Question

Gas contained in a piston-cylinder assembly undergoes a polytropic process for which the relation between pressure and volume is given by p∀ n = constant. The initial volume is 0.04 m3 and the initial pressure is 2 bar. The final volume of the gas is 0.1 m3 , For each of the following cases, determine the final pressure [bar] and the work for the process [kJ]. Show integration for each case. a. n = 0, b. n = 1, c. n = 1.4.

Answer 1

a) P = 2 bar, W = 12 kJ

B)P = 0.8 bar, W = 7.33 kJ

C) P = 0.55 bar, W = 6.14 kJ

Step-by-step explanation:

`pV^n = constant.`

The initial volume
`V_1 = 0.1 m3,`

The final volume
`V_2 = 0.04 m3`,

The initial pressure
`P_1 = 2 bar.</p><p>`

We know that

`P_1 V_1^n = P_2 V_2^n,`

`P_2 = P_1((V_1)/(V_2)^n)`

`= 2((0.04)/(0.1)^n)`

`a) n = 0,
`P_2 = 2((0.04)/(0.1)^0)  = 2 bar`

`W = P_2(V_2 - V_1) = 2*100 kPa * (0.06 m3) = 12 kJ`

b) n = 1,
`P_2 = 2((0.04)/(0.1)^1)  = 0.8 bar`

`W = P_2 V_2 ln((V2)/(V1) = 7.33 kJ`

c) n = 1.4,
`P_2 = 2((0.04)/(0.1)^1.3)  = 0.5542 bar`

`W = ((P_2 V_2 - P_1 V_1))/((1 - n)) = 6.14 kJ`