CHAPTER 13 Transient Temperature Measurement

CHAPTER 13

Transient Temperature Measurement

The True Meaning of a Term Is to Be Found by Observing What a Man Does With It, Not What

He Says About It.

13. 1 GENERAL REMAKS

Because of inertia, no instrument (or anything else for that matter) responds instantly or with perfect fidelity to a change in its environment.

In mechanical systems, mass is the familiar measure of inertia, whereas in electric and thermal systems inertia is characterized by capacitance. We are concerned here with the response of a temperature sensor to a change in its environmental temperature.

The simplified, hence manageable, temperature changes considered here are ：

(1) The ramp change, in which the environment temperature shifts linearly with time from T1 to T2 ;

(2) The step change, in which the temperature of the sensor environment shifts instantaneously from T1 to T2 ; and

(3) The periodic change, in which the environment temperature alternates sinusoidally with time between

+T2 to -T2 (figure 13.1).

We seek answers to the following questions: what is the speed or measure of the sensor response? What is the fidelity or faithfulness of the sensor response?

Thermal response belongs, fundamentally, in the realm of transient heat transfer. The rate of response of a temperature sensor clearly depends on the physical properties of the sensor, the physical properties of its environment, as well as the dynamical properties of its environment.

Amplifying on this, we note that because physical properties normally change with temperature, it follows that the response time of a sensor will vary with the temperature level.

Because heat transfer coefficients are strongly dependent on the Reynolds number, it follows that sensor response will vary with the mass velocity of its environment.

It is common practice to characterize the response of a temperature sensor to a nonisothermal change of state of its environment by a thermal time constant. Although a single time constant can exactly describe the response behavior of only the simplest of systems, it is nonetheless common practice to consider first-order response only.

In Section 13.2, we confine our attention to first-order systems in which the sensor exhibits a rate of change in temperature that is exactly proportional to the temperature difference between the sensor and its environment.

In Section 13. 3, second-order systems are considered. We defer until Section 13. 6 the complications that arise from the additional considerations of conduction, radiation, temperature level, turbulence, and distributed thermal capacities.

13. 2 MATHEMATICAL DEVELOPMENT OF

FIRST -ORDER RESPONSE

A simplified one-dimensional heat balance can be written for a temperature sensor subjected to a time-varying environmental temperature.

We assume that all the heat transferred to the sensor is by convection , and that all this heat is retained by the sensor, that is , the thermal resistance of the system is lumped in the convective heat transfer film around the sensor, and the thermal capacity of the system is lumped in the sensor itself ( Figure 13.2 ) .

c

dThA T T Mc

dt (13.1)

Thus the heat transfer rate through the film to the sensor exactly equals the rate of heat storage in the sensor. Expressed in terms of Newton’s law of cooling and Black’s heat capacity equation , we have

where

h is the convective heat transfer coefficient of the fluid film surrounding the sensor

A is the surface area of the sensor through which heat is transferred ,

TC is the environment temperature at time t ,

T is the sensor temperature at time t ,

M is the mass of the sensing portion ( it can also be expressed as , that is , density times volume ) ,

and c is the specific heat capacity of the sensing portion .

Separating the variables in equation (1 3. 1) yields

V

c

dt dT

Vc hA T T

(13.2)

where the quantity in parentheses is taken to be a lumped constant to have the dimensions of time , and called the time constant . Thus,

tan

tan

Vc thermalcapaci ceofsensor

hA thermalconduc ceoffilm

(13.3)

Since thermal conductance is the reciprocal of thermal resistance, equation(13.3) also indicates that

,which is exactly analogous to the time constant of an electric circuit .

