Development of a thin film heat-transfer gauge for shock-tube flows

FOR SHOCK-TUBE FLOWS

BY

B. W. TAYLOR

, "

','

DEVELOPMENT OF A THIN FILM HEAT-TRANSFER GAUGE FOR SHOCK-TUBE FLOWS

BY

B. W. TAYLOR

.' r'

..

The author wishes to express his thanks to Dr. G.N . Patterson for providing the opportunity to attempt this investigation, to Dr. 1.1. Glass and to Dr. J .G. Hall for their suggestions and criticisms.

Thanks are also extended to Dr. J . H. deLeeuw for his many helpful suggestions .

The financial assistance received from the Defence Research Board is gratefully acknowledged .

'

.

SUMMARY

The use of liquid platinum pairrt for the manufacture of thin film resistance thermometers has been investigated and films so made have been used in the measurement of heat transfer rates in the UTIA 2 x 2 inch strong shock tube. The heat transfer model was a circular cylinder placed normal _to the flow direction and having a thin film drawn at its stagnation line." The response of this thin film to the hot gas flow generated in the shock tube was used to determine heat transfer rates using the results of classical heat conduction theory. These heat transfer rates were compared with th0se predicted theoret-ically by L. Lees in Reference 5 and they were al~o used to calculate a value for the stagnation temperature of the flow.

T ABLE OF CONTENTS

NOTATION

1. INTRODUCTION

II. EXPERIMENTAL METHODS

2.1 2 x 2 inch Combustion Driven Shock Tube 2.2 Heat Transfer Model

2.3 Film Construction 2 . 4 Gauge C alibr a tion 2; 5 Operating Bridge

III. THEORETICAL CONSIDERATIONS 3.1 Thermometer Theory

3.2 Theoretical Heat Transfer Considerations IV. EXPERIME NT AL RESULTS AND DISCUSSION V. CON'CLUSIONS REFERENCES APPENDIX FIG URES 1 - 13 , ' Page ii 1 1 2 2 3 5 6 6 9 11 13 14 16

I R A K

v

E T t

1

h p q k Pr

u

s a ( ii ) NOTATION current

(with subscript) resistance

(without subscript) radius of model heat transfer area of thin film

thermal diffusivity

voltage applied to bridge

voltage measured by oscilloscope temperature

time

thickness of film enthalpy of gas subscripts

gas press ure heat transfer rate thermal conductivity s pecific he at Prandtl number flow velocity T - stagnation w - wall 1 - initial

distance along model surface normal to the axis sound speed

density

f-'

viscosity

'f

ratio of specific heats

For regions designated by subscripts 1, 2, 3 or T applied to flow quantities see Fig. 1.

{ 1 }

1. INTRODUCTION

The development of the high speed rocket has made the problem of aerodynamic heating one of prime importance in recent ye ars. For example. return to the surface of the e arth from a satellite orbit is directly dependent on the ability to manufacture

materials which are able to withstand the high temperatures developed. In order to design acceptable materials it is imperative that there be reliable data from which the temperature environment of the vehicle can be estimated.

In a straight shock tube. flows having a duration of 50 -100 microseconds with stagnation conditions encountered at altitudes of 50.000 to 100.000 feet and velocities up to 20.000 feet per second are readily generated. The development of the thin film heat transfer gauge has been a direct result of the desire to use the flows generated in the shock tube to measure the heat transfer rates and stagnation temperatures occurring at the flight conditions which they simulate.

The gauge is essentially a metallic resistance thermometer mounted on an insulating backing material. The metallic film is made thin enough {about .1 micron} that its response time to temperature changes is of the order of .1 micro-second with the result that the flow times available in the shock tube are ample to obtain temperature'":" time histories of the surface of a model placed in the flow. From these histories, values of the heat transfer rate and stagnation temperatures can be ca1culated by making use of classical heat conduction theory and recent boundary layer theory. In Fig. 1 the flow regions in the shock tube are shown on an x-t diagram; the time interval designated by l:lt represents the duration of the quasi-steady flow w hich is the region useful for testing purposes .

Il. EXPERIMENTAL METHODS

2. 1 2 x 2 inch Combustion Driven Shock Tube

In the present configuration the UTIA 2 x 2 inch

com-bustion driven shock tube consists of a three foot comcom-bustion chamber. anineteen foot channel section and a surge chamber. T he combustion chamber is charged with a stoichiometrie mixture of hydrogen and oxygen diluted with he lium to serve as the driver. T his mixture is ignited by electrically exploding a piece of

#

42 copper wire stretched lengthwise through the centre. of the combustion chamber. The

resulting explosion burst the .035 inch thick scribed brass diaphragm separating the channel from the combustion chamber into four petals _, and produces a shock wave in the channel section. A detailed

description of the operation and instrumentation of this shock tube is given in Ref. 1.

In the present work the shock speed is found using two

wall-mounted thin film heat transfer gauges as shock detectors. The

circuitry connected with this application of the thin films is shown schematically in Fig. 2. The shock tube has a total of 10 ports

spaced at one foot intervals and two of these are used to set the

gauges exactJ.y one foot apart. By connecting the two detectors in series it is possible to record the jump due to the arrival of the

shock wave from each on the same oscilloscope trace so that the time

taken for the shock wave to travel between the detectors can be

measured. This then supplies all the information necessary to find

the wave speed.

