```ME 402
Lab 1: First-Order Time Response of a Thermal System
Q1)
For the experiment, we wanted to find out the time constants for various conditions of
heating and cooling to compare them with our predicted time constants and to also compare
them to each other. Figure 1 shows our experimental results for heating in hot water and
cooling in cold water (room temperature), which we can see conforms to our predicted model.
Figure 2 shows the time response from heating in warm water and cooling in air, which also
conforms to our predicted model with the exception of a few spikes. Figure 3 shows the time
response for heating in warm water and then cooling in cold water, which again depicts our
predicted model. The numerical values of the experimental and predicted time constants are
displayed in Table 1, along with the percentage error for the various conditions.
Hot Water to Cold Water
1
0.9
0.8
Voltage [V]
0.7
0.6
0.5
0.4
0.3
Run1
Run2
Run3
0.2
0.1
0
10
20
30
Time [sec]
40
50
60
Figure 1 shows the experimental time response of heating in hot water and cooling in cold
water.
ME 402
Lab 1: First-Order Time Response of a Thermal System
Warm Water to Air
0.6
0.5
Voltage [V]
0.4
0.3
0.2
0.1
Air1
Air2
Air3
0
-0.1
0
10
20
30
Time [sec]
40
50
60
Figure 2 displays the experimental time response for heating in warm water and cooling in air.
Warm Water to Cold Water
0.8
0.7
Voltage [V]
0.6
0.5
0.4
0.3
Warm1
Warm2
Warm3
0.2
0.1
0
10
20
30
Time [sec]
40
50
60
Figure 3 depicts the time response for heating in warm water and cooling in cold water.
ME 402
Lab 1: First-Order Time Response of a Thermal System
Temp
(deg
C)
Time
Constant
(s)
Trial
Initial
Run 1
26.19
90.64
20.08
1.997
2.331
Run 2
23.5
92.72
20.08
1.489
Run 3
29.97
86.74
19.11
Warm to
Air
Air 1
24.11
55.12
Air 2
35.83
Hot to
Cold
Air 3
Warm to Warm
Cold
1
Warm
2
Warm
3
Peak
Final
Heating
Predicted
tau (s)
Cooling
Heating
% Error
Cooling
Heating
Cooling
2
3
0.170
28.700
2.497
2
3
34.300
20.163
1.740
2.576
2
3
14.929
16.460
36.07
1.733
14.785
2
15
15.420
1.456
54.26
38.52
1.466
15.050
2
15
36.454
0.335
34.24
53.41
37.29
1.289
14.548
2
15
55.195
3.108
24.11
56.34
17.52
1.508
2.922
2
3
32.600
2.687
23.38
54.51
18.49
2.308
2.962
2
3
13.337
1.276
24.11
68.06
18.49
1.645
2.964
2
3
21.603
1.204
Table 1 shows our experimental and predicted results for the various conditions.
Examining our results in table 1, we can see that our percentage errors from the
predicted model to our experimental results were not that great: ranging from 0.170% to only
55.195%.
Figure 4 shows our predicted model, and we can see that it correlates with all 9 of our
experimental runs for the heating, through all the various conditions. In summary, the results
of our experiment show that heating and cooling thermal systems can be modeled by a first
order time response.
ME 402
Lab 1: First-Order Time Response of a Thermal System
Numerical Simulation of First-Order System
100
90
State
80
Temp (degrees C)
70
Numerical
Analytical
60
Time-Dependent
50
40
30
20
0
1
2
3
t/tau
4
5
6
Figure 4 shows our numerical and analytical predicted models we performed in the prelab.
Lab 1: First-Order Time Response of a Thermal System
ME 402
Q2)
Trial
Hot to Cold Run 1
Run 2
Run 3
Warm to
Air
Air 1
Air 2
Air 3
Warm to
Cold
Warm 1
Warm 2
Warm 3
Amplitude
(deg C)
Heating
64.45
69.22
56.77
Time
Constant
(s)
Cooling
Heating
Cooling
70.56
1.997
2.331
72.64
1.489
2.497
67.63
1.740
2.576
31.01
18.43
19.17
19.05
15.74
16.12
1.733
1.466
1.289
14.785
15.050
14.548
32.23
31.13
43.95
38.82
36.02
49.57
1.508
2.308
1.645
2.922
2.962
2.964
Table 2 displays the amplitudes of the step inputs along with the time constants for the
various runs.
The time constant of a LTI, first order system in response to a step input is the same no
matter how large the amplitude of the step input. As shown in table 2, there is no correlation
between the amplitudes of the step inputs and the time constants. For example, the greatest
and smallest time constants for heating (2.308 s and 1.289 s, respectively) corresponded to
two of the smallest amplitudes (19.17oC and 31.13oC, respectively). This effectively shows that
the time constant is not a function of the amplitude of the step input. One may argue that when
looking at the cooling side, the smallest amplitudes (which were for trials with conditions of
warm water to air) display the greatest time constants, so there may be a correlation, but we
must consider that there was a difference between the heat transfer coefficients that could
correspond to the change. After all, air has a much lower overall heat transfer coefficient,
which could be the cause for the greater time constants.
```