The first -order, first - degree linear differential equation expressed by equation (13. 2) has the general solution [1]-[3]

where C is a constant of integration, which is determined by inserting the proper boundary conditions .

th thR C

0

1 tt t teT Ce e T e dt

(13.4)

13.2. 1 Ramp Change Under this condition at t=0,T1=T=C,although in general Tc=T2+Rt

where R represents the rate of change of the environment temperature

Insertion of these boundary values in equation ( 1 3 . 4 ) yields T t

Evaluating equation (1 3. 5), we have

1 2 0 0

1 t tt t t tT T e e T e dt R te dt

(13.15)

00 0

t ttt t tte dt t e e dt

1 2 2t t tT T e T T e Rt R R e

which, expressed in terms of the temperature difference ,becomes

2 1t t

cT T R R e T T e (13.7)

Where the terms involving approach zero, and equation (13.4 )-(13.7) reduce to

According to equation （ 13.8 ) , the time constant for a ramp change can be defined as follows :

If an element is immersed for a long time in an environment whose temperature is rising at a constant rate (i.e., a ramp change), τis the interval between the time when the environment reaches a given temperature and the time when the element indicates this temperature; that is, is the number of seconds the element lags its environment (Figure 13. 1).

T te

cT T R (13.8)

13.2.2 Step Change

Under this condition at ,although in general

Insertion of these boundary values in equation (13.4) yields

which, expressed in terms of the temperature difference, becomes

According to equation (13.10), the time constant for a step change can be defined as follows:

10,t T T C

2cT T

1 2 2t tT T e T T e (13.9)

1t

c cT T T T e (13.10)

If an element is plunged into a constant-temperature environment (i.e., a step change), τis the time required for the temperature difference between the environment and the element to be reduced to l/e of the initial difference, that is, τis the number of seconds for the element to reach 63.2% of the initial temperature difference (Figure 13.1).

13.2.3 Periodic Change

Under this condition , although in general

Whereωrepresents the frequency of the forcing

oscillations of the environment in radians per unit of time.

Insertion of these boundary values in equation (13.4) yields

10,t T T C

2 sincT T t

12 2

1 1 2 22sin tan

11

tT TT T t e

(13.11)

When the last term above approaches zero and the sensor response will lag the environment by the phase angle

which in time units corresponds to a lag of

t

1tanL (13.12)

1tanLt

(13.13)

Whereas the ratio of the sensor amplitude to that of the environment is given by

max

1 222

1

1

T

T

(13.14)

Although no general definition of the time constant for a periodic change is forthcoming, a restricted definition can be given as follows:

If an element is immersed for a long time in an environment whose temperature is varying sinusoidally(i.e., a periodic change), and if the frequency is much less than 1/τthen, to a close approximation,

τis the number of seconds the element lags its environment (Figure 13. 1).

In equation (13.4) and in all equations thereafter, the time constant consistently has represented the lumped constant

Thus it follows that , whenever all conditions stated tender the three separate sections on the forcing functions are met.

Vc hA

step periodic

13.3 SECOND-ORDER RESPONSE

When several thermal resistances are combined along with several capacities, as in the case of a temperature sensor inserted in a thermometer well, an equivalent electric circuit more complex than that shown in Figure 13.2 must be considered.

Also, solutions more complex than those given for first-order systems, namely, equations (13.8) and (13.10) must be used.

13.3.1 Step Change

A second-order thermal system, as a thermometer well-sensor system, and its equivalent circuit are given in Figure 13.3. An appropriate solution describing the temperature-time response of the thermal system of Figure 13.3 to a step change in temperature can be given [4]-[7] as

where T is the sensor temperature at any time t, is the step change in temperature, with the initial temperature normalized at zero, and where

2 21 2

1 2 2

1 r t r tr rT T e e

r r r r

T

2

1 2

4,

2

a a br r

b

(13.16)

and represent the roots of the second-order quadratic, with

0 0s v s s

w s w

a C R R R C R

b C C R

(13.18)

(13.17)

The thermal resistances and capacities, as shown in Figure 13.3, can be defined mathematically as follows: The sensor capacity is

4

)(C

22iopss

s

ddc

)19.13(

The well capacity is

The sum of the well-side resistances is

where

and represents the heat transfer film resistance outside the well,

and

4

)(C

22iopww

w

DDc

)20.13(

wwfo RRR )21.13(

off Dh

R

1 ）（ 22.13

w

Moww k

DDR

2)/ln(

）（ 23.13

represents the well-to-capacity resistance.

