A numerical solution for the free-molecule impact-pressure probe relations for tubes of arbitrary lenght

REPORT NO. 88 Errata Sheet:

Page 4 Page 17

Fig. 5

top line, add "(see Fig. 3)"

5th line, the equation should read

11

lim

'1)-0

7(5,3)) -

Z

CS)

D )

502.

The area under the curve in the bottom diagram is equal to

r

TECHNiSCli

E

i10G:

SCHOOl

DELFT

VLlEGTUIGBOUWKUNDE

B\BU01lIEEK

A NUMERICAL SOLUTION FOR THE FREE-MOLECULE IMPACT-PRESSURE PROBE RELA TIONS FOR TUBES OF ARBITRARY LENGTH

DECEMBER, 1962

by

J. H. deLeeuw D. E. Rothe

VLI GTUIGB UWr-UND!

BIBLlOT IEE!{

ACKNOWLEDGEMENT

This research was supported by the Defence Research Board under Grant No. 9551-02 and by the United States Air Force under Grant No. AF-AFOSR- 62-98 monitored by the AF Office of Scientific Research of the Air Research and Development Commando

The authors wish to express their sincere grahtude to Dr. G. N. Patterson for making this research possible. Thanks are also due , to Dr. C. C. Gottlieb of the University of Toronto Computation Centre for

r

SUMMARY

A numerical analysis was made of the free-molecule impact-.·

pressure probe relations for circular tubes at zero angle of attack. A com-puteJ? program was written, and the calculations were carried out by an IBM- 650 electronic computer for specific speed ratios in the range between

zero and twenty and for tube geometries varying from an orifice to an

infinitely long tube. Graphs and data are presented at the end of this report. The impact-tube relations used were originally developed in UTIA Report No. 52 by Harris and Patterson, who assumed a Maxwellian random motion of the gas molecules entering the tube and diffuse reflection of the molecules from the tube walls.

TABLE OF CONTENTS

- NOTATION _{... v}

1.

INTRODUCTION _L

Il. FREE-MOLECULE IMPACT-PROBE RELATIONS 2

2.;1 Basic Assuro.ptions 2

2.2 The Orifice Probe (D

=

(0) 2

2.3 The Impact-Tube of Finite Length (OL. DL (0) 3 lIl. NUMERICAL EVALUATION OF THE IMPACT-TUBE RELATIO,NS 6

3.1 Calculation of

Y(

(S, D) 3.2 Calculation of

fA-

(S, D)

3. 3 The Case of the Infinitely Long Tube 3.3. 1 General Reduction of the Problem 3.3.2 Determination of

J~o ~

IV .. CONCLUDING REMARKS

4. 1 Representation of Data 4. 2 Experimental Verification 4.3 Limitations of the Theory REFERENCES

APPENDIX A APPENDIX B APPENDIX C TAB LES I to III FIGURES 1 to 14 7 10 12 12 14

17

17

17

18 19 20

23

24

Roman Letters c· ₁ k I m n p t t· ₁ u· ₁ w(

X

,

D) x y NOTATION

constants used in the error-function approximation

(Appendix B).

Sim pson IS R ule constants (Appendix A)

velocity components of random motion

(i = 1, 2, 3)

most probable random speed of a molecule inside diameter of impact tube

error-function (see Eq. 37)

Boltzmann constant

common factor in a geometrie series length of the impact tube

molecular mass

number of molecules per unit volume pressure

sin

~

any particular value of t(i

=

0, 1, 2 ... f)

final limit of t

velocity components of mass motion (i

=

1, 2, 3) Clausing function for the probability that a mole-cule leaving the wall at position

X

=

x/I will ulti-mately reach the outlet.

distance from tube inlet any variabIe

A· ₁ D D·

J

F{Y)

r.:-f

(t) N (S, D) Q R T W(S, D)

x

X· ₁

the ith area increment under the 7(t)-curve

(i :: 1, 2, 3, .... f)

d/l - ratio

any particular value of D

(i ::

0, 1, 2 ... J)

large st value of Dj

D/X l:: d/x

function of Y (see Eq. 18)

function of t (see Eq. 19)

1/

d-ratio

No. of molecules per unit time passing through the tube

Avogadro's number

No. of molèculés per unif time passing from one side through a circular orifice of diameter d from a· gàé at rest with resp~ct to the orifice

No A(s)

(see Eq. 4)

1/(1

+

by)

universal gas constant

molecular speed ratio

absolute temperature

probability function (see Eq~ '13)

x/I

any particular value of X (i :: 0,

1,

2' . . . 1)

smallest vaille of Xi

non-dimensional rectangular co-ordiriate defined

"

Greek Letters: 0(

~

Ç(D)

7(S,D)

f(S,D)

~

7\.(5 )

reD)

(see Eqs. 5and6)

1

,1

'

tan - D V 1 - y2 .

2 [ (1

+

D 2) 3 / 2 - D 3 - 1] / 3D 2 (see Eq. 16)

(see Eq. 16)

spherical polar co-ordinate defined in Ref .. 5 e-S2

+

S

ffF

(1

+

erf S)

2 [{ 1

+

D 2 ' - 1 ] /D2 Subscripts: (see also Fig. 1)

pressure, temperature and nurnber den,sity in the free stream

presSure, temperature and number density inside an orifice gauge

pressur~:. . tem perature and num ber density in the gauge volume behind an impact tube of non-zero length.

I. ;;':;TliJGC""UWKUND!

BJBUOlHEE.

1. INTRODUCTION

The properties of low density gas flows in the free-molecule flow regime are most conveniently determined by measuring the thermo-dynamic properties of the gas brought to rest inside a suitable gauge volume, which is open to the ambient flow via an orifice or an impact tube of diameter d and length

1

(Fig. 1). The pressure inside the gauge volume is generally obtained from molecular number density measurements made by means of an ionization gauge. Pirani gauge. thermistor gauge, or any similar device whose sensing elements can be accommodated within the gauge volume. When equilibrium has been established the pressures and temperatures in the ambient flow and in the gauge volume can be related by equating the mass flows into and out of the gauge volume. The equations for the case in which the impact tube points directly into the flow are more readily solved than those for the case in which the tube axis makes an angle of attack with the direction of the gas flow. In the latter case the axial sym-metry of the molecular distributïon within the tube is lost.

Here only the impact probe at zero angle of attack is con-sidered. 8everal unsuccessful attempts have been made to obtain a general analytic solution valid for all diameter-to-length ratios (D

=

dil') and all

sp eed ratios (8

= ui

cm. where u is the macroscopic velo city of the gas with respect to the probe, and cm is the most probable random speed of the molecules in the free streamL Exact solutions are available for the two special cases:

1. The orifice probe (D = 00) at any value of gas velocity (O~ SL 00) and any angle of attack (Refs. 4 and 8).

2. The "long tube" probe (D« 1) at low speed ratios (8 L 1) only and zero angle of attack (Ref. 5).

The present report treats the general impact tube relations. as derived by Harris and Patterson in Ref. 5. numerically. The calcu-lations· were performed with the help of the university' s IBM- 650 electronic computer.

11. FREE-MOLECULE IMPACT-PROBE RELATIONS 2.1 Basic Assumptions

1. Free-molecule flow exiSts throughout the length of the tube

0

.

e. the Knudsen Number based on the tube dimensions is approximately five or greater). This assures that the molecular flows from the free stream into the gauge and vice versa do not interact and may be considered independently of each other. Only collisions between molecules and the walls and not between molecules themselves are considered.

