Liquid monopropellants and liqued-fuel rocket engines

Liquid Propellents a n d Liquid-Fuel Rocket Engines

109

COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS

IN DROPLETS

By C. SANCHEZ TARIFA, P. PEREZ DEL NOTARIO AND F. GARCIA MORENO Introduction

Theoretical and experimental results of an investigation on the combustion or decomposition of monopropellant droplets in an inert atmo-sphere, and on the combustion of bipropellant systems consisting of fuel (oxidizer) droplets in the vapor of an oxidizer (fuel) are given in the present work.

Combustion of droplets has been mainly in-vestigated for the case of fuel drops burning in air. Theoretical studies1,2,4,7,9,30,11 n a v e be e n carried out by assuming that the reaction rate is infinitely rapid. Such an assumption implies that chemical reaction occurs at a zero-thickness layer, where the mass fractions of both fuel and ozidizer become equal to zero. On the basis of this assumption, which disregards chemical kinetics, it was found that the burning rate is proportional to the droplet radius, and, there-fore, that the square of the droplet radius is a linear function of time. I t was also found that the flame: droplet radius ratio does not depend upon the droplet radius.

For droplets larger than 100 to 200 n in diam and at pressures equal or higher than one atmos, the above-mentioned linear law has been ex-perimentally verified on several occasions,1-4 • 7,9,10,11 a n ci the theoretical and experimental values of the burning rate were in good agree-ment.

On the contrary, experimental values of the flame:droplet radius ratio did not coincide with the theoretical ones and, what is more signifi-cant, the observed values of these ratios were not constant, but they increased as the droplet radius decreased.4,10

Such lack of agreement was at first attributed to the influence of natural convection, but

Kum-agai and Isoda10, u found similar results in a zero-gravity field. These investigators explained the difference between theoretical and experi-mental results by means of an approximated theory on the transient combustion of droplets. However, transient conditions alone cannot ex-plain such effects, because Wise, Lorell and Wood4 obtained, for steady-state combustion, similar laws of variation of the flame-droplet radius ratio as a function of the droplet radius b3^ using the porous sphere technique.

There are no experimental results on the com-bustion of droplets at low pressure and very little experimental evidence on the combustion of small droplets. Hall and Diederichsen3 and Bolt9 found values of the evaporation constant for droplets of about 200 /JL, smaller in diameter than those obtained for large droplets, but these differences were attributed to the experimental techniques.

Lorell, Wise and Carr5 studied the combustion of droplets considering finite chemical kinetics, and solved the problem by integrating numeri-cally the differential equations of the process. However, they considered a fixed value of the droplet radius, which was rather large, and they studied the problem taking equal values of both molecular weights and Lewis-Semenov numbers. For these conditions, results obtained by assum-ing an infinite reaction rate are not very different from those derived considering finite chemical kinetics.

droplet radius or the pressure tends to an infinite value.

In general, results tend rapidly toward their asymptotic values. However, for small droplets or at low pressure, important errors may be in-troduced by assuming that the reaction rate is infinitely fast. For example, theoretical results show that, under certain values of either the pressure or the droplet radius, combustion seems impossible. This important theoretical conclu-sion, which does not appear when chemical kine-tics is disregarded, has been experimentally verified. The errors introduced by disregarding chemical kinetics may be especially important when the molecular weight of the gaseous reac-tant is much smaller than the molecular weight of the liquid reactant, as for the case of the combustion of oxygen droplets in a hydrogen atmosphere.

Combustion of monopropellant droplets has been the subject of several works.6's> 12,13, u

In this process the flame is of the premixecl type. Therefore, the assumption of taking an infinite reaction rate, which may give approximate re-sults for diffusion flames, is not applicable.

In the present work it will be shown that chem-ical kinetics influences decisively the process and that the controlling parameter is the ratio of the activation energy to the heat of reaction. Experimental evidence on the combustion of monopropellant droplets is almost absent. There-fore, an accurate comparison between experi-mental and theoretical results cannot be realized.

Fundamental Assumptions

Combustion of both monopropellant and bi-propellant droplets will be studied under the following assumptions. The droplets are con-sidered to be isolated and at rest, and the proc-ess is assumed to have spherical symmetry. Therefore, the influence of natural convection is disregarded. The process is stationary. Under such conditions the study is only strictly applic-able to the combustion of constant radius droplets fed with a fuel flow equal to the amount of fuel which is evaporated and burned.

