Temperature distribution in a vaporizing droplet with internal heat generation

COMBUSTION A N D F L A M E 25, 79-84 (1975)

79

Temperature Distribution in a Vaporizing Droplet with Internal Heat Generation* MICHEL A. SAAD and GENE J. ANTONIDES University of Santa Clara, Santa Clara, California 95053

The time-dependent temperature distribution within a droplet with internal heat generation and surface vaporization has been studied using a finite-difference numerical method. The heat transfer equations were nondimensionalized for computer calculations, and solutions were obtained for cases when the initial temperature is equal to and different from the surface temperature. Variation of droplet radius due to vaporization was included in the assumed model, and radial temperature distributions, relative to the surface temperature, were calculated under the assumption of constant surface temperature. Representative results are given as temperature and radius histories for a wide range of dimensionless variables. The results are applicable when surface temperature changes with time provided that the rate of change is small compared to the initial temperature transient due to heat generation.

1. Introduction During combustion of hypergolic rocket propellants, unlike-pair injector streams impinge on each other forming droplets of propellant mixture. Attention is focused on the preignition period in which exothermic chemical reaction within the liquid droplet and vaporization at its surface take place. The preignition period may be considered to consist of three periods that overlap to varying degrees: a preheat period, a vaporizing period, and a short high-energy-release period [1]. When vaporization is dominant, the liquid surface temperature of the droplet stabilizes at the evaporative wet-bulb temperature. Most kinetic studies of droplet ignition assume that the temperature of the entire liquid droplet rises uniformly. However, in a reacting, vaporizing droplet, when the energy needed for vaporization comes from the droplet itself, there is a radial temperature distribution that depends on time, droplet size, material properties and initial conditions. A difference in temperature between the center and the surface of a small droplet may seem inconsequential. However, a small increase in tem*Research sponsored by Air Force Office of Scientific Research under Grant No. AFOSR-68-1478C.

perature may produce a large increase in the rate of heat generation since most rates are highly temperature-dependent. In addition, a high temperature at the center of the droplet may cause vapor bubbles to originate inside the droplet due to the higher vapor pressure. These bubbles could have a significant effect on the subsequent behavior of the droplet. Ignition delay time of hypergolic propellants has been studied [2-5] both theoretically and experimentally. Expressions have been developed relating ignition delay time to ambient pressure, but no attempt was made to include the effect of temperature variation within the droplet in the analysis. Temperature gradients within monopropellant droplets were measured by Hall [6] using fine thermocouples. Heat conduction calculations substantiated the presence and the general reliability of the measured gradients in the droplets during the heat-up period. Faeth, Karhan, and Yanyecic [7] investigated the effects of the heating up of monopropellant droplets on combustion zone development, in contrast with most droplet burning studies, which deal with the steady-state burning period. Using a simple heat-up theory of ignition, an expression for the surface temperature was developed. They indicated that there was evidence of large tem-

Copyright © 1975 by The Combustion Institute Published by American Elsevier Publishing Company, Inc.

80

M.A. SAAD and G. J. ANTONIDES

perature gradients in the liquid phase especially at high pressure burning. A thorough search of the literature did not reveal an analytical or a numerical solution for the temperature distribution in a spherical volume with both internal heat generation and changing radius. Awbery [8] in 1927 developed an analytical solution for a similar problem, but without radius change. Carslaw and Jaeger [9] present solutions for a sphere with a surface temperature different from its initial temperature, and for a sphere with internal heat generation, but again without change in dimensions. However, the problem can be solved simply, if not exactly, by expressing the differential equations for heat flow and surface vaporization in finite-difference form and calculating the temperature and radius histories numerically. This method of solution is presented and results are given in the following sections. The Awbery solution for the radial temperature distribution is T=Ts +

G(a2-r2)

OO

(1)

a

where

p%

//

= k

p% Other terms will be defined later. This equation is reproduced here to suggest the impracticality of pursuing an analytical solution to the problem where surface vaporization is included. The equation, however, was used to check the results of the numerical method. The check was made by introducing an artificially high heat of vaporization in the numerical calculation so that the effects of

(2)

where k is the thermal conductivity, r is the radial position, T is the temperature, Q is the rate of heat generated per unit volume, O is the density, Cp is the specific heat, and t is the time.