2.2 Heat Transfer Model

The heat transfer model support (F ig. 3) was designed

to fit the existing ports in the tube. lt is essentially a circular

can-tilevered shaft having one end butted against the side of the tube

opposite the port. To prevent mechanical vibrations of the shaft due

to the impact loading at the arrival of the shock wave this end is

fitted with a lead washer • which is compressed when the support is

fastened in POSitiOll, thus providing a friction support for the

other-wise free end of the cantilever .

The heat transfer model itself is a one inch long

cir-cular glass cylinder having an outside diameter of 12.5 mmo and a

wall thickness of 2 mm. The gauge is a thin platinum film placed

parallel to the axis of the cylinder, the manufacture of which is

described in Sec. 2.3. The glass cylinder is placed on a brass

sleeve which fits over the shaft of the support described in the last

paragraph and this assembly is secured in position by a brass nut

having the same diameter as the model. One end of the film is

con-nected to a wire sealed in a hole through the shaft; the other is

connected to the brass nut and thence to the support which serves as

the ground connection to the gauge. Both connections are made

using silver circuit paint. which proved more convenient than solder

even though the paint required a three to four hour drying period

before it could be used as an electrical conductor. 2.3 Film Construction

In Reference 2. Vidal has constructed a table in which the films commonly used in heat transfer work are compared for

ease of application. durability and uniform ity. His conclusion is

that the Hanovia paints have an advantage in their ease of application and durability. although they may have less uniform thickness.

lt has been found here that Hanovia Liquid Bright

Platinum # 05

-x

gives very good results provided that adequate care

is exercised in its application . The practice is to clean the glass

'f

( 3 )

platinum paint 1/16" wide. about 1" long and as thin as possible is

drawn on the glass. This is then heated in a ventilated oven to about

12500F and removed immediately to prevent distortion of the glass.

If at this stage the film has bonded to the glass it appears quite bright;

if not. it is dull and can be readily rubbed off with a cloth. If the film

has been properly applied. the resistance of the initial coat is about

700 ohms per inch. Three coats similarly applied and individually

fired give a resistance of about 100 ohms per inch, which is the value

used in the present experiments.

When inspected against a strong light. drawn films shovi

fewer colour gradatlóns . than carefully applied painted films. so the

chief objection to using liquid paints. i. e. nonuniformity of thickness

and width. is partially removed by this technique.

2.4 Gauge Calibration

In order to trans lat e the response of the th in film

resist-ance gauge into heat transfer rates it is necessary to know the

therrno-dynamic and electrical constants of the film and its backing material.

If only the heat transfer rate is required then it is s ufficient to know a

composite constant containing the density (

fz.

),

the specific heat

(Cpz. ) and the thermal conductivity CK2) of the backing material. and

the coefficient of resistivity ( 0( ) of the film. If the film temperature

is required. then it is necessary to know the coefficient of resistivity

explicitly.

Because of the interpenetration of the glass and metal at

the interface of the film and backing. and the very shallow layer

affected by the he at pulse. the physical quantities tK.

P )

Cp) 0( ) cannot

be considered to be those of the bulk material. There is also the

possibility that the properties of the glass may vary from sample to

sample. Under these conditions it seems desirabIe to have a method of

calibration that can be applied to each gauge under actual transient

conditions . This may be achieved by applying a known transient heat

input. measuring the gauge response. and thus determining the constant

which relates the heat input to the gauge output.

One method of doing th is is described by RabinoVlicz in

Reference 3. The input to the gauge is obtained from the dis charge

of a capacitor. This capacitor is connected through a mercury switch

to a bridge circuit one arm of which is the heat transfer gauge as

shown in' Fig. 4. The capacitor is selected to give a long time

con-stant of its dis charge through the bridge compared with the test time

required. so that a good approxirnation to a step function input is

obtained. If the bridge is balanced before the step function is applied •

then only the output generated by the change in gauge resistance will

be recorded. The response of the bridge when a stable resistance is

substituted for the gauge and the bridge deliberate~y unbalanced is

the dis charge current is weU described by a step-function. The output wtth a thin film gauge in place. (Fig. '9). is weU described by a parabolic temperature-tirne variation for the first 150

micro-seconds. which is the surface temperature variation obtained from the solution of the one-dirnensional class ic al heat conduction equation. The hook at the beginning of the trace in Fig. 9 is due to the slight difference in inwctahce between the gauge and the other bridge elements; however it does not introduce any additional problems in the calculations. since for times above 40 micro-seconds the quantity ~ is constant indicating a constant heat transfer rate which. as wiU be seen, is the condition necessary for using this method of calibration. Thy numerical value of the constant is determined as follows:

The heat input qo is given by:

2.. l... l... ~ 9

0 := loR'? ~ =Je..fo ~

o

f1 ('1')'),'1- ..94-,19 (m2. Sec

( 1)

where 10 is the initial gauge current in amperes, Ro is the initial gauge resistance and A is the gauge area in square centimeters. It should be noted that for qo to be a true step function the film must have a uniform width and thickness and that the re lative change in resistance must be sm all. The first two conditions make it

necessary to exercise care in applying the film, and the third requires that the voltage (V) on the capacitor be smaU enough that ~ and

~

are of the order of 1-10 or less. 10

Considering the bridge circuit shown in Fig. 4, it can bel seen that the out-of-balance voltage ë.E is related to the change in film resistance by

z.

