Thermocouple TransientResponse Characteristics
Thermal Systems Laboratory
ME 4351 - 141
Instructor: Dr. Jerry R. Dunn
Lab Group
John Burroughs, Group LeaderKim ShinnJay Spikes
Barry Webster
Date dueSept. 30, 1997
i
Executive Summary
The objective of this laboratory was to experimentally determine the effects of flow
conditions and bead diameter on thermocouple transient response characteristics and to
compare the measured response characteristics with predictions from theory. The transient
temperature response was measured experimentally for two type T, exposed junction,
sheathed thermocouple assemblies. The two assemblies had a bead diameter of 1/8 in. and
1/16 in. respectively. The response of each assembly was measured at free stream flow
velocities of 20 ft/s and 40 ft/s and a free stream temperature of 180 ˚F. The initial
temperature was kept at 80˚F for each test. Each thermocouple assembly was suddenly
exposed to a step change in free stream temperature and the response of the sensor
measured at 1 sec. intervals for each set of test conditions. Results were plotted and
compared with theory from convection and transient conduction heat transfer.
Measured time constants ranged from 2.5 to 9.8 sec. These were slightly faster than values
predicted from theory which ranged from 2.7 to 10.7 sec. for the conditions of this test.
The time constant decreased as bead diameter decreased and as free stream velocity
increased. A factor of two decrease in bead diameter had a significantly greater effect on
time constant, a change of 10.7 to 3.7 sec, than a factor of two increase in velocity, a
change of 10.7 to 7.6 sec. This was because of a decrease in bead diameter resulted in an
increase in heat transfer coefficient and a decrease in the mass, thermal capacitance, of the
thermocouple. Flow stream turbulence and the uncertainty associated with the correlation
for heat transfer coefficient were possible causes for the difference between theoretical and
measured values of time constant.
ii
Table of Contents
Page
Executive Summary i
List of Tables iii
List of Figures iv
I. Introduction 1
II. Theory 2
III. Test Setup 6
IV. Test Procedure 8
V. Results and Discussion 9
VI. Conclusions 12
VII. References 12
Appendices 13
Data Sheets and Predicted Theoretical Response
Sample Calculations
Handout
iii
List of Figures
Figure No. Page
2.1 Representation of a bead thermocouple 2
2.2 Theoretical TC Transient response 5
3.1 Schematic of Test Section 6
4.1 Transient Temperature vs Time, D = 1/8 in. 11
iv
List of Tables
Table No. Page
1. Equipment List 7
2. Transient Temperature Response Test Matrix 8
3. Heat Transfer Coefficient Calculation 9
4. Predicted Thermocouple Time Constant 10
5. Thermocouple Response Data, D = 1/8 in. 14
6. Predicted Thermocouple Response, D = 1/8 in. 15
1
I. Introduction
Reliable and accurate temperature measurement is an important and necessary element
in many of today’s engineering problems and designs. Accurate temperature measurements
are typically important to the control, performance, and normal operation of many
engineering processes and operating equipment. Typical examples range from cooking,
heating and cooling, temperature measurement and control in processes such as
combustion, steam generation, and chemical production, and maintaining acceptable
operating conditions for temperature sensitive equipment such as electronics and energy
conversion systems.
Devices used to measure temperature include the basic thermometer, thermocouples,
thermisters, resistance temperature detectors (RTD’s), and optical pyrometers. These
devices have a wide range of temperature measurement capability, accuracy, and
characteristics. The measurement characteristics of each must be understood in order to
select the correct sensor for a given measurement application. This is particularly true
when dynamic response characteristics are important to the given measurement application.
Not only is there a wide difference in response characteristics between devices such as a
thermocouple and a thermometer, but also the transient response of a given device such as a
thermocouple can vary significantly depending on the design and operating conditions of
the sensor.
Therefore, it is the objective of this experiment to determine the transient response
characteristics of a bead thermocouple for various values of thermocouple bead diameter
and flow conditions.
2
II. Theory
The development of the governing equations for the transient response characteristics
of a temperature measurement device begins with the use of the First Law of
Thermodynamics to perform an energy balance on the temperature sensing element. The
development shown below follows the analysis presented by Incropera and DeWitt [1].
Consider a mass, m, shown below in Fig. 2.1, which is at an initial temperature, Ti, and
exchanges heat with the surroundings at a temperature, T∞, by convection. We will assume
that the geometry of the mass is sufficiently small that internal temperature gradients can be
neglected. Therefore, as energy is transferred to or from the mass, we will assume that the
temperature of the mass (our temperature sensing device) changes uniformly throughout the
volume and at any time, τ, has a single value at all points in the mass.
