Aerothermochemical analysis of nonoscillatory and oscillatory characteristics of liquid bipropellant rocket motors - wave model, droplet heating, and evaporation computer progra

Agosta, V. D.

Burstein, S. Z.

English

NASA Technical Reports Server

I .  
r A E R  OTHER MOCHEMIC A L  ANALYSIS O F  1 
NON- OSCILLATOR Y AND OSCILLATORY 
CHARACTERISTICS OF LIQUID 
BIPROPELLAST ROCKET MOTORS: 
WAVE MODEL, DROPLET HEATING, AND 
EVAPORATION PROGRAM COMPUTAT IOhT L 
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IPAOESI c 
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c;z5+/ iCATTGORYl R e p o r t  No, 65-1 (NASA CR OR TMX OR AD NUMBER) 
I 
._ .
15 January 1965 
https://ntrs.nasa.gov/search.jsp?R=19660003794 2020-03-16T21:35:30+00:00Z
. 
- V l T O  D A G O S T A  
AND OSCILLATORY CHARACTERISTICS OF LIQUID 
BIPROPELLANT ROCKET MOTORS: WAVE MODEL, 
D R O P L E T  HEATING, AND EVAPORATION PROGRAM COMPUTATION 
V, D. Agosta  
S. Z, Burs t e in  . 
Prepared For 
Jet P r o p u l s i o n  Laboratory 
Pasadena, Cal i forn ia  
R e p o r t  No. 65-1 
Third Q u a r t e r l y  Repor t  
15 January 1965 
. .  . .  
V l T O  D. AGOSTA 
1 
CONTENTS 
Ab s t r act 
Nomenclature ' 
. I. Introduction 
II. Wave Model 
i 
III. Dro2let Heating and Evaporation 
IV, Description of the Non-Oscillatory Steady State 
C o mp U t e  r Program 
V. Computer P rogram 
Page 
ii 
iii 
1 
2 
3 
12 
14 
. .  
. .  . 
-. 
ii VlTO D .  AGOSTA - ___ - -. - _ - - - - ~  -- 
\ .  
7 '  
AEROTHERMOCHEMICAL ANALYSIS O F  NON-OSCnLATORY 
'h OSCILLATORY CHARACTERISTICS OF LIQUID 
BIPROPELLANT ROCKET MOTORS: WAVE MODEL, 
DROPLET HEATING, AND EVAPORATION PROCRm- COMPUTATION . .  . .  
by. 
V. D, Agosta and S, Z, Burstein 
- - - - -  - - - -  - - z- - -  c- - -- 
L 
- - _  - - - -- + ___ ;ABSTRACT 
\ -- 
. .  . .  This report  is divided into three parts: (1) a description of the wave * 
propagation model, (2) fuel droplet heating analysis, and (3) discussion of . 
fur ther  work done on the non-oscillatory aerothermochemical co.rnputer 
program, 
I 
It is decided to use a wave propagation model in which the wave 
*. 
lengths a r e  coupled to the cavity dimensions, 
I F r o m  the droplet heating analysis, it is found that very la rge  
temperature gradients exist near  the surface,  the interior of the droplet 
remaining relatively unheated. 
ditions for  a (0.375~ 10°3ft) drop is about 40 microseconds. 
In addition, heat-up time to saturation con- 
. 
The computer program presented herein is the modification of an 
earlier code that was designed to test some basic ideas on the theory of evap- 
oration ra te  controlled combustion. It is designed to solve automatically 
and simultaneously the interaction between the nonlinear gas dynamic equa- 
tions and the two components of the  spray, i, e, , fuel and oxidizer, . .  Xn 
addition, the effect of equilibrium dissociating combustion products on 
. -  
a; ---- ' 
. . I  -'. j this - solution is incrude& ., . ,  
. .  
.. : I 
. .  
: . .. 
I . *. 
~~ 
a 
h 
k 
r 
T 
t 
P 
droplet radius 
specific heat 
thermal film coefficient 
thermal conductivity 
radius 
t empe rat ure 
time 
density 
iii 
- - _ _  
V l T O  D. AGO.STA 1 
A E R O T H ~ C H E M I C A L  ANALYSIS OF NON-OSCILLATORY 
AND OSCILLATORY CHARACTERISTICS OF LIQUID 
BIPROPELLANT ROCKET MOTORS: WAVE MODEL, 
DROPLET HEATING, AND EVAPORATION PROGRAM COMPUTATION 
I. INTRODUCTION 
9 
This report  is divided into three parts, These a r e  (1) a descrip- 
tion of the wave propagation model, (2) fuel droplet heating analysis, and 
r t 3  
I 
(3)  discussion of further work done on the non-oscillatory aerothermo- 
