Francis Turbine — Flow Ratio Range What is the typical range of flow ratio (phi = V_f1 / sqrt(2 g H)) used in the design of Francis turbines?

Introduction:
The flow ratio is a non-dimensional parameter representing meridional (radial) velocity at runner inlet normalized by the ideal velocity from the head. It is a primary choice variable in Francis turbine design affecting incidence losses and efficiency.

Given Data / Assumptions:

• phi = V_f1 / sqrt(2 * g * H).
• Francis turbine (mixed/inward-flow reaction).

Concept / Approach:
Empirical design practice places phi for Francis turbines in a modest range to balance entry shock and diffusion losses. Too low a phi leads to throttling and high incidence; too high a phi increases losses and reduces reaction. The commonly adopted design window is roughly 0.15–0.30.

Step-by-Step Solution:
Normalize inlet meridional velocity by sqrt(2 g H).Use empirical range ensuring high efficiency and stable operation.Select phi within 0.15–0.30 for standard designs.

Verification / Alternative check:
Manufacturer guidelines and classical design charts align with this interval.

Why Other Options Are Wrong:
0.4–0.5 and higher ranges: excessive meridional velocity, atypical for Francis.1–1.5: nonphysical for normalized inlet velocity in this context.

Common Pitfalls:
Confusing flow ratio with speed ratio; they are different non-dimensional parameters.

Final Answer:
0.15 to 0.3