Supersonic combustion studies II: Exponential Relationships Between Pressure, Area and Process Length, for Heat Addition in a Non-Constant-Area Duct

CoA REPORT AERO No. 204

VLIEGTUIGBOUW KUNDE BIBLIOTHEEK

THE COLLEGE OF AERONAUTICS

CRANFIELD

SUPERSONIC COMBUSTION STUDIES

IL EXPONENTIAL RELATIONSHIPS BETWEEN PRESSURE,

AREA AND PROCESS LENGTH. FOR HEAT ADDITION

IN A NON-CONSTANT-AREA DUCT

by

CoA Report Aero No. 204 March, 1968.

THE COLLEGE OF AERONAUTICS

CRANFIELD

Supersonic Combustion Studies

II. Exponential Relationships Between P r e s s u r e , Area and P r o c e s s Length, for Heat Addition in a Non-Constant-Area Duct

by

Paul Flanagan, A, F. R. Ae. S. and Roy A. Cookson, B. Sc. , P h . D . , M. A. I. A. A.

SUMMARY

Research into the injection and combustion of gaseous fuels in a super-sonic flow of high-enthalpy air, flowing in a non-constant-area duct, has indicated the need for an analytical description of the processes involved.

The p r e s s u r e - a r e a relationship proposed by Crocco for heat-addition in a duct has been discussed in this report and gasdynamic equations which utilize this relationship have been derived.

Because of the need to include process length in the analysis, an

exponential relationship between p r e s s u r e , area, and length has been proposed and gasdynamic equations based upon this concept have been established. These equations have been used to calculate stagnation and static p r e s s u r e and temperature ratios for an inlet Mach number of 2 and a value for y of 1. 32.

CONTENTS

P a £ Sünamary

List of Symbols

Introduction 1 Gasdynamic Equations Using the Crocco Relationship 3

Exponential Flow P r o c e s s e s 7 Temperature Variation with P r o c e s s Length 10

Ranges of Possible Flows 11 Concluding Remarks 12 Acknowledgement 13 References 14 Table of Ranges of Possible Flows

LIST OF SYMBOLS Flow area

K . Exponential flow constant for area K Exponential flow constant for p r e s s u r e K Exponential flow constant for teraperature M Mach number

M Critical Mach number c

p Static p r e s s u r e

P Total (stagnation) p r e s s u r e q Specific heat addition

R Characteristic gas constant s Specific entropy

t Static temperature

T Total (stagnation) temperature V Gas velocity X P r o c e s s length

X = [i + ( ^ ) M 2 )

= < 1 + (ri + 1)7 M\ 1 + (n+ 1)7 M^

y Is entropie flow index

e Crocco type flow index

n Exponential type flow index

Superscripts

* Referred to state at point where M = 1 ** Referred to state at point where M = M

1. Introduction

The problems associated with extending the range of flight velocities of air-breathing aircraft into the hypersonic range (i. e. greater than Mach No. = 5.), a r e considerable. In the first instance the stagnation p r e s s u r e s and temperatures resulting from such flight speeds are very high. For example, the stagnation p r e s s u r e and temperature associated with a velocity of Mach No. = 8 at an altitude of 120, 000 feet, would be at least 1600 p. s. i. and 3000°K respectively.

From the above values of p r e s s u r e and temperature, which would of pourse occur before any combustion process took place, it can be seen that conventional turbomachinery is unlikely to be of use for reasons of structural integrity alone. Over and above this essentially practical consideration, this type of propulsion unit would be impracticable by virtue of the chemical dissociation which would take place under these severe operating conditions. The phenomenon of dissociation is such that certain of the fuel and air species would break down into their constituent atoms or radicals, and these molecular fragments would c a r r y with them large quantities of energy which would not be released until they recombined. At these high temperatures, recombination is not likely to take place until the exhaust gases mix with the ambient air, by which time the released energy is lost. The proportion of the energy in the fuel which is lost by reason of dissociation, will increase with increasing Mach number until at some flight speed the injected fuel will not provide useful thrust.

The efficiency loss due to dissociation will of course apply to any prop-ulsion unit wherein the incoming a i r - s t r e a m is decelerated to a low subsonic velocity, and for this reason the conventional ramjet can also be excluded.

