What is LMTD?
The Log Mean Temperature Difference (LMTD) is the logarithmic average of the temperature difference between the hot and cold fluids at the two ends of a heat exchanger. It is used to calculate the heat transfer rate when the temperature difference varies along the length of the exchanger.
LMTD Formula
LMTD = (ΔT₁ – ΔT₂) / ln(ΔT₁/ΔT₂)
Where:
- ΔT₁ = Temperature difference at inlet end = T_h,in – T_c,out (counterflow) or T_h,in – T_c,in (parallel flow)
- ΔT₂ = Temperature difference at outlet end = T_h,out – T_c,in (counterflow) or T_h,out – T_c,out (parallel flow)
Heat Transfer Equation Using LMTD
Q = U × A × LMTD × F
Where:
- Q = Heat transfer rate (W or kW)
- U = Overall heat transfer coefficient (W/m²K)
- A = Heat transfer area (m²)
- F = LMTD correction factor (for multi-pass or cross-flow exchangers)
Parallel Flow vs Counterflow LMTD
| Parameter | Parallel Flow | Counterflow |
|---|---|---|
| Hot fluid inlet | Same side as cold inlet | Opposite side to cold inlet |
| LMTD value | Lower | Higher |
| Maximum cooling possible | Limited by T_h,out ≥ T_c,out | T_h,out can approach T_c,in |
| Temperature cross | Not possible | Possible |
| Preferred for | Controlled mixing, viscous fluids | Maximum heat recovery |
LMTD Correction Factor (F)
For heat exchangers that are not pure counterflow (multi-pass shell and tube, cross-flow), a correction factor F (between 0 and 1) is applied. F is determined from standard charts (R and P parameters):
- R = (T_h,in – T_h,out) / (T_c,out – T_c,in)
- P = (T_c,out – T_c,in) / (T_h,in – T_c,in)
For a 1-2 shell and tube exchanger, F ≈ 0.8–0.95 is acceptable. If F < 0.75, a different configuration should be considered.
Worked Example: Double Pipe Heat Exchanger
Problem: Water at 80°C enters a counterflow double pipe heat exchanger and exits at 40°C. Cold water enters at 20°C and exits at 55°C. Calculate the LMTD.
Solution:
Counterflow arrangement: ΔT₁ = T_h,in - T_c,out = 80 - 55 = 25°C ΔT₂ = T_h,out - T_c,in = 40 - 20 = 20°C LMTD = (25 - 20) / ln(25/20) LMTD = 5 / ln(1.25) LMTD = 5 / 0.2231 LMTD ≈ 22.4°C
Experiment in the Lab
The LMTD method is verified experimentally using:
Students measure inlet/outlet temperatures in both parallel and counterflow configurations, compute LMTD, and determine the overall heat transfer coefficient U.
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LMTD Correction Factor for Multi-Pass and Cross-Flow Exchangers
The simple LMTD formula assumes pure counter-flow or pure parallel-flow. Real shell-and-tube exchangers use multiple tube passes and cross-flow arrangements, where the true mean temperature difference is lower than the counter-flow LMTD. A correction factor F is applied:
Q = U × A × F × LMTD_counterflow
The correction factor F (always ≤ 1.0) is read from charts as a function of two dimensionless ratios:
- P (thermal effectiveness): P = (t_out – t_in) / (T_in – t_in) — temperature rise of cold fluid relative to maximum possible
- R (capacity ratio): R = (T_in – T_out) / (t_out – t_in) = (m_c × c_pc) / (m_h × c_ph)
For a well-designed exchanger, F should be ≥ 0.80. If F falls below 0.75, add more shell passes or switch to a counter-flow arrangement.
Solved Example — 1 Shell Pass, 2 Tube Pass Exchanger
Hot oil enters at 120°C and leaves at 80°C. Cooling water enters at 25°C and leaves at 55°C.
LMTD (counter-flow basis): ΔT1 = 120 – 55 = 65°C; ΔT2 = 80 – 25 = 55°C
LMTD = (65 – 55) / ln(65/55) = 10 / 0.1671 = 59.8°C
P = (55 – 25)/(120 – 25) = 30/95 = 0.316
R = (120 – 80)/(55 – 25) = 40/30 = 1.33
From the 1-2 exchanger chart: F ≈ 0.88
Corrected mean ΔT = 0.88 × 59.8 = 52.6°C
If Q = 50 kW and U = 350 W/m²K: A = Q/(U × F × LMTD) = 50000/(350 × 52.6) = 2.72 m²
When to Use LMTD vs. Effectiveness-NTU
| Use LMTD Method When… | Use Effectiveness-NTU When… |
|---|---|
| All four terminal temperatures are known | Outlet temperatures are unknown |
| Sizing a new exchanger (finding area A) | Rating an existing exchanger (finding outlet T) |
| Design calculation | Performance/checking calculation |
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