Experiment 2
Thermometry I
Expansion-Type Thermometers
Objective:
To determine the calibration and response time of certain expansion-type
thermometers.
Apparatus:
a. Dewer flask
b. stirring device
c. Distilled water, ice
b. stopwatch
e. stands, clamps to support thermometers
f. 1 standard, precision mercury-in-glass thermometer
g. 1 mercury-in-glass thermometer
Theory of Liquid-In-Glass Thermometers
The most common expansion-type thermometer is the liquid-in-glass
thermometer. In this type of thermometer, the difference in expansion between the liquid
and the glass container serves as an indicator of the temperature. Although almost any
type of fluid can be used, the most common fluid types are mercury and ethyl alcohol.
Although the coefficient of expansion of ethyl alcohol (1.12 x 10-3 at 20°C) is about 6
times greater than that of mercury (0.1819 x 10-3 at 20°C), mercurial thermometers are
invariably more accurate than alcohol thermometers and are preferred except when very
low temperatures are to be measured (mercury freezes at –38.87°C). The two principal
reasons for the lower accuracy of alcohol thermometers are that (1) alcohol is more
volatile and thus has a greater tendency to evaporate into the space above the liquid
column, and (2) alcohol “wets” the surface of the glass, so that when the temperature
drops sharply a small part of the liquid remains on the wall above the column. The errors
of temperature measurement can be classified into two general categories: (1) those that
are inherent to the particular instrument, and (2) those caused by the lack of thermal
equilibrium between the thermometer and the air at the moment the thermometer is read.
Of considerable interest is the time that it takes for a given thermometer to reach
approximate thermal equilibrium with the surrounding air.
The ability of a thermometer to respond to changes in ambient temperature is
expressed in terms of its “time constant” or “time lag”. Time lag is also referred to as
“response time”, “lag coefficient”, or simply “lag”. The time constant, τ, is defined as the
time required for an instrument to reach within 36.8% (i.e., e-1) of equilibrium after the
sensor has been subjected to a step input change. This definition of the time constant
arises from the fact that the approach of the indicated value to the “true” value will be
proportional to e-t/τ , where t is time. The rate at which the indicated value changes with
time, dT/dt, when a thermometer bulb is placed in a medium of differing temperature can
be expected to be proportional to the rate at which heat flows into the bulb (i.e., dT/dt ∝
1
q, where q is the heat flux). The heat flux in turn will be proportional to the difference
between the environmental temperature, Te, and the temperature of the thermometer, T(t).
Thus, the general dynamic response equation of the liquid-in-glass thermometer can be
expressed in the following manner:
dT Mc p
=
[Te − T (t )]
dt
C
(1.1)
where
M is the mass of fluid that the thermometer comes in contact with in the time
increment dt
cp is the specific heat of the fluid (at constant pressure) that the thermometer is
immersed into
C is the thermal capacity of the thermometer
Te is the environmental temperature
T(t) is the temperature reading on the thermometer at time t
t is time
Integrating equation 1.1 with the assumption that Te is constant, over the limits of T0
(initial temperature) to T(t) results in the following expressions:
Mc p
 T (t ) − Te 
ln 
t
=−
C
 T0 − Te 
(1.2)
Solving 1.2 for T(t) yields:
 Mc p 
T (t ) = (T0 − Te ) exp −
t  + Te
 C 
(1.3)
From equation 1.3, it is evident that the time constant for a liquid-in-glass thermometer
can be determined by setting the exponential term to -1. This results in the following
equation:
 Mc p  T (τ ) − Te
τ =
exp[−1] = exp −
= 0.36788
T0 − Te
 C 
(1.4)
Determining the time constant of a given instrument is very important because failure to
do so can result in significant temperature errors when the environmental temperature
varies in time.
When measurements are intended to be representative of periods of minutes or
more, a relatively long time constant may actually be desirable. However, a short time
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constant is required when the thermometer is in motion or when short-period temperature
oscillations are present. Since the response time of an instrument depends on the heat flux
between the medium and the sensor, the time constant can be modified by changing the
rate of such heat transfer. The rate of heat transfer can be modified by changing the
thermal capacity of the thermometer (i.e., the bulb shape, thermometer length, liquid
type, etc.), ventilating the thermometer (i.e., increasing the mass that comes in contact
with the bulb), or by immersing the thermometer in a fluid of differing specific heat.
Since the choice of immersion fluid and the thermal capacity of the thermometer cannot
usually be modified by the user, modifying the degree of ventilation is the only practical
way to alter the response time of a given thermometer. In general, the response time
decreases very rapidly with even the slightest ventilation.
Task #1: Verify Factory Calibration Using Water Baths
Goal: To verify the factory calibration of the mercury precision thermometer using two
known temperatures (i.e., the freezing and boiling points of water).
a) Use a Dewar flask filled to the top with an ice bath of distilled water to calibrate the
precision thermometer at 0°C. Immerse as much of the thermometer as possible in the
bath, allowing just enough of the scales to protrude to permit reading. Be sure that the
thermometer does not touch the sides or bottom of the flask. Record the measurement in
lab book only after the thermometer has stabilized.
b) Fill the 1000 mL beaker with distilled water and bring to a rolling boil using the gas
burner. Immerse as much of the thermometer as possible in the water making sure that
the thermometer does not touch the sides or bottom of the beaker. Record the
measurement in lab book only after the thermometer has stabilized.
c) Discuss any errors associated with these measurements.
Task #2 Determine Time Constants
Goal: To determine the time constants of the thermometer under unventilated and
ventilated conditions
a) Record the environmental air temperature of the thermometer.
b) Place the thermometers into an ice bath and let it equilibrate. Take the thermometer out
of the water, quickly wipe away the excess water, and place it into still air. Record the
measured temperature at 5-second intervals for ~3 minutes.
c) Repeat procedure (b) except this time place the thermometers in front of the ventilating
fan. Record the measured temperature at 5-second intervals for ~2 minutes.
d) Plot the response of each of these test thermometers as a function of time for both
ventilated and unventilated conditions.
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e) Use the data to determine the time constants for the thermometer under both ventilated
and unventilated conditions.
e.1) Compute the ratio T(t)-Te/Ti-Te and then plot the results on semi-logarithmic
scale to determine the time lag. T(t) is the temperature at time t, Te is the
environmental temperature (temperature of the bath, room), and Ti is the initial
temperature of the thermometer.
e.2) Plot T(t)-Te/Ti-Te as the ordinate and t(time in sec) as the abscissa. Read from
the graph the time lag for each test.
f) Discuss your results in your report.
Task #3 Calibration of Hobo Loggers
Goal: To calibrate each Hobo logger thermistor to one another.
a)
b)
c)
d)
Collect all 13 logger probes and bundle together
Place the bundle into the ice bath (as in Task 1) and let stabilize.
Download the data from the Hobo
After the data is placed into the course data dir plot each hobo on one plot using
Matlab or Excel.
Table 1. Calibration at 0 °C
Time
Standard Thermometer
#1
Test Thermometers
#2
#1
Test Thermometers
#2
Table 2. Calibration at 100 °C
Time
Standard Thermometer
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Table 3. Unventilated Air
Time (s)
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
Te
#1
Temperature C
#2
Ratio T-Te/Ti-Te
#1
#2
#1
Temperature C
#2
Ratio T-Te/Ti-Te
#1
#2
Table 4. Ventilated Air
Time (s)
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
Te
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Table 5. Time Lag in sec
Ice Bath
Air
Thermometer
1
2
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