LAB 4: FIRST-ORDER SYSTEMS
1
Lab 4: First Order Systems
Kerwin S. Dewberry Jr
Lab Partner: Aaliyah M.P.A. Thompson-Mazzeo
AME 300 - SP24 001 (Monday, 2:00)
Date of Experiment: 25 March 2024
LAB 4: FIRST-ORDER SYSTEMS
2
Abstract
The purpose of this lab is to ultimately measure the step response of a thermocouple,
allowing the measurement for the time constant for the test to be found, utilizing a voltage and
temperature relation. Using LabVIEW, the voltage concerning time is recorded, and once
finding and evaluating existing equations between voltage and temperature, the transient
response of the thermocouple is found, and then the time constant is found from the error
function/temperature difference from the calculated values. The time constants were found to 3
test Cold to Hot Water, where t =0.836s, Ambient Temperature to Heated Airjet with t= 10.4s,
and Heated Airjet to Ambient Temperature with t=80.6. After calculating the 3-time constants,
uncertainty always is performed to ensure that the data generally falls between the lines as
expected. Despite some of the data falling outside of the lines, the time constants calculated do
align with the ideas and concepts that were learned in thermodynamics, proving that this
experiment was generally successful in proving the core ideas associated with it. Lastly, some
possible sources of error were discussed as well as some of the possible ways to prevent said
errors.
LAB 4: FIRST-ORDER SYSTEMS
3
Table of Contents
Abstract...........................................................................................................................................2
Table of Contents........................................................................................................................... 3
Nomenclature................................................................................................................................. 4
Experiment 4: First-Order Systems............................................................................................. 5
Introduction....................................................................................................................................5
Required Equations....................................................................................................................... 5
Equations: General.....................................................................................................................5
Equations: Uncertainty Analysis................................................................................................7
Experimental Apparatus and Procedure.....................................................................................8
Cold Water to Hot Water............................................................................................................9
Ambient Temperature to Heater Airjet.................................................................................... 10
Heated Airjet to Ambient Temperature....................................................................................11
Results and Discussion.................................................................................................................11
Results: Cold Water to Hot Water............................................................................................12
Uncertainty Analysis: Cold Water to Hot Water......................................................................14
Results: Ambient Air to Heated Airjet.....................................................................................15
Uncertainty Analysis: Ambient Air to Heated Airjet.............................................................. 19
Results: Heated Airjet to Ambient Air.....................................................................................20
Uncertainty Analysis: Heated Airjet to Ambient Air.............................................................. 24
Discussion: Error Analysis...................................................................................................... 25
Conclusion.................................................................................................................................... 26
References..................................................................................................................................... 28
Appendix.......................................................................................................................................29
LAB 4: FIRST-ORDER SYSTEMS
4
Nomenclature
∆𝑇
Change in Temperature (Equal to the Error Function)
Γ
Error Function
N
Number of SECONDS (whole)
τ
Time Constant
V
Voltage (V)
T
Temperature (C, K)
𝑆𝑦𝑥
Standard Deviation
𝑡95%,∞
Student t-distribution
t
Time (s)
LAB 4: FIRST-ORDER SYSTEMS
5
Experiment 4: First-Order Systems
Introduction
This experiment aimed to measure the step response of a thermocouple using its time
constant, which can be found by using the transient response from computer equipment and
relating the measured values to temperature. A thermocouple is a useful tool that can measure the
transient response by functioning on the principle that a voltage difference is produced when
dismissal metals are joined together, knowing that the voltage being read depends on the
temperature of the junction. By being able to model the temperature in terms of a voltage vs.
time relationship, the response of a thermocouple to a step change in temperature can be
modeled, allowing the time constant, τ, to be solved.
The use of a thermocouple allows for experiments to be performed inexpensively due to
the simple build, and when used correctly, the metal in the thermocouple is relatively durable.
