Chapter 11
Temperature Measurement in
Moving Fluids
It must be obvious that a thermometer, placed in the
wind registers the temperature of the air, plus the greater
portion, but not the whole, of the temperature due to the vis
viva of its motion.
James Prescott Joule and Thomson (1860)
Thermodynamic temperatures are defined for thermal
states of statistical equilibrium only.
These are seldom encountered in practice; thus the usual
temperature concept must be modified. A common nonstatic
condition for which temperature measurement requires
special definitions is that in which a directed kinetic energy
of flow exists in a fluid.
Methods for the practical measurement of temperature in
real moving fluids using real temperature-sensing probes car,
be developed by malting certain necessary modifications of
the concepts of idealized fluids, temperature relations, and
probes.
11.1 IDEALIZED GAS
Because of its simplicity, an expression of the form:
pv=RT
(11.1)
has been used since 1834 as an idealized equation of state of
a gas, and may be accepted as a first approximation to the
equation of a real gas[see equations (2.10), (2.13), (2.15),
(3.4)] [1].
In this connection we note a most important
thermodynamic relationship that holds for all real gases and
has never been contradicted by experiment, namely,
(pv)0=RT
(11.2)
where the superscript0 refers to the zero pressure intercept
[see equation(3. 11 )].
By comparing equations (11.1) and (11.2), we may
observe some of the implications of using the idea equation
of state to represent a real gas [2], [3].
The expression pv = RT serves as an increasingly exact
relation for all real gases as the pressure approaches zero
along the: respective isotherm.
Since at these low pressures (large specific volumes) both
the size, of the molecules and the intermolecular forces are
negligible, the transport properties such as viscosity and
thermal conductivity which depend on molecular size and
interaction must likewise be considered negligible.
11.2 IDEALIZED GAS TEMPERATURESENSING PRORES
An idealized gas temperature-sensing probe is defined
arbitrarily at this point as one that will completely stagnate a
moping gas continuum locally (i.e., ideal geometry) and is
isolated, in terms of heat transfer, from all its surroundings
(i.e., adiabatic).
11.3 IDEALIZED GAS TEMPERATURE RELATIONS
Consider several cases (Figure 11.1a), each involving;
the following: a system with fixed boundaries across which no
mechanical work car heat is transferred, an idealized gas
continuum (pv/T = a constant, cp= a constant), and an
idealized gas temperature-sensing probe (ideal geometry,
adiabatic.
1. For the case in which both the probe and the gas are at rest
with respect to the system boundaries, !he probe will
indicate the gas temperature.
This temperature may be visualized as a measure of the
average random translational kinetic energy of the
continuum molecules (but see, for example, (4)).
2. For the case in which bath the probe and the gas am in
identical motion with respect to the system boundaries,
the probe again will indicate the gas temperature.
However, this local temperature must be distinguished
carefully from the temperature of case 1. The gas
temperature of case 2 is lower than that of case I by an
amount equivalent to that part of the random thermal
energy which now appears in the form of directed kinetic
energy of the gas continuum.
For the case in which the gas is in direct motion with
respect to both the probe and the system boundaries, the
probe will indicate not only the gas temperature (as in
case 2) but, in addition, will indicate a temperature
equivalent to the directed kinetic energy of motion of the
gas continuum.
This latter part of the net indicated temperature is
obtained by stagnating the gas continuum locally. The
probe thus reconverts the directed kinetic energy back to
random thermal effects.
Note that this net indicated temperature is identical to
the gas temperature of case 1.
Three separate temperatures must be distinguished in the
aforementioned three cases (Figure 11.1b)
Static temperature T This is the actual temperature of
the gas at all times (in motion or at rest). It has been
considered as a measure of the average random
translational kinetic energy of the molecules. The static
temperature will be sensed by an adiabatic probe in
thermal equilibrium and at rest with respect to the gas.
2. Dynamic temperature Tb The thermal equivalent of
the directed kinetic energy of the gas continuum is known
as the dynamic temperature.
3. Total temperature Tt This temperature is made up of
the static temperature plus the dynamic temperature of the
gas. The total temperature will be sensed by an idealized
probe, at rest with respect to the system boundaries, when
it stagnates an idealized gas.
1.
That these three temperatures are related in the manner
stated may be seen by applying the steady-flow general
energy equation to a situation in which the idealized gas
continuum is in direct motion with respect to the system
boundaries [5]:
VdV
(11.3)
 Q   W  du  d ( pv) 
gc
or introducing the enthalpy definition h = u + pv, equation
(11.3) becomes:
VdV
(11.4)
 Q   W  dh 
gc
where
δQ = heat transferred across system boundaries
δW =mechapical work transferred across system boundaries
dh = enthalpy change between two thermodynamic states
within system: in general,
(11.5)
v
dh  c p dT  [v  (
T
) p T ]dp
and for the ideal gas,
dh  c p dT [since(v / T ) p T  v]
All are on a per pound mass basis. In the absence of heat
transfer and mechanical work across the system boundaries,
equation (11.4) may be integrated as
V12  V22
C p (T2  T1 ) 
2 Jgc
where J is the mechanical equivalent of heat
(778 ft-lbf/Btu), Thus the general energy relation indicates
that a change in the directed kinetic energy of the gas
continuum is always accompanied by a change in the static
