Log-mean temperature difference (LMTD) versus other means: for heat exchangers with two inlet temperatures, how does LMTD compare to the arithmetic and geometric means across typical flow arrangements?

Introduction / Context:
The logarithmic mean temperature difference (LMTD) is the exact temperature driving force for true countercurrent or cocurrent exchangers with constant specific heats and no phase change. Understanding its relation to arithmetic and geometric means offers quick consistency checks during design.

Given Data / Assumptions:

• Two temperature differences at the ends: ΔT1 and ΔT2, both positive.
• No internal heat sources/sinks; constant properties.
• Standard cocurrent or countercurrent arrangements.

Concept / Approach:
For positive ΔT1 and ΔT2, the logarithmic mean LM(ΔT1, ΔT2) lies between the geometric mean GM and the arithmetic mean AM. Inequality: AM ≥ LM ≥ GM. This stems from the properties of means and the convexity of the logarithm function used in the LMTD definition.

Step-by-Step Solution:
Define LMTD = (ΔT1 − ΔT2) / ln(ΔT1/ΔT2).Recall the inequality among means: AM ≥ LM ≥ GM for positive unequal values.Conclude LMTD is less than AM but greater than GM.

Verification / Alternative check:
Test with ΔT1 = 40, ΔT2 = 20: AM = 30; GM ≈ 28.28; LMTD ≈ 28.85 → indeed GM < LMTD < AM.

Why Other Options Are Wrong:
“Always more than AM”: violates basic inequality of means.“May be either more or less”: incorrect—ordering is fixed for positive ΔTs.

Common Pitfalls:
Using end temperatures instead of end temperature differences; applying LMTD correction factors (F_T) without verifying flow configuration.

Final Answer:
is always less than arithmetic mean value, but more than geometric mean value.