Since the time constant can be easily measured from the response of a thermocouple by utilizing
readily available computer software, a thermocouple is a reliable way to analyze first-order
systems. This lab will test this relationship and will measure the time constant for 3 different
conditions, Cold Water to Hot/Boiling Water, Ambient Air to Hot Air/Heated Airjet, and Heated
Airjet to Ambient Air. The time constant will be calculated using the equations in the next
section.
Required Equations
Equations: General
This experiment's main objective is to use the thermocouple to measure the temperature
of what is being tested, use computer programming to measure the transient response, and then
find the time constant of the thermocouple from the measurement of temperature. This is going
to be accomplished by using a relation between the voltage and temperature, which is where the
thermocouple is primarily used. The thermocouple functions on the principle that a voltage
difference is produced when dissimilar metals are joined together at a junction between the two
metals, and if the voltage is measured at this junction, the resulting thermocouple voltage is a
LAB 4: FIRST-ORDER SYSTEMS
6
function of the temperature difference. The response of a thermocouple in a typical first-order
system to a step change in temperature is given by:
(1)
𝑇∞−𝑇(𝑡)
𝑇∞−𝑇0
Where 𝑇
temperature, (𝑇
∞
∞
(−𝑡/τ)
=𝑒
is the final ambient temperature, 𝑇 is the initial ambient/surroundings
0
& 𝑇0 are measured with a thermometer), 𝑇(𝑡) is the temperature indicated
by the thermocouple at time t (this will be measured during the experiment), 𝑡 is time and τ is
the time constant, which is what is being solved for. To measure 𝑇(𝑡), the following general
form equation is used, which utilizes the voltage, initial temperature, initial voltage, final
temperature, and final voltage, as well as the measured voltage from the thermocouple, where:
(2) 𝑇(𝑡) = 𝑇𝑖 + ([𝑇𝑓 − 𝑇𝑖]/[𝑉𝑓 − 𝑉𝑖]) * [𝑉 − 𝑉𝑖]
The value of these depends on the particular test. For the cold water to hot water test, the
initial temperature, 𝑇𝑖, would be the temperature of the ice water, and the final temperature, 𝑇𝑓,
would be the temperature of the boiling water, both of which are measured with a thermometer.
𝑉𝑓 and 𝑉𝑖 are the voltages measured by the thermocouple for the boiling water and ice water,
respectively. For the Ambient to Heated Airjet test, the initial temperature and voltage will be
from the free air/room temperature measurement and the final temperature and voltage will be
from the thermocouple under the Heated Airjet. Lastly, for the Heated Airjet to Ambient test, the
initial temperature and voltage will be from the Heated Airjet, and the final temperature and
voltage will be from the ambient state. V can also be written as V(t), but this is just a measured
value from the thermocouple.
From the collected voltage data and thermometer measurements, the temperature was
first calculated using equation 2 and 3 graphs were made, Temperature vs. Time,
Ln(ErrorFunction)/ln(deltaT) vs Time, and the Ln(ErrorFunction)/LN(deltaT) vs Time, adjusted
to remove unexpected or invaluable collected data. From the two Ln(ErrorFunction)/LN(deltaT)
LAB 4: FIRST-ORDER SYSTEMS
7
charts, the time constant was calculated, using a modified version of equation 1, which used the
relationship of a semilog plot, where:
(3) 𝑦 = 𝑚𝑥 + 𝑏 => 𝑙𝑛(𝑦) = 𝑚𝑥 + 𝑏 = 𝑙𝑛(Γ)/𝑙𝑛(∆𝑇)
and in this case, ln(y) represents both ln(deltaT) and ln(ErrorFunction), and m represents the
slope of the line of best fit of the for the Ln(ErrorFunction)/LN(deltaT) vs time graphs, where:
(4) 𝑚 =
1
− τ , 𝑥 = 𝑡, 𝑏 𝑣𝑎𝑟𝑖𝑒𝑠
From (4), τ, or the time constant, is calculated, where:
(5) τ =
1
− 𝑚
Equations: Uncertainty Analysis
Typically, it has been found that when working with temperatures, especially the heated
jet, there will always be some amount of uncertainty that is higher than expected. While errors
will be avoided at all costs, some are unavoidable considering time constraints and waiting for
instruments to heat up and cool down, though this will be discussed in more detail during the
error analysis and conclusion. For the uncertainty analysis, there will be 3 calculated
uncertainties, one for each test. Of which, the general form will be followed, using the student
t-distribution 𝑡95%,∞ = 1. 960, the confidence interval is defined by:
(6) 𝑦𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 ± 𝑡95%,∞ • 𝑆𝑦𝑥/ 𝑁
Where 𝑆𝑦𝑥 is the standard deviation of the data and N is the total number of seconds between the
data points..
