A clear guide to heat transfer for engineers — Fourier conduction, Newton convection, radiation, the overall heat transfer coefficient U, the LMTD method, exchanger types, and the impact of fouling.
The Three Modes of Heat Transfer
Heat moves from hot to cold by three mechanisms, and most equipment combines them. Understanding each mode — and how their resistances add up — is the basis for designing every heat exchanger, furnace, and reactor cooling jacket in a plant.
Conduction — Fourier's Law
Conduction is heat flow through a stationary solid or fluid by molecular and electron interaction. Fourier's law for one-dimensional steady conduction is:
Q = −kA (dT/dx)
where k is the thermal conductivity, A the area, and dT/dx the temperature gradient. Metals have high k (copper ≈ 400 W/m·K); insulators are low (fiberglass ≈ 0.04 W/m·K). For a flat wall, this simplifies to Q = k
A·ΔT/L, where L is the wall thickness.
Convection — Newton's Law of Cooling
Convection transfers heat between a surface and a moving fluid. Newton's law of cooling is:
Q = h A (Tsurface − Tfluid)
The convective coefficient h depends strongly on fluid properties, velocity, and geometry. Forced convection (a pump or fan drives the flow) gives much higher h than natural convection (buoyancy-driven). Boiling and condensation give the highest coefficients of all.
Radiation — Stefan-Boltzmann
Radiation transfers energy by electromagnetic waves and requires no medium. The emissive power scales with the fourth power of absolute temperature:
Q = εσA(T₁⁴ − T₂⁴)
where ε is emissivity and σ is the Stefan-Boltzmann constant. Because of the T⁴ dependence, radiation is negligible at near-ambient temperatures but dominates in furnaces, flares, and fired heaters.
Thermal Resistances in Series
Heat crossing from a hot fluid to a cold fluid through a tube wall passes through several resistances in series, exactly like resistors in an electrical circuit:
- inside convective film,
- inside fouling layer,
- tube-wall conduction,
- outside fouling layer,
- outside convective film.
Because they add in series, the overall heat transfer coefficient U obeys:
1/U = 1/hi + Rfi + (wall) + Rfo + 1/ho
The smallest h (largest resistance) controls U — it is the bottleneck. There is no point boosting a coefficient that is already large while the limiting film is untouched.
The LMTD Method
The duty of a heat exchanger is:
Q = U · A · ΔTlm · F
The temperature difference between the two streams changes along the exchanger, so a simple arithmetic average is wrong. The correct average is the log mean temperature difference (LMTD):
ΔTlm = (ΔT₁ − ΔT₂) / ln(ΔT₁/ΔT₂)
where ΔT₁ and ΔT₂ are the temperature differences at each end. The factor F corrects for multi-pass and cross-flow geometries where flow is neither purely co-current nor counter-current; F = 1 for ideal counter-current flow. Counter-current flow is thermodynamically superior to co-current because it sustains a larger average ΔT and can bring the cold outlet above the hot outlet.
Heat Exchanger Types
| Type | Best for | Notes |
|---|
| Shell-and-tube | High P/T, large duty, phase change | Workhorse of the process industry; robust, well-understood |
| Plate (gasketed) | Clean liquid-liquid, compact | Very high U, easy to clean and expand; limited P/T |
| Air-cooled (fin-fan) | Where cooling water is scarce | Uses ambient air; large footprint |
| Double-pipe | Small duties | Simple, true counter-current |
The shell-and-tube exchanger remains the industry workhorse: one fluid flows through a bundle of tubes, the other through the shell around them, with baffles to direct shell-side flow and boost h. Variants serve as condensers, reboilers, and process interchangers.
Fouling and Its Cost
Over time, surfaces accumulate scale, corrosion products, biological films, or polymer deposits. This fouling adds a resistance (Rf) that lowers U and raises pressure drop. Designers build in a fouling factor so the exchanger has enough extra area to meet duty even when dirty. Choosing the fouling allowance is a balance: too little and the unit underperforms before the next turnaround; too much and you pay for a needlessly oversized exchanger. Mitigation includes higher velocities (which sweep deposits), chemical treatment, and scheduled cleaning.
Design Workflow
A practical sizing sequence: define the duty Q from a stream energy balance, estimate the individual film coefficients, add fouling and wall resistances to get U, compute the LMTD and F factor from the terminal temperatures, then solve A = Q / (U·ΔTlm·F) for the required area — and translate that area into a tube count and shell size.