Summary of Relations
5. Summary of Relations
These are four important relations, namely
( ∂ T / ∂ v= ) s −∂∂ ( P / s ) v , ( ∂ T / ∂ P ) s = ( ∂∂ v / s ) P , ( ∂∂ P / T= ) v ( ∂∂ s / v ) T , and ( ∂∂ v / T= ) P −∂∂ ( s / P ) T .
Even though the relations were derived by using thermodynamic relations for closed systems, the derivations are also valid for open systems as long as we follow a fixed mass.
Figure 4: Thermodynamic mnemonic dia- gram.
For a point function, say, P = P(T,v), it can be proven that
(29) Equation (29) is useful to obtain the derivative ( ∂v/∂T) P if state equations are available for the
( ∂P/∂T) v ( ∂T/∂v) P ( ∂v/∂P) T = –1, i.e., ( ∂v/∂T) P =–( ∂P/∂T) v /( ∂P/∂v) T
pressure (e.g., in the form of the VW equation of state). Likewise, if u = u(s,v), ( ∂u/∂s) v ( ∂s/∂v) u ( ∂v/∂u) s = –1.
Using the relation ∂u/∂s = T, ∂s/∂v = P/T, and ∂v/∂u = –P, we find that, as expected, T (P/T)(–1/P) = –1.
e. Example 5 Show that both the isothermal expansivity β P = (1/v)( ∂v/∂T) P and the isobaric com-
pressibility coefficient β T = – (1/v)( ∂v/∂P) T tend to zero as T → 0. Solution
Example 1 shows that ( ∂s/∂T) v → 0 and (∂s/∂P) v → 0 as T → 0. From the fourth of the relations,
( ∂v/∂T) P = –( ∂s/∂P) T , so that ( ∂v/∂T) P → 0. Similarly, using the third relation and the cyclic relations it may be shown that
( ∂P/∂T) v = –(( ∂v/∂T)/(∂v/∂P)) = (∂s/∂v) T → 0 as T → 0. Since ∂v/∂T → 0, it is appar- ent that ∂v/∂P → 0 as T → 0.
Remark The experimentally measured values of β P and β T both tend to 0 as T → 0. Using the
relations the reverse can be shown, i.e., ( ∂s/∂T) v → 0 and (∂s/∂P) v → 0.
f. Example 6 When a refrigerant is throttled from the saturated liquid phase using a short orifice, a two-phase mixture of quality x is formed. We are asked to determine the choking flow conditions for the two-phase mixture, which occurs when the mixture reaches
the sound speed (c 2 =–v 2 ( ∂P/∂v) s ). We must also derive an expression for the speed of sound in a two-phase mixture. Assume ideal gas behavior for the vapor phase and
that the liquid phase is incompressible. Solution
During the elemental expansion of a two phase mixture of a specified quality x from P to P + dP, and v to v + dv,
dh = Tds + v dP, and (A) du = Tds – Pdv.
(B) Since,
dh f = d(u f +Pv f )
For incompressible liquids,
dh f = du f +v f dP.
For a two phase mixture of vapor and liquid,
dh = x dh g + (1 – x) dh f = x dh g + (1 – x)(du f +v f dP).
Assuming ideal gas behavior for the vapor phase, and if du f = cdT, then
(C) Similarly,
dh = x c p,o dT + (1 – x)(c f dT + v f dP).
(D) Considering constant entropy in Eqs. (A) and (B), using Eqs. (C) and (D), dividing by
du = x c vo dT + (1 – x) c f dT.
dT, we obtain the relation
(dP/dT) s = (x c p,o + (1 – x) c f )/(v – (1 – x)v f ), i.e.,
(E)
(F) Dividing Eq. (E) by Eq. (F), we obtain the relation –(dP/dv) s = (x c p,o + (1 – x)c f )(P/(v – (1 – x)v f ))/(x c vo + (1 – x)c f ).
(dv/dT) s = –(x c vo + (1 – x)c f )/P.
