Question

Air (ideal gas) is contained in a cylinder/piston assembly at a pressure of 150 kPa and a temperature of 127°C. Assume that the air is compressed adiabatically in a polytropic process with n 1.45 to a pressure of 450 kPa. Is this process possible? Why or why not? Show all work.

Answer 1

Answer:

The process is not possible.

Explanation:

We know for ideal condition, the work done for isothermal process is

$W_{ideal}$ = $P_{1}.V_{1} ln\frac{V_{2}}{V_{1}}$

and for ideal gas, we know  PV = mRT

Therefore, $W_{ideal}$ = mRT$ln\frac{V_{2}}{V_{1}}$

                                                  = mRT$ln\frac{P_{1}}{P_{2}}$

                                                  =  0.287 x 400$ln\frac{150}{450}$

                                                  = -126.12 kJ/kg (negative sign indicates that the process is compressive, so work input to the compressor is 126.12 kJ/kg )

Now we know for adiabatic compression process

                    P$V^{\gamma }$ = C

We know $\frac{T_{2}}{T_{1}}=(\frac{P_{2}}{P_{1}})^{\frac{\gamma -1}{\gamma }}$

$T_{2}$ = 556 K

For adiabatic work done, $W_{adiabatic}$ = $\frac{P_{1}\times V_{1}-P_{2}\times V_{2}}{\gamma -1}$

                                                                       = $\frac{mR(T_{1}-T_{2})}{\gamma -1}$

                                                                       = $\frac{0.287(400-556)}{1.45 -1}$

                                                                       = -99.49 kJ/kg (negative sign indicates that the process is compressive, so work input to the compressor is 99.49 kJ/kg )

We know that in isothermal process, work input to the compressor is minimum. But in the above adiabatic polytropic process, work input to the compressor is less than the work done in the isothermal process.

Thus the process is not possible.