Reference

NBS

PUBLICATIONS

NAT'L INST. OF STAND & TECH

AlllOb 0 3 M 3bb

IMBSIR 85-3203

Response Behavior of Hot-Wires

and Films to Flows of Different

Gases

William M. Pitts Bernard J. McCaffrey

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

July 1985

Supported in part by: —QC -

100

.1156

85-3203

1985

orce Office of Scientific Research

>rce Systems Command, USAF

ington, DC

v

NATIONAL bureau

OF STANDARDS

LIBRARY

'

NBSIR 85-3203

RESPONSE BEHAVIOR OF HOT-WIRES

AND FILMS TO FLOWS OF DIFFERENT

GASES

William M. Pitts Bernard J. McCaffrey

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

July 1985

Supported in part by:

Air Force Office of Scientific Research

Air Force Systems Command, USAF

Washington, DC

U.S. DEPARTMENT OF COMMERCE. Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

LIST OF TABLES iv

LIST OF FIGURES v

Abstract 1

1. INTRODUCTION 1

2. PAST WORK 3

2.1 Results for Air 3

2.2 Results in Different Gases 15

3. EXPERIMENTAL SYSTEM AND DATA MANAGEMENT 25

3.1 Flow System 25

3.2 Sources of Gases 26

3.3 Gas Properties 26

3.4 Anemometer Electronics 27

3.5 Probes 29

3.6 Experimental Procedure 30

3.7 Data Analysis 30

4. RESULTS 33

4.1 Hot-Wire 33

4.2 Hot-Film 38

5. CORRELATION OF EXPERIMENTAL RESULTS 40

6. DISCUSSION 47

6.1 Correlation of Hot-Wire and Hot-Film

Responses in Different Gases 47

6.1.1 Precision of Correlations 47

6.1.2 Physical Reasonability of the Correlation . . 50

6.1.3 Comparison of Results for Air with

Literature Correlations . . 55

6.1.4 Comparison of Restuls for Different

Gases with Available Literature

Correlations 58

6.2 End Conduction Correlations 60

6.3 Reynolds Number Dependence of Heat Transfer .... 61

6.4 Accommodation Effects 66

6.5 Some Comments on Hot-Wire and Film Calibrations . . 72

6.6 Heat Transfer Studies Using Hot-Wires

or Films 74

ii

7. FINAL REMARKS 75

8. ACKNOWLEDGEMENTS 77

REFERENCES 77

NOMENCLATURE 83

iii
LIST OF TABLES

Page

Table 1. Values found by Collis and Williams [15]

for use in eq. (4) 87

Table 2. Suppliers and purities of gases used

in this work 87

Table 3. Sources for the properties of gases used

in this investigation 88

Table 4. Hot-Wire and film properties used for

calculations 88

Table 5. Values of nQ resulting from fits of

= A' 4- B'Un are listed for the hot-wire

measurements made with different gases 89

Table 6. Values of nQ are listed which result when

the hot-wire data for the different gases are fit

to eq. (3) 90

Table 7. This table lists results for the hot-film

calibration in flows of different gases 91

Table 8. Values of C and n* reported by Hilpert [16]

for use in eq. (46) 91

IV
LIST OF FIGURES

Page

Figure 1 Typical Hot-Wire Response Air

Line: Eq. (29) n = 0.43 96

Figure 2 Sensitivity Analysis for n 97

Figure 3 Hot-Wire Response for Eight Gases

Lines: Eq. (29) 98

Figure 4 Measured Nusselt Number vs. Reynolds

Number to the 0.43 power. 99

Figure 5 Nusselt Number Corrected for End Conduction Losses . . 100

Figure 6 Nusselt Number Corrected for "Classical Rarified

Gas Effects and Finite Aspect Ratio 101

Figure 7 Typical Hot-Film Response

Lines: Eq. (29) n = 0.45 102

Figure 8 Measured Nusselt Number for Hot-Film 103

Figure 9 Response of Film at Onset of Vortex Shedding 104

Figure 10 Hysteresis Near Flow Transition (RE 44) 105

Figure 11 Corrected Nusselt Number for Hot Film 106

Figure 12 Failure of Correlation with Prandlt Number 107

Figure 13 Correlation of Intercept with Viscosity (Hot-Wire). . . 108

Figure 14 Correlation of Slope with Ratio of Kinematic

Viscosities (Hot-Wire) 109

Figure 15 Hot-Wire Results Incorporating All Corrections

Except Thermal Accommodation Effects 110

Figure 16 Hot-Wire Results Including Accommodation

Corrections to He and CH^ Ill

Figure 17 Simple Kinematic Viscosity Correlation

(No Accommodation) 112

Figure 18 Simple Thermal Conductivity Correlation

(No Accommodation) 113

Figure 19 Hot-Film Results with All Corrections 114

Figure 20 Hot-Wire. Comparison to Literature 115

Figure 21 Hot Film. Comparison to Literature 116

v

ABSTRACT

Measurements of the voltage output for hot-wire and film anemometers

placed in flows of nine different gases have been made as a function of flow

velocity. It has been possible to correlate the measurements quite accurately

by treating the data in terms of suitably defined Reynolds and Nusselt

numbers. In order to obtain these correlations it has been necessary to

consider and correct for the effects of probe end conduction losses, temperature dependencies of gas molecular properties, flow slip at the probe

surfaces, and gas accommodation. With the exception of the results for helium

(for which accommodation effects are strong), the most important correction is

shown to be that for the different temperature dependencies of the gas molecular properties. This finding is contrasted with previous studies which have

assumed that the largest effect among different gases was due to variations in

the Prandtl number. The importance of the nature of the flow over the cylindrical devices to the heat transfer behavior is described. A previously

unreported hysteresis in the heat transfer behavior for Re ~ 44 has been

characterized and attributed to the presence or absence of eddy shedding from

the heated cylinder.

1. INTRODUCTION

The simultaneous measurement of concentration and velocity in turbulent

flows of gas mixtures has traditionally been very difficult. Such measurements are necessary before comprehensive models of turbulent mixing in

variable density flows can be developed. Recently, we have demonstrated that

the techniques of Rayleigh light scattering for concentration measurement

- 1 -
[1,2] and hot-wire or film anemometry can be combined in order to make such

measurements in isothermal flows [3,4]. This new technique is capable of

simultaneous, spatially-resolved, real-time measurements of concentration and

velocity in turbulent flows of two gases.

In order to apply this diagnostic technique, it is first necessary to

calibrate the response of the hot-wire or film as functions of concentration

and velocity. This may be done by monitoring the response of the heated

filament as a function of velocity for several different mixtures of gases

covering the concentration range of interest. However, this procedure is

tedious and time consuming. A second method would be to have suitable

empirical laws so that a calibration of wire response for a single gas would

be sufficient to allow the prediction of the wire response for other gases and

mixtures.

With the hope of generating such empirical laws we have measured the

response of hot-wires and films for a variety of gases and binary gas

mixtures. In this paper we will describe the behavior of the wire and film in

pure gases. (Note that air is included as a pure gas.) A second paper [5]

will describe similar measurements for binary gas mixtures.

Since we are primarily interested in calibrating the anemometers for use

in our flow system, we have limited our study to one overheat ratio for the

film and one for the wire and to the velocity range of interest in our experiments (» 50 to » 1000 cm/s).

-2 -
As will be shown, we have been able to generate empirical relationships

for predicting hot-wire and film response variations as a function of changes

in gas composition. In the process we have developed a much better understanding of the behavior of hot-wires and films. Such an understanding is

necessary in order to be able to assess the accuracy of velocity measurements

in flows of varying composition. These new results also allow a more critical

examination of past experiments reported in the literature. In particular,

the effects of thermal conduction along the probes, rarefied gas effects,

thermal accommodation on the probe surface, temperature dependence of gas

properties, and flow transitions on heat transfer behavior are described.

In the following section (2) we will describe past studies which are

relevant to this work. This will be followed by sections describing our

experimental techniques (3) and the results of our measurements (4). Efforts

to develop empirical heat transfer laws which correlate these measurements

will be reported in section 5. In section 6 we will discuss possible

explanations for and implications of our observations concerning the behavior

of the hot-wire and film. Finally, section 7 contains a summary of important

results and conclusions.

2. PAST WORK

2.1 Results for Air

Hot-wire and film anemometry are the most widely used techniques for

velocity measurements in fluid flows. Many books and monographs contain

extensive discussions of the use and applications of these devices (e.g.

,

-3 -
[6-12]). When such anemometers are employed carefully, they are highly

precise and accurate.

As might be expected, the vast majority of gas velocity measurements

using heated filament anemometers have been made for air. There are very few

reports of their use in other gases and still fewer regarding their application in measurements where the gas composition is fluctuating. Here we will

describe the relevant findings for air flows and then describe past studies of

hot-wire and film responses in different gases.

The principle underlying the use of hot-wire or film anemometry is that

heat transfer from an electrically heated surface depends on the velocity of

fluid passing over the surface. If the heated element also has a temperature

dependent resistivity, the heat loss from the surface due to forced convection

and changes in the probe temperature are coupled. By proper choice of

operating parameters it is possible to relate velocity to the measured power

dissipation of the filament.

Most of the hot-wire and film anemometers in use today employ cylindrical

conducting wires or conducting films applied to nonconducting substrates.

These probes are usually operated in the "constant temperature" mode. Since

the resistivity of the probe depends on temperature, in practice this means

the resistance of the probe is held constant and the amount of current flowing

through the probe is measured to obtain the power dissipation within the

probe. If it is assumed that the only heat loss mechanism is due to convection, the power loss of the probe can be related to velocity.

-4 -
Heat transfer laws for Infinitely long cylinders are often used to

predict the behavior of heated wires and films in gas flows. These laws are

usually expressed in nondimensional form by giving the Nusselt number (Nu) as

a function of Reynolds number (Re). Symbols are defined in the nomenclature

table. One of the first heat transfer laws was due to and bears the name of

King [13]. It can be written as

where Pe is the Peclet number which is equal to the product of Re and the

Prandtl number (Pr). King's derivation was based on the assumption of

potential flow which has been shown to provide a poor approximation of real

flows. However, relations based on eq. (1) are often used in the form

Nu = C^Pr) + C

2

(Pr)Re 1/2

, (2)

as a basis for empirical heat transfer laws for hot-wires and films. and

C

2 are parameters which depend on Pr.

For air or other gases this relation can be rewritten as

Nu = A + BRe

11

, (3)

where the exponent, n, is now treated as a parameter to accommodate the experimental observation that n may vary. Andrews et al. [14] have compiled a list

of many of the correlations found in the literature which have been used to

predict Nu behavior as a function of Re.

-5 -
The nondimensional correlation which is used most often is due to Collis

and Williams [15]. It can be written as

Nu (T /T ) a = A + B Re

11

, (4) a

v m oo' a a

where the values of the powers a and n, as well as A

a , and B a have the values

listed in table 1. These measurements were made for overheat ratios (defined

as (T - T l/T ) in the range of 0.1 to 1. The fluid properties necessary for

eq. (4) are evaluated at the mean temperature of the sensor surface and the

ambient gas, Tm

» A modification of heat transfer behavior at Re » 44 was

found. It was abrupt and attributed to the onset of vortex shedding from the

cylinder. A similar observation of this flow transition had been reported

earlier by Hilpert [16].

The subscript a in Nu. refers to a continuum Nusselt number which has

also been corrected for accommodation effects. Collis and Williams [15]

showed that rarefied gas effects are important for small diameter wires. They

derived an approximate correction for these effects which for air can be

written as

1 _ 1 Nu Nu

oo a 2Kn (5)

where Kn is the Knudsen number, X/D. The subscript oo indicates a Nusselt

number for a wire of infinite length, i.e. , no end losses. These authors also

found that buoyancy effects become important in the heat transfer at very low

Re. They concluded that buoyancy effects are negligible when

-6 -
Re > Gr 1/3

, ( 6 )

where both Re and the Grashof number (Gr) are evaluated for air at ambient

conditions. For the above overheat ratios, velocities ought to be larger than

about 2 to 5 cm/s to insure that buoyancy effects can be neglected.

Bradbury and Castro [17] verified the results of Collis and Williams [15]

for overheat ratios of 0.07 to 2.1. These authors reported some temperature

decay measurements for wires heated by current pulses which also supported the

findings of Collis and Williams.

Andrews et al. [14] have also investigated the responses of hot-wires in

air. They have given an excellent discussion of the corrections necessary for

rarefied gas and accommodation effects.

Accommodation effects [18] are treated using the accommodation

coefficient (a), defined as

a (7)

which can be thought of as the ratio of the average increase in energy of the

gas molecules after striking the surface and the increase in energy they would

have if they remained near the surface long enough to come into equilibrium at

the surface temperature. a = 0 if the gas "accommodates" no energy from the

surface and a = 1 if gas molecules striking the surface come into thermal

equilibrium at the surface temperature.

-7 -
Heat transfer to a rarefied gas from a heated surface is treated using

the concept of a temperature jump at the surface [18]. For a heated cylinder,

the temperature jump is defined as

where is the temperature the surface would have if the temperature jump did

not occur and A is the temperature jump distance. Kennard [18] showed that

for a rarefied gas with gas accommodation effects, A can be expressed as

A slip parameter which relates A to the diameter of the cylinder is often

defined as

where 0' = A/ A.

Andrews et al. [14] showed that for an ideal gas the Nu for the gas in

the absence of rarefied effects and with a = 1 could be related to Nu a by d

Nu

(9)

8 = ^

= 0'Kn, ( 10 )

Nu

a 1 - $ KnNu (ID

CO

where,

( 12 )

-8 -
x and y are the exponents which are found by assuming y

« Tx

, k » and 0 ' is

evaluated at Tg

. While not exact, these two expressions do provide a good

approximation for the temperature dependencies of these gas properties. For

eq. (11), Kn is evaluated at Tm. Eq. (12) is derived by assuming is nearly

equal to T

g

. This is generally a good assumption.

Andrews et al. found that 2.23 < $ < 2.67 for air at their experimental

conditions. Substituting <}> in eq. (11) shows that there is rough agreement

between their more exact theoretical treatment and the simplier model of

Collis and Williams [15] as given by eq. (5). In fact, for reasons of

simplicity, Andrews et al. treated their data using eq. (5).

Unlike the work of Collis and Williams [15], Andrews et al. [14] found

that a correction for temperature loading was not required when eq. (5) was

used to correct Nu for rarefied gas and accommodation effects. For large

00 aspect ratios ( n/D) they found that the equation

Nu = 0.34 + 0.65Re°* 45

(13)

a

could be used for 0.02 < Re < 20, 0.02 < (T - Tj/T^ and 0.052 < Kn < 0.1.

Since Collis and Williams [15] also included Kn effects in their study it is

difficult to understand why the results of Andrews et al. [14] differ from

this earlier work, the answer may lie in the different molecular properties

used by the two groups (see the discussion in section 6.1.3).

The work described above is for heated wires of sufficient aspect ratio

that heat transfer from the ends of the wires to the supports can be ignored.

-9 -
In practice, this is often not the case. Andrews et al. [14] found that the

aspect ratio was important for 2,/D < 400. These workers investigated wires

having aspect ratios in the range 24 < £/D < 1300. They employed an effective

Nu

ffl based on observed wire heat loss and did not correct for end losses. Eq.

(3) was used to treat their data. They concluded that the value of n remained

constant, but that the parameters and Bm increased as £/D decreased.

These observations are consistent with earlier experiments reported by Collis

[19] and Champagne and Lundberg [20]. The increase of Bm

was shown to be much

less rapid with decreases in 2./D than the value. This is not unexpected

since the Am

value is generally associated with free convection and conduction

losses while Bm

is believed to be primarily dependent on forced convection.

Several authors (e.g. , [6,7,11,12,21,22]) have discussed the effects of

probe end conduction on measured heat loss and have provided methods for

calculating the predicted heat loss in the absense of conduction. Most of

these derivations are based on a simplified heat transfer differential equation which can be written as the sum of conductive heat losses,

(14)

convective heat losses

= wDhO (15)

and joulean heating

Qj

- i

2

P r (e) (16)

- 10-
The above expressions assume no radial temperature gradiants, no

temperature dependence of probe thermal conductivity (k

g

), the probe diameter

is constant, and radiative heat losses are negligible.

For thermal equilibrium.

Qc +

Qh -

Qj = 0 (17)

The local resistance of the probe is usually written as,

P (0) = p(0) (1 + a 0), (18)

r r p

where higher order terms in 0 are disregarded.

By substituting eqs. (14), (15), (16), and (18) into eq. (17) and

rearranging, the following equation is obtained,

d

2 ~e - C 0 + D = 0 (19)

, A o o

where

C

o

•nPh A k

s s

I 2

p r (°) %

A k

s s

and

D

o

I2

p r (0)

TkS S

( 20 )

( 21 )

- 11 -
Clearly D

Q

is > 0. For practical flows, C

Q

is also found to be greater than

zero.

In solving eq. (19) for a heated probe oriented perpendicularly to the

flow it is usually assumed that the temperature distribution is symmetrical

about the center of the probe (x = 0) which requires that d9/dx = 0 at x = 0.

