Calculus

Supersonic convergent-divergent nozzle is used to accelerate the gas (here air) from subsonic to supersonic velocities. Let V. be the actual (non-isentropic) velocity at the exit area of the nozzle and V is that for isentropic flow conditions. The kinetic energy efficiency n, for the nozzle is defined by: V2 n. n, ranges between 0.80 to 0.999. In convergent-divergent nozzle, the flow is accelerated in the convergent section to sonic velocity which happens at the minimum area (throat) of the nozzle. Then the flow is accelerated further in the divergent section to supersonic velocity. Under isentropic conditions the Mach number at the throat is unity (M2=1) corresponding to maximum flow rate (or choked condition). In real nozzles, the Mach number at the throat is given by: M’ (1,-1)+ M – 2n mo 2-44:1) 37, (6-1 (y – 1) Assuming the specific heat ratio y = 1.4 and 1, = 0.95, calculate the throat Mach number M, using: = 1. The Bisection method 2. The Newton-Raphson method 3. The Newton-Raphson method with the derivative evaluated numerically. 4. The Secant method In each part, carry out two iterations by hand and then use MATLAB or excel. Use an absolute error tolerance of 10-6 You can check your answer by solving the equation analytically. This is probably the only form of the fourth order algebraic equation that can be solved analytically as it can be put in a second order polynomial form.

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