In a shell-and-tube exchanger, suppose overall U scales with the tube-side volumetric flow rate as U ∝ (Q_tube)^0.8. This scaling holds only when what fraction of the total thermal resistance is contributed by the tube-side film?

Introduction / Context:
Overall heat-transfer coefficient U in exchangers is influenced by a series of resistances: tube-side film, wall conduction, fouling, and shell-side film. Empirical exponents like 0.8 relate individual film coefficients to flow rate via Reynolds-number-driven correlations. For U to scale with tube-side flow, the tube-side film must dominate the total resistance.

Given Data / Assumptions:

• U ∝ (Q_tube)^0.8 is observed.
• Other resistances (shell-side film, fouling, wall) are relatively small.
• Fluid properties are not strongly changing with Q over the range considered.

Concept / Approach:
Overall resistance 1/U = R_total = R_tube + R_wall + R_shell + R_foul. If R_tube dominates, then U ≈ 1/R_tube and thus U tracks the tube-side film coefficient h_i. For turbulent internal flow, h_i often scales with Re^m Pr^n, and through velocity–flow relations, with Q^0.8 under fixed geometry, making U ∝ Q^0.8 only when R_tube ≈ R_total (i.e., the ratio R_tube/R_total ~ 1).

Step-by-Step Solution:

Verification / Alternative check:
When shell-side fouling or low shell-side h dominates, changing tube-side flow barely moves U; measured exponents drop far below 0.8, confirming the dependency on resistance dominance.

Why Other Options Are Wrong:

• Infinity or zero fractions are nonphysical as presented.
• 2 or 2^0.8 have no basis as “fractions of total resistance.”

Common Pitfalls:
Assuming U will always respond strongly to tube-side flow; overlooking that fouling or shell-side limitations can cap performance improvements.

Final Answer:
Approximately 1 (tube film accounts for nearly all resistance)