## The Exergy Function

The exergy function is a general work potential function for simple chemical systems. The function evolved from the work of Carnot and Clausius, and is due to Gibbs (1978). The function is:

Es is the maximum work that could be obtained from a sample of matter of energy U, volume V, number of moles (or mass) of each matter species Nc when the sample of matter is allowed to come to equilibrium with an environment of pressure P0, temperature Ta, chemical potential /¿co for each species Nc. The same expression measures the least work required to create such a sample of matter out of the same environment. Various special forms of potential work have been defined to meet specific needs such as Helmholz and Gibbs free energies. A form useful to second law computations for systems in the steady state is:

Equation 2.2 is a flow exergy. For convenience, it is often expressed as the sum of two changes: (1) A change under constant composition {XL) from the state at P and T to a state at a reference PQ and T0. (2) A change under constant P„ and Ta from composition {Xc} to a state at reference {Xc0}. The state at Pm T„ and {Xcn\ defines the reference dead state environment for computing exergy, i.e:

where (.H" - T0 * S°)Po To Xc = (£ (ic * Nc)Po To is used.

### 2.1.1 A derivation of the forms of exergy

A derivation of Equations 2.1 and 2.2 may be obtained with reference to Figure 2.1 by considering mass, energy and entropy balance equations for a simple chemical system. Figure 2.1 shows a system of mass M at pressure P, temperature T and of composition {A^} interacting with a large reference environment at pressure P0, temperature T0, and composition {Xco} where {co} are the system species at the composition of the reference environment {Xco}. The state of the environment is a dead state where no changes happen. Also the system has matter and heat interactions by other systems represented by dm, and dQ. Matter dm, is at T,. P, and Xci. Heat dQ is at Tq. A set of ideal devices, in an abstract sense, is assumed to include expanders, compressors, pumps, selective membranes, chemical reaction cells and heat exchangers. Each device extract the maximum work or asks for the minimum work while exchanging heat dQa at T„ with the dead state environment. Figure 2.1 assumes work extraction. In a time period dt, dmi and dM acquire the pressure, the temperature and the composition of the environment, respectively. The question is how much work can be tapped in the period dtl

|Abstract Ideal Set of Devices dM

dQo at T0

Piston

Dead State Large Environment at P0, T0 {Xco}

### Figure 2.1 An illustration of exergy derivation.

In a time period dt, mass balance, energy balance and entropy balance are: Mass Balance (species and bulk): Stored = In — Out dMc — XCi * dmj — Xce * dme (2.3)

Energy Balance: Stored = In — Out dU = dQ — dQ0 - dWu - P0 * dV + Hi * dm, - ^ Ha, * Xce * dme (2.4)

dV is a volume difference of mass dM at P,T,{XC} and the same mass at T0, Pa, {Xco} Entropy Balance: Stored = In — Out + Created dS = -- — + 8Scr + Si * dm, - V See * Xce * dme (2.5)

Hce, Sce are partial values of exiting species to environment. Multiply Equation 2.5 by Ta and subtract from Equation 2.4

dQ*(I - T0/T) - dWu - T0 * 8Scr + (Ht -T0* Si) * drm

- Y^Hce -To* Sce) * Xce * dm,, = dU + P0*dV-T0*dS (2.6)

Let the chemical potential per unit mass be ¡x. In the term Hce — T„ * Sce) * Xce * dme of Equation 2.6, (Hce —T0* Sce) is a chemical potential ¡xco in equilibrium with ambient and Xce*dme = Xci*dmj — dMc (by Equation 2.3), then

— T0 * Se) * Xce * dme = ^ fico * XCi * dnii — ^ nco * dMc (2.7)

Substituting Equation 2.7 in Equation 2.6 then dQ*(l-T0/T) + (Hi-T0*Si-^i ucoXci) * dm-, - - r0 * &S*

For an ideal set of devices, Ta * SSCT — 0 and dWu is expressed by the remaining three terms of Equation 2.8 which have to be potentials for work or in other words exergies. Let E represents an exergy identified by a superscript, then Equation 2.8 becomes:

Equation 2.9 is an exergy balance equation. In ideal conversions, exergy destruction SD = 0. Each of dEq, dE{, and dEs represents dEw as a work potential in the absence of the other two:

• dEq is due to Carnot and represents the maximum work for a closed system (dE{ = 0) in the steady state (dE" = 0).

• dE{ represents the maximum work if a flowing stream interacts directly with the reference environment and comes in complete equilibrium with it.

• If the stream does not exchange species with the reference environment then dmi = dme = dm all of the same composition and the change from "in" to "out" is dEJ = (dH —T0* dS) * dm.

• dE5 as a decrease represents the maximum work obtainable from a system of mass M as dM comes to complete equilibrium with the reference environment.

• If species are not exchanged then dEs = dU+ P0*dV—Ta*dS (Keenan Availability).

• If the volume remains constant as well then dEs = dU —T0*dS (Helmholz Free Energy).

• If instead the pressure of the system is same P0 then dE? = (dH — T*dS)T0tP0 (Gibbs Free Energy).

