A well-insulated heat exchanger conserves energy exactly: whatever heat Q the hot stream loses, the cold stream gains, so the first-law (energy) efficiency is always 100%. But every real exchanger transfers that heat across a finite temperature difference, and that irreversibility destroys usable work — a fact the first law is blind to. This 2D view adds a temperature-vs-position (T-x) diagram: the shaded gap between the hot and cold lines at every point along the core is exactly the local driving ΔT that generates entropy.
ε-NTU model:
C_min = Cr·C_max, Q_max = C_min(Th,in − Tc,in)
Q = ε·Q_max
Th,out = Th,in − Q/Ch, Tc,out = Tc,in + Q/Cc
Entropy generation (Kelvin):
ΔS_hot = Ch·ln(Th,out/Th,in) (negative — hot stream cools)
ΔS_cold = Cc·ln(Tc,out/Tc,in) (positive — cold stream warms)
Ṡgen = ΔS_hot + ΔS_cold ≥ 0 (2nd law)
Gouy–Stodola theorem:
Ex_destroyed = T0 · Ṡgen
- Effectiveness ε — how close the exchanger gets to the thermodynamic maximum Q_max; a bigger/better core raises ε and lets the two outlet temperatures converge, which lowers Ṡgen since heat moves across a smaller ΔT. On the T-x diagram this pulls the two lines closer together.
- Capacity ratio Cr — how balanced the two streams' heat-capacity rates (ṁ·cp) are. Cr → 1 (matched flows) minimises destroyed exergy for a given ε; a very mismatched pair wastes more available work even at high ε, which shows up as the two T-x lines having very different slopes.
- ηII = Ex gained by cold stream / Ex given up by hot stream — the true "quality" efficiency. Unlike the always-100% energy balance, ηII drops whenever the exchanger forces heat across a large ΔT, exposing losses the first law hides.
- Dead-state (ambient) reference is fixed at T0 = 20 °C (293.15 K) — the temperature at which a stream's exergy is defined to be zero.
Real-world relevance: this exact Gouy-Stodola accounting is how power-plant and process engineers find where a plant's real energy losses actually live — not in the smokestack, but in every finite-ΔT heat transfer along the way.