Thermodynamics Experiment 1: Temperature Measurement
March 28, 2016
A very basic experiment, we conducted this experiment to find the minimum waiting time before a thermometer displays an accurate temperature measurement. Generally, the thermometer measures the heat transfer between the object and the thermometer. For an alcohol thermometer, the alcohol inside the glass expand linearly as the temperature increases.
In theory, for a linear temperature sensor, the rate of change of ΔT=Tf - Ti is directly proportional to the decreasing ΔT. That is,
dΔT/dt = -kΔT,
(1)
where k must be 1/𝝉, and 𝝉 is the minimum waiting time. This is a pretty straightforward derivation using rate of change. Integrating both sides, we will get the simple equation,
The experiment was divided into two parts: heating and cooling. In heating, the final temperature Tf was the temperature of the hot bath and the initial temperature Ti was recorded from the cup of ice. T(𝝉), and succeding time constants, were computed beforehand. The time it took for the temperature change from Ti to T(𝝉), T(2𝝉), T(3𝝉), ... were recorded. The same procedure were implemented for the cooling part except that Tf is the temperature in cup of ice, and Ti the temperature on the hot bath.
Using Equation 2, we linearize the time recorded and get the inverse of the slope to obtain the time constant 𝝉.
T(t) = Ti + ΔT(1-e^(-t/𝝉))
(2)
The experiment was divided into two parts: heating and cooling. In heating, the final temperature Tf was the temperature of the hot bath and the initial temperature Ti was recorded from the cup of ice. T(𝝉), and succeding time constants, were computed beforehand. The time it took for the temperature change from Ti to T(𝝉), T(2𝝉), T(3𝝉), ... were recorded. The same procedure were implemented for the cooling part except that Tf is the temperature in cup of ice, and Ti the temperature on the hot bath.
Using Equation 2, we linearize the time recorded and get the inverse of the slope to obtain the time constant 𝝉.
Figure 1. Sample linearized plot of t vs t/𝝉.
The average 𝝉 obtained is 6.29 for the heating part and 11.39 for the cooling part (not including the outlying values). The total average time constant is 8.84, and it is accepted to wait 3 to 5 time constants
(~30 sec to 2 min). However, the system being observed on the experiment was water. For a real fever, Tf will increase, thus, increasing time for stabilization. And this will lead to a minimum waiting time of at least three and a half minutes[1].
Footnote(s):
1. Huber, Mike, Teaching Differential Equations with Modeling and Visualization- Fever, August 2010. Retrieved from <<http://www.codee.org/library/projects/teaching-differential-equations-with-modeling-and-visualization>>.
(~30 sec to 2 min). However, the system being observed on the experiment was water. For a real fever, Tf will increase, thus, increasing time for stabilization. And this will lead to a minimum waiting time of at least three and a half minutes[1].
Footnote(s):
1. Huber, Mike, Teaching Differential Equations with Modeling and Visualization- Fever, August 2010. Retrieved from <<http://www.codee.org/library/projects/teaching-differential-equations-with-modeling-and-visualization>>.
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