For turbulent flow of a relatively low-viscosity liquid in steel pipes (Re > 2100), an empirical heuristic for the optimum inside diameter is:\n\nGiven symbols:\nQ = volumetric flow rate (ft^3/s)\nρ = fluid density (lb/ft^3)\nDi = optimum inside pipe diameter (inches)\nμ = dynamic viscosity (centipoise, when used)

Introduction / Context:
Early-stage pipe sizing often uses heuristics that relate a near-economic optimum diameter to flow rate and fluid properties. These correlations balance pumping power (friction losses) against installed cost (material and fabrication) and are useful for screening before detailed head-loss economics are performed.

Given Data / Assumptions:

• Turbulent flow regime in steel pipe.
• “Low” viscosity relative to water-like services in preliminary design.
• U.S. customary units as stated in the options.

Concept / Approach:
Optimum diameter scales positively with flow rate; higher density slightly favors a larger diameter due to higher mass flow at a given Q. Strong exponents on viscosity are uncommon in turbulent-flow heuristics because friction factor is only weakly dependent on μ in fully rough regimes. Thus, correlations with extreme viscosity exponents are not realistic for the intended screen.

Step-by-Step Solution:

Verification / Alternative check:
Back-of-the-envelope checks with common flows yield diameters in practical ranges (e.g., 2–12 inches) using the selected correlation, consistent with project heuristics.

Why Other Options Are Wrong:

• (b), (c), (d): Excessive viscosity exponents imply unrealistic sensitivity for turbulent flow, contradicting friction factor behavior.

Common Pitfalls:
Treating heuristics as code; final sizing should use detailed pressure-drop and lifecycle cost analysis for the selected service.

Final Answer:
Di,opt = 3.9 * Q^0.45 * ρ^0.13