Second Law EfficienciesSecond Law Efficienciesandand

Exergy Change of a SystemExergy Change of a System

Both heat engines have the same thermal efficiency. Are they doing equally well?

Because B has a higher TH, itshould be able to do better.

Hence, it has a higher maximum(reversible) efficiency.

The second law efficiency is a measure of the performance of adevice relative to what its maximum performance could be (under reversible conditions).

Second lawefficiency forheat engine A

For heat engine B, ηII = 30%/70% = 43%

The second law efficiency is 100 percent for all reversible devices.

Second Law EfficienciesSecond Law Efficiencies

• For heat engines = For heat engines = ηηthth//ηηth,revth,rev

• For work - producing devices = WFor work - producing devices = Wuu/W/Wrevrev

• For work – consuming devices = WFor work – consuming devices = Wrevrev/W/Wuu

• For refrigerators and heat pumps = For refrigerators and heat pumps = COP/COPCOP/COPrevrev

• WWrevrev should be determined using the same should be determined using the same initial and final states as actual.initial and final states as actual.

• And for general processes = And for general processes = Exergy recovered/Exergy supplied = Exergy recovered/Exergy supplied = 1 – Exergy destroyed/Exergy supplied1 – Exergy destroyed/Exergy supplied

Second Law EfficienciesSecond Law Efficiencies

• For explanations of what For explanations of what these terms mean for a these terms mean for a particular device, see text particular device, see text page 401page 401

Example 7-6 Second law efficiency of resistance heaters.

Thermal efficiency is 100%. However, COP of a resistanceheater is 1.

What is the COPHP,rev for these conditions? = 1/(1-TL/TH)

It works out to be 26.7so second law eff. isCOP/COPrev = 1/26.7or .037 or 3.7%

See now why resistanceheating is so expensive?

Exergy of a fixed mass or closed system. For a reversibleprocess, the system work:

δW = PdV = (P – P0)dV + P0dV = δWb,useful + P0dV

For the system heat through areversible heat engine:

δWHE = (1 - T0/T) δQ = δQ – T0/T δQ = δQ – (-T0dS) which gives:δQ = δWHE – T0dS

Plug the heat and work expressions into:-δQ – δW = dU and integrateto get:Wtotal useful = WHE + Wb,useful

=(U–U0) + P0(V–V0) – T0(S–S0)= Wrev = X (exergy)

Exergy Change of a Closed SystemExergy Change of a Closed System

• ΔΔX = (UX = (U2 2 - U- U11) + P) + P00(V(V2 2 - V- V11) – T) – T00(S(S2 2 - S- S11) + m() + m(۷۷2222 - -

۷۷1122) +mg(z) +mg(z22 - z - z11))

• Can also do it on a per-mass basis, Can also do it on a per-mass basis, ΔφΔφ = = ΔΔX/m.X/m.• The exergy change of a system is zero if the state The exergy change of a system is zero if the state

of the system or of the environment does not of the system or of the environment does not changechange– Example – steady-flow system.Example – steady-flow system.

• The exergy of a closed system is either positive The exergy of a closed system is either positive or zero.or zero.

Even if T<T0 and/or P<P0 the exergy of the system is positive.

In flowing systems, you also have flow energy.The exergy of flow energy is the useful work that would be delivered by an imaginary piston in the flow (xflow = Pv – P0v.

Just like with energy, withexergy you can replace theu’s with h’s and get the exergy of a flowing system.

Just like we use θ for the energy of a flowing system,we use the Greek letter psi, ψ, for the exergy of a flowingsystem.

Example 7-7 Work Potential of Compressed Air in a Tank.Assume ideal gas and ke and pe negligible.Can calculate mass by ideal gas law. Exergy equation:

X1 = m[(u1-u0) + P0(v1-v0) – T0(s1-s0) +V12/2 + gz]

Why? Then use ideal gas law relations and T1 = T0 to get X1.

Exergy Change During a Compression. Change in exergy equation for flow systems:

Δψ = (h2 – h1) – T0(s2 – s1) + (V22 – V1

2)/2 + g(z2 – z1)

Now, with the two states given, findh’s and s’s and calculate Δψ.

This represents the minimum workrequired to compress the refrigerantbetween these two states.

This also represents the maximumamount of work you can get from expanding this gas again betweenthe same two states.