2. The re- emis sion of the' molecules from the wall follows the cosine law of diffuse reflection, such that, after the first collision with the wall of the tube, a free- stream molecule has lost its preferred (macro-scopic) direction of motion.

3. On either side of the impact-tube, in the free stream and

in the gauge volume, the microscopic random motion of the molecules is Maxwellian.

4. A state of equilibrium has been reached with respect to the mass flows into and out of the gauge volume.

2.2 The Orifice Probe (D ;;; co)

The number of molecules entering and the number of mole-cules leaving the gauge through the orifice per unit time are given by (Ref. 8):

(~d?-) ~(S)

= (1) and

N

=

no c

MO ( I/

d2.)

IC(O) _

1'10

C~O

( "

cf')

o

2.

-{ff

4-

-

2.

-{ff

4-(2)

respectively; _{where n1 and no are the number densities and c m1 and c mo} are the most probable random speeds in the free stream and the gauge volume, respectively.

If the chemical composition of the gas inside the gauge volume is the same as that in the ambient flow (i. e. the gauge walls are non-catalytic with respect to dissociation, recombination and other chemi-cal reactions possible in the gas), then the equilibrium mass balance across the orifice reduces to

=

.,

or

Using cm

=

-J

2kT

I

m' and

pi

:3

pressures and temperatures as

(k/m)T this can be written in terms of

)(($ )

'lt.

(0)

7\(3)

( 3)

A true orifice is a mathematical figment which cannot be achieved in practice. Actual Itorifices" are short tubes often in the range

10 L DL 100. and suitable corrections (Fig. 12) should be applied when the orifice relation (3) is used to determine the free- stream properties.

2.3 The Impact-Tube of Finite Length (OL DL 00)

Consider a uniform cylindrical tube of circular cross- section under the conditions shown in Fig. 2. The number of molecules entering the inlet of the tube in unit time is N_s' as given by Eq, 1. The number of molecules, which succeed in passing through the tube in unit time is (Ref. 3,

Eq. 37) where

1

N (8, D) = Nee (8, D)

+

J

N_er(8, X)w(X, D)dX

o

( 4)

= number of molecules per unit time, which pass through the tube without colliding with the wall;

Ncr(S. X)dX = number of molecules per unit time which pass through the inlet and have their first collision within a ring- shaped wall element bounded by X and X

+

dX (Fig. 2), and

w(X,D)

₌

_{probability that a molecule. leaving the wall at position} X

=

x/I from the inlet. will reach the outlet of the tube.

In Ref. 3 Clausing showed that w(X. D) may be given to a

good approximation by a function of the form

where Q is given by

ct(D)

==

3

U

3+ D2(3TfTD)?' -

'3 ] -

flD'2 [i7 +(3

+

fiD)2' -

-fT]

(~)

D

[7(3-tfi'D)

+

(3-f7D)

17t

(3+

_ff

D)2.]_r 9 (3

+

ff'D ),

-(3-ff'D)t

_9tD'(3+fj'D/l

J7+(3+17D)'2.' H9+D'l.(3-tj7D)2.

j .

This relation is exact in the limits

and

lim w(X, D)

=

X (since

D-O

lim a(D)

=

0)

D-O

lim w(X, D) = 1/2 (since lim a(D)

=

1/2)

D-+(X)

(7)

( 8)

Clausing also verified that Eq. 5 is "very accurate" for D ~ 1 and that for

DL 1 the deviation from the exact probability is less than 2%

The number of molecules per unit time. which pass directly from the inlet through a circular cross-section a distance X down the tube. without having first struek the wall, is given by Nee (S, D/X). Therefore,

-the number of molecules striking -the wall for -the first time within a ring-shaped element between X and X

+

dX must be

a

Ncr (S. X)

= -

d

X Nee (S. D/X)

(9)

With the aid of Eqs. 5 and 9 relation (4) may be re-written

1

N(S, D) = Q [N

S + Nee (S,

D)]

+

(1 - 2Q)

J

Nec (S. D IX) dX (10)

o

From a consideration of the tube geometry and the molecular velocity distribution in the free stream Harris and Patterson (Ref. 5) found

N cc (S, D) = No [

?\.

(S) - e-S2

r

(D) -

_~

r;

(S. D)

J

yTï l

where

7

(S, D) and

r

(D) are functions that are defined in Eq. 16.

( 11)

Let the probability that a free-stream molecule (whieh has just entered the tube) wiU eventually emerge through the exit, af ter any number of wall eollisions. be indieated by the expression:

N(S, D)

NS

=

W(S, D)

so that W(S, D) = N(S, D)

No (13)

Here N _o is the number of molecules per unit time, which would enter the inlet of the tube if the surrounding gas were at rest with respect to the probe. Similarly, the probability that a molecule leaving the gauge volume will eventually pass out of the tube into the surroundings is thus given by

N' (0, D) NI

o

I I

=

W(O, D) _{?( (0)}

=

W(O, D) (14)

where N _o and N (0, D) are the number of gauge molecules per second entering the tube from the gauge volume and leaving the tube into the free stream, respectively.

In terms of the W-function as defined above the number of free- stream molecules entering the gauge volume in unit time is

while the number of gauge molecules leaving the tube inlet into the ambient flow in unit time is

Equating these two expressions gives, under similar conditions, arelation of the same form as Eq. 3, namely

=

W(S, D)

W(O, D) (15)

where

r(D)

=

2[~1+D2

i

_]]/D

L

C(D)

=

2[(1+

D2)3/2_

D

3

-1]

/3D

2

1 A 2 '2..1

1

j

.

fr'-

-$

sin cp

?

(S) D)

==

D

cJ

Y )

[1

+

erf

(5

cos

~

)] cos

r}

e

cl

~

,

o

0

Î-'

=

to.n-

1

DYl_y2'

1

f(5,D)

=

J

7

(s, D/X)

dX

o

and Q' is given by Eq. 6. It should be noted that

and thus lim Q'(D) = 1/2, D-oo lim 'f(D) = 0, D-oo lim

7

(S, D) = 0 D--oo lim W(S,D)=/(S)

,

D-oo (16)

which shows that Eq. 15 reduces to the orifice relation (3) as D becomes infinite.

Hl. NUMERICAL EVALUATION OF THE IMPACT-TUBE RELATION In this Section the details of the numerical evaluation of the impact-tube relation as given by Eq. 16, are presented.

3. 1 Calculation of

7

(S, D)

After a change of variabie, t = sin

~

the expression for

7

(S, D) _{can be re-written}

o

where and

7

(S, D) = -

~

J

_{F(Y) dY} 1

J

(1 - y2) / (1 - y2

+

1/ D2) r F(Y)

=

J

:r

(t) dj

o

T(t)

=

[1 + erf (S

1

1 --t 2 ') ] (17) (18) (19)

r--A plot of

.f

(t) versus the new variabie t is given in Fig. 4. The following properties of the T(t)-function may be worth noting:

1. ForOL...SL.oo:

'F

(0) = 1 + erf S

T

(1) = _e -S2

r=-I

f

(0) = 0

r<

1) has no meaning (singularity), since 'T(t) and all its derivatives become complex for t ~ 1.

