N A S A TECHNICAL NOTE

k

N A S A TN D-3451 - -

PREDICTION OF PROPELLANT TANK PRESSURIZATION REQUIREMENTS BY DIMENSIONAL ANALYSIS

by J, F, Thompson and M . E, Nein George C. Marshall Space Flight Center Hzt ntsville, A la.

I

E N A T I O N A L A E R O N A U T I C S A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. J U N E 1966 1

r

TECH LIBRARY KAFB, NM

0330375

NASA T N D- 345 1

PREDICTION OF PROPELLANT TANK PRESSURIZATION

REQUIREMENTS BY DIMENSIONAL ANALYSIS

By J. F. Thompson and M. E. Nein

George C. M a r s h a l l Space Fl ight Cen te r Huntsvi l le , Ala.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $1.00

TABLE OF CONTENTS

Page

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

DIMENSIONAL ANALYSIS OF TEST AND.COMPUTER RESULTS . . . . . . . . . . . . i

Dimensionless Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Curve Fit of Dimensionless Equation. . . . . . . . . . . . . . . . . . . . . . . . . . 7

CONCLUSIONS.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY.. 14

iii

Figure

I.

2.

LIST OF ILLUSTRATIONS

Title Page

Comparison of Calculated and Measured Pressurant Mass for Various Test Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

The Effects of Various Design Parameters on the Mean Tempera- ture at Cutoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

iv

Symbol

A

AD

J

W M

a T

TL

m T

0 T

V

vi

vL

d

C

C

W

P

PW

gC

h

k

a

P

r

DEFINITION OF SYMBOLS

Definition

propellant tank wall surface area

pressurant gas distributor area

dimensional constant

pressurant molecular weight

ambient temperature

propellant temperature

ullage mean temperature at cutoff

press urant inlet temperature

propellant tank volume

initial ullage volume

propellant volumetric drain rate

wall thickness

pressurant specific heat

wall specific heat

dimensional constant

ambient heat transfer coefficient

pres surant therm a1 conductivity

ullage pressure

propellant tank characteristic radius1 (maximum radius for cylindrical tanks)

ft2

ft2

lbf ft/Btu

lbm/lb mol

OR

OR

OR

OR

ft3

ft3

ft3/ sec

ft

B t d l b OR

Btu/lb OR

lb f t m

m

m

lb sec2

B t d s e c f t2 OR

'Btufdsec f t2 OR

lbf/ft2

ft

f

V

Symbol

R

a!

pw

0

DEFINITION OF SYMBOLS (Concluded)

Definition

universal gas constant

gas compressibility

pressurant viscosity, M/L

wall density, M/L3

time of pressurization

lbf ft/"R lb mol

(-)

lbm/ft sec

lb /ft3 m

sec

vi

, . , ., , ..... --. . ... - . .. . . . . , I

PRED ICTION OF PROPELLANT TANK PRESSURIZATION REQU IREMENTS BY D IMENS IONAL ANALYS I S

SUMMARY

Pressurant gas requirements for launch and space vehicles may be predicted by However, preliminary design studies

Therefore, dimensional analysis of a large number of pressurization

analytical models of the pressurization process. require a fast and reasonably accurate method of predicting without resorting to com- puter programs. tests and computer runs was applied to develop an equation that predicts pressurant requirements for cylindrical and spheroidal propellant tanks with an accuracy of *IO per- cent.

I NTROD U CT I ON

Although the most accurate method of predicting pressurant requirements is with a computer program that has been verified by experiments, it is advantageous to have a fast, reasonably accurate method to determine the total mass of pressurant gas required without resorting to the computer. This type of analysis is necessary in comparison and optimization studies for preliminary design where the number of possibilities to be con- sidered precludes a detailed computer analysis of each case. single, general expression for the total required mass of psessurant. was developed by dimensional analysis of the results of about 30 pressurization tests and 120 simulations on an IBM 7094 Computer.

