Laboratory Report: Temperature
Measurement and Heat Transfer
Abstract
The purpose of this report is to discuss an investigation into heat transfer from
experimentation and background reading. As a fluid has been tested, there will be a focus
on convection, both free and forced (stirring). Convection occurs due to hot fluid rising and
cooler denser fluid falling (1). Other thermodynamic principles will also be explored.
Two different methods of measuring temperature were used with both found to give similar
results once appropriated. In addition, it was found that stirring and insulation both had a
positive impact on the rate of heat due to doing work on the fluid and reducing heat
dissipation.
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Contents
Abstract ...................................................................................................................................... 2
Introduction ............................................................................................................................... 4
Method ...................................................................................................................................... 5
Data and Discussion ................................................................................................................... 7
Conclusion ................................................................................................................................ 13
References ............................................................................................................................... 14
3
Introduction
For the experiment, a thermometer and thermocouple were used to record temperature.
Despite the two measuring tools differing, the same underlying relationship should be
observed. This is an example of calibration (2) and can be used to increase confidence in
data.
An analogue thermometer uses the proportional relationship of volume and temperature in
a liquid to give a reading. As the temperature increases, the liquid will expand and rise up a
narrow tube. The height gained is proportional to the increase in temperature and thus can
be used to measure the temperature of a body.
A thermocouple probe is a highly responsive (under 0.05s(3)) digital measurement device
suitable for temperatures between -270 and 3000oC (4). However, they are susceptible to
electrical noise (5, 6). The thermocouple can measure temperature due to three effects: the
Seebeck effect (Equation 1), the Peltier effect (Equation 2) and the Thomson effect
(Equation 3) (7). Although these individual effects will not be further explored in this report,
they offer an appreciation for the physical properties which impact the thermocouple’s
readings.
𝐸𝑒𝑚𝑓 = −𝑆𝛻𝑇
Where
𝐸𝑒𝑚𝑓 is the electromotive force (W)
𝑆 is the Seebeck coefficient (VK-1)
𝛻𝑇 is the temperature gradient
𝑄̇ = (𝛱𝐴 − 𝛱𝐵 )𝐼
Where
(2)
𝑄̇ is the Peltier heat (J)
𝛱𝐴 and 𝛱𝐵 are the Peltier coefficients (V)
𝐼 is the electric current between points A and B (A)
𝑞̇ = −𝜅𝐽 ⋅ 𝛻𝑇
Where
(1)
(3)
𝑞̇ is heat production rate per unit volume (Wm-3)
𝜅 is the Thomson coefficient (VK-1)
𝐽 is current density (Am-2)
When the beaker is stirred, work will be done on the body of water and this will lead to
greater temperature readings (8). This principle is used in industry (9) and thus it is
predicted that the rate of temperature increase will increase when the beaker is stirred. In
addition, the rate of temperature increase will also increase when the beaker is insulated, as
less heat will transfer out of the beaker and be lost to the surrounding air.
4
Method
Materials used:
Equipment needed:
Beaker (1000ml ± 5%)
Thermometer (110°C ± 0.5°C)
Thermocouple
Stopwatch (± 0.01s)
Clamp stand
Clamp
Insulation (sides)
Insulation (top)
Water
Hot plate
Magnetic stirred
Ruler (1m ± 0.5mm)
Figure 1: Equipment
Figure 1 shows an image of the equipment required except for the water (which was taken
from a laboratory tap), magnetic stirrer and the clamp.
Procedure:
Firstly, the uninsulated measurements were recorded. The side insulation was not required
at this stage.
800ml of water was added to the beaker and the beaker was placed on a flat surface to
ensure that the correct amount of water was added - according to the beaker’s calibration
line.
The thermometer and thermocouple were then placed at the beaker’s lowest line, 20mm
upwards from the base. This was achieved by gripping both apparats in the insulation placed
on top of the beaker and pushing each measuring device to the bottom of the beaker. An
appropriate marking should be made on each measuring device to show the height of each
device in relation to the top of the insulation. Using the ruler, a marking 20mm lower can be
made which will show the height the device should be at in the beaker. This procedure is
partially demonstrated in Figure 2.
The hotplate was then heated to 50°C and
left for 5 minutes (timed using the
stopwatch) to allow the system to stabilise.
After, the beaker was placed on the hot
plate and the initial temperature was then
recorded using both measuring devices.
Figure 2: Measuring length methodology
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Additional readings were taken each minute. The final reading occurred 300s after the
initial.
After five minutes and six readings, the hot plate was turned
off and the beaker removed to avoid unintentional injury
from the hot surface. The water was then replaced, and the
experiment repeated with the readings now taken from the
top of the beaker (80mm), using the same process.
Once top and bottom readings had been recorded, the
investigation was repeated using a magnetic stirrer at a
speed of 300rpm. Top and bottom readings were taken at
the same time intervals.
Finally, both unstirred and stirred top and bottom
measurements were taken when the beaker had the side
insulation wrapped around it (shown in Figure 3). All the
required raw data had been recorded.