In the presence of exergy destruction 8D = Ta* SCT, Equation 2.9 becomes:

The actual work is reduced from the maximum by the exergy destruction.

Integrating Equation 2.9d under the constant environment properties P0, Ta, {/¿co}, noting that end state is the environment at zero exergy, Equation 2.1 is obtained.

For a constant flow per unit time M = dmijdt, Equation 2.9b reduces to Equation 2.2 for the flow exergy.

Some useful forms of the flow exergy, Equation 2.2 are listed in Appendix 9.1. The superscript /is dropped, i.e. E — Ef.

### 2.1.2 Exergy balance and entropy balance

A systematic way to reach exergy balance from entropy balance is to multiply the entropy balance equation by sink temperature T0 and subtract from the energy balance equation as done with Equation 2.6. Exergy balance is used when interest is in both exergy and exergy destructions. Entropy balance is used when interest is limited to exergy destructions. Both balances can be performed around a system, subsystem, compound process or elementary process. In the steady state, exergy balance per unit time or per unit reference matter takes the form of the following equation:

{Eh} are exergies entering and leaving at the boundaries of the entity. D is the exergy destruction within the entity

Eh = Eq + Ew + E^ (by heat, shaft work, and flowing matter) (2.1 la)

The following equation is for entropy balance in the steady state:

{Si} are entropies entering and leaving at the boundaries of the entity. Scr is the created entropy sb = Sm + Sq (by flowing matter and heat) (2.12a)

A real dead state environment does not exit but may be idealized as a gas, a liquid or a solid. Atmospheric air, pure water, seawater and abundant minerals at ambient pressure and temperature may serve as natural dead state environments. When a natural environment is selected (e.g. atmospheric air) and the composition of a particular species (e.g. fossil fuel) cannot be set accurately in the environment, an equilibrium chemical reaction is introduced as an intermediate process in which the products of the reaction have their equivalent in the environment. This can also be used to establish an equivalent equilibrium composition of the missing species in the environment.

A working fluid operating in closed circuit such as a refrigerant may have the dead state environment with the working fluid itself being at ambient temperature and suitable pressure, because in this case there is no interest in an interaction with an environment of different composition. The choice of a dead state environment is tied to the interest in the interaction with it. For example a mineral resource may serve as a dead state environment to reveal the minimum work in extracting desired species from the resource.

2.1.4 Fuel resource allocation to processes in a system

All energy conversion systems are driven by one or more energy driving resources. The utilization of the driving resources throughout a system gives a transparent picture of how the system processes share the driving resources. In an ideal system the exergy of a driving resource is converted completely to the exergy of the product(s). All processes are exergy transmitters. In real systems all processes destruct exergy beside transmission. All processes induce resource penalties due to their inefficiencies.

Figure 2.2 displays the exergy destructions throughout the steady state processes of a simple combined cycle as well as the leaving exergy losses for an assumed design point listed in left column. Exergy destructions and leaving exergy losses are all fuel penalties. The figure shows how much of 195 MW input exergy became exergy destructions throughout the system before the net power 88.6 MW is tapped. An ideal system would require only 88.6 MW driving exergy. Different efficiencies of devices, different operating levels of pressures and temperatures and different system configurations give different distributions of system exergy destruction. They also give different distributions of the capital cost of the system.

The computations include the flow exergy at each of the 17 interconnecting stations as well as powers, rate of heat exchanges and mass rates. For clarity these numbers are not included in the figure. For example, the flow exergy at location 3 is 211.9 MW; at location 4 is 50.07 MW; and at location 6 is 32.3 MW. The compressor power is 91.18 MW; the gas turbine power is 154.55 MW and steam turbine power is 25.41 MW. The difference in the chemical exergy of entering air and leaving exhaust gases is only 2.17 MW while the difference in the thermal mechanical exergy is 10.6 MW.

### Design Point

P2 = 132 psia T3 = 1600 F P6 = 600 psia T8 =100 F Pinch= 20 F CondnsrAT, = 10 F Suprhtr ATh = 50 F dp/P loss =0.01,(9) Ad iabatic Efficiencies comp =0.85 GT =0.9 Stn T =0.9 pmps =0.8, (2) Boundary conditions Fuel:natural gas 8.8 lb/s PI = 14.7 psia T1 = 80 F P15 =15 psia T15 =70 F P17 = 15 psia Dry8=satl Dry 10=satl Dryl l=satv

Decisions=3NL-2NP =51 -20=31

Zero exergy reference Po =14.69 psia To = 55 F {Xio}= as exist*

* Note: For {Xio} to be the pure species, add the chemical exergy change from current composition to separated species. For air add (-19) Btu/lb. For combustion gases add (-21.89) Btu/lb. For H20 add nothing. Overall Eff= 0.4176 1st law _= 0.4860 2nd law

D=0.027 MW

Overall Eff= 0.4176 1st law _= 0.4860 2nd law

D=0.027 MW

Figure 2.2 Exergy destruction in a simple combined cycle: a method of allocating fuel to each process.

Figure 2.2, a result of second law analysis, gives a clear energy picture of a system. However, the cost picture of the system is, so far, absent. 