Figure P2.70 shows a gas contained in a vertical piston-cylinder assembly. A ver
ID: 1817671 • Letter: F
Question
Figure P2.70 shows a gas contained in a vertical piston-cylinder assembly. A vertical shaft whose cross-sectional area is 0.8 cm2 is attached to the top of the piston. The total mass of the piston and shaft is 25 kg. While the gas is slowly heated, the internal energy of the gas increases by 0.1 kJ, the potential energy of the piston-shaft combination increases by 0.2 kJ, and a force of 1334 N is exerted on the shaft as shown in the figure. The piston and cylinder are poor conductors, and friction between them is negligible. The local atmospheric pressure is 1 bar and g = 9.81 m/s2. Determine, (a) the work done by the shaft, (b) the work done in displacing the atmosphere and (c) the heat transfer to the gas, all in kJ. (d) Using calculated and given data, develop a detailed accounting of the heat transfer of energy to the gas.Explanation / Answer
at equilibrium, pressure inside the cylinder is equal to pressure exerted by the piston and the shaft and atmospheric pressure
mass of piston and shaft = 25kg, diameter of the piston = 10cm
pressure inside the cylinder = 1atm + 25*9.81/(0.05^2) = 1,31,226.2N/m^2
from 1st law of thermodynamics,
heat supplied = change in internal energy + external work
dQ = dU + dW
potential energy change in the piston is the work done by the gas
given dU = 0.1kJ, dW = 0.2kJ
heat transferred to the gas = 0.3kJ
potential energy = mgh = 0.2kJ
h=0.2/mg
=0.2*1000/(25*9.81)=0.8154m=81.55cm
volume swept (v) = r^2h=*5^2*81.55= 6404.877cm^3 =6404.877*10^-6 m^3
slow heating implies quasi static
work done in displacing the atmosphere = pV=10^5 * 6404.877 *10^-6=640.287J
external force of 1334N is acting on the shaft of area 0.8cm^2
pressure acting on the gas due to external force = F/A= 1334/0.8 * 10^4 = 166.75 bar
total external pressure = atmospheric + 166.75 = 167.75 bar
pressure in side the cylinder = 1.31 bar
knowing any of the intial conditions, the thermodynamic cycle and ideal gas equation, work done by the shaft can be found out using p1v1-p2v2, where p1, p2 ; v1,v2 are the initial and final pressures and volumes respectively.
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