2. For S = 0:

'}"(t) = 1; " I

_t

(t) = r;-II

t

(t) = ••• = 0 3. For S = 00:

'T(t) = 0 except when t = 0, at which point

g:-

(0)

=

2; thus the curve for S

=

00 is also discontinuous at t = O. To integrate Eq. 17 numerically, the integrand F(Y

i) for a particular Yi may be interpreted as the area under the ~(t)-curve between

t

=

0 and t·

= /

(1 _ y. 2) / (1 _ y. 2 + 1/ n2 )'

1

l/

1 1

function F(Y) vanishes, while for Yf = 0

tf =

Dil

1+'02.'

F(O)

=

j

.r(t)

Jt

o

(Fig. 5). For-_Yo

=

1 the

(20)

When the area under the F(Y)-curve is divided into narrow

strips of width

(6

Y)i = _{Yi - Y i _ 1, then to a first approximation the}

integrated value of the F(Y)-function over any such strip may be thought

of as the "volume" formed by the area under the 'j(t) - curve between t =0

and ti multiplied by (

.6

Y)i. One can approximate the area under the

F(Y) - curve between Yi _ 1 and Yi by the area of a trapezoid:

Ti

J

Ft

y)

cl

Y

~ ~

[ F

(y[

-1)

+

F

(yc) ]

(L\

y)

i

Y(-l (21)

=

~ (o}i(~)cH

+o}

iet)

elt ](4Y),

Writing

j

t

l

jti-l

f

tL

'FCt)Jt

=

T(üdt

+

1TOolt

a

0

t.

L-l

and designating the area of the i;..,fu strip under the

'F

(t) - curve by Ai,

Eq. 21 can be restated as

Y:

J

F'(Y)dy

. [fAi

+~AiJ['((t')-'L(tl.')]

J.=1

Yt-l

where

1 - t·₁ 2

Summing over all the Y-incrernents from Y_i =

o

1

jF(Y)

dy .

or 1

7(5,

D)

=

b

jF(Y)dY

o

,

_,

)

1 to Y_f = 0 gives:

,

(22) ( 23) (24)

T

ECHNISCHE HOGESCHOOL DElF

VLIEGTUIGBOUW KUNDE

The calculation indicated by Eq. 24 has to be repeat~ ~

lIEEJ\

each (8, D)-combination. Keeping 8 constant while choosing a new D-value the T(t)-function to be integrated remains the same as before, only the end point of integration tf = Df-./ 1 + D2' shifts. For-small D this end point lies close to t

=

0, and for large D it is close to t '" 1. 8ince T(t) does not dep end on D, it is desirabIe to use the same t-increments for all D-ratios to be considered, in which case the r(ti)- or Ai-values need to be calculated only once. If in addition the increments in t are made equal, the corresponding increments in Y, as obtained from relation (23), are not equal. Instead the Yi-values are much more crowded near Y

=

1 than near Y = 0 (Fig. 6). Furthermore, keeping (Ll t)i constant over the range of D-values considered causes tf, and thus the number of t-increments, to become small for small D-values (long tubes). This tendency has a direct effect on the percentage error, which under these conditions may become significant for long tubes. The spreading out of the Y-increments near Y

=

0 does not have a serious effect on the accuracy of ~ since the F(Y)-curve is very flat in this region.

An expres sion similar to Eq. 24 can be derived using a

curvi-linear approximation (e. g. 8impson's Rule), thereby making possible a reduction in the number of increments necessary to obtain a given accu-racy in

?

However, since the Y-increments are not uniform over the range of integration. the simple 8impson's Rule has to be replaced by Eq. A4 in Appendix A. which makes the resulting expression considerably more complicated than Eq. 24. This in turn more than offsets the apparent gain from fewer increments. Uniform t- and Y-increments can be arbi-trarily selected only if the integration with respect to t is kept completely separate from the integration with respect to Y, which in turn would in-crease the amount of numerical calculation by approximately a factor of two.

The actual form of Eq. 24 used in the calculations is based on the trapezoidal integration scheme and takes the form:

7

(5,D)

=

~ {~AI

(1-Y,)

+

(A,

+

~A2)(Y,

-

YJ

+

[(A

1

+A

z )

+

~A3

J

(Y2 -

Y3)

where

--r •••

+ ...

AL

=

~

[TCtt -

_{L )}

+

TCtJ]!l

t

~ ~ ~[T(-tf-l)+T(tf)](tf-t;_l)

Yi.

=

V

1 -

t

i.2.

(1

+

1/

D

2. )

1 -

t.2..

l 3.2 Calculation of

JA

(S, D) 7 '>

,

In order to perform the integration necessary for the determination of

J<

(S, D) all the

?

(S,:6 i)-values required have to be known (Fig. 7), where the choice of values used for:b i = D/X

i is dictated by:

(a) The D for which

i<

(S, D) is to be found, (b) The choice of X-increments.

( 25)

To limit the computing time to areasonabie figure the

number of

?

-values calculated by Eq. 25 must be minimized. It becomes then imperative to make the choice of the XCvaIues such that the

'?

(S,2)i)-values required for the integration are the same as those already calcu-Iated as useful

Y(

(S, D)-values for the various D-ratios of interest. Designating these D-ratios of interest by Dj' where j = 0, 1, 2, 3 ... J, a complete duplication of

'1

(S, Dj)- and ~ (S, ~ i)-values is achieved when for each i

:iJ

.

=

.Ei..

=

1 _X·

1

where g is any one j-number 0, 1, 2, 3 ... etc.

The above can be realized by letting both the Xi I S and the

D·'s form geometric series:

J i. e. and Then

iJ

i = X.

=

X k-i. 1 0 ' i = 0, 1, 2, 3 ... I if X

=

1 o D. J

=

j

=

0, 1, 2, 3 ... J D kj + i

=

o

,

(26) (27) (i. e. g = j + i)

If for the largest Dj to be considered (j

=

J) one likes to have a predetermined number I of X-increments, then the numBer of

7

-values, which have to be calculated, are J

+

I, a considerable reduc

-tion from J x 1. Furthermore, for Drvalues less than DJ the extra I( -values available can be used to extend the XCseries further towards X '" O. This is of importance as will be discussed in Appendix C.

The use of a geometric series for the Xi's produces incre

-ments which are large near X

=

1 and become increasingly smaller towards

X

=

O. Fortunately, a plot of

rz

(S, D/X) versus X reveals that the main variations in the first and second derivatives of 7.. (S, D/X) occur near X

=

0, while the curve shows very little curvature and change in curvature near X

=

1, so that this change of increment si ze is actually an advantage. Another point to consider is that the Xi-series must be terminated, leaving the final XI short of the limit of integration X = O. Knowing that the limit

lim

re

(S, D/X) =

x-a

lim Y) (S,0) = 0 0-+00 l

this missing portion can be approximated by ?- triangle (Fig. 7). A modi

-fied "Simpson' s Rule" for the case Xi = XOk-1 (instead of equal increments Xi = _{X o} + iL1X), is derived in Appendix A and is given by Eq. A6. Approxi-mating the integration by this rule the f-function may be written:

1

f(S,D)

=

OJ?CS,D/X)dX

I-2

L

fi [6

0

7(S).0;)

+

6

1

re

(5

>

~i+l)+

6

₂

7(5,

Z>i+2)]

L (28)

where i

=

O. 2~ 4 .•. I - 2 and I is an even number.

Once

?

