Reversibility

1. Reversibility Thermodynamics Professor Lee Carkner Lecture 14

2. PAL # 13 Entropy • Work needed to power isentropic compressor • Input is saturated vapor at 160 kPa • From table A-12, v = 0.12348, h1 = 241.11, s1 = • Output is superheated vapor, P = 900 kPa, s2= s1 = 0.9419 • From table A-13, h2 = • Mass flow rate • m’ = V’/v = 0.033 / 0.12348 = • Get work from W = m’Dh • W = (0.27)(277.06-241.11) =

3. Ideal Gas Entropy • From the first law and the relationships for work and enthalpy we developed: • We need temperature relations for du, dh, dv and dP • ds = cv dT/T + R dv/v • ds = cp dT/T + R dP/P

4. Solving for s • We can integrate these equations to get the change in entropy for any ideal gas process s2-s1 = ∫ cp dT/T + R ln (P2/P1) • Either assume c is constant with T or tabulate results

5. Constant Specific Heats • If we assume c is constant: s2-s1 = cv,ave ln (T2/T1) + R ln (v2/v1) s2-s1 = cp,ave ln (T2/T1) + R ln (P2/P1) • since we are using an average

6. Variable Specific Heats • so = ∫ cp(T) dT/T (from absolute zero to T) • so2 – so1 = ∫ cp(T) dT/T (from 1 to 2) • s2 – s1 = so2 – so1 – Ru ln(P2/P1) • Where so2 and so1 are given in the ideal gas tables (A17-A26)

7. Isentropic Ideal Gas • Approximately true for low friction, low heat processes cv,ave ln (T2/T1) + R ln (v2/v1) = 0 ln (T2/T1) = -R/cv ln (v2/v1) ln (T2/T1) = ln (v1/v2)R/cv (T2/T1) = (v1/v2)k-1

8. Isentropic Relations • We can write the relationships in different ways all involving the ratio of specific heats k (T2/T1) = (P2/P1)(k-1)/k (P2/P1) = (v1/v2)k • Or more compactly • Pvk =constant • Note that: • R/cv = k-1

9. Isentropic, Variable c • Given that a process is isentropic, we know something about its final state • Pr = exp(so/R) • T/Pr = vr (P2/P1) = (Pr2/Pr1) (v2/v1) = (vr2/vr1)

10. Isentropic Work • We can find the work done by reversible steady flow systems in terms of the fluid properties • But we know • -dw = vdP + dke + dpe w = -∫ vdP – Dke – Dpe • Note that Dke and Dpe are often zero

11. Bernoulli • We can also write out ke and pe as functions of z and V (velocity) -w = v(P2-P1) + (V22-V21)/2 + g(z2-z1) • Called Bernoulli’s equation • Low density gas produces more work

12. Isentropic Efficiencies • The more the process deviates from isentropic, the more effort required to produce the work • The ratio is called the isentropic or adiabatic efficiency

13. Turbine • For a turbine we look at the difference between the actual (a) outlet properties and those of a isotropic process that ends at the same pressure (s) hT = wa / ws≈ (h1 – h2a) / (h1 – h2s)

14. Compressor hC = ws / wa≈ (h2s – h1) / (h2a – h1) • For a pump the liquids are incompressible so: hP = ws / wa ≈ v(P2-P1) / (h2a – h1)

15. Nozzles • For a nozzle we compare the actual ke at the exit with the ke of an isentropic process ending at the same pressure hN = V22a / V22s≈ (h1 – h2a) / (h1 – h2s) • Can be up to 95%

16. Entropy Balance • The change of entropy for a system during a process is the sum of three things • Sin = • Sout = • Sgen = • We can write as: • DSsys is simply the difference between the initial and final states of the system • Can look up each, or is zero for isentropic processes

17. Entropy Transfer • Entropy is transferred only by heat or mass flow • For heat transfer: • S = • S = • S = • For mass flow: • S = • n.b. there is no entropy transfer due to work

18. Generating Entropy • friction, turbulence, mixing, etc. DSsys = Sgen + S(Q/T) Sgen = DSsys + Qsurr/Tsurr

19. Sgen for Control Volumes • The rate of entropy for an open system: dSCV/dt = S(Q’/T) + Sm’isi – Sm’ese + S’gen • Special cases: • Steady flow (dSCV/dt = 0) • S’gen = -S(Q’/T) - Sm’isi + Sm’ese • Steady flow single stream: • S’gen = • Steady flow, single stream, adiabatic: • S’gen =

20. Next Time • Read: 8.1-8.5 • Homework: Ch 7, P: 107, 120, Ch 8, P: 22, 30