LAB 4: FIRST-ORDER SYSTEMS
8
Experimental Apparatus and Procedure
In addition to the Lab Manual, the following experiment was used in this lab:
SCXI-1000
SCXI-1305 BNC box
Chassis/Aluminum Box
Heated Jet and Hot Plate
Chromel-Alumel
Thermocouple
Thermometer
Beaker of Water
LabVIEW
There are 3 different experimental setups for this lab, one for the Ice Water to Boiling Water test,
one for the Ambient Temperature to Heated Jet (Hot), and one for the Heated Jet to Ambient
Temperature. The thermocouple was set up according to the lab manual, shown below:
Figure 1: Final Experimental Set Up - Thermocouple
The thermocouple used a LabVIEW program to collect data. This program (called VI)
provided the experimenters with a plot of the measured thermocouple voltage versus time, a
table presenting the time series data (time in seconds, and TC voltage), and an output file that
was exported to Excel to create the fitted plots, uncertainties, etc.
LAB 4: FIRST-ORDER SYSTEMS
9
Cold Water to Hot Water
To measure the time constant for cold water to hot water, the first beaker was first filled
with ice, and water was allowed to cool, and then the second beaker was taken, filled with warm
water, and allowed to boil on the heated plate. It is important to note that ideally, the temperature
would be 0 degrees Celsius, however, that is not the case during this experiment. After
measuring the temperature of the ice water inside of the beaker using a thermometer, the
thermocouple was placed inside of the ice water beaker, and after a near-constant voltage was
being read, we rapidly transferred the thermocouple from the cold water to the boiling water.
Ideally, there should be a spike in the Voltage vs. Time graph, of the form is a typical function
with a step input. After allowing the thermocouple to reach a near-steady state temperature in the
hot water, the thermocouple was removed and allowed to cool for the next round of testing.
Then, uncertainty analysis was conducted on the experiment, and error bars were added
accordingly. This general format is followed for all 3 tests, the only thing that changes is the
equation of the line of best fit from Excel, and the initial/final values.
See the graph below for the experimental setup:
Figure 2: Final Experimental Set Up - Cold to Hot Water
LAB 4: FIRST-ORDER SYSTEMS
10
Ambient Temperature to Heater Airjet
The setup for the Ambient Temperature to the Heated Airjet test was first, taking the
thermocouple and placing it on a stand, allowing it to remain untouched, at a 30-degree angle
from the table (or ground), and then leaving the thermocouple alone while it reaches room
temperature. Then, the hair-dried (Heated Airjet) was turned on high and hot and was placed
significantly far away from the thermocouple, ensuring that there was no accidental temperature
reading, affecting the steady state of the thermocouple at ambient temperature. When testing was
ready (the thermocouple had reached a steady state voltage reading and the Airjet was
significantly hot enough) and the LabView was ready to collect data, the Airjet was moved
directly under the tip of the thermocouple to collect the change in voltage data from the
thermocouple over time. Ideally, the voltage would see a relatively sharp spike that does have a
small curve to it, due to the change in temperature being much smaller than the change in
temperature from the cold water to hot water test. After the heated jet reached a response that
was close to a steady state condition ( a constant voltage reading), the heated Airjet was kept on
and under the thermocouple, the test was stopped on lab view, and the file name was changed be
“Hot Air to Ambient Temperature”, and then testing for the next test immediately began. Similar
to the last test, the 3 graphs were made, the time constant was calculated from the two
Ln(ErrorFunction) vs Time graphs, and uncertainty analysis was performed.