However, it has been shown that for large or medium sized droplets the errors introduced by considering steady-state conditions are not im-portant. This is clue to the fact that radial veloc-ity of the droplet surface is small as compared to the diffusion velocities of the species. For

this reason, results obtained bjr using porous spheres of constant diameters are similar to those obtained by burning real liquid droplets. Nevertheless, theoretical results obtained for small droplets will show that the influence of the droplet radius on the process may be very important in certain cases. For these special conditions the steady-state assumption no longer holds.

In order to obtain general conclusions, only reactant species and reaction products will be considered, and the actual chemical kinetics of the process will be approximated by means of over-all reactions of ?ith order. However, the analytical method used to solve the problem is applicable to more complicated kinetic schemes.

8 , 13

For simplicity, average values will be adopted for the thermal conductivity and for the specific heat of the mixture. The gas pressure will be considered constant throughout the process and the heat transferred to the droplet surface through radiation will be neglected.

General Equations

Under the aforementioned assumptions, the general equations of the process are as follows:

CONTINUITY

For each species, we have:

4:irr2 dr

Wi (1)

If v; represents the stoichiometric coefficients of the over-all reaction, molecular weights, Mi, and the reaction rates, vo%, are related as follows,

Wi IVi

vi Mi -v, M, '

we have:

~~v1 Mt

e, — e

= ±

-v,Mi '

(2)

(3)

ENEEGY

The equation of energy is given hy:

m(J2 hi ei — S ha eh + qi)

_47rr2X ? = 0. (4)

COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS IN DROPLETS 1 0 3 7

Expressing the enthalpies as a function of the temperature, we have:

(5)

_4 7 rr2X ^ = 0.

Introducing the heat of reaction by means of relations (3), taking an average value for the specific heats, and referring the equation to the combustion products of Equation (3), we ob-tain:

m[cp(T - T.) -qres+ g j

c/r

(6)

DIFFUSION

Assuming that only concentration diffusion exists, we have:

£

mc%

i^n

\Y3

Y„

<Li%

1 dYt 1 dY: Y i dr Yj dr _

(7)

= 0.

BOUNDABY CONDITIONS

I t will be assumed that liquid phase reactions do not exist and that there is only one liquid chemical species." I t will also be assumed that the chemical reaction goes to completion at infinitjr and that the temperature and composi-tion of the atmosphere at infimtjr are known. Thus, we have:

T = Ts r = rs< eu = 1

e,s = 0 (i^l)

T = ^ (8)

eico = 0 .

= oo< Fl o 0 = 0 Y = Y

There exists one boundary condition in excess

a No distinction will be made between the

species of a monopropellant composed of a mix-ture of fuel and oxidizer.

of the number of differential equations, which gives the burning rate m, or "eigenvalue" of the system.

Solution of the System

Solution of the system is simplified by means of the following change of variables:

= -

P

( T -

T

S

+ P

X = m

4-7TX r

From which:

det dX dd_ dX

c X r

~l~X^

- (6 ~ e3) w.

IKY,

F,

1 dYt + Y% dX where Xs, given b3^:

X , =

AJl

1

dYJ

Y,dX

(9)

(10)

(11)

(12)

(13)

m 47T-X rs '

(14)

is the new eigenvalue of the system. The bound-ary conditions are now as follows:

X = Xs{ eu = 1

els = 0 (i ?£ 1)

X =0

(15)

i i « — €lo

F„ = Y^

The solution of the problem lies on the integra-tion of the nonlinear system of differential Equa-tions (11), (12) and (13) with boundary condi-tions (15). In addition, an expression of to t as a

function of the mass fractions and temperatures should be known.

The law of variation of the functions

e, ( X ) a n d de%

dX

(X),

which was found by means of numerical integra-tion of the equaintegra-tions for several typical cases,8, 13,15 suggested the adoption of an approximate analytical integration method. This method is based upon considering a reaction zone of finite thickness and on approaching the law of varia-tion of e, within such zone by means of two parabolic curves tangent to each other at point (eJQ0 + e,s)/2 and tangent to the lines e% =

e,M and et = els at the boundaries of the

reaction zone.

The reaction zone is determined by the values

Xi and Xn of its hot and cold boundaries

re-spectively, or else, by means of its central point

X* and thickness x = Xn — Xr. The

maxi-mum value of deJdX, which will be called et*, should also be fixed.

The values of x, U* and X* are obtained from the system of equations:

X *

det dX /

x=x-Jo

2 2

Cp Xs 7 X X * 4

*

Wi =

dX dX — 2(e,g — eiao)

(16)

(17)

ctel

dX'< X* = Q-+X*w'%* -too* = 0 (18)

In this system w* is the value of wl{Y%, 8, X)

for X = X*, m which Y% and 6 are obtained

bjr integrating differential Equations (12) and (13). This integration is performed bjr introduc-ing into these equations the followintroduc-ing expres-sions for e* :

0 ^ X ^ X *

-X* - I t X

ZX*); e, = e,

Z\€lta €ls)

X

- [

x* -1

(19)

(20)

x* t x t

x* + I. ,

(21)

+

2(e1 0 0 — ei s)

X* + § - X

X* + | ^ X < x

6 (22)

The integration of Equation (12) is straight-forward and expressions for 6 are then readily derived. The eigenvalue Xs is obtained by in-tegrating Equation (12) between 8S and 0OT .