Figure 3: Insulated testing
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Data and Discussion
Table 1 contains the raw data readings from the experiment and some basic calculations.
The grey cells in the table indicate readings in which the beaker was insulated.
Table 1
The ‘TM’ column shows the thermometer readings and the ‘TC’ column shows the
thermocouple readings. The corrected thermocouple readings were found by plotting the
thermometer’s measurements against the thermocouple measurements and creating a
trendline. These values are shown in the ’CTC’ column. The final set of columns presents the
mean temperature readings (found using Equation 4) for each set of variables.
𝑀𝑒𝑎𝑛 =
Where
𝑇𝑀 + 𝐶𝑇𝐶
2
(4)
𝑇𝑀 is the raw thermometer reading (℃)
𝐶𝑇𝐶 is the corrected thermocouple reading (℃)
The trendline used to calculate the CTC values is shown in Figure 4. The gradient of this line
can be used within Equation 5 to find the corrected values.
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𝐶𝑇𝐶 =
𝑇𝐶 + 2.1929
0.9957
(5)
𝑇𝐶 is the raw thermocouple reading (℃)
Where
It is only possible to correct the thermocouple’s measurements from the more accurate
thermometer’s readings (2) due to the linear trend in Figure 4.
29
y = 0,9957x - 2,1929
27
TC /℃
25
23
21
19
17
15
15
17
19
21
23
25
27
29
31
TM /℃
Figure 4: Presentation of proportionality if thermocouple and thermometer readings
The horizontal error bars shown in Figure 4 were added due to the ± 0.5℃ uncertainty in the
thermometer’s measurements. As the thermocouple could not be read to a precision
greater than 1℃, the vertical error bars also have an absolute error of ± 0.5℃.
As the gradient of Figure 4 is 1.00, once corrected, both pieces of equipment give very
similar results. The difference of 2.1929℃ (shown by the y-intercept in Figure 4) is likely due
to the thermocouple originally being mis-calibrated by this amount.
Figure 5a and 5b will be used to discuss the impact of stirring. The error bars in these figures
were calculated in the same way as those in Figure 4.
Figure 5a presents a graph with the mean bottom temperature of the uninsulated tests
plotted against time. The trendline of this figure has a gradient equal to the average rate of
change of temperature, over time, for the experiment.
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y = 0,0258x + 19,447
Mean temperature /℃
27,0
25,0
23,0
No Stirring
y = 0,0077x + 21,141
21,0
Stirring
19,0
17,0
0,0
100,0
200,0
300,0
400,0
Time /s
Figure 5a: Comparison of uninsulated mean temperatures when stirred and unstirred
The trendline gradients in Figure 5a make it clear that the rate of change of temperature
increases drastically when the water is stirred. This is due to the stirrer doing work and
giving the individual fluid molecules in the beaker more energy, which in turn vibrates the
molecules and increases temperature. As the stirrer is rotating at a constant rate, the
amount of work done per unit time is also constant and this leads to a constant relationship
represented by the linear trendline.
As the hotplate provides constant power, both trends in Figure 5a are linear. The ratio of
trendlines in Figure 5a shows that stirring increase the rate of change by 3.35 times.
Figure 5b shows the bottom temperature measurements when the beaker was insulated.
Mean temperature /℃
27,0
y = 0,0231x + 19,591
25,0
23,0
y = 0,0155x + 19,4
No Stirring
21,0
Stirring
19,0
17,0
0,0
100,0
200,0
300,0
400,0
Time /s
Figure 5b: Comparison of insulated mean temperatures when stirred and unstirred
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A sharper gradient (0.0155 compared to 0.0077) indicates that the rate of change
approximately doubled when the unstirred vessel had insulation as less thermal energy
dissipated to the surroundings. A shallower gradient is observed when comparing the stirred
samples however, with the insulated beaker showing a 10% decrease to the gradient in
comparison to the uninsulated, stirred sample.
Theory dictates that an insulated heated fluid should increase in temperature at a greater
rate than an uninsulated beaker (Equation 6, Newton’s Law of Cooling); although one can
observe this when comparing the unstirred tests, this is not followed for the stirred. This is
likely due to randomised error (e.g. difficulty in reading the thermometer).
𝑞̇ 𝑐𝑜𝑛𝑣 = −𝐴ℎ(𝑇𝑆 −𝑇𝑎 )
(6)
𝑞̇ 𝑐𝑜𝑛𝑣 is the heat transfer rate (W)
𝐴 is the surface area exposed to the cooling fluid (m2)
ℎ is the conventional heat transfer coefficient (W/(m2 K))
𝑇𝑆 is the temperature of the solid body (K)
𝑇𝑎 is the temperature of the fluid far away from the solid body (K)
Where
Data was recorded separately at the top and bottom of the vessel to analyse how the
distance from the heat source influences the temperature.
Figure 6 presents the difference between top and bottom data readings during stirring.
Insulated and uninsulated values have been used. The error bars shown in Figure 6 were
calculated in the same way as those in Figure 4.
31,0
Top Temperature /℃
29,0
y = 0,8166x + 6,6001
27,0
25,0
23,0
y = 0,8962x + 2,395
21,0
Линейная (Not insulated)
Линейная (Insulated)
19,0
17,0
15,0
15,0
20,0
25,0
30,0
Bottom Temperature /℃
Figure 6: Comparison of top and bottom temperature readings, with and without insulation
The top recorded temperature was always greater than the bottom recorded temperature
(Figure 6). Although the insulated beaker was cooler than the insulated beaker at all
instances, the insulated beaker had a greater rate of temperature increase between bottom
and top readings. This is shown by a steeper gradient, 0.8962 compared to 0.8166. The
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lower readings can be explained due to a lower initial temperature. This could be due to the
atmosphere surrounding the apparatus being cooler for the insulated sample than the
uninsulated sample.
Although the top temperature readings are greater than the bottom temperature readings
in absolute terms, it can be inferred that the bottom temperature readings are relatively
greater due to the values in gradients for trendlines. Closer to the heat source, heat energy
will have less distance to travel and thus will have dissipated less to the surrounding area.
The amount of dissipation can be explained by Equation 6 and the side surface area of a
cylinder relative to its height (Equation 7), i.e. cooling will increase as surface area increases.
𝐴𝑆 = 2𝜋𝑟 ⋅ 𝑙
Where
(7)
𝐴𝑆 is the side surface area of the cylinder (m2)
𝜋 is the mathematical constant, pi
𝑟 is the radius of the cylinder (m)
𝑙 is the length of the cylinder (m)
To estimate the heat transfer rate of the hotplate, the bottom data from when the beaker
was both insulated and stirred has been selected. As only work done by the hotplate will be
examined, it is unreasonable to use data from the stirred tests. Furthermore, the bottom
temperature readings for the insulated test have been chosen to minimise energy loss from
dissipation/cooling.
The heat transfer rate from the hotplate can be calculated from the total energy change in
the volume of water divided by the time the beaker was heated for (300s). The total work
done on the beaker of water by the hotplate can be found using Equation 8.
𝑄 = 𝑚𝑐𝛥𝑇
Where
(8)
𝑄 is energy (J)
𝑚 is the mass of the fluid (kg)
𝑐 is the specific heat capacity of the fluid (Jkg-1)
𝛥𝑇 is the change in temperature (K)
The specific heat capacity of water is 4186 Jkg-1. The change in temperature may be
calculated from the difference between the mean temperature initially and after five
minutes. The mean temperature has been chosen to minimise random error by utilising
multiple measurements. The temperature difference was found to be 4.5℃ using this
methodology.
The mass of the water is calculated using Equation 9. Although density will change with
temperature (10), the value will be 1000 kgm-3 to three significant figures (sf) for all initial
temperature values used. The volume used is 0.800m3. As the percentage error of the
beaker is 5%, this value is accurate to two significant figures.
𝑚 = 𝜌𝑉
(9)
11
Where
𝜌 is density (kgm-3)
𝑉 is volume (m3)
Therefore, the mass of the water is 0.80 kg. From these values, the work done on the water
in total can be calculated as 15100J (3sf). The heat transfer rate can then be calculated using
Equation 10, leading to a value of 50.3W for the rate of heat energy transfer from hotplate
to water.
𝑊=
Where
𝑄
𝑡
(10)
𝑊 is power (W)
𝑡 is time (s)
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Conclusion
A difference of approximately 2.19℃ was found between thermometer and thermocouple
readings, likely due to the thermocouple being incorrectly calibrated. Due to the highly
proportional gradient, it can be assumed both measuring devices are suitable and accurate
for temperature measurement. However, there is not complete confidence in this verdict as
it is reliant on the assumption that the thermometer used was entirely accurate and precise.
Stirring and insulation were both discussed; both had a positive effect on the rate of
temperature change with stirring increasing the rate by approximately 3.35 times and
insulation doubling the rate. However, as shown in Figure 5b, there is likely to be an error in
the results as adding insulation to the stirred sample decreased the rate of temperature
change by 10%. Preliminary research suggests this observation to be erroneous and thus the
data linking to the observation should be rerecorded. Stating this however, the rest of the
data gathered supports current scientific theory and allows this report to achieve its
purpose of increasing understanding of heat transfer.
The perpendicular distance from a flat heating source was studied by taking measurements
at two set distances. In absolute terms, the top temperatures were greater than the bottom
temperatures. However, the opposite was true relatively. The relative relationship can be
explained by heat travelling a shorter distance and having less availability to dissipate into
the surroundings. The absolute difference was possibly due to different atmospheric
temperatures causing different initial temperatures.
By comparing the trendlines in Figure 6, one notices that gradient is sharper for the
insulated sample in comparison to the non-insulated. This is due to the insulation being less
thermally conductive than the air and so it is more difficult for heat energy to be lost to the
surroundings.
An increased understanding into heat transfer has been achieved in this report with strong
qualitative comparisons involving insulation and work. Improvements could include using a
more precise volume of water to increase confidence in the quantitative result for heat
transfer and to ensure a constant ambient temperature – to allow for more reliable
comparisons.
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