jeffoshiro lab 1 assignment
TRANSCRIPT
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Jeffrey Oshiro
ME 402
Dr. Bingham
January 24, 2013
Lab 1: First-Order Time Response of a Thermal System
Q1: Comparing Model Prediction and Experimental Measurements
In this procedure the heating and cooling curves for various water temperatures were examined to
determine a time constant for the first-order system. An example of a first-order time response is
shown inFigure 1. This model plots the variable (velocity of a falling object) against time. The results of
the procedure were compared qualitatively to this model. This was to determine if the thermal systems
behaved like a first-order system. The time constants of the results were also compared to the time
constant of a predicted model. This was to determine how well the predicted model accurately
represents the behavior of the actual system. For this procedure, the results for Run1 of the timeresponse for Hot Water to Room Temperate Water inFigure 2 will serve as the predicted model.
The heating and cooling curves for the time response of Hot Water to Room Temperature Water is
shown inFigure 2. The results for Warm Water to Room Temperature Water is shown inFigure 3. The
results for Hot Water to Air is shown inFigure 4.
Figure 1: Example of a first-order system
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The results of the all the procedures look consistent with the example (Figure 1)of how a first-order
system behaves. They all display that characteristic exponential change from the initial state before
reaching the steady-state. However, there are some inconsistencies in several trial runs. In Run1 in
Figure 2 there is a bump in the curve before the 10 s mark. This could be due to shifting the probe in the
hot water during data recording. Run3 ofFigure 2 also has a bump at around . This might be due
to shifting the probe.
ForFigure 2 andFigure 3,the steady-state of the heating curves gets progressively smaller with each
run. This is due to the hot or warm water cooling over time and with each transfer of heat from the
water to the probe. It is also observed that the heating curves forFigure 2 andFigure 3 also become
progressively steeper as it reaches the steady-state and flatten out quicker after reaching the steady-
state. The flattening could be due to the hot or warm water cooling.
The results of the heating curve data of all the procedures are summarized inTable 1. The results of the
cooling curve data are summarized inTable 2. InTable 1 andTable 2,and represents the initial
temperature and steady-state temperature of the respective curves. The time constantwas
determined by calculating 63.2% of the absolute difference betweenand (the amplitude of the
step-unit) and graphically finding the time when that value occurs in the curve. The percent error wascalculated as the absolute difference between the time constant of a particular curve and the time
constant of the corresponding heating or cooling curve of the predicted model (Run1 ofFigure 2)
divided by the time constant of the predicted model.
Figure 2: A plot of the heating and cooling curves for hot water to room temperature water.
In the absence of a predicted model, the curve "Run 1" will serve as the prediction.
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Figure 4: The heating and cooling curve for the time response from hot water to air.
Figure 3: The heating and cooling curves for the time response from warm water to room
temperature water.
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Q2: Defending a Hypothesis with Evidence
The time-constant of the step response of a first-order system is independent of the amplitude of the
step unit. If the time-constant depended on the amplitude of the step inputTable 1 andTable 2 would
show a correlation between the percent error and the amplitudes. However, there doesnt seem to be
any consistency with the percent errors and the amplitudes.
Table 2: Cooling Curve Data Summary
Cooling
Ti(C) Tf(C) Amplitude % Error
Hot to Room
81.25 22.102 0.8787 59.148 N/A
75.6774 20.9288 1.1297 54.7486 28.56493
72.1578 20.9288 1.318 51.229 49.99431
Warm to
Room
53.5196 20.2793 1.0042 33.2403 14.28246
51.3408 20.7263 0.8787 30.6145 4.04E-13
46.8715 22.1788 1.2552 24.6927 42.84739
Hot to Air 86.9204 29.5321 38.5983 57.3883 4292.66
Table 1: Heating Curve Data Summary
Heating
Run Ti(C) Tf(C) Amplitude % Error
Hot to Room
1 26.0126 84.8054 0.6904 58.7928 N/A
2 23.2751 76.6302 1.0042 53.3551 45.45191
3 22.3953 73.3942 0.8159 50.9989 18.17787
Warm to
Room
1 22.1788 53.6313 0.8786 31.4525 27.25956
2 20.6145 51.642 0.6904 31.0275 0
3 19.7765 46.7598 0.6904 26.9833 1.61E-14
Hot to Air 1 21.2221 87.1159 0.5648 65.8938 18.19235