Heat exchangers: for a multi-pass shell-and-tube exchanger, which statement about the log mean temperature difference (LMTD) is correct?

Introduction / Context:
The log mean temperature difference (LMTD) is the effective driving force for heat transfer in exchangers. Its value, when compared to common mathematical means, follows useful inequalities that help sanity-check calculations, especially for multi-pass shell-and-tube configurations.

Given Data / Assumptions:

• Two terminal temperature differences exist: ΔT1 and ΔT2, both positive.
• LMTD is defined as (ΔT1 − ΔT2) / ln(ΔT1/ΔT2).
• Arithmetic mean (AM) = (ΔT1 + ΔT2)/2; Geometric mean (GM) = sqrt(ΔT1 * ΔT2).

Concept / Approach:
Mathematical inequalities for positive numbers state: GM ≤ AM. For distinct ΔT1 and ΔT2, the logarithmic mean lies between the geometric and arithmetic means: GM ≤ LMTD ≤ AM, with strict inequalities when ΔT1 ≠ ΔT2. This remains valid for single- or multi-pass designs; multi-pass influences the correction factor but not these ordering properties.

Step-by-Step Solution:

Verification / Alternative check:
Evaluate a numerical example: let ΔT1 = 30, ΔT2 = 10. AM = 20, GM ≈ 17.32, LMTD = (20)/ln(3) ≈ 18.19, confirming GM < LMTD < AM.

Why Other Options Are Wrong:

• Greater than both means: violates inequalities.
• Less than both: contradicts definitions.
• Equals AM or equals GM: equality holds only when ΔT1 = ΔT2 (degenerate case), not generally.

Common Pitfalls:
Forgetting to use a correction factor for multi-pass arrangements; mixing up the ordering of means; using negative or zero terminal differences.

Final Answer:
LMTD is less than the arithmetic mean and greater than the geometric mean.