The sum of the sensor-side resistances is

where

represents the well-to-sensor resistance, and

represents the air film between well and sensor, and

represents the sensor-to-capacity resistance.

sAwsf RRRR

w

iows k

DDR

2)/ln(

A

oiA k

dDR

2)/ln(

s

Mos k

ddR

2)/ln(

)27.13(

)26.13(

)25.13(

）（ 24.13

The geometric definitions are as follows:

= outer diameter of well

= inner diameter of well

= mean diameter of well

=outer diameter of sensor

=inner diameter of sensor

= mean diameter of sensor

Equation (13.15) is the second-order counterpart of the first-order solution given by equation (13. 10). The roots and of equations (13.15) and (13.16) are reciprocals of the time constants, that is,

oD

iD

)(2

1 22ioM DDD

od

id

)(2

1 22ioM ddd

22

11

1,

1

rr )28.13(

When one time constant dominates the other, that is, when equation (13.15) can he reduced to the form of a first-order solution, namely,

In similar form, equation (13.10), the first-order solution can be written

The conclusion expressed by equation (13.29) is in agreement with Looney[5] and Coon [6] to the affect that in most cases a single, representing the 63.2% definition of the time constant, is adequate to represent even the more complex temperature-sensing systems.

])(1[ 2/

21

1 terr

rTT

)29.13(

]1[ /teTT )'10.13(

13.3.2 Ramp Orange

The same equations for r1 and r2 hold for the ramp

change as for the step change, that is, equations (13.16) to (13.18) and (l3.28). Hence the same time constants apply as well. Just as stated after equations (13.7), after a time lapse of about five dominant time constants, the temperature-sensing systems will follow a temperature-time ramp of the same slope as its environment (Figure 13.1), and the lag for the second-order systems will be

as compared to the first-order equation (13.8).

RLTTc )31.13(

13.3.3 Periodic Change

When the environment temperature is varying sinusoidally, at a frequency below the temperature-sensing system behaves essentially as a single time constant system with an effective equaling the lag of equation (13.30).

13.4 EXPERIMENTAL DETERNUNATION of TIME CONSTANT The ramp change definition of the time constant provides one method for determining τ. The sensor, initially at some uniform temperature, is inserted into an environment whose rate of change of temperature with time is fixed and known.

However, several problems are encountered. The environment temperature must be known as a function of time, and this requires a sensor of known τ or a sensor having an insignificantly small τ. In addition and this is a problem common to all methods of determining τ, the film coefficient of the environment-sensor interface must be known [4]. [8], [9], By the Nunsselt equation for heat transfer by forced convection [see equation (12.29)]

cT

where Nu = Nusselt number=

Re = Reynolds number＝ Pr = Prandtl number=

It follows that, along with the physical properties of the film and sensor, a knowledge of the mass velocity G of the environment relative to the sensor is requited if the film coefficient is to be determined within a reasonable uncertainty.

Pr)(Re,1Nu )32.13(

khD /

/DG

kcp /

The step change definition of the time constant provides the usual method of determining . The sensor is plunged from an initially different temperature into a constant-temperature bath. It does not matter whether the sensor is heated or cooled by the step change, but we should not start timing the response at the instant of the plunge.

0

At least four should elapse first to allow stabilization of the fluid film on the sensor, since during this initial time the response of the sensor is not well approximated by the first-order equation.

As in the ramp change, the film coefficient must be known. In a stagnant bath, reliable repeatable values for the film coefficient cannot be obtained because of the variableness of the natural convection currents set up in the bath by the temperature gradients.

0

mDGhD )(2 )33.13(

Murdock, Foltz, and Gregory [10] have discussed a practical method for determining response times of thermometers in stirred-liquid baths.

Their detailed method is somewhat as follows: Noting that the fluid properties K, μ and Cp can be taken as constants for a given liquid bath, equation (13.32) reduce to

Evaluation of the effective mass velocity around the sensor is complicated by the fact that the liquid swirls in a three-dimensional flow pattern. It varies not only with the rate of agitation but also with the physical location of the sensor in the bath.

