Question

A gas which obeys the equation of state, PV/RT=1 +BP , where B is a known
temperature-dependent parameter, is fed at a steady rate to an insulated valve where its pressure is reduced from P1 to P2. Given that the upstream temperature of the gas is T1 and that the constant-pressure molar heat capacity of the gas in the ideal gas state is given by Cp* = a + bT, where a and b are known constants, derive an expression for the temperature T2 of the gas downstream from the valve. Express your answer in terms of the parameters given above. Neglect the effect of any kinetic energy change across the valve.

(Chemical engineering)

Answer 1

This problem involves throttling of gas which obeys the equation of state

PV = RT(1+BP) ..............(1)

Let, subscript 1 represent the inlet condition and subscript 2 represent the outlet condition

Initial pressure and temperature are denoted by P₁ and T₁ respectively

Final pressure and temperature are denoted by P₂ and T₂ respectively

The molar heat capacity of the gas is given by the equation

C_p = a + bT ............(2)

Take energy balance across the valve

.............(3)

Let,

Q = Heat energy

W = work

K.E. = Kinetic energy

P.E. =Potential Energy

E_in = E₁ = Q₁ + W₁ + K.E.₁ + P.E.₁

E_out = E₂ = Q₂ + W₂ + K.E.₂ + P.E.₂

Substitue in eq.(3)

E = (Q₁ + W₁ + K.E.₁ + P.E.₁) - (Q₂ + W₂ + K.E.₂ + P.E.₂) ...........(4)

As the valve is insulated, there will be no Heat interaction

Hence Q₁ = Q₂ = 0

There is no change in elevation

Hence P.E.₁ - P.E.₂ = 0

Change in kinetic energy is negligible

Hence K.E.₁ - K.E.₂ = 0

rewriting Eq.(4)

E = W₁ -W₂

For a steady state process E = 0

Hence W₁ -W₂ = 0

W_total = 0

Consider the throttling process is carried out in two steps

1) Constant pressure expansion (i.e. at P₁) to change temperature from T₁ to T₂

2) Constant temperature expansion (i.e. at T₂) to reach pressure from P₁ to P₂

Work for constant pressure process is calculated as

W = PV

W₁ = P₁(V₂ - V₁) ................(5)

From eq.(1)

PV = RT(1+BP)

PV - BPRT = RT

P(V - BRT) = RT

P = RT / (V - BRT)

For constant pressure process, we can write

Rearranging above equation we get

V₂RT₁ = V₁RT₂

substitute in eq.(5)

......................(6)

Now, calculate work for constant temperature process

........................(7)

Adding eq.(6) and eq.(7)

..................................................Ans.