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 ) .
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
hA  Tc  T

Mc
dT
dt
(13.1)
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 V , 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
dt
dT

 Vc hA Tc  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,
Vc thermalcapaci tan ceofsensor


hA
thermalconduc tan ceoffilm
(13.3)
Since thermal conductance is the reciprocal of thermal
resistance, equation(13.3) also indicates that
  RthCth ,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]
T  Ce
t 
1
 e

t 

t
0
Te et  dt
(13.4)
where C is a constant of integration, which is determined by
inserting the proper boundary conditions .

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
T t Insertion of these boundary values in equation ( 1 3 . 4 ) yields
T  T1e
t 
t
1
 e

t 

t
t
T2  e dt  R  tet  dt
t
0
0

(13.15)
t
t


0 te dt  t e 0  0 e dt
t
t
t
Evaluating equation (1 3. 5), we have
T  T1et   T2  T2et   Rt  R  R et 
which, expressed in terms of the temperature difference ,becomes
Tc  T  R  R et   T2  T1  et 
(13.7)

Where T   the terms involving et  approach zero,
and equation (13.4 )-(13.7) reduce to
(13.8)
Tc  T  R
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).
13.2.2 Step Change
Under this condition at t  0, T  T1  C,although in general
Tc  T2
Insertion of these boundary values in equation (13.4)
yields
T  T1et   T2  T2et 
(13.9)
which, expressed in terms of the temperature difference,
becomes
Tc  T  Tc  T1  et 
(13.10)
According to equation (13.10), the time constant for a step
change can be defined as follows:
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 t  0, T  T1  C , although in general
Tc  T2 sin t 
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
T  T1 
T2
sin t  tan
2 12
1    


1
  
T2
1   
t 
e
2
(13.11)
When t   the last term above approaches zero and the
sensor response will lag the environment by the phase angle
L  tan1  
(13.12)
which in time units corresponds to a lag of
tL 
tan 1  
(13.13)

Whereas the ratio of the sensor amplitude to that of the
environment is given by
Tmax
1

12
T2
1   2 


(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
 Vc
hA
Thus it follows that    step   periodic ,
whenever all
conditions stated tender the three separate sections on the
forcing functions are met.
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 firstorder systems, namely, equations (13.8) and (13.10) must be
used.
13.3.1 Step Change
A second-order thermal system, as a thermometer wellsensor system, and its equivalent circuit are given in Figure
13.3. An appropriate solution describing the temperaturetime response of the thermal system of Figure 13.3 to a step
change in temperature can be given [4]-[7] as
  r1   r2t  r2   r2t 
T  T 1  
e  
e 
 r  r2 
  r1  r2 

(13.16)
where T is the sensor temperature at any time t,  T is the
step change in temperature, with the initial temperature
normalized at zero, and where
a  a 2  4b
r1 , r2 
2b
and represent the roots of the second-order quadratic, with
a  Cs  Rv  Rs  R0  Cs R0
(13.17)
b  CwCs Rw
(13.18)
The thermal resistances and capacities, as shown in Figure
13.3, can be defined mathematically as follows:
The sensor capacity is
Cs 
 s c ps (d o 2  di 2 )
4
(13.19)
The well capacity is
Cw 
 wc pw ( Do 2  Di 2 )
4
(13.20)
The sum of the well-side resistances is
where
1
Rf 
h f Do
（13.22）
Ro  R f  Rww
(13.21)
and represents the heat transfer film resistance outside the well,
and
Rww
ln(Do / DM )

2k w
（13 .23）
represents the well-to-capacity resistance.
The sum of the sensor-side resistances is
R f  Rws  RA  Rs
where
Rws 
（13.24）
ln(Do / Di )
2k w
(13.25)
represents the well-to-sensor resistance, and
RA 
ln(Di / d o )
2k A
(13.26)
represents the air film between well and sensor, and
Rs 
ln(d o / d M )
2k s
represents the sensor-to-capacity resistance.
(13.27)
The geometric definitions are as follows:
Do = outer diameter of well
Di = inner diameter of well
1
D 
( D  D ) = mean diameter of well
2
d o =outer diameter of sensor
di =inner diameter of sensor
1
d 
(d  d )
= mean diameter of sensor
2
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,
2
M
o
2
i
2
M
o
2
i
r1 
1
1
, r2 
1
2
(13.28)
When one time constant dominates the other, that is,
when equation (13.15) can he reduced to the form of a firstorder solution, namely,

r1
T  T [1  (
)e  t /  2 ]
r1  r2
(13.29)
In similar form, equation (13.10), the first-order solution
can be written
T  T[1  et / ]