See the graph below for the experimental setup:
Figure 3: Final Experimental Set Up - Ambient Temperature to Heated Airjet
LAB 4: FIRST-ORDER SYSTEMS
11
Heated Airjet to Ambient Temperature
Picking up immediately from the last test, the lab view program (with the name being
changed to “Hot Air to Ambient Temperature”) was started, and then the Heated Airjet was
removed quickly from being under the tip of the thermocouple. The lab view should ideally show
a negative exponential climb down the steady state hot temperature down to the steady state at
ambient temperature (room temperature). This will require more measurements to be taken since
going from hot to ambient takes longer than going from ambient to hot without external
influences. After the lab view shows a near-constant voltage reading, indicating that there is little
to no temperature change, the test was stopped. The data was finally exported to Excel to
prepare the 3 graphs, calculate the time constants, and then calculate the uncertainty.
See the graph below for the experimental setup:
Figure 4: Final Experimental Set Up - Heated Airjet to Ambient Temperature
Results and Discussion
First, what needed to be collected was the offset voltage from the voltage measurement
tool that the thermocouple is connected to. That is to say, the “offset” voltage that is recorded
without anything being connected needs to be subtracted from the measured value for the most
accurate measurements. That is done by plugging in the thermocouple, putting the cap on the
LAB 4: FIRST-ORDER SYSTEMS
12
channel where the thermocouple will be connected (channel0), and pressing record on lab view.
From the measurements taken in the lab, an offset voltage of 0.00150V was found, and this
value was subtracted from all of the recorded voltage readings from the thermocouple
(subtracted because the offset is positive).
Results: Cold Water to Hot Water
The cold-to-hot water test produced the following table in LabVIEW, showing a Voltage versus
Time graph:
Figure 5: Voltage Vs. Time - Cold to Hot Water
From this graph, the output file containing the numerical values for the voltage at each measured
time value was exported to Excel for analysis. First, the offset voltage (0.00150) was subtracted
from all of the voltage values. Second, the equation for the voltage/ temperature relationship
(equation 2) was filled in to solve for temperature with respect to time, in this case:
𝑇(𝑡) = 𝑇𝑖𝑐𝑒 + ([𝑇ℎ𝑜𝑡 − 𝑇𝑖𝑐𝑒]/[𝑉ℎ𝑜𝑡 − 𝑉𝑖𝑐𝑒]) * [𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 − 𝑉𝑖𝑐𝑒]
Where T and V ice are with respect to the ice water, T and V hot are with respect to the boiling
water, and V is the measured value in the output file. In this case,
𝑇𝑖𝑐𝑒 = 275. 15 𝐾, 𝑇ℎ𝑜𝑡 = 369. 15 𝐾, 𝑉ℎ𝑜𝑡 = 1. 950, 𝑉𝑖𝑐𝑒 = 0. 001524. The following chart
was created for the Temperature vs Time relation:
LAB 4: FIRST-ORDER SYSTEMS
13
Figure 6: Temperature Vs. Time - Cold to Hot Water
Next, equation 2 was used to calculate the error function, Γ, which is also the change in
Temperature, ∆𝑇. Then, to solve for the time constant, the graph ln(ErrorFunction) vs Time was
created:
Figure 7: Ln(ErrorFunction) Vs. Time - Cold to Hot Water
LAB 4: FIRST-ORDER SYSTEMS
14
To account for the fluctuation region and the extra measurement of the cold water, the graph was
