LMTD Calculator: Log Mean Temperature Difference

What is LMTD?

In a heat exchanger the temperature difference between the hot and cold fluids changes from one end to the other. When you use Q = U × A × ΔT, the right difference to put in is the log mean temperature difference (LMTD):

ΔT_lm = (ΔT₁ − ΔT₂) / ln(ΔT₁ / ΔT₂)

ΔT₁ and ΔT₂ are the temperature differences at the two ends. The arithmetic mean is always larger and makes the required area look smaller than it is. When the two end differences are close, the two means nearly agree.

Counter flow and parallel flow

In counter flow the fluids run in opposite directions: ΔT₁ = hot inlet − cold outlet and ΔT₂ = hot outlet − cold inlet. In parallel flow they run the same way: ΔT₁ = hot inlet − cold inlet and ΔT₂ = hot outlet − cold outlet.

For the same four temperatures counter flow always gives the larger mean difference and so the smaller exchanger. In parallel flow the cold fluid can never leave warmer than the hot outlet. In counter flow it can.

Example

Hot water enters at 90 °C and leaves at 60 °C while cold water warms from 20 °C to 40 °C. In parallel flow the end differences are 70 K and 20 K and the LMTD is 39.9 K. In counter flow they are 50 K and 40 K and the LMTD is 44.8 K. For the same duty the counter flow unit needs about 11 percent less area.

The correction factor F

In shell and tube and cross flow exchangers the flow is neither purely counter nor purely parallel. The counter flow LMTD is then multiplied by a factor F below 1: ΔT_m = F × ΔT_lm. F depends on two dimensionless numbers, P and R. For shell and tube arrangements the tool works F out with the Bowman, Mueller and Nagle relation. For cross flow you read it from a chart and enter it.

If F drops below 0.75 the arrangement does not suit those temperatures: a small change in temperature then cuts F sharply and the exchanger falls short.

Limits

The LMTD method is for sizing problems where all four temperatures are known. If the outlet temperatures are unknown, use the effectiveness and NTU method. Where specific heat varies strongly or phase change and sensible heat occur together, split the exchanger into zones.