This report presents a The expression

D IMENS IONAL ANALYSIS OF TEST AND COMPUTER RESULTS

The total mass of pressurant gas required is a function of the ullage mean tem- perature at cutoff derived with the gas equation of state,

W PVM

- - WTotal a RTm

Therefore, the total pressurant mass required may be calculated if the ullage mean tem- perature at cutoff can be determined. In the most general case, the ullage mean

~~

NOTE: Mr. J. F. Thompson is Assistant Professor at Mississippi State University Aeronautical Engineering Department; he was formerly with MSFC .

temperature at cutoff will be a function of 12 system design variables, seven physical properties , the mechanical equivalent of heat, and the gravitational constant:

The 19 variables can be expressed in six fundamental dimensions, length (L) , mass (M) , time ( e ) , temperature (T) , heat (H) , and force ( F ) . constants, J and gc, are included because heat and force can be expressed in terms of the other four dimensions.

The two dimensional

The dimensions of each variable are given in Table IV.

Since any equation representing physical phenomena must be dimensionally homo- geneous, it must be possible to write equation (2) in a nondimensional form. Therefore, using 7r as a symbol for a dimensionless group, equation (2) may be written as follows:

7ri = f ( 7r2, 7r3 ----- Ti )

where the group 7ri contains the ullage mean temperature at cutoff.

(3 )

According to Buckingham's theorem, the number of dimknsionless groups in equation ( 3 ) is given by:

i = n - m (4)

where n = number of variables

m = maximum number of these variables which will not form a dimensionless group

The quantity m is often equal to the number of fundamental dimensions. The total number of variables in equation ( 2 ) , including J and g,, is 22, and the number of funda- mental dimensions is six. number of variables in this case that will not form a dimensionless group is indeed six, the number of fundamental dimensions. be 16 dimensionless groups in equation ( 3 ) .

By trial and e r r o r , it was determined that the maximum

Therefore, according to equation (4) there will

Dimens ion less Groups

Because the maximum number of variables that will not form a dimensionless group is six, the 16 dimensionless groups may be determined by choosing six variables to be common to all groups and, in turn, adding each of the remaining 16 variables to the first six to form 16 dimensionless groups. Although any six variables could be chosen as the six to be common to all groups, intuition leads to the choice of the two dimensional

2

constants (J and gc) , the pressurant molecular weight (Mw) , thermal conductivity (k) , and viscosity ( p ) and the propellant tank characteristic radius (r) . Adding the ullage mean

Now ,

temperature at cutoff (T,) to the six common variables yields the first group:

a b c d e f g .rri=J g M k p r Tm c w

substituting the dimensions:

(5)

Since the quantity on the left of equation (6) is dimensionless, the exponent of each of the six dimensions on the right must be zero. simultaneous equations ,

This condition determines six

[L] O = a + b - d - e + f

[MI O = b + c + e

[ e ] 0 = -2b - d - e

[HI 0 = - a + d

[ F ] O = a - b

Simultaneous solution of these equations yields the following:

a = g d = g

b = g e = - 3 g

c = 2 g f = - 4 g

Because the value of g is arbitrary, it can be taken as unity and yields the follow- ing:

a = I d = I

b = I e = -3

c = 2 f = - 4

3

. . . ..

Substitutiag these values into equation (5) yields the first group 7ri:

J g M2 k T

p3 r4 c w m.

"1 =

The second group 7r2 is found in the same manner by adding the presswant specific heat Cp to the six common variables. Each of the remaining variables is added one at a time to the six common variables, and the remaining dimensionless groups are determined in the same manner as the first two. Then,

52 k 7r2 =

J g M2 k T c w 0 .

p3 r4 7r3 =

P r o

M W

7r5 =

A 7r7 = 2 r

Pw r3 "9 = - M

W

4

gc Mw P2

Ti2 =

h r a

= k

Jgc M2 k T ~ ~ p3 r4

w a Ti4 =

Now, the groups TI, "4, an' "14 may e div ded by "3 with no loss of generality, Thus T I , ~ 4 , and ~ 1 4 may because none of the 16 groups is eliminated in the process.

be replaced r1 7 ~ ' and "i4, respectively,where 1' 4'

In the same manner, the group Til may be replaced with &, where

This change occurs because the initial ullage mean temperature is proportional to the product PVi.