slightly adjusted, yielding the following chart (notice that the R^2 value is slightly larger):
Figure 8: Ln(ErrorFunction) Vs. Time (adjusted) - Cold to Hot Water
To measure the time constant for this data set, use equation (5) with m being the slope of the
adjusted line of best fit, where:
τ =
1
1
− 𝑚 => τ𝑐𝑜𝑙𝑑−>ℎ𝑜𝑡 =− −(1.1959) = 0. 836 𝑠𝑒𝑐
Uncertainty Analysis: Cold Water to Hot Water
Uncertainty analysis was performed on the data set, and with a calculated standard deviation of
𝑆𝑦𝑥 = 2. 1958 from Excel and N = 8 for 8 total seconds of testing for the adjusted time, the
confidence interval is defined as
𝑦𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 ± 𝑡95%,∞ • 𝑆𝑦𝑥/ 𝑁 =>
=>
Shown below as:
− 1. 1959𝑥 + 8. 1632 ± 1. 960 • 2. 195/ 8
− 1. 1959𝑥 + 8. 1632 ± 1.521
LAB 4: FIRST-ORDER SYSTEMS
15
Figure 9: Ln(ErrorFunction) Vs. Time (adjusted) with Uncertainty - Cold to Hot Water
The results for the Ambient Air to Heated Jet and Heated Jet to Ambient Air tests follow the
same procedure as the Cold Water to Hot Water test, so the results section will be slightly
shortened.
Results: Ambient Air to Heated Airjet
The Ambient Air to Heated Airjet Voltage vs. Time graph generated by Excel is shown below:
NOTE: The original LabVIEW chart was lost, but a second chart was generated
LAB 4: FIRST-ORDER SYSTEMS
16
Figure 10: Voltage Vs. Time - Ambient Air to Heated Jet
From this graph, the output file containing the numerical values for the voltage at each measured
time value was exported to Excel for analysis. First, the offset voltage (0.00150) was subtracted
from all of the voltage values. Second, the equation for the voltage/temperature relationship
(equation 2) was filled in to solve for temperature with respect to time, in this case:
𝑇(𝑡) = 𝑇𝑎𝑚𝑏𝑖𝑒𝑛𝑡 + ([𝑇𝐴𝑖𝑟𝑗𝑒𝑡 − 𝑇𝑎𝑚𝑏𝑖𝑒𝑛𝑡]/[𝑉𝐴𝑖𝑟𝑗𝑒𝑡 − 𝑉𝑎𝑚𝑏𝑖𝑒𝑛𝑡]) * [𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 − 𝑉𝑎𝑚𝑏𝑖𝑒𝑛𝑡]
Where T and V ambient are with respect to room temperature measurements, T and V Airjet are
with respect to the Heated Airjet, and V was the measured value in the output file. For reference,
𝑇𝑎𝑚𝑏𝑖𝑒𝑛𝑡 = 296. 15 𝐾, 𝑇𝐴𝑖𝑟𝑗𝑒𝑡 = 342. 15 𝐾, 𝑉𝐴𝑖𝑟𝑗𝑒𝑡 = 1. 430, 𝑉𝑎𝑚𝑏𝑖𝑒𝑛𝑡 = 0. 4145. The
following chart was created for the Temperature vs Time relation:
LAB 4: FIRST-ORDER SYSTEMS
17
Figure 11: Temperature Vs. Time - Ambient Air to Heated Airjet
Highlighted in red is an unexpected error region, and may have been caused by someone
bumping the table, hitting the Heated Airjet, or even a small fluctuation in the temperature of the
Heated Airjet. Next, equation 2 was used to calculate the error function, Γ, which is also the
change in Temperature, ∆𝑇. Then, to solve for the time constant, the graph ln(ErrorFunction) vs
Time was created:
LAB 4: FIRST-ORDER SYSTEMS
18
Figure 12: Ln(ErrorFunction) Vs. Time - Ambient Air to Heated Airjet
To account for the fluctuation region and the extra measurement of the ambient temperature, the
graph was slightly adjusted, yielding the following chart:
LAB 4: FIRST-ORDER SYSTEMS
19
Figure 13: Ln(ErrorFunction) Vs. Time (adjusted) - Ambient Air to Heated Airjet
To measure the time constant for this data set, use equation (5) with m being the slope of the
adjusted line of best fit, where:
τ =
1
1
− 𝑚 => τ𝑐𝑜𝑙𝑑−>ℎ𝑜𝑡 =− −(0.0957) = 10. 4 𝑠𝑒𝑐
Uncertainty Analysis: Ambient Air to Heated Airjet