U∞
D
T∞
Fig. 2.1 Representation of a bead thermocouple
Equating the energy transfer by convection to the rate of energy change of the control
volume (the mass m), we obtain Eqn. 1.
− − =∞( )h A T T CV dTdc c ρ τ (1)
3
Defining a new variable, θ, as θ = T - T∞ and separating variables, we obtain the
following:
ρ θθ τ
θ
θ τV Ch Ac c
d di
∫ ∫= −0 (2)
Integrating from time, τ, equal to zero, the equation becomes:
ρ τθθ
V Ch A
nc c i
l = −(3)
Taking the inverse of the natural log, we obtain:
θθ
τρi
h AV C
c c= −
exp(4)
Defining the time constant, τc , as
τ ρc
VCh Ac c
=(5)
we finally obtain the equation for the transient response in the form:
θθ
ττi
T TT Ti c
= =−−
∞
∞−
exp(6)
4
We observe that Eqn. 6 has the following characteristics:
* At τ = 0, θ / θi = 1
* At τ = ∞, θ / θi = 0
* At τ = τc, θ / θi = 1 /e = 0.3678
We therefore see that the time constant physically represents the time necessary for the
temperature sensor to reach 63.22 % (1 - 1/e) of the maximum possible temperature
change. Graphically, this response should appear as a decaying exponential and will
approach the final, steady-state value asymptotically as shown in Fig. 2.2.
0
0.25
0.5
0.75
1
1.25
τ/τc
Fig. 2.2 Theoretical TC Transient Response
θ/θi
We can see that after a time equal to three time constants, the sensor still has not reached the
final steady-state value and that the longer the time constant, the longer the time required to
reach steady-state. It is thus desirable to have a small time constant if rapid sensor
response is important in a given engineering process. From Eqn. 5, it is seen that the time
constant decreases as heat transfer coefficient and surface area increases and as mass and
5
specific heat decreases. This information is helpful to the engineer in selecting and
designing temperature sensing devices.
6
III. Test Setup
The apparatus used in this experiment was designed to have the capability to suddenly
expose the bead of an exposed junction, sheathed thermocouple to a simulated step change
in free stream temperature. The response of the thermocouple is then measured as a
function of time to determine the transient response of the specific sensor being tested. The
configuration of the apparatus test section designed to conduct this test is shown in Fig.
3.1. The main air duct provides a conduit for the free stream air flow to which the
thermocouple is to be exposed.
∞U U ∞
Thermocouple
Protection tube
Flow tube
T∞
Ti air flow
Fig. 3.1 Test Section Schematic
The flow rate can be controlled by varying the opening of an air damper to the fan to
provide a range of free stream velocities for the step temperature change. This results in the
7
ability to change the heat transfer coefficient for the conditions of the test. Air flow in the
duct is heated to the free stream test temperature by an electric resistance duct heater located
upstream of the test section. A spring loaded protection tube is inserted into the flow duct
to initially shield the sheathed thermocouple from the free stream flow. An unheated stream
of air is delivered to the protection tube to maintain the thermocouple at the initial
temperature, Ti. The protection tube is then suddenly withdrawn exposing the
thermocouple a step change in temperature and flow conditions. The output of the
thermocouple is monitored as a function of time to provide a record of the transient
response for the conditions of the test. A digital temperature indicator provides a visual
display of the changing temperature during the test period. A second thermocouple is
located in the flow duct upstream of the test section to measure the free stream temperature
for the conditions of the test.
A detailed description and list of the equipment and instrumentation used in this experiment
is given in Table 1.
Table 1. Equipment List
Item Manufacturer Model Specifications
Fan New York Blower AH8-6C 175 cfm
Electric duct heater General Electric DH-1-B 1 kw, 220 V, 5 A
Sheathed T/C Omega GTQSS-18E-12 .125”dia., Type T,exposed junction
Sheathed T/C Omega GTQSS-116E-12 .0625”dia., Type T,exposed junction
Digital temperaturesensor
Omega DP465-TF -199˚ - 752˚ F, +/- .1 ˚ F
Computer Gateway 2000 166 Mh
A-D data acquisitionboard
National Instruments 16 channel, 12digital, 4 analog
8
IV. Test Procedure
A series of four tests were conducted for this experiment. The test matrix for the tests
consisted of the combinations shown in Table 2. The transient temperature response was
measured for two thermocouple diameters and two values of free stream velocity. All tests
were conducted at a free stream temperature of 180˚F.
