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Homework 1

Answer all questions. Show all calculations, define all necessary terms, and show reasoning to receive credit. Absolutely no credit will be given for guessing. No handwritten/scanned work will be accepted. All work must be typed in whatever preferred format and all implemented MATLAB code must be included in your solutions. Submit your homework at the Moodle page.

Problem 1: Euler’s Method and Newton’s Law of Cooling. (Chapra, Problems 1.17, 1.18)

You are working as a crime scene investigator and must predict the temperature of a homicide victim over

a 5-hour period . You know that the room temperature was 10 °C when the victim’s body was discovered .

a.   Use Newton’s law of cooling (see problem  1. 17) and Euler’s method to compute the victim’s body temperature for the 5-hr period using values of k = 0. 12/hr and Dt = 0.5 hr. Assume the victim’s body temperature at the time of death was 37 °C over the 5-hr period.

b.   Further investigation reveals that the room temperature had actually dropped linearly from 20 to 10 °C over the 5-hr period. Repeat the same calculation as in (a) but incorporate this new information.

c.   Compare the results from (a) and (b) by plotting them in the same graph. In this problem(1), you can use either MATLAB (include your code with the solution) or any other worksheet software (EXCEL) as discussed in class. You can follow the example presented in class.

Problem 2: Euler’s Method and Evaporation. (Chapra, Problem 1.15)

Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area:

   = − kA

where V = volume (mm3), t = time (min), k = the evaporation rate (mm/min), and A = surface area (mm2) . Use Euler’s method to compute the volume of the droplet from t = 0 to 10 min using a step size of 0.25 min . Assume that k = 0.08 mm/min and that the droplet initially has a radius of 2.5 mm . Assess the validity of your results by determining the radius of your final computed volume and verifying that it is consistent with the evaporation rate . In this problem (2), you can use either MATLAB (include your code with the solution) or any other worksheet software (EXCEL) as discussed in class . You can follow the example presented in class .

Hint: As the droplet is evaporating, its area and volume are simultaneously changing . Find a relationship between area and volume for a sphere . 

Problem 3: Vectors in MATLAB. (Chapra, Problem 2.8)

If a force F(N) is applied to compress a spring, its displacement x(m) can often be modeled by Hooke’s law:

F = kx

where k = the spring constant (N/m). The potential energy stored in the spring U(J) can then be computed as

U = kx2

Five springs are tested and the following data compiled: 

F, N      14         18         8          9          13

x, m  0.013  0.020  0.009  0.010  0.012 

Use MATLAB to store F and x as vectors and then compute vectors of the spring constants and the potential energies. Use the max function to determine the maximum potential energy

Problem 4: Arrhenius equation and plots in MATLAB. (Chapra, Problem 2.20)

The temperature dependence of chemical reactions can be computed with the Arrhenius equation:

k = Ae − E /(RTa )

where k = reaction rate (s- 1), A = the  preexponential (or frequency) factor, E = activation energy (J/mol), R = gas constant [8.314 (J/mole ×K)], and Ta = absolute temperature (K). A compound has E = 1 x 105 J/mole and A = 7 x 1016 . Use MATLAB to generate values of reaction rates for tempeartures franging from 253 to 325 K. Use subplot to generate a side-by-side graph of:

a.  k versus Ta and

b.  log10k versus 1/Ta .

Employ the semilogy function to create (b). Include axis labels and titles for both subplots (choose different colors for the plot lines). Interpret your results (briefly explain what you observe).