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Question 17.CSGP.79: A convergent nozzle with exit diameter of 2 cm has an air in......

A convergent nozzle with exit diameter of 2 cm has an air inlet flow of 20°C, 101 kPa (stagnation conditions). The nozzle has an isentropic efficiency of 95% and the pressure drop is measured to 50 cm water column. Find the mass flow rate assuming compressible adiabatic flow. Repeat calculation for incompressible flow.

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Convert ΔP to kPa:

\begin{aligned}& \Delta P =50\,cm \,H _2 O =0.5 \times 9.8064=4.903 \,kPa \\& T _0=20^{\circ} C =293.15 \,K \quad P _0=101 \,kPa\end{aligned}

Assume inlet  V _{ i }=0 \quad P _{ e }= P _0-\Delta P =101-4.903=96.097 \,kPa

\begin{aligned}& T _{ e }= T _0\left(\frac{ P _{ e }}{ P _0}\right)^{\frac{ k -1}{ k }}=293.15 \times\left(\frac{96.097}{101}\right)^{0.2857}=289.01 \\& \mathbf{V} _{ e }^2 / 2= h _{ i }- h _{ e }= C _{ p }\left( T _{ i }- T _{ e }\right)=1.004 \times(293.15-289.01)\end{aligned}

= 4.1545 kJ/kg = 4254.5 J/kg    \Rightarrow \mathbf{V} _{ e }=91.15 \,m / s

\begin{aligned}& \mathbf{V} _{ e\,ac }^2 / 2=\eta \mathbf{V} _{ e\,s }^2 / 2=0.95 \times 4154.5=3946.78 \quad \Rightarrow V _{\text {e \,ac }}=88.85 \,m / s \\& T _{ e a c}= T _{ i }-\frac{ V _{ e\,ac }^2 / 2}{ C _{ p }}=293.15-\frac{3.9468}{1.0035}=289.2 \,K \quad\end{aligned}
\begin{aligned}& \rho_\text {e ac }=\frac{P_e}{R T_p}=\frac{96.097}{0.287 \times 289.2}=1.158 \,kg / m ^3 \\& \dot{ m }=\rho A \mathbf{V} =1.158 \times \frac{\pi}{4} \times 0.02^2 \times 88.85= 0 . 0 3 2 3 ~ k g / s\end{aligned}

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