4 E

(Ro

of

Rz.)

(2)

f.1R

::h

V

Rz

and

!

0 is given by Ro f-Rz.

Io

-

Va

•

(3)

If the temperature change is not too large then the resistance change obeys the linear relation

!if

=

/(0 [I -f0(

('Tt

-

ro)

4R

::

T~ -

Tc

0( Ro

where Rf and Tf are the final resistance and temperature of the film .

(4)

The solution of the one -dirnensional heat conduetion equation for a step function input (see Sec. 3) gives for the surface temperature:

( 5 )

T(o,t

j :: (5)

The result of combining equations (1) to (5) is an expression con-necting the physical constants of the gauge to the output of the bridge. which can be written as.

4-4. /911T

IJ

(Ro

rR,..)

4é

2

:f,

Ro

1-

-1/0

3

rr

~

-

(6)

rr;;:~

-For a constant heat transfer rate the quantity

~

is a

constant and as can be seen in Fig. 11 this is so for the calibration curve. and therefore Eq. (6) ean be used to find

vO<

~

.

Any gauge for which this is not so should be discarded.

f"'Cf~

2.

Using a constant temperature oil bath it has been found that at least in the range 200C - 1000C the variation of film resistance with temperature is linear and the measured values of for several films varied betweerr .0020 aIld .0028 per °C. In addition to this it

was found that af ter a few days the value of (0() for a given gauge might increase by as much as 10% which is probably due to oxidation of the film surface. Because of this it was decided that each gauge should be calibrated as described above immediately before its use in the shock tube.

2.5 Operating Bridge

For its use as a heat transfer gauge the film is incorporate into one arm of the Wheatstone bridge shown in Fig. 6. This arrange-ment facilitates the measurearrange-ment of the film resistance when it is at the equilibrium temperature corresponding to the heat input of the

operating current. The balance condition of the bridge in Fig. 6 is

easily seen to be:

I?o

~

.Rv

(Rl. T

R""J

R,

(7)

The oscilloscope used to record the temperature-time history of the film in the uniform flow behind the shock wave is con-nected across the film so that the voltage meas ured by the oscillo-scope is

E

=

IR (8)

where land Rare the instantaneous current and resistance of the

film. Changes in E measured by the oscilloscope will then be given by

To simplify the calculations it is desirabie to make

6.:r

in E q. (9)

vanishingly small, i.e. to operate the gauge at constant current.

The bridge shown in Fig. 4 accomplishes this hy having the total

resistance in each branch much larger than the resistance change

expected in the gauge. This means that the current through the

gauge is practically independent of the variation in the gauge

resis-tance, and is therefore a constant depending mainly on the setting

of Rc which is simply a "dropping" resistance. Eq. (9) then

becomes

6E

~ Iob.R (10)

lIl. THEORETICAL CONSIDERATIONS

3.1 Thermometer Theory

Once having obtained a temperature-time history of the

thin film, the heat transfer rate can be found by solving the classical

heat conduction equation . For a thin narrow film the tangential and

longitudinal heat flows wi11 be much less than the normal heat flow

so that in good approximation the problem is one-dimensional.

One would expect that in the 200 micro-seconds for

which the quasi-steady flow in the shock tube lasts the heat pulse will

not penetrate far into the glass backing, so that assuming the glass

to be semi-infinite in the direction of the heat flow should also be a

good approximation. Since the film is made very thin and since

platinum has a high coefficient of heat conduction one might say

intui-tively that the effect of the film on the ternperature of the glass

should be small. With this in mind it would now seem instructive to

solve the heat transfer equation for a semi-infinite homogeneous body. The system to be solved is: (ReL 4)

q = q(t) q

=

0

K~

_d;,'l.

= ~

_dt' y=O,t~ 0 y=O.t< 0 T = 0 t

<

0

where q is the heat transfer rate; T is the change from the initial

temperature.

A solution to this system is given in Ref. 4 (page 56)

2.

J

-e

~I( ( t - ... 1) I

lt

1'2-T

I .. • 1 -

K

q1tJ \ j l '-j - - b"-~ Til/'%. o

clA

- - I ('é -/I) V'2.. ( 11) ( 12) (13)

- - - - --- - - -- - - -- - - .

( 7 )

1-F;rom Eq. (12) it is seen th at

~t

=

1 provides a measure of the depth of penetration of the heat pulse, for glass and a pulse of 200 micro-seconds duration the depth will be:

,)j ::

j.OOSB x 206 x 10- (" :::: I. I xlo-3 Cm.

Since the wall thickness is 2 X 10 -1 cm., the ass umption that the wall is infinitely thick in the direction of the heat flow is justified.

If it is desired to consider the effect of the film on the solution of the heat conduction equation the model to be considered becomes: METRLLlC f="ILM REG/ON (j) REG/ON @ .à T, = 1\, .,) ~ T

J

t: ,)il. ~ T2. _ K è'1.Ît. TT - z..

hd-L

( 14) T. = 0 Tl.. =- 0 ~ T,

=

_..l.. Q(t)

J't

k, ()

where

f

-

density

T - temperature change from initial conditions

t - time

k - thermal conductivity Cp - specific heat

K - therm al diffus iv~ty

q(t) - arbitrary heat transfer rate

1.