Table 2. Transient Temperature Response Test Matrix
Thermocouple Diameter - D 1/8 in. 1/16 in.
Free Stream Velocity - U∞20 ft/s 40 ft/s 20 ft/s 40 ft/s
Free Stream Temperature - T∞180 ˚F 180˚F
The following sequence was used to conduct the tests:
(1) Insert the test thermocouple into the test section,
(2) Turn on the fan and adjust the vane damper to obtain the desired free streamvelocity,
(3) Turn on the duct heater and adjust the control to obtain the desired free streamtemperature,
(4) Turn on the computer and set the data sampling conditions for the A-D boardas required for the test. Set the A-D board to standby,
(5) Trip the test section mechanism to suddenly remove the protection tube and tobegin recording the thermocouple temperature at the desired sample rate,
(6) Continue the test until the measured temperature is within 5˚F of the freestream temperature.
Repeat steps 1 - 6 for each set of test conditions.
9
V. Results and Discussion
The first step in the analysis of the response characteristics of the thermocouple was to
determine the convection heat transfer coefficients for the geometry and conditions of the
tests. The correlation for a sphere by Whitaker [2], Eqn. 7, was used to determine the
Nusselt number.
NuD D Ds
= + +( ) ( )
2 0 4 5 2 3 0 4
1 4
. Re Re Pr. / .
/µµ
(7)
The results are given in Table 3 for the conditions used in this experiment.
Table 3. Heat Transfer Coefficient Calculation
T∞ (F)= 1 8 0 Ts (F)= 1 3 0µ∞ (kg/m s)= 2 .12E-05 µs(kg/m s)= 1 .96E-05ρ (kg/m3) = 0 .87
Pr∞ = 0 .699 k∞ (w/m K) = 0 .0334
U∞ (ft/s) 2 0 2 0 4 0 4 0D (in.) 0 .125 0.0625 0.125 0.0625
Re 794.3 397.1 1588.6 794.3NuD 15.95 11.53 22.49 15.95
h (W/m2 K) 167.8 242.6 236.6 335.6
It is seen that the heat transfer coefficient varied from 167.8 to 335.6 W/m2˚K as the free
stream velocity and bead diameter changed from 20 ft/s, 1/8 in. to 40 ft/s, 1/16 in. These
results were used in Eqn. 6 to predict the theoretical response for the test conditions. It is
noted that theory predicts a slightly greater effect on the heat transfer coefficient due to
reducing the bead diameter by a factor of 2 than for increasing the velocity by a factor of 2.
Thermocouple time constant can now be calculated using Eqn. 5 and the predicted values of
heat transfer coefficient and thermocouple properties. The predicted values of time constant
are shown in Table 4 for the conditions of this test.
10
Table 4. Predicted Thermocouple Time Constant
ρ (kg/m^3) = 8 5 0 0 C(J/kg K) 4 0 0
D (in.) 0 .125 0.125 0.0625 0.0625
U∞(ft/s) 2 0 4 0 2 0 4 0
h(w/m2K ) 167.8 236.6 242.6 335.6
τc (sec) 10 .7 7 .6 3 .7 2 .7
As expected, an increase in velocity and a decrease in thermocouple size results in a
decrease in time constant. It is noted that the decrease due to a factor of two decrease in
diameter is greater than the decrease due to a factor of two increase in flow velocity. This
is due to the fact that a decrease in diameter both increases the heat transfer coefficient and
decreases the mass of the thermocouple. These values of time constant were then used to
predict the transient temperature response for the conditions simulated in the test. The
results are tabulated in Tables 5 and 6 in Appendix A.1.
Fig. 4.1 compares the predicted theoretical response with measured results for a 1/8 in.
diameter thermocouple. As expected, the response time decreases as flow velocity, and
therefore heat transfer coefficient, increases. A time constant of 9.9 sec was obtained for a
free stream velocity of 20 ft/sec and a diameter of .125 in. This decreased to 7 sec when the
velocity increased to 40 ft/sec. The experimentally measured response was slightly, less
than 1 sec, faster than that predicted by theory. This is possibly due to slight variations in
the bead geometry and an increase in heat transfer coefficient due to turbulence near the
thermocouple bead. Response time also decreased as the bead diameter was decreased by a
factor of 2 from .125 to .0625 in.