(13.10' )
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.
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 temperaturesensing 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
Tc  T  RL
as compared to the first-order equation (13.8).
(13.31)
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
Tc
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)]
Nu  1 (Re, Pr)
(13.32)
where Nu = Nusselt number= hD / k
Re = Reynolds number＝
DG / 
Pr = Prandtl number= c p  / k
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.
The step change definition of the time constant provides the
usual method of determining  0 . 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.
At least four 0 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.
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
hD  2 ( DG)m
(13.33)
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
G  3 N b
(13.34)
Hence equation (13.33) also can be given as
hD  4 Dm N e
(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
(hD)sait  3.11( DN )0.6
(13.36)
where the film coefficient h is in Btu / h  ft 2  F ,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.
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
（hD)salt  3.00(DN )0.665
(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
hD  5 D
m
(13.37)
For a stirred-oil bath at about 4000F, Murdock's data can
be expressed for constant stirrer speed as
(hD)oil  17.5D0.7
(13.38)
Again, this is one particularization of equation (13.35). In
another bath, at a different but unrecorded stirrer speed,
there has resulted
(hD)oil  32D0.51
(13.38')
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 Btu / h  ft 2  F while equation
(13.38') yields h=32 Btu / h  ft 2  F
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
Vs 

4
( Do2  Di2 ) L 
 Do2 L
4
(13.39)
Furthermore, for radial heat transfer the surface area of
the sensor is
(13.40)
As   Do L
Hence
c 2
hD  ( ) D
4
(13.41)
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).
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:
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
13.5 APPLYING THE TIME CONSTANT
The ideal model for the time constant of equation (13.3),
that is,
h 
Vc
A
 cons tan t
has four assumptions built into it.
(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 stainlesssteel 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),
hDo  3.11(800 1/12)0.6  38.646Btu/h-ft- F
hsalt  38.646 / (1/12)  463.8Btu/h-ft- F
though this hsalt 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)
 salt 
 cD
2
o
4hDo
 10.92 s

1 2
)
12  0.003032h
4  38.646
500  0.135  (
More precisely, for the hollow cylinder, following equation
(13. 39)
 salt 
 c( Do2  Di2 )
4hDo
500  0.135  (1  0.0676)

4  463.8 12
 0.002827h  10.18s
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 1200Btu/h-ft 2 - F 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
hsalt
463.8
 steam   salt (
)  10.18(
)  3.9s
hsteam
1200
When we would conclude that the sensor is faster to respond
in steam than in salt.
By equation (13.42),
t99％  4.6  4.6  3.9  17.9s
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 Btu / h  ft 2  F 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
 air
hsalt
464
  salt (
)  10.18(
)  101.8
hair
46.4
2. It further follows that
2

rad / min
10
2 101.8
 L  tan ( 
)  46.8
10
60
2 rad 10 min
t L  46.8  (
)
 1.3min
360 deg 2 rad
a
1

 0.7
2 1/2
A [1  (2 /10 101.8 / 60) ]
1
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/hft2-℉ 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.
The Biot number Bi of heat transfer analysis formalizes
these observations, where
hro
Bi 
k
(13.44)
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.
To make equation (13.43) more realistic, in the general
case. an effective film coefficient heff , can be defined
through the following steps. An effective thermal resistance
is first defined for the series convection-conduction
case as
Reff  Rfilm  Rsensor
(13.45)
Reff
This can be interpreted in terms of the film coefficient as
1
1

 Rsensor
heff As hAs
(13.46)
or, multiplying through by the sensor surface area As , as
1
1
  Rs As
heff h
(13.47)
Thus, the effective film coefficient can be given as
1
heff 
(1/ h)  Rs As
(13.48)
where the sensor thermal resistance can be given for the
usual hollow cylindrical sensor by
ln( Do / Di )
Rs 
2 kL
(13.49)
and, using equation(13.40) for As, we have
ln( Do / Di ) Do
Rs As 
2k
(13.50)
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
 onedim 
 c( Do2  Di2 )
4Do heff

cDo
(13.51)
4heff
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
Example 4
Redo Example 2 in terms of the more realistic equation
(13.51).
Solution
2
h

463.8Btu
/
h

ft
 F
As already defined by Example l , salteff
and  salt  10.18s
The Biot number of equation(13 . 44 ) for this case is
hro 1200 1/ 24
Bi 

 4.464
k
11.2
Since this is much greater than the 0.2 suggested, we
must expect the lumped solution of Example 2 to be invalid.
By equation (13.50),
ln(1/ 0.26) 1/12
Rs As 
 0.005011h  ft 2  F / Btu
2 11.2
By equation (13.48),
hsteameff
1

 171Btu / h  ft 2  F
1/1200  0.005011
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),
 steam
one dim
  salt (
hsalt
hsteameff
463.8
）＝10. 18（
）＝27.6s
171
where we must now conclude that the sensor is actually
slower to respond in steam than in salt. By equation (13.42),
t99%
one dim
 4.6  4.6  27.6  127s
Example 5
Redo part 1 of Example 3 in terms of the more realistic
equation (13.51).
Solution
As already determined by Example 1,
hsalteff  463.8Btu / h  ft 2  F
 salt  10.18s
The Biot number for this case is
46.4 1/ 24
Bi 
 0.173
11.2
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,
Rs As  0.005011h  ft 2  F / Btu
haireff
1