The supersonic combustion ranajet or "scramjet" is one type of a i r -breathing engine, to date the only one, which will overcome the problems outlined above. In the scramjet the inlet air is diffused down to a lower, but still supersonic velocity, before fuel is added and combusted. Hence the maximum possible p r e s s u r e or temperature will not be realized within the engine since a large proportion of the total energy remains in the kinetic form.

Heat release in a ducted supersonic gas stream can be shown to be less efficient than the subsonic case. However, this loss in efficiency may be more than balanced by a reduction in total p r e s s u r e loss, particularly as the aircraft inlet diffusers will not be required to diffuse the incoming air through the troublesome transonic condition.

Because fuel ignition will take place spontaneously, the scramjet comb-ustion chamber will not requii*e any form of flameholder, and will consist simply of a duct of the correct geometry together with a fuel injection system. Hence the main problems associated with the design of this type of combustor, will be to determ.ine a suitable geometry and to develop a viable fuel injector.

The combustor geometry is of considerable importance in the case of supersonic heat release, since quite small variations in the area-length

relationship will modify the flow and hence the combustion process. The fuel injection system is important to the extent that shock-free injection is required, otherwise high temperatures and p r e s s u r e s will again exist behind the shock-wave. In a diffusion-controlled system such as that which exists in the scramjet combustor, the total combustor length will be dependant upon the mixing of the fuel and air. The rate at which the fuel and air mixes will vary with different types of injector, for example, a m i d - s t r e a m injector should produce a more rapid mixing than injection from a wall slot. However, other factors, such as heat-transfer, may influence the choice.

The simplest combustor geometry which could be used, would of course be a constant-area duct, however, the properties of constant-area supersonic heat-addition are such that the flow will choke (i. e reach Mach No. = 1) if sufficient heat is added. It can be shown that the naost efficient method of heat-addition will take place at constant static p r e s s u r e (see reference 1), although the work shown later in this paper indicates that this particular mode of heat release may not be precisely achieved.

An experimental programme is currently being conducted in the

Propulsion Department with investigations into the fuel injection and mixing, and combustor geometry as outlined above. A high-enthalpy facility has been constructed and has provided flow conditions which simulate those which would occur within a scramjet combustor at Mach number = 2 (see ref. 2).

The initial experiments were with the "free-jet" injection into a Mach 2 s t r e a m of air at a stagnation temperature of approximately 1800°K. In these tests hydrogen and methane were injected through a mid-stream injector and a study made of the mixing and combustion p r o c e s s e s . P a r t of this work is included in reference 3 and the remainder will be published at a future date. With the combustion in these tests taking place at ambient p r e s s u r e the overall process relates very closely to a constant-pressure process. However for useful thrust to be produced the combustion must take place under enclosed conditions, and it is the study of the effect of the combustor wall which forms the basis for the next phase of our experimental programme.

A number of com.bustor configurations have been tested to date, either of constant-area or constant divergence, and much data on the influence of the combustor geometry has been obtained. The aim of this series of tests is the design of a combustion chamber with just sufficient length for the heat release to reach completion, and along which the static p r e s s u r e remains as nearly constant as possible. It has already become apparent that the presence of the combustor does modify the supersonic combustion p r o c e s s , at least to the extent of increasing the distance required for complete combustion,

compared that is, with the equivalent length in the free-jet case. The reasons for this modification will not be known until further tests are carried out, but two suggestions a r e a) that the combustor wall is influencing the p r e s s u r e field, or b) that the heat loss to the cooled walls affects the combustion p r o c e s s , possibly by accelerating the flow. One other factor arising from the experimental prograname is that the variation of duct area with duct length is of more importance than the simple ratio of outlet to inlet a r e a s .

For analysis to be made of the variable-area heat addition p r o c e s s , it is necessary to derive gasdynamic relationships which are more general than the conventional constant-area duct equations.

Reference 4 has derived some gasdynamic equations using a p r e s s u r e -a r e -a rel-ationship proposed by Crocco in reference 5, n-amely,

e

p A = constant

or p A = constant.

