NASA TECHNICAL NOTE NASA TN D-6825...1. Report No. NASA TN D-6825 2. Government Accession No. 3....

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NASA TECHNICAL NOTE NASA TN D-6825

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CALCULATION FOR

C O P Y

by Satoaki Omori, Klaus W. Gross, and Alfred Krebsbach

George C. Marshall Space Flight Center Marshall Space Flight Center, Ala. 35812

N A T I O N A L AERONAUTICS A N D SPACE ADMINISTRATION WASHINGTON, D. C. JULY 1972

1. Report No. NASA T N D-6825

2. Government Accession No. 3. Recipient's Catalog No.

George C. Marshall Space Flight Center Marshall Space Flight Center, Alabama 35812

4. Title and Subtitle

Wall Temperature Distribution Calculation For A Rocket Nozzle Contoa

11. Contract or Grant No. I

5. Reno-* Oate July 1972 -- _ _

6. Performing Organization Code

7. Author(s)

Satoaki Omori*, Klaus W. Gross, and Alfred Krebsbach

9. Performing Organization Name and Address

:5:%pple&it&y Notes

Prepared by Astronautics Laboratory, Science and Engineering *National Research Council, National Academy of Science, Washington, D. C. 20546 NASA, Marshall Space Flight Center S&E-ASTN-PPB, Marshall Space Flight Center, Alabama 35812

16. Abstract

The JA"AF Turbulent Boundary Layer (TBL) computer program, applicable to rocket nozzles, requires a wall temperature distribution among other input parameters to determine boundary layer behavior, heat transfer, and performance degradation. The inclusion of a complete regenerative cooling cycle model with associate geometry, material and fluid property data provides a capability to internally calculate wall temperature profiles on the hot gas and coolant flow-side, as well as the coolant flow bulk temperature variation. Besides the regular heat transfer and performance degradation calculations, the new concept can be used to optimize the cooling cycle, coolant flow requirements, and cooling jacket geometry.

8. Performing Organization Report No.

10. Work Unit No.

2. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, D. C. 20546

13. Type of Report and Period Covered

Technical Note - 14. Sponsoring Agency Code

17. Key Words (Suggested by Author(s)) Compressible Turbulent Boundary Layers Regeneratively Cooled Rocket Thrust Chamber Heat Transfer Thrust Chamber Performance Cooling Jacket Geometry

18. Distribution Statement

19. Security Classif. (of this report) 20. Security Classif. (of this page)

Unclassified Unclassified

21. NO. o f Pages 22. Price'

$3.00 114

TABLE OF CONTENTS

Page

SUMMARY .......................................... 1

INTRODUCTION ........ .............. .............. 1

FUNDAMENTAL EQUATIONS FOR THE BOUNDARY LAYER 3 . . . . . EQUATIONS FOR THE REGENERATIVE COOLING CYCLE. . . . . . . 6

INTERNALLY CALCULATED COOLANT BULK TEMPERATURE.. . 12

SEQUENCE OF CALCULATION . . . . . . . . . . . . . . . . . . . . . . . . . 14

TOTALHEATTRANSFERRATE ........................ 18

SAME DIRECTION COOLANT FLOW,. . . . . . . . . . . . . . . . . . . . . 20

DOUBLE PASS COOLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

EXAMPLE ....................................... 21

CONCLUSION 0 . . . a . . . . . e 24

DESCRIF'TION OF PROGRAM INPUT . . - 59

Input Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . h p t Tables . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

60

DESCRIPTION OF PROGRAM OUTPUT 0 6 1

APPENDIX A: DERIVATION OF EQUATIONS (33) and (34) . . . . . . 63

APPENDIX B: TBL MODIFIED COMPUTER PROGRAM LISTING (TBLREG) .. .... ................ ...... 6 5

REFERENCES .................................... 105

iii

L I S T OF ILLUSTRATIONS

Figure Title Page

1 . Regeneratively cooled combus tor flow model . . . . . . . . . . . . 26

2 . Model of temperature profile . . . . . . . . . . . . . . . . . . . . . . 26

3 .a. Flow chart indicating the calculation procedure a t each station x = x .................................. 27

1

3.b. Overall flow diagram ............................ 28

4 . Schematic identifying temperatures used in the coolant flow temperature analysis ......................... 29

5 . Shuttle engine nozzle contour determined by Rao's method . . . . 29

6 . Booster engine contour ........................... 30

7 . 30

8 . Combustor cooling geometry ....................... 31

Input freestream parameters (obtained from TDK analysis) . . .

9 . Input temperatures to initiate calculation ............... 32

10 . Calculated temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11 . Specific heat transfer rate ......................... 33

12 . Velocity and temperature thicknesses . . . . . . . . . . . . . . . . . 34

13 . Momentum and energy thicknesses . . . . . . . . . . . . . . . . . . . 35

14 . Displacement thickness ........................... 35

15 . Displacement thickness (hot wall and cold wall) . . . . . . . . . . 36

16 . Thrust loss due to viscous boundary layer effects . . . . . . . . . 37

Loss of specific impulse due to boundary layer effects . . . . . . 17 . 37

iv

L I S T OF ILLUSTRATIONS (Concluded)

Figure Title Page

18 . Calculated temperatures ........................... 38

19 . Nozzle temperatures calculated ..................... 39

20 . Averaged nozzle wall temperatures ................... 40

21 . Displacement thickness ........................... 41

22 . Momentum thickness ............................. 41

23 . Thrust loss due to viscous boundary layer effects ......... 42

L I ST OF TABLES

Table

1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 .

Title Page

Regenerative Cooling Equations ..................... 43

Input Data for Combustor ........................ 44

C -T Relationship of Combustion Products ............. 48

Physical Properties of Liquid Hydrogen ............... 49

Input Data of Thrust Chamber ...................... 50

Input Data of SSME Booster Nozzle (Down Pass) . . . . . . . . . 51

Input Data of SSME Booster Nozzle (Up Pass) . . . . . . . . . . . . Input Data of SSME Booster Nozzle (Up and Down Passes) . . .

Nozzle Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

P

54

57

Calculated Displacement and Momentum Thicknesses Along

V

DEFINITION OF SYMBOLS

Symbol

tube A

cf

cH

D

H

tube

J

a M

gn

al P

W Q

r P

e R

z T

co U

C P

Definition

Cross-sectional area of each cooling tube o r channel, ft2

Skin friction coefficient

Stanton number

Equivalent tube diameter, f t

Enthalpy, ft2/s2

Conversion factor between thermal and work units ( 778.2) , ft-lbf/Btu

Mach number at boundary layer edge

Mean molecular weight at bwndary layer edge, lbm/mole

Static pressure at boundary layer edge, lbf/ft2

Total heat transfer rate* Btu/s

Prandtl number

Reynolds number

Universal gas constant

Temperature, OR

Velocity at boundary layer edge, ft/s

Specific heat at constant pressure, Btu/lbm OR

Acceleration of gravity (32.174) , ft-lbm/lbf s2

Total enthalpy, ft2/s2

vi

DEFINITION OF SYMBOLS (Continued)

Svmbol

h g

hP

P m

r

t

U

X

Y

a!

6

6 ' r

6*

A

e

9

P

Definition

Heat transfer coefficient on the gas side, Btu/ft2 soR

Heat transfer coefficient on the coolant side, Btu/ft2 soR

Coolant mass flow rate, lbm/s

Specific heat transfer rate, Btu/ft2 s

Specific heat transfer rate into coolant, Btu/ft2 s

Nozzle radius, ft

Chamber wall thickness, ft

Velocity within boundary layer, ft/s

Axial coordinate, ft o r -

Distance normal to wall, ft o r -

Angle between wall and nozzle axis

Velocity thickness, ft

Distance from nth streamline to real wall, ft

Displacement thickness , ft

Temperature thickness, ft

Momentum thickness, ft

Energy thickness, ft

Dynamic viscosity, lbm/ft s

vii

DEFINITION OF SYMBOLS (Continued)

Symbol Definition

P Density, lbm/ft3

A Thermal conductivity, Btu/ft soR

7 Shear s t ress , lbm/s2

77

W

Cooling coefficient for geometry effects

Efficiency (enhancement) factor for surface roughness and turbulence effects 77E

Subscripts

aw Adiabatic wall

C Calculated value o r convection

ETAB

i Section

j Overall iteration number

Final section of input tables

I Coolant

N

r Radiation

W Wal l o r wall material

Final section of input tables

Wg Gas side wall

Wl Coolant side wa l l

m Free stream or boundary layer edge

viii

Subscripts

0

1

2

DEFINITION OF SYMBOLS (Concluded)

Stagnation o r approximated value

Section o r iteration number

Section or iteration number

ix

WALL TEMPERATURE D I STR I BUTION CALCULATION FOR A ROCKET NOZZLE CONTOUR

SUMMARY

A concept is presented which allows the calculation of the temperatures along a thrust chamber nozzle contour on the hot gas and on the coolant flow- side. Also considered is a regenerative coolant flowing in the opposite or the same direction to the chamber reaction products. Coupling of the boundary layer equations fo r the hot gas-side with the regenerative cooling equations provides the results. Since the new analytical model has been integrated into the JA"AF Turbulent Boundary Layer computer program, the thrust degradation, due to the viscous effects close to the wall, is simultaneously obtained. The calculation is started with approximated temperature distributions for the hot gas-side wal l and the coolant flow. Iterations within the computer program are executed until the heat transfer rates from the boundary layer to the wall and from the wall to the coolant are equal. Kinetic inviscid flow conditions for the boundary layer edge a re considered by means of table inputs representing the variation of appropriate parameters. Since the chamber wall thickness and the coolant flow channel geometry are part of the analysis, optimization studies can be performed for these parameters by consecutive computer runs. A sample calculation, utilizing the new concept for a small area ratio high pressure thrust chamber, is included.

INTRODUCT ON

The calculation of various turbulent boundary layer thicknesses in the thrust chamber and the temperatures of the gas-side wall, the regenerative coolant-side wall, and the coolant fluid along the thrust chamber contour is simultaneously made, by considering the heat exchange between the combustion product flow in the thrust chamber and the coolant flow in the cooling jacket. The Turbulent Boundary Layer Computer Program TBL-I [ 11 has been modified to carry out the calculation by using a new concept in which the boundary layer equations are coupled with regenerative cooling equations.

The steady-state conditions that are considered require the temperatures of the combustion products, the chamber walls, and also the heat flux through

the walls to remain constant at any point in time. It is assumed that heat transfer occurs only by convection and conduction from the hot combustion products to the thrust chamber wall, neglecting the radiation. However, inclusion of radiation is not difficult, i f the emissivity of the combustion products and the Stefan-Boltzmann constant can be accurately determined; since the total specific heat flux from the hot gas into the chamber wall is composed

of the convective 4 and the radiant 4 heat flux: C r

The coolant fluid in this analysis flows through the tubes o r channels in the opposite o r same direction to the combustion products, receiving the heat by convection and conduction. The heat exchange takes place simultaneously in many small sections which have an arbitrary length along the contour of the thrust chamber, accounting for the gas phase turbulent combustion product flow, the temperature of the thrust chamber wall material, and the temperature of regenerative coolant flow. The temperature distributions obtained from the first iteration a re internally used as initial values for the second iteration. Iterations are performed until convergence i s obtained. Since the influence of the coolant transport properties on the resulting temperatures is quite signifi- cant, it is important to use pertinent values especially in the supercritical region of the coolant fluid. The empirical relationship of the heat transfer coefficient for computing the heat exchange with the coolant flow significantly affects the results as well as the Stanton number of the combustion products [I].

The methods to calculate the various turbulent boundary layer thicknesses in the thrust chamber a re explained in detail in the documentations of TBL [ 2 ] and TBL-I [ 11, and only the fundamental equations and concepts of the calculations improving the latter TBL-I computer program are outlined in this report. The concept is demonstrated for a regenerative coolant flowing in opposite direction to the combustion products. The alternate equations for the coolant flowing in the same direction a re explained in the section entitled Same Direction Coolant Flow.

