A nozzle is designed to expand air from 689 ππππ and 32β to 138 ππππ. Assume an isentropic expansion and negligible approach velocity. The airflow rate is 1.36 kg/s. Calculate (a) the exit velocity; (b) the proper exit cross-sectional area.
(a) Since the expansion is isentropic
"\\frac{T_2}{T_1} = (\\frac{p_2}{p_1})^{\u03b3-1\/\u03b3} \\\\\n\n\\frac{T_2}{305} = (\\frac{138}{689})^{1.4-1\/1.4} \\\\\n\nT_2 = 192.625 \\;K"
Applying steam flow energy equation to nozzle
"m(h_1 + \\frac{v^2_1}{2000} + \\frac{z_1g}{1000}) + Q = m(h_2 + \\frac{v^2_2}{2000} + \\frac{z_2g}{1000}) + w \\\\\n\nm(h_1 + 0 +0) + 0 = m(h_2 + \\frac{v^2_2}{2000} + 0) + 0"
Neglecting potential energy change
"mh_1 = m(h_2 + \\frac{v^2_2}{2000}) \\\\\n\nmc_pT_1 = m(c_pT_2 + \\frac{v^2_2}{2000}) \\\\\n\n1.36 \\times 1.005 \\times 305 = 1.36(1.005 \\times 192.652 + \\frac{v^2_2}{2000}) \\\\\n\nv_2 = 475.205 \\;m\/s"
(b) Mass flow rate "m =\u03c1_1A_1v_1 = \u03c1_2A_2v_2"
From ideal gas equation
"p_2 = \u03c1_2RT_2 \\\\\n\n\u03c1_2 = \\frac{p_2}{RT_2} = \\frac{138}{0.287 \\times 192.652} = 2.4958 \\;kg\/m^3 \\\\\n\nm=\u03c1_2A_2V_2 \\\\\n\n1.36 = 2.4958 \\times A_2 \\times 475.205 \\\\\n\nA_2 = 1.1467 \\times 10^{-3} \\;m^2"
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