Your chemical plant manufactures a chemical which has a constant heat capacity of CV = 3R, is in the gas phase at all process conditions, and can be modeled as an ideal gas as long as the pressure is below 5 atm. There is a point in the process where the compound is at P=2 atm and T=400 K, and must undergo a reduction in pressure to P=1 atm. Currently, this is done by a steady-state adiabatic throttling valve. Your boss has asked you to examine the possibility of replacing this valve with a turbine, so that some useful work can be obtained from the necessary drop in pressure. The efficiency of the turbine would be 75%.

Question

A) In the current process, what is the temperature of the gas leaving the valve?
B) How much work would the turbine produce for each mole of entering gas?
C) What would be the actual temperature of the gas leaving the turbine?

Answer 1

A) The temperature of the ideal gas leaving the valve is the same as the inlet temperature. This is because an adiabatic valve is an isenthalpic unit operation and the enthalpy of an ideal gas is a function of temperature only. Hence, Tout = 400 K.

B) In order to calculate the actual work the turbine would produce, the definition of the turbine’s efficiency must be applied, relating the reversible work of the turbine to the actual work:



Where W ?_(s,rev) is calculated through an energy balance around the reversible model of the turbine:



Where ??H_rev=C_P^* (T_(out,rev)-T_in ) for the ideal gas

Where? C?_P^*=3R+R and T_(out,rev) of the ideal gas is calculated through a reversible entropy balance:



Plugging T_(out,rev) into the reversible change in enthalpy equation:





D) The actual temperature of the gas leaving the turbine is found by first applying an energy balance around the actual turbine:



Applying the change in enthalpy of an ideal gas equation leads to the calculation of Tout,actual: