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Maxwell Relations Thermodynamics Professor Lee Carkner Lecture 23

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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

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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

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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

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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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

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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

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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

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

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