### 6HA - Applied Mathematics in Chemical Engineering

April - 1997

Part - A (20 X 2 = 40 marks)

1. Convert y = ax3 + b into a linear form.
2. What type of graph paper is used to get a straight line for the equation y = axb?
3. If y = f(x) and dx is the error in x, what is the corresponding error in y?
4. State the principles of group averages in curve fitting.
5. What is meant by mass balance in chemical engineering?
6. If a chemical disintegrates at a rate proportional the square of the amount present at any instant, what is the differential equation governing this rule?
7. Transform y'' + y' + y = x; y(0) = 1, y'(0) = 1 into two simultaneous (equivalent) first order problems.
8. State Newton's backward difference interpolation formula.
9. Obtain the second degree polynomial through the points (0, 2), (2, 1), (1, 0).
10. What is the central difference approximation to Uxx (where U = U(x,y) ) taking h, k as the step sizes, at the point (xi, yi).
11. Find D (x2 -x +1) taking h = 1.
12. Express (12.25)10 in binary system.
13. Write the octal number 156 in the decimal system.
14. Distinguish between floating point and fixed point constants.
15. Write the Runge Kutta algorithm of second order for solving y' = f(x, y), y(x0) = y0.
16. Write the Newton Raphson formula for finding the roots of the equation f(x) = 0.
17. What is the purpose of DO statement in FORTRAN?
18. Distinguish between arithmetic IF and logical IF in FORTRAN?
19. A homogeneous rod of length l, has one end at x = 0 at 0oC and the far end is insulated. Write the boundary conditions in symbolic form.
20. Write a solution by the method of separation of variables of the heat equation Ut = Uxx.

Part - B (5 X 12 = 60 marks)

Answer ALL questions, choosing 'a' or 'b' from each question.

1. (a) Fit a least square exponential curve y = abx to the data.
2. ```x: 0       1       2       3       4       5       6
y:      32      47      65      92      132     190     275```

(b) Write a technical note on treatment of experimental data and interpretation of results in solving through mathematical methods, chemical engineering problems.

3. (a) Random decomposes at a proportional to the account present. If a fraction p of the original amount disappears in 1 year, how much will remain at the end of 21 years.
4. (b) A tank contains 1000 gallons of brine in which 500 lb of salt are dissolved. Fresh water runs into the tank at a rate of 10 gallons/min and the mixture kept uniform by stirring, runs out at the same rate. How long will it be before only 5 lb of salt are left in the tank.

5. (a) Solve using RK fourth order method, dy/dx = 1 -2xy y(0) = 0 at x = 0.1 and 0.2.
6. (b) For the given data below

```x: 1       3       6       10
y:      0.4     9       18      35```

estimate, using Lagrange's interpolation, y(5).

7. (a) Using Newton's Raphson method, solve for a positive root of ex - 4x = 0, to 5 decimals.
8. (b) Write a complete FORTRAN program to solve the problem 24 (a) through computer implementation.

9. (a) Using method of separation of variables, solve the one dimensional heat equation (boundary value problem) Ut = a 2Uxx; 0 < x < p ; t > 0 with Ux(0, t) = Ux(p , t) and U(x, 0) = cos(2x) - 7cos(4x) and a = 1.

(b) Write a FORTRAN program for solving the problem 23 (b).

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