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Calibration of Thermometric Sensors and Comparative Time Response to Ventilation

Virginia Rux, Matthew Rogers, Emily Ramnarine, Tessa Vollrath, Michael McKeirnan

University of Washington—Atmospheric Sciences ___________________________________________________________________________________ ABSTRACT Our cohorts calibrated thermometric instruments to build a relationship between the primary sensors in wide use and the National Institute of Standards and Technology calibration standard. We examine how error propagates through this process and appreciate the multitude of sensor types and the sensitivity of time response to ventilation. The main goal being to alleviate future systematic errors and quantify random ones. Ventilation was simulated by subjecting a thermometer to a wind tunnel operated with a range of velocities (0, 1.8 – 2.4 m/s). We analyzed the response time through the time decay and time constant variables. We discovered through linear regression techniques and error propagation that the Davis weather station probe thermometer generated the largest error of 0.54 degrees Celsius compared to the national standard with the bead thermistor error close to the Davis (0.49°C). Generally, the thermometers responded quickly to winds of higher velocity and that lower velocities lead to longer response times. The examination of this laboratory provided useful insight for why in situ thermometers need to be calibrated and sheltered from radiation, moisture and wind. ___________________________________________________________________________________ Introduction

When instruments such as thermometers are used on a regular basis, it's ability to read an accurate measurement decreases. Accuracy can diminish from handling of the instrument or even from the expansion and contraction of the components while under thermal stress, altering the fluids expansion chamber. It is important to calibrate the thermometers in applications to many career sectors including meteorology for safety reasons and for weather predictions. For example, an uncalibrated thermometer could misrepresent data for a given area. If there are more than one or two uncalibrated thermometers for a spatial area, then the reliability to accurately represent the weather is decreased further. Another consideration to the spatial error, is the initial condition given for predicting the weather. Since the atmosphere is chaotic, any slight errors can alter the ability to predict even a few days out. Prediction can be especially important if there is a possibility for ice or snow as it only takes one or two degrees to change the precipitation type. A warmer thermometer that is not calibrated could mean serious injuries or lost lives if the actual temperature is a couple degrees cooler, and there is ice on the road or sidewalks. These instances often lead to litigation, requiring expert opinion based on the observations.

In order to adequately calibrate a thermometer, we compare the observations of to a certifiable standard. In the United States, the National Institute of Standards and Technology (NIST) serves as the ultimate standard for comparison. It is not practical to have every instrument in use sent to a calibration lab because it can be expensive and it does not account for influences with the environment the instrument will be involved, resulting in unexpected errors. (Brock, pg. 16). We therefore calibrated a common alcohol thermometer to a NIST certified, American Society of Testing and Materials (ASTM) mercury (Hg) thermometer for wider use as a transfer standard to calibrate other thermometric devices. ASTM is a performance standard for a product while NIST is a calibration standard specific for the United States.

In addition to the basic calibration of thermometers, we consider the time response of a thermometer to reach an equilibrium, ambient temperature and the affect on errors read by the thermometer or sensor. In a realistic situation, the environment a thermometer is exposed to is not limited to variable wind, radiation, and moisture which all affect the temperature response, fluctuations, and

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readings. To alleviate these variables and mitigate emergent error, thermometers would typically be located in a shelter or shield. This lab will show how ventilation and wind affects the time response of a thermometer.

There are many devices to measure temperature, we examine a few different ways, a NIST certified ASTM mercury thermometer, a plain alcohol thermometer, a bead thermistor, and a Davis weather station temperature probe/sensor. Understanding how to calibrate thermometric instruments and appreciating its reaction to the environment will allow us to calibrate the Davis sensor to an actual weather station if we were interested.