From these expressions it can be seen that the constant-area and constant-pressure processes can be represented by e values of 1 and 0 respectively. This approach has been developed further in the next section of this report, and a full set of gasdynamic equations based upon the Crocco relationship have been derived. It is clear that this type of analysis has

value, particularly in giving insight into certain flow processes. Unfortunately the p r e s s u r e and area relationships so obtained can only be expressed in

t e r m s of inlet and outlet conditions, and that no cognizance is taken of

process length. Reference 4 has attempted to overcome this disadvantage by applying different e values to successive sections of combustor. and hence optimising the duct geometry.

In section 3 of this report an attemipt has been made to introduce a combustor length dependency into the gasdynanaic equations. The resulting relationships are one-dimensional and frictionless from reasons of simplicity but it is thought that, this analysis will still give useful informiation on the p r o c e s s e s involved,

There would appear to be two necessary conditions for any p r e s s u r e arealength relationship adopted for this type of analysis. The first r e q u i r e -ment is that it should approximately represent the true condition, and the

second, that the relationship should be in a form which allows for an analytical

n2

solution, particularly for the I p. dA t e r m in the momentum equation. F o r the analysis developed in section 3 an exponential form has been adopted for both the pressure-length and the area-length dependencies. This type of relationship is shown to be amenable to analysis and the exponential area-length variation is found to be a very good approximation to the geometry of a typical combustor. All that remains is to obtain experimental confirm-ation of the variconfirm-ation of p r e s s u r e with length.

2. Gas Dynamic Equations Using the Crocco Relationship A p r e s s u r e - a r e a relationship of the form,

p A ^ / < ^ - ^ ) = const. or p A = const.

(1)

has been proposed by L. Crocco (reference 5). In this way a family of general flow processes can be described by various values of e. for example. processes for which e = 0 and e = 1 are constant-pressure and constant-area processes respectively.

If the p r e s s u r e - a r e a relationship as given by equation (1) is substituted in the momentum equation,

p^ A^ + mV^ + pdA = p^A^ + mV^ (2)

we obtain

^ P l ^ i + « i V ^ = CP2A2 + mV^ (3)

and hence, by utilizing the definition of Mach number and the equations of state and continuity,

pA [e + 7 M ] = constant (4) A 2/Ai e + 7 M 2 e + 7M2 1 - e (5) ' 2 / P j e + 7 M e + 7M2 (6) •v - 1 2

If, for simplicity, we substitute X for 1 + (-^—^—)M

P = p x''''^"^ " ^^ and T = tX Therefore e + 7 M e + 7 M ' 1

x"

7 / ( 7 - 1) (7)

M2(e+ 7 M p M^(e + 7M2) (8) and — 1

K -.]

IMJ X, \ 1 1/ e + 7 M ₁ e + 7 M,

and also from the definition of entropy change,

L \

As = s s -2 1 7 - 1 2 In

rv

7 (2 - e) + e 7 ( In e + 7 M e + 7 M, (9) F r o m Damkohlers 2nd ratio, ,2 A _ iV q ^ 2 ^ 2 c t, , ,^2,2 p i [e + 7 M 2 ]

e + 7M^ j

(10) (11)

we can determine the Mach number at which thermal choking will occur (i. e. the "critical" Mach number M ), by equating dq/^j^ to zero and solving for M„ = M

M _{[(^) - (7 - 1) ]} - 1

7(1 - e) + e (12)

Hence if we consider constant-area and constant-pressure heat addition to a one-dimensional, frictionless flow of an ideal gas (i. e . , e = 1 and e = 0 respectively), it can be shown that the critical Mach number M , for the constant-area condition, is unity. This case, of course, is that usually defined as a Rayleigh type flow. However, for the constant-pressure case an interesting conclusion results from equation (12), namely, that the critical Mach number M 0, i. e. , that if heat is added at constant static p r e s s u r e , thermal choking will not occur.

Using the * notation to represent conditions at the point where M = 1. we can derive the following relationships.

