Noncircular ducts: for a square duct of side d, what is the equivalent (hydraulic) diameter to use for heat-transfer calculations?

Introduction / Context:
Engineering correlations for pressure drop and convection in ducts often employ the hydraulic (equivalent) diameter to account for noncircular cross-sections. For turbulent flow and most design work, the same hydraulic diameter is used for both friction and heat-transfer correlations.

Given Data / Assumptions:

• Square duct with side length d.
• Use of standard equivalent diameter definition Dh = 4A/P, where A is flow area and P is wetted perimeter.
• Application to heat-transfer calculations under typical conditions.

Concept / Approach:
Compute Dh using geometry: for a square, A = d^2 and P = 4d. Therefore Dh = 4A/P = 4 d^2 / (4 d) = d. This provides a simple, exact equivalence for the square section. Unless special laminar, developing-flow, or constant-heat-flux corrections are required, Dh = d is used directly in Nusselt, Reynolds, and friction factor correlations.

Step-by-Step Solution:

Verification / Alternative check:
Handbooks list Dh = characteristic dimension for common shapes; for square ducts Dh equals side length.

Why Other Options Are Wrong:

• d/2 or d/√2: not consistent with 4A/P.
• 4d/π: corresponds to circular equivalence that does not apply here.
• 2d: overestimates the scale and would mispredict Re and Nu.

Common Pitfalls:
Mixing hydraulic diameter definitions for laminar fully developed specialized cases; forgetting to use wetted perimeter for partially filled ducts.

Final Answer:
d