-

thlckness- of the film

A solution of the system of Eq~. (14) is best effected using Laplace transforms; the method can be found in any text on the

subject. see for example Ref. 4. The tem perature Tl ean then be found to be g!ven by the

~xpression:

q)

Jt [

2. }

JfI

QIM

-~

E

1{.?)

/?nt+.'fJ

(~1

t

=

..!.. k. b

e

4-k"\(~-~)

dÀ

+.l..~ ê" - - e-4-K\(t-;l.)e.-₄1(\(t:-il À I

-t,

1f (t -/l)lh 1:>, 11 ~ T 15) o n~l where

J4e1

e.

Cf'. -

1

~"'~'\.~""

~

=

J

~, ~

Cf"

+

1

Je 1. .... Cf>- ...

Within the limits of the initial ass umptions this is an

exact expression for Tl and at the expense of a great deal of tedious

calculation could presumably be used to find a value of q (t) if Tl

were known. It should be noted that E q. (15) gives the film

temp-erature as a function of the distance from the surface. whereas the

thin film thermometer measures only the average temperature of the

film and therefore Eq. (15) would have to be integrated over the

interval 0 ~ y ~ 1 to obtain the average temperature before it could

be used to find a value for the heat transfer rate.

noted:

Two important features of the above model should be

1) The thickness of the film is very smal! and hence the

characteristic time ~ of the film is short compared with

the testing time. 1

2) The heat conductivity of the metal is much greater than

that of the glass.

On the basis of these features it can be assumed that the measured

temperature is the surface temperature. so that Eq. (15) simplifies

to:

T(t")

(16)

As Vidal has pointed out in Ref. 2. the solution should be

predominantely that for a hornogeneous glass body. To obtain E q. (16)

in a form showing this explicity he has added and subtracted:

l,Jr:

[z~nJ;r~:)

b

I

to Eq. (16). The result is:

1-t- co t n2.,e

t

Trt)::. I .

~O)

dJ..

'

-

2.

r-'6'n(l!~

I/_e-k,li.;))

~(17)

{fT'

Pz

Cp/i;

t: - À

j

Th P g r",

L;,.

F -

~

L

IC: 11" \I 1'-, n= I

o

or integrating the second term by parts. expanding it and neglecting z

terms of the order

.-e

t

T(tJ

~

1

1

~(À)

d" - q(c)

..f

(1q,8Ço;

-1)

(18)

j1rf ..

Cfl)l1..

Jt-"

IJ '-té,\-1e,FlC,a-2,

o

In Ref. 2. Vidal has considered the effect of the assumptions

made in deriving Eq. (18) and he shows that for a .1 micron thick

platinum film mounted

on

glass. the error introduced by neglecting

( 9 )

that for sufficiently thin films the homogeneous glass solution (Eq~ 13»

1:

Ttt) =_1 -

k

'i{~

dJ.

JTï1(z..\LCp-z' -t - À-

-o

is quite acceptable for reducing experimenta1 data.

For a step function heat input (q = constant), Eq. (13) integrates to give: T(t)

=

zi

ft'

J-rr

F,-

-"e -,. Cp- ... or (19) Ct- _

J

TT ft. CP ... -'fl.... T(t) 0 - 2

.J-t

If the heat transfer rate is not constant, then Eq. {13) can

be inverted and integrated by parts to give: (Ref. 4)

t

=fi

JF.zQ\.{?1. [2TrrJ

+j

tT(tI - TI/I)

d'A}

(20)

~(2)

2

f t

0

(t _

?,.)3/z.

Substituting Eq. (4) and (10) in Eq. (19) and (20) results in the following expressions for the heat transfer rate in terms of film voltage meas urements

(21)

q2)

where L\ E(t) and 4E( À) are meas ured from E = IoRo as the reference

voltage.

In view of the fact that in the shock tube the heat transfer

rate is not constant during the first few micro-seconds. it is necessary

to use Eq. (20) in reducing the results for flow times less than 100 micro-seconds. For flow times longer than this the calculations can be made on the ass umption of a constant heat transfer rate, and Eq.

(19) can be used.

3.2 Theoretical Heat Transfer Calculations

At high gas temperatures where dissociation occurs ~ heat

energy is transported not on1y by conduction but also by mass diffusion of atoms and molecules carrying chemical enthalpy. However, if the

Lewis-Se.minov number (the ratio of the ~ass diffusity to the thermal

diffuaivjty) is unity, it can be shown (Ref. 12) that the energy transported

chemical process. and therefore the chemical enthalpies do not have

to be considered separately. Then the heat transfer to a solid wal!

will depend principally on the enthalpy gradient across the flow. i. e.

b~

-

/';;

fi

(23)

In view of the fact that at the forward stagnation region of

a blunt body the boundary layer thickness is determined by the

kin-ematic viscosity and the velocity gradient, L. Lees (Ref. 5) has put

Eq. (18) in a form suitable for calculation. For the case of a circ ular

cylinder with its axis perpendicular to the flow. he gives the

stag-nation line heat transfer rate as.

YZ

[R

_Uz

_ds

dUeJ

(24)

where

Pr ---- an average Prandtl number. taken as .7 in the present

work

- density

- viscosity

- particle velocity behind the initial shock wave

- stagnation enthalpy

- enthalpy of the gas at the wal! ternperature

- radius of the cylindrical model

- distance along the surface of the model normal to the

axis of the cylinder.