__________________________________________________________

Experimental Methodology At the University of Washington Atmospheric Sciences, our cohorts calibrated thermometers in the laboratory and further investigated the time response of an alcohol thermometer and the Davis weather station thermometer. The Davis thermometer probe and a bead thermistor were each calibrated to an alcohol thermometer, a transfer standard through calibration to a NIST certified ASTM mercury thermometer. Before the calibration process, it was important to document the NIST certificate provided in our calibrated ASTM mercury thermometer case. It included information about the ASTM mercury thermometer’s readings compared to the NIST standard which allows us to calibrate other thermometers and relate any thermometer we calibrate back to the NIST standard. Calibration of the alcohol thermometer

To begin calibration, the dewar (an insulated container, also known as a vacuum flask) was filled with ice and some water to make it easier to stir but not enough to melt the ice. The ASTM thermometer (serial #2Z0299) and the alcohol thermometer (VWR NA Cat. No. 89095-564) were then immersed into the freezing bath. The ASTM has graduation lines of 0.1°C while the alcohol thermometer has graduation increments of 1°C (see Figure 1). We allowed a few minutes for the thermometers to adjust to the new environment. In order to ensure that equilibrium is indeed met, the thermometers were allowed to reach a steady temperature for about minute. Each observer documented the temperature of each thermometer to the next significant figure that the thermometers graduations display. The process was repeated for five trials, each with a new temperature between freezing and 30°C as the ASTM thermometer was limited.

For subsequent trials using the ASTM and the alcohol thermometer, hot water was slowly added to the dewar to raise the bath to a new temperature. Additional stirring was necessary to ensure the temperature would be evenly distributed in the ice/liquid bath and equilibrium temperature could be attained more efficiently. Again, the thermometers were immersed in the liquid filled dewar where they were allowed to steadily achieve an equilibrium temperature in which each observer would commence documentation of the temperatures. The procedure for calibrating the alcohol thermometer to the ASTM mercury thermometer would be repeated for three more trials (totaling five trials/temperatures). In this portion of calibration, we had three observers accounting for each trial. After observations were completed for the ASTM and alcohol thermometers, a best estimate average was calculated to represent each thermometer and for trial. Developing of a relationship between the thermometers allows the alcohol thermometer to be calibrated to the ASTM mercury thermometer (further analysis in this relationship will be apparent in the following sections). Choosing the calibrate the alcohol thermometer is an inexpensive way to preserve the more expensive nationally calibrated thermometer as regular use, expansion and contraction of the thermometer will cause it to need recalibration sooner.

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Calibration of the Davis weather station probe thermometer

We used the alcohol thermometer as a NIST-traceable reference standard for calibrating other devices. To calibrate the Davis thermometer probe, which was obscured by a shield to reduce environmental influence on in situ observations, the temperature station housing (labelled #6: a shelter portion that looks similar to louvers) needed to be disassembled and electronically connected to a Davis weather monitor (#1) to digitally display a temperature reading (see figures 3). We continued calibration techniques with a dewar filled with ice and a little water as described above. This portion of data collection was conducted by four observers. Calibration of a bead thermistor

A bead thermistor is a type of resistance thermometer that uses electrical resistance to determine temperature (Moore, pg 587). The resistance in a thermistor decreases with increasing temperature, making it more sensitive than other resistance thermometers. It is about ten times more sensitive than a platinum resistance thermometer. Using a voltmeter, a quick conversion of the voltage reading of the bead thermistor by 100 corresponds to temperature (0.13 Volts ~ 13°C). A few examples of the practical use of thermistors include digital thermometers, vehicle fluid temperatures, and household appliances (Wavelength Electronics). Calibrating a bead thermistor is as simple as the other calibration techniques we used earlier in that all we needed to do was immerse the beaded wire into the dewar ice bath, along with the reference alcohol thermometer. For this calibration, we had four cohorts present to collect data from each trial. The bead thermistor (Servotron Regulated Power Supply: EE15D25) was calibrated against the alcohol thermometer with water bath (see figure 4). Time Response The time response was observed and analyzed for the alcohol thermometer and the Davis weather station probe. Our wind tunnel apparatus which fits on top of the lab table (see figure 5), was not as variable of wind speed as intended. It would not ventilate a large range of speeds as it was hardly noticeable, but we were able to adjust the speed enough to analyze our data. The group intended to observe wind speeds that were slow and fast. The wind tunnel had difficulty operating below two meters per second, therefore, to obtain more accurate measurements due to the quicker response of temperature, we recorded our data through electronic video devices.

