ChE 401: Computational Methods in Chemical Engineering : Summer 1430/1431 (2010) Prof. Ibrahim S. Al-Mutaz 1 King Saud University College of Engineering Chemical Engineering Department ChE 401: Computational Methods in Chemical Engineering : Summer 1430/1431 (2010) Prof. Ibrahim S. Al-Mutaz Selected Problems –Part1 Problem 1: Solve the following system of equations (a) 4x 1 + x 2 + x 3 = 4 (b) x 1 + x 2 – x 3 = 2x 1 + 4x 2 – 2x 3 = 4 2x 1 + 3x 2 + 5x 3 = – 33x 1 + 2x 2 – 4x 3 = 6 3x 1 + 2x 2 – 3x 3 = 6(i) by the Gauss elimination method with partial pivoting,(ii) by the decomposition method with u 11 = u 22 = u 33 = 1.Problem 2: Show that the following matrix is nonsingular but it cannot be written as the product of lower and upper triangular matrices, that is, as LU. 1 2 3 2 4 1 1 0 2 A = − Problem 3: Show that there is no solution to t he following linear system: 4x 1 – x 2 +2x 3 + 3x 4 = 20 0x 2 + 7x 3 – 4x 4 = -7 6x 3 + 5x 4 = 4 3x 4 = 6 Problem 4: Determine whether the following matrix is singular: 2.1 0.6 1.1 3.2 4.7 0.8 3.1 6.5 4.1 − − − Problem 5: Determine the number of terms necessary to approximate cos (x) to 8 significant figures using Taylor series approximation, cos(x) = 1 - x^2/2! + x^4/4! - x^6/4! + ... etc. Calculate the approximation using a value of x=2 B. Write a program to determine your result. Problem 6: Given the two following equations: 4x 1 – 8x 2 = 4 and x 1 + 6x 2 =9. (a) Solve them by Gaussian elimination; (b) Write Matlab code to solve by left division (backslash) operator. (c) Write Matlab code to solve them by first performing LU decompos ition, and then using the matrices L and U for the solution.
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3x1 + 2x2 – 4x3 = 6 3x1 + 2x2 – 3x3 = 6 (i) by the Gauss elimination method with partial pivoting, (ii) by the decomposition method with u11 = u22 = u33 = 1. Problem 2:
Show that the following matrix is nonsingular but it cannot be written as the product of lower
and upper triangular matrices, that is, as LU.
1 2 3
2 4 1
1 0 2
A
= −
Problem 3:
Show that there is no solution to the following linear system:
4x1 – x2 +2x3 + 3x4 = 200x2 + 7x3 – 4x4 = -7
6x3 + 5x4 = 4
3x4 = 6
Problem 4:
Determine whether the following matrix is singular:
2.1 0.6 1.1
3.2 4.7 0.8
3.1 6.5 4.1
−
− −
Problem 5:
Determine the number of terms necessary to approximate cos (x) to 8 significant figures using
Taylor series approximation, cos(x) = 1 - x^2/2! + x^4/4! - x^6/4! + ... etc. Calculate the
approximation using a value of x=2B. Write a program to determine your result.
Problem 6:
Given the two following equations: 4x1 – 8x2 = 4 and x1 + 6x2 =9.
(a) Solve them by Gaussian elimination;
(b) Write Matlab code to solve by left division (backslash) operator.
(c) Write Matlab code to solve them by first performing LU decomposition, and then usingthe matrices L and U for the solution.
If necessary, make sure to rearrange the equations to achieve convergence.
Problem 15:
Determine the real roots of f(x) = -0.4x2
+ 2.2x + 4.7;
a) Graphically.
b) Using the quadratic formula.
c)
Using three iterations of the bisection method to determine the highest root. Employinitial guesses of xl=5 and xu=10. Compute the estimated error ge and the true error gt
after each iteration.
Problem 16:
Determine the real roots of f(x) = - 2 + 7x - 5x2
+ 6x3;
a) Graphically.
b) Using bisection method to locate the lowest root. Employ initial guesses of xl=0 and xu=1
and iterate until the estimated error ge below a level of gt =10%.
Problem 17:
The derivative of f(x)= 1/(3- 2x2) is given by 4x /(3 – 2x2)2. Do you expect to have difficultiesevaluating this function at x= 1.22? Try it using 3 and 4 digit arithmetic with chopping.
Problem 18:
The saturation of dissolved oxygen in freshwater can be calculated with the equation:
Where Osf = the saturation concentration of dissolved oxygen in freshwater at 1 atm in mg/L and
Ta = absolute temperature in K. Remember that Ta = t + 273.15, where t = temperature inoC.
According to this equation, saturation decreases with increasing temperature. For typical naturalwaters in temperature climates, the equation can be used to determine that oxygen concentration
ranges from 14.621 mg/L at 0oC to 6.949 mg/L at 35
oC. Given a value of oxygen concentration,
this formula and the bisection method can be used to solve for temperature inoC.
a) If the initial guesses are set at 0 and 35oC, how many bisection iteration would be
required to determine temperature to an absolute error of 0.05oC.
b) Based on a), develop and test a bisection program to determine T as function of a given
oxygen concentration. Test your program for Osf = 8, 10 and 14 mg/L. Check your results
For a desired error Ea,d the number of iteration can be found from the following equation: