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Engineering · Mechanical Engineering
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(a) Derive, from the energy equation, the following relationship between the total temperature Tt the static temperature Tand the Mach number M: T, [5 marks] ) Use the energy equation to derive the equation for the steady-flow adiabatic ellipse Here V is the gas velocity, and a and at are the speed of sound based orn the local static and total temperatures respectively. Vmax is the maximum velocity the gas could obtain if it were hypothetically expanded to zero temperature [5 marks] Air is at rest in a large reservoir where the total temperature is 1000 K. Calculate Vmax, at, a*, and V* (where the * superscript refers to critical conditions, where the Mach number is 1). Sketch (rather than plot accurately) the adiabatic ellipse, quantitatively marking the above calculated values [5 marks] ) The air flows steadily from the reservoir through an adiabatic duct of changing area to a Mach number of 3 where it passes though a normal shock. Confirm (within the accuracy of the tables) that both the upstream and downstream conditions, in terms of (V, a), are located on the adiabatic ellipse. Add their location to your sketch above [5 marks] ) Sketch the temperature-entropy diagram for the flow from stagnation conditions to and then through the shock wave, marking salient temperatures and pressures. Determine the increase in non-dimensional entropy As/R across the shock. Assume isentropic flow upstream of the shock. [5 marks]

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