The probe is also assumed to be attached to massive prongs having an infinite

heat capacity and a temperature equal to the ambient flow temperature. This

condition can be written as 9 = 0 at x = ± n/2. The solution of eq. (19) for

these boundary conditions is

cosh (VC x)

e/e =i z=— (22)

cosh ($, /C

q

/2)

where 9 = D /C is the overheat which would exist in the absence of end heat

o o o losses due to conduction.

By integrating along the probe, an expression can be obtained for the

average overheat, 9,

9/e =1 — tanh (% /C ) (23)

° wr u °>

o

In the hot-wire literature the effects of end conduction are often

discussed in terms of the "cold length", £ c> introduced originally by Betchov

[23]. The cold length is related to the above equations by l = 1//C

q

.

The above description of probe thermal conduction effects is related most

closely to the development given by Corrsin [6]. We will use these results in

- 12 -
section 3.7 to develop a method for correcting our experimental results for

end conduction losses.

Collis and Williams [15] found that for large aspect ratio wires it was

necessary to multiply Nu by (T /T ] ^ to correlate their results (see eq.

(4)). Koch and Gartshore [24] reported a study for 5 ym wires having aspect

ratios of 230. Even though these workers did not correct their results for

end conduction losses or accommodation effects, they fit their data to an

equation of the same form as eq. (4). Nusselt numbers based only on experimentally observed heat losses are denoted Num

. In agreement with Collis and

Williams [15], it was found that plots of Numversus Re were fit best for

n = 0.45. As expected, based on the discussion above, their measured values

of A = 0.72 and Bm = 0.80 are larger than found by Collis and Williams (see

table 1) and is increased proportionally more than B

m

. The major surprise

of this work is the measured value of a = 0.67. This value of a has a different

sign and much larger magnitude than found by Collis and Williams [15]. This

difference was attributed by Koch and Gartshore [24] to aspect ratio effects.

Bruun [25] also reported a study of the effect of overheat ratio on the

response of hot-wires when operated in the constant temperature mode. The

hot-wires used in this study had aspect ratios varying from 360 to 420. The

results of this study were not treated in terms of eq. (4), but rather in

terms of the observed voltage drop across the hot-wire,

E

2

W

S' J T \ = A' + B'Un

, (24) Vs TJ

which is proportional to Num

.

- 13 -
Bruun found that a value of n = 0.45 fit his results well for velocity

ranges corresponding to Re < 44. Even though his measurements were not

analyzed in the form of eq. (4), he was able to show that the value of a in

this equation depended quite strongly on velocity. His measured values of a

differed markedly from those of Collis and Williams [15] or Koch and Gartshore

[24].

In treating his data, Bruun mentioned potential problems in obtaining

accurate values of E

g which can arise when utilizing standard constant temperature anemometry circuitry. Such circuits consist of a Wheatstone bridge, for

which one leg is formed by the probe, and a feedback amplifier which responds

to variations in probe resistance in such a way as to maintain a constant

resistance. Usually, these amplifiers also include a small offset voltage

which is designed to force the bridge slightly out of balance in order to

obtain a minimum damping of the amplifier outlet. In this manner the

frequency response of the device is maximized. Due to this offset voltage, a

measurement of the voltage drop across the bridge does not provide a true

measure of the voltage drop across the probe. Bruun overcame this problem by

measuring the voltage drop across different legs of the bridge.

The effects of offset voltage on constant temperature anemometry was

discussed in much greater detail by Morrison [26]. He provided a theoretical

analysis which showed that the offset voltage setting can modify the observed

voltage drop across the bridge and lead to what appears to be a probe overheat

dependence which is in reality an artifact due to the electronics.

- 14 -
Experiments were reported which supported the theoretical conclusions.

Morrison notes how this artifact can result in errors in measured heat

transfer laws and proposed this effect as the source of disagreement between

the measured value of a for the work of Collis and Williams [15] and Koch and

Gartshore [24].

It seems clear from these papers that differences in observed heat

transfer laws for hot-wire anemometers should be expected unless great care is

taken to account for end conduction losses and response behaviors of the

electronics used.

2.2 Results in Different Gases

Most of the relevant studies on hot-wire behavior in different gases have

resulted from efforts to use two hot-wires (and/or films) to make simultaneous

concentration and velocity measurements. As first noted by Corrsin [27], such

measurements are possible when the heat loss of the two heated filaments is

different for a given velocity and concentration. By simultaneously

monitoring the heat loss of each wire, the concentration and velocity can be

determined from suitable calibration curves.

Even though there had been previous efforts to use the ideas of Corrsin

[27] to extend hot-wire anemometry to variable composition flows [28-30], the

first literature reports appear to have been due to Kassoy [31] and Aihara et.

al. [32] who investigated the response of hot-wires in mixtures of helium and

nitrogen as a first step in calibrating hot-wires for making simultaneous

concentration and velocity measurements. They attempted to use a modified

King's law written in the form

-15 -
E - K (25)

( A + BPe

1/2

)

to predict the response of the wire when placed in different gases. By using

measurements in nitrogen as a reference, eq. (25) was used to predict the

voltage required by the hot-wire to maintain a constant temperature in flows

of helium and mixtures of nitrogen and helium for Re » 10. To their surprise,

2 they found that the calculated heat loss, which is proportional to E , was

much less than predicted. For this reason, they performed an extensive

theoretical [31] and experimental [32] study for Re < 0.1. These studies

indicated that the observed discrepancies were due to temperature slip effects

at the surface of the wire. Even though the Kn were in a range (« 0.01) where

such effects are generally considered to be small, the extremely low thermal

accommodation coefficient (a) of helium on tungsten greatly increases the

importance of slip effects. A theory was developed by Kassoy [31] which

predicted the behavior of Nu in terms of the flow parameters, gas properties,

a and the slip parameter (g) as defined by eq. (10).

The above studies were performed for a Re range which is much lower than

generally used in hot-wire studies. Furthermore, the theoretical study [31]

considered infinite cylinders and the experimental work [32] used wires having

large l/D ratios and guard heaters to minimize heat transfer to the prongs.

For these reasons, this early work is not directly applicable to the hot-wires

generally used in practice. However, they do contain several findings which

are necessary to understand the response of practical hot-wires and films in

variable composition flows. As already discussed, the importance of thermal

slip effects was noted. These effects were shown to be much less important

for nitrogen than for helium. In fact, the heat transfer of the tungsten wire

- 16-
in nitrogen could be predicted without including the effects of thermal slip

as shown by the agreement between the predictions of Kassoy's theory [31] and

experiment. Both the measurements and theory were in excellent agreement with

the empirical correlation of Collis and Williams [15]. These papers

emphasized the problems in predicting the effects of thermal accommodation

behavior and the lack of theories for predicting a in mixtures.

A third paper from this same group reported measurements similar to those

discussed above for nitrogen-neon mixtures as well as further measurements of

nitrogen-helium mixtures [33]. It was shown that the thermal accommodation

coefficient of neon on tungsten must be considered in using Kassoy's theory

[31] to predict the Nu for a given flow velocity. A simple gas kinetic

00 theory argument was used to predict the behavior of a as a function of neon

concentration.

Wu and Libby [34] later reported Nu measurements versus Re for a Re

00 range of 0.03 to 10 for air, helium, and mixtures of the two gases. In this

way they were able to compare the results at lower Re with earlier work from

the same group [31-33] and at the same time obtain measurements over a Re

range of more interest for practical hot-wire measurements. These experiments

were done using a platinum wire instead of tungsten. In analyzing their data

these workers introduced a modified version of eq. (4) using the values of A,

B, and n listed in table 1 for Re < 44,

Pr v0.20 'Pr ,0.33

0.24

Pr + 0.56

air ; Pr

Re

0.45

air/

(26)

where x refers to the gas under investigation. The Pr dependence in eq. (26)

-17 -
is taken from the work of Kramers [35]. Kramers deduced his correlation for

heated cylinders (which differs slightly from that of Collis and Williams

[15]) by using data for air, water, and several viscous oils. It is somewhat

alarming that in deducing this relation he dismissed the existence of a flow

transition at Re » 50. This is the same flow transition which is clearly

shown in the work of Collis and Williams [15].

Wu and Libby found measurements of Nu^ as a function of Re for air to be

in excellent agreement with predictions using Collis and Williams’ relationship [eq. (4)]. However, when they investigated pure helium and mixtures of

helium and air, the observed Nu for a given Re was lower than predicted by

00

eq. (26). Differences between the predictions and experiment were attributed

to thermal slip. These workers used the parameter g* to characterize the

effects of thermal slip by an equation which is very similar to eq. (5),

1/Nu - 1/Nu = g'/2, (27)

OO 3

where g’ is treated here as an empirical parameter. Based on experimental

results, they estimated g’ to be 0.274 which gives a = 0.11 from eqs. (9) and

(10). An empirical relationship which assumed that g’ for helium-air mixtures

was a linear function of helium mole fraction gave limited agreement with

experimental results.

Another early study which considered the effects of variable composition

on hot-wire response was done by Wasan et al. [36]. These authors modified a

heat transfer correlation due to Van der Hegge Zijnen [37] to obtain an equation that predicted Nu behavior as a function of Re, Pr, and Gr and which was

OO

-18 -
shown to be valid in Re regions where mixed natural and forced convection were

occurring. Measurements were reported for a constant temperature platinum

wire which showed that this expression gave reasonably good predictions of the

heat loss for a low velocity air flow [37].

These authors also studied flows of air and water vapor. As expected,

the response of the wire was modified when water vapor was added to the air

flow. Using their correlation, they were able to predict the modifications of

the wire behavior quite well. Despite this success, this type of correlation

has not come into widespread use and apparently has not been tested for other

gases. Furthermore, at higher Re this correlation predicts a Nu dependence

00 similar to eq. (3) with n = 0.5. Since n is believed to be 0.45 for Re < 44,

this correlation might be questioned over the Re range of 0.01 to 44.

Later, Wasan and Baid [38] published a study of the response of hot-wires

and films to flows of air, carbon dioxide, and mixtures of these two gases.

These authors chose to try and fit their results to Collis and Williams'

correlation [15] (eq. (4) with the constants of table 1 for 0.02 < Re < 44).

Using their results they were unable to distinguish between 0.45 and 0.5 as

the best value of n to use in eq. (4). By using measurements in air as a

reference, they attempted to predict the heat loss for carbon dioxide as a

function of velocity using eq. (4). They calculated a considerably smaller

Nu

ffi for CO2 than was actually observed. Analysis showed that most of the

error was in the parameter A^ and that Bm

was nearly equal for the two gases.

Note that no dependence on Pr is included in eq. (4) such as that in eq.

(26). However, the Pr numbers of air and CO2 are so similar that any effect

of their difference will be small.

- 19-
Wasan and Baid were able to predict hot-wire and film responses in air,

CO

2 , and mixtures of the two by using experimental values of the parameters A

and B of eq. (4) for gases of two different compositions and performing a

linear extrapolation of the values as a function of gas thermal conductivity.

They concluded, based on this result, that this procedure could be used for

any suitable pair of gases.

Andrews et al. [14] have also investigated the effects of molecular

composition on hot-wire response. Their study included results for measurements of methane/nitrogen mixtures as well as the air measurements already

discussed above. Based on their findings they concluded that eq. (13) can be

used to fit their mixture results as well as those for air and suggested it

would be of value for other gases and mixtures.

McQuaid and Wright [39] have reported a study of hot-wire response in

different gases. Instead of treating the instantaneous response of the wire,

they chose to derive relationships in terms of average values and intensities

of concentration and velocity fluctuations on two different sensors. They

showed that these values could be related to the time-average and root mean

square of the voltage drops across the two hot-wires. For carbon dioxide-air

and argon-air mixtures it was found that the function

where all of the voltage values are recorded for a constant velocity, is

independent of velocity. The overbars indicate time-averages and subscripts 1

and 2 refer to results for pure gases. The lack of a velocity dependence in

eq. (28) requires that calibration equations written in the form

r

(28)

-20 -
E (29) 2 = A' + B'U

n

have a common intersection point for all concentrations of the given gas

pair. Note that A' and B' are concentration dependent and that for constant

temperature operation eq. (29) can be reduced to the form of eq. (3) by using

the relationships between and Nu and U

n and Re11

, respectively. For cases

00 where heat loss to the prongs is important, Nu^ should be replaced by Num

.

These authors found that hot-wire voltage behavior as a function of velocity

could be fit well using n = 0.45. This is in agreement with the findings of

Collis and Williams [15].

For C02~air mixtures, McQuaid and Wright [39] found that ^(Y^q ) was

linearly dependent on Y^q . There was a great deal of scatter in the measurements due to the small difference in heat loss of the hot-wire to air and CO2 which resulted in large uncertainties when the differences in eq. (28) were

taken. In fact, when calibration results for air and CO2 were fit to eq.

(29), the two lines crossed at a velocity of » 8m/s for an overheat ratio of

0.8. Argon-air mixtures also gave results for iJ>(Y ) which were independent

of velocity, but in this case the dependence of on Y^r was nonlinear. Due

to the large differences in the heat loss of the wire in air and argon, the

uncertainties in ^(Y^

r ) were considerably less than found for ^(Y^ ).

These authors also gave criteria to serve as a basis for choosing suitable gas pairs for simultaneous concentration and velocity measurement. A

heat transfer law of the form of eq. (26) was used as the basis for these

criteria. However, it was emphasized that this relation is expected to be

only crudely correct and to only be suitable for the prediction of gross

effects.

-21 -
Simpson and Wyatt [40] have investigated the heat loss of various hotfilms in air, helium, and argon and in helium-air and argon-air mixtures.

Both cylindrical and parabolic sensors were considered. These authors used

the modified Collis and Williams' heat transfer law given in eq. (26) as a

basis for analysis of their results. Since cylindrical hot-films have very

small aspect ratios, it was necessary to allow the numerical constants (0.24

and 0.56) in eq. (26) to be parameters and determine them experimentally.

Using the resulting equation, they were able to obtain an excellent correlation of their experimental results for argon, air, and mixtures of these two

gases.

However, when they considered helium and helium-air mixtures they found

that the modified Collis and Williams' equation overpredicted the convective

heat transfer from the film to the gas. This observation is similar to those

found by Libby and coworkers [32-34] for helium-air and helium-nitrogen

mixtures. In this earlier work, the decrease in Nu from that expected was

00 attributed to thermal accommodation effects on the surface of the platinum and

tungsten wires. However, Simpson and Wyatt [40] noted that a for helium on

quartz (the overcoating used on the film sensor) is expected to be larger than

on the metals used for construction of the wires. The result is that thermal

accommodation effects should be much less important for films than for wires.

For this reason, these authors attribute the decrease in Num

to thermal diffusion effects. Thermal diffusion refers to concentration gradients which are

formed when gas mixtures are placed in temperature gradients such as exist

around a heated filament. These concentration gradients result in variations

in fluid properties of the mixture near the heat source and modify the heat

transfer from that expected in the absence of thermal diffusion. There is

-22 -
apparently no theory which allows the prediction of such effects on hot-wire

behavior. It should be noted that Khalifa et al. [41] have considered the

effects of thermal diffusion in the context of hot-wire thermal conductivity

measurements. We shall discuss the effects of thermal diffusion and develop

an approximate theory to describe these effects in the paper dealing with hotwire behavior in mixtures [5].

Some work has been reported on the heat transfer from hot surrounding

gases to cooled-films [42]. These experiments are obviously related to the

hot-wire studies, but Fingerson and Ahmed [42] argue that the same heat

transfer laws should not hold because of differences in the dynamical

dissimilarity with temperature loading between heating and cooling of rods.

Measurements were reported in heated flows of N

2

, He, CC^, and various

mixtures of these gases [42]. Their studies covered a range of ambient

temperatures from 800 to 1600 K and film surface temperatures from 350 to

525 K. For 5 < Re < 44 they found that their results could be well correlated

by

Nu = f0. 21 + 0.50 Re0,45

] f v /v ) °* 15

(30)

This equation is clearly similar to eq. (4) when the parameters of table

1 for Re < 44 are used. However, the temperature loading factors lead to

different calculated Nu based on the two expressions.

It is interesting that Fingerson and Ahmed [42] have been able to

correlate their measurements of He, CO

2

, N2 , and mixtures of these gases.

This implies that for their experimental conditions such effects as thermal

-23 -
slip and thermal diffusion are absent. This observation is consistent with

the large Kn for the film and the large accommodation coefficient for helium

on quartz as pointed out by Simpson and Wyatt [40].

Brown and Rebollo [43] have given a general discussion of the response of

a hot-wire to variations in the concentration of a constant velocity flow in

connection with their development of an analytical probe for concentration

measurements in binary gas mixtures. This device was designed to operate as a

sonically choked aspirating probe in such a way that flow past the internally

mounted hot-wire would be at a constant velocity as long as the external

pressure was constant. As a part of this study these workers attempted to

predict the wire response to different gases using heat transfer laws similar

to those previously discussed. For a platinum wire they concluded that it is

necessary to consider accommodation effects when helium and argon are used.

Tombach [30,44] has also briefly described the importance of thermal slip

effects on hot-wire measurements and noted a memory effect for hot-wires

placed in helium and then used to measure air flow velocities. These difficulties led Tombach to investigate the alternate diagnostic technique of heat

pulse anemometry for velocity measurements in inhomogeneous flows.

