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Rocket Motor Efficiency This page discusses the two common measures of rocket motor efficiency – Specific Impulse (ISP) and Combustion Efficiency (C*). These terms express the performance characteristics of a given motor, but the equations used to derive them are useful design tools, linking propellant selection and desired performance to chamber geometry and expected losses. Specific Impulse The energy produced by a rocket motor is typically classified in terms of Impulse, which is defined as the thrust of a rocket multiplied by the time of operation. The term of total thrust is useful to show how much force can be exerted on an object, but in order to show how much energy is being used, that force must be expressed in terms of how long that force is being applied. For example, if you push a shopping cart with 10 pounds of force for 20 seconds, then you have exerted an impulse of 200 pound-seconds on the cart. Since we know how much energy was applied to the cart, we can obtain the same rolling velocity using less force if we apply that force for a longer period of time. In a frictionless environment, applying 5 pounds of force to the cart for 40 seconds will get the cart going just as fast as applying 10 pounds of force for 20 seconds. The term Specific Impulse (Is or ISP) is used in rocket propulsion to define the Impulse created per unit weight of propellant. Specific impulse is usually expressed in “seconds”. The ASA rocket motor, for example, has a specific impulse of about 254 sec at sea level. This means that the ASA rocket motor will produce 254 pounds of thrust for every pound of fuel injected into the combustion chamber (per second). This means that in order for the ASA rocket motor to produce 11000 pounds of thrust, we must inject ~43 pounds of fuel per second into the motor. Expressed mathematically: Specific Impulse (in seconds) = Is Thrust (in pounds) = F Fuel Flow (in pounds/second) = w’ -or- Is = F/w’ When comparing two rocket motors, if one motor has a higher specific impulse than the other, the motor with higher ISP will produce the same amount of thrust using less fuel than the motor with lower ISP. That motor is therefore more efficient and desirable. The other way to think about it is that for the same amount of fuel, the more efficient motor will produce more thrust.

So what makes a rocket motor have high ISP? If this topic were simple, then rocket propulsion would not be so complicated. The strongest parameters influencing ISP are the amount of chemical energy in the fuel, the efficiency of combustion, and the shape of the nozzle. Other contributors are the geometry of the internal components (injector, combustion chamber, etc), and the friction of gas and the amount of heat lost to the walls of the motor. Maximizing the ISP of a rocket motor is humorously termed a “black art”, since there is no absolute answer for making a motor efficient – it is a lot of trial and error (and making sure you learn from the mistakes of others!). One parameter that can be readily changed to improve the performance of a rocket motor is the effect that outside air pressure has on the exhaust velocity of the rocket motor. As external air pressure falls, the nozzle “expands” the compressed exhaust gas further than it would have at a higher ambient pressure. This additional expansion translates into increased exhaust velocity and thereby increased motor thrust. This is why you typically see two numbers quoted for a given rocket motor’s thrust, “Sea-Level Thrust” and “Vacuum Thrust”. Typically, nozzles are designed to expand the exhaust flow to a specific pressure (14.7 psi for a “sea level” rocket motor, for example). If a 14.7 psi nozzle is operated at high altitude, then it is not quite as efficient as a nozzle designed to work at the higher altitude. Since simple rocket nozzles can only be designed to maximize efficiency at one point of their flight envelope, the designer must perform trade studies to find the perfect compromise setting for exhaust pressure. Some motor designs can actually change the nozzle shape during flight. These motors maximize performance for all altitude conditions, greatly improving overall efficiency. Some motor designers use an extendable rocket nozzle that can lengthen the nozzle as the outside air pressure falls. Aerospike nozzles and Expansion-Deflection nozzles, however, use ambient air pressure as one of the two surfaces needed contain the burning gases. As the ambient air pressure falls, the containment lessens and the plume grows – maximizing the expansion efficiency for a given altitude. These nozzles are not widely used, but are notably more efficient than standard nozzles.

Combustion Efficiency – The quantification of motor combustion efficiency is called the “Characteristic Exhaust Velocity”, or the term C^* (“cee-star”). Not to be confused with actual motor exhaust velocity, this parameter is independent of nozzle characteristics and is a typical way to express a motor’s efficiency in turning chemical energy into physical energy (combustion chamber pressure and temperature). This parameter is strongly affected by propellant selection and combustion chamber geometry. If the motor characteristics are known, C^* can be expressed mathematically: Combustion Chamber Pressure (psi) = P Nozzle Throat Area (square inches) = At --or-- C^* = P*At/m’ Fuel Flow (slugs/second) = m’ Note that throat area is not a strong nozzle efficiency characteristic. Throat area is a function of the combustion process (chamber pressure, temperature, and combustion products), and geometric characteristics of the combustion chamber. Nozzle throat area is, therefore, more directly linked to combustion efficiency than nozzle efficiency. How does all of this tie together? The ASA rocket motors are designed using the tool REPSOP (discussed on the ASA Rocket Motor page), which integrates all components of the rocket vehicle mathematically. There are a few items that REPSOP cannot model well, however, and one significant item is ISP. Theoretical ISP, or the efficiency of a perfect rocket motor can be calculated using known equations. Actual ISP, however, is much more complicated since much of a motor’s efficiency losses cannot be calculated without CFD modeling. A conservative estimate of the actual motor ISP can be obtained by multiplying the theoretical ISP by a coefficient of overall motor efficiency. In Rocket Propulsion Elements, George Sutton states that rocket motors, in practice, produce 88-97% of their theoretical ISP potential. Sutton also states that one third of this loss in efficiency is typically due to combustion inefficiencies and the majority of the remainder tied to the nozzle. Other sources claim that motor efficiencies of 93-98% can be obtained. ASA’s motor and vehicle performance calculations assume that the rocket motor is 95% efficient. This value forces conservatism into motor and vehicle design until testing reveals the actual ISP of the motor. The following set of calculations demonstrate the equations used to calculate theoretical ISP. Five parameters must be known about the motor to calculate ISP. ASA obtains this combustion information from published data and a third-party software product. T: Combustion Temperature P: Combustion Pressure Pe: Nozzle Exhaust Pressure R: Gas Constant of Exhaust Products k: Specific Heat Ratio C^*: Characteristic Exhaust Velocity (calculated by combustion values alone) C^* = ((k * R * T)^0.5)/( k *((2/( k +1))^(( k +1)/( k -1)))^0.5) Cf: Thrust Coefficient Cf = (((2*( k ^2))/( k -1)* (2/( k +1))^(( k +1)/( k -1)) *(1-( Pe / P)^(( k -1)/ k)))^0.5) + ((p2-p3)/p1)*(A3/A1) ISP: Specific Impulse ISP= C^* * Cf / 32.174 If the combustion parameters are known, then an estimate of actual ISP can be calculated by: Actual ISP=ISP*Coefficient of Motor Efficiency This page reviewed the parameters and calculations used to define the efficiency of a rocket motor. These terms are applicable to both solid rocket motors and liquid rocket motors. See the Rocket Motor Fundamentals page to learn how the ISP calculations shown here are useful in the overall design of a rocket motor. Please check back with this web page for the next propulsion topic, and contact Rob Morehead at rmorehead@asa-houston.org if you have questions relative to the ASA propulsion department. Other ASA propulsion topics include: · Solid-Rocket Motor Propulsion for the ASA Test Launch Vehicles · Liquid Rocket Motor Fundamentals · The ASA liquid-rocket motor · Efficiency comparisons of various types of rocket motors (Liquid, Solid, and Hybrid systems)

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