6 -A s * = s - s p = p * P p * A A* ' t t * T T * S * = ;; e + . e + T Y M ^ 1 e T 2 + (7 - 1) M^ 7 + 1 e + y . e + 7 M^ M(e + 7 ) 1 e + 7M^ J 7 - 1 e + 7 e + 7 M ^ _ € 1 - e 2 j 2M^ + (7 - 1)M^ 7 + 1 Y- 1 2 I n M +

[^

' ( 2 e + 7 . e + 7 M ^

- ^) ^ ^1 in

7 J 2 r e + 7 r l e + 7M^J_ (13) (14) (15) (16) (17) (18)

We c a n now h o w e v e r , define the flow v a r i a b l e s m o r e g e n e r a l l y , r e f e r r i n g t h e m to the c r i t i c a l point a s defined by equation (12). F o r t h i s g e n e r a l

definition of t h e c r i t i c a l flow c o n d i t i o n s w e u s e t h e n o t a t i o n * * . E = p * *

r2te)-(V)i

[(J) - (7-l)j{l +

( J ) M 2 ] (19) ((?) - (7 - 1)1 r -V 1 2^ P - P ^ |(I) . (X.:_l)j - 1 (20) t_ t * * ^' 2 J I 1 + ( 1 ) M 2 J

{(J) - (7 - 1)]

(21) T {I + ( ^ ) M ' ) (22)

r

- 7 A A i

2{(^) - ( ^ ) }

1 - e

l(^) - (7 - i)Ui +

( ^ ) M M

(23) A ï - f O A s = s - s = C T(2 -^e) + e k

2[(J) - ( ^ ) }

U ( ^ ) - ( 7 - 1 ) } ( I + ( ^ ) M 2 ] + C I n P M ^ { ( J ) - (7- 1)] (24) E x p o n e n t i a l Flow P r o c e s s e s It i s p o s t u l a t e d that, P2 = Pi e K X P ₍₂₅₎ w h e r e x is the p r o c e s s length = x - x dp and h e n c e -r- = K . p dx p '^ (26)

S i m i l a r l y an exponential r e l a t i o n s h i p i s p o s t u l a t e d between the a r e a and the p r o c e s s length A^ = A^ e K , x A o r ^ = K , A dx A (27) (28) F r o m (28) and (26) 1 dA _ _ 1 _ dp K„ A ' K p A p "^ K and therefore for s i m p l i c i t y we u s e r^ = =7^ ^ / K p (29) (30)

- 8

= c o n s t a n t (31)

n \ '•I

P

Using t h i s f o r m of p r e s s u r e - a r e a r e l a t i o n s h i p we can now solve the m o m e n t u m equation (2)

T h u s (2) now b e c o m e s ,

- ^ 1 ' P i ^ i - n - T T P i ^ 1 = - ^ 2 -^ P 2 ^ - n - T T P 2 ^ , •

giving p A U l ^ ) + 7 M j = c o n s t (32)

o r P A | ( ^ ^ ) + 7 M ^ ] = c o n s t (33) F r o m (4) and (33) it can be s e e n that e and — a r e a n a l o g o u s . To t e s t

r) + 1 t h i s e q u a l i t y we put e = ( ^) (34) Adp f r o m (26) and (28) pdA + Adp e - 1 - p dA , , . . . = —f- -r- and solving e A dp ^ p A ^ ' ^ ~ = c o n s t a n t which i s the C r o c c o r e l a t i o n s h i p .

T h e r e f o r e we can s u b s t i t u t e the t e r m ( ) for the C r o c c o p r e s s u r e -1 + -1 a r e a exponent e in e q u a t i o n s (5) and (6) t h u s , A , _ 2 A . 1 + ( n + 1 ) 7 M ^ | ^ [ 1 + ( n + 1) 7 M 2 (35a) .2, . , , _ 2 1 ^ A A K ( 1 + 7 M ; ) + K 7 M _ 2 ^ j _ p 1 A 1 [ K ^ + Kp (35^j ""^ A I 2 2 "^1 l K p ( l + 7 M 2 ) + K ^ 7 M 2 p^ f 1 + (r)+ 1) 7 M ^

and ^ - ——-4

\ ^"-^ (36a)