-24 -
3. EXPERIMENTAL SYSTEM AND DATA TREATMENT

3.1 Flow System

The response of the hot-wire and film were investigated as a function of

velocity using a TSI model 1125 flow calibrator*. This device consists of a

flow regulator, baffles and flow straighteners, and carefully machined nozzles

of decreasing size which are designed to provide uniform velocity profiles.

The velocity probe is aligned perpendicularly to the flow direction at a small

distance above the exit of the nozzle.

We have investigated flows of relatively low velocity (« 50 cm/s to

w 1000 cm/s). For these velocities the gases can be considered to be

incompressible to a very high degree of accuracy. In this case, the pressure

drop across the nozzle is related to velocity by the Bernoulli equation which

can be written as

where AP is the pressure drop across the nozzle and p is the density of the

gas which is determined by the molecular weight of the gas, the atmospheric

pressure, and gas temperature.

*Certain commerical equipment, instruments, or materials are identified in

this paper in order to adequately specify the experimental procedure. Such

identification does not imply recommendation or endorsement by the National

Bureau of Standards, nor does it imply the materials or equipment are necessarily the best available for the purpose.

(31)

-25 -
The difference in pressure on either side of the nozzle was determined

using a Datametrics electronic manometer (type 1014A) equipped with a 1 torr

differential head which could be used for pressure measurements to 1.3 torr.

The pressure reading of the instrument is available as a voltage which was

read by a 3-1/2 digit digital volt meter. The performance of the electronic

manometer was checked against a Serta Systems electronic manometer. The two

instruments gave identical readings.

3.2 Sources of Gases

Experiments have been performed for a wide variety of gases. Table 2

lists the sources of the gases and stated purities.

3.3 Gas Properties

A knowledge of the physical properties as a function of temperature is

required for the gases used in this study. Densities have been obtained from

known values at given temperatures and extrapolated to the temperature of

interest assuming ideal gas behavior. In recent years reliable values of p , k

and Cp have become available for most of the gases we have investigated.

Where possible, we have chosen to use the values listed in the Thermophysical

Properties of Matter, The TPRC Data Series [45-47] which is published by the

Thermophysical Property Research Center of Purdue University. The only excep

tion is the thermal conductivity of propane which is taken from a more recent

measurement [48]. The values found in these tables are based on extensive

reviews of the existing literature (up to the early seventies) and critical

analysis of these results. Uncertainties are quoted for the values over

various temperature ranges.

-26-
Physical properties for gases not listed in the TPRC Data Series are

taken from various sources. Table 3 lists references to property data for all

the gases studied here.

During the course of this work it was necessary to calculate mean free

paths for some of the gases investigated. We have chosen to do this using the

relationship between viscosity and the mean free path:

where c is the average velocity of the molecule which for an ideal gas is

equal to

This same relation was used by Collis and Williams in their classic study

A standard commercial anemometer system was used in this study. This

system consists of several modules. A TSI model 1051-2 monitor and power

supply generates the necessary voltages for the anemometer system and also

outputs a voltage equal to that necessary to maintain the sensor bridge

circuit in a balanced condition. This voltage is measured with a 4 digit

voltmeter having a 0. 1 second time constant.

X = 2p/pc (32)

(33)

[15]

3.4 Anemometer Electronics

-27 -
The constant resistance of the sensors is maintained by a balanced 5:1

Wheatstone bridge. This bridge is contained in a TSI model 1054B linearized

anemometer module. Note that the linearizing circuits were not utilized in

this investigation and that the recorded output is the bridge voltage necessary to maintain the balance of the bridge. We have not made corrections for

the voltage offset of the high gain amplifier used in the feedback circuit to

maintain a constant resistance. This value was set as recommended in the

operating manual for the anemometer. Checks showed that the measured bridge

voltage was very insensitive to the setting of the offset voltage for the

present situation of a single temperature setting.

The resistance of the probe leg of the bridge is determined by the

external application of a known control resistance to the model 1054B. A

variable decade resistor (TSI model 1056) is used for this purpose. The

actual resistance of the heated filament (R

g in table 4) is obtained by

subtracting the internal probe and connecting cable resistances from the total

resistance of the sensor leg of the bridge (R^).

The power dissipated in the wire or film is calculated from a knowledge

of the resistance and current flow of the heated filament using P

g = i R

g

.

The current through the sensor is equal to V/R-p where R.p is the total of the

standard resistance in series with the probe leg (RQ ) and R

g

. These resistance values are all available and the total power dissipation within the

heated filament can be written as

P

s (34)

-28 -
3.5 Probes

All of the measurements reported in this work were made for one wire and

one film. The wire measurements employed a TSI 1210-T1.5 tungsten wire which

is coated with a very thin layer of platinum to minimize surface oxidation.

Table 4 summarizes the dimensional data, operating resistance, the operating

temperature of the wire (T_) assuming a uniform temperature along the entire

length of the wire, overheat ratio ((T - T^j/Tj, and Tm

defined as the

average of T and T . The active length of the wire is determined by gold

oo S plating placed on the prong ends of the wire. These wire parameters have been

calculated using data supplied by TSI. The operating temperature is calculated

using the relation

0 (35)

R and R are the resistances of the wire at the operating and ambient gas

O 00 temperatures, respectively, is calculated from TSI calibrations of the

wire resistance at the ice and steam points of water.

The film element is a TSI 1205-20. These probes are manufactured by

depositing a thin film of platinum onto a cylindrical quartz substrate. The

active length of the sensor is then defined by plating a layer of gold

(» 6.4 pm thick) at both ends which are attached to the probe mount. The

layer of gold provides a low resistance path to the platinum as well as a

means for holding the ends of the active area at a temperature very nearly

equal to that of the prongs (T ). Finally, the entire probe is overcoated

with a protective layer of alumina. The exact thicknesses of the substrate

-29-
and coatings of these devices are difficult to measure and are expected to

vary from probe to probe. After consultation with TSI, we have assumed the

following dimensions for our calculations: substrate diameter = 50 y

,

tp

t =0.1 y, and t^ = .8 y. The film parameters listed in table 4 are

obtained in the same manner as those for the wire.

3.6. Experimental Procedure

The following experimental procedure was used for measuring the responses

of the hot-wire and film to varying velocities of the different gases. A

cylinder of the pure gas was connected to the calibrator. The gas was allowed

to flow for a period of time sufficient to flush the entire flow system. The

electronic manometer was then zeroed for a zero flow velocity. Pairs of

measurements of pressure drop across the nozzle (aP) and voltage drop across

the bridge (E ) were then recorded for velocities spanning the range of

interest. Generally, 20 to 35 measurements were taken. The readings were

taken from high to low velocities and back to high velocity to insure the

absence of systematic or time dependent effects. The zero velocity calibration of the manometer was checked often.

3.7 Data Analysis

The results of our measurements on the hot-wire and film have been

analyzed in several different ways. First the data are fit using eq. (29).

The value of E used is the experimentally observed voltage drop across the

bridge and the velocity (U) is calculated from the observed pressure drop

using eq. (31).

-30-
The exponent n is treated as a variable. Linear least squares fits of

the data are obtained using a range of different n values (steps of 0.01).

The goodness of fit for these lines can be quantified by using the parameter

2X - e y com'

(36)

o where the subscripts m and com refer to measured and computed values of E .

For a given n the linear least squares fitting procedure generates a minimum

2 2 value of x • X varies as a function of n and we have defined the optimum

2 value of n (denoted as n

Q

) as that which gives the smallest x

•

A crude estimate of the uncertainty in a single point for the fitting of

2 2

E is obtained by taking the square root of x and dividing by the total

number of measurements. The relative error is estimated by dividing this

2 value by the median value of E .

The measured Nusselt and Reynolds numbers are then calculated for each

point and the results fit to eq. (3). Num

is calculated as

Num

“ V(’k *( T

s ' TJ) <3 7 >

where P

g is given by eq. (34). All of the properties of the gas under study

used to calculate Nu

ffl and Re are taken at the appropriate Tm listed in table 4

for the wire or film. Note that for any one gas, Numis related to E and Re

is related to U by constants. For this reason, the n

Q

for use in eqs. (3) and

(29) are identical.

-31 -
We have also corrected Nu for end conduction effects to obtain Nu . m oo

Eqs. (20), (21), and (23) are used for this purpose. Eq. (23) can be

rewritten as

£_ . JL

D C

o o

2 tanh (fc /C^/2)

(38)

Eq. (21) shows that D

Q is a function of I, p r (0), A

g and k

g

. All of

these values can be measured or estimated for each experimental data point and

a value of calculated. The values of k used for the hot-wire and film are

o s included in table 4. For the wire this is a literature value [53] of tungsten

at 293 K. The hot-film value is obtained by summing together values of

Aqfkqt* ^Pt^Pt’ an<* ^A1kAl and dividing by A

g to give a k^^ for the probe.

Necessary values for thermal conductivities are taken from [53,54]. This

procedure must be considered a crude approximation since it ignores the

effects of radial temperature gradiants on the heat transfer process.

Since the value of 0 = T - T is also known, eq. (38) can be solved for

S oo

a value of C

Q

. In practice we do this by comparing a table of CQ values as a

function of 0/D with the 0/D value determined using 0 from table 4 and D^

o o o calculated from eq (21).

Once a value of C

Q

is obtained, it is substituted in eq. (20) along with

o values of A , k I , p (0) and a to yield a value for irDh where h is the

s s r p heat transfer coefficient for the wire or film when end losses are zero.

Nu is obtained from irDh by dividing by it and k^, for the gas.

in

-32 -
Once values of Nu and Re are available for the entire data set, a least

CO squares analysis which is identical to that described above is used to fit the

results to the form of eq. (3).

Values of Nu are then corrected for thermal slip effects. This is first

00 done assuming that the thermal accommodation of the gas on the probe surface

is perfect (i.e., a = 1). With this assumption, values of Nu

£ are calculated

from Nu using eq. (11) with <j> given by eq. (12) and 0' calculated using

oo I S eqs. (9) and (10). The results of these calculations are then fit to eq. (3)

and values of n

Q , Ac

, and B

Q are determined. For gases for which it is necessary to consider accommodation effects, the above calculations are repeated

using a suitable value of a in eq. (9). The Nu which results from this calculation is denoted Nu

a and the parameters which result from fits to eq. (3) are

denoted as nQ , Aa , and B a

.

The above calculations have been programmed on a minicomputer. In order

to obtain values of A, B, and n it is only necessary to input experimental

values of E and AP along with the physical properties of the gas under study.

4. RESULTS

4.1 Hot-wire

Figure 1 shows a plot of E versus U for the response of the hot-wire in a

flow of air. This data has been fit to eq. (29) using the procedure described

in section 3.7. Equation (29) was found to fit the data quite well. A plot

2

of y (defined by eq. (36)) versus n is shown in fig. 2. It is clear from

-33-
this figure that the value of n which gives the best fit of eq./(29) (denoted

by n

Q

) lies between 0.42 and 0.43. The sharpness of the minimum in the plot

2

of x versus n has been found to depend strongly on the quality of the experimental data. Smaller uncertainties in the data result in much more clearly

defined values of n

Q

. The results shown in fig. 2 are typical of those we

have found for all of our measurements using the hot-wire.

For the data shown in fig. 1 we find that the best parameters for use in

eq. (29) are n

Q = 0.43, A' = 3.618 ± .0004 V

2

, and B' = 0.3649 ± .0003

V^(s/cm)^*^^. The error limits given refer solely to those arising from the

impreciseness of the linear least squares fit to the data and are not intended

to be an estimate of the actual experimental uncertainty in the paramters A*

and B'. The A’, B', and n

Q

values have been used to generate the solid line

shown in fig. 1 which is in excellent agreement with the experimental results.

Using the definition given in section 3.7, the relative uncertainty for a

single point compared to the calculated line is estimated to be 0.0016. An

uncertainty of this magnitude is consistent with the numbers of significant

figures available in the measurements of E and U.

The response of the hot-wire to changes in velocity has been calibrated

for ten different pure gases. These gases include nitrogen and air. The

results for nitrogen and air were so similar that we have only reported the

results for air. Figure 3 shows examples of versus for eight of

these gases. The strong dependence of the hot-wire heat loss at a given

velocity for different gases is clear from this figure. The results for

helium are not included in this figure since these measurements were made over

a wider range of velocity and the bridge voltage is considerably larger than

for the other gases.

-34 -
Table 5 summarizes the results for all of the hot-wire calibration runs

that have been made. Values of n

Q are listed in the first column. These

measurements show that all of the experimental data, with the exception of

that for helium, can be fit accurately to eq. (29) using n = 0.43. For this

reason, the values of A’ and B' reported in Table 5 are fits of eq. (29) with

n “ 0.43.

The measurement results summarized in Table 5 were obtained over a period

of several months. In general, both A* and B' values for a given gas were

found to be reproducible to within » 0.5%. While small, these variations are

considerably larger than the uncertainty in the fit of a set of data recorded

during a single calibration. Comparison of data recorded at different times

show that there are slight systematic variations in the measurements. We

attribute these variations to minor modifications on the surface of the wire

(e.g. , due to dust or oxidation) and environmental influences such as small

temperature changes.

We have also fit our calibration results to eq. (3) in terms of Num using

the calculational procedures described in section 3.7. Figure 4 shows

examples of Num

plotted as a function of Reu * for all of the gases investigated. Note that helium has been included on the plot despite the fact that

the value of n„ for this gas is less than 0.30. Values of A and B for fits

of eq. (3) using n = 0.43 are included in Table 5. These values refer to the

particular data sets which are displayed in fig. 4. The approximate Re range

of the measurements is also included in this table.

-35-
Equations C 20 )— ( 23 ) and (38) have been used to calculate Nu^ for each

bridge voltage reading recorded during a given pure gas calibration. Examples

of the results of these calculations for each gas are shown in fig. 5 where

Nu is shown plotted against Re®*^. The data for each gas have been fit to

00

eq. (3). Table 5 lists the resulting values of A and B . 00 00

The following observations concerning the effects of end conduction

corrections on the calculated Nu behavior are obtained by comparing Nu

ffi and

Nu : 00

1) Nu^ is always less than Num. For this reason, < Am

and B < B_.

oo m

2) The fractional decrease of B compared to Bm is much less than

oo in the corresponding decrease of A compared to A

m>

3) The value of n

Q is increased slightly by the correction, but, in

general, An

Q

< 0.01.

4) The larger the k value for the gas, the smaller is the relative

reduction of Nu from the value of Nu • oo m

5) The relative decrease of Num

at higher velocities is slightly

reduced from that found at lower velocities.

It should be noted that the calculational result that n

Q remains

essentially constant while A

m and Bm

both increase with end conduction losses

-36 -
is consistent with the experimental conclusions of Andrews et al. [14] and

others [19,20]. The relative changes in the magnitudes of Am

and Bm

after

these corrections are also consistent with these past experimental results.

Nu data shown in fig. 5 have been corrected for thermal slip effects

00 assuming a = 1 as described in section 3.7. Values of n

Q along with Ac and B c

for n = 0.45 which result from fits of eq. (3) are listed in table 6. In

general, n

Q values for these fits are slightly lower than 0.45. However, we

have chosen to treat our data in this form to be consistent with Collis and

Williams [15] as well as a large number of other workers. Differences in the

quality of fits using actual values of n

Q or n = 0.45 will be very small for

every gas except helium. Figure 6 shows plots of Nu

Q

versus Re^*^ for the

nine gases studied.

For some gases it has been necessary to consider accommodation effects.

Calculations made for a ^ 1 will be discussed in section 5.

By comparing fig. 3 with figs. 4, 5, and 6 and the numerical results

listed in tables 5 and 6, it is clear that the use of eq. (3) in terms of

either Num, Nua , or Nu c gives a rough correlation of the hot-wire data for

different gases. It is also clear that this correlation is not complete and

that other factors must be considered. This point will be discussed further

in section 5.

Note that the highest Re value reached for the hot wire measurements

(Re = 6.2) was well below that required for the onset of vortex shedding

(Re « 44).

-37 -
4.2 Hot-Film

Calibrations for the hot-film have been performed using procedures

identical to those described for the hot-wire. We have investigated the hotfilm response for eight pure gases. Figure 7 shows examples of the hot-film

data for different gases plotted as versus U^*^. Data for helium has been

omitted from the figure for the same reasons as described above. Comparison

of figs. 3 and 7 shows that there are similarities in relative voltage magnitudes and slopes for the same gas on the hot-wire and film. Further inspection reveals some striking differences. All of the hot-wire data was fitted

quite accurately by using eq. (29) with n = 0.43. A velocity exponent of 0.45

gives an accurate fit for some of the hot-film results (e. g. , air and Ar), but

there is a clear curvature in the plots of some of the other gases (e. g. , CgHg

and SFg) which indicates that other values of n would give an improved fit of

the data to eq. (29).

The data for all of the different gases investigated have been fit to eq.

(29). Table 7 summarizes values of n , A', and B* for all of the hot-film

measurements we have made. A' and B* are the parameters for least squares

fits taken with n = 0.45. Note that there is excellent agreement between the

results of measurements for the same gas made over a period of several weeks.

Comparison of n

Q

values shows that the curvature observed in the data for

different gases in fig. 7 is related to variations in this parameter. For the

hot-wire results, only helium gave an n

Q value which differed significantly

from 0.43. However, for the hot-film, values of n

Q vary from 0.37 to 0.45.