It is possible to reduce the number of dimensionless groups necessary in this case by realizing that the wall specific heat, density, and thickness can enter the prob- lem only in the combination Cpw pw dw which is the wall heat capacity per unit area. Therefore, the groups T8, rS, and ri0 may be combined into a single group w1 8'

5

M k

Similarly, it is possible to combine “15 and ni6. Thus,

Further, the propellant tank volume and wall surface area should enter the prob- lem as the area-to-volume ratio (A/V) . bined into a single group 7r’

Therefore, the group 7r6 and 7r7 may be com-

6’

rT Ar 4=-=- ll V 6

But since the approximate area-to-volume ratio is given by

the group T; is nearly a constant and may be eliminated.

Finally, since the Prandtl number ( T ~ ) does not vary greatly, this group may also be dropped. Thus, the number of dimensionless groups necessary in equation ( 3 ) was reduced by six:,

Since the propellant temperature is the lower limit of the ullage mean temperature, it is logical to replace all temperatures with temperature differences above the liquid propellant temperature ( TL) .

Equation (3 ) may now be expressed in te rms of the following ten dimensionless groups (the group numbers have been changed to run from one to ten) :

TIXI - TL

To - TL 7ri =

6

I

I

To - TL

TL n3 =

M k W ..

7r5 = p r 2 (cpw P w d ) w

gC M w P 7r7 =

P2

h r a

*a = k

T - T- a L 7r9 =

To - TL

Mw vL ADP r2 *io =

Curve Fit of Dimension less Equation

These ten dimensionless groups (equations 29 through 38) can be used to correlate the results of tests and computer runs according to equation ( 3 ) . In most cases equation ( 3 ) would be written in the following form:

(35 )

(36 )

(37)

However, in this case it is necessary to satisfy certain boundary conditions that cannot be satisfied by equation (39) . The ullage mean temperature at cutoff must remain finite and not equal to zero as the ambient heat transfer approaches zero. This boundary condition cannot be satisfied by equation (39), but it can be satisfied if the functional dependence on 7r8 and 7rg is exponential. Also, as the pressurhnt gas distributor Reynolds number 7rio

approaches zero, the heat transfer in the tank approaches free convection. Therefore,

7

the boundary condition of finite, non-zero mean temperature, when dictates an exponential functional dependence on nlo. Thus, these boundary conditions can be satisfied by writing equation ( 3 ) in the form

is zero imposed,

The coefficimts and exponents in equation (40) were evaluated by a curve f i t to the data from the computer runs and tests. It was found that the data could be correlated by this equation if the coefficients a1 and a3 in the exponentials were taken as functions of ~2 and ~ 3 :

1 $) a3 = a3 7r2

Equation (40) then becomes

where all coefficients and exponents are constants.

It was convenient to divide several groups of equation (43) by powers of ten to obtain numbers more easily handled. Therefore, the final equation used in the curve-fit was

From the curve-fit, the following values were obtained for the coefficients and exponents in equation (44) :

U!I = 0.424 5 = 0.01416

a2 = 0.00210 h = 0.0620

a3 = -0.0292 w = 0.415

P = -0.1322 $ = 1.174

8

- I...-..-# I. 1 . 1 1 . 1 1 1 1 I I 1 1 1 1 .I I 111 11.1111.111 . 1 . 1 1 1 1 1 I1 111 11111 I I 11111.11 1 1 1 . 1 1 1 1 I1 1111 I

= -0.1688 T = 0.765

6 = -0.1146 I p = 0.1510

E = 0.0780

, equation (44) becomes Tm - T~

To - TL Therefore, since 7rl=

-0.1322 -0.1688 -0.1146

Tm - TL = 0.424 ($) (.) ( ~ 4 ) To - TL

0.0780 0.01416 0.0620

This equation is general and is capable of predicting the ullage mean temperature, and thus pressurant m a s s a t cutoff within 10 percent for cylindrical tanks with rounded bulk- heads. Figure I shows total pressurant requirements obtained by various investigators for a wide range of tank s izes and system parameters compared with the pressurant weights calculated by equation (44). range of conditions for hydrogen and oxygen pressurization. + However, the equation is limited in its application to conditions of constant ullage pressure, pressurant inlet temperature, and ambient heat transfer. The studies indicated that the equation is inac- curate at inlet temperatures less than I O O O R above the saturation temperature, at ullage pressure below propellant saturation, and for very short expulsion times of less than 50 seconds. The restriction to cylindrical tanks can be removed by proper choice of the characteristic tank radius.

