project for numerical methods
malak22
TermProject-20183.pptxExam-1 Solution and Comments
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Aniline is being removed from water by solvent extraction using toluene. Aqueous and organic phases will form after adding the organic solvent into the water phase. The extraction is carried out in a 10-stage countercurrent tower, shown in the following figure. The equilibrium relationship valid at each stage is, to a first approximation:
where:
Yi = (lb of aniline in the toluene phase)/(lb of toluene in the toluene phase)
Xi = (lb of aniline in the water phase)/(lb of water in the water phase)
(a) The solution to this problem is a set of 10 simultaneous equations. Derive these equations from material balances around each stage. Present these equations using compact notation. (15%)
(b) Solve the above set of equations to find the concentration in both the aqueous and organic phases leaving each stage of the system (Xi and Yi). (25%)
(c) If the slope of the equilibrium relationship is replaced by the expression m = 8 +20 Xi, the solution becomes a set of simultaneous nonlinear equations. Describe a procedure which would solve this problem. (15%)
(d) Solve the problem described in (c) above. (25%)
(e) Prepare a report to summarize the mathematical modeling and numerical solution procedures. Make sure to attach all MATLAB or excel codes used in this project. (20%)
Aniline is an organic compound with the formula C6H5NH2. Consisting of a phenyl group attached to an amino group, aniline is the prototypical aromatic amine. Its main use is in the manufacture of precursors to polyurethane and other industrial chemicals. Like most volatile amines, it has the odor of rotten fish. Aniline is toxic by inhalation of the vapor, ingestion, or percutaneous absorption. It is listed as Group 3 pollutant (not classifiable as to its carcinogenicity to humans) due to the limited and contradictory data available.
Mass balance at the feed stage:
Feed Stage (stage f)
W, Xf-1
F, YF
W, Xf
S, Yf+1
, Yf
F + S =
Solvent flow:
Water flow:
W = W
Mass balance on aniline above the feed stage:
Stage i (i<f)
W, Xi-1
W, Xi
, Yi
Aniline flow in = Aniline flow out
WXi-1 + Yi+1 = WXi +Yi (i= 1, 2, …, f-1)
, Yi+1
Mass balance on aniline around the feed stage:
Aniline flow in = Aniline flow out
Feed Stage (stage f)
W, Xf-1
F, YF
W, Xf
S, Yf+1
, Yf
WXf-1 +SYf+1 +FYF = WXf +Yf
Mass balance on aniline below the feed stage:
Stage i (i<f)
W, Xi-1
W, Xi
S, Yi+1
S, Yi
Aniline flow in = Aniline flow out
WXi-1 + Yi+1 = WXi +SYi (i= f+1, f+2, …, N)
The equilibrium relationship: Yi = mXi
WXi-1 + Yi+1 = WXi +Yi (i= 1, 2, …, f-1)
WXi-1 + (mXi+1) = WXi +(mXi) (i= 1, 2, …, f-1)
-WXi-1 + (W+)Xi Xi+1=0 (i= 1, 2, …, f-1)
When i=0
(W+m)X1 X2= WX0
The equilibrium relationship: Yi = mXi
-WXf-1 + (W+)Xf Xf+1=FYF
WXf-1 +SYf+1 +FYF = WXf +Yf
The equilibrium relationship: Yi = mXi
WXi-1 + S(mXi+1) = WXi +(mXi) (i= f+1, f+2, …, N)
When i=N
-WXN-1 + (W+m)XN =0
WXi-1 + Yi+1 = WXi +SYi (i= f+1, f+2, …, N)
-WXi-1 + (W+m)Xi Xi+1=0 (i= f+1, f+2, …, N)
(W+m)X1 X2= WX0
-WXi-1 + (W+)Xi Xi+1=0 (i= 2, …, f-1)
-WXf-1 + (W+)Xf Xf+1=FYF
-WXi-1 + (W+m)Xi Xi+1=0 (i=f+1, f+2, …, N-1)
-WXN-1 + (W+m)XN =0
The mass balance equations generate a tridiagonal set of simultaneous linear algebraic equations that can be solved by the matrix inversion method.
Solution Procedure for Part (b) (Method 1)
1. Derive all 10 equations
2. Convert them into [A][X]=[C]
3. Solve [X]=[A]\[C]
Solution Procedure for Part (b) (Method 2)
1. Solve [A][X]=[C]
2. Develop a MATLAB script or function m-file to create both coefficient and constant matrices [A] and [C]. This is helpful when N is very large (>20).
3. Solve [X]=[A]\[C]
The equilibrium relationship: Yi = p+qXi
WXi-1 + Yi+1 = WXi +Yi (i= 1, 2, …, f-1)
WXi-1 – (W+p)Xi + pXi+1-qXi2+qXi+12=0 (i= 1, 2, …, f-1)
When i=0
WX0 – (W+p)X1 + pX2-qX12+qX22=0
The equilibrium relationship: Yi = p+qXi
WXi-1 + Yi+1 = WXi +Yi (i= 1, 2, …, f-1)
-WXf-1 + (W+)Xf Xf+1=WYF (i= f)
WYF -WXf-1 + (W+p)Xf - pSXf+1+Xf2 =0
The equilibrium relationship: Yi = p+qXi
-WXi-1 + (W+())Xi Xi+1=0 (i=f+1, f+2, …, N)
WXi-1 + Yi+1 = WXi +SYi (i= f+1, f+2, …, N)
When i=N
-WXi-1 + (W+pS)Xi – pSXi+1+ = 0
-WXN-1 + (W+pS)XN + = 0
The mass balance equations generate a set of non-linear algebraic equations that can be solved by the Newton’s method or MATLAB solve fsolve. Excel solve can also
be used, but probably very tedious to input all the equations.
Solution Procedure for Part (d)
1. Create a function m-file to store all the equations generated from the mass balance.
2. Using the results from Part (b) as the initial values for X.
3. Solve X using Newton.m, newtmult.m or fsolve.