This variation in n

Q complicates the comparison of the hot-film response in

different gases.

-38 -
Values of Num

as a function of Re have been calculated for the hot-film

results using eq. (37). Examples of the results for different gases are shown

in fig. 8 plotted as Numversus Re^*^»

Table 7 includes the values of Am and Bm which give the best fit of eq.

(3) with n = 0.45 for selected data sets. Table 7 also includes the approximate Re ranges over which the measurements for different gases have been

made. A comparison of these ranges with the observed values of n

Q indicates

that those gases for which the data extend to higher Re have smaller values of

n . As will be discussed in section 6.3, we attribute this behavior to

transitions in the flow behavior of the gas as it passes over the heated

cylindrical film.

A dramatic demonstration of the importance of these transitions can be

found by making the hot-film measurements for CgHg, CFgBr, and SFg at slightly

higher velocities than employed for the data reported above. Figure 9 shows

results of measurements made over the Re regime where the flow transition

occurs. A clear change in heat transfer behavior can be seen. Least squares

fits generated from the lower velocity data clearly give poor fits to the data

recorded at flow velocities corresponding to Re which lie above the flow

transition region. In a Re range extending from « 36 to « 55 a distinct

hysteresis is observed which depends on whether the flow velocity is increased

or decreased to the measurement value. Figure 10 shows plots of such

measurements for the three gases. The source of this hysteresis will be

discussed in section 6.3, however, it is worthwhile to point out here that it

occurs over a Re regime where a flow transition (Re » 44) was observed by

Collis and Williams [15] and attributed by these authors to the onset of

vortex shedding.

-39 -
The hot-film results have also been corrected for end conduction losses

and rarefied gas effects using the calculational procedures given in section

3.7. It has been assumed that a = 1. The corrections due to slip effects are

very small. Figure 11 shows examples of Nuc plotted as a function of Re^*^.

Values of A

c and Bc from least squares fits of the data to eq. (3) are

included in table 7.

As is the case for the hot-wire results, it is found that eq. (3) alone

is not sufficient to correlate the hot-film results. The similarities between

figs. 4 and 8 and figs. 6 and 11 are encouraging in that they indicate the

major source of variation in Nu behavior between different gases is due to gas

properties and not to such variables as device response (two different devices

were used), Ts (the wire and film were operated at different T

g

), or Re

regimes (due to the different diameters of the wire and film the Re ranges of

the measurements vary widely).

5. CORRELATION OF EXPERIMENTAL RESULTS

The experimental results shown in figs. 6 and 11 and summarized in tables

6 and 7 show that the hot-wire and film responses in different gases are not

correlated by simply plotting Nu

c versus Ren

. We have tried many schemes for

collapsing our data for different gases to single curves.

Two different philosophies have been employed in the literature to

correlate heat transfer data such as that generated in this study. The first

assumes that an expression in the form of eq. (3) can be used to fit all of

the data if the proper corrections can be found for the coefficients A and

-40-
B. The modification of A and B in eq. (26) to allow for Pr dependencies is an

example of this approach. The second philosophy assumes that a simple correction should be applied to the Nu in order to collapse the experimental data to

a single curve. The temperature correction included in eq. (4) and the v

correction included in eq. (30) are typical examples of this approach. Both

philosophies have been tried in this work as described below.

The first attempt to correlate the hot-wire experimental data assumed an

equation of the form of eq. (26), i.e. , a Prandtl number correction. This

equation has been suggested as a means of correlating hot-wire results for

different gases by several authors [34,39,40]. Figure 12 shows log-log plots

of A

c and B

c as functions of the Pr for the various gases. If eq. (26)

provided a means of correlating the measurements, these sets of data would

form two straight lines. It is clear from fig. 12 that this is not the case.

Apparently, the use of eq. (26) alone is not sufficient to correlate our

experimental results. Note that we are omitting helium from consideration at

this point due to the large accommodation effects expected in heat transfer

from the hot-wire to this gas.

We next considered other means for reducing Ac and B c values for the

different gases to universal coefficients. The approach used is demonstrated

in figs. 13 and 14 which show log-log plots of A

c and B c as functions of

y and v /v , respectively. Note that the B_ value of helium (B_ = 0.410) has m m oo c c been omitted from these plots. If the coefficient data fall on straight

lines, then the dependent variable provides a means to correlate the coefficients. Many different expressions for the dependent variable were tried.

These included km, v^, and X^ as well as those shown in figs. 12-14. Most of

-41 -
these plots indicated A

Q and B c had little or no dependence on these variables. However, as shown in fig. 14, values of B

c were fit quite well by

assuming a power law dependence of B

c on values of v /v^ for the different

gases. Only methane seems to be in disagreement with this conclusion. It

will be shown that this disagreement is most likely due to a small accommodation effect. B was also found to be roughly correlated with k /k , but the

c ° m oo correlation with v /v was slightly better. As shown in fig. 13, values of A„ m oo c were found to be dependent on u . Values of A for all nine gases seem to be

reasonably well correlated using this parameter.

Since the molecular properties of air, Ar, CO2 , and C 3H8 are believed to

be more accurately known than those for the other gases, the results for

measurements in these four gases have been used to calculate expressions for

Ac and B

c in terms of the suitable dependent variable. The results are

fB ) = 0.2606 (v /v l

1 ’ 355

, (39)

c

J x K m oo-'x *

and

(A

c ) x

= 0.0431/(„J°-

222

, (40)

for gas x. Equations (39) and (40) are taken from linear least squares fits

of the logs of the independent and dependent variables. The lines corresponding to these fits are included in figs. 13 and 14.

Equations (39) and (40) have been used to develop a correlation of the

experimental results for the different gases. Instead of using these equa-

-42 -
tions directly, we have normalized the results to those of air by plotting

Figure 15 shows the data plotted in this manner. Its clear that this

calculational procedure provides a good correlation for all of the data except

that for helium and methane.

In the past, deviations in the heat transfer behavior of heated cylinders

in flows of helium have been attributed to the lack of "accommodation” of this

light gas on the surface. This effect can be treated using eqs. ( 9 )—( 12) and

allowing a to be a parameter. Eq. (11) is then used to generate new values of

Nu

a from Nuc where <j> is now a function of a. Values of Nua are then fit to

eq. (3). In order to determine the best value of <f> for use in eq. (11), it

has been assumed that B a equals the B c predicted for helium using eq. (39).

For helium, this was found to be true for a = 0.48.

Significantly, n0 is found to be 0.45 when Nua values for helium are fit

to eq. (3) by the linear least squares fitting procedure. This result is in

excellent agreement with the values of nQ found for measurements in the other

gases. Contrast this value of nQ with that of n

Q = 0.35 which results from a

plot of the helium results as Nu

c versus Re11

.

Nu c " ( kA>x versus Re 0#45 (KB

) x for ( KA^x defined as

(41)

and (KB ) X as

(42)

-43 -
Values of Aa and B a for helium with a = 0.48 are plotted as solid symbols

in figs. 13 and 14. As required by the calculational procedure, B

a lies oq the

line corresponding to eq. (39) which is included in fig. 14. There is also

good agreement between the value of Aa and that predicted by eq. (40) as can

be seen in fig. 13.

As noted earlier, the value of B

c (see fig. 14) for methane is slightly

lower than predicted by eq. (39). We have assumed that this observation is

due to a slight accommodation effect for methane on the probe surface and have

treated the data in the same manner as for helium. In this case, B

a

is found

to equal the predicted B

c for a = 0.61. The value of n

Q = 0.47 found for the

plot of Nua versus Re

11 is a little higher than expected, but is not unreasonably larger than the values of 0.44 and 0.45 found for the other gases.

Values of A

fl and B a for methane are included in figs. 13 and 14 as solid

symbols. As for helium, there is good agreement between these values and

those predicted using eqs. (39) and (40).

Figure 16 shows the effects of correcting Nu

c for accommodation effects

on the collapse of the experimental data to a single curve. Values of

Nu

a - (K^) x with a = 0.48 for helium, a = 0.61 for methane, and a = 1 for the

remaining seven gases are plotted as a function of Re^’^^Kg)^ Collapse of

the experimental data to a single curve is excellent. Corrected values of A_a and B

a for the nine gases are listed in the last two columns of table 6. The

calculational procedure has clearly correlated the heat loss data for the nine

different gases to a high degree.

-44 -
Correlations similar to eq. (30) have also been tried. Ratios of

( v /v ) and fk /k ) have been used as fitting parameters. The exponents v m oo 7 m oo

7

necessary to generate the best fits have been treated as variables. They have

been generated by making log-log plots of calculated values of Nu

c at Re = 1

against (v /v^J for x equal air, CgHg, CF4 , CFgBr, CO 2 , Ar, and SFg. A

linear least squares fitting procedure gives the exponent from the slope of

the line. For v /v the exponent was found to 1.48. This value is considerm 00 ably larger than the value of 0.15 given by Ahmed and Fingerson [42] for a

cooled film. The exponent for use with fk /k ) is calculated to be 0.274.

^ m 00 '

Figure 17 shows Nu c (K where

(K ) = [(v /v )

. /fv /v ) l 1 ' 48

, ^ v'x LV m oo-'air^m <»'x J (43)

plotted against (Re)^*^. As above, the data for the different gases have

been normalized to the results for air. This procedure does give a fairly

good correlation for all of the gases except helium, for which the fit could

be improved using an a M as was done above. By comparing figs. 16 and 17,

it is clear that the calculational procedure which modifies the parameters A

c and B

c as given by eqs. (39) and (40) provides a much better correlation of

the experimental results.

Figure 18 shows Nu

c

(KjiC ) x plotted as a function of Re^*^. (K^x is

defined analogously to fK ) , namely

v x

(\)x = [(Wair^AlJ 0.274

(44)

-45 -
Plotting the experimental data in this way also gives an improved correlation

of the data for the different gases as compared to using Nu c alone. However,

the degree of correlation can be seen to be slightly less than shown in fig.

17, while fig. 16 still seems to provide the best collapse of the data.

Attempts to correlate the results for the hot-film response in different

gases is complicated by the variation of n

Q when the experimental data are fit

to eq. (3). For this reason, it was decided to treat the hot-film data in the

same way as described above for the hot-wire data. The hope was that such a

treatment would collapse the hot-film data for different gases to a single

curve. Figure 19 shows Nu

c - (K^) x versus Re^*^(Kg) x for the eight gases for

which hot-film response measurements have been made. Here (K^)x

and (Kg) x are

given by eqs. (41) and (42) with Tm

taken as 472 K. The data plotted in this

figure show that the hot-film response for different gases is correlated well

by treating the data in this manner. However, the collapse of the data to a

single curve is not as complete as found for the hot-wire data. Note that it

is not necessary, as is the case for the hot-wire, to modify calculated values

of Nu

c of He and CH4 for effects due to a lack of gas accommodation on the

surface of the hot-film.

The fact that the same correlation procedure works for the hot-wire and

hot-film results implies that the procedure is independent of probe surface

temperature effects, aspect ratio, and probe diameter since these parameters

are very different for the two probes investigated here. The implications of

this finding will be discussed further in the next section.

-46 -
6. DISCUSSION

6.1 Correlation of Hot-wire and Hot-film Responses in Different Gases

6.1.1 Precision of Correlations

Figures 16, 17, and 18 show that it is possible to collapse the

experimental data for the response of a hot-wire in different gases to a

single line with a high degree of accuracy. For each of these plots, the data

for different gases are normalized by suitable correction procedures to the

results for air plotted as Nuc (or Nua) against Re^’ 4 "*. Note that K^, Kg,

K

y

, and are all equal to one for air. The line which results from fitting

the air results to eq. (3) with n = 0.45 is

Nu = 0.272 + 0.650 Re 0,45

. (45)

c

Analysis of the data shown in fig. 16 indicates that the largest

deviations from this line for the data of other gases is + 2.0% and - 2.7%.

As can be seen in fig. 16, it is the data for CF^, CFgBr, and SFg which are in

poorest agreement with eq. (45). On the other hand, the data for CgHg, CH^,

Ar, and CO2 all fall within 1% of the values predicted by eq. (45).

A similar analysis shows that the method of correlation used for the data

displayed in fig. 17 gives results which all lie within + 3.6% and - 4.5% of

the line given by eq. (45). Clearly the deviations of the data from eq. (45)

are greater for data calculated by multiplying (Nu Q ) x

by (K ) x than using the

two parameter correction employed for the results shown in fig. 16. Note that

as drawn in fig. 17, the Nu

c values for helium and methane have not been

corrected for accommodation effects. The error limits reported here refer to

-47 -
data corrected for these effects. The correction employed for the data shown

in fig. 17 corresponds to multiplying (Ac ) x and (B c ) x

by (K )

. Comparison of

eqs. (42) and (43) shows that there is very little difference between values

of (K^) x

and (K^) x. For this reason, the increased scatter of the one

parameter correction is due primarily to improper correction of the intercepts

(Ac )x*

In fig. 18 the maximum deviations found from the line given by eq. (45)

are + 2.9% and - 7.3%. As already noted in section 5, the calculational

procedure used for the data displayed in this figure yields the poorest correlation of the three used.

In discussing the precision of the various correlations for the hot-wire

results, it must be kept in mind that there are uncertainties in the values of

molecular properties used in the various calculations. Unfortunately, the

largest uncertainties are found in the thermal conductivities for the

different gases. These values are very important since they play a dominant

role in end conduction loss calculations as well as appearing to the first

power in the Nu. The parameters A and B given in eq. (3) are directly proportional to this variable.

For two of the gases, namely CF^Br and C 3H3 , widely different values of k

are available in the literature. We have chosen to use the literature values

of k which yield values of Nu

c consistent with those found for the other gases

investigated. In the case of propane, this is a value taken from work [48]

published after the TPRC tables [45] had appeared. The later value of k is

5.6% times larger than the TPRC value. Interestingly, an uncertainty estimate

-48 -
of 5% is given in the TPRC for

c

3h8

at the temperature of interest to this

study. In the case of CF 3Br we have used a value of k taken from an earlier

handbook [49] and have ignored a later measurement [55] of which gives

a value which is 4.7% greater.

Given such uncertainties in k values, we conclude that our calculational

procedures yield correlations of hot-wire response data which are as precise

as possible within the limits of our knowledge of gas properties. In fact,

such uncertainties may explain the small variations of the data plotted in

fig. 16 from eq. (45). It is the data for CF^Br, SFfc, and CF 4 which deviate

to the largest degree. These gases are also those for which the largest

uncertainties in k values are expected.

Comparison of the hot-film results plotted in fig. 19 with those for the

hot-wire shown in fig. 16 indicates clearly that the hot-film results are not

correlated quite as well. There are several possible explanations for this

observation. Perhaps the most likely lies in the simplifying assumptions used

to treat end heat conduction losses for the hot-film. This device has a very

complicated physical structure and the assumption of a one-dimensional heat

transfer is certainly naive. Furthermore, as pointed out in section 3.7, k

g must be considered as an approximation of the true one-dimensional heat

transfer conductivity. Since £,/D is much smaller for the hot-film than for

the hot-wire, end loss corrections are expected to be more important in the

case of the hot-film.

We have assumed that the hot-film results can be corrected for gas

property temperature dependencies using the same procedures as for the hot-

-49-
wire (see eqs. (41) and (42)). Even though this assumption should be approximately correct, it is possible that the same correction may not apply if the

hot-film surface has a different surface temperature distribution or heat loss

behavior. Uncertainties in the values of molecular properties may also be

playing a role.

We conclude that correlating our hot-film results in the same manner as

for the hot-wire yields a collapse of the data that is precise to within the

limits defined by our lack of detailed knowledge of heat conduction within the

probe and uncertainties in the values of molecular properties for the

different gases investigated.

6.1.2 Physical Reasonability of the Correlation

In developing the correlations of heat transfer behavior given in section

5 we have made a variety of assumptions and approximations. It seems

pertinent to try and understand the physical basis of these assumptions and

also to relate them to past engineering practice.

The first choice we have made is to treat our experimental results in

terms of nondime ns ional variables. Reduction of experimental data in this way

is standard engineering practice and has its basis in dimensional analysis

[56]. Problems of heat transfer to and from flows of gases are usually

treated by experimentally determining variations of Nusselt number as a

function of flow Reynolds number and other possible nondimensionalized numbers

such as the Prandtl number. One of the major limitations of this approach is

that there are no definitive recipes for deciding how such nondime ns ional

-50 -
variables should be formed. For instance, in this study there were standard

and physically based expressions for generating the appropriate Nu and Re

(i.e., in terms of probe diameter), but there was no standard method for

deciding at which temperature to evaluate the molecular properties which are

used to form the nondimensional variables. A wide variety of different

temperatures have been used in the literature. We finally settled on an

evaluation of properties at Tmnot only because this temperature is widely

used in the literature, but also because it seems reasonable from a physical

point of view. The choice of a median temperature for molecular properties

should minimize the effects of property variation with temperature.

We first attempted to correlate our results in terms of Num

. These .

attempts failed due to the neglect of heat losses at the end of the probe.

