Heat Exchanger Design Basics (LMTD)

Heat Exchanger Design Basics (LMTD)

A heat exchanger is a device designed to efficiently transfer heat from one fluid to another. The Logarithmic Mean Temperature Difference (LMTD) method is the primary analytical tool used for sizing a heat exchanger when the inlet and outlet temperatures of both fluids are known.

1. The Heat Transfer Equation

The overall rate of heat transfer (QQ) in a heat exchanger can be expressed using the overall heat transfer coefficient (UU), the heat transfer surface area (AA), and the mean temperature difference:

Q=UAΔTlmQ = U A \Delta T_{lm}

Where:

  • QQ is the total heat transfer rate (W).
  • UU is the overall heat transfer coefficient (W/m²·K). This accounts for convection on both sides and conduction through the wall.
  • AA is the total surface area for heat transfer (m²).
  • ΔTlm\Delta T_{lm} is the Logarithmic Mean Temperature Difference (K or °C).

2. Log Mean Temperature Difference (LMTD)

Because the temperature difference between the hot and cold fluids changes as they flow through the exchanger, a simple average temperature difference is inaccurate. The true effective mean temperature difference is the LMTD, defined as:

ΔTlm=ΔT1−ΔT2ln⁡(ΔT1/ΔT2)\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}

The definitions of ΔT1\Delta T_1 and ΔT2\Delta T_2 depend on the flow arrangement:

Parallel Flow

Both fluids enter at the same end and flow in the same direction.

  • ΔT1=Th,in−Tc,in\Delta T_1 = T_{h,in} - T_{c,in}
  • ΔT2=Th,out−Tc,out\Delta T_2 = T_{h,out} - T_{c,out}

Counter-Flow

The fluids enter at opposite ends and flow in opposite directions.

  • ΔT1=Th,in−Tc,out\Delta T_1 = T_{h,in} - T_{c,out}
  • ΔT2=Th,out−Tc,in\Delta T_2 = T_{h,out} - T_{c,in}

Counter-flow is thermodynamically more efficient and yields a higher LMTD for the same terminal temperatures, resulting in a smaller required area AA.

3. Correction Factor (F)

For complex flow arrangements like cross-flow or multipass shell-and-tube exchangers, the pure counter-flow LMTD must be multiplied by a correction factor FF (where F<1F < 1):

Q=UAFΔTlm,cfQ = U A F \Delta T_{lm,cf}

Where ΔTlm,cf\Delta T_{lm,cf} is the LMTD calculated assuming a pure counter-flow arrangement. The factor FF is typically read from empirical charts based on temperature ratios.