Frank H. Verhoek
The Ohio State University 43210 Columbus,
Thermodynamics and Rotket Propulsion
O n e of the most interesting applications of the principles of chemical thermodynamics to an important problem concerns the comparison of the merit of one rocket propellant (fuel plus oxidizer) with that of another fuel-oxidizer pair. This paper outlines this application for liquid-fueled rocket motors. The rocket is one of the simplest engines for imparting motion to a vehicle, in contrast to more conventional engines, which subject the chemical energy of the fuel to several transformations as the vehicle moves. Conventional engines first convert chemical energy to heat energy which is used to drive a steam engine, turbine, or internal combustion engine; these in turn drive the vehicle, or, by driving an electric geherator, provide electric power to drive the vehicle. I n a rocket the conversion is much more direct: chemical energy heats matter in a rigid chamber to a high temperature; the matter is then ejected through a nozzle in a specified direction; and the reaction from the jet pushes the rocket forward, without intervention of reciprocating parts nor mechanical or electrical conversion of rotational motion to linear motion. The rocket motor, consisting of chamber and nozzle, converts random thermal energy into a collimated jet in which all the molecules are going in the same direction. While a conventional motor is designed to propel a nearly constant load a t a nearly constant speed, the mass of a rocket steadily decreases as its matter is ejected, and if the rocket is to reach its goal, it must accelerate rapidly enough during the burning time so that its terminal velocity a t burn-out will be sufficient to carry i t along in free flight. The impulse given the rocket, defined as the product of the accelerating force exerted times the time during which it is exerted, is thus the important factor. However, a propellant fuel which gives a large impulse only when a large mass of it is burned will not be as useful as one which gives the same impulse for burning a small mass; since a rocket carries its fuel with it, a rocket carrying the large mass of propellant cannot get off the ground as easily as the rocket carrying the small mass. Hence the criterion of the merit of a rocket fuel is its impulse per unit mass of propellant burned, called the specific impulse. This is closely related to the velocity of the gases which exit from the nozzle, as shown in the next paragraph. The simplest case is that of a rocket operating under ideal conditions with all gas flow perpendicular to the exit plane of the nozzle and the pressure at the exit equal to the surrounding pressure. Then the total force acting to propel the rocket is the reaction force, equal to the time rate of change of momentum d(mv)/dt of the gas leaving the nozzle. For fixed pressures in the chamber and a t the nozzle exit, a steady state exists and the 140
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gas velocity is unchanging with time; hence
If a bum-out time tb a mass of propellant M , is used up, mt, = M,, and Ft, is the impulse delivered. Hence the specific impulse I is
so that the specific impulse, I, is equal to the velocity (relative to the rocket) of the gases leaving the nozzle. Unfortunately, it is the common practice of rocket engineers to express the force in pounds-force and the mass of propellant used in pounds-mass. Recorded values of specific impulse I, are thus given in seconds and this recorded value must be multiplied by the gravitational constant g (for converting slugs to poundsmass) to get the gas velocity; hence When the conditions are not ideal, the force acting to propel the rocket is given by F
= XliLVa
+ ( p a - p.)os
where h is a factor less than unity to correct for nonperpendicular flow at the exit, and (p, - p,) u, represents the "pressure thrust" exerted across the nozzle crosssection n, because of the difference between the pressure of the exhaust gases p, and the atmospheric pressure p,. The magnitude of the pressure thrust is determined by the design of the nozzle, which fixes a, and p,, and is normally much smaller than the "momentum thrust" hmv,, which is determin6d by the chamber temperature reached in the combustion process. Since the chamber temperature is in turn a function of the heat of the reaction and other chemical factors, it is the momentum thrust in which the chemist is interested, and the specific impulse, mve/m, is a proper . measure of the merit of the propellant. The problem is to calculate the specific impulse for eiven urouellant materials (or the exit velocitv. to which the spkciec impulse is propdrtional). The im&se exists because of the high temperature (and pressure) created
-
Rocket chamber and nozzle. Preuure, kmperdure, onthdpy, entropy, and velocity symbols or used in the text.
by the combustion reactions in the rocket chamber; as known data for the calculation we have the thermodynamic properties of the reactants and products in those reactions. To see what needs to be done, let us first consider that we have in the chamber a single working gas, which is heated by some external agency to a high temperature and pressure, and allowed to expand through the nozzle.' From the first law of thermodynamics dp
=
+ ur = dE + pdV = dH - Vmdp
dE
where, a t the third equality, we have substituted dH = dE Vdp pdV obtained by differentiation from the definition of enthalpy (H = E pV), and have chosen one mole as the basis, so that V is the molar volume V,. Then the density is
+
+
+
where M is the molecular weight. We must now introduce one equation from hydrodynamics, the Euler equation,%which states that ideal steady flow of a compressible fluid along a streamline, neglecting viscosity and gravitational effects, obeys the differential equation thus relating the change in velocity of the fluid, do, to the pressure differential, dp. Hence M -dp = -- vdv
v,
and, substituting this value for dp dg = dH
+ V,
M vdv = d(H v." -
+ '/sMvZ)
=
d(H
+ K.E.)
where K.E. represents kinetic energy of the directed flow of the gas out of the chamber. If the flow process is adiabatic, so that no heat is gained or lost by the gas i n the expansion process, dq = 0, d(H K.E.) = 0, and on integration we arrive at the important conclusion that the sum of the enthalpy plus the kinetic energy (of directed flow) is constant in steady adiabatic flow.