2

FUNDAMENTAL EQUATIONS FOR THE BOUNDARY LAYER

The integral momentum and energy equations in axisymmetric form [ 2 , 3 ] for compressible turbulent boundary layer flow are:

1

and

if!= dx 'H [Haw-:] H, - [l+(cb?,;] 1'2

d( P, U, ) 1 d r 1 + - - + rdx H , - H dx -'[ & dx W

where the displacement thickness 6" , momentum thickness €J and energy thickness 9 are identified as follows:

3

ho - H

0

The skin friction coefficient is defined as

and has a form of the Blasius relation [ I ]

0.025 = 0. 25

Reo 9

with the following Reynolds number based upon the momentum thickness

The Stanton number

c = H P,Um (Haw - Hw)

9

is calculated using the formula [ 11

4

Velocity and enthalpy profiles across the boundary layer are assumed to follow the relationships:

U f o r y > 6 - - - 1 a)

U

h, - Hw for y 5 A H, - H W =(m)"

ho - H

Ho - H W

for y > A = 1 W

The definition of enthalpy is

T

0 H = J C dT

P

5

The adiabatic wall enthalpy H is defined as aw

The density p within the boundary layer is obtained from the perfect gas equation, assuming that the pressure is constant across the boundary layer:

m

where the temperature T is calculated via the velocity and enthalpy distributions, equations (11) , (12), (13), (14), and (16). The boundary layer calculations use the Runge-Kutta Gill solution method for given parameters at the boundary layer edge such as x, r, Ma , Pa , Tco , Urn, andm. The only

unknown parameter is the wall temperature T in equation ( 1 7 ) . wg

EQUATIONS FOR THE REGENERATIVE COOLING CYCLE

As shown in Figure 1, the coolant flows in an opposite direction to the combustion products of the thrust chdmber. The regenerative fluid enters

6

downstream with a lower temperature and a higher pressure than at the injector head, since heat is continuously transferred from the combustion products to the coolant through the chamber walls. T

temperature, T the coolant-side wall temperature, and T the coolant

bulk temperature at an arbitrary station x with x = 0 at the throat (Figs. 1 and 2) . We consider the case in which the heat is transferred only by convection from the hot combustion products to the chamber wall and that the direction of heat flow is normal to it. Since steady-state conditions a re treated, the temperatures of the combustion gas and wall and the specific heat flux through the walls remain constant with time at any given point.

denotes the gas-side wa l l wg

W I I

The five fundamental equations representing the cooling cycle, including an empirical relation for the heat transfer coefficient of the coolant, are as follows:

1. Specific heat transfer rate on the gas-side,

where h is the heat transfer coefficient in a gas and T is the adiabatic

wall temperature. g aw

2. Heat transfer coefficient in a gas related to the Stanton number which is calculated in TBL-1,

H - H aw w h = p, U, C

Taw - [Twglj Y

g

H - H _ _ _ _ aw w h = p, U, C

Taw - [Twglj Y

g

where

p is the-free stream density, a3

U is the free stream velocity, W

7

C is the Stanton number, H

H is the adiabatic wall enthalpy, aw

H is the wall enthalpy, W

T is the adiabatic wall temperature, aw

and

is the input wall temperature o r calculated wall temperature. [Twg] j

3. Specific heat transfer rate through the wall by conduction,

w2 = A w t ,

where h is the thermal conductivity of the wall material and t is the wall

thickness . W

4. Specific heat transfer rate into the coolant,

where h is the heat transfer coefficient for the coqlant. B

5. Empirical relation of the heat transfer coefficient for the hydrogen coolant flow [4] is a modified Colburn equation. For any other coolant flow, a similar relationship must be utilized including the effects of curvature, associated turbulence, and surface roughness of the tubes represented by the enhancement factor rj The accuracy of the enhancement factor significantly E'

8

affects the heat transfer calculation and the resulting wall temperatures. Since this effect is coupled with the cooling fluid heat transfer coefficient, i t is evident that the physical property information must be very precise.

0.55 'I ~ 0 . 8 p0.4 (e) 'E

e r I Dtube I I

h = 0.025

The above equation is valid for temperature ratios T

9.2, where the Reynolds number and the Prandtl number of the coolankare defined a s follows:

/TI between 1.44 to W

I

'1 '1 Dtube Reynolds number, Ro =

cLc Prandtl number, P = mpe

AI . r I

Mass flow density, p U = p,(x) UI(x) . Q I

Equivalent tube diameter, Dtube = 2 (A tube /y . Coolant bulk viscosity, pl = y (TI , Pressure) .

9

Coolant bulk specific heat, C , Pressure) . (30)

Coolant bulk thermal conductivity, A I = AI ( Te , Pressure) . (31)

For steady-state conditions, the heat flux through all three realms must be constant ,

= %2 = Q~ = = constant qwl

T , T and TQ . % ’ wg w I

Unknowns in equations (20) through (24) are

In equation (21) h is independently calculated when T is given. Com- g wg

bining equations ( 2 0 ) , ( 2 2 ) , ( 2 3 ) , and (32) results in

hI(l+:)TI+% t h T aw

9

W h

T = I W -

t

( 3 3 )

and

Derivations of the above equations are shown in Appendix A. Thus, the solution can be obtained by considering equations (20), (21), (24 ) , (33) , and ( 3 4 ) , (Table 1). The flow chart to compute T (x) , T ( x ) , TIc(x) , and wgc I W

10

(x) i s shown in Figure 3 where the subscript c denotes the internally

calculated te mpe ra ture . At the beginning of the calculation, the coolant bulk temperature distri-

bution is approximated. The coolant-side wall temperature T at an axial

distance x is obtained according to iterations in statement ( 2 ) of Figure 3a. The gas-side wall temperature T

ever, to differentiate between the input table values of T

c is added in statement (3 ) of Figure 3a. The term 4 which differs from

the % output of TBL-I, should coincide with g?l . after the iteration is com-

plete. In statement (5) of Figure 3a the coolant temperature is calculated by using i ts previous iteration values. The derivation of the equation in statement (5) is shown in the section entitled Internally Calculated Coolant Bulk Tempera- ture.

Q W

is calculated from equation (34) ; how-

. wg w '

wg , the subscript

After obtaining T (x) and T (x) at each table point of x , the wgc I C

values of T (x) and T

mined as follows:

(x) to be input for a successive iteration a re deter- I wg

r 1

and

In repeating the preceding calculation, we obtain values of [TI(x)l and

in a la te r section is achieved.

T (x) 3. This operation is applied until a desired convergence outlined [ w g 1

INTERNALLY CALCULATED COOLANT BULK TEMPERATURE

For simplicity, assume that the inner wall of the thrust chamber consists of a single wall and not of tubes. Let us consider an arbitrary section i in Figure 4 and calculate the coolant temperature at x. which is the distance

along the nozzle axis. Section i contains the surface area between B and D, a s shown in Figure 4,

1'

( 3 7 ) A ' = x. - A x , which is x il x. 1 - 1 1

E ' = x. + A x which is x

i2 ' X i + l 1

where the step sizes A x and A x a re arbitrary. il i2

The inlet temperature of the coolant at section i is T P

and the outlet temperature is TI (xi - %) . The heat transfer rate through

the cylindrical surface area of section i between B and D is

where

Axil + Ax

2 A:. = Y

i 2 1

12

and .I(..) is the angle between the chamber wall and the nozzle axis at x..

The wall radius is r(x.) and $(x.) is the specific heat transfer rate a s

shown in equation (32). The outlet temperature of the coolant at

in section i is calculated by x = x . - -

1 1

1 1

A xi1 1 2

where pi( xi)] and PI( xi + A xi2) a re either previously determined l j

coolant bulk temperatures or initial input values. The value A I ant flow rate and C (x.) the mean specific heat of the coolant between B

and D. Then T ( x ) is approximated a s

is the cool-

$ 1

I C i

9 (42) TI (xi - +)+ 2

2 T ( x ) =

I C i

where the subscript c denotes a calculated value compared with a previously determined value or the initial input number. Combining the above three equations results in

[.l(xi)l + [. (x. + Axi2)] . T r(xi) (xi) A x . ' (43)

1 1, 1 1

14.1 c (x.) cos (Y(Xi) m p e 1

2 =

This is the internally calculated coolant bulk temperature. For real thrust chambers composed of tubes o r channels, a cooling efficiency q should be applied to the second term on the right side of equation (43) to account for the real geometry effect. Thus,

13

T ( x ) = Pc i

However , the cooling efficiency 77 should be equal to one, if the empirical relationship in equation (24) is based upon real thrust chamber data and not upon a single tube experiment.

To start the calculation, a coolant flow temperature distribution must (x.) through iteration by equation (44). be given or approximated to obtain T

The initial value

from

Q c 1 for successive iterations can be obtained internally

Successive iterations a re made until the desired convergency is obtained, i. e., the computation is completed when the total heat transfer rate through the chamber wall on the gas side SUMQDA in TBL-I and that on the coolant side [equation (48)] (represented by SUMQWI in the present computer program) become equal. The specific heat transfer rates through the walls on the gas

side (iw = QW in TBL-I) and the coolant side (6w1 = QWI in the present com-

puter program) at each section are simultaneously equal. Now the coupling of the regenerative cooling cycle and TBL-I [ 11 is completed.

SEQUENCE OF CALCULATION

The numbers of the items that follow in this topic correspond to those in Figure 3a. The calculation sequence at station x = x1 progresses as follows:

1. As shown in the flow chart of Figure 3a, the coolant bulk temperature [‘P(x’] and the gas-side wall temperature distribution

14

i_aput to initiate the computation, where the subscript 0 denotes the first approximated value. The gas-side wall temperature [ T wg(x)] is used to

obtain the heat transfer coefficient on the gas side h (x) at each station

according to the equation below: g

where j = 0 denotes the first overall iteration loop. Each parameter except Fwg(x] , on the right side of the above equation, is calculated by the

eqlLations shown in equations ( 1) through ( 19) , or is input. The velocity Urn (x) is the only input parameter in equation (46) which remains constant

during all iteration at each local station,

2. The wall temperature on the coolant side T and the heat trans- W

I fer coefficient of the caolant flow h I loops, because e@ion (B) is implicit,

are calculated by small internal iteration

(equation 33)

and

h = 0.025 - D hl Roo8 POg4 (f:: j ) ' o 5 l 7E . (equation 24) e r

tube P 1 1

Since each parameter in the previous equations is a function of the axial distance x , the argument x is dropped for simplicity purposes. The subscript j identifies the iteration number with j = 0 indicating the first iteration.

15

3. The new gas-side wall temperature T is obtained from equa- wgc

tion (34)

(equation 34)

where the subscript c denotes a calculated value. The h in equation (34)

[Twg] j * The is still based upon the input wall temperature on the gas side

coolant-side wall temperature T

g

in equation (34) has been obtained previ- Q W

ously . 4. The specific heat transfer rate is obtained by any one of equations

( 2 0 ) , ( 2 2 ) , or (23) because of their equivalence represented by equation (32). Equation (20) is selected here,

(equation 20)

Another specific heat transfer rate based upon the input gas-side wall temperature is obtained from

The h term in both equations is based on the temperature [Twg] j. When

the overall iterations are completed, the following condition must be satisfied: g

. Q = Q' , because T wgc = pwg] j

16

5. The coolant bulk temperature has to be corrected at this point by considering the heat transferred at section i with the respective x = x. and

the input coolant bulk temperatures. 1

Derivation of this equation was shown previously.

6. New temperature approximations for the bulk coolant and the gas- side wall a r e predicted, for use in the succeeding overall iterations, from

and

The above procedure from steps 1 through 6 is repeated at each local and the two total heat transfer rates through the wall are corn- station x = x

pared at the end of every overall iteration loop station x = x

A solution is obtained when the two values fall within a small tolerance,

i’ (Fig. 3b). IXTAB

N , N

- c GW c Qw i = l i = 1

N

i = 1 c Qw

< > - Tolerance Y

17

N

i = 1 The expression ew will be described in equations (47) through (51) and

N - is identified a s SUMQWI in the computer program, whereas 6, is based

i = 1

upon iw and denoted as SUMQGA. A s long a s convergence is not attained, l- -I

and LTi(x)j j + 1 iterations must be continued with new estimates of

Pwg'x)] j + 1'

TOTAL HEAT TRANSFER RATE

The heat transfer rate through section i , between B and D in Fig- . ure 4 is Q (x.) according to equation (39 ) . The surface area of this wall

section is w 1

27r r (x . ) A?. I I

cos cY(x.) 1

Summation of the heat transferred up to section i = N is equal to

(47)

This amount is the heat which is transferreG through the chamber walls

A x

N 2 between x = xi and x = x +- N2 into the coolant per unit time, and not

N' up to x = x

18

The heat transfer rates through the initial and the final section of the contour a r e those through the area between C and D, and B and C y respec- tively, Figure 4. Heat transfer rate through the initial section i = 1

between x = xi and x = xi +- is Ax 2

SXTAB whereas for the final section at x = I

per unit time results SXTAB Integrating the heat transferred from point xi to

in

The coefficient 17 is used to account for surface area geometry effects; i.e., 17 = 0.5 for double pass cooling if only one path is considered. The above amount at x =

from the gas phase final iteration value. The latter amount has been denoted as "SUMQGA" = "SUMQDA*q" whereas the former, equation (51), will be desig- nated as "SUMQWP'. These two values are not the same at an intermediate section x = x. because "SUMQGA" in TBL-I is the amount between x = xi

and x = x whereas "SUMQWI" is obtained between x = xi and

x = x . + - i2 . Iterations are performed until the following convergence is

obtained:

must coincide with the total heat transfer rate calculated X~XTAB

1

i' A x

1 2

19

SUMQGA - SUMQWI

SAME D I RECTION COOLANT FLOW

< Tolerance .

We have considered the case of the coolant flowing in an opposite direction to the combustion products. Now the coolant bulk temperature calculations I

a re described for the coolant flowing in the same direction as the combustion products. Since rocket nozzles have been built with coolant flow passages in either direction and combinations thereof, the up and downstream coolant flow simulation of this new concept provides a capability for sectional treatment of changing the cooling cycle patterns. Equations in the section entitled Internally Calculated Coolant Bulk Temperature which must be replaced for the downpath simulation, a r e shown as follows: In changing the arrow of the coolant flow to point in the same direction as the combustion products in Figure 4, the temperature of the coolant leaving section i can be determined by I

represents the coordinate at the coolant outlet where the argument x. + - location in section i. This equation must replace equation (41).

A xi2 1 2

The coolant bulk temperature to be calculated at x = xi is obtained in

a way similar to equations (42) and (44),

20

with 11 a s the cooling efficiency due to geometry effects. Since the coolant flow temperature in this case inc reases toward the nozzle exit, the temperature input tables must be arranged correspondingly.

