Thrust coefficient

As shown by Eq. (1.65), the maximum thrust is obtained when at a given 

Equation (1.68) can be represented by a simplified expression for thrust in terms of nozzle throat area and chamber pressure  

When the nozzle expansion ratio becomes infinity, the pressure ratio pJPa Iso becomes infinity. The maximum thrust coefficient then becomes 

It is convenient to non-dimensionalize the thrust equation and obtain a dimensionless ratio F/Pc At. This ratio is defined as the thrust coefficient  

One can replace pa /pc by eopt through equation n. A. 8. A plot of Cp vs eopt gives a curve which goes through the maxima in figure n. A. 5. This curve approaches an asymptotic value at infinite expansion ratio or (pa /pc ) = 0. This asymptotic value is the ultimate Cp obtainable and is given by  

It is significant that cF is completely independent of T and fll. Consequently as a figure of merit, it is insensitive to the efficiency of combustion, but sensitive to the nozzle design. In practice the test engineer compares the measured Cp, which is determined from actual measurements of pc, At, and F, with the theoretical Cp computed from equation n. A. 17. or figure n. A. 5. to determine whether the nozzle is functioning efficiently. In this way he can localize to some extent the cause of an unexpected defect in the specific impulse. If no defect is found in Cp then the loss must be in the combustion process. In the next paragraphs a parameter solely dependent upon the combustion efficiency is defined. 

Thus one sees that c is readily determined from experimental measurements. It is immediately evident that  

Many possible expressions may then be developed for c. Some forms are  

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In addition, the specific impulse is given by the thrust coefficient and the characteristic velocity according to... 

The specific impulse of a rocket motor, I, as defined in Eq. (1.75), is dependent on both propellant combushon efficiency and nozzle performance. Since is also defined by Eq. (1.79), rocket motor performance can also be evaluated in terms of the characterishc velocity, c, defined in Eq. (1.74) and the thrust coefficient, Cp, defined in Eq. (1.70). Since c is dependent on the physicochemical parameters in the combustion chamber, the combushon performance can be evaluated in terms of c. On the other hand, Cp is dependent mainly on the nozzle expansion process, and so the nozzle performance can be evaluated in terms of Cp. Experimental values of and Cpgxp are obtained from measurements of chamber pressure, p, and thrust, F ... 

Fig. 14.6 Theoretical and experimental thrust coefficient of AP-RDX-HTPB propellants as a function of (AI). ...

II. A. 5. Variation of rocket thrust coefficient with nozzle... 

Fig. II. A. 5 Variation of rocket thrust coefficient with nozzle area ratio and pressure ratio Pc/Pa for Y = 1.2...

The above procedure gives more information than the optimum area ratio. In the e op, -determination, A t /mis obtained. Since c = Pc A t /m the value of c is calculated readily from the given value of Pc. The theoretical thrust coefficient is obtained as ue /c. ... 

The effect of mixture ratio on characteristic velocity and thrust coefficient... 

If the effect of mixture ratio upon the characteristic velocity and thrust coefficient is examined, additional information is gained over that obtained from consideration of the specific impulse alone. Since the characteristic velocity bears the same dependence upon combustion temperature and molecular weight as does the specific impulse, the optimum characteristic velocity and optimum specific impulse would be expected and are observed to occur at approximately the same mixture ratio, see figure V. A. 6. The failure of these two performance parameters, c andcF. to have maximum values at the same propellant mixture ratio is traceable to their differing dependencies upon the specific heat ratio. The same effect is reflected in the dependency of the thrust coefficient upon the mixture ratio. For optimum expansion and a fixed pressure ratio across the nozzle, the thrust coefficient depends only upon the specific heat ratio. The dependence of the specific heat upon the mixture ratio in turn results in a dependence of the thrust coefficient upon the mixture ratio. Since thrust coefficient increases with decreasing specific heat ratio for a fixed pressure ratio, the maximum thrust coefficient should occur at the mixture ratio of maximum specific heat. The specific heat of the products increases both with temperature and with complexity of the product species. The maximum specific heat is found near the stoichiometric mixture ratio. For equilibrium expansion the specific heat also includes the effect of exothermic recombinations. This later effect is a maximum at the condition of maximum dissociation in the chamber which similarly occurs at the maximum combustion temperature or near the stoichiometric mixture ratio. 

The observations that the thrust coefficient has its maximum value near the stoichiometric mixture ratio, see figure V. A. 6.,is consistent with the foregoing expectations. Since specific impulse is proportional to the product of the characteristic velocity and the thrust coefficient, it is expected and observed that the optimum mixture ratio in terms of the specific impulse should fall between the optimum mixture ratios for the characteristic velocity and for the thrust coefficient. It is noted that the characteristic velocity is the dominant member of the pair, a property which adds further to the utility of the characteristic velocity as a performance parameter to rocket propulsion development. 

C =Vj/CF=gIs/CF=gF/tf (1/Cp) where Vj = effective exhaust velocity Cp= thrust coefficient g=gravitational acceleration (general) Is= specific impulse F = thrust... 

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