described by n vapor-liquid equilibrium ratios Ki, one for each component. Given these K-values, a feed flow rate, and feed compositions, it is possible to find the product flow rates and compositions by finding the root of the following equation in the interval [0,1]:

€ zi(1-Ki) 1+Ψ(Ki -1) = 0 i=1 nΣ,

where the independent variable Ψ is defined as Ψ = V/F. In other words, the system can besolved by finding a value of Ψ that makes the left-hand side of the equation above equal to zero.

Once this value of Ψ is found, L can be found as L = F - V and the compositions in each phase can be found as:

€ xi = zi 1+Ψ(Ki -1) and € yi = xiKi.

Write a program that to find the output flow rates and compositions for such a separation system with an input flow rate (in mol/h), input compositions (in mole fraction) and K-values (dimensionless). Your program should be able to handle any number of components in the feed.

Consider a 7-component system where the feed compositions are: z1 = 0.16, z2 = 0.09, z3 = 0.21, z4

= 0.13, z5 = 0.06, z6 = 0.15, and z7 = 0.20; the K-values are K1 = 3.52, K2 = 1.92, K3 = 1.22, K4

= 0.84, K5 = 0.43, and K6 = 0.08. The total feed rate is 1500 mol/h. Plot the output compositions and flow rates in both phases as functions of K7 as K7 varies from 0.05 to 4.00.

Note: While there are several root-finding routines built into MATLAB as functions, avoid using them in this project. Instead, write your own root-finding routine based the one of the algorithms we studied in class.