chemic a1 computer program. 
It has become apparent as work progressed 02 the instability program 
that the wave equations, the wave energy coupling mechanisms, the method 
of solution for these wave equations, and the injector geometry must be 
considered in formulating and sdving the nox-oscillatory aerothermochemical 
program. Revie-n of the oscillatory data and injector configuration s q p l i e d  
by the Jet Propulsion Laboratory suggested the approach assumed and given 
below. The most significant finding in the analysis of the oscillatory data 
is that the tangential mode of oscillation exhibits the wave characterist ics 
_ .  
of the rocket chamber cavity. 
Mach number agree with those expected for  the classical wave behavior in 
In other words, the wave period and estimated 
the tangential mode. 
behavior in terms of the field equations (e, g. , Moore and Maslen) in 
contradistinction to analyzing wave propagation along a 
circumferential  strip (Priem). 
Thus the conclusion made is to analyze this wave 
one-dimensional" 
The analysis remains nonlinear because of 
the large wave amplitudes observed, 
predominately indicative of the gas dynamic behavior of the waves, since 
the rocket chamber wall continuously modifies the wave shape due to reflec- 
tions as the wave sweeps along the wall, 
this model manifest itself' t o  the previous approach assumed? 
is "very well; 81 the develo2ment of the wave model is given below. 
In this case,  the wave slope is not 
The question a r i s e s  - how does 
The answer 
11. WAVE MODEL 
Essentially, the wave model consists of a resonating cavity in which 
the oscillations a r e  three-dimensional in a two-phase reacting fluid flowing 
in one dimension, i, e. , axially, 
circulatory motion io the fluid, and to cause the liquid phase of the fluid 
flowing to evaporate and react  at ra tes  different than those in the non- 
oscillatory case. 
The effect of these waves i s  to impose a 
These excesses serve as energy sources to drive the 
wave if the transport  relaxation times a r e  in the proper phase, 
reduces to: (1) generating the wave equations in three dimensions, 
(2)  determining the axial droplet distribution f rom the one-dimensional non- 
oscillatory aerothermochemical analysis, (3 )  determining the radial and 
tangential droplet distribution from the injector geometry, (4) determining 
The problem 
the relaxation time f r o m  the non-steady droplet evaporation analysis and 
a diffusion analysis of the evaporated fuel vapor through the film surrounding 
_- -2- . - -- 
the fuel droplet, (5) froLm (2), ( 3 )  and (4) calculating the excess V y term, 
and ( 6 )  applying (5) to the wave dynamic equations of (1) and solving, 
The gas dynamic equations given in the previoKs- Quarterly Report,.  
Re?ort No, 64-3, are general and serve as the basis for  the wave equations. 
.* 
. 
V l T O  D .  AGOSTA 3 
These wave equations will be derived shortly and given in the Fourth Quarterly 
Report. 
in regions near the chamber walls, and thus wi l l  not be included at this time, 
Droplet heating and evaporation is discussed in the following section below. 
It is assumed that viscous and thermal effects a r e  important only 
- - 1  - 
III. DROPLET HEATING AND EVAPORATION 
The droplet heat transfer model is  given in Figure 1. The basic 
equation for  heat transfer is 
-- - 
Tt = KVaT 
a 
a t  where the subscript t means (-). 
Spherical symmetry is assumed; therefore, Eq. (1) is written as 
with the boundary and initial conditions 
h(Tf-.T) = k T r  ; r = a  t Z 0  
T (0 t) is finite 
T ( r ,o )  =Ti 
The solution to this equation is (H, S. Carslaw and J. C. Jaeger,  