By fixing the sensor location in the bath, we can express the effective value of G in terms of the stirrer speed as

Hence equation (13.33) also can be given as

3bG N (13.34)

4m ehD D N (13.35）

Murdock et al. [10] summarize their experimental data for cylindrical sensors in their stirred-liquid salt with variable agitation by the empirical equation

where the film coefficient h is in ,the sensor diameter D is in feet, and the stirrer speed N is in revolutions per minute. Equation (13.36) indicates that the exponents m and c of equation (13.35) are equal.

0.6( ) 3.11( )saithD DN (13.36)

2Btu / h ft F

It is important to note that equation (13.36) is only one particularization of equation (13.35). It cannot apply exactly in the general case. For example, in another stirred-salt bath there has resulted

0.665) 3.00( )salthD DN（ (13.36 ')

It is clear that either equation (13.36) or equation (13.36') yields close estimates to the film coefficient in a stirred-salt bath. If the stirrer speed also is fixed, equation (13.35) reduces further to

5mhD D (13.37)

For a stirred-oil bath at about 4000F, Murdock's data can be expressed for constant stirrer speed as

Again, this is one particularization of equation (13.35). In another bath, at a different but unrecorded stirrer speed, there has resulted

It is clear from a comparison of equations (13.38) and (13.38') that both the stirrer speed and perhaps the bath geometry are of importance in determining the film coefficient by this method. Far example, at D = 1 in, equation (13.38) yields h = 17.5 while equation (13.38') yields h=32

0.7( ) 17.5oilhD D (13.38)

2Btu / h ft F

2Btu / h ft F

0.51( ) 32oilhD D (13.38')

By expressing equation (13.3) for cylindrical sensors in the form of equation (13.37), the unknown function of D can be obtained. That is, for a hallow cylinder and neglecting end effects, the volume of the sensor is

22 2( )

4 4o

s o i

D LV D D L

(13.39)

Furthermore, for radial heat transfer the surface area of the sensor is

Hence

According to the method under discussion, the physical properties of the sensor in equation(13.41) vary almost linearly with temperature and are to be evaluated at the average temperature corresponding to 36.8% of the difference between the initial and final temperature. Thus this method provides an experimental determination of the effective film coefficient of a stirred-liquid bath according to equation (13.36) and (13.41).

s oA D L (13.40)

2( )4

chD D

(13.41)

To determinate τ, one starts with a liquid bath of known physical properties, stirred at known speed, held at a known temperature. A sensor of known physical properties is plunged into a fixed location in the bath and after waiting an appropriate time, one starts timing its response. The timing is stopped at a predetermined percentage of the temperature difference from start to final temperature, where the 63.2% mark yields the first-order time constant directly.

As indicated in Figure 13.1, a first order response plots as an exponential curve on linear coordinates and as a straight line on a semilog grid. Since it is usually desirable to come closer to the final temperature than 63.2%, equation (13.10) can be rewritten as a=(TC-T)/(TC -Ti)=e-t/τ

where a indicates how close the sensor temperature is to the final temperature. Equation (13.42) yields, for a few points, the following table:

13.5 APPLYING THE TIME CONSTANT

The ideal model for the time constant of equation (13.3), that is,

has four assumptions built into it.

a 0.5 0.368 0.2 0.1 0.05 0.01

t 0.7τ τ 1.6τ 2.3τ 3τ 4.6τ

%recover 50 63.2 80 90 95 99

tanVc

h cons tA

(13.43)

These are:

1. All thermal resistance to heat transfer is lumped in the fluid surrounding the sensor-well system.

2. All thermal capacitance of the system is lumped in the sensor-well system.

3. All the heat received through convection is stored in the sensor-well system.

4. The heat transfer is one-dimensional.

In the following three examples, the idealized lumped relation of equation (13.43) is used as the basis of solution.

Example 1.

Predict the idealized first-order time constant for a thermocouple embedded to the center of a hollow stainless-steel cylinder of 1-in outside diameter and 0. 26 in inside diameter when plunged into a salt bath that is stirred at 800 r/min. Assume for the properties of steel those given in Table 13.1

Solution

By equation (13.36),

though this is an effective h, as noted after equation (13.41), because it includes the cylinder thermal resistance as well as the film resistance, it is the h normally used to characterize the bath-sensor film.

