Answer:
Question 1 (a) (i) Explain the term the effective exhaust velocity. (ii) Is it greater or smaller than the exhaust velocity? [2 marks] (b) (i) Calculate the change of velocity, Av, of a spacecraft of mass m and initial velocity v after the ejecting a small mass of propellant Amp with the velocity v relative the spacecraft. (ii) Passing to infinitesimal Amp, Av and integrating the obtained differential equation, derive Tsialkovky's equation. [2 marks] (c) (i) State two possible definitions of the specific impulse? (ii) Explain the words "specific" and "Impulse" in this term? (iii) Which definition is more often used and why? [3 marks] (d) A rocket engine burning liquid oxygen and kerosene operates at a combustion chamber pressure of 30 MPa. The nozzle is expanded to operate at the ambient pressure of 18 kPa. The specific impulse equals 340 s at this ambient pressure. Find its combustion chamber temperature. Adiabatic constant of the exhaust gas is 1.20, its molar weight is 23.2. [2 marks] (e) Find the mass flow rate of this engine described in Q1(d) if it produces 2.4 MN of thrust at the sea level (ambient pressure is 101 kPa). The exit diameter of the nozzle is 1.3 m. [2 marks] (f) A 15,000 kg spacecraft is in Earth orbit traveling at a velocity of 7,900 m/s. Its engine is burnt to accelerate it to a velocity of 11.2 km/s to reach the escape orbit. The engine expels mass at a rate of 125 kg/s and has a specific impulse of 430 s. Calculate the duration of the burn. [3 marks]
(a) (i) The effective exhaust velocity is a notional velocity that measures the efficiency of a reaction mass engine, such as a rocket or jet engine, in creating thrust by using its propellant more effectively. It is the speed at which the engine ejects its propellant, taking into account the mass of the combustion air that is not being accounted for in the calculation of actual exhaust velocity. (ii) The effective exhaust velocity is higher than the exhaust velocity because the latter only accounts for the mass of the propellant being ejected, while the former includes the acceleration of additional mass such as air that the engine has to process. (b) (i) The change of velocity, Av, of a spacecraft of mass m and initial velocity v after ejecting a small mass of propellant Amp with velocity v relative to the spacecraft is given by Av = Amp * Ve * ln(m0/m), where Ve is the effective exhaust velocity and m0 is the initial mass of the spacecraft and its propellant . (ii) Tsialkovky's equation is derived by passing to infinitesimal Amp, Av, and integrating the obtained differential equation. It gives the relationship between the effective exhaust velocity, the specific impulse, and the change in the mass of the spacecraft as it expels propellant. (c) (i) Two possible definitions of specific impulse are: (1) the change in momentum per unit mass of propellant used by the engine, or (2) the amount of time the engine can accelerate its own initial mass at 1g. (ii) The word "specific" means per unit mass of propellant used, while "impulse" refers to the change in momentum experienced by the engine due to its use of propellant. (iii) The first definition is more often used because of its application to the calculation of rocket performance, particularly in terms of the needed delta-v to reach a given destination. (d) The combustion chamber temperature for the given rocket engine can be found using the specific impulse formula, Isp = (g0 * Ve) / (gc * Cstar), where g0 is the standard gravity, Ve is the effective exhaust velocity, gc is the gravitational constant, Cstar is the characteristic velocity, and Isp is the specific impulse. Solving for Cstar and using the given values, we can find the combustion chamber temperature using the formula T1 = (2 * Cstar^2 * M)/(R * (k-1)), where T1 is the combustion chamber temperature, M
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