The experiment required substantial manipulation and high precision measurements, so we needed our cohorts to participate in more than just simple observation. One had to be sure the temperature in the dewar was above 50°C, dry off the thermometer bulb or probe and insert it into the side of the wind tunnel (see figure 5 for apparatus). Another had to steadily capture footage of thermometer responding to the environment. The video was later analyzed to document the time and temperature. In order to get a good illustration of the temperature-time response, it’s necessary to dry the thermometer before inserting into the wind tunnel. A wet thermometer could alter the temperature to be cooler due to evaporation and latent heat transfer from the thermometer and could even exhibit fluctuations at any point in the trial, not lending to an accurate response time.

To begin the time response, we placed both the alcohol thermometer and the Davis weather station thermometer probe into a dewar of hot water above 50°C. After each trial, the thermometers were placed back into the dewar of hot water. A total of five trials were conducted for the alcohol thermometer and five total from the Davis weather thermometer probe including one for each where the wind velocity was zero meters per second. The thermometer’s temperature would drop instantaneously, so it was important

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to get the thermometer warm enough, dry enough and placed into the wind tunnel efficiently. In some cases, especially for no wind velocity, the temperature would not fully reach the ambient temperature. The ambient temperature in our trials were assumed based off the value the thermometers approached. It may have been more accurate to note the actual ambient temperature.

While waiting for the thermometers to heat up for the following trials, we adjusted the velocity of the wind tunnel using the variac dial (see figure 3 & 5). To obtain the velocity, we used a stop watch, reset the needle gauge anemometer and recorded the time it took to complete one revolution on the anemometer. Given that the needle gauge anemometer claimed that one revolution is 100 miles, we calculated the wind velocity. Information obtained from this procedure allowed us to further calculate the average time constant between the initial and final temperatures for each trial and examine a relationship between the time constant and wind speed.

__________________________________________________________ Data Analysis Total Error Propagation

From calculating linear regression, we obtained values of the slope, error in the slope, the intercept, error in the intercept and R2. Using the values from a linear function, 𝑦 = 𝑚 ± 𝜎2 ∗ 𝑥 +(𝑐 ± 𝜎8) to obtain a linear fit, we were able to propagate error.

Error propagation allowed each thermometer to be linked a related back to the NIST standard. We used the values outputted from the linear regression to calculate total error. Although our reference thermometer (NIST, ASTM, and alcohol) is the dependent variable and the observing thermometer is the independent variable when we plot our calibration figures, the linear equation makes x a function y. Even though that is the case, we still want our dependent variable as a function of the observed thermometer.

𝑺𝒐𝒃𝒔𝟐 =?

@AB   (𝑇DEF G − 𝑻𝒇𝒊𝒕(𝒊))B@

G ;  𝑻𝒇𝒊𝒕(𝒊) =MN,OPQA8

2

We iterated five times (N = 5) per thermometer observed to get the square of the average deviation

of the observed temperature (bolded variable). Substituted further into the general equation for the variance of the reference thermometer (bolded variable).

𝑆STUB = 𝑇DEFB ∗ 𝜎2B + 𝜎8B + (𝑚B ∗ 𝑺𝒐𝒃𝒔𝟐)

The variance of the thermometer being calibrated alone was not useful, so we used a series of

calculations to link each thermometer’s total error back to the NIST standard. An example of this process shows the Davis thermometer calibrated against the alcohol thermometer and further propagated to ASTM and NIST.

⇒   𝑺𝒂𝒍𝒄𝒐𝒉𝒐𝒍𝟐 = 𝑇Z[\GF

B ∗ 𝜎2,Z[\GFB + 𝜎8,Z[\GFB + (𝑚Z[\GFB ∗ 𝑆Z[\GFB)

⇒   𝑺𝑨𝑺𝑻𝑴𝟐 = 𝑇[_8D`D_B ∗ 𝜎2,[_8D`D_B + 𝜎8,[_8D`D_B + (𝑚[_8D`D_

B ∗ 𝑺𝒂𝒍𝒄𝒐𝒉𝒐𝒍𝟐)

⇒   𝑆@abMB = 𝑇cbMdB ∗ 𝜎2,cbMdB + 𝜎8,cbMdB + (𝑚cbMd

B ∗ 𝑺𝑨𝑺𝑻𝑴𝟐)

Finally, we took a square root of the final answer and obtained the total error of the Davis thermometer to the NIST standard (see Tables 1 & 2). Based on our total error for each thermometer, Table 2 ranks the order which deviates furthest from the NIST standard.

