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Question 17.CSGP.81: A convergent nozzle is used to measure the flow of air to an......

A convergent nozzle is used to measure the flow of air to an engine. The atmosphere is at 100 kPa, 25°C. The nozzle used has a minimum area of 2000 mm2 and the coefficient of discharge is 0.95. A pressure difference across the nozzle is measured to 2.5 kPa. Find the mass flow rate assuming incompressible flow. Also find the mass flow rate assuming compressible adiabatic flow.

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Assume  Vi≅0,vi=RTi/Pi=0.287×298.15/100=0.8557 m3/kg\mathbf{V} _{ i } \cong 0, \quad v _{ i }= RT _{ i } / P _{ i }=0.287 \times 298.15 / 100=0.8557 \,m ^3 / kg

Incompressible flow:

Ve,s2/2=hi−he,s=vi(Pi−Pe)=2.1393 kJ/kgVe,s=2×1000×2.1393=65.41 m/sm˙s=AVe,s/vi=2000×10−6×65.41/0.8557=0.153 kg/sm˙a=CDm˙s=0.1454 kg/s\begin{aligned}&\mathbf{V} _{ e , s }^2 / 2= h _{ i }- h _{ e , s }= v _{ i }\left( P _{ i }- P _{ e }\right)=2.1393 \,kJ / kg \\& \mathbf{V} _{ e , s }=\sqrt{2 \times 1000 \times 2.1393}=65.41\, m / s \\& \dot{ m }_{ s }= A \mathbf{V} _{ e , s } / v _{ i }=2000 \times 10^{-6} \times 65.41 / 0.8557=0.153 \,kg / s \\& \dot{ m }_{ a }= C _{ D } \dot{ m }_{ s }= 0 . 1 4 5 4 \,k g / s\end{aligned}

Compressible flow:

Te,s=Ti(Pe/Pi)(k−1)/k=298.15(97.5/100)0.2857=296 KΔh=CpΔT=1.0035×2.15=2.1575=Ve,s2/2Ve,s=2×1000×2.1575=65.69 m/s\begin{aligned}& T _{ e , s }= T _{ i }\left( P _{ e } / P _{ i }\right)^{( k -1) / k }=298.15(97.5 / 100)^{0.2857}=296 \,K \\& \Delta h = C _{ p } \Delta T =1.0035 \times 2.15=2.1575= \mathbf{V}_{ e , s }^{2 }/2 \\& \mathbf{V} _{ e , s }=\sqrt{2 \times 1000 \times 2.1575}=65.69 \,m / s\end{aligned}

ve,s=0.287×296/97.5=0.8713 m3/kgm˙s=AVe,s/ve,s=2000×10−6×65.69/0.8713=0.1508 kg/sm˙a=CDm˙s=0.1433 kg/s\begin{aligned}& v _{ e , s }=0.287 \times 296 / 97.5=0.8713 \,m ^3 / kg \\& \dot{ m }_{ s }= A\mathbf{V} _{ e , s } / v _{ e , s }=2000 \times 10^{-6} \times 65.69 / 0.8713=0.1508 \,kg / s \\& \dot{ m }_{ a }= C _{ D } \dot{ m }_{ s }= 0 . 1 4 3 3 ~ k g / s\end{aligned}

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