Closed System (Non–Flow Systems)
6. Closed System (Non–Flow Systems)
In this section we will further illustrate the use of the availability balance equation Eq. (47), particularly the boundary volume changes resulting in deformation work ( Figure 11 ).
a. Multiple Reservoirs For closed systems, ˙ m i = m ˙ e = 0, and the work W ˙ cv = W ˙ shaft + W ˙ u + P dV o cyl / dt . For a closed system containing multiple thermal energy reservoirs, the balance equation assumes the
form
0 cv )/ dt = ∑ j = 1 Q ˙ Rj , ( 1 − T 0 / T Rj , ) − W ˙ cv − I .
(61) If this relation is applied to an automobile piston–cylinder assembly with negligible shaft work
dE ( − TS
cv
( δW shaft =0) and with the inlet and exhaust valves closed, the useful optimum work delivered to the wheels over a period of time dt is δW u. The work
δ W u =− ( dE + T dS 0 ) − P dV 0 +∑ δ Q R ( 1 − T 0 / T Rj , ) w − δ I , i.e.,
W u =− ( ∆ E + TS 0 ∆ ) − PV 0 ∆ +∑ Q R ( 1 − T 0 / T Rj , ) − I (63)
where ∆E=E 2 -E 1 , ∆S=S 2 -S 1 , ∆V= V 2 -V 1 . Dividing the above relation by the mass m,
w u =− ( ∆ e + Ts 0 ∆ ) − Pv 0 ∆ +∑ q R ( 1 − T 0 / T Rj , ) − i .
b. Interaction with the Ambient Only With values for q R = 0, i = 0, and e = u, Eq. (64) simplifies as
w u,opt = φ 1 – φ. (27) When φ 2 = φ 0 ,
w u,opt,0 = φ 1 ′ = φ 1 – φ 0 .
The term φ´ is called closed system exergy or closed system relative availability. Consider the cooling of coffee in a room, which is a spontaneous process (i.e., those that occur without out-
side intervention). The availability is completely destroyed during such a process that brings the system and its ambient to a dead state. Thus, w u = 0 and i = w u,opt,0 = φ 1 – φ 0 .
c. Mixtures If a mixture is involved, Eq. (63) is generalized as,
W u =− ( Σ ( Ne k , 2 ˆ k , 2 − Ne k , 1 ˆ) k , 1 + T 0 ( Σ Ns k , 2 ˆ k , 2 − Ns k , 1 ˆ) k , 1 )
− P 0 Σ ( Nv k , 2 ˆ k , 2 − Nv k , 1 ˆ) k , 1 +∑ Q R ( 1 − T 0 / T Rj , ) − I
where, typically, ˆ e ≈≈ e u and s , ,ˆ k = s o k − R ln p k / P ref for a mixture of ideal gases and P ref =1 bar
f. Example 6 This example illustrates the interaction of a closed system with its ambient. A closed tank contains 100 kg of hot liquid water at a temperature T 1 = 600 K. A heat engine transfers heat from the water to its environment that exists at a uniform temperature T 0 = 300 K. Consequently, the water temperature changes from T 1 to T 0 over a finite time period. What is the maximum possible (optimum) work output from the engine? The specific heat of the water c = 4.184 kJ kg –1 K –1 .
Solution Consider the combined closed system to consist of both the hot water and the heat en- gine. Since there are no thermal energy reservoirs within the system and, for optimum work, I = 0,
dE ( cv − TS 0 cv )/ dt = W ˙ cv opt , , or (A)
R,1 T
R,1 Q
Figure 11. Application of the availability balance for a piston-cylinder assembly.
W cv,opt = (E
– (E
cv –T 0 S cv ) 1 cv –T 0 S cv ) 2 , where
(B) (E cv ) 1 =U 1 =mcT 1 , (E cv ) 2 =U 2 =mcT 2 , and (S cv ) 1 – (S cv ) 2 = m c ln(T 1 /T 0 ). (C)
Substituting Eq. (C) into Eq. (B), we obtain
W cv,opt = m c (T 1 –T 0 )–T 0 m c ln(T 1 /T 0 )=
100 × 4.184 × (600 – 300 – 300 × ln (600/300)) = 38520 kJ. Remarks
If only the heat engine is considered to be part of the system, it interacts with both the hot water and the ambient. In this case the hot water is a variable–temperature thermal energy reservoir. Since the heat engine and, therefore, the system, is a cyclical device, there is no energy accumulation within it. Therefore, for an infinitesimal time period
(D) where the hot water temperature T R,w decreases as it loses heat. Applying the First and
δQ R,w (1–T 0 /T R,w )= δW cv,opt ,
Second laws to the variable–temperature thermal energy reservoir, δQ R,w = –dU R,w =– m w c w dT R,w and δQ R,w /T R,w = dS R,w . Using these relations in the context of Eq. (D) we obtain the same answer as before.
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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