P i I 1 + (ri+ 1) 7 M 2

9

-K

or P 2 j V ^ " ^ ^ l ^ " ^ A ^ ^ l ] ^ ^ I ^

Pi [ K p ( l + 7 M 2 ) + K ^ 7 M 2

(36b)

and once again using X in place of 7 ' l ^ ' ' 1 + {^^-^ ) M' we have, ( 3 7 ) t^ / M ^ l 2 ( 1 + ( n + l ) 7 M ^

"^'^^^ ^ h i ! W + ( . + 1)7M2

(38a) M, o r K (1 + 7 M : ) + K . 7 M , p 1 A 1 ^ i j [ K p ( l + 7M^2) + K ^ 7 M ^ ^ _ (38b) and a s before (39)

a l s o the entropy change can be e x p r e s s e d thus M,

' = = < ^ l - " M 7 ' ^ '

7 ( 2 n + 1) + 1 7 ( n + 1) I n fi + ( n + 1) 7 M ^ l l + (ri+ 1) 7 M ^ o r (40a) 7 R ( 7 - 1) 2 I n M M_ 2' f 2 7 K ^ + ( 7 + DKpl ( K p ( l + 7 M 2 ) + K ^ 7 M 2

1 -^(^A-^^

Kp(l + 7M2) + K ^ 7 M 2

and the c r i t i c a l Mach n u m b e r i s given by ,2 _ : . . - - 1 M 7 n + 1 / K 7 K ^ + K ; I A p , (40b) (41a) (41b)

10

-z =

1 + (n + I ) 7 M ' 1 + (r) + 1) 7 M

2 J and t h e r e f o r e from (25) and (36a)

I n Z a l s o

Kp- ( n + 1)

ï j ^ . ( — T T ) I n Z from (27 and 35a)

K ^ ï) + 1 ' h e n c e

(^A

^ V

I n 1 + (rj + 1) 7 M 1 1 + (1 + 1) 7 M ' (42) 4. T e m p e r a t u r e V a r i a t i o n with P r o c e s s Length F r o m equation (42) 1 (K^ + K ) ^ A p ' I n Z and Z c a n be shown to b e n + 1 _{/ t .} ^ 1

M, from (36a) and (38a)

<^A ^ ^ p ) I n M \ M l (43) h e n c e ( K . + K )x ^ A p ' \ \ 2 / Mj ₍₄₄₎ o r L \ ,^ , 2 2(K^ + K )x M^ ' A p ' (45)

o r

- 11

(K. + K )x . A p '

(46)

Thus it is seen that the ratio of the outlet to inlet static temperatures which result from heataddition in an exponentially related p r e s s u r e a r e a -length duct is not itself exponential but is modified by the ratio of the outlet to inlet gas velocities.

The ratio of the outlet to inlet static temperatures can obviously be written V, K^x (47) where K, = K^ + K t A p

5. Ranges of Possible Flows

Frona equations (12) and (41) the critical Mach number is given as,

M

_{(?) - ^y -']'' - VTTT}

and from this expression we are able to delineate the various ranges of operating conditions, for example,

2 1 7 a) M = 00 when (i) rj = - — ; (ii) e = '—r

c 7 7 - 1

b) M^ < 0 when (i) rj < - - ; (ü) e > "^ v _{c 7 7 - 1}

c) M ^ when (i) n = 00 ; (ii) c

0.

Obviously case b) is unreal while case a) and c) represent the limiting conditions. Table 1 contains calculated values of n . M and e for an assumed

c

7 of 1. 32, and these values are shown diagrammatically on Figure 1. From Figure 1 it is possible to pick out the basic processes, for example the

constant-area process is represented by •^he rj = 0 line, and processes with large r) values tend toward the constant-pressure process.

12

-6. Concluding R e m a r k s

T h e p r e s s u r e - a r e a r e l a t i o n s h i p p " A = constant, p r o p o s e d by C r o c c o in r e f e r e n c e 5, c l e a r l y g i v e s s o m e insight into the flow p r o c e s s e s o c c u r r i n g for the c a s e of h e a t addition in a n o n - c o n s t a n t - a r e a duct. F i g u r e 1 i n d i c a t e s by m e a n s of an e n t h a l p y - e n t r o p y d i a g r a m , the r a n g e of flows which c a n be d e s c r i b e d by t h i s r e l a t i o n s h i p . F o r e x a m p l e e = 1 and e = 0 c o r r e s p o n d to the p a r t i c u l a r conditions of c o n s t a n t - a r e a and c o n s t a n t - p r e s s u r e p r o c e s s e s r e s p e c t i v e l y .

