Shell-and-tube exchanger with square tube pitch: the shell-side equivalent (hydraulic) diameter De for crossflow is given by (P = pitch, d = tube outside diameter):

Introduction / Context:
On the shell side of a shell-and-tube heat exchanger, fluid weaves through the tube bundle and baffle windows. For pressure drop and heat-transfer estimation, engineers use an equivalent or hydraulic diameter that reflects the characteristic length scale of flow passages formed by tube pitch and tube diameter.

Given Data / Assumptions:

• Square tube pitch with pitch P (center-to-center spacing).
• Tubes of outside diameter d.
• Crossflow passages dominate between baffles.
• Small leakage/bypass neglected in the geometric definition.

Concept / Approach:
The equivalent (hydraulic) diameter is defined as De = 4 * (flow area) / (wetted perimeter). For a square pitch array, the net flow area per tube cell is P^2 - (π d^2)/4. The wetted perimeter per cell against which fluid shears is approximately π d. Substituting these into the definition gives a compact expression for De.

Step-by-Step Solution:
Compute net flow area per tube cell: A = P^2 - (π d^2)/4.Compute wetted perimeter per cell: Pw = π d.Hydraulic diameter: De = 4 * A / Pw.Therefore, De = 4 * (P^2 - (π d^2)/4) / (π d).

Verification / Alternative check:
Numerically, 4/π ≈ 1.273 and (π/4) ≈ 0.785, so an equivalent form is De ≈ 1.27 * (P^2 - 0.785 d^2) / d. Both are algebraically identical, confirming correctness.

Why Other Options Are Wrong:

• (P^2 - (π d^2)/4)/d: Missing the factor 4/π; underestimates the true De.
• 4 * (P^2 - 0.86 d^2)/(π d): Uses an incorrect geometric constant (0.86 vs 0.785).
• (π d)/(4 P^2): Inverts dependencies; not derived from hydraulic diameter definition.
• 0.9 * (P^2 - (π d^2)/4)/d: Arbitrary coefficient; not geometry-based.

Common Pitfalls:
Mixing triangular and square pitch constants; forgetting that flow area excludes the tube's circular cross-section area; confusing shell-side De with tube-side diameter.

Final Answer:
De = 4 * (P^2 - (π * d^2)/4) / (π * d)