2~x(Ti-vrr Ts)]

G=~Q~-, a

or

k

+ 1_ E Nue-V2rr2at/a2sin urrr,

Nu = (-1)v[v3rr3a2Ga3

2. Formulation of the Problem The assumed model consists of a spherical volume composed of an exothermically reacting liquid. The heat of the reaction is assumed to be generated uniformly within the sphere at a constant rate. It is also assumed that liquid circulation does not occur within the sphere so that heat flow is by conduction only, and that vaporization occurs only at the outside surface. The lack of internal circulation in small droplets was concluded by Hall [6]. A schematic of the model is shown in Fig. 1. The subscripts i and s will refer to the initial and surface conditions, respectively. As exothermic reaction occurs, the temperature rises non-uniformly depending on radial location. The heat flow equation, in spherical coordinates, relating the temperature at any point and at any time, is expressed by the following differential equation: r 2 ~r

6a

r P=l

vaporization and the resulting change on the temperature distribution were negligible.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Fig. 1. Analytical model..

TEMPERATURE IN A MONOPROPELLANT DROPLET The heat generated within the spherical volume supplies the heat required to vaporize material at the surface. With the assumption of a constant surface temperature, the vaporization rate is determined by equating the heat required to vaporize a thin layer at the surface with the heat conducted to the layer from interior liquid:

pX d a = k (~_T) dt or r=a

a ( l .l

.~a p

=C

c (r-

k(At)

-

(6)

0Cp(Ar)2 ' which must be ~< _1 if the iterations are to converge to a solution. 2 In terms of the nondimensional quantities, the Fourier modulus is expressed as 0=C

(At') .<. 1 .

(~Xr') =

-- -r'

-_=Ot - t '

ai

oX r'

0

(3)

where X is the latent heat of vaporization and a is the instantaneous outside radius. Six dimensionless groups were formed to reduce the number of parameters required for solving the problem. These are:

kX a7 CpQ

81

5

X

Finite differentials are substituted into the heat flow equation, giving an expression for u' in terms of its value and neighboring values obtained from the previous time interval: ' Ui,j+l

'

= u '.. + A t'. [C ui+ 1,1 t,,

1

-2u'..+u'...

(A~)t;L - ,-x_,J + r;,]]. (8)

- r ) = T: X

The temperature gradient at the surface, which is used in the vaporization calculation, is simply: t

p

(OT'] = Ua,/+l - Ua-l,/Z1 "0r' J/ a' Substitution of these new parameters and the new variable u' ---r' T' into the heat flow equation yields: C 02u~' + r ' - Ou' 0/2 Ot'

(4)

The vaporization equation in terms of the new parameters is

&' - c (_aT'~ dt' "Or' "r'=a' '

(7)

2

(5)

where the temperature variable is more simply expressed in terms of T' rather than u'. In the finite difference formulation, the independent variables r' and t' are each divided into a large number of equal intervals. Thus, the spherical volume was divided typically into 20 spherical shells each of thickness Ar', and time is advanced in discrete increments, A t', between calculations. Selection of these intervals is governed by the Fourier modulus:

(9)

Using these finite-difference equations, a computer program was written to calculate temperatures in a droplet as a function of time and radius. The Calculations proceed as follows. At each new t ~_ t . time (t) At]), the temperature of each spherical shell is calculated from Eq. (8). Then the temperature gradient at the surface is found using Eq. (9) and is substituted into Eq. (5) to determine the vaporization rate, da'/dt'. The radius change due to vaporization is (da'/dt') At'/, and the new radius is a)+1 = a) - (da'/dt') Aq. The new smaller sphere is then divided into the same number of, but smaller, increments, and a new time interval for the next calculation is calculated by applying Fourier's modulus with the new Ar'. The computer program advances one time interval and the procedure is repeated until the final time of interest is reached. The effect of the number of shells used was evaluated by comparing results at t' = 0.6 for 10, 20, 50 and 100 shells. There was no discernible difference in the center temperature calculated;

82

M.A. SAAD and G. J. ANTON1DES

however, the radius history was affected as shown in Fig. 2. Comparison of the values of radius a' at t' = 0.6 shows an error of less than 1% when 20 shells are used and somewhat over 1% when 10 shells are used. The time steps required to satisfy the Fourier modulus for any given shell size were small. Consequently, the time interval was never large enough to introduce a significant error. The number of shells had to be strictly limited in order to obtain numerical results in a reasonable time on the computer (HP 2100).

1.00

i

~ ~ . ~

Results Results can be obtained for any value o f the nondimensional quantity C = (kk)/(a~ c Q). For representative values of C equal to 0.01,6.1 and 1.0, Figs. 3 to 5 show the outside radius history and the droplet temperature histories for a set of radial

,o

,.,,

0.92

- i,

3.

I

INO OF SHELLS

TI

0.90--

-,

is

' Cp ( T i - T I )

~ ' ~ J ^

i

v

0.2

0.3

0,SB o

oJ

o.4

05

o.~

t' = O t ~pill

Fig. 2. Effect of number of shells on numerical results.