By equation (13.41)

0.63.11(800 1/12) 38.646Btu/h-ft- FohD

38.646 / (1/12) 463.8Btu/h-ft- Fsalth

salth

22

1500 0.135 ( )

12 0.0030324 4 38.646

10.92

osalt

o

cDh

hD

s

More precisely, for the hollow cylinder, following equation (13. 39)

2 2( ) 500 0.135 (1 0.0676)

4 4 463.8 12

0.002827 10.18

o isalt

o

c D D

hD

h s

Example 2

If the above sensor is to be used in a steam turbine where the film coefficient is estimated, as by the Nusselt equation (12.29), to be what will be the approximate time required lo reach 99% of a given step change in temperature?

Solution. The time constant predicted by equation (13 .43) is

When we would conclude that the sensor is faster to respond in steam than in salt.


21200Btu/h-ft - F

steam

463.8( ) 10.18( ) 3.9

1200salt

saltsteam

hs

h

99 4.6 4.6 3.9 17.9t s ％

Example 3

The sensor in Example 2 is to he used in an air furnace that has a film coefficient, as estimated by the Nusselt equation, of 46.4 and is cycling periodically every 10 min.

1. What is the time constant of the sensor in air?

2. What are the phase angle lag, the time lag, and the amplitude ratio of the sensor response with respect to the air furnace?

Solution

1. The time constant predicted by equation (13.43) is

2Btu / h ft F

464( ) 10.18( ) 101.8

46.4salt

air saltair

h

h

2. It further follows that

1

2 1/2

2 101.8tan ( ) 46.8

10 602 10 min

46.8 ( ) 1.3min360 deg 2

10.7

[1 (2 /10 101.8 / 60) ]

L

L

radt

rad

a

A

2/ min

10rad

The assumptions given under equation (13.43) are not always realistic. For example, at very low values of the film coefficient h~5-50Btu/h-ft2- and for very high values of ℉sensor-well conductivity k~200Btu/h-ft2- , thermal ℉resistance is essentially confined to the film and equation (13.43) is valid.

However, at very large values of h~1000Btu/h-ft2- and far very low values of k~10Btu/h-℉ft2- the sensor-well system contributes substantially to the ℉overall thermal resistance, and large departures from the idealization of equation(13.43) are to be expected.

Thus the Biot number compares the relative magnitude of the film resistance to the sensor resistance. Whenever the Biot number is less than 0.2, the idealized lumped model for the time constant is a good approximation, and conversely.

ohrBi

k (13.44)

The Biot number Bi of heat transfer analysis formalizes these observations, where

To make equation (13.43) more realistic, in the general case. an effective film coefficient , can be defined through the following steps. An effective thermal resistance

is first defined for the series convection-conduction case as

This can be interpreted in terms of the film coefficient as

or, multiplying through by the sensor surface area , as

effh

effR

eff film sensorR R R

1 1sensor

eff s s

Rh A hA

(13.45)

(13.46)

1 1s s

eff

R Ah h (13.47)

sA

Thus, the effective film coefficient can be given as

where the sensor thermal resistance can be given for the usual hollow cylindrical sensor by

1

(1/ )effs s

hh R A

(13.48)

ln( / )

2o i

s

D DR

kL (13.49)

and, using equation(13.40) for As, we have

Thus a more realistic expression for equation (13.43) can now be given in terms of equations ( 13.39 ) , ( 13.40 ) , ( 13.48 ) , and 13.50) as

Examples 2 and 3 will now be done over in terms of equation (13.51) to not the effect on the time constant of considering the effective film coefficient

ln( / )

2o i o

s s

D D DR A

k (13.50)

2 2

dim

( )

4 4o i o

oneo eff eff

c D D cD

D h h

(13.51)

Example 4

Redo Example 2 in terms of the more realistic equation (13.51).

Solution

As already defined by Example l ,

and

The Biot number of equation(13 . 44 ) for this case is

Since this is much greater than the 0.2 suggested, we must expect the lumped solution of Example 2 to be invalid.