(S, D) and ~ (S, D) are known the probability function W(S, D) can be calculated from Eq. 16. For the case of zero mass velocity this relation becomes independent of

?

and

jA-:

w (

0)

D)

=

1 -

0(.

"r(

D) - (

1 - 2

~)

ç (

D )

(29)

3. 3 The Case of the Infin.itely Long Tube 3.3. 1 General Reduction of the Problem

The infinitely long tube represents a special case insofar as N/N - 0 as D-O. Therefore. W(S, D) and W(O. D) both vanish. and the

o ratio

Pt ff,1

--

-

-P1 Tt W(S, D) W(O. D)

becomes indeterminate. A more detailed investigation is thus required to find the pressure ratio for this special case. The following easily derived limits will become useful in this investigation

lim

't

(D)

=

1

,

D ... O lim

t

(D) = 1 ( 30) D-O lim Cl' (D) = 0 D--O

and lim d'r 0

=

D-O dD

,

,

( 31) lim d~ -2/3

=

D-O dD lim dQ,

=

2/3 D-O dD

As the tube is made longer and longer one expeets from physieal arguments that the ratios Nee/N

_s

and N/NS tend to vanish as D-O for a finite value of

S.

While the vanishing of the ratio Nee/NS is self- evident, the following argument may be presented in support of the statement that the ratio N/NS goes to zero as the tube length becomes infinite: Eor a given tube diameter and speed ratio the number distribution of first impacts of the incoming molecules with the tube wall is a fixed function of the distance x from the inlet. If the tube is now allowed to become infinitely long the fractional distance X - x/

I ,

over whieh first collisions can geometrically oeeur for a finite speed ratio, shrinks to zero. Thus, aeeording to Eq. 7, a free- stream molecule whieh is emitted from the wall af ter its first collision has a zero chance of ever passing through the exit.

With the help of Eqs. 11 and 16 the following relations must then be true:

lim [ / \ (8) - e -8 2

't

(D) -

~

YJ

(8, D)]

=

0

D-O

nr (

,

and

Inserting the limiting conditions (30) into the above yields lim

?

(s, D) =

71

(1

+

erf 8) D-O 4 and ( 32) r -lim

r

(8, D) = 11 (1

+

erf S) D-O 4

Applying l'Hospital's Rule to the limit as D -

°

of W(S, D)/ WeD, D) this limit ean be written:

lirn

W(S) D)

D"'O

weD)

D)

It wil! be shown later that

but lim D-+Q lim D-O

~

dD

₌

°

~:f

dD

°

(Fig. 9)

Using the limits (30l (31) and (34) Eq.

:

33

ean be redueed to

lim W(S, D)

D-.+O W(O,D) = e -S2

+

S1"

Rfl

2(1

+

erf S)

-(33)

(34)

d~

1

(35) dD

.

The sueeess in obtaining a limiting value for Pt-iTi/P1-1Tt for the infinitely long tube thus hinges about the possibility of fin ding the slope of the)< -funetion at D

=

0.

3.3.2 Determination of lim

D-+O

~

dD

Reeapitulating from Eq. 16 the

?

1

(3

7(5

,V)

=

~ J~Y

J[l

+

er{CS

cos

~)] cos~

o

0

-funetion ean be written

) (36)

where

r

=

tan-1 D .; 1-y2: Within the limits of integration

°

f:.yf:.

1 and

°

~ rf; ~f3

the limitation D

«

1 is analogous to making the restrietions

(3«

1

and

~«

1 .

With th is in mind and the further eondition th at S be smal!, the trigonometrie and exponential functions ean be approximated by the first few terms of their eorresponding Taylor expansions about D

=

0,

f3

=

O,and

cos

f

-

1 -

i!

+

- -

~4-2 ₂₄ sin

p

f -

~3

rj;S

-

- - + 6 120 -8 2 sin2...1 2 4 4 e >" == 1 - 8 2 s in

~

+

~

8 s in

~

( _8 2

+

_8 4 ) rI t ;" 1 - 8 2 ~ 2

+

'I'

erf (8 cos

cf )

=

erf 8 +

3 2 2

ft

1

8 cosrt 2 e- t dt 8 (Ref. 5) [ 8

-82cos2~

-82COS2~]

. f 8 2 e ( d _ 1)- 2 83cos rf> e (cos" _1)2

=

er + ,r;::::" 0 + - , cos 'P

2!

jl.I

I"

1.

8 -82 ; e r f 8 -

~ [~2+(282_ ~ )~4-]

IN 12

where all terms of higher than fifth order in D,

p,

or r/> have been neglected. 8ubstituting these approximations back into Eq, 36 and carrying out the inte-grations results in the following series expansion for

7

(8, D), valid for small 8 and D

«

1:

7

(8, D) ) (37 ) where f 1 (8)

=

'ïï

4 (1 + erf 8), f 2 (8)

=

11

[(282 + 3) (1 + erf 8) + 2Se-S2 ] 32

_ff

2 r -[ 2 2 28 e-8 11

]

f3 (8)

=

(484 + 20 8 + 15) (1 + erf 8) - (48 - 9)

1if

256 and

Equation 37 yields the already derived result

lim

7

(~,

D)

=

D--O (1

+

erf S);

4 it also proves that

lim ~

=

0

D-O dD

because there is no linear term in D.

Now making the substitution

;:0

=

D/x

in the expression for JA (S, D)

results in /

D

fot (

S ,

D)

= -

J

D

~

CS,

t1»

d:IJ

/ c c , 1 ) 2 Therefore,

D

00

J

7

(S ,

~

) -

?

(S)

D)

ol

tJ

0

2

and taking the lim i t as D _ 0, lim D-+O

~

dD

=

.

,

Z(S,D)

D

(38) The integration of Eq. 38 requires the knowledge of

ë

(S, ~ ) for the range 0 L

:6

L

co.

An analytic solution of

lim ~

D-O dD

thus involves making an approximation of

?

(S, D) for large as weU as small

values of D, whereas expansion (37) is only good for very small D. It turns

difficult as the solution of

f<

(S, D), and thus W(S, D), itself. However, once sufficient

r;

(S, D)-values have been calculated numerically, it becomes a small task to also evaluate the integral (38) numerically. From Eq. 37

and

lim

D--O =-f2 (S)

Also for

0

=

02)1» 1

the numerical value of

?

(S,~

I)

«

~ ~

J

7

(S

,2» -

?

(s) 0)

d

~ ~

-

f

(5)

J

d

ft)

~2 1 R)2

~I

~I

and

This makes it possible to re-write Eq. 38 as

1)r

lim

~

= _

J

:heS) -

?(S

,tJ)

dtJ

D-+O dD ~ '2.

o

(39)

which can easily be evaluated numerically knowing that the integrand is equal to f₂(S) at

iJ

=

O. An outline of the machine calculations is given in Appendix C.

IV. CONCLUDING REMARKS 4. 1 Representation of Data:

The results of the numerical analysis for the impact tube of arbitrary length are presented in graphical form in Figs.. 11 to 14. To show up the variation with D more clearly and to allow for interpolation be-tween the curves, the probability ratio W(S, D)/W(O, D) has been divided by the orifice function / ( (S) (Fig. 10):

W(S, D) Pt

_~

P1

_Fa

Pt W(O, D)

~(S)

=

~ ~ ~

Po

1

~

=

Po

(40)

Graphs shown for speed ratios not specifically used in the computations were determined by graphical interpolation.

