Compute the specific speed of a turbine (metric) A hydraulic turbine develops 10,000 kW under a head of 25 m and runs at 135 r.p.m. Determine the specific speed N_s (metric), choosing the nearest value.

Introduction:
Specific speed is a similarity parameter that helps select the turbine type suited to the site. In the metric definition most used in textbooks, it combines rotational speed, power, and head into a single index. This problem tests correct application of the formula and basic arithmetic handling of exponents.

Given Data / Assumptions:

• Power P = 10,000 kW.
• Head H = 25 m.
• Speed N = 135 r.p.m.
• Metric definition: N_s = N * sqrt(P) / H^(5/4).
• Steady operation and geometric similarity implied by the definition.

Concept / Approach:
The metric specific speed relates families of turbines at different scales but similar flow patterns. The exponent 5/4 on head arises from dimensional analysis using power scaling with head and discharge for dynamically similar turbines. Careful evaluation of H^(5/4) is essential to avoid exponent mistakes.

Step-by-Step Solution:

Verification / Alternative check:
Sanity check by order of magnitude: for H = 25 m and moderate speed, a value in the low hundreds is expected (Francis range). The computed ≈ 241.5 r.p.m. fits selection charts for medium-head turbines.

Why Other Options Are Wrong:

• 175.4 or 215.5: result from incorrect exponent handling or rounding.
• 275.4 or 305.0: typically arise from using H^(1/2) instead of H^(5/4), or from arithmetic slips.

Common Pitfalls:
Using discharge instead of power in the formula; replacing H^(5/4) with H or sqrt(H); forgetting consistent units (kW and metres) that the metric N_s convention assumes.

Final Answer: