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Air is compressed in an axial-flow compressor operating at steady state from $27^{\circ} \mathrm{C},$ 1 bar to a pressure of 2.1 bar. The work required is 94.6 kJ per kg of air flowing. Heat transfer from the compressor occurs at an average surface temperature of $40^{\circ} \mathrm{C}$ at the rate of 14 kJ per kg of air flowing. The effects of motion and gravity can be ignored. Let $T_{0}=20^{\circ} \mathrm{C}, p_{0}=1$ bar. Assuming ideal gas behavior, (a) determine the temperature of the air at the exit, in $^\circ C,$ (b) determine the rate of exergy destruction within the compressor, in kJ per kg of air flowing, and (c) perform a full exergy accounting, in kJ per kg of air flowing, based on work input.

Ambient air at 100 kPa and 300 K is compressed isentropically in a steady-flow device to 0.8 MPa. Determine (a) the work input to the compressor, (b) the exergy of the air at the compressor exit, and (c) the exergy of compressed air after it is cooled to 300 K at 0.8 MPa pressure.

Air is compressed in an axial-flow compressor operating at steady state from $27^{\circ} \mathrm{C}, 1$ bar to a pressure of $2.1$ bar. The work input required is $94.6 \mathrm{~kJ}$ per $\mathrm{kg}$ of air flowing through the compressor. Heat transfer from the compressor occurs at the rate of $14 \mathrm{~kJ}$ per kg at a location on the compressor's surface where the temperature is $40^{\circ} \mathrm{C}$ . Kinetic and potential energy changes can be ignored. Find

(a) the temperature of the air at the exit, in ${ }^{\circ} \mathrm{C}$ .

(b) the rate at which entropy is produced within the compressor, in $\mathrm{kJ} / \mathrm{K}$ per $\mathrm{kg}$ of air flowing.

Question

Air is compressed in an axial-flow compressor operating at steady state from $27^{\circ} \mathrm{C}, 1\ \text{bar}$ to a pressure of $2.1\ \text{bar}$ . The work required is $94.6 \mathrm{~kJ}$ per $\mathrm{kg}$ of air flowing. Heat transfer from the compressor occurs at an average surface temperature of $40^{\circ} \mathrm{C}$ at the rate of $14 \mathrm{~kJ}$ per kg of air flowing. The effects of motion and gravity can be ignored. Let $T_0=20^{\circ} \mathrm{C}, p_0=1\ \text{bar}$ . Assuming ideal gas behavior, determine (a) the temperature of the air at the exit, in ${ }^{\circ} \mathrm{C}$ , and (b) the rate of exergy destruction within the compressor, in $\mathrm{kJ}$ per $\mathrm{kg}$ of air flowing,

Solution

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$\text{\color{#4257b2} a)}$ In this part of the problem we need to calculate the final temperature $T_2$ . We will use the energy balance equation with the given assumptions that we can neglect motion and gravity effects and the ideal gas model for air. We will also need the given initial temperature $T_1=300\,\text{K}$ , work done $w=94.6\,\text{kJ/kg}$ and heat transfer $q=14\,\text{kJ/kg}$ .

\begin{align*} 0&=q - w + (h_2 - h_1) \\ 0&=q - w + c_p \cdot (T_2 - T_1) \\ 0&=14\,\frac{\text{kJ}}{\text{kg}} - 94.6\,\frac{\text{kJ}}{\text{kg}} + 1.01\,\frac{\text{kJ}}{\text{kg K}} \cdot (T_2 - 300\,\text{K}) \\ T_2&=\boxed{\color{#c34632}380\,\text{K}} \\ \end{align*}

$\text{\color{#4257b2} b)}$ In this part of the problem we will calculate the exergy destruction $\dot{e}_d$ using the exergy rate balance equation. For the calculation we will also need the given surface temperature $T_b=313\,\text{K}$ and ambient temperature $T_0=293\,\text{K}$ and pressures $p_1=1\,\text{bar}$ and $p_2=2.1\,\text{bar}$ .