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Derivation of Time Constants

The alcohol and Davis probe thermometers were heated and removed from a hot water dewar (T > 50°C) and were placed in a wind tunnel. The thermometers responded to the ventilation by a decline in temperature by 1/e = -1/τ, where τ is the time constant. We defined 1/e decay time as the time between the thermometer’s initial and ambient temperature readings to reach 37% (i.e., small values of τ denotes a quick response, large τ a slower response). The ambient temperature for the Davis probe thermometer was 22.2°C. The ambient temperature for the alcohol thermometer readings was 22.9°C. Our wind tunnel was set to a new wind speed, including no wind speed for five trials. The range was not very large, so almost every time response decay, the temperature dropped about the same rate (see figures 10 & 11). We recorded our temperatures in time intervals of approximately one second for the Davis thermometer probe and approximately 5-10 seconds for the alcohol thermometer.

Using non-linear regression and taking the difference between the initial temperature and the ambient temperature, and also a temperature difference between each recorded temperature in time in relation to the ambient temperature, we were able to rearrange an equation to find τ, the time constant, the 1/e decay, and the error in tau.

𝜏 =−𝑡

ln  ( ∆𝑇∆𝑇2[k

)

We were particularly cautious that of any temperature differences equaling zero, or our equation

would output infinity or undefined. Augmentation of the data would have been from the points in the very beginning or the end. Once we had all our tau values, we took a mean of the tau for each thermometer and plotted it against speed as a scatter plot. To examine the linearity of response time to wind speed, we applied the linear regression as we did for the calibration in thermometers trials and generated a relationship between the time constants of the Davis thermometer probe and the alcohol thermometer.

__________________________________________________________ Results Calibration

Propagating the total error through each calibration experiment, Davis weather station

thermometer probe had the largest total error, with a standard deviation to the NIST standard of 0.54°C. The thermistor had an error of 0.49°C similar to the Davis but was slightly more accurate. However, the alcohol thermometer had a lower total error than I personally expected, about 0.33°C (see table 2).

Once the observed thermometer was calibrated to the reference (NIST standard, ASTM thermometer, or alcohol thermometer), a best estimate average was calculated for each trial and for each thermometer. The corresponding temperature pair was then plotted and a linear regression was applied to these points to gather information about the relationship to obtain values which help to propagate error. At the end of each of thermometric analysis, error was propagated back to the NIST standard for each thermometer. For the ASTM mercury thermometer, a best estimate average was not calculated because we essentially had the NIST certificate indicating the ASTM thermometer’s observed measurements against their standard (see figure 6). However, an average of the five data points proved to be useful for calculating the total error when comparing other thermometers to the NIST standard. Thereafter, each calibration trial used a best estimate average to plot the corresponding pairs in addition to the linear

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regression calculation and fitting a line to the corresponding data points (see table 1). Our team discovered that the relationships were highly linear when calibrating the thermometers.

The ASTM data were very well fit with an R2 value of one. The alcohol thermometer to the ASTM and the Davis probe and thermistor to the alcohol thermometer were also represented by a strong linear relationship (see figures 6-9). Time response Our wind tunnel did not operate in a large range of the velocities due to the limitations of the wind tunnel apparatus, therefore the decay of temperature with time fell similarly across all velocities except for no wind (see figures 10 & 11). The trend towards the ambient temperature was expected with little apparent error.

As expected, the general trend was when wind speed increased, the time constants decreased (see figure 12) from 145.2 seconds to 20.1 seconds for the alcohol thermometer and from 112.8 seconds to 42.2 seconds for the Davis thermometer. At lower wind velocities (< 1 m/s), the Davis thermometer had smaller time constants, suggesting that the weather station thermometer responds quickly under low ventilation whereas at higher wind velocities (> 1 m/s), the alcohol thermometer responded quickly to ventilation. The decoupling of time response grew at higher wind velocities compared to low velocities between the alcohol thermometer compared to the Davis.

