In this experiment we attempted to observe the similarities between thermal resistance and capacitance with their electrical counterparts (resistors and capacitors). This was accomplished by heating an enclosed insulated box with a light bulb, and measuring the temperature inside with respect to time. Comparing the results with what one would expect from an electrical circuit suggested that the thermal characteristics of the box indeed functioned and could potentially be modeled by an "equivalent" electrical circuit.
The insulated enclosure can be represented as a simple first order system with heat flowing in, some capacity to store the heat, and heat flowing out. The electrically equivalent first-order circuit consists of a voltage source, capacitor, and resistor connected in parallel.
The thermal capacitance of the insulated box is analogous to the capacitor in the circuit, as it is capable of holding some heat, while the capacitor holds charge. Thus, heat and charge are analogous quantities also. As the charge builds up in the capacitor up to the maximum capacity, the heat also builds up in the box up to the maximum heat until the thermal resistance allows the heat to leave at the same rate that it enters.
The thermal resistance is analogous to the resistor in the equivalent circuit because it blocks the heat inside (high temperature) from flowing out (to a lower temperature). The heat flow through the insulation (equivalent to charge flow through the resistor) increases as the temperature inside the box increases. Also, the units of resistance (Ohms) and thermal resistance (Kelvins/Watt) are analogous.
The heat provided by the light bulb is analogous to the current flow in the electrical circuit, and the voltage is analogous to temperature, because it measures a potential energy difference in both situations.
Because the behavior of two systems can be modeled by the same forms of equations, the graphs of the temperature decreasing as a function of time after turning off the light bulb should exhibit the same exponential decay as given by a discharging capacitor. Both systems thus have a time constant, and the time constant is equal to the resistance times the capacitance (either thermal or electrical) in both cases.
Note that the ground connection in the circuit is equivalent to the ambient temperature in the laboratory. Both specify a zero reference point from which other potential measurements are made (either voltage or temperature).
The procedure followed in the laboratory did not deviate from the procedure described in the experiment description, which can be found at: http://palantir.swarthmore.edu/maxwell/classes/e12/S04/labs/lab01/
The equation governing the electrical system is:
For the thermal system, the rate of energy accumulating in the box can be described be the following relations:
1. Rate of energy accumulated = rate of energy in - rate of energy output
2. Rate of energy accumulated = heat in - heat out
Mathematically, this can be represented by: m * C_p * dT/dt = P - (T - T_0) / R
In this equation, m, C_p, and T are the mass, specific heat, and temperature of the air in the box. T_0 is the ambient laboratory room temperature, P is the power supplied by the light bulb, and R is the thermal resistance. The parallel between the electrical circuit's equation is clear.
The term m*C_p*dT/dt is equivalent to C*dV/dt, since m*C_p and C both are describing the physical properties of the systems, and as discussed before, temperature and voltage are analogous quantities because they both describe a difference in potential energy. Also, the expressions I_in - V/R and P - (T - T_0)/R are equivalent.
In order to smooth out the data acquired by the data acquisition system, we convolved the input data with a Gaussian filtering function to average adjacent data values. The filtering involved a 1-4-6-4-1 averaging scheme. (Note that the 1-4-6-4-1 scheme was arrived at by convolving the 1-2-1 filter with itself.) This means that for some data point d_10, the filtered data point is given by:
f_10 = ( d_8 + 4*d_9 + 6*d_10 + 4*d_11 + d_12 ) / 16.
This method was used to filter all the data to arrive at better exponential fits.
| Run Number | Time Constant (tau) | Thermal Resistance | Thermal Capacitance |
|---|---|---|---|
| 1-unfiltered | 170.4622 | 0.68485 | 248.904 |
| 1-filtered | 170.3200 | 0.68467 | 248.762 |
| 2-unfiltered | 169.1589 | 0.72372 | 233.733 |
| 2-filtered | 159.0305 | 0.72362 | 219.769 |
| 3-unfiltered | 142.1787 | 0.68812 | 206.617 |
| 3-filtered | 142.8979 | 0.68777 | 207.768 |
The results obtained coincide with what one would expect from the nature of the runs. For run 2, the box was better insulated because of the books and coat, and thus the thermal resistance should be greater than for the other two runs, which it is. The time constant is lowest for the last run in which the box was open and cooling off. This make some sense because while heating the box for the other two runs, heat was lost through insulation errors, meaning that it would take longer to heat the box to the final temperatures. The thermal capacitance values range between 200 and 250 Joules / Kelvin, whereas the calculated value was approximately 45 Joules / Kelvin. This could be because the inside of the box was lined with a reflective aluminum layer, which would reflect much of the heat that would normally be lost through the insulation back into the box. As a result, it would appear that the box had a much higher thermal capacitance. The thermal capacitance values for the two heating runs are both greater than the values for the cooling run, which makes sense because the top of the box was open for the cooling run, rendering the box less capable of storing heat.
Some problems with the model used that would lead to errors would be holes in the insulation.
The time constant values obtained (between 140-170 seconds for the various runs) accurately describe the amount of time required for the exponential fit to decay to 67% of its initial value. This is analogous to the time constant work we did with first order electrical circuits in previous labs.
Despite some explainable discrepancies, the lab conveniently summarized the analogues between the electrical and thermal systems. The theoretical values for thermal resistance were relatively close to the measured values, and while the theoretical values for thermal capacitance were not close to the measured values, the differences could be accounted for.
Future work will involve more labs in the future.
Much of the text of this writeup is more or less directly taken from the original lab description, which is available at:
http://palantir.swarthmore.edu/maxwell/classes/e12/S04/labs/lab01/
We replaced the light bulb with a hair dryer implement as the heat source to observe the differences in the heating and cooling characteristics of the box. The graphs of the runs are given below.
It is clear from the acquired data that heating the box with the hair dryer did indeed result in an exponential curve, however the cooling process doesn't seem at all to fit an exponential. The only explanation we can think of is that we did something horribly wrong with the cooling down process.
The calculations for the hair-dryer heated run are below:
tau = 1 / 0.070479 = 14.188 seconds
Thermal resistance = 51.154 / (wattage of hair dryer) = ? K/W
Thermal capacitance = tau / R = 14.188 / ? = ? J/K
Since the exponential fit for the cooling down was so bad, no calculations were attempted.
Once we know the watt output of the hair dryer that was used, we will post the remaining calculations and provide a discussion of the resulting R and C values with respect to the heating implement used.