Thermodynamics · the second law, drawn live

Where your engine loses its work

Every real cycle pays a strip of the T–s diagram to the sink. 800 K over 300 K allows 62.5%; this engine gets 50%.

Dr. Mara Lindqvist · fictional lecturer

workheat rejectedlost work

Carnot, 1824 · live

The ceiling no design beats

η_max = 1 − T_CT_H

Only two temperatures enter. The working fluid never appears.

Hot source T_H800 K Cold sink T_C300 K

62.5%Carnot ceiling

625 Jat most, per 1000 J in

375 Jat least, to the sink

Live · Gouy 1889 · Stodola 1898

The receipt

Hot source T_H800 K Cold sink T_C300 K η_s0.85

In 1000 J, out 531 J; ceiling 625 J. Gouy–Stodola prices the gap: T_C·S_gen = 300 K × 0.313 J/K = 94 J to the sink.

62.5%

ceiling

53.1%

actual

0.313 J/K

S_gen

94 J

lost work

Clausius, 1865

One definition, three cases

Fix the endpoints, invent a reversible path, integrate.

dS = δQ_revT

Ideal gas expands

ΔS = nR ln r

n, molr = V₂/V₁

5.76J/K

Heating copper

ΔS = mc_p ln(T₂/T₁)

m, kgT₁ → T₂, K

110.8J/K

Mixing two gases

ΔS = −R Σ n_i ln x_i

n_A, moln_B, mol

11.53J/K

A fitter installing a blade on a steam turbine rotor in a Siemens works: rows of blades on a bright steel shaft, the man in blue overalls reaching up — photo: Siemens Pressebild, CC BY-SA 3.0, via Wikimedia Commons

Real data · Çengel & Boles Ex. 10-1 / 10-7

Where a steam plant loses

Exergy destroyed per kg of steam — tap a bar. 3 MPa, 350 °C → 75 kPa; furnace 1600 K, sink 290 K.

1524

kJ/kg destroyed

31.8%

second-law efficiency

State	h, kJ/kg	s, kJ/kg·K
1 · condenser out	384.44	1.2132
2 · pump out	387.47	1.2132
3 · boiler out	3116.1	6.7450
4 · turbine out	2403.0	6.7450

The design lever · live

Raise T_H, or lower T_C?

T_H800 K T_C300 K

One kelvin at the sink is worth 2.7 kelvin at the source.

+1.06 pts

sink 10 K colder

+0.40 pts

source 10 K hotter

53.1%

this engine

Where it shows up

Same strip, three industries

Power plants

73%of Ex. 10-7's loss is the boiler

Reheat and regeneration raise the mean heat-addition temperature.

Refrigeration

COP ≤ 11275 K box, 300 K room

Ceiling T_C/(T_H − T_C): every kelvin of lift costs work.

Heat exchangers

250 Jlost per kJ crossing 400 → 300 K

S_gen = Q(1/T_C − 1/T_H); a small pinch is bought with area.

Sources

Real figures, fictional lecturer

DATAÇengel & Boles, Thermodynamics: An Engineering Approach, 8th ed. — Ex. 10-1 / 10-7, Table A-3. Steam-table digits shift between editions.
1824Carnot, Réflexions sur la puissance motrice du feu
1865Clausius, Annalen der Physik 125, 353
1889 · 1898Gouy; Stodola — W_lost = T₀S_gen
HEREEvery knob-driven number is computed in this page. Dr. Mara Lindqvist is fictional.