Conduction of Heat in Solids, n Oxford at the Clarendon P r e s s ,  1950): 
V l T O  D. AGO'STA 
.j . 4 
. .  
F/GI/R E / .  
. 
V l T O  D .  AGOSTA 5 
and 
p + ana cot ana = 0 
F o r  50/50 hydrazine-UDMH blend a t  about 650°R: 
1 p N 53 lbm/ft3 
C 2 0.694 Btu/lbmoR 
k 
P 
0.42 x lo-* Btu/sec ft OR 
K = 1.15 x ft2 /sec. 
Fo r  O / F  2.20 , the flame temperature near the injector end of . 
the chamber is Tf p 5800°R and h =  1.5 B/ft2secoR to a droplet 
a = 0.375 x lo -=  ft. 
Reference is now made to Eq. (2). Fo r  short time intervals, i. e. , 
lo-' < t < 
Therefore, the rate  of convergence of this solution for short t ime intervals 
e - a n 2 ~ t -  - e0 = 1, In addition, s in%a< 1 and s i n s r <  1, - - - -  
is dependent on the factor 
and this converges very slowly, making the solution of no practical use 
in the time range of interest, Thus, a short t e r m  solution is sought. 
The method of solution again is found in H. S, Carslaw and J, C. 
Jaeger ,  "Operational Methods in Applied Mathematics , Dover Publica- 
tions, Inc, , 1963 , and was used by Richard F, Parisse in "Ablation of a 
, 
. 
* 
V l T O  0. AGO'STA 
Solid S2here of a Low Conductivity Material, 
Polytechnic Institute of Brooklyn, 1964, 
reproduced below, and modified to the extent that the te rms  apply to a 
liquid fuel droplet. 
PIBAL Report No. 6 8 3 ,  
The heating up solution is 
A short-time solution for the temperatur-e distribution in a liquid 
sphere subjected to a point symmetric heat input is obtained by utilizing 
~- - 
5. 
Laplace transformations, f The appro2riate fo rm of the heat conduction 
equation is 
with the conditions 
T( r,o) = T. 
1 
T( o,t) is finite 
where the symbols a r e  defined as 
1 
a initial radius of sphere 
h heat transfer coefficient 
k thermal conductivity 
r radial coordinate 
t time 
T temperature 
heated surface temperature 
Ti initial temperature of material 
Ta 
6 
. .  
I 
Tf 
K thermal diffusivity 
stagnation temperature of flow field 
Eq. ( 7 )  cam be transformed into a more convenient form by the 
substitution 
u = r( Tf- T) 
The result i s  
and Eqs.( 8) become 
u(r,o) = r(Tf-Ti) 
u(0,t) = 0 
where p = (ah-k)/k. 
The Laplace transformation is now applied to Eq. (9)  
and by using Eq. (lob) , the resulting subsidiary equation is 
( loa) 
I .  
V l T O  D. AGOSTA 
.l 
where 
Eqs, (loa) and (1Oc) become 
8 - _ _  
A general solution to  Eq. (11) is 
where C1 and C2 are'constants, 
Eqs. (12a) and (12b). 
These constants a r e  evaluated by applying 
The result  in s e r i e s  form is as follows: 
F o r  a short-time solution, Eq. (13) can be truncated after the n=O term. 
F o r  the range of parameters  considered in this report, the higher order  
terms of Eq. (13)  have been shown to be negligible. . . - _ _  -- -
The inverse transformation of Eq. (13) for  n=O is 
! V l T O  D. AGOSTA 
K t  t! =- 
a2 
9 
r f  = r / a  
- T- Ti 
T =-. 
Tf- Ti 
Aplot w a s  made of the heating u? of a fuel droplet, (50 /50  N,H,/UDMH, 
f o r  a = 0.375 x 
short  time the surface of the d r q  heats up. 
vapor pressure-temperature data, a saturation temperature of 480 F was 
assumed. 
UDMH and 235OF for  N,H,; and at the cri t ical  pressure of 1696 psia, 
Ti = 634 F. 
ft)  and is shown in Figure 2. It is seen that in a very 
In the absence of saturation 
0 
0 Note the saturation temperature at p = 14.7 psia i s  146 F fo r  
0 . For  this temperature of 480°F, the heat-u? t ime of the 
droplet before evaporation occurs is about 40 microseconds. 
The temperature distribution in the droplet was determined for the 
case  where the surface temperature was equal to the evaporation tempera- 