However, there was one time constant (at 1.85 m/s) in the Davis probe and the alcohol thermometer that was peculiar. For the trial going from 1.85 m/s to 1.97 m/s, the corresponding time constants for both thermometers did not necessary decrease but rather increased by four seconds for the alcohol thermometer and three seconds for the Davis (see table 4). It is possible that because the wind speeds were very close together, there might have been a random error in calculating the actual wind speed (see the discussion section).

__________________________________________________________ Discussion

To reduce the impact of systematic, analytical errors, from parallax while reading the graduation lines on the ASTM and alcohol thermometers, we took best estimate averages. We also used the same thermometers and conducted the experiments in the same location throughout the lab to reduce systematic bias differences between the specific instruments. The representation of the linear fit showed that systematic and analytical errors were hardly detectable for this lab. Any other systematic errors within the specific thermistor or Davis thermometer were not detectable but it could have contributed to slight errors that arose from total error propagation.

The wind tunnel apparatus that we used did not have the desired range of velocities to measure time response. It may have made our results for the time constant more apparent and could have possibly helped to make our scatter plots more linear. The representation was not bad, but the systematic error could have been better.

We attempted to alleviate possible random and analytical errors during the wind tunnel experiences by recording the data onto an electronic device in order to have the opportunity to slow down the response times and gather more precise readings. On one hand, it was to reduce observational judgment errors from reading a thermometer that was responding too quickly and possibly be misrepresented.

A few considerations for random error which may have been apparent in the time response portion of our lab, we had one data point where the value of 1/e (or time constant) was higher than expected compared to the wind speed. It is possible that the reaction time of the stop watch was a potential source of error in representing the wind speed. One way to reduce this kind of error is to perhaps make a few

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attempts to calculate the wind speed and take a best estimate average of that trial’s wind speed. During the time response, if the thermometer was not dried off well enough, the response time would fluctuate. We did not see any apparent fluctuations as we attempted to have dry thermometers being placed into the wind tunnel. Nonetheless, it is a consideration that would lead to random error. Not waiting for the thermometers to properly settle to equilibrium could be an example of observational judgment that would lead to random error. We attempted to reduce judgment errors by waiting long enough for equilibrium. In the time response, we assumed that the ambient temperature was the temperature that the values seemed to approach, errors could arise from this, but it did not seem to be a problem.

__________________________________________________________ Conclusions This laboratory allowed us to examine the magnitude error can propagate. Interestingly, the alcohol thermometer as a transfer standard to the NIST had comparatively more error (about 0.33°C) than a thermometer sent to the laboratory for calibration. Fortunately, calibrating the alcohol thermometer to the NIST calibrated ASTM mercury thermometer was inexpensive and would have exceeded the cost of the thermometer alone. We saw that the alcohol thermometer had a better representation of the true (NIST being considered “true”) temperature compared to the Davis temperature probe and the thermistor (0.54°C and 0.49°C, respectively). The calibration plots for our thermometers provided a clear representation of how the data was linearly related although it did not explain much about error. Overall, the data showed that ventilation was less linear, with R2 = 0.97 for the alcohol thermometer and 0.85 for the Davis thermometer than the calibration portion of our laboratory where the coefficient of determination was very close one.

While the temperature calibration goals were accomplished, the time response portion of the laboratory was mostly satisfied. The response time provided evidence that a thermometer is sensitive to ventilation in that higher velocities induced quicker responses. Without ventilation, the time constant for the alcohol thermometer, 145.22 seconds, was larger than the Davis, 112.82 seconds, which provided insight that the Davis thermometer is more sensitive and quicker at responding to it’s environment at little to no wind velocity (< 1 m/s) compared to the alcohol thermometer. At higher velocities however, the sensitivity switched and decoupled with the Davis not as sensitive to ventilation than the alcohol thermometer. Clearly, lower time constants were observed for the alcohol thermometer, near 20 seconds for velocities greater than 2 m/s compared to about 50 seconds for the Davis probe thermometer—over twice the time constant! However, precision of the sensitivity of how the Davis thermometer compared to the alcohol thermometer respond to ventilation would have been more effective had the apparatus been able to obtain a wider range of wind speed.