These conduction heat losses not only vary with the flow parameters, but are

also dependent on the thermal conductivity of the gas. End heat loss corrections are discussed further in section 6.2 where we conclude that the mathematical procedure used is physically realistic for the hot-wire and is only

approximately correct for the hot-film.

Comparison of figs. 4 and 5 shows that not only are values of Nu for the

00 different gases greatly reduced from Nu

ffi by this correction, but the relative

spacings and orderings of the data for different gases are changed. This is

particularly true in the case of helium which has an unusually large value of

km

. For this reason, values of Num

for this gas are reduced proportionally

less by this correction than found for the other gases. Similarly, CF 3Br has

an extremely small value and the reductions of its Num

values following end

loss corrections are proportionally greater than found for the other gases.

-51 -
Correction of Num

for end conduction losses leads to a Nusselt number

(Nu ) which corresponds to the Nu for convective heat transfer from an

00 infinite cylinder. It is not surprising that the use of Nu^ yields a better

correlation of experimental results than Num

.

Values of Nu are next corrected for what might be called "classical"

00 thermal slip. By this we mean noncontinuum effects on heat transfer, but

assuming perfect accommodation of the gas on the surface of the probe.

Equations (9-12) with a set to 1 are used for this purpose. Figure 6 shows

plots of the data for the nine gases plotted as Nu c versus Re^*^. Comparing

values of Nu_ shown in fig. 6 with the corresponding values of Nu shown in

fig. 5 indicates that the correction for noncontinuum effects has very little

effect on the relative ordering or spacing of the results for different gases.

Note that there is a significant increase in value of n

Q following the correction for thermal slip. As might be expected, the largest effect is for helium

(Kn = 0.075) which has the largest Kn of any of the gases investigated here.

Even for this gas, correction of our measurements for "classical" thermal slip

only Increases values of Nu by » 10% as compared to Nu .

As figs. 6 and 11 show, plots of Nu

c as functions of Re^*^ for the

different gases do not lead to a complete correlation of the results.

Clearly, some additional correction must be made. As discussed in section 5,

multiplying Nu

c by either (^m

/k

oo ) x or (v /v^) raised to suitable powers

increases the degree of correlation of the results (see figs. 17 and 18). Is

this physically reasonable? The answer is yes. By multiplying Nu

c by either

of these ratios, we are partially correcting values of Nu

c for variations in

the temperature dependencies of the molecular properties of the gases. Heat

-52 -
transfer from a heated cylinder actually occurs along a temperature gradiant,

therefore, it is not surprising that the temperature dependence of the molecular properties is important in the heat transfer. Since variations in k

and v for different gases are roughly related, it is expected that correlations in terms of either k /k or v /v will give nearly the same collapse of m °o m <® the data. This is found to be the case.

The necessity of providing corrections for physical property temperature

dependencies is well known. The method used here is consistent with

approaches which have been employed in past engineering studies [57].

Modification of (Nu

c

) x

by multiplying by either (K^^ or (K gives a

much better collapse of the data than is found for data simply plotted as

(Nu

c

) x

. However, by using equations of the same form as eq. (3) and looking

for ways to correlate the values of Ac and B

c individually, it has been

possible to obtain an improved correlation of the results.

The parameter B

c was found to be dependent on (v /v^) raised to a power

(see fig. 14 and eq. (39)). This dependence is similar to that described

above for the corrections of Nu

c for the temperature dependence of gas

properties. Since the parameter B in eq. (3) is generally thought to be

determined by forced convection, it is reasonable to expect this parameter to

show a similar dependence on molecular property temperature variation.

On the other hand, the parameter (A

c

) x

has been shown to be primarily

dependent on values of (vm) x raised to a power (see fig. 13 and eq. (40)).

While not exactly correct, this parameter is expected to be determined

-53-
primarily by buoyant and conductive heat transfer processes. It has not been

possible to relate such heat loss mechanisms to a dependence on y^. At this

time the physical basis of the observed dependence of (A

c

) x

on (y m) x must be

considered as unknown and perhaps coincidental.

It has been necessary to make one further correction of the data in order

to obtain a satisfactory correlation of the experimental results for all of

the gases. This final correction allows for accommodation effects [18]. This

effect will be discussed in greater detail in section 6.4. Eqs. (9), (11),

and (12) which have been used here to correct Nu

c for the effects of accommodation provide a good approximation of the dependence of heat transfer on Kn

and a which are found in more detailed theories of this effect.

In general, it is found that lighter gases show more pronounced accommodation effects than heavier gases [58]. This is due to their smaller sizes

and higher average molecular speeds. In this study we have only found it

necessary to modify the hot-wire values of Nu

c measured in helium and methane

in order to account for changes in transfer behavior due to small values

of a. This finding does not imply that small accommodation effects are not

present for the other gases investigated, but only that such corrections are

similar for measurements made in the other gases. See the discussion in

section 6.4.

In the case of helium it is highly significant that the correction for

accommodation effects also changes the observed value of nQ from 0.35 for the

data in terms of Nu to the value of 0.45 found for the data treated as Nu . c a This is a clear indication that the correction of Nu c for accommodation is

-54 -
behaving properly since the new value of n

Q is in complete agreement with

those found for the other gases when plotted in terms of Nu c

. Further verification of this conclusion is provided by the results of the hot-film measurements in helium. For this case, no accommodation effects are required in

order to correlate the helium data with that found for the other gases (see

fig. 19). This observation is in accord with the expectation that helium will

be accommodated much better on a heated alumina surface (film) than on a

heated platinum surface (wire) [58]. The reduction in accommodation effects

in the heat transfer of helium for a hot-film as compared to a hot-wire has

been discussed by Simpson and Wyatt [40].

When developing the types of corrections for heat transfer data we have

used here it must be remembered that very simple equations are being used to

approximate very complicated processes involving complex boundary layers and

heat transfer at surfaces. Despite this, after correction of the experimental

data for physically realistic heat loss processes and known molecular effects

a very good correlation of the heat transfer data from heated filaments is

found for a wide range of gases. This finding is the best justification for

the various approximations and simplifications which have been employed.

6.1.3 Comparison of Results for Air With Literature Correlations

Most of the past work on correlations of heat transfer from anemometric

probes has been reported for measurements made in air. Figure 20 shows the

line corresponding to our results for air plotted as Nu Q versus Re^*^. Note

that the actual value of n

Q

in this work was determined to be 0.44. This

small change in n

Q has very little effect on the fit of the data to the equa-

-55 -
tion. On the same plot the corresponding curves are drawn for results taken

from Collis and Williams [15] (eq. (4)), Andrews et al. [14] (eq. (13)), and

Fingerson and Ahmed [42] (eq. (30)). For eq. (4) values of Tmand correspond to those used in this study. Similarly, values of and used in eq.

(30) are taken for the conditions of the hot-wire in our study. The two

literature hot-wire studies employed probes having large aspect ratios while

the cooled-film used by Fingerson and Ahmed had a small aspect ratio.

Apparently, the latter results have not been corrected for end heat conduction

effects.

Figure 20 shows that the present experimental results for Nu

Q as a

function of Re^*^ iie very close to the curve given by Collis and Williams.

There is a difference of * 9% in slope and » 6% in intercept. Given the

various corrections applied to our finite aspect ratio hot-wire measurements,

this agreement must be considered excellent. As discussed in section 2.1,

variations in heat transfer laws might also be expected due to the use of a

feedback amplifier in our electronic circuitry [25,26].

The correlation reported by Andrews et al. [14] is in poor agreement

with our results (» the same slope and * 25% higher in intercept) as well as

that given by Collis and Williams [15]. A possible explanation for this

discrepancy can be found by comparing the thermal conductivity values for air

used in the two earlier papers. Andrews et al. report that their k and p

values were taken from the International Critical Tables [59] published in

1928 while Collis and Williams have used a formula for k given by Kannuluik

and Carman [60] and have taken their v values from the tables of Goldstein

[61]. The values of kinematic viscosity employed in these two earlier studies

-56-
are nearly identical to those used in this study. However, the values of k

for air at Tm obtained from the literature sources for the three studies are

7.51 X 10“5

, 8.34 X lO"5

, and 8.38 X 10“5 cal/cm » K • s for Andrews et al.

,

Collis and Williams, and this work, respectively. The value of k used by

Andrews et al. is 10% smaller than that used in this study or the work of

Collis and Williams. The effect of using the more recent value of k on the

results of Andrews et al. is shown in fig. 20 where their results have been

redrawn (dotted line) using our value of (k ) . . It can be seen that the

results for all three studies of heat transfer from hot-wires in flowing air

now lie relatively close together. On this basis, we conclude that our

measurements of Nuc in air are consistent with past literature measurements.

The correlation of cooled-film results reported by Fingerson and Ahmed

[42] lies considerably below those measured for the hot-wires. This apparent

difference between the most suitable correlations for the hot-wire and cooledfilm results has already been noted by Fingerson and Ahmed and attributed to a

dynamical dissimilarity between heating and cooling of cylinders.

Figure 21 shows the line corresponding to our hot-film results in air

plotted as Nu

c versus Re®*^» Lines corresponding to the results of Collis

and Williams [15] and Andrews et al. [14] (both as given by the authors and

corrected for k values as described above) are shown on the same plot. It is

clear that there is relatively poor agreement between the hot-film results and

the two earlier correlations obtained for hot-wires. Both the slope and

intercept for our data are higher. Using a larger value of k g for the thermal

conductivity of the film would result in decreased values of Nu

c when Nu

m

is

corrected for end conduction losses and noncontinuum effects. Such a

-57 -
modification would give better agreement with the hot-wire correlations.

However, as already noted in section 4.1, the primary effects of the end

conduction corrections are to generate a relatively large decrease in Ac and a

much smaller decrease in B c

» Test calculations indicate that while reducing

k

g does indeed increase the agreement of our hot-film results with those of

Collis and Williams and Andrews et al. , this correction alone is not

sufficient to give agreement for both Ac and B

c

. We attribute this

observation to the simplifications which have been made by assuming a onedimensional heat conduction along the film. Improved agreement between heat

transfer laws for hot-films and hot-wires will require a better theoretical

treatment of heat conduction losses for the hot-film.

6.1.4 Comparison of Results for Different Gases with

Available Literature Correlations

There are several studies available which have investigated the responses

of hot-wires and films to changes in gas composition [14,27-40]. Most of

these studies have involved no more than three different gases and usually one

of these was air.

The most common correlation suggested for the treatment of hot-wire and

film measurements in different gases is the combination of Collis and

Williams’ [15] relationship for air with Kramers' [35] results for heat

transfer from cylinders placed in different fluids to give eq. (26). This

equation has been used or suggested by at least three groups of workers

[34,39,40]. Figure 12 shows that there is very little relationship between

values of Ac or B c for the various gases and the corresponding Pr. Perhaps

more importantly, given the small variations in Pr among different gases, the

-58-
exponents required on the Pr ratios in eq. (26) would have to be many times

larger than the values of 0.20 and 0.33 in order to reduce the data for the

gases investigated here to a common curve. We conclude, based on these observations, that while it is possible that forced convective heat transfer from

cylinders to gases may have a weak dependence on gas Pr, this dependence is

hidden by the much stronger effect of molecular property temperature variations on the heat transfer.

Fingerson and Ahmed [42] are the only other workers of which we are aware

who have used a temperature dependent property ratio to correlate heat

transfer results for anemometers placed in different gases. Their correlation

for cooled—films is given by eq. (30). The ratio of v /v used in this correIB oo lation is similar to one of the corrections we have used to modify Nu c (see

eq. (43)). As already noted, the exponent for the property ratios are very

different for the two studies, but it is significant that both of these

studies have required such a ratio to correlate heat loss results in different

gases. Interestingly, these authors have reached the same conclusion as given

above concerning the inability to discern an effect of Pr on the heat transfer

behavior.

It is also of interest to compare our results with the conclusions of

Andrews et al. [14] who reported measurements of hot-wire response in air and

mixtures of methane and nitrogen. These workers correlated their results for

pure gases and mixtures in terms of Nua

by correcting large aspect wire

results for thermal slip effects. The agreement was quite good, but in light

of the results given in this paper must be considered fortuitous. As fig. 6

shows, values of Nu

c for methane and air plotted as functions of Re^*^ do not

-59-
lie on the same curve. The discrepancy in the findings of these two studies

may be due to differences in the values of thermal conductivity used.

6.2 End Conduction Corrections

In section 3.7 we have given a recipe for the correction of measured Nu

made for finite aspect ratio probes. This correction compensates for heat

losses to the prongs supporting the probe due to heat conduction along the

probe. The resulting Nu corresponds to that for heat transfer from an

infinite probe at the same average probe temperature. In making these calculations we have made several assumptions and approximations as discussed in

section 2.1. Additionally, a literature value of thermal conductivity of

tungsten was used for k despite the fact that the probe is known to have a

layer of platinum deposited on its surface. Despite these approximations, we

have obtained results for air which are in excellent agreement with measurements of hot-wire response where end conduction effects are known to be absent

[14,15,17,34]. Barring the accidental cancellation of errors, this finding

validates the procedure used to correct for heat losses by conduction along

the hot-wire.

It is extremely encouraging that the correction procedure leads to

conclusions 1-5 listed in section 4.1. These calculated effects of changes in

heat loss behavior with aspect ratio are in complete agreement with experimental findings. This agreement lends further support to the model used here

to make the end loss corrections. The importance of making such corrections

is clear from the finding that no correlation of our results in terms of Num

could be found.

-60 -
In the case of the hot-film measurements, it is clear that our correction

procedure only provides an approximate physical picture for end conduction

heat losses. It is not surprising that our approximations fail in this case

since heat transfer along the hot-film is not expected to be one-dimensional.

A more highly developed heat transfer expression for conduction heat losses

along the heated film needs to be developed in order to allow more exact

correlations of our hot-film data in different gases.

6.3 Reynolds Number Dependence of Heat Transfer

It has been shown that the hot-wire results for different gases can all

be fit extremely well in terms of Nua using a value of n = 0.45 as the

exponent for the Re. Table 5 lists the Re ranges over which these measurements have been made. All of the measurements taken together cover a Re range

from » 0. 1 to ** 6.2. Note that the results for some of the gases fall at the

lower end of the range while those for others fall at the upper end. The fact

that observed values of n

Q are so similar for all of the different gases

requires that there be very little change in heat transfer behavior over this

limited Re range.

The situation for the hot-film is very different. In this case, the Re

range covered is » 1.0 to » 43. Gases which fall in the lower portion of this

range (e.g. , air, Ar, and CH^) have nQ values of « 0.45. These values are

consistent with those found for the hot-wire. On the other hand, the three

gases for which results lie at the high end of the Re range (C 3 H8 , SF 6 , and

CF 3Br) give nQ values in the 0.36 to 0.39 range. Significantly, the measurements for CO2 , which lie in an intermediate Re range, give a nQ value of 0.42

-61 -
which is between the n

Q values found for the measurements made at the extremes

of the Re range. Note that all of the measurements are made for velocities

low enough to insure that the sharp flow transition at Re » 44 has not

occurred.

The results of measurements made over the transition region mentioned in

the last paragraph have been described in section 4.2. Figures 9 and 10 show

clearly that there is an abrupt transition in heat transfer behavior for this

Re range as described by Collis and Williams [15] and much earlier by Hilpert

[16]. Collis and Williams used hot-wire anemometry in the wake of the hotwire to show conclusively that the Re for which the transition occurs is

identical to that for which eddy shedding commences from the heated probe.

The sharpness of this flow transition is demonstrated by the data plotted

in fig. 10. A clear hysteresis is observed in the heat transfer behavior

depending on whether the velocity of the gas is increased or decreased to the

measurement value. This observation implies that there is a narrow bistable

Re regime which lies between the flow velocities where eddies are not shed

from the cylinder and those where such shedding always occurs. Closer inspection reveals that for the three cases investigated, the transition from no

eddy shedding to the onset of shedding is extremely abrupt and occurs within a

Re change of - 0.6. The results for propane also show a sharp transition from

flow with eddy shedding to flow without eddy shedding. Interestingly, the

results for SFg and CF 3 Br indicate that flows of these gases have a smooth

transition in heat transfer behavior on going from the eddy shedding to no

eddy shedding flow conditions. The differences in transition behavior for SF^

and CF 3Br as compared to C 3HQ result in the bistable region of the former two

-62-
gases extending over a wider range of Re. At the present time we can offer no

firm explanation for the differences in transition behavior between C 3Hg and

the heavier molecular weight gases. It is intriguing to speculate that there

may be a molecular weight effect on this flow transition behavior.

There are some additional points which should be made concerning the

onset of vortex shedding. First, note that the Re range covered by the

transition regions extend from Re * 36 to « 55 which is nearly centered on the

value of Re * 44 quoted by Collis and Williams [15] as the Re for the onset of

eddy shedding flow behavior. In their study of cooled-films , Fingerson and

Ahmed [42] also report a flow transition between Re = 40 and 55. These three

studies have been made for a wide range of temperatures varying from room

temperature to 1700 K. The fact that the onset of eddy shedding is found at

Re a 44 in all of these studies is a strong recommendation for the use of T

ffl in determining gas properties for use in the nondimens ional variables. If

values of v at different temperatures (e.g. , Tfi or T ) were used, the observed

values of Re for the transition would vary widely. Minor variations are

observed in the exact Re ranges over which the hysteresis occurs for the

different gases as can be seen in fig. 10. We attribute these to uncertainties in the values of v used to calculate Re and to differences in the temperature dependencies of v. In analogy with the discussion given in 6.1.2, such

variations are expected when T

ffi is used to determine values of v for use in

calculating Re.

