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Combustor Performance with Instantaneous Mixing
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W. H. AWRY
AND
R. W. HART
Applied Physics Laborafory, The Johns Hopkins University, Silver Spring, Md.
URING the past 10 years a concentrated effort has been made a t several laboratories to obtain fundamental understanding and engineering design data on combustion under ramjet operating conditions. It has been shown that good performance of ram-jet power plants requires heat release a t a rate greatly exeeeding that available in conventional furnaces. I n addition, stable combustion must be maintained at net gas velocities of several hundred feet per second, and over a pressure range which may vary from a fraction of an atmosphere to 10 or more atmospheres. The complexities of the aerodynamic and combustion phenomena discouraged previous efforts to derive design data from an analysis of the physical and chemical behavior of the system. I n particular, the high level of turbulence in the burning zone prevented an assessment of the relative importance of mixing and chemical kinetics in determining combustion rates. Search for means of solving design problems in a fundamental manner led the authors to consider what behavior might be expected in a combustor in which mixing was perfect and instantaneous, but heat evolution was governed by the usual kinetic laws. The maximum space rate of heating of such a burner may be readily expressed in terms of the reaction kinetics and thermodynamic properties of the fuel. The concept also has implications with regard to burning stability and combustor geometry. A thorough examination of the consequences of the hypothesis has been carried out which has led to several basic conclusions about the principles underlying ram-jet combustor performance. I n the discussion in this paper it is assumed that a homogeneous mixture of air and fuel a t stoichiometric proportions enters the ram-jet combustor and that a stable source of ignition, or pilot flame, is provided a t the upstream end of the combustion chamber. It is assumed that the amount of material introduced a t any stage into the burning zone may be controlled as a function of distance downstream from the pilot, and that as soon as combustible mixture enters the zone a t any station i t is mixed instantaneously with all of the burning material a t the station. The maximum mass rate of burning with instantaneous mixing will not occur a t the adiabatic isobaric flame temperature of the reaction because the heat necessary for warming incoming fuel to the reaction temperature must be supplied from the heat content of the material in the reactor. At some temperature below the flame temperature a maximum mass rate of burning will exist, This is the result of the presumed exponential decrease in reaction rate with temperature, while the heat available for warming incoming gas increases only linearly with decrease of temperature below the flame temperature. In the systems considered in this paper the reactor is designed so that the flow velocity and pressure will not change significantly with length. I n this case the optimum temperature will be the same for all stations downstream of the pilot-that is, the combustor operation will be isothermal. The physical method of introduction of unburned material into the flame zone is not specified. I n a later publication the
instantaneous mixing concept will be applied to the design of a combustor in which the pressure and flow conditions upstream of the combustor determine the rate of flow into the reaction zone. Analysis for Instantaneous Homogeneous Mixing Indicates Combustor Characteristics
The burner is shown schematically in Figure 1. The combustor is assumed to be a piloted adiabatic can type in which a partially burned fuel-air mixture leaves the pilot in the plane a t xo and additional combustible mixture is added downstream and mixed instantaneously a t each station with instantaneous heat transfer, It is assumed that heat conduction and diffusion may be neglected.
Figure
1.
Schematic Burner
Diagram
of
The mass flow from the pilot is mo,the temperature of the gas emerging from the pilot is T , and the temperature of the added combustible mixture is T,. The combustible mixture is introduced parallel to the main stream and with the stream velocity. The heat of reaction is H and the reaction rate equat.ion (for a static system) is of the form _ -dcF =
dt
C$CiB'e--E/RT =
(a/f)SC;B'e-E/RT
(11
where CF = concentration of fuel in grams per cc., C A the concentration of air in grams per cc., a l f is the air to fuel mass ratio of the unburned mixture, k , 8,n, and B' (cc./gram)"-l X second-1 are constants for a given mixture, E is the activation energy of the reaction, R is the gas constant, and T is the temperature. This assumption is made in lieu of experimental data on reaction rates a t high temperatures.
