atw International Journal for Nuclear Power | 04.2020

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atw Vol. 65 (2020) | Issue 4 ı April FEATURE | RESEARCH AND INNOVATION 192 made of a uranium/tungsten carbide material that allowed them to operate at temperatures somewhat higher than those achievable in NERVA. As a result, the RD-410 was slightly more efficient than the NERVA engines. Nuclear Rocket Engine Operational Characteristics In chemical rocket engines, a fuel (normally in conjunction with an oxidizer) combusts to form the gaseous propellant which is expelled through a nozzle to generate the thrust. When speaking about chemical engines, therefore, the terms fuel and propellant are often used interchangeably since the fuel (that is, the substance that supplies the energy to heat the propellant) and the propellant are one and the same. In nuclear engines, however, the propellant is simply the working fluid heated by the nuclear reactor to produce the thrust. The fuel in this case is actually the fissioning uranium in the nuclear reactor. When speaking about nuclear engines, therefore, the term fuel is used to describe the fissioning uranium in the nuclear reactor and the term propellant is used to designate the working fluid being expelled through the nozzle. It is interesting to note that because of the extremely high energy density characteristic of nuclear fuels, an entire crewed Mars mission may be accomplished through the fissioning of less than 200 g of 235 U. The efficiency of rocket engines depends upon, among other things, the temperature of the engine propellant exhaust gases (e.g. the higher the temperature, the higher the rocket efficiency). In chemical engines, the temperature of the exhaust gases is limited by the amount of energy that may be extracted from the fuel and oxidizer as they react. Thus, chemical engines are said to be energy limited in their efficiency. An NTR engine, on the other hand, operates by using nuclear fission to heat the propellant to high temperatures. Because the energy released from the fissioning of nuclear fuel is extremely high as compared to that available from chemical combustion processes, the propellant in an NTR can potentially be heated to temperatures far in excess of that possible in chemical engines. The main limitation of these engines results from restrictions on the rate at which this heat energy can be extracted from the nuclear fuel and transferred to the propellant. This rate of energy transfer is limited by the maximum temperature the nuclear fuel can withstand. It is this temperature limitation that puts an upper limit on the maximum efficiency attainable by solid core nuclear thermal rocket engines. As such, NTR engines are said to be power density limited in their efficiency. To determine the efficiency of a rocket engine, be it chemical or nuclear, a characteristic value called Specific Impulse (or I sp ) is used which is analogous to liters per 100 km for an automobile. Specific impulse is calculated by dividing the rocket thrust in Newtons by the propellant mass flow rate in kg/sec. It has units of seconds and physically represents the length of time a rocket engine can produce one newton of thrust from one kilogram of propellant. It is also proportional to the propellant exhaust velocity (υ e ). (1) Also note that: (2) Equation 2 simply illustrates that at a given power level specific impulse is inversely proportional to thrust. Specific impulse can be related to the temperature of the exhaust gas by performing a heat balance and using the First Law of Thermodynamics. (3) where: Q = Reactor power h = Enthalpy of propellant c p = Specific heat of propellant T = Temperature of propellant In the above equations, the subscript “c” stands for the conditions at the exit of the reactor and the subscript “e” stands for the conditions at the nozzle exit. Rearranging terms from Equation 3 then yields for the specific impulse. (4) Noting the following specific heat relationships for ideal gases: (5) where: γ = Specific heat ratio of propellant R u = Universal gas constant Also recalling that for isentropic flow temperature and pressure can be related by: (6) where: P = Pressure of propellant By substituting Equations 5 and 6 into Equation 4 for the specific impulse, it is found that: (7) Assuming that the nuclear engine is operating in space with an infinitely large nozzle, the pressure ratio goes to zero and we are left with an expression which represents the specific impulse only in terms of the chamber temperature and the propellant thermodynamic properties: (8) where: F = Thrusting force ṁ = Propellant mass flow rate g c = Gravitational acceleration constant Plotting Equation 8 for a chemical engine using liquid oxygen and liquid hydrogen as propellants and for a nuclear thermal rocket using only hydrogen as the propellant, it can be seen in Figure 2 that the nuclear Feature Nuclear Rockets for Interplanetary Space Missions ı Dr. William Emrich
atw Vol. 65 (2020) | Issue 4 ı April | Fig. 2. Specific Impulse Comparison between Nuclear and Chemical Rocket Engines. engine provides about twice the specific impulse at a given temperature as compared to a chemical engine. This variance is due almost entirely to the difference in molecular weight between the two exhaust gases. So how does this efficiency increase using nuclear thermal rocket engines translate into improved interplanetary mission profiles? First, noting that most interplanetary missions using high thrust propulsion systems such as what would available using nuclear thermal propulsion do not apply thrust for the entire flight, but rather execute a series of thrusting maneuvers near the departure and destinations planets with relatively long coast periods between the planets. Normally, at least four major propulsive maneuvers are required for round trip missions. These main propulsion system burns include: 1) a departure acceleration burn from home planet, 2) an arrival deceleration burn at the destination planet, 3) a departure acceleration burn from the destination planet and, 4) an arrival deceleration burn back at the home planet. Second, because the planetary alignments are continually changing, these propulsive maneuvers cannot be performed anytime one wishes, but only during certain windows of time when the planetary alignments are favorable. The various thrusting maneuvers described above may be added together to yield what is called the total mission velocity that describes the total velocity increment that must be delivered to the spacecraft in order to complete the mission. This velocity increment is a function of the engine specific impulse and the vehicle mass fraction that is defined to be the ratio of the spacecraft unfueled mass to its fueled mass. Applying the principle of conservation of momentum, this high velocity propellant exhaust flow has the effect of forcing the spacecraft forward as is illustrated in Figure 3. Thrust is defined to be the force produced by the rocket engine due to the time rate of change of momentum of the exhaust gas | Fig. 3. Rocket Thrusting and the Conservation of Momentum. Expanding the above equation and rearranging terms then yields: (10) Note that in the above equation U + V = υ e = g c I sp , therefore, taking the limit of the above equation as time goes toward zero and applying Newton’s Second Law of Motion along with the definition for specific impulse (11) By integrating the above equation, the total change in spacecraft velocity possible for a given vehicle mass fraction and specific impulse may be determined. (12) where: V f = Final velocity of the rocket (e.g. or total mission velocity) m 0 = Initial mass of rocket (fully fueled) m f = Final mass of rocket (fuel expended) f m = Vehicle mass fraction (ratio of to ) The above equation is commonly known as the rocket equation and its solution yields the maximum velocity increment attainable by a space vehicle in terms of the vehicle mass fraction and the engine specific impulse. For example, if a spacecraft has a fairly doable mass fraction of 0.15 along with a nuclear engine having a specific impulse of 900 sec., the total change in velocity that the vehicle is capable of achieving is about 16.8 km/sec. FEATURE | RESEARCH AND INNOVATION 193 (9) where: F ext = External forces acting on the rocket (normally assumed to be zero in space) p = momentum t = time m = mass U = velocity of exhaust propellant V = velocity of rocket | Fig. 4. Mars Mission Characteristics. Feature Nuclear Rockets for Interplanetary Space Missions ı Dr. William Emrich
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atw International Journal for Nuclear Power | 04.2020

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