DOUBLE PASS COOLING

In carrying out the calculation for a double pass cooling jacket with coolant flowing downstream initially and upstream afterwards, we assume, at first, that the nozzle wall consists only of down-pass tubes engaged in the heat transfer process. A correction is made to the analysis by a cooling coefficient q which represents the surface area exposed to the hot gas covered by the downstream cooling tubes, compared to the total surface area. Then, the upstream pass calculation is executed in the same fashion neglecting the downstream coolant flow part. With each heat transfer calculation process, a wall temperature profile is provided. In order to determine the real temperature profile for the nozzle wall on the hot gas side, an average from the two temperature profiles can be determined.

The cooling coefficient q is usually less than unity for the double pass cooling jacket. For coolant flowing in one direction, the cooling coefficient may exceed a value of one, since the wall surface area per unit length may be greater than the circumferential area due to the ripples formed between adjacent cooling tubes.

In the computer program an option indicator will identify which type of coolant flow direction should be considered in the analysis:

IDUMP = 0 Coolant flow upstream

IDUMP = 1 Coolant flow downstream

Modifications made to the existing TBL program a re shown in Appendix B.

EXAMPLE

In this section of the paper the previously described new concept is applied to a thrust chamber nozzle similar to the Space Shuttle's main engine. A common chamber down to an area ratio of E = 7 is coupled with different

2 1

nozzle extensions expanding the combustion products to an area ratio of E = 35 o r E = 150 depending on low altitude or vacuum operating conditions, Figure 5. The nozzle contours were optimized according to Rao's method [ 5,6] to pro- vide maximum performance. Since a common chamber, Figure 6, was considered f o r both engines, the orbiter contour had to be modified a s indicated by the dotted l ine in Figure 5. In the thrust chamber liquid hydrogen and oxy- gen react at a mixture ratio of 6. 0 at a pressure of 3020 psia (212.33 kgf/cm2), resulting in a stagnation temperature of 6600"R (3667°K). The free stream inviscid flow parameters serving a s boundary layer edge conditions such a s Mach number Ma, , static pressure Pa , static temperature Too and mean

molecular weight 312, were obtained from the Two-Dimensional Kinetics (TDK) computer program [ 71.

I

I

First , only the combustion chamber expanding the reaction products to a n area ratio of E = 7 is considered. regeneratively cooled with liquid hydrogen which flows in an opposite direction to the combustion products. The input data for the modified computer program a r e shown in Table 2 and Figure 7. The cross-sectional area variation of an individual cooling tube, assumed values for the gas-side wall temperature, and coolant bulk temperature as functions of the axial nozzle length, a r e presented in Table 2 and Figures 8 and 9. When a study is performed to optimize the cooling jacket geometry, the cross-sectional area in Table 2 and Figure 8 must be changed in each separate analysis. From such a parametric analysis, the best cooling tube geometry can then be selected. In the present example, however, the jacket geometry is fixed. Table 3 represents the relationship between the specific heat at constant pressure and temperature of the combustion products in the boundary layer. In order to determine the coolant flow heat transfer coefficient, the specific heat, thermal conductivity and viscosity for an expected pressure range between 4500 psia and 6000 psia (316.38 kgf/cm2 and 421.84 kgf/cm2) fo r the coolant fluid must be established a s functions of temperature. The input data based upon References 4 and 8 a re specified in Table 4. Additionally required input data can be found in Table 5. The calculated temperature distributions on the hot gas-side, liquid coolant-side and the coolant a r e plotted in Figure 10. The total heat transferred through the chamber w a l l without considering an enhancement factor is 10 580 kcal/sec (42 000 Btu/sec) , whereas the local specific heat flux is exhibited in Figure 11. The velocity and temperature boundary layer thicknesses a r e presented in Figure 12 and the momentum and energy thicknesses are plotted in Figure 13.

In this section the chamber wall is

The most important result from a performance aspect is the boundary layer displacement thickness 6* , Figure 14. This parameter, significantly

22

affected by the wall temperature, reveals by how much the wall contour, iden- tical to the inviscid flow border streamline, must be displaced to allow the same mass flow condition. A negative sign of 6" means a displacement of the inviscid-flow contour towards the thrust chamber centerline.

If the density across the boundary layer is constant, the profile of the mass flow density p u is in principle s imilar to the velocity profile, Figure 15a. However, i f the density varies the mass flow density overshoots its free stream value p,Ua , especially when the wall is highly cooled, Figure

15b. The dotted line in either schematic denotes the mass flow density profile for inviscid flow. Results from the present analysis indicate that the displacement thic'mess 6" is negative for the most part of the combustion chamber to compensate for the strong cooling effect, Figure 14. The performance deficiency represented by a thrust loss, Figure 16, down to an expansion ratio of E = 7 is already quite large according to the equation [ 1,2,31

The corresponding loss in specific impulse is shown in Figure 17.

To investigate the effect of variable and constant properties necessary to calculate the coolant flow heat transfer coefficient, an additional analysis was performed using constant values for the specific heat C

lbmoR, thermal conductivity Aa = 0.0000288 F%u/ft s OR and the dynamic

viscosity p = 0.0000065 lbm/ft s which represents mean values between

the temperatures of 50"R and 550"R. In comparing the results in Figure 18 with the ones obtained for variable properties in Figure 10, it is evident that the wall temperatures a re higher at the throat and lower at an expansion ratio of E = 7. This study clearly outlines that most accurate input data must be used to perform a reliable analysis.

= 3.75 Btu/ pe

I

Only the chamber section down to an area ratio of E = 7 has been dis- cussed. area ratio E = 7 to E = 35 is treated. For convenience, this nozzle contour has been selected, although an analysis for the orbiter nozzle contour would be similar. The booster nozzle wall is also cooled by the hydrogen in a

Now, the nozzle extension for the booster engine ranging from an

23

i I

double pass cycle. The coolant enters 564 tubes of an area ratio of E = 7, flows toward the nozzle exit area ( E = 35) and is then turned upstream. The wall thickness of each tube varies from 0.18 to 0.25 mm toward the nozzle exit. All required input data for the downstream and upstream analysis a r e shown in Tables 6, 7 , and 8. The resulting w a l l temperature distributions presented in Figure 19 are considerably different for both cooling paths and exhibit a minimum in the down-pass section, where the coolant bulk temperature reaches a value of approximately 140°K (250"R). At this state the hydrogen possesses a maximum specific heat or highest cooling capacity. In the real nozzle the temperature differences between the down and up-pass cooling tube wi l l come to an equilibrium temperature through lateral heat transfer at each local station. Therefore, an arithmetic mean of the different temperatures will represent the real nozzle temperature more realistically, Figure 20. The individual displacement and momentum thicknesses a r e presented in Table 9, whereas their averaged values a re plotted in Figures 21 and 22. The total performance degradation, expressed in thrust and specific impulse loss at the nozzle exit, resulted in A F = 4.742 tons (10 470 lbf) B. L. and A ISP = 7.687 s ( Fig. 23). Heat absorbed by the coolant fluid between the injector face and the nozzle exit ( E = 35) amounts to 27 000 kcal/s (107 000 Btu/s). This method was also applied to identify the area of ice formation (wall temperatures less than 460"R) inside the 5-2 engine; since deposition of ice crystals along the nozzle exit periphery were observed during altitude simulation test firings.

CONCLUSION

A new method has been presented by which the hot gas-side and the coolant flow-side wall temperature distributions, as well a s the coolant fluid temperature variation of a regeneratively cooled thrust chamber, can be determined. The analytical formulation is based upon a coupling of the boundary layer equations with the heat transfer process through the nozzle wall and the coolant flow heat absorption. The new concept has been incorporated into the existing JANNAF Turbulent Boundary Layer (TBL) computer program. A sample case showing the application of the new calculation process for a thrust chamber similar to the Space Shuttle booster engine, has also been outlined. Since several e m p i r i d relationships such as the friction coefficient of the hot gas-side wall, the Stanton number, and Colburn's equation for the

1. Analytical Prediction of Ice Formation Inside the 5-2 Engine Nozzle Contour (200 K Thrust Level). Memorandum &E-ASTN-PP (72M-5) NASA, Marshall Space Flight Center, January 1972.

24

coolant flow heat transfer coefficient were used and no adjustments for the coolant flow turbulence and channel curvature were made, the results are only approximate. In addition, this new model could serve as a convenient tool for the design of an optimum cooling path and channel geometry concept.

25

- I %- ----- INJECTOR THROAT --

I

COMBUSTION PRODUCTS - -

COOLANT INLET

COOLANT OUTLET

Figure 1. Regeneratively cooled combustor flow model.

THICKNESS

Figure 2. Model of kmperature profile.

26

I I x = x i

X W h W h l + - ) [ ~ j I j + - T,, 1' thg t

TWl = x w x

h 1 + - ) + 2 thg t

(2)

'1 0.8 0.4 ITP j 0.55

(7) 'E Dtube e p 'e we h = 0.025 - R P e

I I

I

Figure 3. a. Flow chart indicating the calculation procedure at each station x = xi .

2 7

x = x.

i = l

rr t

I (SEE FIG. 3a)

7

i = 1 Q W

< x = x IXTAB >

NO

r"l FINAL SOLUTION

Figure 3. b. Overall flow diagram.

28

- 7 - COMBUSTION PRODUCTS

I I I

! I i A I I

Figure 4. Schematic identifying temperatures used in the coolant flow temperature analysis.

101.5 in ,-c(

[ {- (572.8 225.5in em)

Figure 5. Shuttle engine nozzle contour determined by Rao' s method.

29

E= 35

7000 -

T,

6000 -

- 0" 5000- -

4MIo-

3000 -

Tw UP Tw DOWN

Figure 6 . Booster engine contour.

Figure 7. Input freestream parameters (obtained from TDK analysis).

30

0.00006

- 0.00005

0.00004

0.00003

.c -

I I 1 I

- 0.006

-0.005 - AREA OF EACH -0.004 -

- cy

E CROSS-SECTIONAL -

COOLING TUBE -

- 0.003

Figure 8. Combustor cooling geometry.

0.005

0.004

0.003

0.002

0.001

31

1.5

*-\ /-a-l.o ; - - - WALL THICKNESS

7 I I I I a

1

- 0.5

GAS-SIDE WALL

[Twglo

2000

1500

- K 0,

1000

500

0

-20 0 20 40 I I I I I I I

- -

- - - - - - - - - - - - -

COOLANT BULK

[TRIO

- 1 I I I

-1 .o -0.5 0 0.5 1 .o

Figure 9. Input temperatures to initiate calculation.

32

000

Y 0-

io0

3

X (cm) n

0.04

-

0.03 -

-

-

/ - 1.0 /

/ /

/ -

/ / ,//- /

- TEMPERATURE THICKNESS: A

I I :/ I

0.5 i I 1 1 I l o

0 0.5 1 .o 1.5 -1 .o -0.5

x (ft)

Figure 12. Velocity and temperature thicknesses.

34

8 v) v) W z Y

I c > 0 K W z W

..

0

I r

. I

N 8 0

8 8

rn m

[D

Q 5 k a,

03 rl

a, k 5 M k .d

3 5

Y

Y

Twg

6"

(a 1 HOT WALL

I 6"

COLD WALL

0

0

Figure 15. Displacement thickness (hot w a l l and cold wal l ) .

36

0 d

0 N

I - E, X

0

0 U

Q) m

37

X (cm)

---- -20 0 20 40 ZOO[

150c

- K lOOC 0-

500

COOLANT PROPERTIES ARE CONSTANT

C

x (ft)

1000

Y 0-

500

0

Figure 18. Calculated temperatures.

38

c r,

x

C 6

rc: r

C r

ur C

0 0 00

0 0 e4

W a

a - 2

P Q .. c

W n

a

- 2 z 4 0 8 ..

