This is a thermo question please help show steps or show lin

This is a thermo question. please help show steps or show link to solve. Will rate quickly!

In a vortex tube, air enters perpendicular to the tube axis and flows in a swirling motion around the inside periphery of the tube. Because of centrifugal forces, the air is separated into a hot and a cold stream with one stream exiting at the top and the other exiting at the bottom. It is proposed to construct a vortex tube to use as a refrigeration system. Air enters at 20 degree C and 300 kPa and exits from the top at 60 degree C and 280 kPa and from the bottom at 0 degree C and 280 kPa. The tube is perfectly insulated. Is it possible for such a device to operate as described?

Solution

As per the second law of thermodynamics states that the total entropy of an isolated system always increases over time, or remains constant in ideal cases where the system is in a steady state or undergoing a reversible process.

So I will check Entropy of this system.

P₁ = 300 kPa, T₁ = 20 C or 293 K

P₂ = 280 kPa, T₂ = 0 C or 273 K

P₃ = 280 kPa, T₃ = 60 C or 433 K

Entropy Change = C_p ln(T_2/3/T₁) – R ln(P_2/3/P₁)   [C_P = 1.00 kJ/kg.K and R=0.287 kJ/kg.K ]

S = S₁ + S₂

S₁ = (1.00) ln (273/293) – (0.287) ln (280/300)

    = (-0.0707) – ( -0.0198)

   = -0.0509

S₂ = (1.00) ln (433/293) – (0.287) ln (280/300)

= (0.3905) – (-0.0198)

   = 0.4103

S = (-0.0509) + (0.4103)

= 0.3594

So yes this is possible as Del S is positive.