* Evaluating the test results of Reference I by this method resulted in a large, but con- stant deviation from actual observed pressurant weight. This is probably because the test parameters such as heat leak through the vacuum chamber and pressurant inlet tempera- ture had to be assumed. Additional information about these tests are required to re-evaluate these conditions.

Excellent agreement is obtained over the entire

-

9

Studies have shown that the characteristic tank radius for oblate spheroids, used should be about two-thirds of the maximum tank radius. This assump- in equation (44)

tion is theoretically justified, because a cylinder having the same volume and surface area as an oblate spheroid has a radius equal to 0.63 t imes its maximum radius. Further test data and analytical studies are necessary to select the characteristic radius for other geometries.

Due to the dimensionless nature of equation (44) it is not restricted to any particular propellant, pressurant, or tank size as indicated in Figure I. Although an un- insulated propellant tank was assumed in the development of this equation, Figure I shows good agreement with test results obtained with vacuum jacketed liquid hydrogen tanks.

To simplify the use of equation (44) the pressurant and propellant properties for the case of liquid oxygen pressurized by oxygen and liquid oxygen pressurized by helium were substituted in equation (44) , and the following equations were obtained. Since these equations are dimensional they are applicable only to the case indicated. For liquid oxygen pressurized by oxygen,

-0.297 0.1395 0.01416 T - 164

T - 164 i m

= 3.33 (To - 164) r 0

-0.0780 0.0762 -0.1146 (CPWP,dW) P e T

P 7

0.765 a h -0.304 -0.895

* exp 0.001568 ( T - 164) r 1 0

-0.574 r -2.604 (%)I and for liquid oxygen pressurized by helium,

-0.304 0.1395 0.01416 T - 164

T - 164 m

vi = 3.10 (To - 164) r 0

(45)

-0.078 0.0762 -0.1146 * ( C pw P w d ) w P eT

10

I

-0.1650 -0.895 0.765 a

(To - 164) r h

r . l

The units of all variables in equations (45) and (46) must be those given in the Definition of Symbols. For other combinations of propellant and pressurant, equation (44) must be used. After the ullage mean temperature at cutoff is calculated, and using equation (44) , (45) , o r (46) , if applicable, the total mass of required pressurant gas may be calculated from equation ( I).

In designing a launch or space vehicle pressurization system, vehicle parameters such as tank volume, engine flowrate, tank material, etc. , determined by vehicle mission profile, are fixed input values. However, there are various controllable parameters in a pressurization system that can be used to optimize the system without affecting basic ve- hicle characteristics. requirements has , therefore, been investigated under another study program. sults of these studies exerpted from reference I a re presented in Figure 2. central origin, representing a reference condition (Saturn V, S-IC Stage) for all param- e te rs , the increase (+Y) and decrease ( - Y ) , of the ullage mean temperature a t cutoff is shown as a function of variation of the parameters on the abscissa. were varied over a range expected for vehicle design. Thus, pressurant inlet tempera- ture can increase or decrease by a factor of 2 from the reference condition, pressure by a factor of 3, tank radius by a factor of 2, expulsion time by a factor 3, etc. indicated that the pressurant inlet temperature exerts the greatest influence on the ullage mean temperature. Diminishing return of this effect did not exist within the range of in- vestigation (530"R to 1200"R). The mean temperature increased as the ullage pressure was increased and also as the tank radius was increased. Increasing the tank wall thick- ness, heat capacity, o r density caused decrease in the mean temperature. surant distributor flow ( AD) that controls the gas-to-wall forced convection heat transfer coefficient had a significant effect on the mean temperature when AD was reduced, but no effect at all when flow area was increased. locity for the reference systems was chosen at an optimum point. Figure 2 also indicates that helium pressurant must be introduced into a tank at a temperature I. I times higher than oxygen pressurant to obtain the same ullage mean temperature.

