Tube-side hydraulics in a shell-and-tube exchanger: how do heat-transfer coefficient and pressure drop scale with tube-side mass velocity?

Introduction / Context:
Preliminary exchanger sizing often uses correlations that relate tube-side heat-transfer coefficient and pressure drop to mass velocity G. Recognizing the approximate scaling helps in choosing tube counts and velocities to meet both thermal and hydraulic constraints.

Given Data / Assumptions:

• Turbulent single-phase flow in smooth tubes.
• Dittus–Boelter-type heat transfer correlations and Darcy–Weisbach pressure drop relations.
• Fluid properties held constant for scaling discussion.

Concept / Approach:
For turbulent flow, Nu ≈ 0.023 Re^0.8 Pr^n, so h ∝ Re^0.8 ∝ (G)^0.8 for fixed properties. Pressure drop per length ΔP/L ∝ f * (G^2), with friction factor f varying weakly with Re; thus, ΔP scales roughly as G^2 for design screening.

Step-by-Step Solution:
Adopt Dittus–Boelter: h ∝ Re^0.8 → h ∝ G^0.8.Use Darcy–Weisbach: ΔP ∝ f * (ρ v^2) → for constant area, v ∝ G/ρ → ΔP ∝ G^2.Therefore, statements (a) and (b) hold concurrently.

Verification / Alternative check:
Design tools and vendor charts show h increases sublinearly with velocity, while pressure drop rises roughly quadratically—driving the classic trade-off between thermal performance and pumping cost.

Why Other Options Are Wrong:
(d) contradicts standard correlations; (e) overstates h scaling for turbulent flow and ignores established 0.8 exponent behaviour.

Common Pitfalls:
Extrapolating beyond turbulent regime; neglecting property changes with temperature; ignoring entrance, fouling, and fin effects.

Final Answer:
Both (a) and (b)