F i g u r e 1 a l s o i l l u s t r a t e s the concept of s u b c r i t i c a l and s u p e r c r i t i c a l h e a t addition, which specifies the e x i s t e n c e of a c r i t i c a l Mach n u m b e r M ,

c below which heat m a y not be added without modification of the duct inlet flow conditions. T h i s p r o c e s s of modification, known a s t h e r m a l choking, o c c u r s at M = 1 for the usual Rayleigh flow in a c o n s t a n t - a r e a duct, and at

M = 0 (i. e. will n e v e r o c c u r ) for the c o n s t a n t - p r e s s u r e c a s e .

A s e t of g a s d y n a m i c equations b a s e d upon the above p r e s s u r e - a r e a r e l a t i o n s h i p have b e e n d e r i v e d in s e c t i o n 2 of t h i s r e p o r t .

E x p e r i e n c e and intuition t e l l s us that a given flow p r o c e s s m u s t be defined in t e r m s of the flow phenomena o c c u r r i n g between inlet and outlet, a s well a s the inlet and outlet conditions. Hence s o m e r e f e r e n c e to p r o c e s s length i s r e q u i r e d and in section 3 of t h i s r e p o r t an exponential length dependency h a s been a s s u m e d . F o r the p u r p o s e of this r e p o r t both duct a r e a and s t a t i c p r e s s u r e a r e a s s u m e d to v a r y exponentially with length, a s follows and o r P2 = ^ 2 = Ap-'^ K X . / - " = constant w h e r e _K By using a r e a - l e n g t h and p r e s s u r e - l e n g t h r e l a t i o n s h i p s a s d e s c r i b e d above, the m o m e n t u m equation can be r e a d i l y evaluated. S i m i l a r l y , m o s t p r a c t i c a l g e o m e t r i e s and d i s t r i b u t i o n s can be r e p r e s e n t e d a p p r o x i m a t e l y by an exponential r e l a t i o n s h i p . F o r example a typical c o m b u s t o r used in o u r e x p e r i m e n t s h a s a length of 30 i n c h e s with an inlet d i a m e t e r of 2 inches and an outlet d i a m e t e r of 3 i n c h e s , the d i v e r g e n c e being l i n e a r . T h e s e v a l u e s when substituted in the exponential a r e a r e l a t i o n s h i p , yield a value for the constant K of 0. 324. The m a x i m u m difference d i s c o v e r e d between

13

-the exponential shape corresponding to this value and -the true conical section is only 0. 030".

The range of flows which can be described by the rj parameter is also illustrated by Figure 1, with r) = 0 and rj = oo representing the constant-area and constant-pressure, processes respectively.

Gasdynamic equations for all of the normal flow variables are derived in section 3 and illustrated for the specific case of inlet Mach number = 2 by Figures 2 to 5. Similarly the calculated values used in Figures 2 to 5 are based upon a value for 7 = 1 . 32. The conditions M = 2 and 7 = 1. 32 are those currently operating in our experimental programme.

In section 4 the variation of static temperature with length is shown to be represented by the product of an exponential term (with K = K + K ),

and• the ratio of outlet and inlet velocities. P Although the analysis used in this report is one-dimensional, it can be

seen that the exponential form of pressure-length and area-length relationship results in gasdynamic equations which are readily usable and experimentally reasonable. Thus it should now be possible to use our experimental results to calculate the optimum duct shape.

7. Acknowledgement

We would like to thank Mr. J. R. Palmer of the Propulsion Department for his considerable assistance in checking the derived gasdynamic equations.

8. R e f e r e n c e s 14 1. F e r r i , A. Cookson, R. A. Penny, G. S. 4. Billig, F . S. C r o c c o , L. D o b r o w l s k i , A.

