Interpolation practice problems PDF

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Some practice problems enhancing your enterpolation skills...

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PRACTICE PROBLEMS

37

Chapter 10 Interpolation Practice Problems Use Excel and VBA to solve the following problems. Document your solutions using the Expert Problem Solving steps outlined in Table 1.2.

1. Consider the vapor-liquid equilibrium mole fraction data below for the binary system of methanol and water at 1 atm. X 1 0.882 0.765 0.653 0.545 0.443 0.344 0.25 0.159 0.072 0

Y 1 0.929 0.849 0.764 0.673 0.575 0.471 0.359 0.241 0.114 0

2. Determine the vapor mole fraction of methanol (y) corresponding to the liquid mole fraction of methanol of x = 0.15 by linear interpolation and a quadratic Lagrange interpolating polynomial. Compare your results to the experimental value of y = 0.517. 3. Find the vapor mole fraction of methanol for a liquid mole fraction of 0.15 using a cubic-spline interpolation. 4. A flash drum is used to separate methanol from water by concentrating the methanol in the vapor phase and the water in the liquid phase exiting the flash drum. The exit streams are assumed to be in equilibrium. 5. For the following schematic V, y

F, z

L, x

Determine the composition of the vapor and liquid streams for a feed composition of methanol z = 0.5 when 60% of the molar feed leaves in the liquid phase. a. Set up a third order Lagrange interpolating polynomial worksheet in Excel and use the Solver to determine the exit stream mole fractions. b. Set up a cubic spline interpolating function in Excel and determine the exit stream compositions using GoalSeek. Xmethaol

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ymethanol

0.000

0.418

0.579

0.665

0.729

0.779

0.825

0.870

0.915

0.958

1.000

Answers: x = 0.362, y = 0.707 6. Refer to the Pressure-Volume data. Use a four-point Lagrange interpolating polynomial to estimate the volume, V, at P = 1.2 atm. T = 298 K, Rg = 0.0821 L-atm/gmol-K.

38

PRACTICAL NUMERICAL METHODS

PV A A  1  1  22 R gT V V P(atm) 0.969 1.090 1.341 1.606

V(L) 25.0 22.2 18.0 15.0

7. Interpolate the following data to estimate the pressure at 3 minutes using a 3-point Lagrange polynomial. Compare your result to the interpolated value obtained from a 2nd order polynomial. Show your work. Time/min Pressure/mmHg 0.0 7.5 2.5 10.5 5.0 12.5 10 15.8 15.0 17.9 20.0 19.4 8. Use a 3rd order Lagrange interpolating polynomial to predict y at x = 1.25 for the following data. x 1 1.1 1.2 1.3 1.4

y 0.14 0.21 0.33 0.54 0.92

9. Refer to the data in the following table. Interpolate the data from x = 1 to 10 at increments of 0.5 using the following two methods. Plot the interpolated values and data together using each of the following methods (create a new plot with the data for each method). You may use Excel for this problem. Label your axis on the plots. a) linear interpolation b) cubic spline interpolation 2

3

4

6

8

10

y 2 2.5

x 1

7

10.5

12.75

13

13

10. Create a VBA user function for Lagrange polynomial interpolation of smooth data. Test your function using results from previous problems....