### 6HA - Applied Mathematics in Chemical Engineering

April-1999

Part A - (20 x 2 = 40 marks)

1. Convert y = axb into linear form.
2. For which type of equation the points (x, y) plotted in a semi-log graph sheet will lie on a straight line?
3. Write the normal equations to fit y = a + bx by the method of least squares.
4. If p and q are the relative errors in x and y respectively, what is the relative error in the quotient x/y?
5. If a : 2b : 3c = 3 : 2 : 1, find (a - b) : (b - c).
6. If y = nx and S = R(c - x)/y, find y when S = 4R, n = 0.25 and c = 0.2.
7. A chemical substance disintegrates at a rate proportional to the amount then present at any instant. Write the differential equation governing this law.
8. Solve rdq + 2qdr = 0.
9. Solve dy/dx = y/x + ey/x .
10. If y is a function of x and z = log(x), express xdy/dx in terms of y and z.
11. If y = 2 when x = -2 and y = 8 when x = 1, find y when x = 0 using linear interpolation.
12. Find D 3 (x2) taking h = 1.
13. What is hybrid computer?
14. Express the binary number 1100 in octal.
15. Express 0.001728 in floating-point.
16. What is a data?
17. What is an algorithm?
18. Write a FORTRAN statement to convert Fahrenheit (F) into Celsius (C).
19. Write the solution of u/ t = a 2 2u/ x2 such that u is finite when t is tending to infinity.
20. State Laplace equation.
21. Part B - (5 x 12 = 60 marks)

Answer (a) or (b) in each question.

22. (a) The heat capacity c of a material (in calories per gram per degree Celsius) could increase with temperature T (in degree Celsius) according to a relationship of the form c = aT2 + bT + k. Determine the constants a, b and k by the method of least squares from the following data:
23. T    0     50    100     150      200

c     0.13200      0.14046      0.15024      0.16134      0.17376

(b) Fit a least square exponential curve y = aebx to the following data:

x     4      9      14      23

y     27      73      197      1194

24. (a) In a chemical reaction involving two substances, the velocity of transformation dx/dt at any time t is known to be equal to the product (0.9 - x), (0.4 - x). Express x in terms of t, given that when t = 300, x = 0.3.
25. (b) A tank of volume 0.5 m3 is filled with brine containing 40 kg of dissolved salt. Water runs into the tank at the rate of 1.5 x 10-4 m3/sec and the mixture, kept uniform by stirring, runs out at the same rate. How much salt is in the tank after two hours?

26. (a) Find the value of c when T = 40 from the following data:
27. T     -50     -20     10     70

c     0.125      0.128      0.134      0.144

(b) Using Runge-Kutta method of fourth order find y(1), y(2) and y(3) given that dy/dx = 4e0.8x - 0.5y,     y(0) = 2.

28. (a) Use the Newton-Raphson method to estimate the root of e-x = x employing an initial guess of x0 = 0 correct to six decimal places.
29. (b) Write a FORTRAN program for the problem given in question 24 (a).

30. (a) A bar of length 30 cm has end temperatures 0o C and 100o C until steady state conditions prevail. Suddenly the temperature at the first end is raised to 50o C from 0o C and thereafter maintained. Find the temperature distribution at any point of the bar at any subsequent time.
31. (b) Write a FORTRAN program for the problem given in the question 23 (b).