The stagnation point velocity gradient is found in Ref. 5

in terms of the flow quantities behind the incident shock by making use

of a Newtonian pressure distribution. In Ref. 3, this has been

expressed so that the velocity gradient is given in terms of the

density ratio across the bow wave. Usin~ an average value of't

=

1.3

and neglecting terms of the order of

(*)

the result is:

[

R

d

Ue.

1

,;.=. _ /

)2

P'2.. ( _

"'''''St;L)

(25)

U

d

R.l.:!.

II .00

z S

B

3

In using Eq. (24) to calculate heat transfer rates. it is

assumed the product (

f

f"-) is constant across the boundary layer.

In Ref. 3. Eq. (19) has been plotted for values of the product (

P

f<')

based on:

I

1) the temperature at the outer edge of the boundary layer.

2) an average of this temperature and the wal! temperature.

3) and an average of the enthalpy at the wall and the enthalpy

at the outer edge of the boundary layer.

These curves are reproduced here as Fig. 7. It can be seen that

( 11 )

rate obtained is noticeably dependent on which of the above assumptions is made regarding the 'value of the product (

F

f- ).

Fay and Ridell~ ~Ref. 6) have numerically integrated the boundary layer equations and obtained an equation similar to Eq. (24), but with an additional term to include the effect of the Lewis -Seminov number. It turns out that for Mach numbers below 10 the additional term has very little effect on the heat transfer rate and was

conse-qu~ntly omitted for this work.

o

Below Mach numbers of 5 (tem peràture of about 2500 K) the value of viscosity given by Sutherland's semi-empirical formula

~

J-L=/4-5.8 T

xlo-

7

/

//o.4-t-T

(26)

(where the viscosity is in grams / cm. -sec)

agrees very weU with experimentaUy found values. For higher temp-eratures a better approximation is found in Ref. 7. Here the Lennard-J ones potential for the inter-molecular force field is applied in the calculation of the viscosity to arrive at the expression:

).Lx/o7 : 266.7..3JmT

I \,:2

o where m is the molecular weight

~ = KT

-r

Y (2, 2. L ) is second collision integral J ( 1; ) is a correction factor.

For air from Ref. 7.

K

= 97.0

Y_o = 3.617

~

(2'1)

The values of J (

1:' )

and Y (2.2,

-r )

are tabulated in Ref. 7. It is stated in Ref. 3 that the use of Eq. (27) gives good agreement with available experimental work.

IV • EXPERIME NTAL RESULTS AND DISCUSSION

Figl,lre 10 shows the voltage -time trace obtained at the stagnation line of the cylindrical model when placed in a flow for which the primary shock Mach number was 8.29. The jump at the beginning of the trace corresponds roughly to the "contact" tempera-ture or the surface temperatempera-ture which results when the hot gas is

brought into contact with the cold model surface. This sudden jump in

surface temperature corresponds to an infinite heat transfer rate at

the surface for t

=

0 which is also the result found when t

=

0 is

sub-stituted in Eq. (21) or (22). Following this initial jump. the film

temp-erature rises smoothly until the arrival of the contact region with its

hot combustion -products. It will be noted that the test time available

for this run is about 50 micro-seconds. On the basis of ideal shock

tube theory with the model 18 feet from the diaphragm station. and

channel press ure of 25 mm. of Hg and Ms = 8.3. the testing time would

be about 300 micro-seconds. however if the same calculation is made

considering real gas effects (see Ref. 10). the useful flow time is found to be about 60 micro-seconds. which agrees with that shown in Fig. 10.

The details of calculating the heat transfer rates and the stagnation temperatures in the uniform flow region behind the bow shock

from the oscilloscope traces are described in the appendix. The

results of these calculations are shown in Figs. 7 and 8. It will be

noted that in Fig. 9 the heat transfer rates are plotted as q

J

R /P1

against shock Mach number. The factor · lR/P1 serves to remove the

effects of cylinder radius and initial pressure from Eq. (19) so that the curve can be easlly used for estimating the heat transfer rate to be

expected under conditions differing from those used in the experiments.

From Fig. '7 it is seen that the experimentally found heat

transfer rates fall generally between the values ca1culated from Eq. (19)

when the product

(f

fA' )

is based on the first tw 0 possibilities s uggested

in Section 3. 2 .

The points near the upper range of Mach nurrbers investi-gated show a tendency to fall more towards the theoretical curve based

on the temperature at the edge of the boundary layer than do the points

at lower Mach numbers. At high Mach numbers the conductivity of the

gas becomes of the sam-e order of magnitude as that of the thin film

heat transfer gauge with the result that the effective res~stance of the

gauge is decreased and the voltage measured across the film is no

longer directly related to its temperature. Experirnents designed to overcome this difficulty (s uch as the calorimeter gauge described in Ref. 8) are necessary to determine whether or not the energy going into

ionization and dissociation of the gas leads to a very marked decrease

in the heat transfer rates at higher shock strengths .

Figure 8 shows the result of substituting the experimentally

found heat transfer rates in Eq. (24) and thus ca1culating the enthalpy

difference (

h, - hw).