The integration of expression (13) is more in-volved, because the equations are not linear. However, when the molecular weights of the species as well as the Lewis-Semenov numbers are equal, equation (13) reduces to:

cm

dX

=

- £ ( F ,

- O

(23)

and the integration can readily be performed. An approximate method has been developed15 to integrate Equations (13) for the general case. This method is based on the series expansion of the expressions for Yt, assuming that x is small,

which is the case that normally occurs in prac-tice.

Several solutions8 , 1 3 , 1 6 for typical cases of systems11"13 given by the approximate analytical method have been compared to those obtained by the numerical integration of the equations. Comparisons were performed for a wide range of variation of the characteristic parameters of the process and for different reaction rates.

These comparisons showed that the approxi-mation furnished by the analytical method was excellent in all practical cases.

Applications and Discussion of Results

MONOPKOPELLANTS

The case corresponding to a first-order reaction of the form:

Ax - > A3 ; Mi = Mz ; (24)

will be considered. Therefore:

eis = 1; 63s = 0; eiOT = 0;

e3M = 1; 1 ico — 0; 13oo = 1 The chemical kinetics of the process will be approximated by means of the over-all reaction rate:

ws = —it>i = BpiX — F3)e ElBT

COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS IN DROPLETS

1039

0.05 0.10 0.1 S

DROPLET RADIUS 1 0_ 5 ^ r , .

FIG. 1. Combustion of nionopropellant droplets within an inert atmosphere. Burning rates, evapora-tion constants [ k ~—] and burning rates per unit surface.

where:

A,ir — Bps Ts t

- 2 " P

q,-^

cPE

Rqr

qi - cp

T

s

(26)

(27)

(28)

Some results are shown in Figures 1, 2 and 3. In Figure 1, burning rates m ~ rs Xs,

evapora-tion constants k ~ m/rs ~ Xs and burning

rates per unit surface m/rs ^ Xs/rs are plotted in function of the cliiiiensionless droplet radius for several values of the temperature at infinity. It may be observed that m/ra is not Gonstant,

0 0.05 0.1 0.15 DROPLET RADIUS 10"5'V^M rs

FIG. 2. Combustion of nionopropellant droplets within an inert atmosphere. Temperatures, flame: droplet radius ratio and reaction zone thickness.

law of variation of the square of the droplet radius as a function of the time does not alwaj^s hold for monopropellants.

For large values of rs, the burning rate per

unit surface tends toward a constant value, and when rs tends toward zero the limiting value

of Xs is given by:

( X . )

r

^ 0 = l o

g

p (29)

OS

which corresponds to the case of evaporation of droplets in the absence of combustion.

COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS IN DROPLETS 1041

toon 100

- * - -1 8o =0.0557

100 800

- * - -1 8o =0.0557 -80

800

- * - -1 8o =0.0557

600 - * - -1 8o =0.0557 60

600 400 300 200 100 -40 400 300 200 100 30 20 10 400 300 200 100

\©(\

s^

30 20 10 400 300 200 100

^ >

_K*

^

c

S w c o 30 20 10 400 300 200 100 30 20 10

80 - 8

80

60 fc 6

60

II 6

i.0

30

20

10

- x -. 4

i.0

30

20

10

x

-3 . 2 i.0 30 20 10

£

3 . 2 i.0 30 20 10 in i o 1 i.0 30 20

10

_i

D X w f c _o u - X

-E|a H

1 8

_i

_D

X w f c _o u - X

-E|a H

•0.8 8

_i

_D

X w f c _o u - X

-E|a H

•0.8 6

i

D X w f c _o u - X

-E|a H

0.6 6

i

D X w f c _o u - X -E|a H 0.6 lt

i

D X w f c _o u - X -E|a H 0.1 • 03 3 2 1

i

D X w f c _o u - X

-E|a

H

i?

. -\ \.eU N

0.1 • 03

3

2

1

i

D X w f c _o u - X

-E|a

H

i?