4. 2 Experimental Verification:

The computed data for long tubes and low speed ratios agree with the isolated values calculated by Harris for D

=

0.04. Harris and Patterson (Ref. 5) and Muntz (Ref. 7), using the low density tunnel and the rotating arm apparatus, have shown the theory to be in good agreem ent with

experiments for long tubes and low gas velocities. The orifice probe theory is weU substantiated by experiment (ReL 4). Further experimentation is re-quired using impact tubes having diameter-to-length-ratios

for SL1

and

oL

DL

00 _for

to verify the shapes of the curves in Fig. 11. Experiments are presently being carried out using the UTIA low density tunnel to test impact probes over a large range of diameter-to-length-ratios and to investigate the varia-tion in gauge pressure produced by rotating finite length probes through various angles of attack.

4.3 Limitations of the Theory:

Although the calculated data are believed to correspond to the true values of the mathematical functions (Eq. 16) to within 1/10 of a per cent, it must not be forgotten that in the derivation of Eq. 16 assumptions and

approximations were made. The limitations and assumptions were stated in Section 2. 1. An approximation was made in Eq. 5 which may weU cause the calculated W(S, D)-value to deviate from the exact value by one or two per cent for long tubes (ReL 3).

The assumption of diffuse reflection from the waUs may not be entirely justified when S becomes very large. For large S the molecules hit the tube walls at very shallow angles on the average, and a deviation fr om the diffuse reflection model may be expected. If this is the case, then e~iments

in high speed flows should indicate a higher pressure ratio Pt-/Ti/P14Tt than predicted by theory; i. e. it should be between W(S, D)/W(O, D) and ?(S)/W(O, D).

1. Baron, M. L. Salvadori, M. G. 2. Chambre, P. L. Schaaf, S. A. 3. Clausing, P. 4. Enkenhus, K. R. 5. Harris, E. L. Patterson, G. N. 6. Hastings, C. 7. Muntz, E.P. 8. Patterson, G. N. 9. Patterson, G. N. REFERENCES

Numerical Methods in Engineering. Prentice-Hall, 1952.

The Theory of the Impact Tube at Low Pressures. Jour. Aero. Sci., Vol. 15 No. 12, Dec. 1948. Ueber die Stroemung sehr verduennter Gase durch Roehren von beliebiger Laenge.

Ann. der Physik, Vol. 12, 1932, p. 961. Pressure Probes at Very Low Density. UTIA Report No. 43, Jan. 1957.

Properties of Impact Pressure Probes in Free Molecule Flow. UTIA Report No. 52, April 1958.

Approximations for Digital Computers, Princeton N. Y., 1955.

Pressure Measurements in Free Molecule Flow with a Rotating Arm Apparatus. UTIA T.N. 22, May~ 1958.

Theory of Free-Molecule, Orifice-Type, Pressure Probes in Isentropic and Non-Isentropic Flows. UTIA Report No. 41, Nov. 1956.

Molecular Flow of Gases. Wiley & Sons, New York, 1956.

APPENDIX A

/

)("2 )('"1 x'"z.

Approxirnation of f(x)dx

=

I

f(x)dx

+ /

f(x)dx

=

L:l

A by the Area Vnder

~o xo )("~

aQuadratic Going Through the Points (xo, yo), (xl, YI)7and (X2, Y2)

(Sirnpson's Rule for Unequal Incrernents of the Independent Variable)

If the curve Y

=

f(x) between Xo and x2 is approxirnated by a quadratic y*

=

E x 2

+

F X

+

G, th en Let )("2

Ll

A =

J

(E x2

+

F x

+

G) dx =

~ (x~

)(0 then y 0

=

E Xo 2

+

F Xo

+

G YI

=

E xl 2

+

F xl

+

G Y2

=

E x2 2

+

F x2

+

G )

EquationsA2 can be solved sirnultaneously for E, F, and G. The result is (A2)

E

=

F -G= Yo(X2- X1> - YI (x2 - x o >

+ Y2 (xl - xo>

(x2 - xo> (x2 - xl> (xl - x o > 2 2 2 2 2 Yo (x2 - x12 > - YI (x2 - Xo >

+

Y2 (xl - X_o ) (x2 - x o > (x2 - xl> (xl - xo >

YoXIX2 (x2 - xl> - xoYïX2(x2 - x o >

+

_{x ox1Y2(xl-xo >} (x2 - x o > (x2 - xl> (xl - x o >

Substituting for E, F and G in Eq.

:

AI:

one obtains af ter simplification:

LlA=

Eq. A4 reduces to Simpson's Rule:

For the special case in which

I

xl =

k

X_o

etc.

follow a georn etric series, one can write 1 - k2 X 2 - Xo

=

k2 Xo 1 - k xl - _Xo

=

x (A3> (A5>

and Eq. A4 reduces

to

where b

=

' 0 b _{2 -} -[(2k-l) k-2] .. k .. 1 6k3 [(k

+

2)k2 - 2J k - 1 ûk 3 v Ü2-k) k + 1 ] k - 2 6k 2

.,

,

(A6)

APPENDIX B

Approxim ation of the Error Function

On page 169 of Ref. 6 Hastings shows that in the range

o

Lx

L

00 the error function

2-

J

~

-

z2.

fif

e

dz

, 0 may be approximated by where 2 erf y ;; 1 - (a1Q + a2 Q2 + a3 Q3 + a4Q4 + a5Q5 ) e-y

Q= 1

1

+

by

+

with an error of less than - 0.000 00015. The eonstants are given as

al

= 0.2548 2959

a2

= -0.2844 9674

a3

= 1.4214137

a₄

=

-1. 453 1520 a5

=

1. 061 4054 b

=

0.3275 9110

For optimum programming it is of advantage to express the eomplementary error function as follows:

eerf y

=

1 - erf y

=

(f[<a 5Q +a4) Q +a3l Q +a2} Q+a1) Q/ey2

l

E

C

_{HNISCHE HOGESCHOOL DE ft}

APPENDIX C VLIEGTUIGBOUW KUNDE

BlBlI01ll

EE

K

C. 1 Debugging the Program

The computer program ._{was written in SOAP II symbolic}

language* for use on an IBM-650 electronic compute'_{r. Based on Eqs. 25 and}

28 the program was set up in such a way, th at for each particular speed ratio

the functions

'?

(S, D),

)A

_{(S, D),and W(S, D) would automatically be} calcu-lated for a whole range of diameter-to-length-ratios, which in themselves were allowed to vary in a geometric series (Eq. 27). Data input cards were used to feed the relevant parameters

into the machine memory, "I" being the number of X-increments and the "b_{j "} being Simpson's Rule constants (Eq. _{A6). Vnder this system an arbitary} selection of the number of integration increments was possible.

A program check to test the logic of the program was made

using the input:

S=l,Llt=t, k=2, Do=l, 1=4, b_{o =0}. _{1815, b1=0.5625, b 2 =0} and

S

=

30,

_"

_"

_"

_"

_"

"

The output data were in complete agreement with previously calculated an-swers, using desk calculators.