2463.8Btu / h ft Fsalteffh

10.18salt s

1200 1/ 244.464

11.2ohr

Bik

2ln(1/ 0.26) 1/120.005011h ft F / Btu

2 11.2s sR A


Where we observe now that most of the thermal resistance is offered by the sensor itself, while the film resistance is negligible. By equation (13.51),

where we must now conclude that the sensor is actually slower to respond in steam than in salt. By equation (13.42),

21171Btu / h ft F

1/1200 0.005011steameffh

dim

463.8( 27.6s

171salt

steam saltone steameff

h

h

）＝10. 18（）＝

99%dim

4.6 4.6 27.6 127one

t s

Example 5

Redo part 1 of Example 3 in terms of the more realistic

equation (13.51).

Solution

As already determined by Example 1,

The Biot number for this case is

Since this is less than the suggested 0.2, we must expect the lumped solution of Example 3 to be approximately valid.

As already determined by Example 4,

2463.8Btu / h ft Fsalteffh 10.18salt s

46.4 1/ 240.173

11.2Bi


where we observe that in the case of low values of h, the film does indeed offer the most thermal resistance to heat transfer.

2137.6Btu / h ft F

1/ 46.4 0.005011aireffh

20.005011h ft F / Btus sR A

2137.6Btu / h ft F

1/ 46.4 0.005011aireffh


as compared with the idealized of equation ( 13 . 43 ) of 101.8 s.

A two-dimensional heat transfer analysis, with axial symmetry, provides a still more realistic approximation to the time constant. This is because it includes not only the finite metal resistance to heat transfer, but it includes axial heat transfer as well as the one-dimensional radial heat transfer.

dim

463.8( ) 10.18( ) 125.6

37.6salt

air salt effone air

hs

h

airt

This can be expressed most readily in terms of the Biot number Bi of equation(13.44) and the Fourier number Fo defined as

where αis the thermal diffusivity of the metal.

2 2o o

t ktFo

r cr

(13.52)

It is interesting to note in this connection that the exponent of e in equation (13. 10) is simply related to BiFo. Two-dimensional numerical solutions to an infinite solid cylinder have been presented in graphical charts by Schneider [11], one of which is given in Figure 13.4. These solutions can be represented, at the 63.2% response, by the empirical fits

For ,and by

for

2log 0.124 0.63(log ) 0.23(logFo Bi Bi ） (13.53)

0.1 4.0Bi

0.5Fo

Bi (13.54)

0.001 0.1Bi

Examples l , 2 , and 3 will be done once again in terms of equations ( 13.44 ) , ( 13.52 ) , 'and ( 13.53 ) to note the effect on the time constant of considering two-dimensional heat transfer . Example 6 Redo Examples l, 2, and 3 in terms of two-dimensional heat transfer.Solution The thermal diffusivity of the cylinder is

11.20.166

500 0.135

k

c

The Biot number for salt is

while we have previously determined that

Since all three Biot number are greater than 0.1, we can determined the Fourier number by equation (13.53), and hence the two-dimensional time constants by equation(13.52), as

463.8 1/ 241.725

11.2saltBi

4.464steamBi

0.173airBi

0.5492saltFo 2

dim

0.001740.5492 20.7

0.166o

salttwo

rFo s

0.3666steamFo dim

0.010482 0.3666 13.8steamtwo

s

3.0913airFo dim

0.010482 3.0913 116.6airtwo

s

The results from these six examples are summarized as follows:

We can conclude that

Fluid τideal τone-dim Τtwo-dim

Salt 10.2 － 20.7

Steam 3.9 27.6 13.8

Air 101.8 125.6 116.6

dim dimideal two one 13.55（）

With the following rationale: is the shortest because it assumes, that no resistance is offered to heat transfer by the

metal. is the longest because it include metal resistance but confines heat transfer to the radial direction.

is between these two, longer than because it includes metal resistance, shorter than , because it includes axial heat transfer.