QJH—<

^ c ; * * • 0 2

3

2

1

i

D X w f c _o u - X -E|a H (>•<* ^ c - 01

08

friv

08

friv

t*_(

SQ =19.2; - 0.08

0.6 i^^> 1U ^ c T , *S* SQ =19.2;

0.6

SfVMs*5 0.06

0M 03

0.2

n 1

" \ 0 0.06

0M 03

0.2

n 1

0.03

- 0 02

0M 03

0.2

n 1

0.03

- 0 02

0M 03

0.2

n 1 _{0.05 0.10} . nm

DROPLET RADIUS l O- 5' ^

0.15

FIG. 3. Combustion of monopropellant droplets within an inert atmosphere. Influence of parameter

6a , dimension!ess ratio of the activation energy to the heat of reaction.

represented in function of the droplet radius. For large droplets, the maximum temperature #** is close to the equilibrium temperature for adiabatic combustion and the flame: droplet radius ratio tends toward unity. When the drop-let radius tends toward zero 0* and 0** tend to-ward the temperature of the surrounding at-mosphere, and the flame: droplet radius ratio tends toward mfmity, which also correspond to the case of evaporation without combustion.

Figure 3 shows the fundamental influence of parameter 8a (dimensionless ratio of the

activa-tion energy to the heat of reacactiva-tion). From this figure and from the work cited in reference 13 it may be seen that, for an average sized droplet,

when 8a is small combustion is rapid, the flame

is close to the droplet surface, and the maximum temperature is almost equal to the equilibrium temperature for adiabatic combustion. On the contrary, when 8a is large, combustion is slow,

the flame is far from the droplet surface and the maximum temperature is close to the tempera-ture of the surrounding atmosphere. However, when the droplet radius tends toward zero, the limiting value of the burning rate does not de-pend on da , because it tends toward the limiting

value given bj^ Equation (29).

-£SLZ_1

09 _{9oo =}

09 _{9oo =}

= 0.9

^ e *

Qm = = 0.8

0.8 0.8

< 2

ct

o

Q.

I

1

<a«

r ^

"* /

K

DROPLET RADIUS

0:01 ' 0.02 ' ' ao3 0.04 FIG. 4. Combustion of hydrazine droplets within an inert atmosphere. Theoretical results

reaction rate obtained by applying the steady- and the chemical kinetics of the process will be state assumption for the radicals to the de- approximated by means of the over-all reaction composition reaction model for hydrazine pro- rate:

posed by Adams and Stock. 2M% \ . e a ° , v

Mi Mi cp [d — do)2

BIPBOPELLANTS . , . , „ , a . u T? A- AT7\

m which 6a and 60 are given hj liquations (27)

The following over-all reaction will be con- and (28) and As is defined by sidered:

Ai + A2 - > 2AX (30)

COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS IN DROPLETS 1 0 4 3

Results obtained in function of the droplet radius showed that all solutions lie between two limiting cases:

When rs —> 0 results correspond to the case

of evaporation of droplets in the absence of combustion, as for the combustion of mono-propellant droplets. Therefore, Xs is given hy Equation (29), 6* tends toward 6W and r*/rs

tends toward infinity.

On the other hand, when rs tends toward

infinity, the limiting values are equal to those obtained by assuming an infinite reaction rate. The maximum temperature 0**, the eigenvalue

Xs, the radius ratio r*/rs, etc., tend toward

asymptotic values which can also be directly derived by taking an infinite reaction rate in systems (11), (12), (13), which implies that Equation (11) be disregarded and that the existence of a zero thickness flame may be as-sumed.15 For example, the asymptotic value for Xs is given by:

( Xs)r s^ c = lOg - — I — 1

U V e 3

" ~

i y

(33)

7T (e3°o 9°o)

Results will be shown for two representative cases:

1. Equal molecular weights and Lewis-Semenov numbers equal to unity

{Mx = M2 = ilf3 = 20;

£12 = £23 — £ 1 3 = 1 ) .

Therefore:

€lw ~ 0 ; €2M = 1 ; 6300 = = 2 ;

Cls = l j 62s = 63s = 0 .

2. Molecular weight M% of the gaseous re-actant surrounding the droplet much smaller than molecular weight M\ of the reactant from the droplet.

The following values have been taken:

Mi = 38; Mi = 2 ; M% = 2 0 ; £i 2 = £2s = 0.45; £13 = 1.18 eico = 0 ; e2M = - 0 . 0 5 2 ;

e3oo = 1.052; els = 1; e2s = e3s = 0 .

In order to compare results, the mean mole-cular weights of the mixtures have been taken to be equal for both cases, and the

Lewis-Semenov numbers £ y have been assumed to be proportional to

y/MiMi/iMi + Mj).

Figures 5, 6 and 7 show results for the first case. Because parameter AB is usually very large

(>101 0 cm-2) the results are generally close to their asymptotic values, provided that the droplets are not too small. This explains why the assumption of an infinite reaction rate leads to rather good results in many cases. However, if the value of parameter 6a is large (weak

reac-tions or high activation energies), the results may be very far from their limiting values, even for large droplets, as shown in Figure 7. 1

Figure 8 shows results for the second

well as a comparison of the values of Xs for both cases. I t may be observed that for this case the results tend much more slowly toward their asymptotic values, which means that the assumption of an infinite reaction rate may in-troduce important errors when M% is much smaller than Mi.

I t is doubtful that solutions close to the case of pure evaporations (rs —» 0) actually represent

real combustion processes, because the tempera-ture profiles are very flat and the reaction zones are very wide.15 Therefore, the results suggest the possibility that there ma3^ be a minimum size for the droplets below which combustion is not possible. If this is so, the minimum radius should be a very sensitive function of parameter

Influence of Pressure

Influence of pressure is practically disregarded when assuming an infinite value of the reaction rate. On the contrary, results obtained consider-ing chemical kinetics show that the influence of pressure may be very important for certain conditions.

Parameters AM and AB are proportional to pn, where n is the order of the over-all chemical

reaction. Therefore, all results have been ob-tained as a function of the dimensionless product

pn'2 X rs (n = 1 for monopropellants and n = 2

for bipropellants). I t follows that for a droplet of given size results depend upon the pressure as they depended upon the droplet radius at constant pressure.

1.100

900

16-K - - 7 0 0 ^

12

I L » 10 •• 500

<

a. co 8 3 Q

<

a: _{6 • 300}

4 •

2- 100

0X

--gST"""

--gST""" •1.4

- i . J . i * .1 2W C D LU

/

1

-i MAXIM L PERAT U • 2.2 s m

o n s t a n t

• 2 c D 8 -1.8 * E|UW II c a in c o u X /E II II I/) -1.6 X c a in c o u X /E II

•<-i

-1.4 c a in c o u X /E II / <> 1 <J

/ \

t •1.2 X I -1 LU 1

-<

en a z a :

OQ _r*

rs

NG S0LU1 E REACT

riON OR ION RATE

M1= M2= M3 = 20

LIMITI INFINIT

NG S0LU1 E REACT

riON OR ION RATE

<

ea

-90 = . ^12 =

0.05

: -^23 = -C

')'2co- »

1 3 = 1 |

100 200 300 400 500 600 700 800

DROPLET RADIUS "V^g rs

FIG. 5. Combustion of fuel or oxidizer droplets within the vapors of an oxidizer or fuel. Results for finite reaction rates and limiting solutions for an infinite reaction rate.

to their asymptotic values even for small values of product pn'2 X rs . Therefore, for such

condi-tions the influence of pressure might be un-noticeable, unless very small droplets were studied (Fig. 9).

combus-COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS IN DROPLETS

Y2co = 1 o

LL)

1045

1.4

1.3

a. 1.2

Y2oo = 0.75

___—• '

3 a > Z ^ _ _

^ ^

\^X-^^

___^,^--«==rrircrr

Y2oo = ^ _ _ _ _

-^C^o-

5

15

r- 13

11

2 9

Q

<

oc 7

Y2 c o =0-5 /

/ Y, = 0.75

/ /CD

V>»=

1

-,—

200

DROPLET RADIUS \RB rs

FIG. 6. Combustion of fuel or oxidizer droplets. Influence of F2oo mass fraction of the oxidizer or fuel at great distance from the droplet.

tion process, especially when the gaseous com-ponent surrounding the droplet has a low molecular weight.

For a droplet of constant size, as pressure decreases burning rate, evaporation constant and maximum temperature also decrease, while ratio

r*/rs and flame thickness increase until a

mini-mum value of the pressure is reached under which combustion seems no longer possible.

Experimental Results

MONOPKOPELLANTS

COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS IN DROPLETS 1 0 4 7

600

500-400

£ II xw300

1-<

a.

200

100

0l Ek"

200 300

DROPLET RADIUS V&Q rs

FIG. 8. Combustion of fuel or oxidizer droplets within the vapors of an oxidizer or fuel. Results ob-tained when the gaseous component surrounding the droplet has a low molecular weight.

observe combustion of droplets for different propellant combinations at variable pressure

(from TV up to 10 atmos). The facility was also suitable for studying combustion of oxidizer droplets in a hydrogen atmosphere. Hydrogen was selected because of its low molecular weight, which prevents natural convection effects and makes the influence of chemical kinetics more noticeable.

The research facility is shown in Figure 13.

Droplets suspended from a quartz fiber are in-troduced into a glass tube filled with C 02, closing simultaneously the tube inlet. The tube is then moved downward by means of an electro-magnetic actuator, and the droplet is ignited by two electrodes which are placed in position by the motion of the tube.

2.2

2.0

1.8

1.6

c 1.4

1.2

1.0

0.8

0.6

0.4

0.2

•S /

///

^

^

/ ^

9/ / / /

// /

<*><?/ 1

t^

t>

1

\

I \

\

\

\

\ \

*>

I

\

\ \

\ \

\

" " " • • • — .

1 1 / / / - — - ~ ~

0

- — - ~ ~

0oo= 0.5

e

o o = 0 . 4

ea = 10 0O = 0.05 * 2 . =

-M2 = 1 M2 = M:

1 = 2 0

J3 = 1 100 200 300 400 500

DROPLET RADIUS V ^ B rs

600 700 800

PIG. 9. Combustion of droplets. Eigenvalue Xs in function of the droplet radius. When 6m is small

there exist three possible values of Xs , for each value of rs , which correspond to three mathematical

solutions of the process.

kinetics of the hydrogen-bromine reaction is known.

I t was verified that for droplets with an initial radius rsi of about 0.5 mm, combustion did not

occur up to pressures of the order of 4 atmos. Slopes of curves r] = f(t) were measured by means of a kinematographic camera, but the flame radius could not be measured because the flame was hardly visible.

By applying the steady-state assumption for the radicals to the hydrogen-bromine reaction model proposed by Campbell and Hirschfelder,6 an over-all reaction rate of f order was obtained, which was utilized to study the combustion of bromine droplets in hydrogen by applying the analytical method previously described. In

COMBUSTION OF LIQUID MONOPROPELLANTS AND B1PROPELLANTS IN DROPLETS 1049

Manometer

Monopropellant Droplet

_. Hypodermic needle 5 jana quartz fiber

-Electric furnace Quartz window

77^7777777777777777 \ - \ Time indicator

Thermocouple \ Temperature control

777777777777^7

D C Electric motors

FIG. 10. Schematic diagram of apparatus used to observe combustion of monopropellant droplets within a mixture of oxygen and nitrogen.

200 400 600

FURNACE TEMPERATURE T °C

FIG. 11. Combustion of hydrazine droplets. Experimental restilts

t ~ 0 t — xV sec t = xS6 sec t = Tb sec t = T46 sec

FIG. 12. Hydrazine droplet burning in a mixture of 90% N2 and 10% 02 Furnace temperature Ta

650°C.

FRONTAL V I E W

Electromagnetic Protection actuator

FIG. 13. Schematic diagram of apparatus used to observe combustion of oxidizer droplets within hydrogen atmosphere.

Combustion of bromine droplets was very fast (a droplet of 1 mm in diameter burns in TV of a second), which produced an important scatter-ing of results. This prevented the formulation of any law of variation of the evaporation constants as a function of pressure However, the experi-mental and theoretical values were of the same order of magnitude.

Combustion of nitric acid droplets in hydrogen was also observed. No precise data were avail-able on the combustion mechanism of this

reac-tion. I t was not possible, therefore, to compare theoretical and experimental results.

Photographs of a burning droplet are shown in Figure 15, in which it may be observed that the flame is spherical. This allows a precise measurement of the flame diameter, assumed to be equal to the illuminated zone of the photo-graphs. Some experimental results are shown in Figure 16. Curves rl = f(t) were approximated

hy means of straight lines, and the resulting

COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS IN DROPLETS

COMBUSTION OF B R O M I N E D R O P L E T S IN H Y D R O G E N

T h e o r e t i c a l and E x p e r i m e n t a l R e s u l t s

F

A | I

18

A _EXf 5E R I M E N

S U I T S ( TAL 18

RES

5E R I M E N

S U I T S ( A )

16

U

12

z _ .

p

z

£

I

-- > — o & CD CD , , :

° ~ 1 c

I z £

- O LO

5 —

16

U

12

z _ .

p

z

£

I

-- > — o & CD CD , , :

° ~ 1 c

I z £

- O LO

5 — i u 16 U 12 o

z

1.

I z £

- O LO

5 —

ASYMPTOTIC VALUE FOR p — • 00 •»»

16

U

12

1

E u 16 U 12

1

o

10

i

THEORETICAL VALUES

10

i

/

E c Li"

CM

10

1

j

r

II

8

1

•a T3

8

1

i

II ₆

i

_E

-£

x: 6

_i

- E -_£

4 2

1

i

4 2

i

i

4 2

i

i

4 2

i

i

5 7 9 11 13

PRESSUREJ A T M O S P H E R E S

15

FIG. 14. Combustion of bromine droplets in hydrogen. Theoretical and experimental results

Results for the flame: droplet radius ratio r*:rs

show that its value is not constant but increases as the droplet radius decreases in a manner similar to that predicted theoretically.

Combustion in air at low pressure of several types of fuel droplets were observed also. In Figure 17 photographs of n-heptane droplets burning in air at low pressure are shown. As the pressure is reduced, the flame becomes larger and darker and its shape is almost spherical, which means that the combustion temperature

is lower according to the results calculated theoretically.

In Figures 18 and 19 results obtained for the evaporation constants and for the flame: droplet radius ratio are shown. The form of functions

k = f(p), k = f(p°-%),e r*:r, = /(rs) and r*:rs = f(p°-6rs) are in excellent qualitative

agreement with the general results obtained

t ~ 0 t = Yt sec t — x2 sec t

FIG. 15. Nitric acid droplet burning in hydrogen. Natural convection effects do not exist. Magnifi-cation ratio 5/1.

85 °/o NITRIC ACID

98 % NITRIC ACID

k= 5.9x10 -3 cm sec -3 c m^ 1.2

4

1

-°

g

°-

9

<

g 0-8 CO

co 0.7

5 0.6

0.5

0.4

1.5

^ ^

> c % ^ x

^ ^

* \ . ""-^

v .

^ ^

^

^ ^

^ \

^ * v *

^

u

1.3

1.2

1.1

1.0

0.25 TIME

0.50 SEC.

0.75

• _^ •

0.6 0.7 0.8 0.9 DROPLET RADIUS r

1.0 1.1

F I G . 16. Combustion of nitric acid droplets within hydrogen. Experimental results.

theoretically for t h e combustion of bipropellant droplets.

Finallj'', in Figure 20, m i n i m u m values of t h e

droplet radius for combustion rs0 are shown as a

function of t h e pressure for n - h e p t a n e a n d e t h y l alcohol droplets. Such radius rs 0 represents t h e m i n i m u m values for which ignition could n o t be achieved b y means of electric s p a r k s , a n d some-times t h e extinction values of t h e radius after a short combustion of t h e t y p e shown in Figure 17b.

T h e influence of t h e ignition mechanism on t h e process a n d t h e possible influence of t h e stability of t h e flame on t h e extinction process p r e v e n t e d t h e formulation of airy precise law of variation of t h e extinction radius as a function of pres-sure. H o w e v e r a p p r o x i m a t e d results m a y be o b t a i n e d from t h i s figure.

Nomenclature

AM , AB constants of the reaction r a t e equations

for monopropellants and bipropel-l a n t s , respectivebipropel-ly

B frequencjf factor

cp specific h e a t at constant pressure

cp average value of cp for t h e mixture

Di, diffusion coefficient E activation energy hi specific e n t h a l p y k evaporation constant m burning r a t e

Mt molecular weight

p pressure

qi l a t e n t h e a t of evaporation

qr h e a t of reaction

r radius

COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS IN DROPLETS 1 0 5 3

Left: p = 1 atmos Center: p « f atmos Right: p « f atmos

F I G . 17a. Combustion in air of w-lieptane droplets of equal size at different pressures. Magnification ratio 4 : 5 / 1 .

Center: t = \ sec Left: t « 0

F I G . 17b. Combustion in air of a n-heptane droplet at J of atmosphere. Magnification ratio 4 : 5 / 1 .

Right: t = \ sec

(Extinction of t h e flame)

R

t T

Wi

X

Yi

X

A

£,•/

gas constant time

absolute t e m p e r a t u r e reaction r a t e

= cpm/4n1\r, dimensionless coordinate

= cpm/4:ir\rs , eigenvalue of t h e system,

proportional to t h e evaporation con-s t a n t

mass fraction

ratio of flux of mass of species i to t o t a l mass flow

thermal conductivity

average value of X for t h e mixture = %/pDijCp , Lewis-Semenov number stoichiometric coefficients of chemical

reaction gas density

dimensionless t e m p e r a t u r e constant

reaction zone thickness at t h e X co-ordinate system

actual reaction zone thickness

SUBSCRIPTS

1

3

I, II

r e a c t a n t species

gaseous r e a c t a n t from t h e droplet (fuel, oxidizer or monopropellant vapors) gaseous r e a c t a n t surrounding t h e droplet

(fuel or oxidizer) reaction products

limiting points of the assumed finite reaction zone

0.002

0.0015

N - H E P T A N E

EVAPORATION CONSTANT k = - - _ r - ~ - 7L IN FUNCTION OF PRESSURE

E 0.001

0.0005

a» _____

A • «

/ a

a / a /

/*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p ( A t m )

