When gas expands in a cylinder with radius r , the pressure P at any given time
ID: 2843995 • Letter: W
Question
When gas expands in a cylinder with radius r, the pressure P at any given time is a function of the volume V: P = P(V). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: F = ?r2P. The work done by the gas when the volume expands from volume V1 to volume V2 is W.
In a steam engine the pressure and volume of steam satisfy the equation PV1.4 = k, where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Calculate the work W done by the engine during a cycle when the steam starts at a pressure of120 lb/in2 and a volume of 300 in3 and expands to a volume of 600 in3. (Round your answer to three significant digits.)
W = ft-lb
Explanation / Answer
Using initial conditions we find the value of k
P = 120lb/in^2
V = 300 in^3
k = PV^1.4 = 352493.4
Now, P = k/V^1.4
So, W = Integral(kdV / V^1.4) from 300 in^3 to 600 in^3
W = -k/ 0.4 * V^0.4
applying limits, V1 = 300 in^3 and V2 = 600 in^3 and replacing value of k, we get we get
W = 21.792 x 10^3 lb-in = 1816.063 ft-lb
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