The behavior of heat transfer from heated cylinders as a function of Re

are related to the values of n0 required to fit eq. (3) for Nu c (or Nu a

) as a

function of Ren. The findings of this study can be broken down Into the

following four Re regimes:

-63 -
I. Re « 0,1 to « 5: n

Q remains essentially constant at » 0.45 for

measurements falling into this Re range,

II. Re » 5 to « 35: n

Q decreases as the Re increases,

III. Re « 35 to » 55: This is the bistable region determined by the

presence or absence of eddy shedding from the heated cylinder.

Very low values of nQ (< 0.35) are associated with gas flows

for which no eddies are being shed. Large values of n

Q

(> 0.50) result when eddies are present behind the cylinder,

and

IV. Re > 55: Values of n

Q which fit the data in this Re regime are

_> to 0.50. We have not made extensive measurements in this Re

regime.

It is interesting to discover that Hilpert [16] divided his heat transfer

results for heated cylinders in air into three regimes roughly corresponding

to I, II, and IV above with a break between II and IV at Re » 40. This author

fit his experimental results to an equation of the form

Nu - C [ Re ( T /T ) 1 / 4

] n ’ (46)

where values of C and n' appropriate for the given Re ranges are listed in

table 8. If Hilpert's results for Nu as a function of Re are fit to an

00 equation of the form of eq. (3) for Re = 4 to 40, n

Q is found to be 0.39 which

is in good agreement with our experimental results for gases falling in this

-64 -
Re range. Unfortunately, a similar calculation for the 1 to 4 Re range gives

a still smaller value of ng which is the opposite of our experimental finding.

In general, Collis and Williams [15] found good agreement with Hilpert's

results [16], but argued that the transition at Re » 4 does not actually exist

but is an artifact of the small number of experimental measurements used by

Hilpert to deduce its existence. As shown by eq. (4) and table 1, Collis and

Williams only required two Re flow regimes to fit their experimental measurements which extended from Re » 0.02 to » 140. A careful examination of the

data given in their paper shows that the slow variation in n

Q

we find just

before the flow transition at Re » 44 may not have been large enough to be

discerned by these authors.

It is possible to speculate on the physical basis for a modification of

heat transfer behavior on going from Re regime I to II. This can be done by

looking at the flow patterns which exist around an infinite cylinder in a gas

flow [62,63]. At Re < 4 the flow past a cylinder does not display a wake and

the flow moves over the cylinder as a nonseparating flow. At Re « 4 the flow

separates from the cylinder, a stagnation region develops behind the cylinder,

and a wake appears. As the Re is increased, the stagnation point moves slowly

forward on the cylinder and recirculation eddies develop in the rear of the

cylinder. These eddies grow and intensify until Re « 44 at which point

alternate shedding of vortices occurs from the sides of the cylinder.

The effects of these transitions can be seen in a plot of the drag

coefficient (Cp) as a function of Re [64]. The modification of C D due to the

change in flow behavior at Re » 4 appears larger than that due to the transition at Re » 44 for the data plotted on a log-log plot.

-65-
It seems clear that the variations in n

Q observed over Re ranging from

* 5 to “50 and for which vortex shedding is absent are due to the formation

of stagnation points and the growth of recirculation zones behind the heated

cylinder. For heat transfer from the cylinder this is a gradual and continuous change as reflected in the slowly decreasing values of n

Q as the Re range

of the measurements is increased. For Re < 5 the fluid flow corresponds to

nonseparated flow and no variation of n

Q

with Re is expected. This conclusion

is in agreement with experimental findings.

The above discussion indicates that the existence of Re regimes 1, 2, and

4 can be identified with the known flow behavior of gases over a cylinder.

Apparently, the hysteresis in heat transfer behavior which is the distinguishing characteristic of regime III has not been reported previously in the heat

transfer or hot-wire literature. However, such bistable regions are not

uncommon in fluid mechanics and the existence of a hysteresis for vortex

shedding from a cylinder should not be a surprise.

6.4 Accommodation Effects

In section 5 we argued that it is necessary to consider accommodation

effects in correlating our hot-wire measurements in flows of helium and

methane. For the remaining seven gases investigated a was assumed equal to

1. It seems advisable to question these assumptions as well as the plausibility of the values of a (a 0.48 for helium and a = 0.64 for methane)

determined.

-66-
Thermal accommodation behavior is poorly understood, but is thought to

arise on a molecular level due to the incomplete transfer of heat from a

heated surface to a molecule or atom striking the surface. The species

colliding with the surface only "accommodates" a fraction of the energy available from the surface with the result that the Boltzmann temperature of the

gas phase species near the surface is lower than expected based on the surface

temperature. Theories which treat this effect are usually based on the

temperature jump which exists at the boundary of the solid and gas. The

result of this temperature jump is that heat transfer is reduced from the

value which would be found if it did not exist.

The primary impression which results from a review of the literature on

thermal accommodation is the degree of complexity of the effect. In

particular, a great deal of variability in measured values of accommodation

coefficients is observed unless extremely clean and well-characterized

surfaces are employed [65]. This variability has severely limited attempts to

understand and predict values of a. The effects of such important system

parameters as surface temperature, gas pressure, and gas molecular weight on

thermal accommodation are poorly characterized and understood.

Despite the confusing and sometimes contradictory experimental findings

concerning the characteristics of thermal accommodation, the following general

observations are believed to be correct [58]. The degree of surface "cleanness" is critical to measured values of a. Cleaner surfaces result in lower

values of a. Values of a for polycrystalline or ordinary industrial matrices

"which are macros copically clean but contaminated with miscellaneous physically and chemically absorbed materials" [66] are expected to be substantially

-67 -
larger than those found for carefully prepared surfaces. Increasing molecular

weight and complexity are associated with increasing values of a. There is an

apparent effect of surface temperature on the degree of thermal accommodation,

but the experimental results are contradictory and no definitive trends are

discernible [58]. The effect of pressure on observed values of a is believed

to be small except for cases where the pressure determines whether or not gas

layers are present on the surface.

The extreme sensitivity of thermal accommodation to so many system

parameters makes prediction of values of a nearly impossible except for experiments made under highly controlled conditions. Measurements with hot-wires

and films certainly do not fall into this class. The surface structures of

these probes are not characterized well and are expected to contain very

uneven and variable surfaces due to the manufacturing processes used in their

construction. Furthermore, these probes are usually operated in environments

which are expected to continuously contaminate and modify the probe surface.

In this study involving the use of several different gases, the nature of

gases absorbed on the surface of the probe might also be expected to vary from

experiment to experiment.

The above discussion makes it clear that one cannot expect to use a

theoretical development to predict values of a for use in hot-wire anemometry.

The best one can hope to do is determine values of a phenomenologically and

compare the experimental results with the general characteristics of thermal

accommodation.

-68-
In the case of the work reported here, the first assumption which should

be questioned is the use of a = 1 for all of the gas flows over the hot-wire

and film with the exception of the wire results for helium and methane.

Andrews et al. [14] have listed selected values of a for a wide range of

surfaces and gases. The variability of results found in measurements of a is

quite clear from this table. Despite this, it is possible by using the

reported values of a along with the expected modifications due to the hot-wire

probe surface conditions to estimate that the value of a will lie between 0.9

and 1 for air on the hot-wire. Based on the molecular and complexity arguments described above, the same estimate will apply to all of the higher

molecular weight gases investigated (i.e. , all of the gases investigated

except helium and methane). Values of a should be very nearly equal to one

for all of these gases on the alumina surface of the hot-film.

These estimates of a can be used to evaluate the effect of thermal accommodation on the correlations of our hot-wire and film results, a enters our

calculations of Nu

a in eq. (9) as a factor in the thermal slip parameter

(A). In calculating A the value of what we have referred to as the

"classical" slip parameter is multiplied by (2-a)/a. For an a value of 0.9

this corresponds to an increase of 22% in the value of A over that expected in

the absense of thermal accommodation effects. Since values of A enter directly into eq. (12) the parameter <[> will also increase by 22%. <j> is used in eq.

(11) to correct Nu

&

for noncontinuum effects. For air the value of <j> is

calculated to be 1.85 if a = 1. By substituting this value of <J> into eq. (11)

along with the Kn for air one finds that a value of Nu

c = 1.050 is calculated

assuming a value of Nu = 1.00. The corresponding value of <J> is 2.260 for

00

a = 0. 9 which gives a value of Nu_ = 1.062 when Nu is corrected for non- a oo

-69 -
continuum effects. The value of Nu This result

fl is only 1% larger than Nu

c

»

shows that the use of a 1 gives results for Nu

e which differ only slightly

from values of Nu which would be calculated if the actual values of a for air

a were known. This conclusion will be true for the hot-wire measurements made

in air, Ar, CO2

, CgHg, CF^, SFg, and CF^Br. It justifies our conclusion that

it is possible to correlate the results for these gases (see figs. 16-18) in

terms of Nu

c

« The errors introduced in the correlation by the use of Nu

c instead of Nu

a are considerably smaller than those arising from other sources

such as uncertainty in molecular property values. This approximation will be

even more correct for the hot-film for which values of a should be even closer

to one than for the hot-wire.

The effect of accommodation on hot-wire anemometry measurements has been

investigated primarily for helium. Libby and coworkers [32-34] were the first

researchers to note the importance of the small accommodation coefficients for

this gas on the metal surfaces of hot-wire probes. For tungsten wires, values

of o = 0.030 [32] and a = 0.040 [33] were reported. A later study [34] of the

heat transfer from a platinum hot-wire gave a = 0.11. In all three cases the

values of a were shown to fall within the ranges reported in literature

measurements of helium on the appropriate surfaces. For the results on

tungsten, Baccaglini et al. [33] indicate that researchers in the area of gassurface interactions had questioned the very small values of a determined due

to the contaminated nature of the hot-wire surfaces. These workers [33] went

on to state that they "have sufficient confidence in our technique to take the

view that reasons for the low values of a require further study by researchers

concerned with gas-surface interactions”.

-70-
Our experimental determination of a for helium on the platinum surface of

the hot-wire gave a = 0.48. We feel that this result is an accurate estimate

of a not only because it yields values of Nu

a for helium which are well correlated with our results for seven other gases, but also because it gives a

value of nrt for data plotted as Nu_ versus Re11 which is in excellent agreement

with those found for measurements in other gases. The value of n

Q is highly

dependent on the magnitude of a as it changes from 0.35 for a = 1 to 0.45

for a = 0.48.

Our value of a for helium on the platinum hot-wire surface is consistent

with values found in the literature. Devienne [58] lists an average value

of a = 0. 54. However, as noted above, the reader is cautioned that the experimental results quoted by Devienne covered an extensive range of values.

It is difficult to suggest a reason as to why our value of a for helium

is so much larger than those found by Libby and coworkers [32-34]. The most

obvious possibility is that the surfaces of the probes are really different

enough to account for the variations. Even though the earlier works were not

corrected for the temperature variations of fluid properties, these variations

are so similar for helium and air that they can account for only a small

fraction of the differences observed.

Our results indicate that there is also a small accommodation effect

(a = 0.61) for methane. This is reasonable since methane is lighter than air

and has high energy vibrational modes. For these reasons, this molecule is

expected to have a value of a which is intermediate between that of helium and

air as observed.

-71 -
One other study has reported accommodation effects on hot-wire measurements. Brown and Rebollo [43] considered such effects in their development of

a concentration probe which incorporated a hot-wire. These workers indicate

that it is necessary to account for accommodation effects for Ar on platinum

in correlating their experimental results. This disagrees with our findings.

However, these workers did not correct their measurements for end heat loss

effects or the temperature dependence of gas properties. The neglect of such

effects can lead to erroneous conclusions concerning the importance of accommodation effects on heat transfer behavior.

Based on the above discussion, we feel that, while neither justified by

past practice or a strong theoretical basis, our procedures for treating

accommodation effects are sufficiently accurate for correlation purposes. Our

general findings concerning this effect are in good agreement with the trends

expected (e.g. , with regards to the effect of molecular weight).

6.5 Some Comments on Hot-Wire and Film Calibrations

There are a number of opinions in the literature concerning procedures to

be used in calibrating the response of hot-wire anemometers [8]. Disagreements concern such general questions as whether universal correlation laws

should be used at all and more specific questions such as which exponent to

use in fitting calibration data to equations of the form of eqs. (29) or (3).

While we do not wish to join this ongoing controversy, a few comments based on

the findings of this work are relevant. Perhaps the major finding is that

hot-wire and film responses to flows of different gases can be correlated. If

accommodation effects are expected to be absent and molecular property data

- 72 -
are available, it should be possible to predict the response of the probe in a

second gas following calibration in another gas. This finding could provide a

considerable savings of time and effort for experiments in which the flow

velocity of several gases are to be investigated. In generating these correlations it is necessary to correct the measurements for end heat loss effects,

temperature variation of properties, and "classical" noncontinuum effects.

The question of whether to use an equation of the form of eq. (3) often arises

in calibrating hot-wires. Variations of the value of n

Q with Re are often

discussed [7], Our work indicates that these variations are due primarily to

changes in the flow behavior with Re and to noncontinuum effects for small

probes and light molecular weight gases. The choice of an equation to fit

experimental calibration data should be based on the Re at which the probe

will be operated. Our findings indicate that either eq. (3) or eq. (29) will

be appropriate if the probe Re falls in the 0.5 to 6 range. The best value of

n to use will depend on exactly how the data is treated. For instance, all of

our hot-wire data, with the exception of helium, can be fit quite accurately

by assuming a value of n = 0.43 and using either eq. (29) or eq. (3) in terms

of Nu . Note that these forms will not allow a correlation of data for m

different gases. On the other hand a value on n = 0.45 is more suitable for

data treated as Nu

c or Nua

. In making such assessments, care must be taken to

identify noncontinuum flow effects. If measurements are made in the Re regime

from 6 to 40 the use of eq. (3) becomes questionable as we have shown that n

Q

values vary over this range. Measurements in the bistable range of Re from

« 35 to 55 should be avoided if at all possible.

-73-
6.6 Heat Transfer Studies Using Hot-Wires or Films

The hot-wire and film investigated here are employed primarily as

velocity measurement devices and this study is designed to investigate this

use of these devices. However, it is worthwhile to point out that these

devices are quite sensitive to changes in convective heat transfer and are

therefore able to provide quite detailed measurements of heat transfer from

cylinders.

This study has generated information on the behavior of heat transfer in

the Re range from 0.5 to 55. The results have allowed us to reach conclusions

concerning the effects of flow transitions on heat transfer behavior. A very

clear hysteresis in heat loss behavior has been observed and associated with

the presence or absence of eddy shedding from the cylinder. Use of a large

number of different gases have allowed us to identify the importance of fluid

property temperature dependence when the nondimensional parameters are calculated in terms of gas properties evaluated at T

m> All of these effects were

poorly characterized in the literature. Clearly, our investigation of heated

velocity probes has generated a great deal of new information of interest to

workers in the area of heat transfer.

It is also of interest to note that with further development measurements

of the type reported here might yield values of gas thermal conductivity or

viscosity. For instance, if the types of correlations we have used here are

shown to be reproducible and universal, it would be relatively simple to

determine a value of k for a gas if the corresponding value of v was available. Since accurate values of v are easier to measure experimentally, it

-74 -
will often be true that such values will be available when k values are not.

Measurements of this type should be possible which have a precision, and

perhaps accuracy, of » 1%.

Measurements of this type also seem to offer the promise of a method for

measuring accommodation coefficients for gases on surfaces in a relatively

straightforward and rapid manner. It may be possible to develop a technique

based on hot-wire anemometry for studying this complex and poorly understood

phenomenon.

7. FINAL REMARKS

This paper has described hot-wire and film calibrations which have been

made for a wide variety of gases. The use of gases having a broad range of

gas molecular properties has allowed us to isolate the corrections which must

be made in order to correlate the data in the usual form of a Nu as a function

of flow Re. These corrections include end heat losses due to conduction along

the probe, a very strong effect due to the temperature dependence of gas

molecular properties, "classical" noncontinuum effects, and accommodation

effects on heat transfer from the probe surfaces to the gases. All of these

effects are known and each has been used separately in the literature in

attempts to correlate hot-wire or film responses. This is the only study of

which we are aware that has investigated the effects of each correction on

heat transfer behavior and has been able to develop a systematic procedure

which allows the correlation of such a wide range of experimental data.

-75 -
In the case of the hot-wire the resulting Nu (Nu

c or Nu

fl depending on

whether or not accommodation effects are important) has a dependence on Re in

air which is very close to that found for previous hot-wire measurements where

end heat losses are absent and noncontinuum effects have been accounted for*

This observation offers the possibility that universal calibration laws may be

possible to describe the response of a hot-wire. The agreement for the hotfilm results is not nearly as complete. We have attributed this to the

approximation of a one-dimensional heat loss used to treat the heat losses at

the end of the probe. It should be possible to develop better models to treat

this process.