temperature of the gas [b]. Furthermore, if the subscript 2 in
equation (13.6) refers to the stagnant condition, we note that
the temperature in a stagnant gas (T2 = Tt) is always greater
than the temperature in a moving gas (T1 = T) by an amount
equivalent to the directed kinetic energy of the gas
continuum, that is,
V2
T1  T 
 T  Tc
2 JgcC p
and thus the dynamic temperature is particularized as
V2/2JgcCp.
11.4 IDEALIZED LIQUIDS
In the absence of heat transfer across the boundaries of a
system in which a frictionless (in viscid), incompressible
fluid flows, the internal enemy remains constant throughout
any process. This may be seen by applying the first law of
thermodynamics to such a situation.
 Q   F  du  pdv
(11.7)
Consequently the temperature of an idealized liquid must
also remain constant throughout any process, This follows
from the basic assertion that the internal energy, generally
u = f(T, v); is a function of temperature only for an
incompressible fluid[7].
Thus it is incorrect to speak of the various temperatures
(static, dynamic, total) of an in viscid, incompressible fluid;
the liquid temperature T, is the only one of any significance.
No change in the directed kinetic energy of a liquid
continuum will effect a change in this liquid temperature,
and any adiabatic probe will indicate this liquid temperature.
In summary, if the realized equation of state (pv=RT)
well represents the thermodynamic quantities (p, v, T) of a
particular gas, and if the specific heat capacity cp of the gas
nay be considered constant over the temperature range
involved, the total temperature will be constant in any
adiabatic, workless change in the thermodynamic state of
the gas.
Furthermore, any adiabatic probe that completely stagnates
this gas locally will indicate the total temperature, that is,
Tpi= Tt= T+T0, (idealized gas and idealized probe), where
Tpi is the equilibrium temperature sensed by a stationary,
ideal-geometry, adiabatic probe.
If a liquid can be considered in viscid and incompressible,
its temperature will remain constant in any adiabatic,
workless change in the thermodynamic state of the liquid.
Furthermore, any adiabatic probe will indicate this liquid
temperature.
11.5 REAL-GAS EFFECTS
Idealized relations are useful in that they allow us to draw
certain broad conclusions with a minimum of effort, but in
the measurement of temperature in moving fluids we must
also consider many perturbation effects, especially those
arising because of a deviation of the fluid characteristics
from assumed idealized relations.
Departures from ideal conditions in gases are immediately
encountered, for we do not always test at near-zero
pressures.
Thus in general, both the size of the molecules and the
intermolecular forces become important.
Along with these realities, the associated transport
properties (i.e., the viscosity and the thermal conductivity)
must also be considered [8]-[10].
A second approximation to the true equation of state of a
real gas was given in 1873 by Van der Waals as
a
( p  2 )(V  b)  RT
V
(11.9)
where the term a/v2 was introduced to account for
intermolecular forces and tine term b for the finite size of
the molecules [11].
The actual equation of state of a real gas is naturally more
complex than any of its approximations. Yet even with the
Van der Waals gas, and even with a constant specific heat
capacity cp, the static temperature no longer remains
constant in an isenthalpic (i.e., constant-enthalpy), workless
change of state.
This phenomenon, wherein the static temperature changes
in a constant enthalpy (i.e., throttling) expansion of any real
fluid, is known as the Joule-Thomson effect [12] (Figure
11.2).
Furthermore, at a print we cannot, in general, realize the
total temperature, although the real gas (or even the van der
Waals gas) is completely stagnated locally by an isolated
temperature-sensing probe.
This is because of the combined effects of aerodynamic
stagnation, viscosity, and thermal conductivity, which set up
temperature and velocity gradients in the fluid boundary layers
surrounding the probe [13].
The consequent rise in temperature of the inner fluid layers
that results from a combination of viscous shear work on the
fluid particles and the impact conversion of directed kinetic
energy to thermal effects is necessarily accompanied by a
heat transfer through the gas, away from the adiabatic probe.
These opposing effects tend to upset the simple picture of
total temperature recovery previous given.
The usual thermodynamic simplification introduced to
allow the continued definition of total temperature in this
situation is the assumption of an isentropic process of
deceleration, signifying a reversible, adiabatic stagnation.
However, by whatever designation, the implications are
unrealistic in practical monnometry.
The term adiabatic is meant to indicate absence of heat
transfer, both to and from the probe and to and from the
fluid, but we have seen that a temperature gradient must
exist in the gas.
The term reversible is reserved for quasistatic (slow) or
“frictionless” processes, but the stagnation of a moving gas
is not a quasistatic process.
Furthermore, the viscous shear forces that are present are
synonymous with friction forces in a fluid.
Thus boundary layer effects, associated with viscosity and
thermal conductivity, conflict with the isenthalpic assumption,
and the total temperature simply cannot be realized by a
momentum in a system.
Note that viscosity and conductivity are simply efforts on
the main flow, and their relative importance is indicated by the
Prandtl number.
The Prandtl number is a ratio of the fluid properties
governing transport of momentum by viscous effects(because
of a velocity gradient) to the fluid properties governing
transport of heat by thermal diffusion(due to a temperature
gradient), that is,
cp 
Kinetic viscosity
Pr 