Uncertainty analysis was performed on the data set, and with a calculated standard deviation of
𝑆𝑦𝑥 = 1. 1538 from Excel, and using N=40 (40 seconds of total testing for the limited data), the
confidence interval is defined as
𝑦𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 ± 𝑡95%,∞ • 𝑆𝑦𝑥/ 𝑁 =>
=>
Shown below as:
− 0. 0957𝑥 + 6. 1925 ± 1. 960 • 1. 1538/ 40
− 0. 0957𝑥 + 6. 1925 ± 0.357
LAB 4: FIRST-ORDER SYSTEMS
20
Figure 14: Ln(ErrorFunction) Vs. Time (adjusted) with Uncertainty - Ambient Air to Heated
Airjet
NOTE: The data that is outside of the confidence interval is where the error region is on
the Temperature vs. Time graph.
Results: Heated Airjet to Ambient Air
A portion of The Heated Airjet Voltage to Ambient Air vs. Time graph generated by LabVIEW
is shown below:
LAB 4: FIRST-ORDER SYSTEMS
21
Figure 15: Voltage Vs. Time - Heated Airjet to Ambient Air
From this graph, the output file containing the numerical values for the voltage at each measured
time value was exported to Excel for analysis. First, the offset voltage (0.00150) was subtracted
from all of the voltage values. Second, the equation for the voltage/temperature relationship
(equation 2) was filled in to solve for temperature with respect to time, in this case:
𝑇(𝑡) = 𝑇𝐴𝑖𝑟𝑗𝑒𝑡 + ([𝑇𝑎𝑚𝑏𝑖𝑒𝑛𝑡 − 𝑇𝐴𝑖𝑟𝑗𝑒𝑡]/[𝑉𝑎𝑚𝑏𝑖𝑒𝑛𝑡 − 𝑉𝐴𝑖𝑟𝑗𝑒𝑡]) * [𝑉𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 − 𝑉𝐴𝑖𝑟𝑗𝑒𝑡]
Where T and V ambient are with respect to room temperature measurements, T and V Airjet are
with respect to the Heated Airjet, and V was the measured value in the output file. For reference,
𝑇𝑎𝑚𝑏𝑖𝑒𝑛𝑡 = 296. 15 𝐾, 𝑇𝐴𝑖𝑟𝑗𝑒𝑡 = 342. 15 𝐾, 𝑉𝐴𝑖𝑟𝑗𝑒𝑡 = 1. 430, 𝑉𝑎𝑚𝑏𝑖𝑒𝑛𝑡 = 0. 4145 (the same as
the Ambient Air to Heated Airjet). The following chart was created for the Temperature vs Time
relation:
LAB 4: FIRST-ORDER SYSTEMS
22
Figure 16: Temperature Vs. Time - Heated Airjet to Ambient Air
Next, equation 2 was used to calculate the error function, Γ, which is also the change in
Temperature, ∆𝑇. Then, to solve for the time constant, the graph ln(ErrorFunction) vs Time was
created:
LAB 4: FIRST-ORDER SYSTEMS
23
Figure 17: Ln(ErrorFunction) Vs. Time - Heated Airjet to Ambient Air
To account for the fluctuation region and the extra measurement of the steady-state Heated Airjet
Temperature, the graph was slightly adjusted, yielding the following chart:
LAB 4: FIRST-ORDER SYSTEMS
24
Figure 18: Ln(ErrorFunction) Vs. Time (adjusted) - Heated Airjet to Ambient Air
To measure the time constant for this data set, use equation (5) with m being the slope of the
adjusted line of best fit, where:
τ =
1
1
− 𝑚 => τ𝑐𝑜𝑙𝑑−>ℎ𝑜𝑡 =− −(0.0124) = 80. 6 𝑠𝑒𝑐
Uncertainty Analysis: Heated Airjet to Ambient Air
Uncertainty analysis was performed on the data set, and with a calculated standard deviation of
𝑆𝑦𝑥 = 1. 146 from Excel, and using N=315 (315 seconds of total testing for the limited data), the
confidence interval is defined as
𝑦𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 ± 𝑡95%,∞ • 𝑆𝑦𝑥/ 𝑁 =>
=>
Shown below as:
− 0. 0124𝑥 + 0. 1038 ± 1. 960 • 1. 146/ 315
− 0. 0124𝑥 + 0. 1038 ± 0.126
LAB 4: FIRST-ORDER SYSTEMS
25
Figure 14: Ln(ErrorFunction) Vs. Time (adjusted) with Uncertainty - Heated Airjet to
Ambient Air
NOTE: The data that is outside of the confidence interval is when the Heated Airjet was
being moved away from the tip of the thermocouple.