This enthalpy difference can be used to

cal-culate the enthalpy ratio across the bow wave and the temperature ratio corresponding to this enthalpy ratio can be found using the curves of

( 13 )

For purposes of comparison the temperature that would be calculated using Sutherland's formula for the viscosity in Eq. (24) are also plotted in Fig. 8. It is se en that the effect. at the highest shock Mach numbers investigated, is to give stagnation temperatures that are about 50/0 higher than those found using Eq. (27) for the viscosity determination .

V. CONCLUSIONS

The following conclusions can be drawn from the present investigation:

a) Metallic paints when applied with care can be used success-fully for the production of thin film heat gauges.

b) In the shock Mach number range 5 ~ Ms ~ 8.5 at a channel press ure of P1 = 25 mm. H g., the theory of Lees (Ref. 5) gives a satisfactory solution to the heat transfer rate and stagnation temperature of the gas for a right

cir-cular cylinder.

c) In the above shock Mach number range, the thin film when mounted on the shock-tube wall and properly instrumented provides a reliable method of measuring wave speed

REFERENCES 1. 2. deLeeuw , J .H., and Waldron, H.F. Vidal, R.J. 3. Rabinowicz, J. 4. Carslaw, H.S., Jaeger, J oC. 5. Lees, L. 6 • F ay, J 0 A • ~ Riddell, F.R. 7. Hirschfelder, J. O. , Curtiss, C.F., Bird, R.B .• Spotz, E. L. 8. Rose, P .H., Stark, W. 1. 9. Feldman, S. 10 . Glas s, 1. 1.

The Combustion Driven Shock Tube at the Institute of Aerophysics and Their Instrumentation, UT IA Report (to be published)

A Resistance Thermometer for Trans-ient Surface Temperature Measure-ments, Presented at the American Rocket Society Meeting, Sept. 1956, Buffalo, N.Y.

Aerodynamic Studies in the Shock Tube, GALCIT Hypersonic Research Project, Memorandum No. 38, June 10, 19~7

Conduction of Heat in Solids, Oxford University Press, 1948

Laminar Heat Transfer Over Blunt-Nosed Bodies at Supersonic Flight Speeds, Jet Propulsion, Vol. 26, No. 4, April 1956, pp. 259-269

Theory of Stagnation Point Heat Tran-sfer in Dissociated Air, J .A.S. Vol. 25, No. 2, Feb. 195'8, pp. 73-85, p. 121

Section D (The Transport of Gases and Gaseous Mixtures) "Thermodynamics

and Physics of Matter'~ edited by F. P. Rossini, Princeton University Press,

1955

Stagnation Heat Transfer Measure-ments in Dissociated Air, J. A. S. Vol. 25, No. 2, Feb. 1958, pp. 86-97 Hypersonic Gas Dynamic Charts for Equilibrium Air, AVCO Research Laboratqry, Jan. 1957

Shock Tubes: Part I, Theory and Performance of Simple Shock Tubes, UTIA Review No. 12, May 1958

11. Vidal, R.J.

12. Lees, L.

13. Hall,J.G.,

Hertzberg, A.'

( 15 )

Model Instrurnentation Techniques for Heat Transfer and Force M easure-rnents in a Hypersonic Shock Tunnel, C ornell Aero. Labs .• Report DD

917-A-1, Feb. 1956

Convective Heat Transfer w ith Mass Addition and C hernical Reactions ,

paper presented at the Third Corn-bustion and Propulsion Colloquirn, AGARD, Palerrno, Italy, March 17-21, 1958

Recent Advances in Transient Sur-face Ternperature Therrnornetry, Jet Propulsion, Vol. 28, Nov. 1958, No. 11, pp. 7 19 - 722

APPENDIX

C a1culations

In Fig. 9 and 10 the calibration and heat transfer traces are given for a heat transfer experiment in which

Ms

=

8.29

P1 = 25 mm. of Hg.

The result of measuring the calibration trace is given in Table 1.

and Fig. 11 shows a plot of Cl. ~ '2.. VB

t .

The fact that this curve is a straight line indicates that the heat transfer rate is constant during

the calib'ration time.

From Fig. 11

z

3 4E =.562

x

/0 1; Vo/fsZ Sec /).1: - ?~7

,rT -

~~.

The operating conditions for this

Vof1-.s

SëC~

gauge were:

Ro

=

110.4 ohms R2

=

50 ohms

10 = 10 ma

voltage gain - .2 volts / cm sweep time - 20/sec/cm area = .0908 Cm

Then from Eq. (6)

4

=

4.19..f7i7){. 0908 x{/. 604) X 108 3 1- .2 4-Zx4-.5x/o x5.0 )(1.104- x/a

v

=

45 volts ': .035 y% 2 Sec Cm. Ca!.

The result of measuring the heat transfer trace is shown

in

Table II. From the table it is seen that

~"l..

does not settle down

to a constant value in the test time available. This makes ~t

neces-sary to use Eq. (22) to find the heat transfer rate. Fig. 12 show s the

heat transfer trace as plotted from the measurement of the

oscillo-scope trace shown in Fig. 10. The practice is to carry out the

numerical integration with the units as centimeters and to convert to

volts and seconds in the final step .