(11.10)
thermal diffusivity
k
By replacing the isentropic assumption with the assumption
of a Prandtl number of 1, we need not discount the effects of
conductivity or viscosity both may be actively present, but
they will be counterbalancing effects.
This is the same requirement as for the Reynolds analogy,
where heat and momentum are transported in the same manner
in a fluid.
Thus we have avoided the mental stumbling block that an
isentropic still will not be sensed by an idealized probe, at rest
with respect to system boundaries, when it stagnation a real
gas, even if the Prandtl number is 1.
11.6 REAL-LIQUID EFFECTS
With liquids, as with gases, we never encounter a real
fluid of zero viscosity or zero thermal conductivity. We
must therefore modify our assertion that was based on an
idealized liquid to the effect that there is only one
significant temperature Tl in a liquid.
The first consideration is the Joule-Thomson effect. For
liquids the Joule-Thomson coefficient is generally negative
(true for water below 450℉).
As pressure drops isenthalpically, therefore, temperature
rises(Figure 11.2), in contrast to the case for most gases.
This means that the liquid temperature will not remain
constant in any throttling change of state of a liquid.
Furthermore, for reasons previously given concerning the
interplay between viscous shear work and heat transfer in
the liquid boundary layers surrounding any probe immersed
in any real liquid, we cannot in general realized the liquid
temperature.
Even when the Prandtl number of the real liquid is 1, the
adiabatic probe will not sense the idealized liquid
temperature Tl.
In summary, the static temperature will not be constant in
an isenthalpic, workless change in the thermodynamic state
of any real fluid.
This is explained by the Joule-Thomson effect.
Furthermore, even if the Prandtl number of a gas is 1, the
adiabatic probe that completely stagnates such a gas locally
will still fail in indicate the local total temperature.
An adiabatic probe immersed in a real liquid, even if its
Prandtl number is 1, will also fail to indicate the local
temperature.
11 .7 THE RECOVERY FACTOR
The fluids we test are not always characterized by a
Prandtl number of l (e.g. ,the Prandtl number of air varies
between 0.65 and 0.70, the Prandtl number of steam varies
between l and 2, and the Prandtl number of water varies
between 1 and 13: see Table 11.1).
Therefore we must again alter the simplified picture of
temperature recovery. Total temperature in a gas and liquid
temperature in a liquid are seen more and more in their true
light, as idealized concepts or devices rather than as
physically measurable quantities.
Joule and Thomson [12], as early as 1860, noted: “. . . it
must be obvious that a thermometer placed in the wind
registers the temperature of the air, plus the greater portion,
but not the whole, of the vis viva of its motion….”
Of course they were close to the idea of a recovery factor,
which we now introduce to account for deviations from
previously stated ideal conditions in real fluids.
11.7.1 At a Stagnation Point
Even when we limit our attention to the fluid stagnation
point, the total temperature of a gas and the liquid
temperature of a liquid ran never he measured directly.
This assertion can be substantiated by the following
development. For an isentropic workless process, equation
(11.4) yields
dhs
VdV
 
