Thermal Systems I


Question 1: What happens if we increase C and Rth?
Change C from 1000 to 2000
Eunice and I predicted that if the heat capacity (the 'C' parameter) is increased, the cooling rate of the coffee -- or rate at which heat leaves -- decreases. We also predicted an increased thermal resistance (the 'Rth' parameter) would cause the same result -- a lower cooling rate and a longer time period for which the coffee stays warm. We implemented these changes separately and saw that our predictions were correct.

Question 2: Calculate P to heat the coffee up to 84 deg. Celsius?
Change Rth from .85 to 2
When it reaches its target temperature at 357 K, it hits equilibrium and there should be no change in temperature (dT = 0) at t = 2000 seconds. Restructuring the equation, we find P = (T-T_air)/Rth = 75.29

Working Backwards: Find C and Rth
Given T_final = 357, T_air = 293.
We calculated this by using two scenarios: 1) Where the temperature reaches equilibrium at 84 degrees Celsius and 2) Where the coffee first begins heating at t = 0.
  1. At equilibrium there should be no change in temperature (dT = 0) and dE also equals 0. The 'C' parameters cancel out and, therefore, Rth = .84
  2. At t = 0 seconds, using the Rth value we discovered in the 1st Scenario, we solved for C by identifying dT/dt is the slope of the graph. After finding the slope of the line from t = 0 sec (dT/dt = .07), we found C = 75/.07 = approx. 1071 J/K*
*Variance in calculations of the slope could result in slightly different answers.

Feedback and Control
1. Bang-Bang Control
Bang-Bang (BB) is appropriate for most thermal systems; it is response-based and reaches the desired temperature. This system though would be undesirable if the change between desired temperature and "normal temperature" is a very small, because BB would overshoot it every time. As seen in our graph, it overshoots regularly when the temperature is turned on (T < 357).

2. Proportional Control
Proportional control (PC) is not as efficient, because, conversely, it never hits the required 84 degrees Celsius (357 Kelvin) and undershoots regardless of the "gain" in the heat-power equation. We can credit the cause to the fact that the heat added becomes extremely small as it gets closer to the desired temperature.

3. Adding a Delay
This exercise was the most challenging! Thanks to the help of Juliette and Amy, we were able to make significant progress. The objective was to create a time delay between the temperature the system reads and the temperature the coffee actually is, in both BB and PC systems.
   
Please clearly indicate the source and give credit if you use the following codes or parts of them.

Comments

  1. UnknownApril 6, 2014 at 4:27 PM

    Good job on all your codes! Your graphs all look really good, especially the delay ones!

    ReplyDelete

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