EEE-244

Spring 08

MATLAB Project #3 (10%)

Instructions:

  • Due date: 5/19/2008 4-5:00 pm (RVR-5025C).
  • Please include all three MATLAB projects in a single folder.
  • The project reports should include all necessary printouts and plots for the numerical work as well as all necessary equations and derivations for the analytical work.
  • Organize the materials in an easy-to-follow manner.
  • A professional quality report is expected for project #3.
  • You are allowed to share ideas but do not let anybody copy your work.

Objective:

Mathematical models of the thermal processes of a typical water heater are given. Your task is to obtain the solution using both analytical and numerical techniques.

Problem Statement:

Most modern water heaters are controlled by a thermostat so that the water temperature is cycled betweentwo preset temperatures. In this project you are required to compute the water temperature during the heating and cool-down of a typical water heater and from the resultsdetermine the duty-cycle of the heater.

Mathematical model:

Two competing thermal processes, heat generation from the heating elements and heat loss through the insulation of the heater walls, are considered and the corresponding one-dimensional ordinary differential equations (ODEs) are developed.

Heater off: (1)

Heater on: (2)

where:t– time

T– Water temperature

Tam– Ambient temperature = 20 C

A– Total surface area of the tank 1.42 m2

P– Power consumption = 5000 Watts

– Heating efficiency of the heater  70%

L– Thickness of the insulation of the heater 0.25 inch

k– Thermal conductivity of the insulation material 0.024 J/secmC

c – Specific heat of water = 4213.3 J/kgC

M– Mass of the water in the tank 80 kg

Solving the ODEs we will obtain the water temperature as a function of time for both the cool-down (heater off) and the heating+cool-down processes (heater on).

Assignments:

  1. MATLAB solutions:
  2. Assume Tmax= 95 C and Tmin = 90 C.
  3. Use MATLAB(ODE23 or ODE45, p. 771 of the text) to solve equation (1)numerically and plot the water temperature as a function of time when the heater is off.
  4. Use MATLAB (ODE23 or ODE45, p. 771 of the text) to solve equation (2) numerically and plot the water temperature as a function of time when the heater is on.
  5. From the solutions determine the heating time and cool-down time.
  6. Compute the duty-cycle of the heater.
  7. Repeat the above computations for Tmax = 95 C andTmin = 85C.
  8. Compare the duty-cycle for the cases:

(Tmax = 95 C and Tmin = 90 C vs. Tmax = 95 C andTmin = 85 C)

Which case is more efficient?

  1. Analytical solutions:
  2. Solve the ODEs analytically – you can solve them by hand or use the dsolve command from the Symbolic Mathematics Toolbox in MATLAB.
  3. From the solutions derive the formulas for the heating and cool-down times as functions of Tmax and Tmin.
  4. Compute the heating and cool-down times for Tmax = 95 C and Tmin = 90 C.
  5. Compute the duty-cycle of the heater.
  6. Repeat the above calculations for Tmax = 95 C andTmin = 85C.
  7. Compare the analytical results to the numerical results.
  8. Do the analytical results and the numerical results match? If not, speculate the reason(s).
  9. Conclusion:
  10. What did you learn from this project?
  11. Future work (MS project using MATLAB???)
  12. Anything else you would want your instructor to know?

Appendix: The MATLAB dsolve command

ODE:

To solve the above ODE analytically you can type:

EDU» y=dsolve('Dy=-a*(y-ymin)','y(0)=y0')

y =

ymin+exp(-a*t)*(-ymin+y0)