Of course, is the recommended time constant because it most realistically approximates conditions in a sensor-well combination.

ideal

dimone

dimtwo ideal

dimone

dimtwo

Example 7

A copper temperature probe of di=0.0078 ft, do=0.005193 ft

and Di=0.05307 ft, Do=0.0833 ft is inserted in a copper

thermometer well of hf=241 BTU/h-ft2-℉. Using physical

properties from Table 13.1, find the sensor temperature after

100s if the system is subjected to a step change from 0 to 200

Solution

By equation (13. 19),


2 2 o578 0.1003 / 4(0.05193 0.0078 ) 0.120018Btu/ FsC

2 2 o578 0.1003 / 4(0.0833 0.05307 ) 0.187705Btu/ FwC




o(1/ 241 0.0833 3600 / 57.08088s- F/BtufR s h ）

o[ln(0.0833/ 0.06984) / 2 212] 3600 0.47628s- F/BtuwwR

o57.08088 0.47628 57.5572s- F/BtuoR





o[ln(0.0833/ 0.05307) / 2 212] 3600 1.21824s- F/BtuwsR

o[ln(0.05307 / 0.05193) / 2 0.02777] 3600 448.02s- F/BtuAR

o[ln(0.05193/ 0.03713) / 2 212] 3600 0.90648s- F/BtusR

o1.21824 448.02 0.90648 450.145s- F/BtuiR

TABLE 13.1 Physical Properties of Selected Materials Properties Copper Stainless AirThermal conductivity 212 11.2 0.0277 Specific heat capacity 0.1003 0.135 Density 578 500

Source: After Keyser[7].





0.120018 507.702 0.187705 57.5572 71.737a s

20.120018 0.187705 57.5572 450.145 583.679b s

11

12

(71.737 53.0236) /1167.3582 0.10687

(71.737 53.0236) /1167.3582 0.01603

r s

r s

0.01603 0.10687200[1 1.1765 0.1765 ]t tT e e

At t=100s equation (13.15) becomes

Note that the time constants, according to equation (13.28), are

Hence and the system can be represented by equation (13.19), namely,

o200(1 0.23682 00000.000004) 152.64 FT

1 29.357163 , 62.383032s s

2 1

200(1 0.23682)T

Example 8

For the same temperature-sensing system as in example 7, find the temperature error after 5 min if the environment is changing at are average rate of 0.6 /s.℉

Solution


By equation (13.31),9.357 62.383 71.74L s

o0.6 71.74 43 Ferror eT T T

13.6 MODIFYING CONSIDERATIONS

Many experimental and analytical studies have been made concerning factors that influence the time constants of temperature sensors in addition to the factors and h previously discussed. These include Mach number, size of temperature change, axial conduction, radiation, fluid turbulence, and installation.

Below a Mach number of 0.4, the time constant is hardly affected by the Mach number, and this effect can be minimized by basing physical properties on the local temperature and total pressure of the fluid.

, , ,c V A

The size of the temperature change affectsτbecause physical properties are not necessarily linear functions of temperature. Wormser[12] notes, a 25% variation inτfor a 400% change in the size of ΔT.

Axial conduction, for example, along bare thermocouple wires from the measuring junction to the supporting probe definitely affects the time constant. Under usual conditions, in which Tsupport , approaches Tjunction , in the steady state,

conduction effects cause an increase inτ.

The thermal linkage of junction and support by conduction causes an increase in the effective mass of the junction. Wormser[l2] indicates thatτcan increase by as much as 80% from this source at the lower mass velocities. He presents the equation

(13.56)

To calculate this effect where signifies the time constant for heat transfer by conduction and convection, τ,. is the time constant for convection alone. α is the thermal diffusivity of the wires (i.e. )and L is the distance from the support to the junction.

2

, 2(1 )

4c

k c c L

,k c

Example 9

The convective time constant of a Chromel-Alumel thermocouple that has 1/8 in of bare wire from junction to support is 0.5 s. What is the percentage increase in the time constant because of conduction effects?

Solution

From Table 13.2


Thus, percentage increase in

20.0077 0.01030.009in /s

2 2Ch AI

2

, 2

0.5 0.0090.5(1 0.86

4 (1/ 8)k c s

, 100 72%k c c

c

Documents

CHAPTER 13 Transient Temperature Measurement