Exergy Destruction in a Heat Exchanger

Stream Conditions

Hot inlet T_h,in 120 °C

Cold inlet T_c,in 20 °C

Heat Exchanger Design

Effectiveness ε 0.70

Capacity ratio C_r = C_min/C_max 0.60

Energy Balance (1st Law)

Heat transferred Q

— kW

T_h,out / T_c,out

— / —

Exergy Balance (2nd Law)

Entropy gen. Ṡ_gen

— W/K

Exergy destroyed

— kW

2nd-law efficiency η_II0%

Ex_dest = T0 · Ṡgen

How it works

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.

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.
Capacity ratio C_r — how balanced the two streams' heat-capacity rates (ṁ·c_p) are. C_r → 1 (matched flows) minimises destroyed exergy for a given ε; a very mismatched pair wastes more available work even at high ε.
η_II = Ex gained by cold stream / Ex given up by hot stream — the true "quality" efficiency. Unlike the always-100% energy balance, η_II 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.

Hot stream (flows →) Cold stream (flows ←)

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