-3 3 c

n 3

I It Y 4 3

c z 0

m

4 8 I I .. f r

0 0 c

cd 0 rn a, k 5 cd k u

# a, u a, 4 N

Q) -4

0

40

0 8 8 00 8 fs 8 U N

I I I 1 I

Q)

N N 0 G

d

% M cd

I I I I

F r

0 0 0 Q ln

0 0

0

0.02

X (m)

1 2

I I I I I I

-

0

-0.01

(ft’

-0.02

-0.03

DISPLACEMENT THICKNESS

- - - - - - - -

COLD WALL

HOT WALL

I I I I 1 6 I I I I -1 0 1 2 3 4 5 6 7 8 9

X (ft)

0

2

4

(mm)

6

a

-10

3

4 - E E -

2

0

41

N

- E

X

r

C

42

k a,

rn 5 0 0 rn

.Fi > 0 42

a, 5 m rn rn 0 4

Q) k 5 M

TABLE 1. REGENERATIVE COOLING EQUATIONS

H - H aw w

aw - [Twg] j H T h = p,Um C

g

% I Re h = 0.025

Dtube B

T W h

ha (1 + +) [ ~ g ] j -+ t aw

- t + hI [ + z ) T = I W

W 1

T h T + - I V J

g a w t w B

W x

h + - g t

(equation 2 0 )

(equation 21)

(equation 24)

(equation 33)

(equation 34)

4 3

1

1 2

I 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 36 37 38 3 9 40 4 1 42 43 44 45 46 47 48 49 50

TABLE 2. INPUT DATA FOR COMBUSTOR

A

-0.950725 -0.914726 -0.842728 -0.770730 -0.734732 -0.662734 -0.590736 -0.554738 -0.518739 -0.446740 -0.410741 -0.374742 -0.302744 -0.266745 -0.230746 -0.194747 -0.158749 -0.122750 -0.086751 -0.057200 0.000000 0.003573 0.007256 0.011012 0.014858 0.018786 0.022785 0.030987 0.035185 0.043767 0.048149 0.057091 0.061650 0.070945 0.080477 0.095255 0.110036 0.125698 0.150196 0.186436 0.222098 0.257291 0.292175 0.326872 0.361499 0.396 154 0.430941 0.465973 0.501365 0.537202

Y

0.802083 0.801277 0.794779 0.781616 0.772440 0.748573 0.7 16937 0.699832 0.682727 0.648511 0.631406 0.614301 0.580085 0.562980 0.545875 0.529072 0.515094 0.504252 0.496347 0.491945 0.488583 0.4886 18 0.488720 0.488901 0.489160 0.489507 0.489942 0.491104 0.491842 0.493650 0.494735 0.497290 0.498775 0.502210 0.506314 0.513926 0.522159 0.53 1066 0.545269 0.566752 0.588215 0.609630 0.631064 0.652552 0.674113 0.695748 0.7 17480 0.739363 0.761452 0.783776

Mach Number

0.220976 0.221523 0.225972 0.235435 0.242420 0.26227 1 0.292080 0.309721 0.329208 0.375117 0.402511 0.433805 0.513023 0.565406 0.628830 0.696138 0.766393 0.839565 0.915600 0.980160 1.193460 1.227370 1.260930 1.293320 1.325760 1.358830 1.392170 1.459980 1.494590 1.565680 1.602360 1.678320 1.7 17660 1.799740 1.887080 2.005430 2.016620 2.035810 2.056660 2.087750 2.108640 2.133830 2.157400 2.183230 2.206210 2.228720 2.251510 2.274500 2.297430 2.321400

Pressure

422396.766 422122.793 421622.242 420527.219 419691.812 417197.340 413116.426 410519.758 407496.906 399765.176 394762.754 388714.000 371954.164 359875.809 344318.844 327004.102 308263.379 288302.387 267383.355 249697.223 193860.369 185645.486 177689.357 169966.758 162435.506 155083.424 147900.947 1.14013.490 127297.204 114299.943 108015.929

95879.731 90029.727 78774.162 68127.430 55677.686 54371.307 52354.768 50226.030 47251.451 45296.231 43109.654 41165.610 39155.333 37436.992 35824.066 34260.629 32744.594 31295.269 29851.381

Static Temuerature

6476.894 6476.810 6475.952 6474.074 6472.632 6468.317 6461.215 6456.660 6451.318 6437.497 6428.419 6417.295 6385.611 6361.932 6330.417 6293.651 6251.739 6204.368 6151.280 6 103.208 5926.078 5888.241 5850.416 5814.319 5777.151 5738.689 5699.396 5617.924 5575.459 5486.489 5439.801 5431.590 5289.898 5180.797 5063.229 4903.150 4891.896 4868.906 4845.773 4810.424 4788.942 4760.864 4734.703 4704.890 4678.748 4652.945 4626.427 4599.258 4571.973 4543.057

44

TABLE 2. (Continued)

- 1

1 2 3 4 5 6 7 8 9 LO 11 12 13 14 15 16 17 18 19 10 11 22 23 14 15 16 17 18 29 30 31 32 33 34 35 36 37 38 39 LO 41 12 13 14 $5 16 47 18 19 50

-

-

X

-0.950725 -0.914726 -0.842728 -0.770730 -0.734732 -0.662734 -0.590736 -0.554738 -0.518739 -0.446740 -0.410741 -0.374742 -0.302744 -0.266745 -0.230746 -0.194747 -0.158749 -0.122750 -0.086751 -0.057200 0.000000 0.003573 0.007256 0.011012 0.014858 0.018786 0.022785 0.030987 0.035185 0.043767 0.048149 0.057091 0.061650 0.070945 0.080477 0.095255 0.110036 0.125698 0.150196 0.186436 0.222098 0.257291 0.292175 0.326872 0.361499 0.396154 0.430941 0.46 5973 0.501365 0.537202

Y

0.802063 0.801277 0.794779 0.781616 0.772440 0.748573 0.716937 0.699632 0.662727 0.648511 0.631406 0.614301 0.580085 0.562980 0.545675 0.529072 0.515094 0.504252 0.496347 0.491945 0.488583 0.486618 0.486720 0.488901 0.489160 0.489507 0.489942 0.491104 0.491842 0.493650 0.494735 0.497290 0.498775 0.502210 0.506314 0.513926 0.522159 0.531066 0.545269 0.566752 0.568215 0.6 09630 0.631064 0.652552 0.674113 0.695748 0.717480 0.739363 0.761452 0.783776

Velocity

1175.830 1178.720 1202.300 1252.440 1289.430 1394.480 1551.980 1645.040 1747.700 1988.940 2132.440 2295.920 2707.390 2977.420 3301.920 3643.090 3995.360 4357.770 4729.150 5040.040 6035.780 6186.780 6334.890 6473.390 66 12.360 6752.420 6891.970 7170.700 7310.530 7592.250 7734.490 8022.470 8168.540 8456.700 8770.780 9166.640 9205.220 9270.930 9342.800 9449.400 9522.320 9607.770 9686.870 9771.880 9847.080 9919.870 9992.570 10065.200 10136.800 102 10.6 00

Molecular Weight

13.590460 13.597353 13.597646 13.598282

13.600157 13.602585 13.604074 13.605855 13.610415 13.613397 13.617002 13.627159 13.634497 13.646259 13.658344 13.672035 13.687345 13.704345 13.719516 13.765060 13.768245 13.771604 13.790009 13.799884 13.809909 13.820396 13.842066 13.852492 13.872702 13.883344 13.905751 13.915809 13.936370 13.958197 13.987167 13.993980 13.996067 14.000566 14.003944 14.006823 14.009449 14.012815 14.015822 14.018885 14.021893 14.024915 14.027241 14.029441 14.031649

13.598768

Coolant Area

0.000053330 0.000053330 0.000053330 0.000053000 0.000052500 0.000051100 0.000049800 0.000049100 0.000048500 0.000047100 0.000046500 0.000045800 0.000044400 0.000043700 0.000043000 0.000042400 0.000041700 0.000041000 0.000040868 0.000040868 0.000040868 0.000040868 0.000040868 0.000040868 0.000040868 0.000040868 0.000040868 0.000040868 0.000040868 0.000041000 0.000041100 0.000041400 0.000042000 0.000042400 0.000043200 0.000044000 0.000045200 0.000046000 0.000047700 0.000050000 0.000052500 0.000055000 0.000057200 0.000059400 0.000061800 0.000064000 0.000064000 0.000064000 0.000064000 0.000064000

Coolant Temperature

460.000 450.000 430.000 410.000 398.000 380.000 366.000 350.000 340.000 320.000 312.000 303.000 287.000 280.000 270.000 262.000 255.000 247.000 240.000 232.000 222.000 221.000 220.000 219.500 219.000 218.000 217.000 216.000 215.000 214.000 213.000 211.000 210.000 209.000 207.000 205.000 202.000 199.000 194.000 188.000 182.000 178.000 172.000 167.000 163.000 160.000 155.000 152.000 150.000 144.000

Wall Temperature

1360.000 1360.000 1360.000 1360.000 1360.000 1360.000 1360.000 1360.000 1360.000 1360.000 1360.000 1360.000 1364.000 1368.000 1370.000 1372.000 1374.000 1380.000 1380.000 1375.000 1320.000 1310.000 1300.000 1295.000 1285.000 1280.000 1272.000 1255.000 1250.000 1235.000 1225.000 1220.000 1210.000 1200.000 1188.000 1180.000 1170.000 1155.000 1145.000 1145.000 1160.000 1180.000 1180.000 1170.000 1164.000 1150.000 1140.000 1125.000 1113.000 1100.000

45

n a a, 7 c c 0 u

.r( 42

W

N

w I4

4 b

a

46

. . . . . . . . . . . . . . . . .

h

0

d

a a, E

m .r(

w F=’

E a, 0

4 a,

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * * * * * * * * * * d * * * * * * w ~ w c D w w w w ~ w w w ~ w w w w 0 0 0 ~ 0 0 0 0 0 0 0 Y 0 ~ Y Y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 ~ 0 0 0 0 0 0 0 ~ 0 ~ ~ ~ 0

d d d d d d d d d d d o o o o o o . . . . . .

k 0 0 d W

u

47

TABLE 3. C -T RELATIONSHIP OF COMBUSTION PRODUCTS P

48

I

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Specific Heat (Btu/lbm s)

0.6199999973

0.6550000012

0.6799999997

0.6950000003

0.7049999982

0.7199999988

0.7282169983

0.7282169983

0.7282169983

0.7282169983

0.7282169983

0.7282169983

0.8833 189979

0.8843249977

0.8854160011

0.8893399984

0.8902480006

0.8906119987

0.8907269984

0.8920999989

Temperature (OR)

400.000

800.000

1200.000

1400.000

1600.000

2000.000

2500.000

3000.000

4000.000

5000,000

5850.000

5926.078

6103.208

6151.280

6204.368

6403.409

6451.318

6470.760

647 6.894

8000.000

u

0 0 o c occ OF: a 0 o c o c o c o c o c d c

4 5- 0 0 u

0 0 0 0 0 0 a0

0 0 0 0 0 0 0 0

0 0

3 2

- 0

0 0

cv Eo 0 0 0 0 0

0

0 0 0

0 0 0 0 0

0

2

0 0 CD 4 In 0 0 0 0 0

0

0 0 (D rl Lo 0 0 0 6 0

0

0 0 * 6 3 ccp 0 0 0 0 0

0

0 0 (0 b m 0 0 a 0

a

d

0 0 cv 4 W 0 0 0 0 0

0

0 0 4)

a 0 0 0 0

0

s 0 0 Eo a W 0 0 0 0 0

0

0 0 cv m t? 0 6 0 0 0

0

0 0 co a b 0 0 0 0 0

0

0 0 cv Q) c- 6 0 0 0 0

0

0 0 a F h m 0 0. o 0 0

0

0 0 0

0 0. 0 0 0

0

a! 0 0 W tc Gu 0 0 0 0 0

0

o o u o o o o o o o o o o o o o o o o c o o o o o o o o o o o o o o o o o o o c 0 ~ 0 o 0 0 0 0 a O ~ E o 0 * * 0 * * 0 C ~ m ~ c a a ~ o ~ ~ c ~ ~ ~ ~ m o o c ~ ~ ~ ~ ~ a n r n * ~ m o o 4 ~ q 4 m b m 4 ~ * W c Q o m 5 3 m N w c ? J m m m m m m m m * * * * * m c 0 0 u ~ a O O O O O o O O O O O O O O C o u o o o o o o o o o o o o o o o o o c O U ~ O O O O O O O O O O O O O O O O C Q 0 O O O O O O O O o O O O O O O O O C

* . . . . . . . . . . . . . . . . I

d ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C

3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30000000000000000000 30000000000000000000

0 l.n 0 a 0 u3 0 l.n 0 u3 0 v3 0 10 0 d 4 6 l m l c 9 m r J 4 * m m ~ W F b a a m m o ; d d d G d d d G d G d d d d d d d d d n 0 In 0

49

o o o o o o o o o o o o o o + + + l l l l l l + + + + l O O O O O O O O O O O O O M O O O O O O O O O O O O O m o o o o o o o o o o m o o 0 7 o m o o o o o o o o o o ~ m 0 m r n 0 7 0 0 0 0 0 0 r - m r - m o ~ l - l m o o o o o o M ~ ~ m c D m d m m 0 0 0 0 0 7 l r - ~ ~

d G ~ A & g A S i A 7 i ? i 7 i g G 4 0 hl

0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 h l ~ 0 0 0 ~ r n m o o m m C \ I m C \ I

o A o ; & o 2 . c ;

II I I II I I II I1 II I1 II II II II II I I II II II II II II II II II II / I II

II II I1 II II II I I II I I II II II / I I I I I II I1 II II II II II II I1 II

n c- II W

0

k 0 0 a,

U

U

3 .e

c 0

0 a, &I

.rl U

El a,

rn 0 a

U .rl

8 a,

d bo c 0 0 0 a, > cd h a, c a, bo

5 .d

.d l-4

.rl U

2 v

II II II II II II II II II I I II

n h 0 II

n 0 II

n Y

b W h

II II II II II II I/

n

0 I!

W

E c

II I1 II II II II II I1

II II 1 1 I1 II II II II I I II II I1 II II / I II II I1 I1 II II I1 II II II II

- 0

M 0

0

u; Y

L=

II W

€3 0 k icl

a, c

k a, m 0

Y

8 2 cn rn Y

5 1

52

O l - t * w m m Q , r l O c - 0 0 Q , r - i W C D m m N o c - * o t - o c - m * d c - r l ~ O ~ * m * ~ r l 0 0 O m c - 0 0 Q , r l C D 0 0 c n r l N M s l m m * * * * * ~ C D w C D t - L - c - L - t ? c - c - c - c - c - L - c - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o * m m Q , C D * * N m t - N m 0 0 m w m Q , r l

