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# An adiabatic air compressor compresses $10 \mathrm{~L} / \mathrm{s}$ of air at $120 \mathrm{kPa}$ and $20^{\circ} \mathrm{C}$ to $1000 \mathrm{kPa}$ and $300^{\circ} \mathrm{C}$. Find $(a)$ the work needed by the compressor, in $\mathrm{kJ} / \mathrm{kg}$, and (b) the power needed to drive the air compressor, in $k W$.

Solution

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We have the following data:

Inlet

$P_1$ = 120 kPa

$T_1$ = $20 \text{\textdegree}$C + 273 = 293 K

$\dot{V}_1$ = 10 $\dfrac{L}{s}$ $(\dfrac{1 \ m^{3}}{1000 \ L})$ = 0.01 $m^{3}$

Outlet

$P_1$ = 1000 kPa

$T_1$ = 300 $\text{\textdegree}$C + 273 = 573 K

These are the assumptions in compressors for this problem:

Negligible heat transfer, $\dot{Q}$ = 0 (well insulated)

Negligible changes in potential energy, $\Delta$pe = 0

Negligible changes in kinetic energy, $\Delta$ke = 0

$\textbf{a) Work required, W}$: The work required can be calculated by applying the energy balance equation by net energy to the system. The energy balanced equation for a steady- flow process compressor including the assumptions can be expressed as

$E_{in} = E_{out}$

$W_{in} + h_1 = h_2$

$W_{in} = h_2 - h_1$

Since enthalpy, $\textcolor{#4257b2}{h = c_pT}$ then

$W_{in} = C_p(T_2 - T_1)$

Where

$C_p$ = heat capacity of air @ average of $T_1$ and $T_2$

$T_{avg}$ = $\dfrac{T_1 + T_2}{2}$ = $\dfrac{293 + 573}{2}$ = 433 K

As per TABLE A-2, Ideal-gas specific heats of various common gases

@400 K, specific heat $C_p$ = 1.013 \ $\dfrac{kJ}{kg \cdot K}$

@450 K, specific heat $C_p$ = 1.020 \ $\dfrac{kJ}{kg \cdot K}$

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