Air flows steadily at the rate of 0.4 kg/s through an air compressor,
entering at 6 m/s with a pressure of 1 bar and a specific volume of
0.85 m3/kg, and leaving at 4.5 m/s with a pressure of 6.9 bar and a
specific volume of 0.16 m3/kg. The internal energy of air leaving is 88
kJ/kg greater than that of the air entering. Cooling water in a jacket
surrounding the cylinder absorbs heat from the air at the rate of 59
kJ/s. Calculate the power required to drive the compressor and the
inlet and outlet pipe cross-sectional areas.
"\\dot m =0.4 \\; kg\/s"
"C_1 =6\\; m\/s, \\quad p_1=100\\times10^3 \\;Pa, \\quad v_1=0.85 \\;m^3\/kg"
"C_2 =4.5\\; m\/s, \\quad p_2=690\\times10^3 \\;Pa, \\quad v_2=0.16 \\;m^3\/kg"
"\\Delta U=88\\times 10^3 \\; J\/kg"
"\\dot Q=-59\\times 10^3 \\; J\/s"
a) "\\dot W -?\n\\;\\;"
b) "A_1 -?"
c) "A_2 -?"
Solution:
"C" - velocity, "p" - pressure, "v" - specific volume, "Z" - height, "U" - internal energy, "A" - cross-sectional area.
a) The steady flow energy equation:
"\\dot Q-\\dot W= \\dot m\\Delta H+\\frac{1}{2} \\dot m(C^2_2-C_1^2)+\\dot mg(Z_2-Z_1)" (*)
"\\Delta H=(U_2+ p_2v_2)-(U_1+p_1v_1)=\\Delta U +p_2v_2-p_1v_1"
"\\Delta H= 88\\times 10^3+690\\times10^3\\times 0.16-100\\times10^3\\times0.85= 113.4\\times 10^3 \\;J\/kg"
"\\frac{1}2 \\dot m(C_2^2-C_1^2)= \\frac{1}2 \\times0.4 \\times (4.5^2-6^2)=-3.15\\; J\/kg"
"Z_1=Z_2"
(*): "-59\\times 10^3 -\\dot W=0.4\\times 113.4\\times 10^3-3.15"
"\\dot W=-104.4 \\;kW", negative sign indicates that the work is done on the system.
b,c) The continuity of mass equation:
"\\dot m=\\frac{C_1A_1}{V_1}=\\frac{C_2A_2}{V_2}"
"A_1=\\frac{\\dot m V_1}{C_1}=0.0566 \\; m^2"
"A_2=\\frac{\\dot m V_2}{C_2}=0.0142 \\;m^2"
Answers:
a) "\\dot W=-104.4 \\; kW"
b) "A_1=0.057 \\;m^2"
c) "A_2=0.014 \\; m^2"
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