Derivation of the thrust equation from conservation of energy

Abstract

The thrust equation for rockets and air-breathing jets, conventionally derived by use of the momentum theorem of fluid mechanics, is derived here from the first law of thermodynamics. The principle of conservation of energy, applied alternately to a system and to a control volume, yields the thrust equation quite directly. The particular control volume specified is compared to two others which are commonly used in derivations based on the momentum theorem. Intermediate relations developed in the analysis are shown to be useful for study of the over-all energy balance of propulsion systems and for interpretation of several definitions of propulsive efficiency used for air-breathing systems. Nomenclature A a = exit area of thrust producer / = fuel-air mixture ratio by mass = m//m 0 F = thrust g = acceleration resulting from gravity h = enthalpy per unit mass ra = mass flow rate p = pressure Q = heat added per unit time u = absolute vehicle velocity ue — velocity of exhaust fluid relative to thrust producer and in the direction of vehicle motion v = absolute velocity in the direction normal to vehicle motion V = velocity, absolute or relative as specified W = work accomplished per unit time z = elevation above datum level f]p = propulsive efficiency Subscripts a = air entering system or control volume or freestream condition e = exit of thrust producer / = fuel at fuel inlet conditions n = direction normal to vehicle motion u = direction of vehicle motion