1. Maxwell Relations Thermodynamics Professor Lee Carkner Lecture 23

2. PAL #22 Throttling  Find enthalpies for non-ideal heat pump  At point 1, P 2 = 800 kPa, T 2 = 55 C, superheated table, h 2 = 291.76  At point 3, fluid is subcooled 3 degrees below saturation temperature at P 3 = 750 K  Treat as saturated liquid at T 3 = 29.06 - 3 = 26.06 C, h 3 = 87.91  At point 4, h 4 = h 3 = 87.91  At point 1, fluid is superheated by 4 degrees above saturation temperature at P 1 = 200 kPa  Treat as superheated fluid at T 1 = (-10.09)+4 = -6.09 C, h 1 = 247.87

3. PAL #22 Throttling  COP = q H /w in = (h 2 -h 3 )/(h 2 -h 1 ) = (291.76- 87.91)/(291.76-247.87) =4.64  Find isentropic efficiency by finding h 2s at s 2 = s 1  Look up s 1 = 0.9506  For superheated fluid at P 2 = 800 kPa and s 2 = 0.9506, h 2s = 277.26   C = (h 2s -h 1 )/(h 2 -h 1 ) = (277.26- 247.87)/(291.76-247.87) = 0.67

4. Mathematical Thermodynamics   We can use mathematics to change the variables into forms that are more useful   Want to find an equivalent expression that is easier to solve   We want to find expressions for the information we need

5. Differential Relations  For a system of three dependant variables: dz = (  z/  x) y dx + (  z/  y) x dy   The total change in z is equal to the change in z due to changes in x plus the change in z due to changes in y

6. Two Differential Theorems  (  x/  y) z = 1/(  y/  x) z  (  x/  y) z (  y/  z) x = -(  x/  z) y   e.g., P,V and T  May allow us to rewrite equations into a form easier to solve

7. Legendre Differential Transformation  For an equation of the form:  we can define,  and get:  We use a Legendre transform when f is not a convenient variable and we want xdu instead of udx  e.g. replace Pd V with - V dP

8. Characteristic Functions  The internal energy can be written: dU = -Pd V +T dS  H = U + P V dH = V dP +TdS   These expressions are called characteristic functions of the first law  We will deal specifically with the hydrostatic thermodynamic potential functions, which are all energies

9. Helmholtz Function  From dU = T dS - Pd V we can define: dA = - SdT - Pd V  A is called the Helmholtz function   Used when T and V are convenient variables   Can be related to the partition function

10. Gibbs Function  If we start with the enthalpy, dH = T dS + V dP, we can define: dG = V dP - S dT   Used when P and T are convenient variables   phase changes 

11. A PDE Theorem  dz = (  z/  x) y dx + (  z/  y) x dy  or dz = M dx + N dy  (  M/  y) x = (  N/  x) y

12. Maxwell’s Relations  We can apply the previous theorem to the four characteristic equations to get: (  T/  V ) S = - (  P/  S) V (  S/  V ) T = (  P/  T) V  We can also replace V and S (the extensive coordinates) with v and s 

13. König - Born Diagram  Use to find characteristic functions and Maxwell relations  Example:  What is expression for dU?   plus TdS and minus Pd V   (  T/  V ) S =-(  P/  S) V H G A U S P V T

14. Using Maxwell’s Relations  Example: finding entropy   Using the last two Maxwell relations we can find the change in S by taking the derivative of P or V   Example: (  s/  P) T = -(  v /  T) P 

15. Clapeyron Equation  For a phase-change process, P is a function of the temperature only   also for a phase change, ds = s fg and d v = v fg, so:  For a phase change, h = Tds: (dP/dT) sat = h fg /T v fg

16. Using Clapeyron Equation  (dP/dT) sat = h 12 /T v 12   v 12 is the difference between the specific volume of the substance at the two phases  h 12 = T v 12 (dP/dT) sat

17. Clapeyron-Clausius Equation  For transitions involving the vapor phase we can approximate:    We can then write the Clapeyron equation as: (dP/dT) = Ph fg /RT 2  ln(P 2 /P 1 ) = (h fg /R)(1/T 1 – 1/T 2 )sat  Can use to find the variation of P sat with T sat 

18. Next Time  Test #3  Covers chapters 9-11  For Friday:  Read: 12.4-12.6  Homework: Ch 12, P: 38, 47, 57