. . . . . . . . . . . . . . . . . . d N N N m m m m m m m m m m m m m m m

W N 0 c- 0 m

00 Q, L- m

m E2 0 0 Qo

c- W rl m CD Q,

Q, m CD d m 0

m 00 c- N

m m m t- m N

c- c- 0 m 03 M

N

00 00 CD

x W 0 00 L- m c-

0 c- 0 CD 00 e-

m m Q, N m 00

0 t? Q, Q,

3

m 0 t- m rl W

m Eo W 00 m N

rl Q, t- In 00 W

0 0 0 0 m N

m Q, 0 m W m

m c- rl m 3

rl N m W c- 00

m CD W m m *

* m 03 Q, Q) 00

s 0 N M m

rl Q, 00 Q, Q, c-

=s 8 0 m

0 00 N m m 00

c- N m rl E- O

m .d

w F

Y

W

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

W * h l o b h l Q , W d * b a 3 olno00

d d r + r + I l r + $ - l d d r l d

d d d d d d d d d d d d d d G d d d d * * * * M M N h l ~ r l O Q , ~ % 8 0 0 c - c - W

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

a o d m T y W c Q ~ r 4 M u 3 0 0 Q , I l m u a W 0 0 0 d d F l d d d r + N a h l h l h l M m M m M *

G 4 d d G < d d ; 4 d 4 G d d d < & d

0 0 0 0 v) 0 0 0 0

0 0 s 0 0 0 0

0 0 0 0 W 0 0 0 0

0 0 0 a3 W 0 0 0 0

0 0 0 00 b 0 0 0 0

0 0 0 d Q, 0 0 0 0

0 0 0 00 Q, 0 0 0 0

0 0 0 00 0 Il 0 0 0

0 0 0 00 rl rl 0 0 0

0 0 0 rl 03 d 0 0 0

0 0 0 c-

0 0 0

2

0 0 0 W W d 0 0 0

0 0 0 00 b Il 0 0 0

0 0 0 0 Q, l-l 0 0 0

0 0 0 In 0 hl 0 0 0

0 0 0 0 hl hl 0 0 0

0 0 0 00 03 hl 0 0 0

0 0 0 00 ua hl 0 0 0

0 0 0 W W hl 0 0 0

d d d d d d d d d d d d d d d d d d d

M 0 c- Q, Il W

Q, W W 00 M hl

Il ua b ua 00 W

0 0 0 0 M hl

m Q, 0 ua CD In

d hl M W c- 00

ua W W

* 3 0 W b u3 hl d

z M Q, Q, 00

* hl 0 hl M M

Il Q, 00 Q, Q, c-

ss 3 0 m

0 00 hl ua u3 00

53

II II II II I1 II I I I I I1 II II I1 II / I II II II II II I / II II / I II II II

n

F; W

m I4

ffi w 3

n rl II

w W

m

2 m w a n d o a

J c b

E Y n

F; 4 W

3 s k L4

m m z w

ffi D b -4 ffi w e,

w ffi

- h

w;' U PI X w E

0 k

W

z b u c ffi w b z

9

H

z w b M

El 0 0 u UJ

.A 4

U u Frc 0 E z 5 2 Frc Frc w 0 u

w I4 F4

Q)

Q 5 0

4

h

PI w B m

4 b

U Q) El M El W

.r(

w w ffi Frc

Y

X k Q)

UJ 0

42

2i II II I I II II II II I / II II II I/ II I / II II I/ I I II II II II I1 II II II I

54

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o * m m a w - + * N m L - N m a m c o m a d

55

56

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0

a * m J o b N a a d * b m m N w m d G d d d G G d d G m i G d G d G m i d d * * & & M M N N N d O a 3 % ~ b b w w

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

o o o o o o o o o o o o o o o o o o c m o o o o o o o o o o o o o o o o o c c - o o m o o m o m J o o w o d o o o o c M m . 4 t - m m N o a o N b a ~ ~ o m o ~ ~ ~ 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 c

o o o o o o o o o o o o o o o o o o c o o o o o o o o o o o o o o o o o o c

. . . . . e . . 4 d ~ d d d d d d d d o o o o o o o o c

h 0

.r(

a a,

7 rn E

3 rn

.r(

k 0 + 0 cd +I

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * 0 w w * w N w N w N m N ~ 0 w 0 w w O @ J d ~ ~ N t - ~ * t - a m w a a d l t - o m ~ e a w m v ) m w m w w w w t - t - ~ - w m m m a w d o o o o o o o o o o o o o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

d d d d d d d d d d d d d d d d d d d d

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O N 0 0 0 0 0 0 W 0 W W 0 * * 0 1 * 0 0 ~ m m w w o w w t - ~ t - m o o m ~ ~ ~ ~ m m m ~ t - w o o ~ N ~ m ~ ~ a ~ N ~ ~ w o ~ ~ ~ ~ ~ ~ m m m m m m m m * * * ~ ~ v ) m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m m v) LQ 0 0 m 0 0 0 0 v) m 4 0 0 w c- w w m w v ) a ~ ~ o a m t - w v ) m m m m ~ + ~ +

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v ) o m o m o m o m o v ) o m o m o m o m o d d ~ ~ m m ~ ~ m m w w t - t - m m m m ~

. . . . . . . . . . . . . . . . . . . .


0 0 0 0 0.0 0 0 . . . . . . . 0 0 0 0 0 0 0 0

. . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

r l w L - N w c a * * 0 0 0 0 . . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

. . . . . . . . . . . . . . . . . . . O O O O O Q O O O O O O O O O O O O O I I I I I I I I I I I I I I I I I I I

MZETA = n

IPRINT

IXTAB

ICTAB

ITWTAB

TO = To

PO = Po

GAM0 = yo

ZMUO = Po

Z MVIS

ZNSTAN

DXMAX

THETAI = e i

PHI1 = $I

EPSZ

RBAR

F J = J

G = g

DESCR PTiON OF PROGRAM INPUT

i n p u t Data

Exponent in velocity profile power law

Print option at every calculated point (= 1) or at input intervals (= 0)

Number of points in x = f (y) and x = g(Ma) tables

Number of points in C = f (T) table

Wall temperature option = 1 (must be input)

P

Stagnation temperature, OR

Stagnation pressure, lbf/ft2

Stagnation specific heat ratio

Stagnation viscosity, lbm/ft-s

Exponent of viscosity temperature law

Boundary layer interaction exponent

Maximum step size

Initial value of momentum thickness, f t

Initial value of energy thickness, f t

Geometry option - Axisymmetric = 1. Plane = 0.

Gas constant a t stagnation, ft-lbf/”R-lbm

Conversion factor between thermal and work units = 778.2, ft-lbf/Btu

Acceleration of gravity = 32.174, ft-lbm/lbf-s2

59

Contour scale factor SCALE

IT ZTAB

D U M P

FLOWRT

MASSL= A1

RAMDW = A

COEFCL = q

TUBEN

W

cpe' Al Number of points in temperature versus

and p table Q

Coolant flow option Same direction = 1 Reverse flow = 0

Combustion chamber mass flow rate, lbm/s

Coolant mass flow rate, Ibm/s

Thermal conductivity of the chamber wall, Btu/ft-s"R

coo~ing coefficient (surface area effect)

Number of cooling tubes

Input Tables

(i) Specific C (CPTAB) versus temperature T (TITAB)

(ii) XITAB Axial distance, ft P

YITAB Radius, ft

ZMTAB Mach number Ma at boundary layer edge

PETAB Static pressure Pa at boundary layer edge, Ibf/ft2

TETAB Static temperature Tm at boundary layer edge, "R

UETAB Velocity Urn at boundary layer edge, ft/s

SMTAB Mean molecular weight 311 at boundary layer edge

ALTAB Cross-sectional area of each cooling tube, ft2

TLTAB Assumed coolant temperature

0

60

TWTAB

THITAB

(iii) TZTAB

CPLTAB

RAMTAB

ZMYTAB

Assumed wall temperature on the gas-side T

Wall thickness of the cooling jacket, ft

Coolant temperature table used to obtain C and 1-1 OR

Coolant specific heat C , Btu/lbm-"R

Thermal conductivity of coolant A Btu/ft-soR

Viscosity of coolant 1.1 lbm/ft-s

[ w g l a , O R

pe' ha I '

pe

I '

I '

DESCRIPTION OF PROGRAM OUTPUT

1 The following parameters a re printed out in addition to the original TBL computer program results [ 31 :

RBAR =z/m

PRANDT = Pr

00 GAME = y

SMOL = %Z

COSAL = cos a ( x )

DELFA = AFB. L.

THRUST = F

DEFTHR = AF/F X 100

TBLISP = -AI SP

THRUSA

VISP = I sPvacuum

Specific gas constant, ft-lbf/lbmoR

Prandtl number of the free stream

Specific heat ratio at the boundary layer edge

Mean molecular weight, lbm

Cosine of the wall angle

Thrust degradation due to turbulent boundary layer effects downstream of the throat only, lbf

Vacuum thrust, lbf

Percent of thrust degradation

Specific impulse loss due to turbulent boundary layer effects, s

Thrust at sea level, lbf

Vacuum specific impulse downstream of the throat only, s

6 1

AISP = I "sea level

DMASSL= p U

HL= hQ

P Q

QWI = $

REYL = R el

SUMQGA

SUMQWI

TEMPRL= T /TP W Q

TLCA= T Pc

TWGCA= T wgc

TWL= T W .e

DIATUB = 2 4 A /n tube

THICK= t

Specific impulse at sea level of the throat only, s

Mass flow density of the coolant fluid, lbm/ft2-s

Heat transfer coefficient of the coolant fluid, Btu/ft2- so R

Specific heat transfer rate based upon calculations for the coolant flow side, Btu/ft2-s

Reynolds number of the coolant fluid based upon tube diameters

Total heat transfer rate, Btu/s

Total heat transfer rate (includes cooling flow calculation), Btu/s

Temperature ratio

Calculated coolant temperature, OR

Calculated wall temperature on the gas side, O R

Calculated wall temperature on the coolant side, OR

Equivalent diameter of the cooling jacket, f t

Chamber wall thickness (input value), f t

George C. Marshall Space Flight Center National Aeronautics and Space Administration

Marshall Space Flight Center, Alabama, February 11, 1972

62

APPENDIX A

DERIVATION OF EQUATIONS (33) AND (34)

Using equations (20) and (22) with equation (32 ) , we obtain

Rewrite the above equation, a s

W Aw T

aw t h

h + - g t

I h T +-

W

T = g wg

Equations (20) and (23) reduce to

hg(Taw - T wg ) =hI(TwI - T I ) 9

so that

Substitute equation (34) into the above equation, then

(equation 34)

63

[%+(hg+%):]T w I = - I W t T aw + (hg+%): g TI

Therefore,

N h

hrn c +%) TI + 7- Taw

W h

T =

wa - + h t I (l++)

(equation 33)

64

APPENDIX B

TBL MODIFIED COMPUTER PROGRAM L IST ING (TBLREG)

65

T P F S b ANCON

C C -- BARCON -- C O N T R O L L I N G S U B R O U T I N E

SUBHOUTINE B A R C O N BARCOOOI

BARCOO02

C O O L I N b # A L L TEMPERATURE I T E R A T I O N

BARCOO31 BARCOO32 B A R C 0 0 3 3 8 A R C 0 0 3 4 B A R C 0 0 3 5 B A R C 0 0 3 6 B A R C 0 0 3 7 BARCOO38 t 3 A R C 0 0 3 9

BARCOOSO BARCOO4 1

66

OW 0.Q HG = o * o I X P Q S = l I T P O S = 1 I C X = 0 I E X = 0 I R X = 0 I S X = 0 l u x = 0 I L X = 0 I M X = O 1 T X = O

I Y X I O lT IYX=O DXRHO=O. I B E G = 2 CFAGT = e002 I S T A R T = O I F ( T H E T A 1 . L E * O e O ) G O TO 2 ZETA = ( P H I X / T H E T A I ) r o R M Z E T A 60 TO 3

1 ~ x 8 0

2 C A L L START

3 CFAGP ;I CFAGT I S T A R T = I

I F ( I C O O L .EO. 0 ) GO TO 4 P E L X U L m 0.0 DELXNE = A B S ( X I T A B ( 2 ) - X I T A B ( I 1 )

A L A L T A B ( 1 j T I 1 T L T A B ( 1 ) T H I C K = T H I T A B ( 1 ) 7 1 2 - 1 T L T A B f Z ) TLO = T L I C A L L XNTERP ~ T L ~ ~ ~ H Y U L I Z P , I Z X ~ T L T A B ~ Z M ~ T A B ~ I T Z T ~ B S C Z ~ B I T P O S ~ I T P O S I 1 z x U I A T U B = 2 m O * S Y R T ( A L / P I E ) REYL H A S S L O D I A T U B / ( A L . T U B E N * ~ M Y U L )

T H E T A = T H E T A I XIBASE XITABtI) XIEND = X I T A B ( 1 X T A B )

DXRHD = ( X l T A B ( 2 ) - X I B A S E ) / I O * O

C A L L B A R P R O ( 5 ) TWGTAB(I) T W G C A T L C T A B ( 1 1 * T L C A C A L L X N T E R P ( x, YR, V R P B I Y x , X I T A B , Y I T A B , I X T A B , C Y X . I H X I

