MU Computer Programming and Numerical Methods - December 2015 Exam Question Paper

MU Chemical Engineering (Semester 3)
Computer Programming and Numerical Methods
December 2015

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Explain how to use 'for' loop in SciLab with appropriate example.

5 M

1(b) Use Crank-Nicholson Scheme to solve,
u_xx = u_t 0≤x≤1 t>0

h = \frac{1}{4}

$h=\dfrac{1}{4}$

k = \frac{1}{4}

$k=\dfrac{1}{4}$
Given u(x, 0) = 0, u(0, t) = 0, u(1, t) =50t Compute u for one step in t-direction.

5 M

1(c) Show progress of bisection method using graphical representation.

5 M

1(d) Solve following system of equation,
x+y+z=7
z+2y+3z=16
z+3y+4z=22

5 M

2(a) Liquid Molar volume of n-butane at 350 K and 9.4573 bar may be calculated using Redlich-Kwong equation as given below

V^{l} = \frac{z R T}{p}

$V^l =\dfrac{zRT}{p}$
where,

z = β + z (z + β) (\frac{1 + β - z}{q β})

$z=\beta +z(z+\beta)\left ( \dfrac{1+\beta-z}{q\beta} \right )$

β = 0.08664 \frac{P_{r}}{T_{r}} P r = \frac{P}{P_{c}} T_{r} = \frac{T}{T_{c}} q = 6.6.48

$\beta=0.08664\dfrac{P_r}{T_r}\ \ \ Pr=\dfrac{P}{P_c}\ \ \ T_r=\dfrac{T}{T_c}\ \ \ q=6.6.48$
for n-butane, T_c = 425.1K_i, p_c = 37.96bar
Calculate liquid molar volume for n-butane at given condition using Newton-Raphson method starting with z=β.

20 M

3(a) Solve following set of equations using Gauss-Seidel and Gauss-Jordan Method
2x₁-3x₂+x₃=-11
3x₁+4x₂-3x₃=-34
x₁+5x₂-2x₃=-17

14 M

3(b) Write Laplace equation and express it in difference form using Taylor's series expansion.

6 M

4(a) A chemical reactor that has a single second order reaction and a outlet flowrate that is a linear function of height has the following model

\frac{d V C}{d t} = F_{i n} C_{i n} - F C - k V C^{2}

$\dfrac{dVC}{dt}=F_{in}C_{in}-FC-kVC^2$

\frac{d V}{d t} = F_{i n} - F

$\dfrac{dV}{dt}=F_{in}-F$
where, F = βV.
The parameters and variables are as given below.
F_in = inlet flowrate, (2 LPM)
     C_in = inlet concentration, (1 gmol/lit)
     k = reaction rate constant, (2 lit/(gmol-min))
     β = 1 min^-1
V = reaction mixture volume, (at t = 0, 1 lit)
     C = concentration in reactor, (at t = 0, 0.5 gmol/lit)
Find the concentration and volume after one minute using Runge-kutta second order method.

20 M

5(a) Friction factor is commercial pipe for turbulcnt flow can be calculated using Colebrook equation. If roughness factor(k) for carbon steel pipe is 0.00015 m for a pipe with ID (D) 0.315 m, using suitable numerical method calculate friction Colebrook equation,

\frac{1}{\sqrt{f}} = - 2.0 log (\frac{k / D}{3.7} + \frac{2.51}{R e \sqrt{f}})

$\dfrac{1}{\sqrt{f}}=-2.0\log\left ( \dfrac{k/D}{3.7}+\dfrac{2.51}{Re\sqrt{f}} \right )$ .

12 M

5(b) Find the root of the function

f (x) = \frac{3 x^{2}}{16} - \frac{27}{4}

$f(x)=\dfrac{3x^2}{16}-\dfrac{27}{4}$ using Regula-Falci method. Consider the span [0, 10].

8 M

6(a) Solve the following system by Gaussian Elimination with and without partial pivoting and comment on the results:

$\begin{bmatrix} 2 & 1 & 1 & -2\\ 4 & 0 & 2 & 1\\ 3 & 2 & 2 & 0\\ 1 & 3 & 2 & 0 \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ x_3\\ x_4 \end{bmatrix}=\begin{bmatrix} 0\\ 8\\ 7\\ 3 \end{bmatrix}$

10 M

6(b) Solve the following system by LU decomposition:

$A=\begin{bmatrix} 4 & 0 & -1 & 3\\ 2 & 1 & -2 & 0\\ 0 & 3 & 2 & -2\\ 1 & 1 & 0 & 5 \end{bmatrix}\ \ \ \ b_1=\begin{bmatrix} 0\\ 1\\ 4\\ -2 \end{bmatrix}$

10 M

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