LMTD Formula
The Log Mean Temperature Difference (LMTD) represents the temperature driving force for heat transfer in flow systems.
For Counter-Flow:
- ΔT1 = T_Hot In - T_Cold Out
- ΔT2 = T_Hot Out - T_Cold In
For Parallel-Flow:
- ΔT1 = T_Hot In - T_Cold In
- ΔT2 = T_Hot Out - T_Cold Out
Frequently Asked Questions
Determine the temperature differences at both ends of the exchanger (delta T1 and delta T2). The LMTD is the difference between these two values divided by the natural logarithm of their ratio.
The Log Mean Temperature Difference (LMTD) method: Q = U × A × ΔTlm × F. Where Q is heat duty (W), U is the overall heat transfer coefficient (W/m²·K), A is heat transfer area (m²), ΔTlm is the log mean temperature difference (°C), and F is a correction factor for non-pure counterflow arrangements (F=1.0 for pure counterflow; 0.7–0.95 for shell-and-tube). The LMTD is: ΔTlm = (ΔT1 − ΔT2) / ln(ΔT1/ΔT2).
The NTU (Number of Transfer Units) method: NTU = U×A/Cmin. Effectiveness ε = Q_actual/Q_max = Q_actual/(Cmin × (Th,in − Tc,in)). This method is preferred when outlet temperatures are unknown or for performance rating of existing units. For a counterflow heat exchanger: ε = (1 − exp(−NTU(1−Cr))) / (1 − Cr×exp(−NTU(1−Cr))), where Cr = Cmin/Cmax.
Fouling factor (Rf, m²·K/W) accounts for thermal resistance added by deposits (scale, biofilm, corrosion) on surfaces over time. The overall heat transfer coefficient including fouling: 1/U = 1/ho + Rfo + tw/kw + Rfi + 1/hi. TEMA standards specify fouling factors: cooling water — 0.0002 m²K/W; steam — 0.0001; crude oil — 0.0003–0.0005. Fouling can reduce U by 20–50%, so adequate cleaning access and water treatment are essential.
Typical U values for common services: Water-to-water (shell-and-tube) — 800–1,500 W/m²·K; Steam condensers — 1,000–6,000 W/m²·K; Water-to-oil — 100–400 W/m²·K; Air-to-water heat exchangers — 25–250 W/m²·K; Gas-to-gas — 10–50 W/m²·K. Air-side coefficients are low, which is why air-cooled exchangers require large fin surfaces to compensate.
In counterflow, hot and cold fluids flow in opposite directions — the temperature difference is more uniform along the length, maximising heat transfer for a given area. In parallel flow, both fluids enter at the same end — the temperature difference is large at entry but shrinks toward zero at exit, limiting effectiveness. Counterflow achieves higher effectiveness and is used in most industrial shell-and-tube, plate, and double-pipe heat exchangers.