It is desired to find the relationship between mass flow and distance along the can that must exist to satisfy the requirement that the temperature be maintained a t the constant value, T, throughout the length of the can. The temperature that corresponds t o the maximum rate of addition of combustible mixture is also to be determined. Consider the zone of thickness, dx,and area, A , Bhown in Figure 1. Burning material enters from the left a t rate tk and additional combustible material is added at z a t the rate, din. By conservation of energy (assuming constant stream velocity),
1634
INDUSTRIAL AND ENGINEERING CHEMISTRY
August 1953
(m3- dm)CT
- Tu)= dq
dlit C ( T
+ (vi f dm) $
= H6h
where a. is the fraction of unburned mass at the exit of the pilot. If the pilot burns the same fuel-air mixture as the main burner and discharges gas a t temperature T , the fraetion burned is a function only of the temperature
(2)
where d
1635
heat liberated per second by burning in the zone l! heat of reaction per unit mass of fuel-air mixture 6 m = mass burned per second in the zone C = heat capacity (assumed independent of temperature and composition) v = stream velocity
I
If the pilot burns a different mixture, the fraction unburned, a, depends on mass flow as well as temperature. Letting CL = K (K'7h0/7h)and substituting Equations 7 and 12 in Equation 10,
-
P
If a is the fraction of unburned mass in the combustor entering the zone a t station x,
+ d& - 6lit
-
li~a
(h
6m = d[h(l
+ dh)(a + da)
- a)]
K'
4- -, K then d?h
(3)
Let
lj2/7ho
Complete burning is defined to be the conversion of a stoichiometric mixture of air and fuel into combustion products. Thus, the nitrogen in the air is "converted" into products of reaction as the oxygen is consumed.
and [(I
+ Ke)r + K'/KIn-'dr/rn
Since CF =
(a/f) ~
+1
pa
d zX [&(l-
where B = B'
[cam
and 6 m = -Adx(l
CY)]
=
p
5
h.dr
= Kn B --(a/f)s X
+ -a/f)dCF dt
(a/f)s(pa)"
+ 1In-'and
=r
(4)
For a second-order reaction, n = 2. Substituting these values in the integrated equation
is the density of the mixture.
We note that m = Apv
P
=
(equation of continuity)
pRT/M
(equation of state)
(6)
If K' = 0
(6)
where P is the pressure and M the molecular weight and
where MUis the molecular weight of unburned fuel-air mixture, Mb is the molecular weight of combustion products, E =
and
pljdv = -dP
(conservation of momentum)
(8)
If v is maintained constant and the added material is injected a t the pressure in the combustor, then P is constant. Substituting A = rit/pv, m d[lit(1 - a ) ] = ; (a/f)sp"-lanBe-E'RTdx (9) Substituting d[lit(l
p
Mixing and Heat Transfer Affect Combustor Volume, Performance, and Efficiency
The preceding theoretical study discloses several points that warrant further comment because of their bearing on the design of practical ram-jet combustors. To simplify the discussion it is assumed that the combustion reaction is of the second order kinetically. Studies of the dependence of the thickness of the Bunsen flame zone on pressure indicate that the rate law is of this form (4, 6,7). Maximum Rate of Heat Release. The temperature, ToDt. which gives the maximum burning per unit volume is that value for which
= MP/RT,
- a)] = mB(a/f)S(~P/nT).-'(or./v)e-E"Tdx
This may be obtained by combining Equations 4 and 14 and differentiating. The result is
From Equations 2 and 3 d [ d z ( l --a)] = d m (C
Since H / C = T f
(10)
HTu)
L__
- Tu,where T f is the flame temperature, Equations 4 and 14 may be combined to yield
Integrating Equation 20, along with Equations 2 and 3,may be used to estimate a numerical value for the maximum space rate of heating.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1636
Since the reaction-rate constants a t temperatures near the flame temperature are unknown, the rates must be determined by extrapolation or inference. This leads to a rather large uncertainty in the calculation, but i t is felt that an approximate value is useful in indicating the extent to which practical combustors approach the performance which would be obtained with perfect mixing. Two methods of estimating the rate constants have been employed, extrapolation of kinetic data and application of the Semenoff expression for laminar flame speeds. EXTRAPOLATION OF KINETICDATA. I n this method it is assumed that data on the rate of oxidation of paraffin hydrocarbons obtained a t temperatures near 800" K. may be extrapolated to the temperatures existing in the combustor. Some justification for this procedure may be adduced from the fact that the paraffin hydrocarbon oxidation reaction at this temperature appears to be beyond the anomalous region of cool flame phenomena, nearly independent of hydrocarbon type (excluding methane), and proceeding with a reasonable temperature coefficient (6). Quantitative data are available for the butane-oxygen reaction ( I ) , for which the rate is found to be
1013
e
--21,000 RT moles/(cc.)(sec.)
