Peng-Robinson

1. Peng-Robinson

2. Calculation Needs • To calculate a complete P-H relationship diagram you need: • An EOS (for example: Peng-Robinson or better). • A vapor pressure equation. • An “ideal gas” heat capacity equation.

3. EOS • EOS allows one to compute the specific volume from P and T, the pressure from V and T or the temperature from V and T. • EOS does not hold in the two phase region • At P’s above the vapor pressure in the vapor region or at P’s below the vapor pressure in the liquid region • Care must be taken to get the correct V for a combination of T and P. • V is near Videal gas in the vapor region. • V is near (1.2 – 2 x greater than b) in the liquid region

4. Vapor Pressure • A good relationship between the vapor pressure and temperature in the two phase region is needed. • Antoines equation • Others • Pvap vs Tsat equation can be used to find the heat of vaporization or entropy of vaporization at a particular temperature.

5. DSvap • Pick a temperature and find Vvap and Viq • Numerically find the derivative of the Pvap vs Tsat equation at that temperature. • Multiply the derivative by the difference in volumes. This gives DSvap

6. DHvap • To get DHvap, multiply DSvap by the temperature you picked.

7. Example Find the enthalpy of vaporization of HFC134a at 0 C (273 K). • Solution: • Find Pvap at 0 C using the vapor pressure equation. You will need the parameters for the vapor pressure equation. • At P= Pvap and 0C, find the saturated vapor and saturated liquid specific volumes at 0 C. • Find the derivative of the Pvap equation and evaluate it at T = 0 C. • Multiply the derivative value by the difference in specific volumes. The answer is DSvap at 0 C. • Multiply the value of DSvap at 0 C by 0 C (273 K). The answer is DHvap at 0 C.

8. Enthalpy Changes Outside the 2 Phase Area • Needs: • EOS • Ideal gas heat capacity equation • Calculation of DH is done by calculation of DU and addition of D(PV). • Must find V2 from P2 and T2 (P-R equ), Must find V1 from P1 and T1 (P-R equ). • DU is calculated with a 3 step process

9. Heat Capacity Equation • The heat capacity is generally a function of both the temperature and pressure. • Only the heat capacity at low pressure (large specific volume) has been measured. This is called the ideal gas heat capacity and is a function of temperature only. • The constant pressure ideal gas heat capacity is reported. One must subtract R to obtain the constant volume ideal gas heat capacity needed to calculate changes in U.

10. 3 step Process for DU • Must involve changes in specific volume at constant temperature. (Steps 1 and 3) • Must involve change in temperature at ideal gas specific volume. (Step 2) • Steps • 1. DU for V2 V=∞ at T2 + • 2. DU for T2T1 at V = ∞ + • 3. DU for V=∞  V1 at T1.

11. Picture of 3 step Process U1(T1,V1) Ideal gas state V U2-U1 U2 (T2,V2) T

12. Example Find the change in enthalpy for HFC134a going from vapor at -14.9 F, 1 atm to 220 F and 300 psia. • Solution • Use the P-R equation to find V2 at 220, 300 and V1 at -14.9 and 14.69 psia. • Calculate DU at 220 F with upper limit of V2 and lower limit of V=inf. • Calculate DU at V = inf with upper limit of T2 (220) and lower limit of T1 (-14.9). • Calculate DU at -14.9 F with upper limit of V = inf and lower limit of V1. • Add DU’s together for complete DU. • Add D(PV) = (P2V2 – P1V1) to DU to give DH

13. Entropy Changes outside the 2 phase Region • Involve changes in specific volume at constant temperature • Involves change in temperature at infinite specific volume. • Steps • 1. DS for V2 V=inf at T2 + • 2. DS for T2T1 at V = inf + • 3. DS for V=inf  V1 at T1.

14. Example Find the change in entropy for HFC134a going from vapor at -14.9 F and 1 atm to 220 F and 300 psia. • Solution • Use the P-R equation to find V2 at 220, 300 and V1 at -14.9 and 14.69 psia. • Calculate DS at 220 F with upper limit of V2 and lower limit of V=inf. • Calculate DS at V = inf with upper limit of T2 (220) and lower limit of T1 (-14.9). • Calculate DS at -14.9 F with upper limit of V = inf and lower limit of V1. • Add DS’s together for complete DS.

15. Liquid Region • In the liquid region approximate calculations are often used instead of exact calculations with the P-R equation. The heat capacity used is the “liquid” heat capacity and is assumed to be volume and pressure independent.

16. Example Enthalpy Calculate the enthalpy difference between saturated liquid water at 20 C and (subcooled) liquid water at 1 atm and 40 C. • Solution: • Calculate the enthalpy difference between liquid water at 20 C and 1 atm and 20 C @ saturated pressure. (use integral of VdP limits of 1 atm and sat’d pressure at 20 C. • Calculate the enthalpy difference between liquid water at 40 C and 20 C at 1 atm pressure. (use integral of CpdT limits 40 C and 20 C) • Add the two values together.

17. Example Entropy Calculate the entropy difference between saturated liquid water at 20 C and (subcooled) liquid water at 1 atm and 40 C. • Solution: • Calculate the entropy difference between water at 20 C and 1 atm and 20 C saturated pressure. (for constant V there is no change) • Calculate the entropy difference between liquid water at 40 C and 20 C at 1 atm pressure. (Use integral of CpdT/T limits of 40 and 20) • Add the two values together.

18. Summary • With an EOS, vapor pressure equation, and heat capacity equation for the ideal gas state, one can calculate any change in U,H,S and therefore A and G, as well as P, V, T relationships inside or outside the two phase region.