N - H E P T A N E

d rs2 m

EVAPORATION CONSTANT k = - — ~ — - IN FUNCTION OF PRODUCT p

Q002

0 6x 7S ' 5

p Q002

Q0015

0

/ • • •

«

a

»

Q0015

•

f •

>

/•

• 0.001 • 0.001

/ 9

c

•|

0

0.1 0.2 0.3 0.4 p0 6x rs ( A t m0 6x m m )

0.5

COMBUSTION OF LIQUID MONOPROPELLANTS AND BIPROPELLANTS IN DROPLETS 1 0 5 5

N-HEPTANE

F L A M E / DROPLET RADIUS RATIO IN FUNCTION OF DROPLET

RADIUS

.. 7

06

rs

0 7 ( mm

)

p

=

0 928 A t m p

=

0 692

_"

p

₌

0 508

_"

p

=

0 399

_"

p

=

0 336

N - H E P T A N E

F L A M E / DROPLET RADIUS RATIO IN FUNCTION OF THE

PRODUCT p°<sx rs

p0 6x rs( A t m ° 6 , mm)

MINIMUM VALUES OF DROPLET RADIUS FOR COMBUSTION IN FUNCTION OF P R E S S U R E

ETHYL ALCOHOL N - H E P T A N E

N- HEPTANE I o Burned Droplet

o Not Burned Droplet

I ETHYL ALCOHOL

+ Burned Droplet I x Not Burned Droplet

0.3 0.4 p (Atm)

F I G . 20. Combustion of n - h e p t a n e and ethyl alcohol droplets in air. E x p e r i m e n t a l values of t h e mini-m u mini-m droplet radius for comini-mbustion.

SUPERSCRIPTS

* center of t h e reaction zone (point a t which dti/clX is a maximum) ** point of maximum value of t h e t e m

-p e r a t u r e

' derivative with respect t o X 0 s t a n d a r d s t a t e

E E F E R E N C E S

1. GODSAVE, G. A. E . : Fourth Symposium

{In-ternational) on Combustion, T h e Williams

& Wilkins Company, Baltimore, 1953.

2. G O L D S M I T H , M . , AND P E N N E R , S. S.:

Cali-fornia I n s t i t u t e of Technology, 1953.

3. H A L L , A. R . , AND D I E D E R I C H S E N , J . : Fourth

Symposium {International) on Combustion,

T h e Williams & Wilkins Company, Balti-more, 1953.

4. W I S E , H . , L O R E L L , J . , AND W O O D , B . J . :

Fifth Symposium {International) on Com-bustion, Reinhold Publishing Corporation,

New York, 1955.

5. L O R E L L , J . , W I S E , H . , AND C A R R , R . S.:

Proceedings of t h e Gas Dynamics Sympo-sium on Aerothermochemistry, 1956. N o r t h w e s t e r n University, E v a n s t o n 111.

6. B A R R E R E , M . , AND M O U T E T , H . : E t u d e

ex-perimentale de la Combustion de Gouttes de Monergol. La Recherche A e r o n a u t i q u e , No. 50, 1956.

7. AGOSTON, A. G., W I S E , H . , AND R O S S E R , W.

A.: Sixth Symposium {International) on

Combustion, p . 708. Reinhold Publishing

Corporation, New York, 1957.

8. SANCHEZ T A R I P A , C , AND SALAS LARRAZABAL,

J. M . : I N T A , A R D C C o n t r a c t N o . A F 61(514)-997, Madrid, 1957.

9. B O L T , J . A., AND SAAD, M . A . : Sixth

Sympo-sium {International) on Combustion, p . 717.

Reinhold Publishing Coi'poration, New York, 1957.

10. K U M A G A I , S., AND ISODA, H . : Sixth Symposium

{International) on Combustion, p . 726.

Reinhold Publishing Corporation, New York, 1957.

11. ISODA, H . , AND K U M A G A I , S.: Seventh

Symposium {International) on Combustion, B u t t e r

-w o r t h a n d Company, L t d . , London, 1958. 12. W I L L I A M S , F . : California I n s t i t u t e of

Tech-nology, T R N o . 21, 1958.

13. SANCHEZ T A R I P A , C , AND P E R E Z D E L N O T A R I O ,

P . : I N T A AFOSR D o c . N o . T N 58-1038, M a d r i d , 1958.

14. P E R E Z DEL N O T A R I O , P . , AND SANCHEZ T A R I F A ,

C : I N T A A R D C C o n t r a c t N o . A F 61 (514)-997, M a d r i d , 1959.

15. SANCHEZ T A R I P A , C , AND P E R E Z DEL N O T A R I O ,