The first trial production runs were made with the following in-put S

=

0.2 } S = 0. 5 t = 0.01, k = S

=

5 = 0.05, I = 30, b o _{b1 = O}= 0.031 555 222, _. 115 965 44 , b2 = O. 026 033 057.

In each case 100

7

_{-vales were calculated for D-ratios from} Do = 0. 05 to D99 = 626._{391, thirty of these being used for the determination} of each of seventy

)A

_{-values and W-values (Fig. 7). To gain an idea of the} accuracy obtained, the number of integration increments used in the evalua-tion of ~ and

f"

_{were alternatively doubled by making}

_.6

t = 1/200 and

k =

-{f:1',

respectively. _{The separate effects on the accuracy were estimated}

*

_{Programs written in SOAP II symbolic notation are automatically} trans-lated into machine language by the IBM-650. In _{so doing the computer assigns}

drum addresses to the program commands in an optimum sequence to

mini-m ize the programini-m running timini-m e.

by comparing the more accurate

7

and

JA-

-values with the previous ones. Increasing the number of X-increments from 30 to 60, by dividing each incre-ment shown in Fig. 7 approximately in half, produced discrepancies in

JA

in the fifth and sixth significant digits only, with the somewhat greater deviations towards small D. Changing the size of the t-increments from 1/100 to 1/200

produced deviations in

7

in the fifth and sixth significant digits for large D and in the third and even second significant digits for very smal! D. Agree-ment in the third significant figure occurred for D rvO. 3. The deviations did not vary much with different speed ratios.

From this it was concluded that k = 1. 1 gave a sufficient accuracy in

jA

*,

while the accuracy obtained using

LJ

t = 1/100 in the integration for

7

was insufficient for D-ratios less than 0.5. This had been caused by the small number of t-increments between t

=

0 and t

=

tf (Fig. 6). resulting in an equally smal! number of Y-increments. For example, tf == 0. 05 for Do

=

0.05,

resulting in five t- and Y -increm ents.for

L1

t

=

1/100, ten for

6

t

=

1/200.

The corresponding numbers of intervals for D = 0.5 are 45 and 90, respectively. Judging from the magnitude of the last term f 3(S)D4 in Eq. 37 values of

"7

(S, D). accurate to at least five significant figures, can be ob-tained from this expansion for S

<

2 and

DL o.

1. For a separate check several

7

-values were calculated with the help of the computer for a few 'distinct (S, D)-combinations, using a Simpson's Rule sub-routine nested within a second Simpson's Rule routine. These routines used a built-in accuracy criterion which could be arbitrarily specified. The integrals were auto-maticaUy and repeatedly determined for finer and finer subdivisions of the independent variabie until the desired accuracy was reached. Table I lists these

1

-values, their accuracy, their computing times, and the number of increments used in each Simpson's Rule routine to obtain the specified accuracy. These values agree very weU with those approximated by Eq. 37. The

re-latively long computing times should be noted, which compare with 1/2 min. to 2-1/2 min. per

7

-value ca1culated in the main program. Furthermore, only four equal t-increments were needed to obtain the given accuracy, while at the same time several hundred equal Y -increments had to be used. This seems to indicate that the choice of equal t-increments is a good choice, while the same is not true for equal Y -increm ents. It is believed that the unequal Y i -distribution (Fig. 7). resulting naturally from Eq. 23 for equal t-increments, is in fact desirabie.

*

Going to k =

-{f:1'

would produce an unnecessary fine grid in D-values at the expense of the range in D-ratios covered. To cover the same range as before more than twice as many

r;

-values would have to be calculated. in-creasing the computation time by a factor of two or three.

In the first part of the main production run

L1

t was 1 / 500. k was 1. 1 and I was 30. The

7

-values obtained for very small D-ratios still differed from the accurately known ones by about 0.3%. while. on the other hand. the W(S. D)/W(O. D)-ratios for D = 0.05 differed by about 4% from the ones calculated by Harris for D

=

0.04 (Ref. 5). using an entirely different approach. but the same fundamental equations. For small S this difference was many times larger than could be accounted for by the difference in D and the known error in

7

The error was finally traced to the triangular approximation used when integrating over X to find

JA-

(S. D), By comparing Fig. 8 with Fig. 7 it becomes evident that the triangular approximation is very exact for large D. but not so for very small D; in fact for D = 0.01 a rectangular approximation may be closer. This was easily remedieà. since in the range of sm all D extra

?

-values were available. which could be used to extend the Xi-series further towards X

= O.

For example. if 90

7

-values were calculated and stored on the memory drum. then for Do an extra 60 increments could be put into the triangle shown in Fig. 8. 59 extra X-increments for Dl, 58 extra onesfor D 2• etc.

C.2 Running the Program

The final calculations we re carried out for speed ratios:

S = 0.000 S = 1. 0 S = 0. 02 S

=

1. 5 S

₌

0.05 S

=

2.0 S = O. 10 S

=

3.0 S

=

0. 20 S

=

4.0 S

=

0.30 S

= 5.0

S

= 0.50

S

= 10

S

=

0.75 S

=

20

Every third W(S. D)/W(O. D)-ratio calculated for the above speed ratios is tabulated in Table 11. These numbers are believed to be accurate to four significant figures. For large D the accuracy is probably " better than this. The final production run was completed in two sessions for reasons outlined below:

Session 1, Part A:

For diameter-to-length-ratios from Do = 0.05 to D29 = 0.793155 the calculations were carried out using

Á'l = 1/500

No. of

f

(t)-values .calc.ulated and stored

=

500 No. of. ~ -values calculated and stored

=

60

[ start,ing. with

'7

(S. Do)

J

J>Io. of X:"increments

=

30

No. of

?

,)-l,and W-values printed out'" 30 Computing time for each speed ratio = 1~ hr. Session 1. Part B:

For diameter-to-length-ratios from D20 :: 0.336 375 to D89

=

241. 501 the calculations were c.arried out using:

At:: 1/100 k = 1. 1

No. of

g::

(t}-values calculated and stored ~ 100 No. of

7

-values calculated and stored ::: 100

[ starting with

'r(

(S, D20) ] No. of X-increments = 30

No., of

'1'1'"

and W-values printed out:: 70 CÖnJputation time for each speed ratio

=

50 min.

Pa~ts A and B allowed for an overlap between D20 and D29' The less accurate values of

7

near the end of this overlap differed from the more accurate ones by less than 0. 05% for all speed ratios.

Session 2:

The inaccuracy caused by the triangular approximation in the integration for

!'

was realized only af ter session 1 had been completed. The out2ut cards of session 1 wer.e then used as input cards reading into the com-puter memory the 90 distinct

7

-values available. For S L 3 the f~rst twelve

'7

-values were replaced by more accurate approximations obtained from Eq. 37 rather than Eq. 25; similarly for S

=

4, 5, 10,' 20 the first 9,

4,

0.0

?

-values were replaced. respectively. . !he ~ and W -values in the range from Do

=

0.05 to D44

=

3.313 204 were then re-calculated using all

Y(

-values

available. The nurpber of X-increments thus varied from 90 for Do to 45 for D44': _ The computa"tion time for each speed ratio was about ten minutes.

A separate program was written för the evaluation of ~he

prb-babili~y ratio W(S, D) /W(O.