(21)
It may be assumed that all paraffin hydrocarbons react at approximately the same rate. This is indicated because the flame speed is nearly constant in the seriw propane to decane, and ram-jet performance does not depend significantly on the specific paraffin hydrocarbon fuel used (8). Substituting the value of the rate constant for butane oxidation in Equation 20, the maximum space heating rate, H s , is H s Z 2.6 kcal./(cc.)(sec.)(sq. atm.) SEMIEXOFF RELATION BETWEEN FLAME SPEEDAND RATECONThe Semenoff relation between laminar flame speed and reaction-rate constant (with known flame speed) may be used as the basis for an alternative method of approximating the rate constant. The Semenoff theory (as Semenoff pointed out) is not strictly applicable to second-order reactions with E / R T j 5 10. STANT.
where
pm =
flame speed
T o = temperature of unburned (cold) mixture
k = heat conductivity coefficient D = diffusion coefficient nl/n2 = moles reactant per moles products
Subscript f indicates evaluation of the parameter a t the flame temperature, subscript o a t the temperature of the unburned gas. Dugger (2) has applied the equation in the interpretation of the dependence of flame speeds in propane on temperature. Although experimental data are not available on transport properties of propane-air mixtures a t temperature near the flame temperature, Dugger, using the observed temperature dependence for air, was able to show that for a value of E of about 33,000 calories per mole the Semenoff theory predicted the observed dependence of flame speed on the initial temperature of the mixture. (This value is estimated to give agreement between the thermal theory and experiment by interpolation between the values of 38.000 and 25,000 calories per mole from Dugger, 3.) This value of E is used as well as the relations of Dugger
D a TI.67
?, a 7'0.84
C, a To.09
T j Z 2200' K.
Dugger used this experimental value for T j in preference to the theoretical value 2390" K. Substitution of these temperature functions in Equation 22 yields Equation 23.
Vol. 45, No. 8
(23)
Letting To equal 300" K. and substituting the transport coefficients for air, O C.) C,, T o = 273" K.
ho = 5.68 X IO-5 cal./(sec.)( ho poCp0Do
I.&
a/f = 15
=
0.24 cal./(g.)(
p L n=
O
C.)
42 cm./sec.
Then
B'
=
8.8 X 1 O I 2 cc./(g.)(sec.) = 3 X 1014cc./(mole)(sec.) (24)
With this value of B' the heat release rate is 1.0 kcal. per (cc.) (second) (square atmosphere). The average value is then 1.8 kcal. per cc. or roughly one billion B.t.u. per (cubic foot) (hour) (square atmosphere). Values approaching this have been reported in experimental ram-jet engines. For example, work by Mullen and coworkers (8)gives a value of about 0.09 kcal. per (cc.) (second) at 1 atmosphere. The close agreement of the two methods of calculation is fortuitous since the uncertainty in the kinetic data for reactions proceeding a t temperatures near 2000' K. does not, of course, permit really quantitative calculations. However, it seems entirely possible that combustion rates under ram-jet conditions of maximum heat release may be determined by kinetic factors rather than mixing times. Further experimental justification for this hypothesis is presented later in this paper. Combustor Shape. I n the theoretical study, the pressure and temperature are specified to be constant throughout the combustor. An immediate consequence of the isobaric, isothermal requirements is that the density and velocity must. be constant also. It was shown that the rate of mass flow, m, consistent with these conditions is given by Equation 18. The cross-sectional area of the combustor is determined, therefore, by Equation 18 and by the condition of mass flow continuity, & = Avp. The density and velocity are constant, so that the rate of increase of area with distance must be proportional to the rate of mass flow increase with distance-Le., the area increases exponentially with length when K' = 0. I n conical combustors the area increases with the square of the distance, and, to apply the instantaneous, homogeneous mixture analysis to such combustors, the cross-section area would be selected (as an independent variable) to conform to the actual area dependence. Extension of the analysis to cover this case will be the subject of a separate report. Combustor Length. The minimum length requirement for a combustor of the isobaric-isothermal type is shown in Equation 18 to be determined by the kinetic constants and flow parameters and by the ratio of total mass burned to that burned in the pilot. If it is assumed that 10% of the mass burned burns in the pilot, that the flow velocity is 200 feet per second, that the pressure is 0.2 atmosphere, and that the space heating rate is one billion B.t.u. per (hour) (cubic foot) (square atmosphere), the calculated minimum length is about 2 inches. Combustor Behavior as a Function of Pressure, Flow Velocity, and Inlet Temperature. The effect of changes of pressure, flow velocity, and input temperature is shown by Equation 18. The mass injection rate is directly proportional to the pressure and the time that the gas spends in the combustor-Le., inversely proportional to the flow velocity. The physical reason for the former proportionality is that since the relative amouut of mass reacted per unit length is proportional to the pressure, a linear increase in concentration of reactant results from an increase in pressure, Pressure dependence of the flame temperature is an additional factor. The mass burned per unit length (per second) increases roughly 25% for a 100" K. increase in inlet temperature.