.The first step in integrating Eq. (22) is to evaluate the

quantity

t1 E It:) - -Cl E

()..J

( 17 )

Table III and Fig. 14 give the results of this evaluation for t = .6 cm in Fig. 12. For t = À (A) approaches the value

g- .

however it has been shown (Ref. 11) th at

A -.. 0 as t - . À

In order to deal with the inaccuracies which arise in the numerical evaluation of (IV as t~ À. the practice is to terminate the curve (Fig. 14) at t-À = .02 cm and to assume that

1:

/h--eo... !lBC D

+

J CDF;:

=

r

tJ.E(t) - anI--.)

d;..

2

jo

(t. -

À) 2>/2

For Fig. 13 the result is:

t

J

4E(t) - ÓEOJ

=

.246

Cir/2..

o

(t -

À)3/2 . 1.i'rom Fig. 12 at t

=

.6 cm Then 2 tJ.Eft)

fT

2 tJE(t) .

JT

2x.354-fT"

B

At this point (B) is put in the proper units. Oscilloscope sweep time = 20 microseconds per divisionon record. Voltage gain = .2 volts per division on record. The travelling microscope gives 1 division on record = .4133 cm. the convers ion factor then becomes

then

.2

vo/I-s dlV t

J

em' .4/33 dlv l ( -_ _ _ _ -4133 On dlV

~

ft -

i)1h

2 t:. E (t)

+i

ÁC{r) - tJ.é().)

d A

ó

Substituting this in Eq. (19)

Vo/fS 69.6

Sec 'Il. Cm 'Iz.

80.7

Vo/I's

sec Y-r.

b

=

_I

JTI

JRCp'l.

k,

)<

Bo.

7

loRo 2 c<

But we have

~

.

_ '095

se~'hctn'Lfrom

the operating conditions

10 = 10 ma

~f.8'

....

'el. - Cq I '

•

•

•

f

= -/-/

O-.-~-X-I<-'()---=2

/,71 1 x: x 80.7 680 eq/s. L/??'l. Sec

If this same operation is performed for t = .7 cm in Fig.

, 12. it is found that

'so that for two significant figures the heat transfer rate is constant

in this interval.

Using a travelling microscope the time between the two

timing pulses for this run is measured to be 106 microseconds. The

shock Mach number is then:

Ms

-=

~

=

I =

8.2

9

ctl 113'1x/06 x

l'o-'-Calculation of Stagnation Ternperature

The initial conditions in the shock tube were:

o

Tl

=

26 C,

and we have calculated

~=680

P 1 = 25 mm. Hg.

, Ms= 8.29

From gas dynamic charts for equilibrium air we can find:

,

fT

=

36

PI

we have Eq. (24) and Eq.

32.9-~z

X .24/ (1- .885)< .24-1)

For the purposes of the calculation we shall assume that:

p~.

-=.7

and is constant across the boundary layer and is constant across the boundary

( 19 ) and also that h i = hw'

is calculated from the equation of state

f

=-

.!3..

=.0388 X / 0 -3 ZI"ms

,

RT,

ems

Then

The viscosity can be calculated from Sutherland's form ula by assuming a stagnation temperature, say 40000K.

Then

T

3/2

-7

~ T == /'15. 8 x/O I/O·4-f T ~97 _3 9rfl> == . 0 X /0 ..-Cm ~ec

fTftr

=

Yz

5.

068

X/{}

~

_Sec 2.

JU;

=

fA=

[prJ

2

13

.B05

Cm

"2.

.788

But hw

=

hl

=

72.04 so that and ratio is

IJ

= 2020 CQIs T r~m

h'-r __

2020

=

28

h,

72.04-The temperature ratio corresponding to this enthalpy .Th

=

/6.3

TI

and therefore ()

TT'= 4900 oK

This temperature can then be used to iterate to find a new value for the viscosity. ~z.

.ûc;J /4-8.5 ~ 4900 = /.02 x /0-3

9

f /Y)

I SOlO eh? sec...

J(f;4-),

=

/./9

x/t>-3l;Z;(?c,

(hTr·l/820

+

1

= 26.2

Which corresponds to a ternperature ratio of 15.7 so that Til)::.. 4-100"

K ..

T Iterating again This gives o

This process has thus converged giving 4700 K as the

stagnation ternperature for this run.

If a similar iteration is carried out using E q. (24) and the

tables in Ref. 7 for the viscosity deterrnination the enthalpy ratio is

fOlL."1d to be

hT = 25.6

h.

which corresponds to a ternperature of

-r,.

=

4500

oK

In Fig. 8. va lues of the ternperature found using both

( 21 ) TABLE I fj E 2 ( Vo /fs) 2.

t

(r

5ec ) 0.00 0.00

_,

2.52 .00283

_\

4.50 .00420 \ 13.80 .00726 20.13 .0107' " 29.71 .0161 37.94 .0212 47.52 .0267 59.76 .0334 75.78 .0405 86.23 .0489 95.62 .0544 101. 8 .0578 118.9 .0670 133.9 .0755 147.6 .0829 168.0 .0941 179.8 .1002 187.3 .1051 195.5 .1099 202.4 .1134 TA-BLE II

2-.ot ..6.E ..6.E / t

0.00 0.00 0.00 0.03? 0.223 1. 55 .078 .258 .853 .143 .277 .536 .251 .291 r337 .343 .308 .276 .451 .. 328 .238 .539 .346 .222 .609 .352 .2Q3 .694 .352 .1ä~ .777 .383 .189

TABLE IIl

Evaluation of the integral in Eq. (22) for t = .6 in Fig. 12

(t-~)