P)

for the infinitely long tube based on Eqs: 35 and 39. The already calculated

7

-values were re:"used. Extra

'1

-values

iri

the range 0 ~;() L. O. 05 were calculated from Eq .. 37. The integral was evalu'ated by themodified Simpson's Rule (A6) up to a lim.iting value:;() I

=

D88

=

219.546;

t

he

remaining area under the taU of the function was approximated by f1(S}/D88' The results are gi ven in Table III~

TABLE I

S D ~ (S, D) Probable Computing No. of No. of

Error Time _ t-increments Y -increments

0. 2 0. 05 0.9592 ~ _{0. 0002 8 min. 4 128} 0 .. _2 0.492487 0. 8784

!

O. 0002 8 min _{4 128} 0_5 0.05 1.19275

!

O. 00005 20 min 4 256 0.5 0.156921 1. 18051 ~ 0. 00002 35 min 4 512 5, 0 0. 05 1. 54522 :: O. 00005 55 min 8 512

TABLE SHOWING RESULTS OF A NUMERICAL EVALUATION OF A FEW SPECIFIC

TABLE II

W(15)D)

/W(O,D)

D

cx:(D)

W(O,D)

_s

₌

_0·02

_S=-

_0.05

_S=

_0.1

_S;;.

0.2-0. 050000 0.030 632 0. 061 367 1. 0409 1. 1044 1. 2165 1.4650 O. 066 550 0. 039 669 0. 079 535 L 0408 1.1_, 1044 1. 2165 1. 4649 0. 088 578 0; 050935 O. 102 309 1. 0408 1. 1043 1. 2163 1.4644 0. 117897 0; 064 689 O. 130 320 1.0408 1. 1042 r 1. 2162 1.4640 0. 156921 0.081 037 0. 164 051 1. 0408

.

1.

1042 1. 2160 1. 4635 O. 208 862 0. 099 835 O. 203 677 1.0407 1. 1040 1. 2157 1. 4627 0. 277 996 0. 120 671 0. 249 045 1. 0407 1. 1039 1. 2154 1. 4620 0.370 012 O. 143 050 0. 299 834 1. 0406 1. 1037 1. 2150 1. 4611 0 .. 492487 O. 166' 909 0. 3.55790 1. 0405 1. 1035 1. 2145 1. 4598 0. 655 500 0. 193134 0. 416 713 1. 0404 1. 1032 1.2137 1. 4577 Oj 872 470 0.223 051 0. 481 741 1. 0402 1.1026 1. 2125 1. 4546 1. 161.~.6 O. 256 762 0. 548 766 1. 0399 1. 1018 1. 2107 1. 4502 1. 545 63 0. 292 422 0.614 986 1. 0395 1. 1009 1J2086 1. 4450 2.057 24 0. 327 391 0.677 841

i.

0391 1. 099.8 1.2062 1. 4391 2.738 19 0;·359 545 0. 735 367 1. 0387 1. 0987 I - .1. 2037 _{1. 4329} 3.644 52 0.387 698 0.786 257 1. 0383 1. 0975 1. 2011 1.4266 4.850 86 0.411 464 0.829 888 1. 0379 1. 0964 1. 1986 1.4208 6. 456 50 0.430 975 O. 866 283 1. 0375 1. 0954 1. 1964 1. 4156 8. 593 60 0.446 656 0.895950 1. 0371 1. 0945 I, ~945 1. 4112 11. 4381 0. 459 054 0. 919 688 1. 0369 1. 0938 1 j 1930 '1-.4075 15 .. 2241 0_, . 468 735 O. 938 402 1. 0366 1. 0932 .1. 1917 1.4045 20. 2633 0.476221 O. 952 988 1. 0365 1. 0927 1. 1907 1. 4021 26. 9704 0.481970 O. 964 256 1. 0363 1. 0923 1. 1899 1. 4003 35.8976 0.486 360 O. 972 901 1. 0362 1. 092~ L 1892 1. 3988 47. 7797 0.489 698 0.979 501 1. 0361 1. 0918 1. 1888 . 1. 3977 63. 5948 O. 492 230 0. 984 520 1. 0361 I 1. 0917 1: 1884 1. 3969 84. 6446 0.494 145 O. 988 324 1. 0360 1. 0915 1. 1881 1. 3962 112.662 0.495 591 0. 991 202 1. 0360 1. 0914 1. 1879 1..3957 149. 953 O. 496· ·682 0. 993 375 1. 0359 1. 0914 1. 1877 1. 3954 199. 588 0.A 97 504 0. 995 015 1. 0359 1. 0913 1. 1876 1. 3951 I

TABLE II (cont1_d)

W(S,D)/

W(O,D)

D

$=0.3

s=

0.5

5-;0.75

S:;

1.0

S=

1.5

8=2-0

0.050 000

1.

7472

2.4189

3

.

4702

4.7647

8.0859

12.361

O

.

066 550

1. 7469

2

.

4180

3

.

4675

4.7586

8

.

0647

12.308

0

.

088 578

1. 7461

2

.

4158

3.4625

4.7484

'

8

.

0336

12

.

,

234

O

.

117 897

1.7453

2.4137

3

.

4571

4.7370

7

.

9965

12

.

144

0

.

156 921

1. 7443

2

.

4110

3

.

4502

4.7225

7

.

9502

12

.

031

0

.

208 862

1. 7429

2

.

4075

3

.

4415

4.7045

7

.

8927

11.891

O

.

277 996

1. 7414

2

.4

036

3.4317

4.6836

7

.

8245

11. 723

0

.

370 012

1. 7397

2.3989

3

.

4198

4.6580

7

.

7405

11. 517

0.492 487

1. 7372

2

.

3924

3

.

4033

4.6233

7

.

6298

11. 253

0

.

655 500

1. 7333

2

.

3823

3

.

3787

4.5731

7

.

4784

10

.

909

0

.

872 470

1.7274

2

.

3675

3

.

3438

4.5038

7

.

2804

10

.

483

1. 161 26

1. 7194

2

.

3478

3

.

2984

4.4165

7.0451

10

.

005

1. 545 63

1. 7098

2.3246

3.2463

4

.

3183

6.7938

9.5216

2.057 24

1. 6991

2

.

2992

3

.

1905

4.2160

6

.

5461

9.0691

2

.

738 19

1. 6878

2

.

2731

3

.

1347

4.1164

6

.

3171

8

.

6691

3.644 52

1. 6767

2.2478

3.0821

4.0249

6

.

1162

8

.

3303

4.850 86

1.

6664

2.2248

3.0353

3

.

9452

5. 9472

_

8.0528

6

.

456 50

1. 6573

2

.

2047

2. 9953

3.8781

5.8091

7.8309

8.593 60

1. 6495

2. 1878

2

.

9621

3

.

8234

5. 6990

7

.

6567

11. 4381

1. 6431

2

.

1741

2. 9354

3.7798

5

.

6127

7

.

5215

15

.

2241

1. 6380

2

.

1632

2

.

9143

3.7455

5

.

5458

'

']

.

4176

20.2633

1.6339

2

.

1546

2

.

8978

3.7189

5.4943

7.3381

26.97

'

04

1. 6307

2.1479

2

.

8851

3.6985

5

.

4549

, 7~ 2778

35

.

8976

1. 6283

2

.

1427

2.8753

3

.

6828

5

.

4250

7

.

2320

47

.

7797

1. 6264

2

.

1387

2

.

8678

3

.

6709

5

.

4023

7

.

