This example shows how to use the relationship obtained to solve a problem in which variations in both solvent and solute concentrations must be considered. The initial concentration of the A component (albumin) is significant to the extent that the assumption that xB = xB0, which was made in Numerical Example 1, is not valid. Without this simplification, an analytical solution for the concentration profile of A is not possible. Thus, MAPLE will be used to develop a trancendental equation in z and xA, which will be solved using a Newton's method algorithm in MATLAB.

Data Given

The data given in this example are the same as in the previous example (Numerical Example 1), with the exception of:

mole fraction of albumin at z = 0, xA0 = .05

Solution

> restart;

Begin with Equation 18.5-18 from BSL (with xB = 1-xA)

> eq:=(xA(z)/xA0)^VBbar*((1-xA0)/(1-xA(z)))^VAbar=exp((VAbar*MB-VBbar*MA)*gom*z/(R*T));

> txA(z):=solve(eq,xA(z));

Without the assumptions made in the first numerical example, we cannot solve analytically for the concentration profile of A. MAPLE just points out the fact that the solution involves the root of a trancendental equation.

> MA:=45000*g/mol:

> MB:=18*g/mol:

> VAbar:=MA/rhoA:

> rhoA:=1.34*g/cm^3:

> VBbar:=MB/rhoB:

> rhoB:=g/cm^3:

> T:=(75+460)/1.8*K:

> R:=8.314e7*g*cm^2/(s^2*mol*K):

> gom:=50000*980.665*cm/s^2:

> xA0:=.05:

> eq;

This is a trancendental equation in xA and z. We can solve this using MATLAB's fzero function (solve for xA given a z). The following MATLAB code was used to generate a graph of the concentration profile of A.

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function y = gotozero(xA, z)

y = ((20.*xA).^(18))*((.95./(1-xA)).^(33582.08955)) - exp(-407.8106102.*z);

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clear;

z = [0:.01:1.0];

for i=1:length(z)

xA(i) = fzero('gotozero', .05,[],[],z(i));

end;

plot(z,xA)

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