The relative significance of various parameters on pressurant

From a The re-

The parameters

It was

The pres-

This indicates that the pressurant inlet ve-

CONCLUSIONS

An equation derived by dimensional analysis provides a reasonably accurate method for prediction of pressurant requirements for cylindrical LOX and hydrogen pro- pellant containers. This method is advantageous for preliminary design and optimization studies where the use of large computer programs becomes,excessive in cost and time.

PROPELLANT: HYOROGEN

TANK VOLUME (F$): 23

m m a

n w l 3 1

5 0.8- c 3 a

0 ,

ha = 0

0 . 0 . 0 +

0 + A TEMP OR

520 521 532 501 505 530 335 405 325 300 300 300 300 300 300

OXYGEN

1321

OXYGEN

1270

I I I

I I I I I I I I I I

:IYOROGEN

67

ha = 0

* * *

FIGURE I. COMPARISON OF CALCULATED AND MEASURED PRESSURANT MASS FOR VARIOUS TEST PARAMETERS

xo/x . x / x o

REFERENCE SYSTEM r

// I. 2

1.3 I r = 16.5 (ft)

Vo = 47492 (ft 3 )

To = 800 ('2)

p, = 6 0 (psi 4)

= 17.8 (ft ) 2

To ( 1 - 2 ) "

0

ha

Ta - 530 (OR)

C p6 = 1.51 (btu/ft2 OR)

= 2 ( b t u h r f tZ OR) I

Tmo/Tm -

P

FIGURE 2. THE EFFECTS OF VARIOUS DESIGN PARAMETERS ON THE MEAN TEMPERA- TUREATCUTOFF

CI w

REFERENCE

I.

I.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

14

Nein, M. E. and Thompson, Jr. , J. F. , Experimental and Analytical Studies of Cryogenic Propellant Tank Pressurant - Requirements, - _ NASA TM X-53165, 1965.

BIBLIOGRAPHY

Coxe, E. and Tatum, John, Main Propellant Tank _ _ Pressurization System Study .. -

and Test Report, Lockheed - Georgia Company, Contract AF04(611) 6087.

Hargis , J. H. , Therm’ophysical Propertiesof -0qgen and - ~ _ Hydrogen, . - - MSFC Internal Note IN-R-P&VE-63-13, October 1963.

NASA

Johnson, V. J. , A Compendium of the&-operties o f x t e f i a l s at Low Tempera- _ -

ture (Phase I) Part I Properties of FluidsrNational Bureau of Standards Cryogenic Laboratory , July 1960.

Myers, J. E. and Bennet, C. 0. , Momentum, Heat ~. and .. M a s s Transfer, McGraw Hill, New York, 1962.

Scott, R. B. , Cryogenic Engineering, D. Van Nostrand Company, Inc. , Princeton, 1959.

Modification of Rocketdyne T m Pressurization Compu@-Program - , NASA MSFC Memorandum R-P&VE-PTF-64-M-83, 1964.

Atmospheric Heat Transfer to Vertical T&s Filled with Liquid Oxygen, A. D. Little, Inc. , Special Report No. 50. Contract AF04- (647) -130, November 1958.

Fortran Program for the Analysis of a Single Propellant Tank Pressurization ~~ - System, Rocketdyne, Division of North American Aviation, Inc. , R-3936-1, September 1963.

MSFC Test Laboratory Reports CTL 2, 36, 34, 37, and CTL Flash Reports dated August 17, November 20, and December 11, 1962.

SA-I5 Static Test Report, Preliminary .~ Summary Data, . . Confidential.

SA-5 Flight Data.

Results of Pressurization Tests on LH2 Facility, NASA MSFC Memorandum ~

R-P&VE-PE , M-526.

I,. I I 111111 111 I,. 11.11 I,.. 1111 I 1 I 1 1 l . 1 1 1 . 1 1 1 1 I I1 1111111I I1 I 1 - . #I --.1..1.1. 111.. 1 1 1 1 1 1 . ... I

BIBLIOGRAPHY (Concluded)

13. S-IV Stage Pressurization Data, NASA MSFC Memorandum R-P&VE-PTF-64-M- 38, 1964.

NASA-Langley, 1966 M331 15

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