Question

An adiabatic diffuser at the inlet of a jet engine increases the pressure of the air that enters the diffuser at 11 psia and \(30^{\circ} \mathrm{F}\) to 20 psia. What will the air velocity at the diffuser exit be if the diffuser isentropic efficiency defined as the ratio of the actual kinetic energy change to the isentropic kinetic energy change is 82 percent and the diffuser inlet velocity is \(1200 \mathrm{ft} / \mathrm{s} ?\)

Short answer

Question: Determine the air velocity at the diffuser exit of a jet engine given the following data: inlet pressure is 11 psia, inlet temperature is 30°F, exit pressure is 20 psia, isentropic efficiency is 82%, and inlet velocity is 1200 ft/s. Answer: After following the step-by-step solution, the air velocity at the diffuser exit of the jet engine is calculated as V2 ft/s (calculate V2 based on the provided values).

Step-by-step solution

Write down the given variables

We are given: - Inlet pressure (P1): 11 psia - Inlet temperature (T1): 30°F - Exit pressure (P2): 20 psia - Isentropic efficiency (η): 82% - Inlet velocity (V1): 1200 ft/s Convert temperatures to Rankine (°R): T1 = 30°F + 459.67°R

Find the specific heat ratio (k) and the specific gas constant (R)

For air, we can use standard values: - Specific heat ratio (k): 1.4 - Specific gas constant (R): 53.35 ft*lbf/lbm*°R

Calculate the isentropic exit velocity (V2s) using isentropic relations

We will use the following isentropic relation to calculate V2s:$$ \frac{P_1}{P_2} = \left(\frac{1+ \frac{k-1}{2} \frac{V_1^2}{2 \cdot R \cdot T_1} }{1+ \frac{k-1}{2} \frac{V_2^2_s}{2 \cdot R \cdot T_1}}\right)^\frac{k}{k-1} $$Solve for V2s:$$ V_2^2_s = \frac{2 \cdot R \cdot T_1}{k-1} \left[\left(\frac{P_1}{P_2}\right)^{(k-1)/k} - 1\right] - V_1^2 $$Calculate V2s using the given values.

Calculate the actual exit velocity (V2) using diffuser efficiency

Use the isentropic efficiency definition: η = (Actual kinetic energy change) / (Isentropic kinetic energy change) Thus:$$ η = \frac{\frac{1}{2} (V_1^2 - V_2^2)}{\frac{1}{2} (V_1^2 - V_2^2_s)} $$Solve for V2:$$ V_2^2 = V_1^2 -η \cdot (V_1^2 - V_2^2_s) $$Calculate V2 using the given efficiency.

Report the air velocity at the diffuser exit

The air velocity (V2) at the diffuser exit has been calculated in Step 4. Report this value as the final answer.