Heat Exchanger LMTD Method
Calculate heat exchanger performance using Q = UA x LMTD, where LMTD is the log mean temperature difference between fluids.
The Formulas
The LMTD (Log Mean Temperature Difference) method is the standard way to design and analyze heat exchangers. It accounts for the fact that the temperature difference between the hot and cold fluids changes along the length of the exchanger.
The LMTD gives an effective average temperature difference that can be used with the overall heat transfer coefficient and surface area to calculate the total heat transfer rate.
Variables
| Symbol | Meaning |
|---|---|
| Q | Rate of heat transfer (in watts, W) |
| U | Overall heat transfer coefficient (in W/(m²·K)) |
| A | Heat transfer surface area (in m²) |
| LMTD | Log mean temperature difference (in °C or K) |
| ΔT₁ | Temperature difference between fluids at one end of the exchanger |
| ΔT₂ | Temperature difference between fluids at the other end |
Counter-Flow vs Parallel-Flow
| Configuration | ΔT₁ | ΔT₂ |
|---|---|---|
| Counter-flow | T_hot,in − T_cold,out | T_hot,out − T_cold,in |
| Parallel-flow | T_hot,in − T_cold,in | T_hot,out − T_cold,out |
Counter-flow exchangers are more efficient because they maintain a larger average temperature difference.
Example 1
A counter-flow heat exchanger has: hot fluid enters at 150°C and leaves at 90°C; cold fluid enters at 30°C and leaves at 70°C. U = 500 W/(m²·K) and A = 10 m². What is Q?
ΔT₁ = T_hot,in − T_cold,out = 150 − 70 = 80°C
ΔT₂ = T_hot,out − T_cold,in = 90 − 30 = 60°C
LMTD = (80 − 60) / ln(80/60) = 20 / ln(1.333)
LMTD = 20 / 0.2877 = 69.5°C
Q = U × A × LMTD = 500 × 10 × 69.5
Q = 347,500 W ≈ 347.5 kW
Example 2
A parallel-flow heat exchanger has: hot fluid 200°C to 120°C, cold fluid 20°C to 80°C. What is the LMTD?
ΔT₁ = T_hot,in − T_cold,in = 200 − 20 = 180°C
ΔT₂ = T_hot,out − T_cold,out = 120 − 80 = 40°C
LMTD = (180 − 40) / ln(180/40)
LMTD = 140 / ln(4.5) = 140 / 1.504
LMTD ≈ 93.1°C
When to Use It
The LMTD method is used throughout thermal engineering.
- Designing shell-and-tube heat exchangers in chemical plants
- Sizing radiators and condensers in HVAC systems
- Power plant condenser and boiler design
- Automotive radiator sizing
- Food and beverage pasteurization equipment
- Oil refinery process heat recovery
For multi-pass or cross-flow exchangers, a correction factor F is applied: Q = U × A × F × LMTD. The correction factor is found from charts based on the specific exchanger geometry.