P E L X B A I [ D E L X O L + D E L X N E J / Z . O

9 P H l = PHI1

I F L I X T A B .LE. 1) LO 1TD 15

15 C A L L W R P R O ( I )

C C S A V E I N I T l A L Y AND P E L S T R i C

D E L = DELSTR Y N l N = YR

ONOC = S O R T ( I * + YRP YRP 1 C

BARCOO42 BARCOO13 BARCOO44

BARCOOY5 BAUCOOq6 BARCOO47 BARCOO1B BARCOO49 B A R C Q 0 5 0 B A R C O 0 5 1 B A R C 0 0 5 2 B A R C 0 0 5 3

B A R C 0 0 5 6 B A R C 0 0 5 7 BARCOO58 BARCOO59

B A R C 0 0 6 2 BARCOO63

B A R C O 0 6 6 B A R C 0 0 6 7

B A R C 0 0 6 8 B A R C 0 0 6 9 B A R C 0 0 7 0 B A R C 0 0 7 1 BA R C O 0 7 2 B A R C 0 0 7 3 B A h C Q 0 7 1 BARCOO75

67

B A R C O 0 8 2 B A H C 0 0 0 3

B A K C 0 0 0 6 B A R C O O I ? B A R C O O 8 8 B A R C 0 0 8 9 B A R C O O 9 0 B A R C 0 0 9 I B A R C O O 9 2 B A H C 0 0 9 3 B A R C O O 9 4 B A R E 0 0 9 5 B A R C O O P I B A R C O O 9 7

0 A H C O 1 0 4 B A R C O I O S

B A H C 0 1 0 8

B A R C 0 1 1 3 B A R C O I I S

8 A R C O l I 7 B A R C O I 18

BARCO I 2 1

63 FOHMAT ( 4IH **BARCON F A I L U R E * * . A X I A L O I S T A N C E X * I P E I S o 7 * B A R C O I Z Z 1 5 X . I l H t t A C h NO. * E14.7r 2 X * 6 H T t l E T A I = e € 1 4 - 7 . Z X C B A R C O l 2 3 z 6 H P t i I I = a E14.7 / 4 4 H T H E T A 1 O R P H I 1 CUMPUTLO AS N E G A T I V E B A R C O l 2 4 3 OR L E R O / 6 Y H *CHECK CONTOUR AND MACH NURBER U l S T H l B U t I O N T A B L E S B A R C 0 1 2 5 4 FOR ERRORS./ l l 0 W *HORE I N P U T P O I N T S M A Y BE R E P U I A E U T O A D E O U A T E B A k C 0 1 2 6 5 L Y O C S C R I B E D E R I V A T I V E V A L U E S ALONG THE CONTOUR A T T H I S P O I N T . / B A R C 0 1 2 7 6 96t4 *A SMALLER RUNGE-KUTTA STEP SIZE M A Y BE H E O U I H E D 7 0 A n F Q l ~ A T 8 A ~ C O l 2 8

C C C C c

C

C c C C

C

C A L L B A H P R O ( 5 1 3 0 C O N T I N U E

C A L L X N T L R P t x. Y R * Y R P ~ IYX. X I T A B . Y I T A B ~ I X T A B ~ C Y X . i n x ONOC = S w H T ( I * + YRP Y H P 1 X C C P ( 1 ) = A + DELSTR * YRP / ONOC Y C C P I I ) = Y K - O E L S T R / ONOC I F ( I P H I N T . b T * 0 ) G O TU 20 C A L L B A R P R O ( 5 1 T W ( r T A B ( i ) T J ~ G C A T L C T A B t 1 1 T L C A

20 C O N T I N U E

Y M I N 0 n l N l n c l M Y V A L U E F O H N O Z Z L E * DEL = 3 E L S T R CORRESPONDING T O M I k I M U f l Y ( T H R O A T ) . KPOT TI+€ P O T E N T I A L T r t R O A T RADIUS.

B A R C 0 1 3 4 B A R C 0 1 3 5 0ARCO I36 B A R C 0 1 3 7 0 A R C 0 1 3 8 0 A R C 0 1 3 9 B A H C O I 4 O B A H C O I 4 2

B A H C O l 4 5 0 A R C O l 4 6 t l A R C O l ' t 7 BARCO 1 9 8 B A R C 0 \ 4 9

B A H C 0 1 5 3 B A R C O l 5 Y B A K C O l 5 5 B A k C 0 1 5 7 B A H C 0 1 5 8 B A R C 0 1 5 9 B A R C O l b O B A R C O l 6 1 BARCO I62 B A R C O I 63 B A k C O l 6 9 B A R C O l 6 5 B A R C O l 6 6 B A R C 0 1 6 7 B A R C O l 6 8 BAHCO I69 B A k C O I 7 0

B A H C 0 1 7 3 B A R C O l 7 4

69

nr, I 3

OAKS 1

/ C S E V A L / / C S E V A L / / C S E V A L /

.-

/ 1 NPUT/

/ I N P U T /

/ T A B L E S / / T A B L E S /

/ I N P U T /

B A K S 2 4 UAAS 2 5 8 A R S 26 B A U S 2 7 B A R S 2 8 B A K S 2 9 B A R S 3 0 B A R S 3 1

BARS 34

R A n S 3 9 B A R S 40 B A R S ’41 B A R S Y2

B A R S SI bARS 5 2 B A R S 53

BARS 57 B A R S 5 0 B A R S 59 B A R S 60 BARS 61 B A R S 62 B A R S 63 BARS 64 B A R S 65 BAHS 66 B A R S 6 7 B A R S 6 0

B A R S 7 3

76

CFEV

CFEV CFEV CFEV CFEV CFEV

Rt ; F

CFEV

C F E v C F L V CFEV CFEV CFEV CFEV C F E v

C F E v CFEV

i

2 3 .-

1 5

1 7 18 1 9 20 2 1

23

26 2 7 28 2 9 3 0 3 1 32

35 36

79

D I R E C T

C SUBROUTINE D I R E C T

10 CALL R E A D I N CALL BARSET CALL BARCON

END GO To 10

80

D I R E 1

D I R E 2 D I R E 3 D I R E 4

D I R E 7

, - _ _ ---- - -

I FllF - _ -

FUNCTION FlIF ( S I

FIIF 4

FIIF 7 FllF 8

FIIF 13 FIlF 14 FlIF 15 FlIF 16 FIIF I 7 FIlF 18

81

r c 2 t l E 2 = Z N E o Z M E GETP PROOI -Z . /RBAR/Z f lEZ /G CETP O E N M Z ~ l e + G M l 0 2 * Z M E Z L E T P T€=TO/DENHZ CE TP I F ( I C T A B eGT. 0 ) GO T O 20 PE=PO/DENM2**606Hl CETP

T 1 = T E 6E7P RETURN GETP

T O L = T O L Z M E / L H E 21 TEO=TE6 GETP

TCO-TC CCTP 1 ~ 6 . 1 ~ GETP C A L L SEVIL(I,TE,CPL.HE) GETP GAwE=CPE/ ICPE-ROJ) CETP TC. fnO-Ht ) /GAME*PRODI CETP 1F ( A B S ( ( T C - T E t / T E t * L E D T O L ) GO T O 3 0

15 P I = P E

2 0 I T E R - o

IF ( I T E R . G T e 0 ) GO T O 2 0 TE tZ .O*TE + T C ) / J * O 6 0 T O 2 8 GETP

W R I T E ( 6 . 2 6 ) L M E D T C D T C O B T E O T E O 24 I F ( I T E R .LED 50) G O T O 2 7

26 FORMAT ( 31HO** ( rETPT F A I L U R E . * * MACH NO. D I P E 1 4 * 7 / 1 Y X i GETP 1 17HT ( C A L C U L A T E D ) D 2E16 .7 / I Y X , 1 7 H T ( b U E S S E D ) = ,GETP 2 2 E 1 6 . 7 / / ) GETP

G O TO 30 GE TP 2 7 Z K = ( T C - T C O ) / ( T E - T E O ) GETP

T E = ( T C - Z R * T E ) / ( I B - L K ) GETP LF [ A B S I t T E T E G ) I T E h DLTS T O L ) G O T O 2 9 IF ( I T E R * L T . IO) 6 0 TO 2 8 1F ( A B S [ L T E - f E O ) / T E ) D L T . T O L I GO T o 2 9

28 I T ~ = I T E R + I 6ETP G O TO 2 1 6 f T P

29 TE=(TE+TE6)/2e CE TP

GO T O 15 hETP END CLTP

30 C A L L S E V A L ( - ~ , T E ~ P E ~ S O )

82

9 LO 11 12

IS

18 19

23 2 4 25 26 27 28

32

35 36 3 7 38 39 ' (0

' (9 sa

I

1 1 12 13 111 15 16

19 20 2 2

25 26 27

30 31 32 33

35 36

39

42 43 44

46 47 48 49 50

53 511

I k T L 63 INTZ 64

I N T Z 65 I N T Z 66

I N T Z 6 9

I N T Z 7 6 I N 1 2 77 I N 1 2 7 8

IN12 8 0 I N T Z 8 1 I N 1 2 8 2

I N 1 2 8 5

INTZ r v

I N T Z 8 8 I N T Z 8 9 I N T Z 9 0 I N 1 2 91 I N T Z 9 2 I N T Z 9 3 I N T Z 9L)

I N T Z 97 I N 1 2 9 8 I N T Z 9 9 I N T Z 100 IN12 I 0 1

84

HAlNTB _ _ - . C I C H P (r REFERENCE PROGRAH T B L C DECK SEQUENCLO BY SUBROUTINE C

COMMQN./I.NPUr/ L D X M A X . I C T A B ~ ~ P R I N T ~ ~ T W T A B . ~ Z E T A ~ ~ M Z E T A ~ D X H A X ~ /INPUT/

K T H E T A l ~ J O L C C A ~ T O L L E T ~ l O L ~ M E ~ Z M U O ~ ~ M V l S ~ Z N S T A N /INPUT/ A E P S Z ~ F J ~ C t G A M O ~ P O ~ P H I l m P l E ~ P R A N D T ~ R B A H m S ~ A L E m T O ~ / I N P U T /

C I D X H A W = 0 CALL DIRECT TBL 1 EblD TBL 3

85

/ Q U I T S

C

C

C

C

C

C

C

C

C

C

C Q U I T 7 a u i ~ e

86

R E A D I N

R E A D 0 0 2 6 R E A 0 0 0 2 7 R E A 0 0 0 2 8 R E A D 0 0 2 9 R E A 0 0 0 3 0 R E A 0 0 0 3 1

R E A P 0 0 3 2 R E A D 0 0 3 3 READ0031) R E A D 0 0 3 5 R E A 0 0 0 3 6

R E A D 0 0 3 8

R E A 0 0 0 4 3

R E A 0 0 0 4 5

89

414 IDXMAX = o RE A D 0 0 4 6 415 DXlYAX = ( ( x I T A B ( I X T A B I - X I T A B ( 1 ) ) 1 100.0 1 * S C A L E RE A D O O w

416 IF ( E P S Z .LE* 0101 * R I T E (6e7) 7 FORNAT ( / / / / / / / / / / 5 6 X r 1 9 H * * * I N F O R M A T I O N * * * / / 3 0 X * ' t 4 H I m T H I S CASE

I CONSIDERS T W O - P I M E N S I O N A L F L 0 1 / / 3 0 X , 3 Z H 2 r THE N O Z L L E r I O T H 1s ON 2 E F O O T / / 3 0 X * S P H 3 m THE S I D E WALLS ARE ASSUMED T O BE A D I A B A T I C AND 3 1 N V I S C I 0 / / 3 0 X ~ b B H ' t o H E A T TRANSFER OCCURS ONLY THROUGH THE ONE FOO 11 h l O E CURVED * A L L S / / 3 0 X * 5 9 H S * THE C A L C U L A T E D THRUST L O S S 1 5 B A S E 5 D O N TWO CURVED W A L L S / / 3 0 X , 6 S H 6 . THE C A L C U L A T E O ThRUST IS B A S E D 0 6 N AN AREA OF ONE BY Z * Y R F E E T / / 3 0 X , 5 5 H 7 0 CHECK THE I N P U T V A L U E S F 7 0 R FLOWRTI MASSLI ANU T U B E N / ]

WRITE ( 6 ~ 3 1 T I T L E 3 FORMAT l l H 1 0 2 7 X * I 3 A 6 / / )

1 ERROR 8 0 R E A D 0 0 5 0 W R l T E ( 6 . 1 0 2 ) MZETA R E A D 0 0 5 1

1 0 2 F O R N A T ( S S H MZETA = V E L O C I T Y P R O F I L E P O l E R L A N f X P O I v € N T 2 7 X l H = 1 ~ 1 R E A D 0 0 5 2 I F (PIZETA mGEm 0 ) GO T O 25 * H I Tf 16,300 1

300 FORMAT ( 4 7 H * *ERROR** VALUE MUST BE GREATER THAN Z E R O I0)m / / I R E A 0 0 0 5 5 IERROR = I R E A D 0 0 5 6

2 5 * R l l E ( 6 ~ 1 0 3 1 I P H I N T R E A D 0 0 5 7 1 0 3 F O R M A T ( 7 3 H I P R I N T = P R I N T AT EVERY C A L C U L A T E O P O I N T ( = I ) OR A T I N P U R E A D 0 0 5 8

I T I N T E R V A L S ( = O l 814) R E A D 0 0 5 9 I F ( I P R I N T . E O * I *OR. I P R I N T * E Q m 0 ) G O T O 513 W H I T E (6,5021

5 0 2 F O R M A T ( q5H * * E ~ ? H O R * * VALUE MUST BE ZkRO ( 0 ) O K ONE 1 1 ) . / / ) R E A D 0 0 6 3 R E A 0 0 0 6 4

5 2 H I X T A b NUMBER CJF P O I N T S I N X ~ V S I Y *VSm M T A B L E S 2 O X l H E A D 0 0 6 7

1 ERNOR = I 513 * R I T E ( 6 , 1 0 4 ) I X T A B

1 0 4 F O R M A T I H = l Y )

