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 TEMPERATURE-SENSING 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)

1. 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.

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]:

(11.3)

or introducing the enthalpy definition h = u + pv, equation (11.3) becomes:

(11.4)

( )c

VdVQ W du d pv

g

c

VdVQ W dh

g

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)

and for the ideal gas,

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

[ ( ) ]p p

vdh c dT v T dp

T

[since( / ) ]p pdh c dT v T T v

2 21 2

2 1( )2p

c

V VC T T

Jg

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,

and thus the dynamic temperature is particularized as V2/2JgcCp.

2

1 2 cc p

VT T T T

Jg C

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.

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].

Q F du pdv (11.7)

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

where the term a/v2 was introduced to account for intermolecular forces and tine term b for the finite size of the molecules [11].

2( )( )

ap V b RT

V (11.9)

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,

Kinetic viscosityPr

thermal diffusivitypc

k

(11.10)

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

(11.11)

and equation (11.8) yields

(11.12)

sc

VdVdh

g

0 ( )sdu pdv

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

`

When the general definition of enthalpy change, equation (11.5), is combined with equation (11.13), this results in

(11.14)

which, when combined with equations (11.11) and (11.13), yields

(11.15)

s sdh vdp

( ) ss p

p

dpVdT T

T c

[ ( ) ]s pc p

T V VdVdT

V T Jg c

(11.3)

On integrating equation (11.15) between the stagnation and free-stream conditions, we obtain

where the stagnation factor

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.

stagnation freestream dynamicT T S T

( ) p

T VS

v T

(11.16)

(11.17)

However, for the idealized gas, since

the stagnation factor does equal unity, and hence

(Tstagnation) ideal gas does reduce to Tt, as already indicated.

( / ) / /V T R p V T

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 , 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),

( / ) 0pV T

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),

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.

2

2tc p

VT T

Jg c

Hence,

By equation (11.16),

where S=stagnation recovery factor =1 from Figure 11.3.

Hence,

In this example, the difference between the measured Tstagnation and the idealized Ttotal is 0 .℉

660 20.8 680.8iT total temperature R

stag cT T ST

660 (1 20.8) 680.8stagT R

Example 2.

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),

where T=400+460=860°R and

T=5002/2×778×32.174×0.58=250,000/29,106.5=8.59°R. Hence,

t cT T T

t cT T T

860 8.6 868.6tT R

By equation (11.6),

where S=1.28 from Figure 11.3, hence,

In this example, the difference between the measured Tstagnation and the idealized Ttotal is 2.4 .℉

stag cT T ST

860 1.28 8.59 871.0stagT R

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),

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

and the difference between the measured Tstagnation and the

idealized liquid temperature is 0.002 . ℉

stag cT T ST

660 0.26 0.008 660.002stagT R

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

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.

flatplateadiabatic freestream dynamicT T r T

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

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.

1/ 2min Prla arr

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.

expacc Since we conclude that 100 measurements are adequate for the required precision in the result.

99.95%

exp

1.99(1%)0.199%

100

1%2

0.1%0.2%

0.5acc

t S X

X N

S CI

X XR R

sensitivityX

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.

expacc Since we conclude that 20 measurements are not adequate to yield the required precision in the result.

exp

20×1000.126 0.25

1000

0.2%

N

acc

w

X X

R R

sensitivityX

＝ ％

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

Briefly, the procedure is as follow: From a small sample of size N1, obtain S1 according to (10.17).

1

2

1, 1N ptotal

acc

t SN

(10.46)

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.

N2=49-10=39 additional measurements required.

12 100

0.2%

×0.2%0.06

1004

1.33.078

2.262 1.349

0.06

acc

acc

T

R R

sensitivityX

X

wS

d

N

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.