 37.6Btu / h  ft 2  F
1/ 46.4  0.005011
By equation (13.48),
haireff
1

 37.6Btu / h  ft 2  F
1/ 46.4  0.005011
where we observe that in the case of low values of h, the
film does indeed offer the most thermal resistance to heat
transfer.
By equation (13.51),
 air
one dim
hsalt
463.8
  salt (
)eff  10.18(
)  125.6s
hair
37.6
as compared with the idealized tair 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.
This can be expressed most readily in terms of the Biot
number Bi of equation(13.44) and the Fourier number Fo
defined as
t
kt
Fo  2 
(13.52)
2
ro
 cro
where αis the thermal diffusivity of the metal.
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
2
log Fo  0.124  0.63(log Bi)  0.23(log Bi）
For
0.1  Bi  4.0 ,and by
0.5
Fo 
Bi

for
(13.53)
0.001  Bi  0.1
(13.54)
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

k
11.2

 0.166
 c 500  0.135

The Biot number for salt is
Bisalt
463.8 1/ 24

 1.725
11.2
while we have previously determined that Bisteam  4.464
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 Biair  0.173
equation(13.52), as
Fosalt
ro2
0.00174
 Fo 
 0.5492  20.7s
salt
 0.5492  two

0.166
dim
Fosteam  0.3666  steam  0.010482  0.3666  13.8s
two dim
 0.010482  3.0913  116.6s
air
Foair  3.0913  two
dim
The results from these six examples are summarized as
follows: 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
 ideal   twodim   onedim
We can conclude that
（13.55）
With the following rationale:  ideal is the shortest because
it assumes, that no resistance is offered to heat transfer by the
metal.  onedim is the longest because it include metal
resistance but confines heat transfer to the radial direction.
 twodimis between these two, longer than  ideal because it
includes metal resistance, shorter than  onedim , because it
includes axial heat transfer.
Of course,  twodim is the recommended time constant
because it most realistically approximates conditions in a
sensor-well combination.
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),
Cs  578  0.1003  / 4(0.051932  0.00782 )  0.120018Btu/ o F
By equation (13. 20),
Cw  578  0.1003  / 4(0.08332  0.053072 )  0.187705Btu/ oF
By equation (13. 22),
Rf  (1/   241 0.0833） 3600s / h  57.08088s-oF/Btu
By equation (13. 23),
Rww  [ln(0.0833/ 0.06984) / 2  212]  3600  0.47628s-oF/Btu
By equation (13. 21),
Ro  57.08088  0.47628  57.5572s-oF/Btu
By equation (13. 25),
Rws  [ln(0.0833/ 0.05307) / 2  212] 3600  1.21824s-o F/Btu
By equation (13. 26),
RA  [ln(0.05307 / 0.05193) / 2  0.02777] 3600  448.02s-o F/Btu
By equation (13. 27),
Rs  [ln(0.05193/ 0.03713) / 2  212]  3600  0.90648s-oF/Btu
By equation (13. 24),
Ri  1.21824  448.02  0.90648  450.145s-oF/Btu
TABLE 13.1 Physical Properties of Selected Materials
Properties
Copper
Stainless
Air
Thermal conductivity
212
11.2
0.0277
Specific heat capacity
0.1003
0.135
Density
578
500
Source: After Keyser[7].
By equation (13. 17),
a  0.120018  507.702  0.187705  57.5572  71.737s
By equation (13. 18),
b  0.120018  0.187705  57.5572  450.145  583.679s 2
By equation (13. 16),
r1  (71.737  53.0236) /1167.3582  0.10687 s 1
r2  (71.737  53.0236) /1167.3582  0.01603s 1
By equation (13. 15),
T  200[1 1.1765e0.01603t  0.1765e0.10687t ]
At t=100s equation (13.15) becomes
1  9.357163s, 2  62.383032s
Note that the time constants, according to equation (13.28), are
T  200(1  0.23682  00000.000004)  152.64 F
o
Hence  2  1 and the system can be represented by equation
(13.19), namely,
T  200(1  0.23682)
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 (13.30),
L  9.357  62.383  71.74 s
By equation (13.31),
Terror  Te  T  0.6  71.74  43o F
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  , c, V , A
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.
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
 k ,c   c (1 
 2 c
2
4L
)
(13.56)
To calculate this effect where  k ,c 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.
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

By equation (13.56),
 Ch   AI
2
0.0077  0.0103

 0.009in 2 /s
2
 k ,c  0.5(1 
 2  0.5  0.009
4  (1/ 8)
Thus, percentage increase in
 k ,c   c

100  72%
c
2
 0.86s