Discussion: Error Analysis
Overall, this experiment yielded some very interesting points for analysis. Even though
the percent error analysis cannot be directly performed without theoretical values, the overall
uncertainty analysis proved that there were some significant places for error to occur. For all of
the tests, towards the end of the experiment where the error function starts to get closer to zero
(the area where ln(ErrorFunction) gets more negative), there is a fluctuating region of
measurements, which requires the data to be adjusted for actual valuable analysis. The reason the
adjustment can be made is that the time constant is dependent on the rate of change between the
two temperatures, so as long as the data between the two temperatures are able to be salvaged to
some extent, a pretty general idea of the time constant can be found.
LAB 4: FIRST-ORDER SYSTEMS
26
In a specific sense, the Cold Water to Hot Water test has some areas of error when it
comes to the line of best fit due to the R^2 value. Seeing that the value is less than .9 for the
adjusted plot is concerning, indicating that there will be a higher uncertainty to account for the
lower relation between the line of best fit and the actual data. For the Ambient to Heated Airjet
test, there was a sudden change in the slope of the graph between the two temperatures. This
was highlighted in the “Error Region'' area and is an issue that arose due to human error of some
sort. The more still the thermocouple, Heated Airjet, and its surrounds are, the better the data will
be, and there is a chance that either the Heated Airjet caused the thermocouple to slip slightly out
of the clamp, someone hit the table on their way out of the classroom by accident or even the
Heated Airjet had a slight fluctuation in temperature (which can happen with older equipment.
All of these factors could have caused the error region to occur, which is why there is a sudden
jump out of the uncertainty range, but the data eventually corrected itself and fell within the
uncertainty, shown by a better R^2 value. Lastly, the Heated Airjet to Ambient temperature was
quite a good test, seen by the high R^2 value and low uncertainty, as the only source of error that
can be seen is towards the beginning where the Heated Airjet has moved away from the
thermocouple, creating a small jump in the ln(ErrorFunction) vs Time graph, but that quickly
corrected itself.
Conclusion
Despite the errors in the test, in general, the experiment still yielded some valuable
information that the group expected. Even though the numbers themselves may not be
completely accurate, it was found that the time constant for the Cold Water to Hot Water test was
τ=0.836s, the Ambient to Heated Airjet was found to be τ=10.4s and the Heated Airjet to
Ambient temperature was τ=80.6s. The idea that it takes much less time for something cold to
heat up than it takes for something warm to cool down is commonly accepted in the field of
thermodynamics, specifically using the second law of thermodynamics, which essentially states
heat naturally flows from a cooler object to a hotter one until equilibrium is reached. When
something cold is heating up, it absorbs energy from its surroundings, but when something hot is
cooling down, it loses heat to its surroundings.