The finding that the heat loss behavior of these devices (as reflected in

the value of n

Q

for fitting data to the form of eq. (3)) is strongly dependent

on the nature of the flow around the probe should also allow a much better

prediction of probe response for a given Re range. These results should be

considered when choosing a probe for a particular application.

Only one overheat ratio has been used for the hot-wire. A slightly

larger value has been used for the hot-film. For this reason, this study

offers no conclusions regarding the effect of overheat ratio on heat transfer

behavior. It is impossible to conclude whether the probe temperature dependence noted in past experiments is due to property temperature dependencies or

to other possible effects such as buoyancy, etc. The role of offset voltage

setting on constant temperature hot-wire anemometry may also be important.

Further studies should be done to clarify this point.

-76 -
Our major goal for this work was to be able to understand and perhaps

predict the response of a heated cylindrical probe to convective flows of

different gases. The experiments reported here have discussed our findings

for pure gases.' For these gases great strides have been made toward our

goal. A second paper [5] will describe the results for mixtures of different

gases.

8. ACKNOWLEDGEMENTS

This research has been partially sponsored by the Air Force Office of

Scientific Research, Air Force Systems Command, USAF, under Grant Number

AF0SR-ISSA-00005. The U. S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation

thereon. The authors would like to thank Dr. Howard Baum of the Center for

Fire Research, NBS for many helpful discussions and Dr. Roger Simpson of

Virginia Polytechnic Institute and State University for sending us a copy of

unpublished work. Thanks are also due to Dr. John Rockett for a thorough

review of this rather long manuscript.

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-77 -
[4] Pitts, W.M. , McCaffrey, B.J. and Kashiwagi, T. , to be published.

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[9] Comte-Bellot, G. , Hot-Wire Anemometry, Ann. Rev. Fluid Mech. 8, 209-231

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[10] Comte-Bellot, G. , Hot-Wire and Hot-Film Anemometry, in Measurement of

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[12] Perry, A.E., Hot-Wire Anemometry, Oxford University Press, New York

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[13] King, L.V. , On the Convection of Heat from Small Cylinders in a Stream

of Fluid: Determination of the Convection Constants of Small Platinum

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373-432 (1914).

[14] Andrews, G.E., Bradley, D. and Hundy, G.F., Hot Wire Anemometry

Calibrations for Measurements of Small Gas Velocities, Int. J. Heat Mass

Trans. 15, 1765-1786 (1972).

[15] Collis, D.C. and Williams, M.J. , Two-Dimensional Convection from Heated

Wires at Low Reynolds Numbers, J. Fluid Mech. 6, 357-384 (1959).

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Forsch. Gebiet. Ingen. 4, 215-224 (1933).

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for Fine Wires, J. Fluid Mech. 51, 487-495 (1972).

[18] Kennard, E.H., Kinetic Theory of Gases, McGraw-Hill, New York (1938).

[19] Collis, D.C., Forced Convection of Heat from Cylinders at Low Reynolds

Numbers, J. Aero. Sciences 23, 697-698 (1956).

[20] Champagne, F.H. and Lundberg, J.L. , Linearizer for Constant Temperature

Hot Wire Anemometry, Rev. Sci. Ins. 37, 838-843 (1966).

-78 -
[21 J Lowell, H.H. , Design and Applications of Hot-Wire Anemometers for

Steady-State Measurements at Transonic and Supersonic Airspeeds,

National Advisory Committee for Aeronautics Technical Note NACA-TN2117

(1950).

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Heated Cylinders, Proc. Royal Soc. 280A, 486-527 (1964).

[23] Betchov, R. , L'influence de la conduction thermique sur les anemometres

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also Betchov, R. , Theorie non-lineaire de I’anemometre a fil chaud,

Proc. Kom. Ned. Akad. Wetenscha. 52, 195-207 (1949), translated in

Nonlinear Theory of a Hot-Wire Anemometer, National Advisory Committee

for Aeronautics Technical Memorandum NACA-TM1346 (1952).

[24] Koch, F.A. and Gartshore, I.S., Temperature Effects on Hot Wire

Anemometer Calibrations, J. Phys. E. Sci. Ins. 5, 58-61, (1972).

[25] Bruun, H.H. , On the Temperature Dependence of Constant Temperature HotWire Probes with Small Wire Aspect Ratio, J. Phys. E. Sci. Ins. 8,

942-951 (1975).

[26] Morrison, G.L. , Errors in Heat Transfer Laws for Constant Temperature

Hot Wire Anemometers, J. Phys. E. Sci. Ins. 9, 50-52 (1976).

[27] Corrsin, S. , Extended Applications of the Hot-Wire Anemometer, National

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[28] Conger, W.M. , The Measurement of Concentration Fluctuations in the

Mixing of Two Gases by Hot-Wire Anemometry Techniques, Ph.D. Thesis,

University of Pennsylvania (1965).

[29] Baid, K.M. , Measurement of Velocity in Gas Mixtures, M.S. Thesis,

Illinois Institute of Technology (1967).

[30] Tombach, I.H. , Velocity Measurements with a New Probe in Inhomogeneous

Turbulent Jets, Ph.D. Thesis, California Institute of Technology (1969).

[31] Kassoy, D.R. , Heat Transfer from Cylinders at Low Reynolds Numbers.

I. Theory for Variable Property Flow, Phys. Fluids 10, 938-946 (1967).

[32] Aihara, Y. , Kassoy, D.R. and Libby, P.A. , Heat Transfer from Cylinders

at Low Reynolds Numbers. II. Experimental Results and Comparison with

Theory, Phys. Fluids 10, 947-952 (1967).

[33] Baccaglini, G. , Kassoy, D.R. and Libby, P.A. , Heat Transfer to Cylinders

in Nitrogen-Helium and Nitrogen-Neon Mixtures, Phys. Fluids 12,

1378-1381 (1969).

[34] Wu, P. and Libby, P.A. , Heat Transfer to Cylinders in Helium and HeliumAir Mixtures, Int. J. Heat Mass Trans. 14, 1071-1077 (1971).

[35] Kramers, H. , Heat Transfer from Spheres to Flowing Media, Physica 12,

61-80 (1946).

-79 -
[36] Wasan, D.T., Davis, R.M. and Wilke, C.R., Measurement of the Velocity of

Gases with Variable Fluid Properties, AIChE J. 14, 227-234 (1968).

[37] Van der Hegge Zijnen, B.G. , Modified Correlation Formulae for the Heat

Transfer by Natural and by Forced Convection from Horizontal Cylinders,

Appl. Sci. Res. 6A, 129-140 (1956).

[38] Wasan, D.T. and Baid, K.M. , Measurement of Velocity in Gas Mixtures:

Hot-Wire and Hot-Film Anemometry, AIChE J. 17, 729-731 (1971).

[39] McQuaid, J. and Wright, W. , The Response of a Hot-Wire Anemometer in

Flows of Gas Mixtures, Int. J. Heat Mass Trans. 16, 819-828 (1973).

[40] Simpson, R.L. and Wyatt, W.G. , The Behavior of Hot -Film Anemometers in

Gas Mixtures, J. Phys. E. Sci. Ins. 6, 981-987 (1973).

[41] Khalifa, H.E., Kestin, J. and Wakeham, W.A., The Theory of the Transient

Hot-Wire Cell for Measuring the Thermal Conductivity of Gaseous

Mixtures, Physica 97a, 273-286 (1979).

[42] Fingerson, L.M. and Ahmed, A.M. , Operation and Application of Cooled

Film Sensors for Measurements in High-Temperature Gases, in Measurements

in Heat Transfer, Eckert, E.R. and Goldstein, R. J. , eds. , 579-596,

McGraw-Hill, New York, 579-596 (1976).

[43] Brown, G.L. and Rebollo, M.R. , A Small, Fast-Response Probe to Measure

Composition of a Binary Gas Mixture, AIAA J. 10, 649-652 (1972).

[44] Tombach, I.H. , An Evaluation of the Heat Pulse Anemometer for Velocity

Measurement in Inhomogeneous Turbulent Flow, Rev. Sci. Ins. 44, 141-148

(1973).

[45] Touloukian, Y.S., Liley, P.E. and Saxena, S.C., Thermophysical

Properties of Matter, The TPRC Data Series, Vol. 3, Thermal

Conductivity —Nonmetallic Liquids and Gases, IFI/Plenum, New York

(1970).

[46] Touloukian, Y.S. , Saxena, S.C. and Hestermans, P., The Thermophysical

Properties of Matter, The TPRC Data Series, Vol. 11, Viscosity,

IFI/Plenum, New York (1975).

[47] Touloukian, Y.S. and Makitas, T. , Therraophyiscal Properties of Matter,

The TPRC Data Series, Vol. 6, Specific Heat —Nonmetallic Liquids and

Gases, IFI/Plenum, New York (1970).

[48] Ehya, H. , Faubert, F.M. and Springer, G. S. , Thermal-Conductivity

Measurements of Propane and N-Butane in the Range 300 to 1000 Deg K,

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[49] Thermophysical Properties of Refrigerants, American Society of Heating,

Refrigeration, and Air Conditioning Engineers, New York (1976).

[50] Hellemans, J.M. , Kestin, J. and Ro, S.T., The Viscosity of CH^, CF^ and

SF^ Over a Range of Temperatures, Physica 65, 376-380 (1973).

-80-
[51] Bakulin, S.S. and Ulybin, S.A. , Thermal Conductivity of Sulfur

Hexafluoride at Temperatures of 230-350°K and Pressures Up to 50 MPa,

High Temper. 16, 46-52 (1978), Translation of Tep. Vys. Temper. 16,

59-66 (1978).

[52] Adler, L.S. and Yaws, C.L. , Physical and Thermodynamic Properties of

Sulfur Hexafluoride, Sol. State Tech. 18(1), 35-38 (1975).

[53] Touloukian, Y.S., Powell, R.W. , Ho, C.Y. and Klemens, P.G.,

Thermophysical Properties of Matter, The TPRC Data Series, Vol. 1,

Thermal Conductivity—Metallic Elements and Alloys, IFI/Plenum, New York

(1970).

[54] Touloukian, Y.S., Powell, R.W. , Ho, C.Y. and Klemens, P.G.

,

Thermophysical Properties of Matter, The TPRC Data Series, Vol. 2,

Thermal Conductivity —Nonmetallic Solids, IFI/Plenum, New York (1970).

[55] Geller, V.Z., Gorykin, S.F., Zaporozhan, G.V. and Voitenko, A.K. , Study

of Thermal Conductivity of Freon-13Bl and Freon-23 Over a Wide Range of

State Parameters (in Russian), J. Eng. Phys. 24, 581-588 (1975).

[56] Kreith, F. , Principles of Heat Transfer, 3rd Ed., Harper & Row, New York

(1973).

[57] Kays, W.M. and Crawford, M.E. , Convective Heat and Mass Transfer, 2nd

Ed., McGraw-Hill, New York (1980).

[58] Devienne, F.M. , Accommodation Coefficients and the Solid-Gas Interface,

in The Solid-Gas Interface, Vol. 2, Flood, E.A. , Ed., 815-828, MarcelDekker, New York (1967).

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Technology, Vol. 5, McGraw-Hill, New York (1929).

[60] Kannuluik, W.G. and Carman, E.H. , The Temperature Dependence of the

Thermal Conductivity of Air, Aust. J. Sci. Res. 4, 305-314 (1951).

[61] Goldstein, S. (Ed.), Modern Developments in Fluid Dynamics, Vol. I,

Oxford, London (1938).

[62] Whitaker, S. , Fundamental Principles of Heat Transfer, Pergamon Press,

New York (1977).

[63] Whitaker, S. , Introduction to Fluid Mechanics, Prentice-Hall, Englewood

Cliffs (1968).

[64] Schlichting, H. , Boundary Layer Theory, 4th Ed., Translated by

Kestin, J. , McGraw-Hill, New York (1960).

[65] Thomas, L.B., Thermal Accommodation of Gases on Solids, in Fundamentals

of Gas-Surface Interactions, Saltsburg, H. , Smith, J.N. , and Rogers, M.

,

Eds., 346-369, Academic Press (1967).

-81 -
[66] Flood, E.A. and Hobson, J.P. , Simple Kinetic Theory and Accommodation,

Reflection and Adsorption of Molecules, in The Solid-Gas Interface,

Vol. 2, Flood, E.A., Ed., 829-846, Marcel-Dekker, New York (1967).

-82 -
NOMENCLATURE

A

As

B

c

C

CD

C

i

C

P Cv

C

o

D

D

o

E

g

Gr

h

I

k

KA

kB

K

v Kk

Kn

exponent of temperature loading factor or temperature dependence

of transport property in Nusselt expression, see eq. (4)

intercept in linear Nu vs. Re expression, see eq. (3)

cross sectional area of sensor

slope in linear Nu vs. Re expression, see eq. (3)

molecular speed, see eq. (33)

parameter used in eq. (46)

drag coefficient for flow around a cylinder

i - 1,2 symbolic constants (replaced by more specific A and B), see eq. (2)

specific heat at constant pressure

specific heat at constant volume

coefficient in heat loss equation, see eq. (19)

sensor diameter

coefficient in heat loss equation, see eq. (19)

voltage drop across wheatstone bridge of anemometer system

acceleration due to gravity

Grashof number. defined as

convective heat transfer coefficient

current through probe leg of anemometer circuit

thermal conductivity

see eq. (41)

see eq. (42)

see eq. (43)

see eq. (44)

Knudsen number, defined as X/D

-83 -
active length of wire or film

cold length parameter used in calculating end loss, equal

to 1//C“

o molecular weight of gas

exponent of Reynolds number in expression for Nusselt number

exponent for use in eq. (46)

2 optimized value of n which gives a minimum y

hD

Nusselt number, defined as -

k

pressure drop used for velocity measurement, see eq. (31)

electrical power dissipation

Peclet number equal to Re x Pr

Prandtl number, defined as C y/k

P

heat

gas constant

resistance of standard resistance placed in series with the sensor

electrical resistance of sensor

total electrical resistance of sensor log

Reynolds number, defined as

V thickness

temperature

gas velocity

axial distance along probe from center and exponent of

temperature dependence of viscosity, see eq. (12)

exponent of temperature dependence of thermal conductivity, see

eq. (12)

accommodation coefficient, see eq. (7)

thermal coefficient of probe electrical resistance

slip parameter, see eq. (10)
6* parameter associated with thermal slip, see eq. (27)

Y Cp/Cv

A temperature jump distance, see eq. (8)

e total thermal energy of molecule, see eq. (7)

0 temperature difference, T - T

00

0 o overheat for sensor with infinite £/D, see eq. (22)

0' temperature jump, A/X

X mean free path of gas molecule

u dynamic viscosity

V kinematic viscosity

P density

P r (9) probe resistance as a function of 0

rarefied effects parameter, see eqs. (11) and (12)

2X goodness of fit parameter for linear least square fits of

results to eq. (3) or eq. (29)

hot-wire response parameter defined as in eq. (28)

Subscripts

a corrected for sensor finite aspect ratio (if necessary),

"classical" rarefied gas effects, and accommodation effects

A1 aluminum

c conductive heat transfer, see eq. (14), or Nu corrected for

sensor finite aspect ratio and "classical" rarefied gas effects

com computed

h convective heat transfer

i incident on surface, see eq. (7)

j joulean resistive heating

m measured (uncorrected); such as Num m property determined at mean of sensor, s, and ambient

temperatures, »; such as Tm, vr

, etc.

-85-
platinum

quartz

reflected from surface, see eq. (7)

sensor

gas under investigation

ambient, such as T

00 or corrected for end conduction loss, as in Nu , i.e.

00 gas 1

gas 2
Table 1. Values found by Collis and Williams [15] for use in eq. (4)

0.02 < Re < 44 44 < Re < 140

n 0.45 0.51

A 0.24 0

B 0.56 0.48

a - 0.17 - 0.17

Table 2. Suppliers and purities of gases used in this work

Gas Supplier Purity

Air Air Products Dry

n

2

In house > 99.5%

He Air Products > 99.995%

ch4 Matheson > 98%

Ar Air Products > 99.998%

co

2 Roberts Oxygen > 99.9%

c 3h8 Matheson > 99%

cf 4 Dupont > 99.7

%

sf 6 Matheson > 99.8%

CF3Br Dupont > 99.7%

-87 -
I

j

Table 3. Sources for the properties of gases used in this investigation

Gas k

Air [45] n2 [45]

He [45]

ch4 [45]

Ar [45] co

2 [45]

C 3H

8 [48]

cf 4 [49]

CF 3 Br [49]

sf 6 [51]

y S

[46] [47]

[46] [47]

[46] [47]

[46] [47]

[46] [47]

[46] [47]

[46] [47]

[50] [49]

[49] [49]

[50] [52]

Table 4. Hot-wire and film properties used for calculations

D(cm) £(cm) R (n)

s T

s <K) Tm

(K) a (K l

) P

2 k (cal/s»cm »k)

s

Wire 0.0004 0.125 12.02 569 431 0.00303 0.33

Film 0.0051 0.10 9.60 650 472 0.00232 0.0075

-88-
Table 5. Values of n

Q resulting from fits of E

2 = A’ + B’Un are listed

for the hot-wire measurements made with different gases.