3/2 LJEfé) - 4E(~) À i1F(i\) lJé{t} - l:.E(Ä) (t' _ A) $/2 -0 0.00 .354 .465 .761 2 .200 .154 .442 .348 4 .. 232 .122 .420 .290 6 .248 .106 .397 .267 8 .258 .096 .377 .255 10 .266 .088 .354 .249 12 .273 .081 .332 .244 16 .282 .072 .292 .247 20 , .288 .066 .253 .261 24 '.295 .059 .216 .273 28 .301 .053 .181 .293 32 .308 .046 .148 .311 36 .315 .039 .118 .331 40 .322 .032 .089 .359 44 .328 .026 .064 .4Q6 48 .335 .019 .042 .455 50 .338 .016 .032 .506 52 .342 .012 .023 .526 54 .345 .009 .0148 .608 56 .348 .006 .0080 .750 58 .350 .004 .0028 1. 42 60 .354 .000 .000 0

®

DIAPHRAGM STATION

~t

is the 'testing' time at

X~

t

Q)

o

_x

-

S

CD

MODEL

POWER

SUPPLY

THIN

FILM

~

GAUGES

~

J.-"

4--1

20!;-SEC.

DELAYED TRIGGER

10NIZATION

r

ROBE

HI"

I

1

---1

~

FLOW

1

.005

VOLTS

1

FIG 2 SCHEMATIC REPRESENTATION OF THE DETECTOR CIRCUIT AND

DUCTS FOR

ELECTRICAL

LEAD

TUBE WALL PLU G

I NSULATING WASHER

THIN FILM

BRASS SLEEVE

LEAD

WASHER

FIG 3 EXPLODED VIEW OF THE HEAT TRANSFER MODEL AND lTS SUPPORT

MERCURY

SWITCH

SUPPLY\

r - - - - ;

900 JL.-FD.

FIG 4

,----,.- THIN F rL M

RESISTANCE

---+---r~THERMOMETER 1'---,

OSCILLOSCOPE

a

GALVANOMETER

/

~---"--CALIBRATION BRIDGE

SWEEP TIME

=

20 fL-SEC!CM.

SWEEP TIME

=

10

0

p.-

SEC/CM.

FIG 5 CALIBRATION BRIDGE RESPONSE WITH A STABLE RESISTANCE IN THE FILM POSITION

SUPPLY

MIL L I A M ME TER

-1'R'mo)

~-+-

GALVANOMETER

~---,.oLTHIN

FILM

SCILLOSCOPE

100

~

L'

/ /

/

/ / /

AVERAGE

_/L/

ENTHALPY

-...~

V/

50

/h

v

P

AVERAGE

~

TEMPERATURE

V

EXTERNAL

~

TEMPERATURE

0 -

EXPERIMENT

10

I I

VI

J

/

IJ

5

II

Ij

Ij

1

2 3

₄

₅

₆

₇

₈

₉

~ o

6000

5000

a..

4000

~ W

I-3000

2000

1000

I

/

p.

= 25

mm

o

Hg.

0 ~ TI

=299 K

V

"

/

v-

PA' E 90 REF. (9)

o POINTS eALeULATED FROM H.T.

MEASUREMENTS WITH

JI.

FR OM REF. (T

~V

- : - POINTS CALeULATED WITH

po

FROII SUTHERLANDS FORIIULA

~

1/

₃

₅

₇

₉

₁₁

13

FIG 8 STAGNATION TEMPERP,TURE FOR A RIGHT CIRCULAR CYLINDER

0,8

en 0,6

....

....J oO~

>

Iïiii

11 11 11 '11

I~

~;

11

11 ~~

I.

11

41

::I '

.

'

...

-=

~ ... in

-=

~ .... u u ...

...

I: ~ . IU IU

11""

.,111: ::.'I..,.,

.

lal _WolK:

.,

..

..

...

uI&I 11

....

lil' ';7'-

...

.

-.

_" l

0.2

f::l _-=~ ;;;;;;;;;;

0

..:::...

-

I1 I1

5::1

I " Ilo I I, I,

...

I.,;".;.;j

=

0

40

₆₀

_}L-SEC~O

100

FIG 9 CAUBRATION TRJ\CE

o.a

en

0.6

....

..J 0

>0.4

0,2

0

0

40

₆₀

_{p._}

_SEC.ao

100

.10

N Cl) :;.08

o

>

NI W

<J06

.04

.02

o

-/

V

o

20

/

/

V

I

V

"

/

V

V

SLOPE. 562

'im.I!

. /

• SEC •

V

V

. / ! i

40

60 80 100 120

140

160 180

200

TIME ,.,. - SEC.

.4

17'

CM.

= .484

VOLTS

-~ .3

·

~ o I

lU.2

<1

.I

o

V

I

I

J

-.L1I

v

, CM.

=

4 8 . 4 i - SEC. \

.0

.I

.2

.3

.4

.5

.6 .7 TIME - CM.

FIG 12 THIN FILM RESPONSE AS DRAWN FROM FIG 10

Ett) - E(À)

.&(t -

Xl

CM~t

1.4

1.2

1.0

.8

.6

.4

.2

o

J

A

/

/

\

"-

_..,,--

~

V

AREA

=

.246 (CM ).! 2 B

I

C F

o

• I

.2

.3

.4

.5

.6

~ (CM,)