Thermal Systems Part I

After reading and trying several exercises on MATLAB, we were given a packet on thermal systems to work on.


Question 1: How does the cooling behavior change if we vary the parameters Rth
and C? Figure this out using intuition and the above equations, and then vary these
parameters in your program to confirm your conclusions.

Answer:

By looking at the equation, dT/dt = -(T - Tair)/(Rth * C), one can see that as Rth increases, it requires more "effort" required to lose energy. Similarly, as C (heat capacity) increases, it becomes "harder" for the coffee to lose energy (or heat), shown by a smaller change in temperature (measure of transfer of heat).

I varied these parameters in my program to confirm my conclusions. I changed Rth to 0.95 (versus 0.85) and noticed T-Tair or delta T was smaller in magnitude. I noticed similar results when I increased the value of C.

Program Used:

Question 2: Calculate a good value for P if we want our coffee to heat up to the
Starbucks ideal 84°C?

Answer:

Working backwards. Given a plot like the one above, can you deduce the thermal

parameters C and Rth ?

One could determine Rth because at a given T (dT/dt = 0),one knows Tair, P is given, C cancels out, and T can of course be read from the plot.

One could also determine C because one can find an approximate value of dT/dt by looking at the slope of the curve in the beginning (because dT/dt is clearer there). Also, P is given, T can be read from the plot, Tair is given, and Rth is given. 

1) Modify the above programs to simulate a temperature controller that uses

bang-bang control to reach and maintain the desired temperature. Bang-bang

control is a very common approach for thermostats. Why is bang-bang control

appropriate for many thermal systems? When might it be insufficient?

Bang bang control is appropriate for many thermal systems because bang bang control either turns on or off the power whenever a limit is reached (temperature is too high or too low). For many thermal systems (like the ones inside our homes), this oscillation is not significant. It is okay for our houses to be a couple of degrees off from the temperature we want. However, for thermal systems that must be more exact (with little to basically no oscillations), bang bang control would be inefficient.

2) Create a program that uses proportional control to reach and maintain the

desired temperature. How does this approach compare to bang-bang control?

Using proportional control makes a system run a lot "smoother" without any noticeable back and forth (power on and off) movements or oscillations. First, it take some time to reach the desired the temperature, but the system stays at that temperature. Bang bang control makes sure the temperature (at the moment) stays within a range. When a limit is reached (the inequality), the system tells the power to either turn on or off completely. With proportional control, the feedback is in proportional (directly related to) to the error or the degree of divergence from the target (what we want). 

3) Suppose there is a delay between the time the coffee reaches a given

temperature and when the temperature sensor records that temperature. Modify

both of your programs to include this effect, along the lines of the programs on the

next few pages that I showed in class, which explore the effect of adding a “sensor

delay” to a simulation of moving a SciBorg lego car a particular distance. What is

the impact of this “sensor delay” on your system in each case? What other delay(s)

might you expect in your thermodynamic system, apart from sensor delays?

Bang-bang control with delay:

Bang bang control with sensor delay makes the system take longer (in terms of time) to turn on or off the power to stay within a range. Thus, the oscillations are more spread out. 

Proportional control with delay:

Proportional control with sensor delay has a small bump on the plot because the delay causes the system to keep increasing or decreasing the temperature until it realizes (after the delay time) that the temperature has passed the desired/target temperature. Then, the system decreases or increases to adjust to the target temperature.

Other delays may include delay for the heater to actually begin the heating process and the delay from the thermometer (to measure the temperature at that moment and report the temperature value).