Burning of a Liquid Droplet - ACS Publications

Burning of a Liquid Droplet HENRY WISE and GEORGE A. AGOSTON

Downloaded by UNIV ILLINOIS URBANA on May 6, 2013 | http://pubs.acs.org Publication Date: January 1, 1958 | doi: 10.1021/ba-1958-0020.ch010

Department of Chemical Physics, Stanford Research Institute, Menlo Park, Calif.

The wide use of spray injection for the introduction of reactants into combustion chambers has pointed to the need for an analysis of the processes which govern combustion of liquid aerosols. This review presents the theoretical and experimental aspects involved in the burning of a single droplet. The application of the results obtained for a single droplet to the burning characteristics of liquid sprays remains a problem of fundamental importance in combustion research.

In general, two cases of droplet combustion are of interest. A heterogeneous monopropellant flame involves a single reactant system i n which a single component evaporates from the liquid surface and then decomposes exothermically at a rate which is a function of temperature and composition. O n the other hand, a bipropellant flame requires the interaction of two reactants, fuel and oxidizer, one of which evaporates from the droplet surface and diffuses into the gas phase containing the other reactant. As a result of the interdiffusion of the reactants, a flame is established at some distance from the droplet. Experimental measurements have shown that the mechanism of droplet vaporization at high heat flux, such as prevails during combustion, is different from that observed at low temperature gradients near the liquid surface. Under the latter conditions the evapora tion rate is governed b y diffusional processes (4, 5), whereas at high temperatures the rate of heat transfer from the flame to the liquid has been found to play a dominant role. Based on a mechanism of heat conduction through a stagnant film between concentric spherical shells, a useful approximation of the burning process may be devised. Thus i t can be shown that the mass burning rate is proportional to the droplet radius and i n versely proportional to the heat of vaporization. I t is apparent, however, that such a model neglects the mass-diffusional process which occurs during the combustion of a liquid droplet and thereby modifies the rate of energy transfer. T h e approximate derivation also implies knowledge of the location of combustion radius with respect to the liquid surface. Such considerations stimulated a new approach to the problem of droplet combustion during the past few years (7-9, 11,17,20, 23, 25, 37, 42, 44, 46, 51, 52, 54, 59, 60, 62).

Theoretical Analysis Heat Transfer by Conduction. I n the theoretical analysis of steady state, hetero geneous combustion as encountered i n the burning of a liquid droplet, a spherically symmetric model is employed with a finite cold boundary as a flame holder corresponding to the liquid vapor interface. T o permit a detailed analysis of the combustion process the following assumptions are made i n the definition of the mathematical model : The droplet and flame surfaces are represented b y concentric spheres. Combustion occurs under isobaric, steady-state conditions. 116

In LITERATURE OF THE COMBUSTION OF PETROLEUM; Advances in Chemistry; American Chemical Society: Washington, DC, 1958.

117

WISE AND AGOSTON—BURNING OF LIQUID DROPLET


The reactants yield product gases by exothermic reaction at a rate governed by the chemical kinetics. The source of one reactant is a liquid surface at its boiling point, and that of the other is located in the gas phase at an infinite distance from the liquid sphere where the weight fraction of gaseous reactant and temperature are specified. The heat released by chemical reaction is removed by thermal conduction and mass transport—i.e., radiation and thermal diffusion are neglected. The effect of radiation has been considered in some detail (23). The transport parameters and the specific heat of the system are considered to be independent of temperature and composition. Average values are used for these parameters and the ratio of Schmidt to Prandtl number is taken as unity throughout the system.

Figure 1.

Model of burning droplet

Dynamic effects on the droplet and interaction between droplets are neglected. As shown schematically i n Figure 1, the gases flow from a cold boundary η outward towards infinity r i n a coordinate system fixed with respect to the cold boundary. A steady state is established in which fuel vaporizes from the liquid surface and flows toward the flame zone while oxidizer diffuses against the flow of products into the flame front. Because the bipropellant system involves the material transfer of two reactants interdiffusing from opposite directions, the exothermic chemical reaction is limited to a region i n which both reactants are present. Although a small w eight fraction of the oxidizer diffuses to the liquid surface, where the weight fraction of fuel is high, the extent of chemical reaction i n this region is very small because of the low temperatures prevailing near the liquid surface. The transport of matter and enthalpy i n such a system is given by the equations for conservation of mass and energy; conservation of momentum is implied in the assumption of an isobaric process. In generalized form (86) these equa tions may be written as 0

r

m, = Kfj + mzj = Apv(Yj

-

DjYi'/v)

j = 1, 2, 3, . . . η

η

i-i

riijhj — A\t' — constant


(1)

(2)

118

ADVANCES IN CHEMISTRY SERIES

f

*

I

Y

Z

(SUBSCRIPT

0.06

1.2

0.05

1.0

1

1

i) 50,000 cal. 25,000 cal. 0 cal.