gc
(11.11)
and equation (11.8) yields
0  (du  pdv)s
(11.12)
Where the subscript s signifies an isentropic process. By
equation (11.12) the enthalpy change (generally,
dh= du + pdv+vdp) also can be expressed as
(11.3)
dhs  vdps
`
When the general definition of enthalpy change, equation
(11.5), is combined with equation (11.13), this results in
dTs  T (
dps
V
)p
T
cp
(11.14)
which, when combined with equations (11.11) and (11.13),
yields
T V
VdV
dTs  [ ( ) p ]
V T
Jgc c p
(11.15)
On integrating equation (11.15) between the stagnation
and free-stream conditions, we obtain
(11.16)
Tstagnation  Tfreestream  S  Tdynamic
where the stagnation factor
S 
T V
(
)p
v T
(11.17)
is often considered a constant, or its mean value may be used
[14].
The factor S is shown graphically; in Figure 11.3 for air,
water, and steam.
It is clear from a comparison of equations (11.7) and
(11.16) that Tstagnation does not equal Ttotal for real gases.
However, for the idealized gas, since (V / T )  R / p  V / T
the stagnation factor does equal unity, and hence
(Tstagnation) ideal gas does reduce to Tt, as already indicated.
Since equation (11.16) applies equally well to liquids,
gases, and vapors, it is likewise clear that Tstagnation does not
equal the liquid temperature Tl which, of course, is the same
as the free-stream temperature. However, for the idealized
liquid, since (V / T ) p  0 , the stagnation factor equals
zero, and hence Tstagnation does reduce to Tl, as already
indicated.
The stagnation factor S, as given by equation (11.17) and
as used in (11. 16). we now define as die recovery factor at
a fluid stagnation point. Any probe that is designed so that
the temperature-sensing portion is located at an isentropic
stagnation point will be characterized try a recovery factor S
and swill yield the stagnation temperature (Figure 11.4),
Example 1.
A temperature stagnation probe, as shown its Figure 11.4a,
is exposed to atmospheric air flow of 500 ft/s, where
cp = 0.24 and the static temperature is 200℉.
Find the deviation between the probe stagnation
temperature and the thermodynamic total temperature.
Solution. By equation (11.7),
V2
Tt  T 
2 Jgc c p
where
T = static temperature = free stream temperature
=200 + 460=660°R, and
Tc= dynamic temperature = 5002/2×778×32.174×0.24
=250,000/12,015 = 20.8°R.
Hence,
Tstag  T  STc
By equation (11.16),
Ti  total temperature  660  20.8  680.8R
where S=stagnation recovery factor =1 from Figure 11.3.
Hence,
Tstag  660  (1 20.8)  680.8R
In this example, the difference between the measured
Tstagnation and the idealized Ttotal is 0℉.
Example 2.
Tt  T  Tc
A turbine blade, arranged as in Figure 11.4b, is exposed to
a steam flow of 500 ft/s, where cp =0.58, the static
temperature is 400℉, and the static pressure is 150 psia.
Find the difference between the air foil stagnation
temperature and the thermodynamic total temperature.
Solution.
By equation (11.7),
Tt  T  Tc
where T=400+460=860°R and
T=5002/2×778×32.174×0.58=250,000/29,106.5=8.59°R.
Hence,
Tt  860  8.6  868.6R
By equation (11.6),
Tstag  T  STc
where S=1.28 from Figure 11.3, hence,
Tstag  860  1.28  8.59  871.0R
In this example, the difference between the measured
Tstagnation and the idealized Ttotal is 2.4℉.
Example 3.
For the stagnation probe Shown in Figure 11.4, what is the
expected difference between the stagnation temperature and
the liquid temperature when water flows at a velocity of
20ft/s at a temperature of 200℉?
Solution. By equation (11.6),
Tstag  T  STc
where T = 200 +460=660°R and S=0.26 from figure 11.3,
and Tv=202/2×778×32.174×1=400/50,062.7=0.008°R.
Hence
Tstag  660  0.26  0.008  660.002R
and the difference between the measured Tstagnation and the
idealized liquid temperature is 0.002℉.
11.7.2 Over a Flat Plate
Much work has been done on the viscous frictional
recovery factor that characterizes the flow of gas over a flat
plate. Patterned directly after equation (11.16), we have
Tadiabatic  T freestream  r  Tdynamic
flatplate
However, although the recovery factor is expressed as
some percentage of the dynamic temperature, there is no
implication that the recovery factor indicates only the
degree of conversion of directed kinetic energy to thermal
effects, or that a recovery of l is the maximum attainable.
For example, for gases having Prandtl numbers below 1,
thermal-conduction effects overshadow viscous effects, and
the adiabatic flat plate will sense a temperature less than the
total temperature (i.e., r<1).
Conversely, for Prandtl numbers above 1, the flat plate will
come to equilibrium at a temperature greater than the total
temperature (i.e., r > 1). The effects of various recovery
factors are shown in Figure 11.5.
For liquids, although not commonly discussed, the recovery
factor r might still be; applied, where r accounts not at all for
the degree of conversion of directed kinetic energy to thermal
effects, but for the net thermal effect resulting from the
viscous shear work and heat transfer in the boundary layers.
Several particular recovery factors have been distinguished
for gases [15].Very little information on recovery factors for
liquids flowing over flat plates appears in the literature.
The frictional recovery factor has been found to be very
nearly independent of the Mach and Reynolds numbers.
Pohlhausen, in 1921, found that r=f(Pr)=0.844 for flat plates
in a laminar flow of air. Emmons and Brainerd[16] in 1941,
and Squire[17] in 1942, found that
rla min ar  Pr1/ 2
For flat plates in a laminar flow of fluid having a Prandtl
number between 0.5 and 2, and for Mach numbers ranging
between 0 and 10. For air, Pr1/2 yields 0.846 for the recovery
factor.
Example 20.
Before a test, find if 100 measurements of X lead to an
acceptable precision in a result of 0.1%. The sensitivity factor
is 0.5, the mean value of X is 50, and the 95% confidence
interval of measurements is estimated to be 1.


