Air is flowing in a duct with velocity of 7.62 m/s and a static pressure of 2.16 cm water gage. The dud diameter is 1.22 m, the barometric pressure is 99.4 kPa and the gauge fluid temperature and air temperature are 30 °C. What is the total pressure of air against which the fan will operate?

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Ans.\begin{array}{l}V=7.62 \mathrm{~m} / \mathrm{s} . \\P_{1}=2.16 \mathrm{~cm} \text { water 2age. } \\D=1.22 \mathrm{~m} . \\P_{\text {baro }}=\text { atmospheric pressure }=99.4 \mathrm{kpq} . \\T_{1}=30^{\circ} \mathrm{C} .=30+273.15=303.15 \mathrm{k}\end{array}absolute siatic pressure.\begin{array}{l}P_{1 a b s}=P_{1}(g a 4 b e)+p_{b a r o} \\P_{1 a b s}=(e . g h)+99.4 \times 10^{3} \\P_{1 a b s}=1000 \times 9.81 \times\left(\frac{2.16}{100}\right)+99.4 \times 10^{3} \\P_{1 a b s}=211.896+99.4 \times 103 \\P_{1 a b s}=99611.896 \mathrm{~Pa}=99.611 \mathrm{kpq}\end{array}\begin{array}{l}\text { Velocity of sound }(C) \Rightarrow \\\qquad C=\sqrt{K R T_{1}}=\sqrt{1.4 \times 287 \times 303.15}\end{array}c=349 \mathrm{~m} / \mathrm{s} /Mach Number\begin{array}{l}M_{a}=\frac{V}{C} \\M_{a}=\frac{7.62}{349} \\M_{a}=0.021\end{array}Now total pressure \left(P_{0}\right)\begin{array}{l}\Rightarrow \frac{P_{0_{1}}}{P_{1 b_{s}}}=\left(1+\frac{k-1}{2}\left(M_{a}\right)^{2}\right)^{\frac{K-1}{k}} \\\Rightarrow \frac{P_{0_{1}}}{P_{1 a_{s}}}=\left(1+\frac{0.4}{2} \times(0.021)^{2}\right) \frac{1.4}{0.4} \\\Rightarrow P_{0_{1}}=1.003087 \times 99.611 \\\Rightarrow P_{0,}=99.64 \mathrm{kpq}\end{array}Ans. ...