A radial-flow pump must deliver 2000 gpm against a head of 950 ft. The motor driving the pump is a DC pump with variable speed. Find the minimum speed the motor needs to run at if the pump is to have high efficiency.
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:
The exercise encourages you to realize that radial-flow pumps operate at high efficiency only in a well-defined specific speed range from about 0.2 to 1.8, and encourages you to use a calculation of specific speed to infer what properties the motor should have. -
ENGINEERING CONTEXT:
In pump selection, matching the pump’s operating point to its efficient specific speed range is essential; here the task is to ensure that the pump is not operated at a specific speed below 0.2, which is known to yield low efficiency for dynamic (radial‐flow) pumps. -
PHYSICAL CONTEXT:
This calculation uses an engienering rule of thumb, not physics. The physics is related to the impeller and volute design typical of radial-flow pumps, and whether they can be designed to impart the amount of energy required to get to a very low specific speed–but that physics is not used for the problem. -
ASSUMPTIONS:
Assume steady, incompressible flow with water (\rho=1000\,\mathrm{kg/m^3}). -
SOLUTION TECHNIQUE:
For centrifugal pumps the nondimensional performance is expressed via the pump specific speed
\omega_p^*=\frac{\omega\,Q^{1/2}\rho^{3/4}}{\Delta p_0^{3/4}},
and high efficiency in a radial (centrifugal) pump is typically maintained when
\omega_p^*\ge 0.2
In SI units. Thus, the minimum speed is obtained from
\omega_{\min}=0.2\,\frac{\Delta p_0^{3/4}}{Q^{1/2}\rho^{3/4}}.
Because the given conditions are in US customary units, first convert to SI:
- Flow rate: Q=2000\,\text{gpm}, with 1\,\text{gpm}\approx6.309\times10^{-5}\,\mathrm{m^3/s}, so
Q\approx2000\times6.309\times10^{-5}\approx0.1262\,\mathrm{m^3/s}\,.
- Head: H=950\,\text{ft}, with 1\,\text{ft}=0.3048\,\mathrm{m}, so
H\approx950\times0.3048\approx289.6\,\mathrm{m}\,.
The pressure rise is
\Delta p_0=\rho gH\approx(1000\,\mathrm{kg/m^3})(9.81\,\mathrm{m/s^2})(289.6\,\mathrm{m})\approx2.84\times10^6\,\mathrm{Pa}\,.
Compute the terms:
- Q^{1/2}\approx\sqrt{0.1262}\approx0.355\,\mathrm{(m^3/s)^{1/2}},
- \Delta p_0^{3/4}\approx(2.84\times10^6)^{0.75}\approx6.92\times10^4\,\mathrm{Pa^{3/4}},
- \rho^{3/4}\approx(1000)^{0.75}\approx178\,.
Thus,
\omega_{\min}\approx0.2\,\frac{6.92\times10^4}{0.355\times178}\approx0.2\,\frac{6.92\times10^4}{63.19}\approx0.2\times1095\approx219\,\mathrm{rad/s}\,.
For practical interpretation, convert to rpm using 1\,\mathrm{rpm}=\frac{2\pi}{60}\,\mathrm{rad/s} or
\mathrm{rpm}=\omega\,\frac{60}{2\pi}\,,
so
\mathrm{rpm}\approx219\times\frac{60}{2\pi}\approx219\times9.55\approx2090\,\mathrm{rpm}\,.
- REFLECTION:
The analysis indicates that to achieve high-efficiency operation, the motor must run at a minimum of approximately 219\,\mathrm{rad/s} (or roughly 2100\,\mathrm{rpm}). Although the calculation involves several conversions, the key insight is that the pump’s specific speed must exceed 0.2 for a radial-flow design.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 2.1: Turbomachines Part 3 Part 1 (HIGHLY RELEVANT)
- Chapter 2.5: Combining and Resizing Turbomachines Part 2 (HIGHLY RELEVANT)
- Chapter 6.3: Pump types Part 2 (HIGHLY RELEVANT)
- Chapter 2.3: Selecting a Turbomachine Class (HIGHLY RELEVANT)
- Chapter 2.2: Nondimensional Turbomachine Parameters Part 2 (HIGHLY RELEVANT)
Disclaimer:
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