t99.95% S X
1.99(1%)
 


 0.199%
 
N
100
 X exp
S CI

 1%
X 2X
R R
0.1%
 

 0.2%
  
0.5
 X acc sensitivity
Since  acc  exp we conclude that 100 measurements are
adequate for the required precision in the result.
Example 21.
Before a test, find if 20 measurements of X lead to an
acceptable precision in a result of 0.1%. The sensitivity factor
is 0.5, the mean value of X is 1000, and the 95% range of
measurements is estimated at 20.
 
w
 20×100 
    N    0.126 
＝0.25％
 X exp
X
 1000 
R R
 
 0.2%
  
 X  acc sensitivity
Since  acc  exp we conclude that 20 measurements are
not adequate to yield the required precision in the result.
In the recent literature[10],[11],[15],the suggestion has been
made to take a small number of measurements and perform a
quick analysis, to determine whether the number of
measurements taken was adequate or how many more
measurements are still required to satisfy test objectives. This
type of analysis is based on (10.15), solved for N as
N total
 t N1 1, p S1 


  acc 
2
(10.46)
Briefly, the procedure is as follow:
From a small sample of size N1, obtain S1 according to
(10.17).
Establish an acceptable confidence interval of the mean.
Estimate NT via (10.46).
then the size of the remaining number of measurements
required is simply N2= NT －N1。
This approach is sometimes called the Stein method.
Example 22.
From 50 measurements it is determined that S1=160.how
many more measurement should be taken to ensure an
acceptable δ of 302 S1=160 based on N1=50
 acc . =30
by(10.46),
NT=(2.01*160/30)2=115
N2=115－50=65 additional measurements required.
Example 23.
For ten initial measurements, the range is determined to be
4. if the mean value is 30 and the sensitivity factor is 0.5,find
the number of additional measurements required to ensured a
△R/R of 0.1% at the 95% confidence level.
R R
 
 0.2%
  
 X  acc sensitivity
X ×0.2%
 0.06
100
w
4
S1 

 1.3
 d 2 100 3.078
 acc 
2.262 1.3 
NT 
 49
0.06
N2=49-10=39 additional measurements required.
Critique
The main fault of these approaches (i.e., Before Test and
During Test) is that they are essentially limited to time
variations only. But variations with space and installation may
far outweigh random variations with time. In such cases, the
numbers predicted by the Before and During approaches may
be of academic interest only. For example , we may be
predicting the need for 100 measurements to reduce
 t to 0.1% when at the same time, all unexamined，
 s and / or
 i are on the order of 2%.
After a Test
One should, of course, use (10. 26 and 10. 34) with the
experimentally determined confidence intervals for the
random errors, and / or the estimated uncertainty intervals for
the systematic errors, to decide if the number of measurements
taken were adequate. Such procedures are illustrated by
Examples 9一19.