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