(G) Using the definition of the sound speed,
c 2 = –v 2 (dP/dv) s , where v = xv g + (1 – x)v f ,
(H) Eq. (G) can be written as
(I) Since v g = RT/P,
c 2 =v 2 (xc p,o + (1 – x)c f ) P/((v – (1 – x)v f )(xc vo + (1 – x)c f .
v = (xRT/P+ (1 – x)v f ), and
c 2 = RT(x + (1 – x)(Pv f /(RT))) 2 (xc p,o + (1 – x)c f )/(x(xc vo + (1 – x)c f )). (J) If x =1, then, as expected,
x, quality
Figure 5: Sound speed of a two phase mixture for R –134a Figure 5: Sound speed of a two phase mixture for R –134a
c 2 = RT(Pv f /RT) 2 /x → ∞.
Using tabulated values for R–134a,
c f = 1.464 kJ kg –1 K –1 , and c p,o at 298 K = 0.851 kJ kg –1 K –1 .
Using the values for P = 690 kPa, R = 0.08149 kJ kg –1 K ,c vo = 0.7697 kJ kg K , v f = 0.000835 m 3 kg –1 , and γ = 1.1. For the conditions x = 1, T = 298 K,
c = 163.4 m s –1 .
A plot for c with respect to x is illustrated in Figure 5 . The expression for the speed of sound in a solid–vapor mixture is similar except that c f must be replaced by c S , spe- cific heat of solid.
Parts
» COMPUTATIONAL MECHANICS and APPLIED ANALYSIS
» Explicit and Implicit Functions and Total Differentiation
» Exact (Perfect) and Inexact (Imperfect) Differentials
» Intermolecular Forces and Potential Energy
» Internal Energy, Temperature, Collision Number and Mean Free Path
» Vector or Cross Product r The area A due to a vector product
» First Law for a Closed System
» First Law For an Open System
» STATEMENTS OF THE SECOND LAW
» Cyclical Integral for a Reversible Heat Engine
» Irreversibility and Entropy of an Isolated System
» Degradation and Quality of Energy
» SINGLE–COMPONENT INCOMPRESSIBLE FLUIDS
» Evaluation of Entropy for a Control Volume
» Internally Reversible Work for an Open System
» MAXIMUM ENTROPY AND MINIMUM ENERGY
» Generalized Derivation for a Single Phase
» LaGrange Multiplier Method for Equilibrium
» Absolute and Relative Availability Under Interactions with Ambient
» Irreversibility or Lost Work
» Applications of the Availability Balance Equation
» Closed System (Non–Flow Systems)
» Heat Pumps and Refrigerators
» Work Producing and Consumption Devices
» Graphical Illustration of Lost, Isentropic, and Optimum Work
» Flow Processes or Heat Exchangers
» CLASSICAL RATIONALE FOR POSTULATORY APPROACH
» Generalized Legendre Transform
» Van der Waals (VW) Equation of State
» Other Two–Parameter Equations of State
» Compressibility Charts (Principle of Corresponding States)
» Boyle Temperature and Boyle Curves
» Three Parameter Equations of State
» Empirical Equations Of State
» State Equations for Liquids/Solids
» Internal Energy (du) Relation
» EXPERIMENTS TO MEASURE (u O – u)
» Vapor Pressure and the Clapeyron Equation
» Saturation Relations with Surface Tension Effects
» Temperature Change During Throttling
» Throttling in Closed Systems
» Procedure for Determining Thermodynamic Properties
» Euler and Gibbs–Duhem Equations
» Relationship Between Molal and Pure Properties
» Relations between Partial Molal and Pure Properties
» Mixing Rules for Equations of State
» Partial Molal Properties Using Mixture State Equations
» Ideal Solution and Raoult’s Law
» Completely Miscible Mixtures
» DEVIATIONS FROM RAOULT’S LAW
» Mathematical Criterion for Stability
» APPLICATION TO BOILING AND CONDENSATION
» Physical Processes and Stability
» Constant Temperature and Volume
» Equivalence Ratio, Stoichiometric Ratio
» Entropy, Gibbs Function, and Gibbs Function of Formation
» Entropy Generated During an Adiabatic Chemical Reaction
» MASS CONSERVATION AND MOLE BALANCE EQUATIONS
» Evaluation of Properties During an Irreversible Chemical Reaction
» Criteria in Terms of Chemical Force Potential
» Generalized Relation for the Chemical Potential
» Nonideal Mixtures and Solutions
» Gas, Liquid and Solid Mixtures
» Availability Balance Equation
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