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Exam 1
IMPORTANT: For each problem you will upload a) all the .m files used and b) the output of
your code obtained using the “Publish” option. In the .m files you need to indicate clearly in
the form of a comment how I should be able to run your code and generate your output. This
is how you will be graded.
Problem 1
The Van der Waals equation of state (VdW EoS) expresses a relationship between the P, T, and molar volume (𝑣𝑣) of a “real” gas. A likely form of the equation is: �𝑃𝑃 + 3 ��𝑣𝑣 − 1� = 8 𝑇𝑇. The
VdW EoS predicts under certain (P,T) conditions either one or multiple likely values of the molar volume 𝑣𝑣 which is calculated, for a given P, T, as a solution to the VdW EoS. You are given that the temperature is T=0.9 (units are arbitrary, so you do not need to worry) while P=0.6 (again in arbitrary units). Calculate the molar volume, 𝑣𝑣, at this (T,P) by solving the VdW EoS. Solve the equation four time using each of the following initial guesses for 𝑣𝑣 = {0.5, 1.0, 1.5, 2.0} and report the solution you obtain for 𝑣𝑣 for each initial condition. Repeat the same calculations at P=0.6, T=1.1 and report again the solution for each initial guess. What do you observe?
Problem 2
Consider the process flow diagram shown below. The material balances can be written as follows (where m’s denote flow rates):
𝑚𝑚 =𝑚𝑚 +𝑚𝑚 +𝑚𝑚 +𝑚𝑚 𝑚𝑚 =𝑚𝑚 +𝑚𝑚 +𝑚𝑚 0.84𝑚𝑚 =𝑚𝑚 +𝑚𝑚 0.2𝑚𝑚 =𝑚𝑚
1 2 3 4 5 12 4 7 11 12 4 7 9 10
𝑚𝑚2 =𝑚𝑚9 +𝑚𝑚10 +𝑚𝑚11 𝑚𝑚1 =100 0.7𝑚𝑚1 =𝑚𝑚2 +𝑚𝑚3 0.85𝑚𝑚2 =𝑚𝑚9 +𝑚𝑚11 𝑚𝑚5 =𝑚𝑚8 +𝑚𝑚7 +𝑚𝑚6 𝑚𝑚5 =5𝑚𝑚8 0.55𝑚𝑚1 =𝑚𝑚9 +𝑚𝑚12 3.2𝑚𝑚6 =𝑚𝑚7 +𝑚𝑚8
Solve the system of equations to obtain the values of the flow rates.
𝑣𝑣2 33
Problem 3
A vapor-liquid mixture of ethylbenzene (EB) and toluene (Tol) has a partial pressure of ethylbenzene 𝑃𝑃 = 250 𝑚𝑚𝑚𝑚 𝐻𝐻𝐻𝐻 and partial pressure of toluene 𝑃𝑃 = 343 𝑚𝑚𝑚𝑚 𝐻𝐻𝐻𝐻. In
𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇𝑇𝑇
thermodynamics you have or will learn that the liquid phase mole fractions of ethylbenzene, 𝑥𝑥𝐸𝐸𝐸𝐸,
and toluene, 𝑥𝑥𝑇𝑇𝑇𝑇𝑇𝑇, satisfy the following equations:
𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑠𝑠
𝑥𝑥 𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠 − 𝑃𝑃 = 0 (please note that 1 − 𝑥𝑥 = 𝑥𝑥 ) with the saturation pressures, 𝑃𝑃
(1−𝑥𝑥 )𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠−𝑃𝑃 =0 𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇
for each chemical given by Antoine’s equation as log10 𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠 = 𝐴𝐴 − 𝐸𝐸 , where A, B and C are
𝑇𝑇+𝐶𝐶
constants and T is the temperature. For ethylbenzene and Toluene you are given that the Antoine constants are:
Chemical
Toluene
A
6.95719
B
1414.255
C
Ethylbenzene
213.21
6.95464
1344.8 𝐸𝐸𝐸𝐸 219.48
Given this information, calculate the liquid mole fraction of ethylbenzene 𝑥𝑥 and temperature T.
You can use as initial guess (𝑥𝑥𝐸𝐸𝐸𝐸, 𝑇𝑇) = (0.6, 100).