6.0

1.2

5.0

I.O o,=(3_ Oi

T-"

4.0

0.8

3.0

0.6

2.0

0.4

I

o I,-

1.0

.02

0.2

0.1

1.0

t'. Qt,'pk

I0

Fig. 3. Temperature histories (for r/a i = 0, 0.5, 0.8, 1.0) and radius history when C = .01. locations (r/a i = 0, 0.5, 0.8 and 1.0) when Tt. = Ts. Solutions can also be obtained by interpolating between two existing figures. To do this, first calculate the quantity C for the given problem, then using the two figures which bracket the C value of the problem, read, at the time t' of interest, the droplet radius a' and also the nondimensional tem-

perature difference ( T ' - T~) for the radial position(s) of interest. An interpolation between the values obtained from these two figures yields the required solution. Of course, a large number of figures, which include intermediate values of C, are needed to get more accurate results. Figure 6 gives the radius history and center

TEMPERATURE IN MONOPROPELLANT DROPLET

83

I.Z

r~

I~0

o i

0.8

0.6

m

I

i-

0.4

0.2

.02

.10

1.0

a,/pX

,',

I0

Fig. 4. Temperature histories (for r/a i = 0, 0.5,0.8, 1.0) and radius history when C = 0.1.

24

i

Kx .,.o ,fc, o

[ c.20

,

I

t

a,=O

1.2

i

1.0 a ' = _Q_ Qi

I

1

.16

=

.12

0,8

i-" I .08

0.4

.04

0.2

0

I

.02

'

0.1

,.

1.0

0

t'= Q t / o ) t

I0

Fig. 5. Temperature histories (for r/a i = 0, 0.5, 0.8, 1.0) and radius history when C = 1.0.

84

M.A. SAAD and G. J. ANTONIDES

6.0

'.2

5.0

.0 0 I.

0 i

4.0

).8

~o

).6

2.0

).4

1.0

).2

n J

"~

0

-I.0 .02

)

0.2 0.1

1.0

t ' - O t "¥^ /---

10

Fig. 6. Center temperature history (r/ai = 0) and radius history when C = .01 and T~- T's = -1, 0, 1. temperature difference history when C -- 0.01 for droplets whose initial temperature differs from its surface temperature. The curves are given for the following values of temperature difference: Cp ( T i - Ts)/k = (T~ = T~) = - 1 , 0 and 1. The subscript 0 in Fig. 6 refers to the center of the sphere. The determination of interior temperatures depends on knowing the surface temperature, since in this analysis only the difference in temperature between some point and the surface is obtained. The results of this paper are applicable when the surface temperature changes with time provided that the rate of change is small compared to the temperature transients due to heat generation. A straightforward extension of the numerical method can be made to account for temperaturedependent properties, including the rate of heat generation. Properties in the form of tables or temperature-dependent equations can be included in the computer program and new values determined at the start of each calculation step.

2.

3.

4.

5.

6.

7.

8. References 1. Sutton, R. D., Propellant Spray Combustion Processes during Stable and Unstable Liquid Rocket Combustion, First Phase Final Report, Rocketdyne Division,

9.

North American Rockwell Corp., AFOSR TR 702714, October 1970. Saad, M. A., and Goldwasser, S. R., Role of Pressure in Spontaneous Ignition, AIAA Z 7 (8) 1574-1581 (Aug. 1969). Saad, M. A., and Goldwasser, S. R., Time-Temperature Simulation in Low-Pressure Ignition of Hypergolic Liquids, AIAA J. 12 (1) 11-12 (Jan. 1974). Corbett, A., et al., Hypergolic Ignition at Reduced Pressures, TR AFRPL-TR-64-175, AD 610 144, 1964, Air Force Rocket Propulsion Lab., WrightPatterson Air Force Base, Ohio. Spengler, G., Lepie, A. H., and Bauer, J., Measurements of Ignition Delays of Hypergolic Liquid Rocket Propellants, 1964 Spring Meeting, Western States Section, The Combustion Institute, Stanford Univ., April 1964. Hall, A. R., Experimental Temperature Gradients in Burning Drops, Seventh Symposium (lnternationalJ on Combustion, Butterworth's Scientific Publications, London, 1958, pp. 399-406. • Faeth, G. M., Karhan, B. L., and Yanyecic, G. A., The Ignition and Combustion of Monopropellant Droplets, AIAA 3rd Propulsion Joint Specialist Conference, Washington, D.C., July 17-21, 1967, Paper No. 67-480. Awbery, J. H., The Flow of Heat in a Body Generating Heat, Philosophical Magazine, 629-638 (1927). Carslaw, H. S., and Jaeger, J. C., Conduction of Heat in Solids, Oxford University Press, London, 1947. Received 25 July 1974; revised 3 February 1975