I F ( I X * R I T E

304 F O R M A T 1 F O U H

1 ENKOR

R E A 0 0 0 6 8 AB rGEm 4 .AND* I X T A B .LE. LOO) GO T O 30 6,304) ( / 2 X , 1 0 4 H * o E H H O R e * VALUE MUST B E . G R L A T t N THAN OR E Q U A L T O 4 1 OR L L ~ S T H A N OR L Q U A L T O ONE HUNDRLD i1oo). 1 1 )

R E A D 0 0 7 6

37 I06

5 1 2

5 2 3 1 1 1

90

R E A D 0 0 7 7 R E A D 0 0 7 8

HE A D 0 0 8 9 R E A D 0 0 9 0 R E A D 0 0 9 1 R E A D 0 0 9 2

R E A u O O 9 6 R E A 0 0 0 9 7 R E A 0 0 0 9 8

I O N TEMPERATUHE 2 ~ x ~ H = I P R E A O O l O l H E A O O l 0 2

I E H R O R = I R E A 0 0 1 0 5 4 1 h R I T E ( 6 , 1 1 2 ) PO READO I06 I12 F O R M A T ( S O H PO a F R E E STREAM S T A G N A T I O N PRESSURE 2 2 X l H = R E A D O l 0 7

l l P E 1 5 . 7 ) R E A D 0 1 0 8 IF ( P O . G T . 0 . 0 ) GO T O 43 * R I T E (6,300) I E R R O R . I R E A D 0 1 1 I

4 3 h R I T E ( 6 r l 1 3 ) GAM0 R E A 0 0 1 12 1 1 3 F O R M A T ( q 9 H GAM0 I S T A G N A T I O N R A T I O OF S P E C I F I C HEATS28X~H=lPEIS.READOll3

1 7 ) READO I 14 I F ( I C T A B *NE. 0 ) G O T O 4 7 I F ( G A M 0 .GT. 1.0) GO T O 9 7 VYRITE ( 6 . 5 9 1 1

591 F O R M A T ( 9 8 H * * E H h O R * * V A L U E MUST BE G R E A T E R T H A N O N E ( 1 . 0 ) . / / ) R E A D 0 1 1 8 I E R R O R = I R E A D 0 1 19

'47 R R I T E (6,115) ZMUO 115 F O N M A T ! 3 8 H Z M U O S T A G N A T I O N V I S C O S I T Y 39XIW.1PE lS .71 R E A 0 0 126

I F ( L M U O .GT. 0.01 G O T O 51 W R I T E (6,300) I E R K O R = I R E A 0 0 1 29

5 1 L R l T E ( 6 * 1 1 6 ) ZVVIS R E A 0 0 1 3 0 1 1 6 F O R r l ~ l ( q 7 H Z M V I S = EXPONENT OF V I S C O S I T Y - T E H P E H A T U R E L A W 2 5 X l H = l P E R E A D O l 3 1

1 1 5 * 7 ) R E A 0 0 132 W R I T E ( 6 , 1 1 7 ) L N S T A N R E A D 0 1 3 3

1 1 7 F O H H A T ( q 5 H Z N S T A N I ~ O U N O A R Y L A Y E R i N T E R A C T I O N € X P ~ ~ € N T ~ ~ X J H L I ~ E I S R E ~ O ~ ~ ~ ~ 1 . 7 ) R E A D 0 1 35

W R l T E ( 6 , 1 1 8 ) DXMAX R E A D 0 1 3 6 118 F O H M A T ( ~ ~ H DXHAX a M A x I M U M S T E P S I Z E 4 1 X l H 1 l P E I 5 . 7 ) R E A D 0 1 3 7

I F ( T H E T A I .LT. 0 . 0 ) G O TO 't9 k R I T E (6,119) T H E T A I

119 F O R M A T ( q 9 H T H E T A I = I N I T I A L V A L U E OF MOMENTUM T H I C K ~ L S S Z J X I H = I R E A D O ~ ~ O REhOO I4 I

1 2 0 F O R l r l A T ( q 7 H P H I 1 I I N I T I A L V A L U E OF ENERGY T H I C K N E S S 2 5 X I H ~ I P E R E A O O l 9 3

4 4 1 V R I 1 € ( 6 , 1 2 1 ) E P S L 121 F O W M A T [ S I H E P S Z L GEOMETRY,.. A X I S Y M M E T R I C ( m 1 . ) . P L A N E ( I O . ) ~ I X I H R E A D O I ~ ~

l = l P E 1 5 r 7 ) R E A D 0 1 9 8

I P E 15.7 )

N R I T E ( b t 1 2 0 ) P H I 1

1 1 5 . 7 ) R E A D O l ' t ' !

I F ( E P S Z . E Q * 0.3 . O R . E P S Z .EQ. 1 . 0 ) G O T O 533 W R I T E (6.502) I E H H O R = I R E A D 0 1 5 2

533 k R l T E ( 6 . 1 2 2 ) R B A R 122 F O R M A T ( lX, j ' f ,HRBAR I GAS CONSTANT A T S T A G N A T 1 O N t 3 6 X . l n r . l P E 1 5 . 7 )

IF ( R B A R o G T o 0 . 0 ) G O T O 53 IR1 T € ( 6 . 3 0 0 ) I E H H O R - 1 R E A D 0 1 5 8

53 *RITE(6,123) F J R E A 0 0 I59 1 2 3 F O R M A T ( S L H FJ = C O N V E R S I O N 8ETYVEEN T H E R M A L *NO hURK U N I T S 2 1 X 1 H R E A 0 0 1 6 0

l x l P E 1 5 * 7 ) R E A D 0 1 6 1 I F ( F J .GT. 0 . 0 ) 60 TO 55 * R I T E (6,300) I E H R O R = I R E A D 0 1 64

5 5 * H I T E ( ~ , I ~ ~ ) G R E A D 0 1 6 5 1 2 4 F O " A T ( 5 7 H G I P H O P O R T t O N A L I T Y C O N S T A N T I N E O U A T I O N -- F = M / G * R E A 0 0 1 6 6

lA15XlH~lPE15.7) R E A D 0 1 6 7 I F ( C r r G T . 0 . 0 ) 60 T O 4 2 0

91

W R I T L (6,300) IERROR - 1 REAOO 170

420 w R f T E ( b , ’ t 2 1 ) S C A L E R E A 0 0 1 7 1 4 2 1 FORMAT ( 3 0 H S C A L E CONTOUR S C A L E F A C T O R D 92X. IH-0 I p E 1 5 . 7 R E A 0 0 1 7 2

I F ( T O L C F A * E O * 1 a O E - ’ I ) GO T O ’ IO2 YvRlTE ( 6 , 9 0 1 ) TULCFA

4 0 1 FORMAT { 4 7 H T O L C F A I T O L E R A N C E FOR SKIN F R I C T I O N I ‘ T L R A T I O N , 2 5 x 1 R E A 0 0 1 7 5 1 1 H - a I P E 1 5 . 7 1 REAOO I 7 6

4 0 2 IF ( T O L Z E T . E Q * a.oao3) G O T O 40s * R I T E (6,4041 T O L Z E T

4 0 9 FOKNAT ( 4 9 H T O L Z E T TOLERANCE FOR SHAPE PARAMETER I T E R A T I O N . R E A 0 0 1 7 9 I 2 3 X , Iha, I P E 1 5 . 7 ) R E A 0 0 1 8 0

405 I F (TOLZME .EQ* 1 o O E - 7 ) G O TO 205 * R I T E ( 6 , 9 0 7 ) TOLZME

4 0 7 FORHAT ( 65H TOLLHE a TOLEHANCE F O R W A C n NO. - T E M f t N A T U R E R E L A T I O R E A 0 0 1 8 3 R E A 0 0 1 8 9 I N I T E R A T I O N , 7X. 1H.e l P E 1 5 . 7 1

205 RRITE ( 6 , 9 0 0 ) I T Z T A B D I D U W P , F L O W H T . M A S S L , R A M O W ~ C U E F C L , T U B E N ~ 1 T O L I T E ~ l C O O L

P O 0 FORMAT ( L X I ~ ~ W I T Z T A B NUMBER OF P O I N T S I N T . V S o C P L . V S . RAWDL 1 V S . LMYUL T A B L E s , J X , I H = , I ’ I / I X , ~ ~ H ~ O U M P COOLANT FLOW O P T I O N -- s ZAME D / R E C T I O N ( = I ) , R E V E R S E ( - O ) , 8 X l l H - , l ~ / ~ X , 5 2 H F L O R ~ T = COMBUSTION

YOOLANT MASS F L O ~ NATE I L B M / S E C ) ~ ~ O X , I H ’ , E I ~ O ~ / I ~ ~ ~ ~ H ~ A M O W = H E A T S C O N O U C T I V I T Y OF TnE CHAMbER # A L L ~ 2 5 X . l H - , E 1 5 . 7 / 1 X . 3 I H C O E F C L COEF < F I C l t N T OF C O O L I N G ~ ~ O X D I H ~ D E I S ~ ~ / ~ X ~ ~ O W T U B E N TUBE N U M B E R o S I X e 7 I H I , E l 5 . 7 / 1 X ~ 5 2 ~ T O L I T E = TOLERANCE FOR TOTAL HEAT TKANSFER ITEHAT ~ I O N ~ I 9 X , l H ~ ~ ~ l 5 . 7 / 1 X ~ 6 9 ~ I C O O L I C O O L I N G O P T I O N -- L I T H C O O L I N G ( m 1 9 ) * r l T H O U T C O O L ~ N G ( - O ) B ~ X , ~ H - , I ~ ~

3 CHAMBER W A S S FLOW R A T E I L B W / S E C ) . ~ ~ X . I H = ~ I P E I S . ~ / ~ X , ~ I ~ M A S S L - c

I F ( I C T A Y O L E O U * A N D O I T L T A B O L E . 0 ) GO TO I I WRITE (6,1311

1 . 3 1 FOHNAT I / / Z X o l H l , S X , l 3 H S P E C I F l C H E A T , S ~ D I I H T E H P ~ R A T U ~ ~ , ~ ~ . I I ~ H C O O L A N T TEMP,SX,LOHCOOLANT c P , 5 X n I ~ H C O N O U C f l V I T Y ~ 5 X ~ 2 9 h V I S C O S I T Y )

L W A X A h A x l ( I C T A d , l T L f A B ) D O 1.33 1 1 , l N A X I F ( I * L E . I C T A B .ANU. I .LE. I T Z T A B ) (10 T O I30 I F ( I C T A B eGT. I T Z T A I ) G O TO 132 L R l T E ( 6 , I l 1 ~ 1 2 T A B ( I ) , C P L T A B ( I ) ( R I M T A B ~ l ~ ~ ~ M Y T ~ B ( I I F O R M A T G O T O 133

I ( I3 v 9 1 X c F 9 . 3 B 6X I F 1016, SX * F I 2 0 IO D 3X B F 1 2 . 1 0 )

130 WRITE ( 6 1 4 1 I , C P T A B ( I I , T C T A B ( I ) , T Z T A B ( I ) , C P L T A B ~ I ) B ~ A M T A B ( I ~ , 1 L M Y T A I ( 1 )

4 FOKMAT 1 1 3 ~ 5 X ~ ~ I J ~ 1 C ~ 6 X ~ F 9 r 3 ~ 8 ~ ~ F 9 ~ 3 ~ 6 ~ ~ F 1 0 o ~ ~ 5 ~ ~ F 1 2 o 1 0 ~ 3 X ~ F 1 2 * l 0 ~ G O T O 133

1.32 * R I T E (6.5) I , C P T A B ( I ) D T C T A B ( I )

1 3 3 C O N T I N U E 5 FOHMAT ( I 3 ~ 5 X , F 1 3 e 1 0 , 6 X v F 9 . 3 )

I F ( I C T A B .LE. 0 ) 6 0 TO I 1 I 1 I C T A B - I REAOO I 9 3 U O 5 9 I = I . I 1 REAOOIP ’ I I F ( T C T A B ( I + I ) .CT. T C T A B ( 1 ) ) G O TO 59 WRITI: (6,310)

313 FONMAT I / Z X , V P H * * ERROR * * T A B L E OF S P L C l F l C HEATS - TEMPERATUHC V IALUES WusT B E I N M O N A T O N I C A L L Y I N C R E A S I N G ORDER. / / )

I E K R o R - 1 R E A 0 0 2 0 1 5 9 CONTINUE R E A 0 0 2 0 2

92

.IF ( T C J A B I I ) 9 G T . O n O ) GO T O 6 1 R E A D 0 2 0 5 * R I T k ( 6 . 3 l Z )

311 F O R H A T ( / 2 X 0 8 7 H * * ERROR * * T A B L E OF S P E C I F I C H E A T S - TEMPERATURE V I A L U E S MUST BE GREATER THAN Z E R O 1 0 ) . / / )

I E R R O R s 1

IF ( C P T A 8 l I ) * b T o 0 . 0 ) 60 T O 63 d R I T E ( 6 0 3 1 3 ) R E A 0 0 2 1 1

61 D O 63 I m I , I C T A B

313 F O R M A T ( / 2 X o a 9 H * * E R R O R * + T A B L E OF S P E C I F I C H E A T S - SPECIFIC HEAT I VALUES MUST BE G H E A T E R THAN ZERO ( 0 ) * / / )