It was interesting how only 2 m/s winds (realistically not very strong wind) could affect the response of a thermometer. In situ, the thermometer would not likely be above 50°C and then have experience sudden winds, but it displayed an exaggerated representation of temperature and the environmental influences of an improperly sheltered thermometer. In improper shield along with high variability would create unreasonable error thus unreliability to truly represent the ambient temperature of the environment.

If our total errors were just a bit larger and temperatures were being used to make judgmental decisions such as weather predictions or verifying conditions which could become a potential lawsuit or defense, we could see how unreliable our data is at face value. It would be important to make adjustments to our thermometric readings with this kind of knowledge. Our team can now make these considerations from the total error used from this laboratory to calibrate our weather station to the NIST standard. ___________________________________________________________________________________

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___________________________________________________________________________________ References (APA) Brock, F. V., & Richardson, S.J. (2001). Meteorological Measurement Systems. New York: Oxford University Press. Moore, J. & Davis, C., & Coplan, M. (2009). Building Scientific Apparatus. Cambridge University Press. Wavelength Electronics. Thermistors. http://www.teamwavelength.com/info/thermistors.php

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APPENDIX

Figure  1.    Calibrating  the  alcohol  thermometer  (white)  to  the  ASTM  mercury  thermometer  (yellow)  in  a  dewar  filled  with  ice  water.     In   the   background,   the   Davis   weather   station  thermometer   probe   is   concealed   inside   of   the   louver-­‐‑style  shelter.    

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Figure  2.    Adding  warm  water  to  the   dewar   while   calibrating   the  Davis   weather   thermometer  probe  to  the  alcohol  thermometer.    The   temperature   for   the  Davis   is  read   using   the   Davis   Weather  Monitor.  

Figure   3.     Allowing   the  thermometers   to   come   to  equilibrium.    The  Davis  monitor  is  more   visible.     Also,   in   the  background,   the   Variac   dial   is  visible.     That   device   controls   the  wind   velocity   through   the   wind  tunnel.  

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Figure   4.     The   bead   thermistor   being  calibrated   to   the   alcohol   thermometer  and   a   voltmeter   reads   the   resistance  from  the  thermistor.      Conversion  is  0.01  Volts  =  1°C.  

Figure  5.    Wind  tunnel  apparatus  with  the  alcohol   thermometer   and   the   Davis  thermometer   sitting   in   a   dewar   of   hot  water  before  calculating  the  time  response  for   the   thermometers   to   reach   an  equilibrium  temperature.  

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Figure  6.   (above).    The  ASTM   thermometer   calibration   to   the  NIST  standard.    Each  value  was  tested  was  from  the  calibration  certificate.  Figure   7.   (below).  The  alcohol   thermometer   calibration   to   the  ASTM   thermometer.    Each   data   point   represents   a   best   estimate   average   of   each   trial.   The   reference  thermometer  on  the  y-­‐‑axis,  the  observed  thermometer  on  the  x-­‐‑axis.  

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Figure  8.  (above).    Davis  thermometer  calibration  to  the  Alcohol  thermometer.    Each  data  point  represents  a  best  estimate  average  of  each  trial.  The  reference  thermometer  on  the  y-­‐‑axis,  the  observed  thermometer  on  the  x-­‐‑axis.  Figure   9.   (below).   Bead   thermistor   calibration   to   the  NIST   standard.     Each   data   point  represents  a  best  estimate  average  of  each  trial.    The  reference  thermometer  on  the  y-­‐‑axis,  the  observed  thermometer  on  the  x-­‐‑axis.    

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Thermometer Slope Slope error Intercept (°C) Intercept error (°C) R2

ASTM mercury 1.00 0.00 0.01 0.01 1.00 Alcohol 0.97 0.00 0.04 0.04 0.99 Davis 1.03 0.01 -0.01 0.08 0.99 Bead Thermistor 1.05 0.01 -0.52 0.13 0.99

Thermometer Total Error (°C) Rank ASTM to NIST 0.02 1 Alcohol to NIST 0.33 2 Davis to NIST 0.54 4 Bead Thermistor to NIST 0.49 3

Table  1.    The  linear  regression  output  for  each  thermometer  based  on  the  calibration  data  in  figures  6-­‐‑9.    R2  shows  how  well  our  data  was  represented  by  the  linear  fit  relationship.  

