I
C. C. MIESSE Armour Research Foundation, Chicago, 111.
A Theory of Spray Combustion A method is presented for determining the optimum design of
a combustor with
E x i s t i n g information on the effect of atomization of performance of liquid propellant combustors (7) has been limited to the beneficial effects of finer atomization (7, 6, 77, 72) and the detrimental effects of liquid viscosity (2, 70). T h e roles of the intermediate processes of evaporation and chemical conversion were also investigated (4, 5, 73). The applicability of these experimental data to this problem was given previously (8), in which the present theory was\ outlined. The dimensionless groups which include the above effects and should facilitate correlation of experimental data were established by Damkohler (3), and were subsequently applied to the problem of scaling combustors by Penner ( 9 ) . Analysis The present theory is based on three assumptions regarding the chemical kinetics of the system: The propellants react only in the vapor phase and as a single chemical substance; the system is isothermal; and the propellant vapors mix immediately upon formation. Thus the law of mass action is expressed as
where dw is incremental per cent of propellant converted in time; dt, 7 is relative reaction rate or "reaction frequency" ( 9 ) in fraction of propellant converted per second; and W is the per cent of unreacted propellant available. Further assumption of a uniform average velocity permits the expression of time in terms of distance
given
performance requirements
where x is distance downstream from the injector, U is average axial velocity, and L is length of the chamber (or portion thereof). Configuration (Figure 1) denotes penetration distance of droplets by L I ,length of vapor section as La, and total chamber length as LT. Atomized Propellants. For those heterogeneous portions of the combustion chamber, in which atomized propellants are not completely vaporized, the propellant vapor increases because of evaporation, and decreases because of chemical reaction. Then, the propellant vapor, W , at any axial position becomes
w = 1 - $12 - w
(3)
where 5 is the square of the d i m ~ t e r ratio of the droplet (DID#, and DO is the original diameter of injected droplet. As previous experimental evidence has shown that s decreases linearly with time, the Percentage reacted at the end Of the atomizing section, for a first Order reaction (' = '1 becomes w1 = 1
-3 26
(*
2z/b
erf
-
+ ( b / u ) (du/ds)
where b = 7D02/h (assumed to be constant to facilitate solution) is identified as Damkohler's (3) second similarity group, representing the ratio of reaction rate to rate of supply. Variation of performance within the atomizing section can be determined
(6)
where u =
4;[ I ~ / ~ ( K P+/ PB) L a / s ( K S V ~ )(]7 )
and 1216 and 1--2/5are modified Bessel functions of the positive and negative 2 / 5 order, respectively. The arbitrary constant, B, is determined by the boundary condition that dv/ds = for = Vaporized Propellants. For i hose homogeneous portions Of the combustion chamber in which the atomized liquid propellants are completely vaporized, the available propellant can be represented by the equation
W = l - w and the equation for
w
=
1
(8)
becomes
- (1 - wl)e-azfor
n = 1
(9)
and
-
+ l]-U*-u for n = 1
(4)
LT
w = 1
w = 1 - (1 w,) [a(l - w , ) ( n - l)z
& - e--b
Figure 1. This idealized thrust chamber configuration depicts parameters used in analysis
For a second order reaction ( n = 2 ) , the equation for w becomes
(10)
where a =
rZLz/Uz = rzTg
(11)
z is measured from the end of the atomizing section; 72, LP,and U Zare the reaction frequency, length, and average gas velocity, respectively, in the vapor section; and To is the gas residence time in the vaporizing section. The parameter a can be identified as first similarity group (3),and is the ratio of reaction rate to rate of discharge of gases. Representative variations of performance with distance for seyeral values of parameters a and b , determined by Equations 5 and 9, show that a rapid rise in performance occurs for the homogeneous gas mixture case, and that the rate of increase grows with increasing values of a, i, e for more rapid propellant conversion. The heterogeneous case exhibits much slower increases in performance, as expected. These theoretical curves agree well with the experimental data reported by Ross ( 7 7).