ture. It is seen in Figure 3 that the heating effect is concentrated in the 
outer one percent of the droplet radius. . This agrees with results from 
the previous aerothermochemical analysis , namely, that the drop, con- 
sidered a t  constant temperature , does not heat up significantly, 
The next step in this analysis is to determine the evaporation ra te  
fo r  the droplet, An equation has been derived and, unfortunately, it is 
highly nonlinear, 
c u r l  ently being investigated. 
Several techniques for the solution of this equation are 
. - 
0- . 
1 
? 
. . .  
1.0 
; I 
k ' .8 
I I  
0 -  
.-- _-..  . . . . . - - ._ -_- . .. . . _. 11 .. , --1 - - - -  -. 
I 
i .  
. 
12 VlTO D .  AGO'STA 
- 
- 
1 
N, DESCRIPTION OF THE NON-OSCILLATORY STEADY STATE 
COMPUTER PROGRAM 
The computer program, which is presented in this quarterly report ,  
is a modification of an ear l ier  code that was designed to tes t  some basic 
ideas on the theory of evaporation ra te  controlled co,mbustion, The ear l ie r  
version, although successful , only considered evaporation of either fuel o r  
oxidizer. The final solution obtained was dependent on the accuracy of the 
guess of the rate  of the evaporation of the most volatile constituent of the 
fuel- oxidizer combination. 
the same rate  , this program could be used but only in the most awkward 
manner, i, e, , iterations between fuel trajectories and gas dynamics and 
oxidizer trajectories and gas dynamics could be handled, but one must also 
c a r r y  out some manipulations of intermediate results manually. 
For coinponents which evaporate at ap;,roximatefy 
These deficiencies appear to be resolved by the present code, It is 
designed to solve automatically and simultaneously the interaction between 
the nonlinear gas dynamic equations and the two components of the spray, 
fuel and oxidizer. 
-
In addition, the effect of the equilibrium dissociating 
combustion products on the solution is included. 
feature comprising the Weber number effect on fuel and oxidizer (droplet 
breakup) i s  included, It must be stated though, that the present version, 
although essentially complete , is not totally debugged. 
be presented in the fourth quarterly report. 
An additional optional 
The final code wi l l  
Its utility will be demonstrated 
with some solutions pertaining to steady flows in a J. P, L, combustion 
chamber (Resonant Chamber 1; referenced to drawings D9 11 1263, 
1 
' D9 11 1264, D9 11 1248) using N2H4- C2H8 as fuel and N204 as oxidizer, 
1 
V lTO 0. A G d S T A  .13 
A glossary of t e rms  wil l  be given when the final version is presented, 
Although the present version considers the oxidizer ballistics to behave 
similarly to the fuel ballistics, an additional program will be presented in 
the fourth quarterly report. 
evaporation as contained in the first  and second quarterly reports, This 
version has  the property that it can predict the transient behavior during 
startup conditions in the combustion chamber. 
two approaches to oxidizer evaporation will be presented, ' 
It w i l l  contain the formulation of oxidizer 
Comparison between these 


Aerothermochemical analysis of non-oscillatory and oscillatory characteristics of liquid bipropellant rocket motors - Wave model, droplet heating, and evaporation program computation  Third quarterly report no. 65-1

Aerothermochemical analysis of non-oscillatory and oscillatory characteristics of liquid bipropellant rocket motors - Wave model, droplet heating, and evaporation program computation Third quarterly report no. 65-1

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