APPLICATION TO BOILING AND CONDENSATION
C. APPLICATION TO BOILING AND CONDENSATION
We will illustrate an application of stability criteria to boiling and condensation (i.e, the formation of two phases) through the following example.
j. Example 10 Water is contained in a piston–cylinder–weight assembly that is immersed in a con- stant–temperature bath at 593 K. Assume that the fluid obeys the RK equation of state irrespective of its phase, i.e.,
P = RT/( ¯v – ¯b )– ¯a /(T 1/2 ¯v ( ¯v + ¯b ))
(A)
Obtain a plot for P vs. ¯v ; Using the relation dg T = v dP, obtain a plot for ¯g vs.
¯v . Solution
This problem was solved in Chapter 7. A brief summary is presented. Using the values ¯a = 142.64 bar m 3
0.5 K –1 kmole , ¯b
3 Figure 10: Variation of = 0.02110 m ∆G with mole fraction in a binary mix- k mole –2 , and T =
ture.
593 K in Eq. (A),
a plot of P vs. ¯v is readily obtained as shown in Figure 11 . The first term in Eq. (A) occurs due to the collisions of high velocity molecules, while the second term appears due to attractive forces that result in a pressure reduction. As the fluid is compressed from state B, the pressure increases along the path BECGKN. If the fluid at state N is compressed fur- ther, it instantaneously condenses into a liquid state L. Similarly, the fluid that is ini- tially at state L can be expanded to a low pressure along the path LFARM. If the fluid at M is expanded further, it vaporizes instantaneously. Since d g_ T = ¯v dP, it is possible to integrate Eq. (A) between the limits v and v ref ,
i.e., ∫d ¯g T = ¯g (T, ¯v ) – ¯g (T, ¯v ref )= ∫ ¯v dP, to obtain
¯g (T, ¯v ) – ¯g (T, ¯v )=P ¯v – P ref ref ¯v ref –( R T ln(( ¯v – ¯b )/( ¯v ref – ¯b )) –
ref )( ¯v ref – ¯b )/( ¯v – ¯b )). We will now arbitrarily set ¯v
¯a/( 1/2 ¯b T )) ln (( ( ¯v / ¯v
= 4.83 m 3 kmole ref –1 (so that P ref = 10 bar at 593 K) and ¯g ref = 0. This enables us to produce a plot of ¯g vs. ¯v (cf. Figure 11 ). Using the same values of v, one can obtain g vs. P as shown in Figure 12 .
Remarks The fluid is in a saturation state at states G and F at which the vapor and liquid coex-
ist. The Gibbs free energy for both phases is equal. (The saturation pressure according to the RK equation is 133 bar at 593 K.) Path QBECGN is a stable vapor branch, since ∂P/∂ ¯v <0 (or ∂ ¯v /∂P <0). The lowest
value of ¯g at specified values of T and P (cf. Figure 12 ) indicates that P < P sat = 133 bar, i.e., the vapor has a lower free energy when it is compared to the liquid curve QRAF. Path FL is a stable liquid branch since ∂P/∂ ¯v <0 (or ∂ ¯v /∂P <0).
Path GKN represents metastable vapor (i.e., an equilibrium condition with a finite constraint). The state N represents an intrinsic stability limit for the vapor at which ∂P/∂v = 0.
The state M is an intrinsic stability limit for the liquid at which ∂P/∂v = 0.
Path MRAF corresponds to metastable liquid with intermediate values of ¯g . Path NDHJM is an unstable branch since ∂P/∂ ¯v > 0 and the highest values of ¯g are
to be found here.
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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