Applications
For a first order reaction, substitution of Equation 4 in Equation 9 yields the VOL. 50, NO. 9
SEPTEMBER 1958
1303
4 Figure 2. Variation of performance with length of vapor section i s shown for variousw values of atomi- f: zation parameter e L W &
b Figure 3. Variation of performance with drop size is shown for various values of total chamber length
k and the appropriate value for c representing the length of the chamber. A general decrease in performance is also noted as the drop size increases. Example. The effect of chamber length on combustion efficiency is illustrated by a typical example, for which the conditions are assumed to be; = 200
rz
=
Do2 Eu4poRATION
USTANCE
FROM
INJECTOR, FT.
Figure 4. Theoretical variation of performance with distance from injector, for typical first-order reaction, is determined by Equations 4 and 9
following equation for performance
(12) where w z is the value of the propellant conversion a t the end of the vaporized section. Figure 2 shows the variation of performance with length of vapor Section, for various values of the atomization parameter b. The variation of performance with total length of chamber L T can be determined directly by plotting performance us. the parameter
which indicates the relation between the parameters a and b. Variation of performance with drop size, for various values of the parameters a, c, and k, can be determined by crossplotting the data obtained from Figure 2. Figure 3 shows this variation for k = ,O, 1, and 2. Variations for any value ,of k can be obtained by connecting the points (a, b ) which are related by ,Equation 13, with the given value of
304
U1 = 100 f.p.s. U Z = 400 f.p.s. X = 0.033 sq. cm./sec. 1.67 X 10-4sq. cm.
sec.-l 600 set.-'
rl
=
The theoretical variation of performance with distance from the injector as determined by Equations 4 and 9 is shown in Figure 4, which also shows the corresponding performance curve for completely vaporized propellants. From these curves, the vaporized system predicts an increase in efficiency of about ZOY0over the atomized propellants system for chamber lengths equal to the spray penetration distance; however, the difference in performance of the two systems be comes negligible for chamber lengths greater than twice the penetration distance. The variation of performance with drop size for a combustor with a value of 2 feet for L Z is determined by using the conditions of the previous example, and varying Do from to IOF3 sq. cm. The results (Figure 5) indicate that the performance w z decreases gradually with D Ountil the penetration distance exceeds the chamber length, at which time the performance decreases rapidly due to the loss of unvaporized liquid propellant.
Dozi
IO',
CM'
Figure 5. This shows variation of performance with droplet diameter, for a typical combustor of given dimensions
(4) Foster, H. H., Ingebo, R. D., Natl. Advisory Comm. Aeronaut. RM E55K02, 1956. (5) Ingebo, R . D., Ibid., TN 3265, 1954. (6) Koenig, R. J., Dandois, M., Ibid., RM 7L17. 1948. (7) Midsse, C. C., Advances in Chemistry Ser. No. 20, 243 (1958). (8) Miesse, C. C., Sixth International Symposium on Combustion, p. 732, Reinhold, Kew York, 1957. 191 Penner. S. S.. Combustion Researches and Reviews (AGARD), p. 140, Butterworths Scientific Publications, London, 1955. (10) Ricci, R. R., Aeronautical Engine Laboratory Rept. AEL-1282, Naval Air Material Center, Philadelphia, Pa. (11) Ross, C. C., Scaling of Liquid Fuel Rocket Combustion Chambers, AGARD Symposium on Combustion, Liege, Belgium, 1955. (12) Sharp, J. G., Aircraft Eng. 23, 2 (1 \ -951) .--,
literature Cited (1) Bellman, D. R., Humphrey, J. C., Male, T., Natl. Advisory Comm. Aeronaut. Rept. 1134, 1953. (2) Bransford, C. K., Horstman, W. W., Wood River Research Laboratory Rept. 1312, Shell Oil Co., Inc., May 28, 1948. (3) Damkohler, G., Z. Elektrochem. 42, 846 (1936).
INDUSTRIAL AND ENGINEERING CHEMISTRY
(13) Trent, C. H., IND.ENG. CHEM.48, 749 (1956). RECEIVED for review September 18, 195' ACCEPTED May 5, 1958 Presented before American Rocket Society, San Francisco, Calif., June 1957. Work begun at Aerojet-General Corp. and sponsored by Wright Air Development Center, TJ. S. Air Force.
