Absolute and Relative Availability Under Interactions with Ambient

1. Absolute and Relative Availability Under Interactions with Ambient

Consider a piston–cylinder assembly in which steam (at an initial equilibrium state (P 1 ,T 1 )) expands to a final equilibrium state (P 2 ,T 2 ) during an irreversible process, i.e., the system properties may be non-uniform and the temperatures at its boundary and in the ambi- ent could be different during the process (cf. Figure 2a ). It thereby loses heat Q 0 to its ambient and produces work.

* e.g., in an automobile engine the temperature at the cylinder walls and heads is different from that at the piston surface.

C.S.2 at T o

C.S.1 at

b Q rev o T Q Q opt, o

System

C T o ,P o

State 1, (PS)

State 1,

P 1 ,T 1 ,V 1 P 1 ,T 1 ,V 1 W E CE

W u,opt

C.S.3

State 2 , Piston

Ambient (SS) at T o ,P o

(a)

(b)

Figure 2: (a) Irreversible process (for control surface 1 T b ≠T 0 , whereas for control surface 2 T b = T 0 ). (b) Reversible process with uniform properties within the system

We denote the matter within the c.s.1 of the piston–cylinder assembly as M and that in the ambient as A, and allow M to undergo an arbitrary change in state from (U 1 ,V 1 ) to (U 2 ,V 2 ) so that the energy in A changes from U 1,0 to U 2,0 , and the corresponding volumetric change (in A) is V 0 = – (V 2 – V 1 ). The total energy E = U (of M) + U 0 (of A) + PE 0 (of A), total volume V = V (of M) + V 0 (of A), and the total mass of the isolated system consisting of both M and A are unchanged, but irreversible processes within the isolated system result in entropy generation. A reversible process (cf. Figure 2b ) that involves work transfer W re, heat transfer Q rev across the boundaries of M which is then used to run a Carnot engine that, in

turn, rejects Q 0 (amount of heat to the ambient) and produces a work of W CE , can also change the initial state of M to the same final state as shown in Figure 2a , but in this case without al- tering the entropy of the isolated system. For the latte r case, the reversible work done by M,

i.e., W rev , and the work delivered by the Carnot engine W CE can be combined so that W opt = W rev +W CE . As the Carnot engine absorbs heat from M ( Figure 2b ), the temperature of M changes and, consequently, the Carnot efficiency continually changes. Hence consider an infinitesimal reversible process:

δW rev = δQ rev – dU. (19a)

If δW rev = 50 kJ, dU = -100 kJ, δQ rev will be -50 kJ; if dU is fixed at –100 kJ, δW rev = 0 kJ, then δQ rev = -100 kJ. The higher the work delivered by the mass, the lower the amount of heat transfer for the same value of dU. The heat δQ rev is supplied from M to the Carnot engine. Since the heat gained by the Carnot engine is (– δQ rev ), the work done by the engine is

δW CE =– δQ rev (1– T 0 /T).

(19b)

Furthermore, since entropy change of matter M, dS = δQ rev /T, the above equation assumes the form

(20) Adding Eqs. (19a) and (19b) and considering an infinitesimal state change,

δW CE =– δQ rev +T 0 dS.

(21) The higher the work δW rev delivered by the matter, the lower the amount of heat

δW opt =– dU + T 0 dS.

transfer for the same value of dU and the lower the value of δW CE . However, δW opt = δW CE + δW rev remains independent of how much work δW rev is delivered by the matter M within the system. Integrating Eq. (21) between initial and final states, respectively, denoted as 1 and 2,

W opt =U 1 –U 2 –T 0 (S 1 –S 2 ).

(22) This is the net work delivered by the matter through the heat transfer Q rev (i.e., W CE ) and Wrev

during change of state from state 1 to 2. However, the work through the piston rod is less since

a part of W opt is used to overcome atmospheric resistance (P 0 (V 2 –V 1 )). The useful or external optimum work during the process is represented by the relation

W u,opt =W opt –W 0 =W opt –P 0 (V 2 –V 1 ), and

(24) This is the same expression as Eq. (11) and represents the optimum useful work delivered by

W u,opt = (U 1 –U 2 ) –T 0 (S 1 –S 2 )+P 0 (V 2 –V 1 ).

the matter M. It is more appropriate to refer to W u,opt as the external work delivered rather than as the useful work, since for compression processes the term “useful” can be confusing to readers. For a compression process W u,opt is the external work required to compress the fluid. Based on a unit mass basis

(25) For an expansion process, the term w u,opt represents the maximum useful work for the

w u,opt = (u 1 –u 2 )–T 0 (s 1 –s 2 )+P 0 (v 1 –v 2 ).

same initial and final states of M. Both processes are represented on the T–s diagram contained in Figure 3 . The term T 0 (s 1 –s 2 ) in Eq. (25) is the unavailable portion of the energy (represented by the hatched area DEGF in Figure 3 ). We denote φ as the absolute closed system availability, i.e.,

φ=u–T 0 s+P 0 v, so that

(27) (The term φ is known as “availability” in the European literature.) The availability is e

w u,opt = φ 1 – φ 2 .

x- pressed in units of kJ kg –1 in the SI system and in BTU lb –1 in the English system. The term φ 1 represents the potential to perform work in a closed system. It is not a property, since it also

depends upon the environmental conditions surrounding a system. If, during a process, the state of a system is known, the availability can be determined and the optimum work can be compared with the actual work being produced.