When gas expands in a cylinder with radius r , the pressure P at any given time

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 done by the engine during a cycle when the steam starts at a pressure of 120 lb/in2 and a volume of 300 in3 and expands to a volume of 700 in3. (Round your answer to two decimal places.)
W =  ft-lb

Explanation / Answer

We have given PV1.4 = k and P=120 lb/in2,V1=300 in3 and V2=700 in3

120*(300)1.4 = k

k=352493.41045

P=k/V1.4

W=integration of (V1=300 to V2=800)(PdV)

=integration of (V1=300 to V2=800)(k/V1.4dV)

=k*[integration of (V1=300 to V2=800)(V(-1.4)dV)]

=k*[V^(-1.4+1)/(-1.4+1)] from V1=300 to V2=800

=k*[-1/(0.4*V(0.4))] from V1=300 to V2=800

=k*[(-1/(0.4*(800)0.4))-(-1/(0.4*(300)(0.4)))]

substitute k=352493.41045

=352493.41045*[(-1/(0.4*(800)(0.4)))+1/(0.4*(300)(0.4))]

=352493.41045*[-0.17246620768+0.2553239219]

=352493.41045*(0.08285771422)

=29206.7982675

converting in-lb to ft-lb

=2433.89985465144 ft-lb

W=2433.89 ft-lb

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