11
75
100
125
150
175
200
T (
F)
0
10
20
30
40
50
Time (sec)
Fig. 4.1 Transient Temperature vs time, D = 1/8 in.
T - exp, 40ft/s
T - theor, 40 ft/s
T - exp, 20 ft/s
T - theor, 20 ft/s
12
VI. Conclusions
Tests to determine transient thermocouple response characteristics clearly showed that the
response time of a bead type thermocouple decreases as bead diameter decreases and as
heat transfer coefficient increases. For the conditions of this test, the effect of decreased
bead diameter was greater than that of increased flow velocity. While time constants as low
as 2.7 sec. were predicted, times as long as 12.5 sec. were required to reach within 1˚ of
the steady-state temperature for similar conditions. Predicted thermocouple response was
slightly slower than that measured for corresponding conditions.
VII. References
1. Incropera, Frank P. and DeWitt, David P., Fundamentals of Heat and Mass Transfer ,
John Wiley and Sons, Fourth Edition, 1996.
2. Whitaker, S., AIChE J., vol. 18, pg 361, 1972.
13
Appendices
I . Data Sheets andPredicted Theoretical Response
II. Sample Calculations
III. Handouts
14
Appendix I.Table 5. Thermocouple Response Data, D = 1/8 in.
D = 0.125 in. D = 0.125 in.U ∞ = 20 ft/s U ∞ = 40 ft/s
h = 167.8 W/m2K h = 236.6 W/m2Kτc = 10.7 sec. τc = 7.6 sec.
time Theor. Exper. time Theor. Exper.(sec) T(τ) (sec) T(τ)
0 80.0 8 0 0 80.0 8 01 88.9 90.5 1 92.3 94.12 97.0 98.2 2 103.1 105.23 104.4 106.1 3 112.6 114.54 111.2 113.4 4 120.9 123.35 117.3 120.2 5 128.2 130.66 122.9 126.3 6 134.6 1 3 77 128.0 131.8 7 140.2 142.88 132.7 135.4 8 145.1 147.59 136.9 139.6 9 149.4 152.3
1 0 140.7 143.8 1 0 153.2 1 5 61 1 144.2 147.2 1 1 156.5 159.31 2 147.4 150.1 1 2 159.4 162.21 3 150.3 153.6 1 3 161.9 164.81 4 153.0 1 5 6 1 4 164.2 167.41 5 155.4 158.2 1 5 166.1 169.21 6 157.6 160.1 1 6 167.8 170.61 7 159.6 162.7 1 7 169.3 1 7 21 8 161.4 163.9 1 8 170.6 173.21 9 163.1 165.4 1 9 171.8 174.32 0 164.6 1 6 7 2 0 172.8 174.92 1 166.0 168.1 2 1 173.7 175.52 2 167.2 1 6 9 2 2 174.5 176.12 3 168.3 169.8 2 3 175.2 176.72 4 169.4 170.4 2 4 175.7 177.12 5 170.3 171.2 2 5 176.3 177.42 6 171.2 172.3 2 6 176.7 177.52 7 172.0 173.1 2 7 177.1 177.82 8 172.7 174.2 2 8 177.5 178.12 9 173.3 174.6 2 9 177.8 178.33 0 173.9 1 7 5 3 0 178.1 178.53 1 174.5 175.5 3 1 178.3 178.73 2 175.0 175.8 3 2 178.5 178.83 3 175.4 176.1 3 3 178.7 1 7 93 4 175.8 176.6 3 4 178.9 179.23 5 176.2 1 7 7 3 5 179.0 179.33 6 176.5 177.2 3 6 179.1 179.43 7 176.9 177.6 3 7 179.2 179.53 8 177.1 1 7 8 3 8 179.3 179.53 9 177.4 178.3 3 9 179.4 179.64 0 177.6 178.8 4 0 179.5 179.7
15
Table 6. Predicted Thermocouple Response , D = 1/8 in.D = 0.125 in. D = 0.125 in.