Values of A' and B’ are reported for n = 0.43. Also included

are selected slopes and intercepts resulting from least square

fit of Numand Nu

fl to eq. (3) with n = 0.43. These sets of

data correspond to those seen in the various figures. The Re

ranges over which the various measurements were made are also

listed.

Gas n

o A'

Air 0.43 3.749

0.42 3.716

0.43 3.716

0.45 3.768

0.43 3.742

0.43 3.723

ch4 0.44 5.667

0.43 5.667

0.43 5.651

Ar 0.41 2.852

0.42 2.865

C

3

H

8 0.44 4.058

0.43 4.051

0.43 4.051

c°2 0.42 3. 180

0.43 3.167

0.43 3.206

0.44 3.184

0.44 3.200

He 0.26 14.92

0.29 15.06

sf6 0.42 2.575

0.41 2.656

CF 3Br 0.42 2.212

0.42 2.192

cf4 0.42 2.987

B’ Re Range Am

0.355 0.09-1.4

0.340 0.12-1.3 0.427

0.336 0.16-1.2

0.339 0.23-1.2

0.339 0.18-1.2

0.339 0.19-1.2

0.494 0.21-1.1 0.423

0.488 0.17-1.1

0.497 0.19-1.1

0.240 0.12-1.4 0.487

0.241 0.12-1.4

0.658 1.28-1.8

0.658 1.27-1.8 0.492

0.655 00 •

CM 1 •

0.370 0.21-2.4 0.478

0.368 0.23-2.2

0.369 0.24-2.2

0.366 0.27-2.2

0.365 0.26-2.2

0.431 0.18-0.7 0.323

0.423 0.10-0.8

0.517 1.3-5.

5

0.519 1. 6-6.1 0.489

0.382 0.94-5.1

0.380 1.3-6. 2 0.538

0.431 0.50-4.0 0.470

Bm

A00 B

00

0.664 0.258 0.617

0.663 0.301 0.624

0.671 0.250 0.622

0.825 0.292 0.743

0.741 0.263 0.691

0.372 0.275 0.356

0.765 0.229 0.717

0.744 0.213 0.719

0.711 0.246 0.665

-89 -
Table 6. Values of n

Q are listed which result when the hot-wire data for

the different gases are fit to eq. (3). Values of Ac , B c , Aa and

B

fl are reported using n 0. 45 in eq. (3). The last two columns

list the values of A and B which result when Aa and B

a are modified using the (K^)x and (Kg)x values as defined by eqs. (41)

and (42), respectively.

Gas n

o Ac B

c no Air 0.44 0.272 0.650 0.44

ch4 0.46 0.318 0.635 0.47

Ar 0.44 0.261 0.660 0.44

G

3

H

8 0.45 0.322 0.725 0.45

co2 0.44 0.282 0.698 0.44

He 0.35 0.289 0.410 0.45

sf

6

0.43 0.264 0.695 0.43

CF^Br 0.43 0.252 0.690 0.43

cf 4 0.44 0.270 0.659 0.44

Aa

B

a Aa (Ka^x Ba /(KB )

0.272 0.650 0.272 0.650

0.312 0.679 0.282 0.640

0.261 0.660 0.274 0.644

0.322 0.725 0.274 0.651

0.282 0.698 0.272 0.643

0.281 0.630 0.284 0.646

0.264 0.695 0.256 0.640

0.252 0.690 0.244 0.654

0.270 0.659 0.269 0. 636

-90-
Table 7. This table lists results for the hot-film calibration in flows of

different gases. Values of n

Q

for fits of the data to eq. (29) are listed along with the corresponding values of A* and B’ found

for n = 0.45. The Re range covered by each set of measurements

is also included. The last four columns contain selected Am, Bm, A and B values which result from fits of the data in terms of

Num

and §u

c to eq. (3) with n = 0.45.

Gas no

A’ B' Re Range ^m ®m Ac B

c

Air 0.44

0.45

0.45

12.04

11.82

11.69

1.387

1.398

1.392

1.3-13.5

1.6-

13.4

1.6-

12.5 1.089 0.854 0.434 0.807

ch4 0.42

0.44

17.36

17.15

2.284

2.251

2.6-12.9

2.1-11.2 0.963 0.892 0.529 0.847

Ar 0.45 9.91 0.981 1.4-14.8 1.362 0.860 0.416 0.814

c 3h8 0.37

0.38

0.39

14.99

14.46

14.50

2.833

2.816

2.859

7.0-

42.8

4.3-43.3

4.0-

42.8

1.247 0.962 0.592 0.911

co

2 0.41

0.41

11.00

11.00

1.547

1.551

1.5-25.0

1.9-23.8 1.272 0.926 0.457 0.872

He 0.39 40.34 2.550 1.0-2.

5

0.694 0.730 0.506 0.738

SF

6 0.37

0.35

10.68

10.15

2.083

2.086

14.4-40.2

7.9-38.4 1.447 0.899 0.448 0.843

CF ^Br 0.38 8.91 1.568 8.2-34.0 1.734 0.935 0.414 0.875

Table 8. Values of C and n' reported by Hilpert [16] for use in eq . (46)

1 < Re < 4 4 < Re < 40 40 < Re < 400

C 0.891

n' 0.330

0.821

0.385

0.615

0.466

-91 -
FIGURE CAPTIONS

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

The response of the hot-wire to varying velocities of an air flow is

plotted as E(volts) versus U(cm/s). The solid curve is that

predicted based on a least squares curve fit of the experimental

data to eq. (29). A value of n = 0.43 has been used.

Values of ^ (defined by eq. (36)) are plotted as a function of the

value of n used to fit the experimental results of fig. 1 to eq.

(29).

Representative data are shown for the response of the hot-wire to

flows of eight different gases. The experimental results are plotted as E^(volts^) versus U^* ^(cm/s )^* Solid lines correspond to least squares curve fits of the data to eq. (29) with

n = 0.43.

Representative values of measured Nusselt number (Num) calculated

using eq. (37) are plotted as a function of Re^*^. Results for

nine different gases are included. With the exception of the helium

results, the data used to calculate Num

are those shown in fig. 3.

The results shown in fig. 4 have been corrected for the finite

aspect ratio of the hot-wire. The resulting Nusselt numbers (Nu )

00 are plotted as function of Re^*43 #

The results shown in fig. 5 have been corrected for "classical”

rarefied gas effects. The resulting Nusselt number (Nu

c

) corresponds to that expected for an infinite hot-wire placed in a continuum gas. Accommodation effects are assumed to be negligible. Values of Nu

c are plotted as a function of Re^*^.

-92 -
Figure 7

Figure

Figure

Figure

Figure

. Representative data are shown for the response of a hot-film to

flows of seven different gases. The experimental results are plotted as E^(volts^) versus U®* ^(cm/s )®# Solid lines correspond to linear least squares curve fits of the data to eq. (29)

with n = 0.45. Note the curvature in the plots for SF^, CF 3Br, and

C^Hg. The curvature indicates that 0.45 is not the best value for n for fitting data of these gases to eq. (3).

8. Representative values of measured Nusselt numbers (Num

) for the

response of the hot-film in different gases are plotted as a function of Re®*^5

. The values of Num

correspond to the experimental data shown in fig. 7 plus a data set recorded for flows of

helium gas.

9. Hot-film results are plotted as Num

versus Re®*^ for CgHg, SFg, and

CFgBr. These results span the Re range (Re » 44) where the onset of

vortex shedding from the heated cylinder is known to commence. This

flow transition results in the markedly different heat transfer

behavior observed in the high and low Re regions.

10. Values of Num

versus Re®*^ are plotted for C^Hg (a), SFg ()> and

CFgBr (O) over the Re range where the onset of vortex shedding from

cylinders is known to occur. The measurements show a clear

hysteresis in the data depending on whether the gas flow velocity is

increased to the measurement value (denoted by filled symbols) or decreased to the measurement value (denoted by open symbols). For reasons of clarity, Num

values for SFg have been offset downward by

0.188 and those for CF^Br have been offset upward by 0.063.

11. Values of Nu

c versus Re^*^“* are shown for the hot-film. The results

given in fig. 8 have been corrected for finite aspect ratio and

rarefied gas effects to give the data shown here.

-93-
Figure 12 . The parameters A

c (a) and BQ (0) which result from fits of the hotwire data for nine different gases to eq. (3) are plotted on a loglog plot as a function of Pr. Note that the value of B

c for helium

is not included on the plot. This figure shows that there is no apparent dependence of these parameters on Pr.

Figure 13. The parameters Ac (a) and B

c (0) found for the hot-wire measurements

made in nine different gases are shown on a log-log plot as a function of f u 1 . Note that B_ for helium has been omitted. This

plot indicates a dependence of A

Q on (vm) x

» The straight line is

the result of a linear least squares curve fit of log(Ac

) versus log(ym) x

. Results of this fit are used to give eq. (40).

The filled symbols correspond to values of Aa and B a calculated for

helium and methane.

Figure 14. The parameters A

c (a) and B

c (0) found for the hot-wire measurements

made in nine different gases are shown on a log-log plot as a function of (v /v ]

. Note that B_ for helium has been omitted.

This plot indicates a dependence of B

c on (v The straight

line is the result of a linear least squares curve fit of log(B

c

) versus logfv /v ) . Results of this fit are used to give eq.

' m oo y x

(39). The filled symbols correspond to values of A

a and B a calculated for helium and methane.

Figure 15. The correlation of the experimental results which is obtained when

the values of Nu

c for the hot-wire in different gases are corrected

for variations in (v /v^) and y m is shown. Values of Nu c-(KA) X

are plotted as a function of Re°*^ 5 (KB ) x

. (KA ) X and (Kg) x are given by

eqs. (40) and (39), respectively. With the exception of the results

for helium, an excellent correlation of the data is obtained.

Figure 16. This correlation of experimental results is the same as that given

in fig. 15 with the exception that the helium and methane results

have been corrected for accommodation effects. The correlation of

results for measurements made in all nine gases is now found to be

excellent.

-94 -
Figure 17. Values of Nu measured for the hot-wire are multiplied by (K ") (as

given by eq. (43)) and plotted as a function of Re^*45 # The results

for helium have not been corrected for accommodation effects.

Figure 18. Values of Nu c measured for the hot-wire are multiplied by (Kj

c

) x

(as

given by eq. (44)) and plotted as a function of Re^*^^. The results

for helium have not been corrected for accommodation effects.

Figure 19. Values for Nu

c ~(K^)x versus Re^*^(Kg)x are plotted for the eight

different gases investigated with the hot-film. Expressions for

(Kb ) x and (K^) x

are same as those used for the hot-wire (i.e.,

eqs. (41) and (42)).

Figure 20. Correlations reported by Andrews et al. [14], Collis and Williams

[15], and Fingerson and Ahmed [42] are employed to give the solid

lines shown (A, C, and D; respectively) as Nu versus Re®»45 for the

hot-wire placed in a flow of air. Tm

is assumed to be 431 K

and T is assumed to be room temperature. The correlation which has

00 been obtained in this work (eq. (45)) is also included in the plot

(line B). The dashed line is the result which is found if the

correlation of Andrews et al. [14] is modified to reflect more recent literature measurements of the thermal conductivity of air

than that used in [14] (see the discussion in the text).

Figure 21. Correlations reported by Andrews et al. [14], Collis and Williams

[15], and Fingerson and Ahmed [42] are employed to give the solid

lines shown (A, C, and D; respectively) as Nu versus Re^*^ for the

hot-film placed in a flow of air. Tmis assumed to be 472 K

and T is assumed to be room temperature. The correlation which has

00 been obtained in this work for the hot film is also included on the

plot (line B). The dashed line is the result which is found if the

correlation of Andrews et al. [14] is modified to reflect more recent literature measurements of air conductivity than that used in

[14] (see the discussion in the text).

-95 -
3

2. ,

FIGURE 1: Typical Hot-Wire Response

Air. Line: Eq. (29) n«0.43

NO C\l CN CNI (S||0A) 3

96

U

(cm/s)
FIGURE 2: Sensitivity Analysis for n

CMX

97

0.38

0.40

0.42

0.44

0.46

0.48
FIGURE 3: Hot-Wire Response for Eight Gases

Lines: Eq. (29)

?(s*|oa) z 3

98

U°.43

(cm/s)

0

* 43
FIGURE 4: Measured Nusselt Number vs. Reynolds Number to the 0.A3 power

o

CN

if)

in

o

E

D

99

Re

0

- 43
FIGURE 5: Nusselt Number Corrected for End

Conduction Losses

O

<N

in

in

o

100

Re

0

- 43
2.0

FIGURE 6: Nusselt Number Corrected for

"Classical" Rarified Gas Effects

and Finite Aspect Ratio

CN

CD

in

o

K)

O

oD

101

Re

0

- 45
FIGURE 7: Typical Hot-Film Response.

Lines: Eq. (29) n=0.45

i(sroa) z3

102

U

0

' 45

fcm/s)

0

- 45
FIGURE 8: Measured Nusselt Number for Hot-Film

E

=3

103
FIGURE 9: Response of Film at Onset of

Vortex Shedding j

in

o

<D O'

104
FIGURE 10: Hysteresis Near Flow Transition

(Re ^44)

CM

CD

O

CD

00

Lfj

CD

in

in

CM

in

o

in

oo

E

3

ino

<1) CC

105
FIGURE 11: Corrected Nusselt Number for Hot

Film

=3

106

Re

0

- 45
FIGURE 12: Failure of Correlation with

Prandtl Number

fr~r

xK> o

-O

IW

• • o o

<o CO O

cs o

o

O

°o

\v i—i—i—i—i—i

, r*i L «n—r~^i

00 Xto O

<1

CJ

O

X

o

<1

cs o

o

<1

o

00

oo

CD

''t

CM

CL

mto U_

o

O

D

o

d> X

<1

L_ »—D

<1

CD

to Lx. o

<1

O

oo

CD

<

i & i 4\\ I"- o

to

o

to

o

CO

o

<

1

! 1

i i i i 1i i to o> m ro to CM eg CM

d o o o o

CD

<D

o m

o

107
FIGURE 13: Correlation of Intercept with

Viscosity (Hot-Wire)

CD <

108

(gs

_1
FIGURE 14: Correlation of Slope with Ratio

of Kinematic Viscosities.

(Hct-Wire)

CD <

109

1.90

1.94

1.98

2.02

2.06

2.10

2.14
2.0

FIGURE 15: Hot-Wire Results Incorporating All

Corrections Except Thermal

Accommodation Effects

110

Re°-

45 (K

)B
FIGURE 16: Hot-Wire Results INcluding

Accommodation Corrections to He and

CHA

V>0-Dn N

111

Re°«(K

)B
FIGURE 17: Simple Kinematic Viscosity

Correlation (no Accommodation)

W"N

112

Re

0

- 45
FIGURE 18: Simple Thermal Conductivity

Correlation (no Accommodation)

K)

x Cx)0n N

113

Re

0

- 45
FIGURE 19: Hot-Film Results with All

Re°-

45 (K

)B
FIGURE 20: Hot-Wire. Comparison to Literature

• • • o o o

in

o

<1) OL

o

3

115
FIGURE 21: Hot-Film. Comparison to Literature

116
NBl'l 1 4 A ihi- v, g.BOl u,». OpH r, Q p comm, BIBLIOGRAPHIC DATA

SHEET (Sue Instruction s)

1. PUiLIGATIQN ©A

REPORT NO.

N18IR-85/3203

7.'

"' P f©'"m

pi g Organ. Repart Ne.

July 1985

4. title AND SUBTITLE

Rosponse Bghavler ef Hot-Wiirt§ ind Film§ to Flew§ of Difftrant Gi§§§

5. AUTHOR(S)

William M. Pitts and Bernard J. McCaffrey

t. PERFORMING ORGANIZATION (If Joint or other than NBS, see Instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. AFOSR-ISSA-00005

8 . Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

Air Force Office of Scientific Research

Air Force Systems Command, USAF

Washington, DC

10. SUPPLEMENTARY NOTES

| | Document describes a computer program; SF-185, FIPS Software Summary, Is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant Information, If document Includes a significant

bibliography or literature survey, mention It here)

Measurements of the voltage output for hot-wire and film anemometers placed in flows

of nine different gases have been made as a function of flow velocity. It has been

possible to correlate the measurements quite accurately by treating the data in terms

of suitably defined Reynolds and Nusselt numbers. In order to obtain these

correlations it has been necessary to consider and correct for the effects of probe

end conduction losses, temperature dependencies of gas molecular properties,

flow slip at the probe surfaces, and gas accommodation. With the exception of the

results for helium (for which accommodation effects are strong), the most

important correction is shown to be that for the different temperature dependencies

of the gas molecular properties. This finding is contrasted with previous studies

which have assumed that the largest effect among different gases was due to variations

in the Prandtl number. The importance of the nature of the flow over the cylindrical

devices to the heat transfer behavior is described. A previously unreported

hysteresis in the heat transfer behavior for RE 44 has been characterized and

attributed to the presence or absence of eddy shedding from the heated cylinder.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons

Accommodation coefficients; convection; cylindrical bodies; heat transfer;

hot wire anemometry; molecular effects; reynolds Number; Vortex Shedding.

13. AVAILABILITY

[X] Unlimited

[33] For Official Distribution. Do Not Release to NTIS

[X] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

|~xl Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

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15. Price

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