INDUSTRIAL AND ENGINEERING CHEMISTRY
August 1953
Variation in Composition at Constant Temperature Unless
Main Burner Matches Pilot. Equation 13 shows that even for
*
a n isothermal, isobaric combustor the composition (fuel unburned to total fluid) varies with mass flow (and, consequently, axial position) unless the pilot is matched to the combustor by having the special composition pertinent to a constant temperature, constant pressure combustor. If this matching condition is not satisfied, a measurement of the amount of unburned fuel (or air) does not serve as a measure of temperature, because equilibrium concentration is not established. Stability Limits. A stationary-state, constant pressure, constant temperature combustor may exist, if the mass is injected into an exponential horn combustor a t a rate specified by Equation 18, subject to the approximations previously indicated. I n order to determine under what conditions of pressure, velocity, and temperature this combustor may operate in a stable stationary state, let the mass injection rate R (the ratio d h / d z ) be altered by a n amount AR. Because of the assumption of instantaneous mixing and energy transfer, the instantaneous effect of increasing the injection rate is to decrease the temperature throughout the combustor. The combustor will be stable against injection rate changes if the rate of heat production a t the reduced temperature is sufficient to maintain combustion. The heat generated per second per unit length a t the reduced temperature must be sufficient to raise the temperature of the injected mass (at the new rate) to the new temperature-i.e., the condition for injection-rate stability is that
where Am = @ -
? h R
I n the limit, Tu + 0,
-A =a _
1 - a
e + 0,
AR R
-AT AR T - T , = R
this inequality reduces to
Conclusion
This analysis has been developed for instantaneous homogeneous mixing and heat transfer and indicates characteristics of a combustor in which the controlling rate process is a reaction kinetic rate. The analysis indicates how much might be gained in terms of performance, combustion efficiency, and combustor volume by improved mixing and heat transfer. The calculations emphasize the need for quantitative data on reaction rates a t high temperatures. Instantaneous homogeneous mixing may not be a n optimum situation. Although increased mixing homogeneity tends to cause an increase in the mean temperature of the reactant, leading to increased specific reaction rate, increased mixing homogeneity reduces the concentration of the reactant by diluting i t with combusted mdterial, thus tending to reduce the over-all reaction rate. Acknowledgment
The authors wish to acknowledge many helpful and stimulating discussions of the material presented in this paper with their colleagues engaged in the Bumblebee propulsion program. Discussions with J. P. Longwell of Standard Oil Development co. and with P. Rosen of this laboratory have been particularly helpful. Literature Cited (1) Appleby, W. G., Avery, W. H., Meerbott, W. K., and Sartor, A. F., J. Am. Chem. SOC.,75, 1809 (1953). (2) Dugger, G. L., Ibid., 72, 5271 (1950). (3) Ibid., p. 5274, footnote. (4) Fristrom, R., Prescott, R., Neumann, R., and Avery, W. H., presented at the Fourth Symposium on Combustion, Cam(5)
Under these conditions the combustor is stable against smaIl changes in mass injection rate only if the tempgrature is greater than that corresponding to T,,t.. This minimum stable temperature is relatively insensitive to small values of T , and e. The combustor is stable for arbitrary changes in mass flow rate that do not instantaneously decrease the temperature below this amount.
1637
bridge, Mass., September 1952. Xlaukens, H., and Wolfhard, H. G., Proc. Bog. SOC.,A193,
512 (1948). (6) Lewis, B., and Von Elbe, G., “Combustion Flames and Explosions,” pp. 152-3, New York, Academic Press, 1951. (7) Longwell, J. P., Frost, E. E., and Weiss, M. S . , IBD. ENG. CHEM.,45, 1629 (1953). (8) Mullen, J. W., 11, Fenn, J. B., and Garmon, R. C., Ibid., 43, 195 (1951). RECEIVED for review January 12, 1953. ACCEPTED May 28, 1953. This work wa8 supported by the Bureau of Ordnance under Contract NOrd 7386.
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High Temperature
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Combustion Chamber
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ALEXANDER WEIR,
JR.
University o f Michigan Aircraft Propulsion laboratory, Ypsilanfi, Mich.
A
GOOD jet combustor requires, in addition to high efficiency, low internal drag, and a wide range of operating conditions, that the combustion be completed in a minimum of space. This requirement has presented many problems and considerable effort has been expended during recent years on the problems of ram-jet combustion. One of the major problems is maintaining adequate flame stability in the combustion chamber. Current
ram-jet practice utilizes a bluff body, inserted in the high velocity gas stream, as a stabilizer or flame holder. A familiar example of a flame holder is the grid or screen on a Mdker laboratory burner. This grid allows more fuel and air to be burned than in an ordinary Bunsen burner before the flame lifts or blows off. Flame holders, however, have certain limitations. Previous work (16,17) has indicated that increasing the size of geometri-
