the reaction turbines at hoover dam installation have rated capacity of 11500 hp at 180 rpm under head of 587 ft. diameter of each turbine is 11ft and discharge is 2350 cfs. Evaluate the speed factor, unit speed, unit discharge, unit power, specific speed
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
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PURPOSE OF THE EXERCISE:
Demonstrate how to nondimensionalize turbine operating conditions by computing the “speed factor” and related nondimensional (“unit‐”) quantities so that one may judge whether a given turbine’s geometry and operating point lie in an efficient design regime. The question is a bit problematic because these terms are defined so differently in so many different sources, that it is hard to give an answer and be confident that it is actually helpful. So I will focus on the specific speed, which is covered in my source material. -
ENGINEERING CONTEXT:
A reaction turbine at Hoover Dam is specified by its rated power, operating speed, head, runner diameter, and discharge. In practice such nondimensional numbers guide the turbine designer in assessing performance and choosing a design class (e.g. Francis) for high‐efficiency conversion of hydraulic energy into rotational power. -
PHYSICAL CONTEXT:
The dominant effects are the conversion of hydraulic head (gravitational potential energy per unit weight) into mechanical power via rotor–fluid interactions, with the losses appearing as dissipation. -
ASSUMPTIONS:
Assume steady, incompressible flow with water (density \rho=1000\,\mathrm{kg/m^3}) and that hydraulic losses may be lumped in overall “efficiency” factors; these are standard assumptions in high–Reynolds–number turbine analyses. -
DRAWINGS:
No detailed drawing is required here; however, a schematic showing the rated head, flow passage from the forebay to the runner, and the rotor geometry (with diameter D) would reinforce the control–volume ideas implicit in these nondimensional groups. -
SOLUTION TECHNIQUE:
Unit speed, unit discharge, and unit power are defined in many different ways and it is not clear which definition to use. Professor Kirby’s text uses specific speed and discharge and power coefficients, are you asking about those?
The turbine specific speed is defined in Professor Kirby’s text, so I show that below.
Following the definition for turbines,
\omega_t^*=\frac{\omega\,P^{1/2}\,\rho^{3/4}}{(\Delta p_{0})^{5/4}}\quad.
We have
\omega=18.85\,\mathrm{rad/s}\,,
P^{1/2}\approx\sqrt{8.58\times10^6}\approx2929\,\mathrm{W^{1/2}}\quad,
\rho^{3/4}\approx1000^{0.75}\approx177.8\,,
and
(\Delta p_{0})^{5/4}\approx(1.76\times10^6)^{1.25}\approx5.97\times10^7\,.
Thus,
\omega_t^*\approx\frac{18.85\times2929\times177.8}{5.97\times10^7}\approx\frac{9.8\times10^6}{5.97\times10^7}\approx0.164\,.
- REFLECTION:
The turbine specific speed $\approx0.164$—quantify how the turbine’s geometry and operating conditions compare with known efficiency design regions. In particular, the computed specific speed is characteristic of a Francis turbine designed for relatively high head and low flow.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 2.4: Selecting a Turbine Class (HIGHLY RELEVANT)
- Chapter 2.1: Turbomachines Part 3 (HIGHLY RELEVANT)
- Chapter 2.2: Nondimensional Turbomachine Parameters Part 2 (HIGHLY RELEVANT)
- Chapter 2.3: Selecting a Turbomachine Class (HIGHLY RELEVANT)
- Chapter 2.5: Combining and Resizing Turbomachines Part 2 (HIGHLY RELEVANT)
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