I E R R O R - 1 R E A 0 0 2 14 63 C O N T l N U E R E A D 0 2 1 5

I 1 D O 65 I = I 0 l ~ T ~ t ) I F ( Z M T A B ( 1 ) * G T * 0 0 0 ) GO T O 6 5 f i n l i t ( 6 * 3 1 q )

3 1 4 FOKMAT ( / Z X , 9 7 H * * ERKOR * * T A B L E OF MACH NUM8ER DISTHIBUTION - h A C IH NUMBER VALUES MUST B E b R E A T E R THAN ZERO t0)0//1

I L R R O R = I H E A D 0 2 2 3 65 C O N T I N U E R E A 0 0 2 2 4

00 67 I 1 1 I 1 R E A D 0 2 2 6 I F ( x I T A B ( I + I ) * G E o X I T A B ( 1 ) ) G O T O 6 7 Y R I T t (603161

I I I X T A B - i REA00225

316 FORMAT ( 4 O H * * L R R O R * * T A B L E OF CONTOUR ~ E S C H I P f l O N o / 6 9 H A X I A L D R E A D 0 2 2 9 IISTANCE V A L U E S ( X ) MUST B E I N MONOTONICALLY I N C H E A S I h G ORDER. / / ) R E A D 0 2 3 0

R E A 0 0 2 3 1 R E A D 0 2 3 2 R E A 0 0 2 3 3 REA00234

3

l E R R O R = I 67 C O N T I N U E

I F ( I T w T A B ) IYi13012 1 2 00 69 1 * l a I X T A B

IF ( T W T A B ( 1 ) O C T O 0.01 G O T O 69 * R I T E (6,317)

7 FOHNAT ( / 2 X 0 1 0 2 H * * ERROR * 9 TABLE OF * A L L 1 - T E M P E H A T U R E VALUES MUST B E GREATER THAN

I E R R U R = I 6 9 C O N T I N U E

G O T O 1 4 13 I F ( T W T A B L I ) * G T * 0 . 0 ) G O T O 1 4

Go T O 424

1 ) S C A L E 1 1 S C A L E 40 1 T E M P E R A T U R E = , F 2 0 * 8 )

ENPEnATUr (E O ~ S T R I B U T I O N ZERO t o ) . / / )

R E A 0 0 2 3 9 R E A 0 0 2 4 0 R E A 0 0 2 4 I

R E A 0 0 2 4 5

93

c C D E F I N ~ ; THE F U N C T I O N R O U T I N E T O BE U S E 0 B Y S E V A L C

6 A P F ( T s G ~ A l s B l T = A A A = 88 E = CC

I F I F t O 1-2 I 3 B 1 , 1 B = B / F J ( ~

I F ( I C T A t ) e G T . 1 B / C P O A CPO 6 0 T O 6 0 0

5 5

/ C S E V A L /

/ C S E V A L /

/ I N P U T / / I N P U T / / I N P U T /

/ C S E v AL /

S E V A SEVA SEVA S E V A S E V A

16 i9 2 0 2 1 2 2

0 ) G O TO 3 5

S E Y A 2 7

SEVA 32

SEVA 38

S E V A L)3

SEVA 5 0 S E V A 5 1

S E V A 541

S E V A 5 7

95

S f V A 6 1

SEVA 6 8

SEVA 7 1

SEVA 7 4

SEVA 8 0

SEVA 8 7

SEVA 9 9 SEVA 95

SEVA 9 7 SEVA 98

SEVA 101 S E V A 102 SEVA 106 S E V A I 0 7 SEVA 108 SEVA 109 S E V A 110 SEVA 1 1 1

96

S T A R T

98

S T A R

S T A R S T A R S T A R S T A R

DOES N O T I N C

S T A R S T A R S T A R

S T A R / I Z H T A B l I ) 5 T A R

S T A R S T A R S T A R S T A R S T A R S T A R S T A R

C M X . I M X 1 S T A R

35

37 38 39 4 0

43 44 4 5

4 9 50 5 1 5 2 5 3 54 55 56 5 7 5 8

61

25 2 6 6 4

67 6 8 69 7 0

7 2 73 7 9 75 7 6 7 7 78

B O 8 1 0 2 8 3 0 Y 85. 0 6 8 7 138 0 9 9 0 91 9 2 9 3

9 6 9 7 9 8 9 9

103 I os

108 i 09 110 1 1 1

C F G .: CFA STAR 112 THETA = CFG CTHET STAR 113

STAR 11q CR II C R T 2 * THETA THETA CFB = C F E V A L ( C R ) STAR 115 TTAm = 1.0 + E R A S E I * S Q R T ( C F B / 2 . 0 ) + E R A S E Z * c F B / Z . O I F (TTAIN * G T * 0 . 0 ) G O T O 1 2 0 CFA * 3.OOCFG

105 L4 m CFA STAR 119 STAR 120

STAR 123

2 2 = CFG 110 I F ( J E .LE. 50) GO T O 85

1030 FORMAT ( / t X * 7 4 H * * STANT F A I L U R E * * I N I T I A L U‘ALULS FUR PI411 AND T H E Y R I T t ( 6 * 1 0 3 0 )

I T A 1 CANNOT B E C O M P U T C D I / ~ X * Z I H * CHECK S K I N F R I C T I O N U A T A v / / ) STAR 1 2 4 STAR 125

GO T O 140 120 CFA m CFI) / ( E K A S E 3 T T A l * * Z M V I S 1

I F ( A B S ( ( C F A - C F G ) / ( C F A + C F G ) ) * L E * T O L C F A ) G O T O 1 4 0 I F ( J L .LT. 2 1 60 T O 105 2 3 = 2 4 L l 2 2 STAR 129 2 4 m CFA S T A R 1 3 0 42 m CFG STAR 1 3 1 LS5 ( 2 4 - 2 3 ) / ( 2 2 - 2 1 ) CFA * ( Z 4 - Z S 5 * Z 2 ) / ( l r O - Z S 5 ) G O T O 110 STAR I 3 4

1 4 0 T H E T A 1 = CFA*CTHET P H I 1 * T H E T A 1 CFAGT .: CFA STAR 138 ZETA 1. STAR 139 Y R l T E ( 6 s l O Y O ) X ~ Y H I T H E T A I I P H I I

1040 FORHAT ( 9 4 H O I N I T I A L VALUES FOR ENERGY ( PHI11 A N D I iUMENTUH ( T H E T S T A R 30 I A I ) T H I C K N E S S E S C A L C U L A T E D AT THKOATv.. / 5 H x I IPE14.7. 5 x 1 STAR 3 1 2 4HY = * L 1 4 . 7 1 S x I ~ H T H E T A I * I E 1 4 . 7 ~ 5 X * 6HPHil € 1 4 . 7 / / STAR 32

HETURN STAR 1 4 1 E N 0 STAR I42

100

XMTE Y XNTE 5 XNTE 6

XNTE 2 1

XWTE 22 X N T L 2 Y XNTE 25

XNTE 30 XNTE 3 1 XNTE 32 XNTE 33 XNTE 3 Y

XNTE 3 7 XNTE 36 XNTE 3V XNTE Y O XNTE 4 1

XNTE Y V

XNTE S I

XNTE 53 XNTE 5 4

XNTE 58 XNTE 59 XNTE 60 XNTE 61 XNTE 62 XNTE 63 XNTE 6 Y

XYTE 67 XNTE 6% XNTE 69 XNTE 7 0 XNTE 7 1 XNTE 72 XNTE 73

% N T E 7 4 XNTE 7 5

X h T E 7 9 XNTE 8 0 XNTE 8 1 XNTE 82 XNTE 83 XNTE 81) XNTE 8 5 XNTE 86 XNTE 07

XNTE 90 XNTE 9 1

XNTE 9 4

XNTE 9 7 XNTE 98

XNTE 1 0 2

XNTE 106 XNTE 107 XNTE 108 XNTE 109 XNTE I10 XNTE 111 XNTE 1 1 2

XNTE 1 4 7

XNTE 1 1 9 XNTE 1 2 0 XNTE 121 XNTE 1 2 2 XNTE 123 XNTE I2'l XNTE 125

i 02

Z E T A 1 T S U B R O U T I N E Z E T A I T

c

C

C

ZETA 1

/ C O F I I F /

/ I N P U T / / I N P U T / / 1 N P U T /

/ I N T E R / / I N T E R / / I N T E R /

/ O U T P U T / /OUTPUT/ / O U T P U T /

/ S A V E D /

Z E T A ZETA ZETA Z E T I ZETA ZETA

Z E T A Z E T A ZETA ZETA ZETA ZETA ZETA ZETA ZETA ZETA Z E T A ZETA Z E T A ZETA ZETA ZETA Z E T A ZETA ZETA ZETA ZETA Z E T A

16 1 7 18 19 20 21

24 25 26 27 28 29 3 0 3 1 32 33 34 35 36 37

39 YO 41 42 93 44 '(5

313

Z E T A 50 ZETA 51 ZETA 52

103

z i = z z Z E T A LY=ZETA ZETA z Z * LET AG ZETA Z 5 ~ ( L ' 4 - 2 3 1 / ( Z 2 - L 1 1 ZETA LETA~IZY-ZS*ZZ)/(lr-ZS) ZETA

30 C O N T I N U E ZETA W R I T E ( 6 a 3 4 ) X , Z M E l T H E T A I PHI Z E T A

1 9 . . / 22HO A X I A L D I S T A N C E X IPELY.7m 5 X e I l H M A C H NO. m Z E T A 2 €1'4.7, 5 x 1 8 H T H L T A I = a E I Y . 7 , 5 X . 6 H P H I l - 1 E l Y . 7 I ZETA

W R I T E ( 6 . 5 0 ) Z I l Z 2 1 Z E T A ( 23. 1 4 L E T A

I - m 2 6 1 6 . 7 / / Z E T A

H N I N T = M Z E T A Z E T A AF 1 NT-A Z E T A B F 1 N T = B / Z E T A ZETA C f IN1.C ZETA Z E T A T f l = Z E T A * * Z f l Z E T A ZETA I F ( L E T A o G E o 1.01 60 T O 3 7 C A L L I N T L E T ( O . D Z E T A ~ L ~ ~ I

BF I N T - 0 . Z f T I

3'4 F O N M A T ( 5 7 H O * * L E T A l f f A I L U R E * e o SHAPE PARAMLTER I T k H A T l O N F A I L U R E Z E T A

50 F O R ~ ~ A T t ZOH Z E T A ( G U E S S E D 1 . i l P J E 1 6 . 7 / 20H Z E T A ( C A L C U L A T E O I Z E T A

35 I F I N T = z

A F I N T = A + B Z E T A

C A L L I N T L E T ( Z E T A S I . ~ L I ~ I Z E T A E R A S E 2 = 2 1 4 + 2 1 5 Z E T A ~ ~ L S O T ~ ~ O O M Z E T - L 1 6 - Z 1 7 ) / E R A S E 2 Z E T A U C L T A = THE T A / 2 M Z E T'A / E RASE 2 ZETA G O T O 38 Z E T A

3 7 CALL I N T Z E T l O ~ m l ~ r Z I Z I Z E T A M M l N T = M Z E T A N ZETA A F I N T = A + C ZETA C f INT.0. ZETA C A L L I N T L E T ( I * i Z E T A i L 1 3 1 ZETA D E L T A = T ~ ~ T A / Z M L E T A / L ~ l Z E T A D E L S O T ~ ~ L E T ~ T f l / Z M Z E T A ~ ~ 1 3 - 2 ~ 2 ~ / ~ 1 ~ Z E T A

38 BDELTA = ZETATM*OELTA D E L S T R - T H E T A * D E L S O T Z E T A RETUHN Z E T A E N D Z E T A

53 544 5 5 5 6 5 7 58 59 60 61 62 63 6 Y 65

6 8 6 9 7 0 7 1 7 2

7 5 7 6 7 7 7 8 7 9 8 0 81 8 2 8 3 8q

86 8 7 88

a s

91 9 2 9 3

1.

2.

3.

4.

5.

6.

7.

8.

REFERENCES

Omori, S. ; Krebsbach, A. ; and Gross, K. W. : Boundary Layer Loss Sensitivity Study Using a Modified ICRPG Turbulent Boundary Layer Computer Program. NASA TM X-64661, May 12, 1972.

Omori, S. ; Krebsbach, A. ; and Gross, K. W. : Supplement to the ICRPG Turbulent Boundary Layer Nozzle Analysis Computer Program. NASA TM X-64663, May 17, 1972.

TBL, ICRPG Turbulent Boundary Layer Nozzle Analysis Computer Program, developed by Pratt & Whitney Aircraft, ICRPG, 1968.

Investigation of Cooling Problems at High Chamber Pressures, Final Report. Rocketdyne, A Division of North American Aviation, Inc., Canoga Park, California, NAS8-4011, May 1963.

Rao, G. V. R. : Exhaust Nozzle Contour for Optimum Thrust. Je t Propulsion, vol. 28, no. 6, June 1958, pp. 377-382.

Nickerson, G. A. ; and Pederson, D. M. : The Rao Method Optimum Nozzle Contour Program. TRW/Space Technology Laboratories, Thompson Ram0 Wooldridge, Inc., Redondo Beach, California, 1967.

TDK, ICRPG Two-Dimensional Kinetic Nozzle Analysis Computer Program, developed by Dynamic Science, ICRPG, 1970.

Properties of PARA-HYDROGEN. Report No. 9050-68, Aerojet- General Corporation, El Monte, California, 1963.

1972 - 28 I 0 5

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