(SUBSCRIPT 2) (SUB SCRIPT S)

^\

0.32


/ 0.04

0.8

0.03

0.6

0.02

0.4

0.01

0.2

S*

-

f

0.31

/! /

\

0.128

Figure 2.

0.130

0.132

0.134 0.136 POSITION

0.138 0.140 r (cm)

0142

0.144

0.30 0146

Flame structure surrounding fuel droplet burning in oxygen at various activation energies

where i ¥ „ the mass flow rate of species j at the liquid surface, represents a constant characteristic of the system of equations and the boundary conditions. T h e notation used is given i n the section on nomenclature (the prime notation refers to time deriva tives and the dashed quantities are space derivatives). T h e parameter Zj represents the contribution b y chemical reaction to component j ; thus Zi -

f

(3)

(Ri/v) dr

F o r a four-component system composed of fuel, oxidizer, products, and inert gas, four conservation-of-mass equations describe the mass flow rate of each component. The mass flow rate of inert gas is zero because no " s i n k " exists within the system for this species of molecules. W i t h the appropriate boundary conditions (36), the solution of the resulting differential equation constituting a fourth-order system for the bipropellant, heterogeneous flame requires numerical integration (38). F r o m such a computation detailed information may be obtained concerning the temperature and composition profiles surrounding the burning droplet. A series of solutions for a representative fueloxidizer system is shown in Figure 2. The values of the parameters used in the numerical Table I.

Parameters for Fuel-Oxygen System [36, 38)°

C (cal./gram° K.) D (sq.cm./second) % L (cal./gram) a (cal./gram) λ (cal./cm.second E.) ρ (gram/ml.) 0

R

0.38 0.8 2.09 238 6694 1.2 X KTf 0.4 X H T

3

\O pY Yx u

F

exp.[-E/K(t+

TÙ]

Boundary Conditions ri S B 0.02 cm. Τι = 351.5° K . To = 291.5° K . Υχο =1.00 (pure oxygen) Averaged at 1500° K .


119

WISE AND AGOSTON—BURNING OF LIQUID DROPLET

integration are listed i n Table I. Of special interest is the effect of the reaction rate on the structure of the flame zone. A decrease i n the activation energy of the process is associated with a diminution in the flame-zone thickness (Table II) until the system attains the flame structure observed i n the limit for zero activation energy. A t the same time the maximum temperature i n the flame zone increases slightly; the highest tempera ture gradient occurs at the liquid surface. This heat sink absorbs energy for vaporization and heating of the liquid phase. Table II.

Thickness of Flame Front (38)


Activation Energy E, Kcal. 0 25 50

Flame Thickness, M m . 0 0.047 0.099

Under the conditions of zero activation energy a flame zone of infinitesimal thickness results at which the weight fractions of both fuel and oxidizer attain zero. The mass flow rates of oxidizer and fuel into this mathematical surface are taken to be of stoichio metric mixture ratio—i.e., mx/mF = —i. The system is thereby divided into two regions, denoted by / and χ (Figure 1) separated by a step function. I n the /-region the extent of chemical reaction is zero (z — 0); i n the z-region, reaction has gone to com pletion (z = 1). Such a simplified analysis of the combustion process which neglects the contribution of chemical kinetics has been applied i n the early theories of heterogeneous combustion (7, 17, 20, 23, 42, 51, 62). The analysis is of particular interest because i t gives rise to a system of first-order differential equations which may be integrated i n closed form. Thus explicit solutions may be obtained for the mass flow rate of fuel, the temperature of the combustion surface, the radius of the combustion surface, and the weight fraction of fuel at the liquid surface, as shown by Equations 4 through 7. M = irrnpD In (1 + B) 1 1 +

F-

Yn = 1 -

(5)

+

jjj = In (1 + B)/\n (l

(4)

1 +

^

+ -^p)

[ ( l + ^ p ) / d + B)

(6) (7)

The parameter Β is given by * - τ ( ψ

+

Burning of a Liquid Droplet - ACS Publications

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