Heat Exchangers

Convection, conduction, convection in series: 1/U = 1/h₁ + t/k + 1/h₂. The design question is which term dominates — thickening a copper wall changes nothing, while improving the worse fluid side changes everything.

In counterflow the hot fluid at its inlet meets fluid that has already been warmed, so every square metre works as hard as the first. That is why almost every industrial exchanger is counterflow, and parallel flow survives mainly where inlet thermal shock is the concern.

Because ΔT varies along the exchanger, the correct average is the log mean — it emerges from integrating the exponential approach, not from a modelling choice. It is always smaller than the arithmetic mean.

Using the arithmetic mean overestimates the driving force and undersizes the equipment, and the error grows as the two ends differ more. Cross-flow and shell-and-tube arrangements apply a correction factor F, which is read from charts.

• Sizing problem — all four temperatures known, find the area. LMTD is direct.
• Rating problem — area known, find the outlet temperatures. LMTD needs iteration, so ε-NTU is used instead.

Effectiveness is actual heat transfer divided by the thermodynamic maximum, which is set by the stream with the smaller capacity rate ṁc_p. Plotted against NTU — dimensionless size — it rises with sharply diminishing returns.

Past NTU of about 3 or 4, extra area costs more in metal than it saves in energy. That is why exchangers are rarely built larger.

Scale, corrosion products and biofilm build a thin layer of very poor conductor, adding a fouling resistance that grows over years. A millimetre of scale can halve U, because scale conducts about as badly as water does.

Exchangers are therefore designed with a fouling allowance — deliberately oversized when clean, adequate when dirty, and cleaned on a schedule. They run inefficiently for their first years by design.

Raising velocity increases h roughly as v^0.8, shrinking the exchanger — and increases pressure drop as v², costing pumping power for the plant's whole life.

The optimum is an economic calculation balancing capital against running cost over decades, which is why the thermal analysis is only half of a heat exchanger design.