Table  2.    The  total  error  for  each  thermometer  based  on  the  calibration  data  in  figures  6-­‐‑9.    Each  thermometer  calibration  was  ranked  from  the  least  amount  of  error  to  the  largest  error  respective  to  the  NIST  standard.  

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Figure  10.  (above).    The  exponential  decay  of  five  trials  using  the  alcohol  thermometer  subjected  to  ventilation  in  seconds.        Figure  11.   (below).  The  exponential  decay  of   five   trials  using   the  Davis   temperature  probe  subjected  to  ventilation  in  seconds.      

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Figure  12.    The  time  constant  of  five  trials  using  the  alcohol  thermometer  and  the  Davis  temperature   probe   subjected   to   ventilation   in   seconds.     Linear   regression   fitting  was  generated  to  see  how  linear  the  relationship  was  for  the  time  response  on  wind  speed.    The   time   constant   decreases   as   the   wind   speed   increases.     Changes   in   response   are  notable  around  1  m/s  when  the  response  sensitivity  switches  from  the  Davis  being  more  responsive   than   the   alcohol   thermometer   to   the   alcohol   thermometer   being   more  responsive  at  higher  ventilation.  

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Thermometer Slope Slope error Intercept (s) Intercept error (s)

R2

Alcohol -55.91 5.38 142.05 9.94 0.97 Davis -26.62 6.35 116.35 11.91 0.85

Thermometer Wind Speed (m/s)

Time constant, τ (s)

Error  𝝈𝝉 (s) 1/e (s-1) Rank

Alcohol 0 145.22 41.52 -0.01 1 1.85 25.26 5.35 -0.04 3 1.97 29.79 2.43 -0.03 2 2.21 21.06 2.17 -0.05 4 2.36 20.06 2.19 -0.05 5 Davis 0 112.82 33.00 -0.01 1 1.85 75.44 231.24 -0.01 3 1.97 77.58 227.09 -0.01 2 2.18 51.31 76.13 -0.02 4 2.36 42.23 53.62 -0.02 5

Table  3.    The  linear  regression  output  for  each  the  time  constant  in  figure  12.    R2  shows  how  well  our  data  was  represented  by  the  linear  fit  relationship.    The  alcohol  had  a  better  linear  relationship  compared  to  the  Davis  thermometer.  

Table  4.    The  ranking  from  largest  to  smallest  of  the  1/e  decay  for  each  thermometer.    It  was  not  quite  perfectly  linear  with  speed  but  it  was  very  close  to  linear.    The  time  constant  for  the  alcohol  thermometer  shows  how  at  little  to  no  wind  velocities,  the  response  time  is  large  but  at  higher  velocities,  the  response  time  is  quite  low  compared  to  the  Davis  thermometer.  

Rux,  V.      |      18  

SYMBOLS MEANING 𝑹𝟐 a coefficient of determination: how well the regression line represents the real data 𝑵 number of observations in a sample 𝒊 index or iteration 𝒎 the slope generated from the linear regression 𝒄 the intercept generated from the linear regression

𝑻𝒊,𝒐𝒃𝒔 observation temperature at the iteration 𝑻𝒇𝒊𝒕 the temperature output given by the linear regression 𝑺𝒐𝒃𝒔𝟐 the variance between the observed temperature and the linear regression 𝑺𝒓𝒆𝒇𝟐 the variance of the the reference temperature given the linear regression and the

variance of the observed thermometer 𝑻𝒐𝒃𝒔 the mean temperature of the observed thermometer for all trials to one reference 𝝈𝒎 standard deviation of the slope from the linear regression 𝝈𝒄 standard deviation of the intercept from the linear regression 𝝉 time constant 𝒕 time seconds ∆𝑻 change in the temperature reading to the ambient temperature

Tambient ambient temperature ∆𝑻𝒎𝒂𝒙 change in the initial temperature to the ambient temperature °C temperature in degrees Celsius m meters s seconds

Table  5.    Symbols  from  mathematical  formulas  and  figures.