Heat Exchanger LMTD
Log mean temperature difference for heat-exchanger sizing. Counterflow and parallel-flow analytical solution. Required UA = Q / (F × LMTD), where F is the geometry correction factor (1.0 for true counterflow / parallel; less than 1 for shell-and-tube and crossflow).
Defaults: water-cooled steam condenser-style — hot 180→120°F, cold 60→100°F. Test that ΔT_2 > 0 (cold out < hot out) — otherwise temperature crossover means your geometry can't deliver this duty.
Why LMTD, not arithmetic mean?
Heat transfer is proportional to ΔT instantaneously. As fluids exchange heat along the heat exchanger, ΔT changes — usually a lot. The simple arithmetic mean of inlet and outlet ΔT doesn't account for the curvature of the temperature profile. The integral ∫dq / ΔT(x) gives a logarithmic mean, hence LMTD.
For ΔT1/ΔT2 < 2, LMTD ≈ AMTD within 4%. For larger temperature differences, LMTD is significantly less than AMTD. Always use LMTD for design — using AMTD undersizes by 10–25%.
Counterflow is best
For the same approach temperatures, counterflow gives the smallest LMTD difference, which means the smallest required UA, which means the smallest heat exchanger. That's why all economically-sized heat exchangers are counterflow or close to it.
Parallel flow is rarely optimal. It's used only when:
- Very high inlet ΔT can damage the cold stream — parallel flow gradually warms it.
- Mechanical/process constraint forces it (rare).
F factor for shell-and-tube
Real heat exchangers are rarely true counterflow. Shell-and-tube has tube-side fluid making multiple passes through the same shell, mixing flow directions. The F factor — multiplied against LMTD — captures this. Bowman, Mueller, and Nagle (1940) charts give F vs P (effectiveness ratio) and R (capacity ratio).
This calculator uses approximate F formulas:
- 1 shell - 2N tube passes: F = (√(R²+1) ln((1−P)/(1−PR))) / ((R−1) ln((2−P(R+1−√(R²+1)))/(2−P(R+1+√(R²+1)))))
- Crossflow, both unmixed: smaller F than 1-2 shell-and-tube; tabulated.
F < 0.75 is poor design — temperature crossover, requires larger or different geometry. Aim for F > 0.85.
Temperature crossover
If Tc,o > Th,o in a parallel-flow exchanger, you've requested impossible heat transfer (cold gets hotter than hot at exit). For counterflow this is permitted and is the source of efficiency. Watch for ΔT_2 ≤ 0 in the formulas — that's the tipoff.
Overall heat transfer coefficient U
U is the reciprocal of the sum of resistances: 1/U = 1/hi + Rfouling + (wall resistance) + 1/ho. Typical values:
- Water-water shell-and-tube: U = 250–500 Btu/(hr·ft²·°F) [1400–2800 W/m²·K]
- Air-water (process gas cooler): U = 5–25 Btu/(hr·ft²·°F)
- Steam-water condenser: U = 800–2500 Btu/(hr·ft²·°F)
- Refrigerant evaporator: U = 200–700 Btu/(hr·ft²·°F)
Reference: Bowman, R.A., Mueller, A.C., Nagle, W.M. (1940). "Mean Temperature Difference in Design." Trans. ASME, 62. Cengel, Y.A., Ghajar, A.J. (2014). Heat and Mass Transfer, 5th ed., McGraw-Hill, ch. 11.