Understanding Log Mean Temperature Difference (LMTD)

The Log Mean Temperature Difference (LMTD) is a crucial parameter in heat exchanger design and analysis. It represents the logarithmic average of the temperature difference between the hot and cold fluid streams at the two ends of the heat exchanger. The fundamental equation for heat transfer in a heat exchanger is given by:

$ Q = U \times A \times LMTD $

where Q is the rate of heat transfer, U is the overall heat transfer coefficient, and A is the heat transfer surface area.

Comparing Flow Arrangements: Parallel vs. Counter Flow

Heat exchangers can be designed with different flow arrangements for the hot and cold fluids. The two most common are parallel flow and counter flow.

• Parallel Flow: In this arrangement, both the hot and cold fluids enter the heat exchanger at the same end and flow in the same direction.
• Counter Flow: In this arrangement, the hot and cold fluids enter the heat exchanger at opposite ends and flow in opposite directions.

LMTD Calculation and Difference

The formula for LMTD is the same regardless of the flow arrangement:

$ LMTD = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)} $

However, the values of the temperature differences at the two ends, $\Delta T_1$ and $\Delta T_2$, depend on the flow configuration:

• For Parallel Flow: Let $ T_{h,in} $ and $ T_{h,out} $ be the inlet and outlet temperatures of the hot fluid, and $ T_{c,in} $ and $ T_{c,out} $ be the inlet and outlet temperatures of the cold fluid. The temperature differences are:
  • End 1: $ \Delta T_1 = T_{h,in} - T_{c,in} $
  • End 2: $ \Delta T_2 = T_{h,out} - T_{c,out} $
  In parallel flow, $ \Delta T_1 $ is always greater than $ \Delta T_2 $.
• For Counter Flow: The temperature differences are defined as:
  • End 1: $ \Delta T_1 = T_{h,in} - T_{c,out} $
  • End 2: $ \Delta T_2 = T_{h,out} - T_{c,in} $
  In counter flow, it's possible for $ \Delta T_2 $ to be greater than $ \Delta T_1 $ if the cold fluid outlet temperature approaches the hot fluid inlet temperature, but typically $ \Delta T_1 > \Delta T_2 $ still holds for effectiveness. Crucially, the temperature difference between the fluids tends to be more uniform throughout the exchanger compared to parallel flow.

When the inlet and exit temperatures of the fluids are the same for both arrangements (i.e., $ T_{h,in}, T_{h,out}, T_{c,in}, T_{c,out} $ are fixed), the terminal temperature differences in the counter flow setup ($ T_{h,in} - T_{c,out} $ and $ T_{h,out} - T_{c,in} $) are generally larger than those in the parallel flow setup ($ T_{h,in} - T_{c,in} $ and $ T_{h,out} - T_{c,out} $).

Consider a case where $ T_{h,in} > T_{c,in} $ and $ T_{h,out} > T_{c,out} $. For the same temperatures, the counter flow differences ($ T_{h,in} - T_{c,out} $ and $ T_{h,out} - T_{c,in} $) result in a larger LMTD compared to the parallel flow differences ($ T_{h,in} - T_{c,in} $ and $ T_{h,out} - T_{c,out} $). A larger LMTD implies a more efficient heat transfer for a given heat exchanger size (area A) and overall coefficient (U).

Conclusion on LMTD Comparison

Therefore, for the same inlet and exit temperatures of the hot and cold fluids, the Log Mean Temperature Difference (LMTD) is greater for a counter flow heat exchanger than for a parallel flow heat exchanger.