Why LMTD is used instead of AMTD in a heat exchanger?

In this post, we will explore why the logarithm mean temperature difference (LMTD) is preferred over the arithmetic mean temperature difference (AMTD) when determining the overall heat transfer coefficient in a heat exchanger.

• The overall heat transfer coefficient (U_overall) is calculated as:

\mathrm{U_{overall}=\frac Q{A\times\triangle T_{LMTD}}}

Where Q is the heat transfer rate, and A is the heat transfer surface area.  

Understanding LMTD and AMTD:

 LMTD:

It is a mathematical representation of the average temperature difference between the hot and cold fluids across the length of the heat exchanger.

• It is given as:

\mathrm{ \triangle T_{LMTD}=\frac{\triangle T_1-\triangle T_2}{\ln\left({\displaystyle\frac{\triangle T_1}{\triangle T_2}}\right)} }

Where ΔT₁ is the temperature difference at one end of the heat exchanger, and ΔT₂ is the temperature difference at other end.

• In the heat exchanger, the temperature difference between hot and cold fluid does not remain constant but decreases exponentially along its length.
• LMTD accurately captures this variation in temperature difference.

Special Case: When ΔT₁ = ΔT₂

If both ends of the heat exchanger have the same temperature difference (i.e., ΔT₁ = ΔT₂), then LMTD simplifies as follows:

\mathrm{ \triangle T_{LMTD}=\frac{\triangle T_1-\triangle T_1}{\ln\left({\displaystyle\frac{\triangle T_1}{\triangle T_1}}\right)}=\frac00\Rightarrow Undefined }

• In such cases, we assume that the temperature difference remains constant throughout the length of the heat exchanger.
• Thus, LMTD calculation is not required, and LMTD is directly taken as ΔT.  

AMTD:

It is the simple arithmetic average of the temperature difference at the ends of a heat exchanger.

• It is given as:

\mathrm{ \triangle T_{AMTD}=\frac{\triangle T_1+\triangle T_2}2 }

• AMTD assumes a linear variation of temperature difference along the length of the heat exchanger, which is not valid in most practical cases.

Temperature Profile in a Heat Exchanger

Co-current Flow Arrangement:

• In a co-current heat exchanger, both hot and cold fluids flow in the same direction.
• The initial temperature difference is large, but it decreases exponentially along the length of the heat exchanger.
• So, the variation in temperature difference along the length of the heat exchanger is nonlinear.

Counter-current Flow Arrangement:

• In a counter-current heat exchanger, hot and cold fluids flow in opposite directions.
• This configuration maintains a more uniform temperature gradient, improving heat exchanger efficiency.
• However, the temperature difference still varies non-linearly along the length of the exchanger.

Comparative Analysis: LMTD vs. AMTD

The following table presents experimental calculations for both co-current and counter-current flow heat exchangers.

LMTD Calculation (Co-current flow)
Sr. No.	\mathrm{T_C^{In}(℃) }	\mathrm{T_C^{Out}(℃) }	\mathrm{T_H^{In}(℃) }	\mathrm{T_H^{Out}(℃) }	ΔT1	ΔT2	LMTD (°C)	AMTD (°C)	% deviation
1	26	48	64	60	38	12	22.56	25	10.82
3	26	39	56	52	30	13	20.33	22	8.21

• For the first set of values

\mathrm{ LMTD=\frac{\triangle T_1-\triangle T_2}{\ln\left({\displaystyle\frac{\triangle T_1}{\triangle T_2}}\right)}=\frac{38-12}{\ln\left({\displaystyle\frac{39}{12}}\right)}=22.56\;℃ }

\mathrm{ AMTD=\frac{\triangle T_1+\triangle T_2}2=\frac{38+12}2=25\;℃ }

\mathrm{\%\;deviation=\frac{AMTD-LMTD}{LMTD}\times100=\frac{25-22.56}{22.56}=10.82\;\% }

• This shows AMTD overestimates the temperature difference.
• This overestimate increases as the difference between ΔT₁ and ΔT₂ increases.

LMTD Calculation (Counter-current flow)
Sr. No.	\mathrm{T_C^{In}(℃) }	\mathrm{T_C^{Out}(℃) }	\mathrm{T_H^{In}(℃) }	\mathrm{T_H^{Out}(℃) }	ΔT1	ΔT2	LMTD (°C)	AMTD (°C)	% deviation
1	30	50	95	85	45	55	49.38	50	1.26

• Here, the deviation is significantly lower. AMTD errors are lower in counterflow but still not negligible.

Conclusion

• LMTD is preferred over AMTD because it accurately represents the nonlinear temperature variation along the heat exchanger length.
• AMTD assumes a linear temperature difference, leading to errors in heat transfer calculations, especially when the inlet and outlet temperature differences vary significantly.
• The error is more pronounced in the co-current flow arrangement compared to counter-current flow.

Thus, LMTD remains the standard for designing and analysing heat exchangers to ensure precise and reliable heat transfer calculations.