Overall heat transfer — In the relation Q = U * A * Δt, the appropriate mean temperature difference Δt for steady-state heat exchangers is:

Introduction:
For steady-state exchangers with constant overall coefficient U, the driving force varies along the flow path. The correct way to collapse this varying driving force into a single effective value is the logarithmic mean temperature difference (LMTD).

Given Data / Assumptions:

• Steady operation and no phase change (or accounted separately).
• Constant specific heats and U along the length.
• Counterflow or parallel-flow configurations.

Concept / Approach:
LMTD is defined from the end-point temperature differences: ΔT1 and ΔT2. The expression is ΔT_lm = (ΔT1 − ΔT2) / ln(ΔT1/ΔT2). It correctly weights the varying driving force along the exchanger length and is valid for both parallel and counterflow (with appropriate ΔT values).

Step-by-Step Solution:
Identify ΔT at the two ends of the exchanger.Compute the log-mean using ΔT_lm = (ΔT1 − ΔT2) / ln(ΔT1/ΔT2).Use Q = U * A * ΔT_lm to size or rate the exchanger.

Verification / Alternative check:
Integrating dQ = U * dA * ΔT(x) over area and dividing by total area leads to the LMTD form, confirming that simpler arithmetic or geometric means are not generally valid.

Why Other Options Are Wrong:

• Arithmetic/geometric means are not derived from the governing energy balance with variable ΔT(x).
• Average bulk temperatures ignore the countercurrent temperature approach near one end.
• U does not “fix” an incorrect ΔT choice.

Common Pitfalls:
Using AMTD for convenience; forgetting to apply LMTD correction factors for multipass or crossflow arrangements.

Final Answer:
The logarithmic mean temperature difference (LMTD)