COR R ESPON DE NCE
Specific Impulse and Heat of Combustion Edward A . Fletcher
E
xception must be taken to a rather small, but fundamentally important, point that appeared in a recent review article by Essenhigh and Howard (7). I t is necessary that this be done because the point, although technically correct, is misleading. It is misleading because, in an article which purports to be concerned with combustion and the application of combustion science to propulsion, it depends for its correctness on a definition of an efficiency which is not really an efficiency at all in the sense in which the word is usually used, and which is in fact, the result of a conglomeration of factors. The efficiency defined as Essenhigh and Howard have defined it, in addition to being misleading, is useless since it can have values which range from zero to infinity, and its values can depend on such various factors that it, in itself, is not a useful measure of any inefficiency. I refer to the internal efficiency vc, which is defined by the authors and by others (2) to be c2/2JAH and is said to fall usually betweei 0 and 507,. The deceptiveness of this definition is well evidenced by the fact that the authxs, themselves, have fallen into the trap in assigning it a finite range of values. Defined in this way, internal efficiency alpears to be a useful combustion parameter because it is the ratio of the kinetic energy per unit mass of the exhaust gas (c2/2) to the heat of combustion per unit mass of the propellant, implying that the velocity of the exhaust gas (and thus the specific impulse) has a AUTHOR Edward A . Fletcher is Professor of Mechanical Engineering and Director of the Combustion Laboratory of the Dekartrnent of Mechanical Engzneering, Institute of Technology, University of Minnesota, Minneapolis, M i n n .
direct functional dependence on the heat of reaction. This is, of course, not true. Molecular hydrogen being simply permitted to expand through a nozzle from room temperature to a temperature of about 60’ K. will give a specific impulse that is of the same order of magnitude as that of the combustion products of hydrocarbon-oxygen mixtures although the “heat of combustion’’ of the former is zero, and the efficiency is therefore infinitely high. The enthalpy difference that is important, insofar as specific impulse is concerned, is the difference between the enthalpy of the chamber gas and that in the exhaust nozzle exit plane. Thus,
I”
=
1 -d2(h, g
- he)
where h, and h , are the specific enthalpies of the chamber gas and the exhaust gas. This enthalpy difference is related only in an indirect way to the heat of reaction. I t depends on the combustion efficiency insofar as the combustion efficiency determines the composition of the exhaust gas and thus the temperature dependence of this enthalpy, but it also depends, to a very great extent, on the expansion process, which may or may not be at all related to the combustion process. Thus, the introduction of the combustion intensity factor, I,, into the specific impulse and thrust equations by elimination of the heat of reac,tion seems rather contrived and pointlesr. REFERENCES (1) Essenhigh, R . H;, Howard, J. B., IND.END.CHEM.5 8 (l), 14 (1966).
(2) Sutton, G . P., York, 1963.
Rocket Propulsion Elements,” 3rd ed., p. 28, Wiley, S e w
Reply Ly R. H. Essenhigh and J . B. Howard
F
letcher‘s criticism of the definition of internal efficiency employed in our paper appears to be without support. Since we were concerned with combustion systems, his example of molecular hydrogen expanding through a nozzle is of course irrelevant. The efficiency parameter in question is used mainly in studies and discussions rather than in design work. Nevertheless, its use is justified by the mere fact that it evidently serves a valuable purpose. In addition to this, instead of being misleading as Fletcher claims, this parameter serves as an indicator of the fraction of the “chemical energy’’ in the reactants appearing later as useful energy-Le., kinetic energy associated with the axial component of velocity-in the rocket exhaust. One of the main sources of error in Fletcher’s argument appears to be his premise that the heat of combustion is relatively unimportant with regard to specific impulse. Reason for rejecting his claim is outlined below. As a result of the behavior described by the first law of thermodynamics and the principle of conservation of momentum, the specific impulse, c/g, depends upon i, the properties (temperaLure, pressure, specific heat, and number of moles of each species) of the burned gases before and after expansion, and ii, the change in entropy of the gases during expansion. However, the expansion process is usually approximately isentropic, and, in ordinary practice at least, it can be shown that the specific impulse depends mainly on the adiabatic flame temperature and the molecular weight and specific heat of the 76
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
burned gases. Since contributions to the enthalpy of the burned gases from energy sources and sinks (e.g., heating of fuel and vaporization and heating of the oxidizer) other than the chemical reaction are usually negligible in comparison with the enthalpy change due to the chemical reaction. the adiabatic flame temperature is found to be influenced strongly by the heat of combustion. Therefore, the use of the heat of cornbustion and/or the combustion intensity is clearly not misleading, but is instead both meaningful and valid. I t should be appreciated that the general expression for specific impulse which we employed reduces to the equation which Fletcher accepts4.e.) ( g I J 2 = 2 (h, - he)-in the specific case defined by the following conditions: (1) that the flow from the combustion chamber be adiabatic and reversible; (2) that there be no component of exhaust velocity except that in the axial direction; and (3) that the nozzle be so designed that the burned gases are expanded to atmospheric pressure. I t thus appears that Fletcher’s equation ties in well with our general formulation. AUTHORS Robert H. Essenhigh is Associate Professor in the Department of Fuel Science, College of Mineral Industries, Pennsylvania State University. Jack B. Howard is now Assistant Professor of Chemical Engineering at Massachusetts Institute of Technology.