U ∞ = 20 ft/s U ∞ = 40 ft/sh = 167.8 W/m2K h = 236.6 W/m2Kτc = 10.7 sec. τc = 7.6 sec.
time Theoretical Results time Theoretical Results(sec) τ/τc θ/θi T(τ) (sec) τ/τc θ/θi T(τ)
0 0.00 1.00 80.0 0 0.00 1.00 80.01 0.09 0.91 88.9 1 0.13 0.88 92.32 0.19 0.83 97.0 2 0.26 0.77 103.13 0.28 0.76 104.4 3 0.39 0.67 112.64 0.37 0.69 111.2 4 0.53 0.59 120.95 0.47 0.63 117.3 5 0.66 0.52 128.26 0.56 0.57 122.9 6 0.79 0.45 134.67 0.65 0.52 128.0 7 0.92 0.40 140.28 0.75 0.47 132.7 8 1.05 0.35 145.19 0.84 0.43 136.9 9 1.18 0.31 149.4
1 0 0.93 0.39 140.7 1 0 1.32 0.27 153.21 1 1.03 0.36 144.2 1 1 1.45 0.24 156.51 2 1.12 0.33 147.4 1 2 1.58 0.21 159.41 3 1.21 0.30 150.3 1 3 1.71 0.18 161.91 4 1.31 0.27 153.0 1 4 1.84 0.16 164.21 5 1.40 0.25 155.4 1 5 1.97 0.14 166.11 6 1.50 0.22 157.6 1 6 2.11 0.12 167.81 7 1.59 0.20 159.6 1 7 2.24 0.11 169.31 8 1.68 0.19 161.4 1 8 2.37 0.09 170.61 9 1.78 0.17 163.1 1 9 2.50 0.08 171.82 0 1.87 0.15 164.6 2 0 2.63 0.07 172.82 1 1.96 0.14 166.0 2 1 2.76 0.06 173.72 2 2.06 0.13 167.2 2 2 2.89 0.06 174.52 3 2.15 0.12 168.3 2 3 3.03 0.05 175.22 4 2.24 0.11 169.4 2 4 3.16 0.04 175.72 5 2.34 0.10 170.3 2 5 3.29 0.04 176.32 6 2.43 0.09 171.2 2 6 3.42 0.03 176.72 7 2.52 0.08 172.0 2 7 3.55 0.03 177.12 8 2.62 0.07 172.7 2 8 3.68 0.03 177.52 9 2.71 0.07 173.3 2 9 3.82 0.02 177.83 0 2.80 0.06 173.9 3 0 3.95 0.02 178.13 1 2.90 0.06 174.5 3 1 4.08 0.02 178.33 2 2.99 0.05 175.0 3 2 4.21 0.01 178.53 3 3.08 0.05 175.4 3 3 4.34 0.01 178.73 4 3.18 0.04 175.8 3 4 4.47 0.01 178.93 5 3.27 0.04 176.2 3 5 4.61 0.01 179.03 6 3.36 0.03 176.5 3 6 4.74 0.01 179.13 7 3.46 0.03 176.9 3 7 4.87 0.01 179.23 8 3.55 0.03 177.1 3 8 5.00 0.01 179.33 9 3.64 0.03 177.4 3 9 5.13 0.01 179.44 0 3.74 0.02 177.6 4 0 5.26 0.01 179.5
16
A.2 Sample Calculations
I. Theoretical Temperature Distribution
θθ
ττi c
= −
exp
ττ
θθc i
= = −{ } =1 1 0 3679, exp .
II. Reynolds Number
Re = ∞
∞
ρµU D
T F T F D in U ft kg ms∞ ∞ ∞= = = = =180 80 0 125 20 87 3, , . ., / sec, . /ρ
P kPa E kg m s E kg m ss∞ ∞= = = − = −88 8 0 699 21 2 6 19 6 6. , Pr . , . / , . /µ µ
Re. / * / sec* . .*. / .
. /=
−87 20 0 125 0254
21 2 6
3kg m ft in m in
E kg m s
Re .= 794 3
III. Nusselt Number
Nus
= + +[ ]
∞2 0 4 5 2 3 4
25
. Re Re Pr. / .
.µµ
NuE
E= + +[ ] −
−
2 0 4794 3 794 3 0 69921 2 619 6 6
5 2 3 4
25
. . . ...
. / .
.
Nu = 15 95.
IV. Heat Transfer Coefficient
k W m K= . /0334
hk
DNu
W m K
in m inW m K= = =. /
. . *. / .. . /
0334125 0254
15 95 167 8 2
17
V. Time Constant
τ ρ ρc
V C
h A
C D
h= = / 6
τ c
kg m J kg K m
W m K= 8500 400 125 0254
167 8 6
3
2
/ / . * .. / *
τ c = 10 7. sec
18
Page for Experiment Handout