Air Turbine with Significant Exit Kinetic Energy: 04-BS-10, December 2016, Q8

Question 8: Air Turbine with Significant Exit Kinetic Energy (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Inlet: $P_1=500$ kPa, $T_1=900$ K, $V_1\approx0$. Exit: $P_2=100$ kPa, $T_2=600$ K, $V_2=100$ m/s. $\dot m=10$ kg/s.

Find. (a) $\dot W$ [kW]; (b) $A_2$ [m$^2$].

Approach

Include the exit kinetic-energy term in the steady-flow energy balance since the inlet velocity is negligible but the exit is not. The exit area then follows from the mass-flow/continuity relation $\dot m=\rho_2A_2V_2$, using the ideal-gas specific volume at the exit state.

Energy balance (part a). $h_1=h(900\ \text{K})=1059.40$ kJ/kg, $h_2=h(600\ \text{K})=733.42$ kJ/kg. $$w=(h_1-h_2)-\frac{V_2^2}{2}=(1059.40-733.42)-\frac{100^2}{2\times1000}=325.99-5.00 =\boxed{320.99\ \text{kJ/kg}}.$$ $$\dot W=\dot m\,w=10\times320.99=\boxed{3209.9\ \text{kW}}\quad\textbf{(a)}.$$
Exit specific volume and area (part b). Ideal gas at the exit state: $$v_2=\frac{RT_2}{P_2}=\frac{0.287\times600}{100}=1.722\ \text{m}^3/\text{kg}.$$ $$A_2=\frac{\dot m\,v_2}{V_2}=\frac{10\times1.722}{100}=\boxed{0.1722\ \text{m}^2}\quad\textbf{(b)}.$$

Quantity	Result
(a) $\dot W$	3209.9 kW
(b) $A_2$	0.1722 m²

Concept & Theory

Topic. Steady-flow air turbine with non-negligible exit kinetic energy, sized for exit flow area.

Key relations. Energy balance with a kinetic-energy term, $w=(h_1-h_2)-V_2^2/2$; continuity $\dot m=A_2V_2/v_2$ rearranged for $A_2$.

Why this works. The exit KE term ($5.00$ kJ/kg) is small here relative to the enthalpy drop ($325.99$ kJ/kg) but is explicitly flagged as non-negligible by the problem statement and must still be subtracted from the available enthalpy drop before it becomes shaft work.

Common pitfall. Forgetting to convert $V_2^2/2$ from J/kg to kJ/kg (divide by 1000) before combining it with the enthalpy terms, which are already in kJ/kg.

Source. Cengel & Boles, Thermodynamics: An Engineering Approach, 8th ed., Ch. 5 (Mass and Energy Analysis § Steady-Flow Devices).

Question 8 of 9: Air Turbine with Significant Exit Kinetic Energy

Question 8: Air Turbine with Significant Exit Kinetic Energy (15 marks)

Approach