" S u p e r s o n i c Combustion Technology". AGARD L e c t u r e S e r i e s on Supersonic T u b o - M a c h i n e r y , V a r e n n a , 1967.

" S u p e r s o n i c Combustion Studies, I. Design, C o n s t r u c t i o n and P e r f o r m a n c e of a High-Enthalpy F a c i l i t y " , College of A e r o n a u t i c s Report

A e r o No. 200, Nov. 1967.

"Diffusion Controlled S u p e r s o n i c Combustion Utilizing a High-Enthalpy Blow-Down S y s t e m " , College of A e r o n a u t i c s , T h e s i s 1967,

"The Design of Supersonic C o m b u s t o r B a s e d on P r e s s u r e - A r e a F i e l d s " . 11th Symposium (Int.) on Combustion, Aug. 1966.

" O n e - D i m e n s i o n a l T r e a t m e n t of Steady G a s D y n a m i c s " , F u n d a m e n t a l s of Gas D y n a m i c s , Vol. Ill of High Speed A e r o d y n a m i c s and J e t P r o p u l s i o n , P r i n c e t o n U n i v e r s i t y P r e s s , 1958. " A n a l y s i s of N o n - C o n s t a n t - A r e a Combustion and Mixing in Ramjet and R o c k e t - R a m j e t Hybrid E n g i n e s " . L e w i s R e s e a r c h C e n t r e , NASA TN-3626, 1966.

T A B L E 1: RANGES OF VALUES FOR GAS 7 = 1. 32 n 00 4 3 2 1 0. 5 0. 33 0 . 2 5 0 - 0 . 29 - 0 . 3 8 - 0 . 4 4 - 0 . 5 0 - 0 . 6 7 - 0 . 7 5 1 1 < -7 = - 0 . 76 M c 0 0 . 4 0. 45 0. 62 0 . 6 6 0. 78 0. 83 0. 87 1.0 1. 27 1. 41 1. 56 1. 72 2. 89 1 0 . 0 M ^ < 0 c e 0 0 . 2 0. 25 0. 33 0. 5 0. 67 0. 75 0 . 8 1 . 0 1.4 1. 6 1.8 2. 0 3 . 0 4 . 0 - - '>' 7 - 1 = 4 . 1 2 5 J 1 C O M M E N T S C o n s t . S t a t i c P r e s s u r e F l o w K = 0 p F l o w s w i t h : -' A -' -' p i A p F l o w s w i t h : -A p R a y l e i g h F l o w K ^ = 0 1 F l o w s w i t h

K.l <

| K

1

' A ' p i No T h e r m a l C h o k i n g

Subcritical

ENTROPY

F I G U R E I. AN E N T H A L P Y - E N T R O P Y DIAGRAM I L L U S T R A T I N G THE RANGE O F F L O W S WHICH CAN B E D E S C R I B E D BY BOTH THE ( AND n P A R A M E T E R S a UJ cc a. UJ a. 2

5

2 0 -3-0 2 0 OUTLET MACH NUMBER M j

FIGURE 2. T H E RATIO O F T O T A L T E M P E R A T U R E S VERSUS T H E O U T L E T MACH NUMBER C A L C U L A T E D FOR AN INLET MACH NUMBER = 2

so

Tri.-i-3'

FIGURE 3. THE RATIO OF TOTAL PRESSURES VERSUS OUTLET M A C H N U M B E R CALCULATED FOR A N INLET M A C H N U M B E R = 2

FIGURE 4. THE RATIO OF STATIC PRESSURES VERSUS OUTLET M A C H N U M B E R CALCULATED FOR AN INLET M A C H N U M B E R = 2

3 0 - a-5-o 20 < a. ut a. =) i-< o: UJ o. 5 ut y

i

0-5 tmax when:- _ i .

'^2={er(T]+')}

- 0 - 9 9 8, 1 1 1 1 0

7 ?

/ ^

1

II

1

4 0 3 0 2 0 1 0 OUTLET MACH NUMBER M ,

FIGURE 5. THE RATIO OF STATIC TEMPERATURES VERSUS O U T L E T MACH NUIMBER CALCULATED FOR AN INLET MACH NUMBER = 2