LAB 4: FIRST-ORDER SYSTEMS
27
The temperature difference for the Cold to Hot Water was much higher, indicating a very
high rate of change, producing the lowest time constant. Since the Ambient to Heated Airjet test
and Heated Airjet to Ambient temperature test both had the same temperature difference, the
difference in the test is that in the Ambient to Heated Airjet test, a cooler temperature was being
heated, causing a higher rate of change in temperature, thus a smaller time constant than the
Heated Airjet to Ambient temperature test, as expected. Overall, the uncertainty analysis seemed
to be generally correct outside sudden jumps due to changes and human error after adjusting the
data to remove some of the fluctuating values and prolonged ambient measurements that seemed
to cause the formulas to fail. Due to this, this experiment would be considered a general success
with some room for improvement when it came to preventing human error and external factors.
LAB 4: FIRST-ORDER SYSTEMS
28
References
Figliola, R. S., & Beasley, D. E. (2010). Theory and Design for Mechanical Measurements (5th
ed.). Wiley.
Lab Manual 4: First-Order Systems
LAB 4: FIRST-ORDER SYSTEMS
29
Appendix
A-1: Figure 1: Final Experimental Set Up - Thermocouple, provided by the lab manual
A-2: Figure 2: Final Experimental Set Up - Cold to Hot Water
LAB 4: FIRST-ORDER SYSTEMS
30
A-3: Figure 3: Final Experimental Set Up - Ambient Temperature to Heated Airjet
A-4: Figure 4: Final Experimental Set Up - Heated Airjet to Ambient Temperature
LAB 4: FIRST-ORDER SYSTEMS
31
A-5: Figure 5: Voltage Vs. Time - Cold to Hot Water
A-6: Figure 6: Temperature Vs. Time - Cold to Hot Water
LAB 4: FIRST-ORDER SYSTEMS
32
A-7: Figure 7: Ln(ErrorFunction) Vs. Time - Cold to Hot Water
A-8: Figure 8: Ln(ErrorFunction) Vs. Time (adjusted) - Cold to Hot Water
LAB 4: FIRST-ORDER SYSTEMS
33
A-9: Figure 9: Ln(ErrorFunction) Vs. Time (adjusted) with Uncertainty - Cold to Hot Water
A-10: Figure 10: Voltage Vs. Time - Ambient Air to Heated Jet
LAB 4: FIRST-ORDER SYSTEMS
34
A-11: Figure 11: Temperature Vs. Time - Ambient Air to Heated Airjet
A-12: Figure 12: Ln(ErrorFunction) Vs. Time - Ambient Air to Heated Airjet
LAB 4: FIRST-ORDER SYSTEMS
35
A-13: Figure 13: Ln(ErrorFunction) Vs. Time (adjusted) - Ambient Air to Heated Airjet
A-14: Figure 14: Ln(ErrorFunction) Vs. Time (adjusted) with Uncertainty - Ambient Air to
Heated Airjet
LAB 4: FIRST-ORDER SYSTEMS
36
A-15: Figure 15: Voltage Vs. Time - Heated Airjet to Ambient Air
A-16: Figure 16: Temperature Vs. Time - Heated Airjet to Ambient Air
LAB 4: FIRST-ORDER SYSTEMS
37
A-17: Figure 17: Ln(ErrorFunction) Vs. Time - Heated Airjet to Ambient Air
A-18: Figure 18: Ln(ErrorFunction) Vs. Time (adjusted) - Heated Airjet to Ambient Air
LAB 4: FIRST-ORDER SYSTEMS
38
A-19 Figure 19: Ln(ErrorFunction) Vs. Time (adjusted) with Uncertainty - Heated Airjet to
Ambient Air
A-20: TA Signature