+
H
+ K.E. = eonstant
Applying this conclusion to gas in a chamber (suhscript c ) and external to the chamber (subscript e ) , H.
+ 1/2Mu,z= H, + '/2Mu,2
and recognizing that within the chamber there is no directional velocity so that v, = 0, we have
Nothing that we have done so far would altered if more than one chemical species was present, except that A t would be replaced by an average molecular weight E, and
' SEIFERT,H. S., MILLS,M. M., J . Phys., 15, 1 (1947).
AND
SUMMERFIELD, M., Am.
A derivation is given, for example, in SHEPI~ERD, DENNISG., "Elements of Fluid Mechanics," New York, Hareourt, Brace and World, Inc., 1965; pp. 117-23, 162.
To increase our insight, however, let us continue as if only a single gas were present, and make the further assumption that it is a perfect gas. Then integration of heat capacity equation
gives H = C,,T
+ Ifo
The difference between chamber and exhaust is
I n terms of the ratio of specific beats, constant
y,
and the gas
and in the reversible adiabatic expansion of a perfect gas
Making the substitutions
Thus we see that the effectiveness of a rocket fuel depends upon (1) The specific heat ratio, 7 = C,/C, (2) The chamber temperature, T, (3) The ratio of the exhaust pressure to the chamber pressure, P.lP0 (4) The molecular weight of the exhaust gases, M.
I n choosing a rocket fuel, not all four of these are entirely under our control. (1) The ratio of specific heats is much the same for all simple gases, so we cannot make a choice on that basis. (2) We want the chamber temperature to he high, and we can choose reactants which will produce a high temperature. Hardware limitations appear, however, since a high temperature can lead to excessive heat losses through the walls; but, more importantly, it means a high pressure. Thus it requires stronger and hence thicker and, most objectionably, heavier, walls for the chamber; it leads to uncomfortable requirements for the pumps injecting fluid into the chamber, etc. The engineers, therefore, are in command of the chamber pressure; their design selects a chamber pressure as a limit for the chemists. (3) Since the exhaust pressure is fixed by the atmosphere, selection of a chamber pressure also fixes the pressure ratio. The calculations in the table which follows are made for p, = 1000 Ih/in2 and p, = 14.7 lb/in2. A nozzle actually designed for this exit pressure will be somewhat less efficient for the lower pressures of the upper atmosphere, but the defect is partially compensated by an increase in the pressure thrust, so the overall rocket performance is not proportionately changed. (4) We want the molecular weight low, and this is in the hands of the chemists. The chemists' problem then is, essentially, to find the fuel-oxidizer propellant pair which gives the highest chamber temperature at the engineers' allowed chamber pressure for the lowest molecular weight in the exhaust. Since many gases are present, however, the general equaVolume
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March 1969
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tion, eqn. (I), rather than eqn. (Z), must be used, and the problem becomes one of calculating the enthalpy difference H , - H , and the average molecular weight a t the nozzle exit. The problem situation is described as one in which the propellant ingredients, consisting of fuel plus oxidizer, are introduced into the rocket chamber at a known ambient temperature and in a definite mass and in a definite ratio to one another; combustion occurs, and heat of the reactions heats the combustion products to such a temperature that the design-selected chamber pressure is reached. To limit the problem several assumptions are made. Assumption I . Chemical and thermal equilibrium exists among the product gases and thus the composition of the mixture in the chamber is the equilibrium mixture for that temperature and pressure. This means that possible kinetic effects, such as a delay in the rates a t which reactions establish equilibrium, are neglected. This appears to be a safe assumption for the liquid fuels discussed here, hut in the case of solid fuels, special attention must be given to burning rates if the chamber pressure is to be maintained a t a constant value. Assumplion I I . All gases behavp as perfect gases, a reasonably safe assumption a t the temperatures concerned. Assumption I I I . When the chamber gases, a t the established T , and p,, expand through the nozzle to the established exit pressure, the expansion is adiabatic, with no heat loss through the walls, and reversible in the thermodynamic sense. The simpler calculation presumes that the composition remains fixed ("frozen") in this expansion; the more precise suggests that the mixture com~ositionadjusts to the ecluilibrium a m r o ~ r i a t eto the temperatureif the gases a t the exit, making us