1973

63

.

5948

1. 6249

2

.

1357

2.8621

3.6618

5

.

3851

7

.

1712

84

'

.6446

1. 6238

2

.

1334

2

.

8578

3

.

6550

5

.

3722

7

.

1515

112

.

662

1. 6230

2. 1316

2.8545

3

.

6498

5

.

3624

7. 1366

TABLE II (cont'd)

W(S)D)/W(O,D)

.D

_$=

_3.0

_.$=4.0

_3=5.0

_S=10

_$=2D

0

.

050 000

23

.

666

38

.

474

56

.

593

--

190

.

34

607.76

O

.

066 550

23

.

466

37

.

9

.7

5

55

.

581

181. 57

548

.

42

0.088 578

23

.

205

37

.

319

54

.

229

170

.

83

483.33

0

.

117 897

_

22

.

874

36

.

474

52

.

547

158

.

14

417

.

40

O

.

156 921

22

.

454

35

.

440

50.497

143

.

88

355

.

67

O

.

208 862

21. 947

34

.

195

48.062

128

.

84

301. 49

0

.

277 996

21. 339

32

.

726

45

.

258

1

1

4

.

07

255. 94

0

.

370 012

20

.

608

31.016

42

.

128

100

.

42

218

.

48

0

.

492 487

19

.

721

29

.

065

38.766

88

.

301

187

.

88

O

.

655 500

18

.

665

26

.

928

35.344_

77

.

777

162

.

82

0

.

872 470

17

.

490

24

.

758

32

.

084

68

.

839

142

.

42

1. 161 26

16

.

306

22

.

732

29. 180

61. 464

126

.

06

1. 545 63

15

.

214

20

.

965

26.724

55

.

540

113

.

18

2

.

057 24

14

.

265

19

.

490

24

.

717

50

.

864

103. 16

2

.

73819

13

.

473

18

.

2

9

1

23

.

111

47

.

213

95

.

419

3

.

64452

12

.

829

17

.

337

21. 845

44

.

388

89

.

474

4

.

850 86

12

.

319

16

.

5

9

0

20

.

861

42.219

84

.

934

6

.

456 50

11. 919

16

.

011

20

.

103

40

.

563

81. 484

8.59360

11. 610

15.567

19.523

3

9.

306

78

.

872

11.4381

11.373

15

.

228

19

.

082

38.355

76

.

899

15

.

2241

11. 193

14

.

970

18.748

37

.

636

75.412

20

.

2633

11. 056

-

14

.

776

18.495

37

.

094

74

.

292

26

.

9704

10. 952

14

.

628

18

.

305

36

.

686

73

.

449

35 .

.

8976

10

.

874

14

.

517

18.161

36.379

72.816

47

.

7797

10

-

.

815

14

.

434

18.053

36

.

148

72.339

63

.

5948

10

.

770

14.371

17. 971

35.975

71. 981

84.6446

10

.

736

14.323

17

.

910

35.844

7

1

.712

112

.

662

10

.

711

14.288

17

.

864

35.746

.

71. 510

149

.

953

10

.

692

14

.

261

17

.

829

35

.

672

71. 358

199

.

588

10.678

14.241

17

.

803

35

.

617

71. 243

S 0 0.02 0.05 O. 10 0.20 0.30 0.50 0.75 1. 00 1. 50 2.00 3.00 4.00 5.00 1 O. 0 2 0. 0 TABLEIII ,

PoiTj:;Pl'JT~

Pt

.J Tl/

P

11

T

~

Pt/po

=

')(S)

llim

df<

I

::: W(S, 0) _ W(S, 0) D-+O dD W(O, 0) W (0, 0)

7(

(S) 1. 000 00 0.66765 1. 000 00 1. 000 00 1. 035 84 0.68548 1. 040 93 1. 004 91 1. 091 12 0.71255 1. 104 61 1.01237 1. 187 24 0.75844 1.21701 1. 025 08 1. 394 23 0.85276 1.466 18 1. 051 61 1. 620 41 0.94974 1. 794 42 1.07962 2.126 31 1. 14836 2.424 39 1. 140 19 2.844 50 1. 39920 3.48332 1. 224 58 3. 633 97 1. 64849 4.791 11 1.31842 5.33268 2.14535 8.167 75 1. 531 27 7. 091 54 2.656 96 12.5491 1. 769 59 10.6347 3.74413 24 .. 3290 2.287 69 14. 1796 4.87837 40.1177 2. 829 ~5 17.7245 6.03283 59.9172 3.38046 35.4491 11. 8953 219.061 6.179 61 70.8982 23. 6285 835.306 11.7818

PRE;:SSURE RATlOS FOR THELIMITING CASES

P =

00 AND D ::; 0

s

U

PI

Tl

n

_l S ~ V FIG .. 1

ct

L

I

t

ct

1 Gauge Volume

(a) ORIFICE PROBE

Gauge Volume

Pt Tt nt

(b) IMPACT TUBE OF FINITE LENGTH

PI Tl

~

hl 1:::::..,

S

Cl

+-!

11 o

r-Element of Inner Wall Surface

=

Tï

d(dx)

/,~

7'

/

x I I I 1 - \ • - I

~

I IH

--~~

-~dx

I

\

FIG. 2 _{THE CYLINDRICAL IMPACT TUBE AT ZERO ANGLE OF ATTACK.}

x

= x/I; D = dil; L

=

lid

Pt Tt

t

nt

0.0 0.30 0.25

t

«.(D) 0.20 O. 15 0.10 0.05 0.00 0.0 0.2 0.2 0.4 0.6 0.8 1.0 ot 0(. (L) ot= CX(D) 0.4 0.6 0.8 1.0 1.2 1.4 0.45 0.40 0.35 .

t

o(.(L) 0.30 0.25 0.20 1.2 1.4

2.0 1.6

t

1. 2 7(t)

0.8

0.4 0.0

n

~

I I I I I I

~

,

-~

\

"\

~

""

\

1\

~

""-~

t-1\

\

I---\

1\

--

~

t-~

\

_\

V

S ::: 0.1

"'-

,

\

\

,

r-

\

\'

i"-S

=

0

\

S

=

1

-,

_\

I'<:S

~

2

,..-'\

~

-\

\

,

_"'~

-

4s=~

..----S :: 5

~

-S

=

00

"0..~

.

~

--....L I I 0.0 0.2

0.4

O. 6

t~ I I I

-S

=

0.5

-~

~

-A

t\..

"\

'"

-I\.

'\

-"-

_...

~

-I I ~ 0.8 1.0 2 2

FIG. 4 VARIATION OF T(t)

=

[1

+

erf

(S~l

- t2 )] e-S t

t

2T---~

T

(t)-curve for constant S 'J(t) 1 Increasing D

t

F(Y)

o

to

o

1 F(Y)-curve for constant S

--

_--

-tf

--

...

t ~ Increasing D

----1- - -- - ---

F(Y f ) Decreasing D

~~

Area

=

D (S, D) ~

Y

"

t

T(t) 1

t

F(Y)

o

FIG.

9

2~----

______________________________________

~ 1 o~~~~~~~~~~~~+---

____________

~ to t 1 t2 t3

o

t₁ · F(Y)-curve

...

Y

DISTRIBUTION OF t- AND Y ~ INCREMENTS FOR A TYPICAL (S, D ) -COMBINATION