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$\begin{array}{l}{\text { An air compressor compresses } 6 \mathrm{L} \text { of air at } 120 \mathrm{kPa} \text { and } 20^{\circ} \mathrm{C} \text { to } 1000 \mathrm{kPa} \text { and } 400^{\circ} \mathrm{C} \text { . Determine the flow work, }} \\ {\text { in } \mathrm{kJ} / \mathrm{kg} \text { , required by the compressor. }}\end{array}$

$A-1 \rightarrow R=0.287 \frac{kPa*m^3}{kg.k}$

$W_{flow}=P_{2} v_{2}-P_{1} v_{1}$

$=R\left(T_{2}-T_{1}\right)$

$=0.287(400-20)$

$=109 \mathrm{kJ} / \mathrm{kg}$

$\begin{array}{l}{\text { Air flows steadily in a pipe at } 300 \mathrm{kPa}, 77^{\circ} \mathrm{C}, \text { and } 25 \mathrm{m} / \mathrm{s} \text { at a rate of } 18 \mathrm{kg} / \mathrm{min} \text { . }} \\ {\text { Determine (a) the diameter of the pipe, }(\mathrm{b}) \text { the rate of flow energy, }}\end{array}$

$\begin{array}{l}{\text { (c) the rate of energy transport by mass, and (d) the error involved in part (c) }} \\ {\text { if the kinetic energy is neglected. }}\end{array}$

$R=0.287 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}$

$C_{P}=1.008 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{k}$

(a) $P{v}=R T$

$\rightarrow v=\frac{R T}{p}$$=\frac{0.287 *(77+273)}{300}=0.3349 \mathrm{m}^{3} / \mathrm{kg}$

$A=\frac{\dot{m} v}{V} \rightarrow \dot{m}=\frac{AV}{v} \rightarrow \frac{18/60*0.3349}{25}$

$=0.004018 \mathrm{m}^{2}$

$A=\frac{\pi D^{2}}{4} \rightarrow D=\sqrt{\frac{4 A}{\pi}}=\sqrt{\frac{4 * 0.004018}{\pi}}=0.0715 m$

(b) $\dot{W}_{flow}=\dot{m} P_{V}=\frac{18}{60}(300)(0.3349)=30.14kw$

(c) $\dot{E}_{\text {mass }}=\dot{m}\left(h+k{c}\right)=\dot{m}\left(C_{p} T+\frac{1}{2} V^{2}\right)$

$=\left(\frac{18}{60}\right)\left(1.008 *(77+273)+\frac{1}{2}(25)^{2} \frac{1}{1000}\right)$

$=105.94 \mathrm{Kw}$

$\dot{E}_{mass }=\dot{m } h=\frac{18}{60}\left(C_{P} T\right)=\frac{18}{60} * 1.005*(77+273)=105.84 kw$

$\text { Error } \% =\frac{|105.84-105.94|}{105 .94} * 100=0.09 \%$