A turbine model with a 260mm impeller is tested and shows 90pct efficiency. What efficiency might be expected from a turbine scaled up to 1.2m impeller?
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 scaling a turbine while preserving geometric similarity (i.e. homologous design) allows one to predict the performance in terms of efficiency. The exercise tests an understanding of affine pump/turbine scaling relations and the role of viscous (Reynolds‐number) effects in determining efficiency. -
ENGINEERING CONTEXT:
In designing turbines, the performance (e.g., efficiency) observed in a model test can be preserved or even slightly improved when the machine is scaled up, provided that all operating conditions (including Reynolds number effects) are appropriately maintained. The key takeaway is that a properly scaled turbine can be expected to produce nearly the same useful energy conversion as its model, enabling confident design of full‐scale devices. -
PHYSICAL CONTEXT:
The dominant physical effects include inertial energy conversion in the turbine runner along with viscous losses that decrease with increasing Reynolds number in a scaled-up (larger) design. -
ASSUMPTIONS:
Assume geometric similarity and that the full-scale operating Reynolds number is sufficiently high such that viscous losses are lower than in the model; this is reasonable because scaling up typically increases Reynolds number and reduces relative losses. -
DRAWINGS:
A schematic showing two turbines—one with a 260 mm impeller and one with a 1.2 m impeller—with notes indicating that all nondimensional geometric ratios are maintained and that the operating point remains similar relative to the efficiency “hump” would assist in visualizing the scaling. -
SOLUTION TECHNIQUE:
For homologous designs, non-dimensional performance coefficients such as specific speed or diameter remain invariant. In scaling a turbine, the affinity laws imply that if the geometry is preserved, then the overall efficiency is dictated by the same nondimensional operating point and efficiency curve (e.g. as indicated via the Cordier diagram for pumps/turbines). Moreover, since viscous losses decrease as the Reynolds number increases (which happens when scaling from a 260 mm to a 1.2 m impeller), a full-scale turbine may even exhibit slightly reduced internal losses. Therefore, if the model turbine (260 mm) operates at 90% efficiency, the full-scale turbine (1.2 m) operating under similar nondimensional conditions is expected to deliver an efficiency near or slightly above 90%. Quantitatively, one might anticipate an efficiency increase on the order of a few percentage points (for example, approximately 91–92%) if all other factors are properly scaled. -
REFLECTION:
This analysis shows that when scaling up a turbine design under conditions of geometric similarity and proper similarity of dynamic conditions, the performance coefficients remain nearly constant. Moreover, the reduction in relative viscous losses at larger Reynolds numbers may even cause a slight increase in efficiency. In an engineering design context, this scaling insight allows for confident extrapolation of model test results to full-scale applications.
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 2.5: Combining and Resizing Turbomachines Part 2 (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)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
In my text I say that turbomachines are approximately Reynolds-number-independent. Using the argument that it is Reynolds-number-independent, a scaled homologous turbine would have exactly the same efficiency – 90%. That is what I’d expect one of my students to say in intro fluids.
In practice, losses are a bit lower as the size increases, this is indicative that turbomachines are approximately Reynolds-number-independent but not totally. That is what PK Bot is referring to, without showing any equation.
Some common correlations give
1-\eta = C\, \mathrm{Re}^n
where n is often between 0.1 and 0.25, which is a weak dependence. This is equivalent to writing
\eta_2 = 1 - (1-\eta_1)\left(\frac{D_1}{D_2}\right)^n
If we assume \mathrm{Re}-independence, we are assuming n=0.
PK Bot, could you perform the calculation assuming n=0.25?
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:
Below is the calculation using the expression
\eta_2 = 1 - (1-\eta_1)\left(\frac{D_1}{D_2}\right)^n\,,
with the given variables:
• Model efficiency, \eta_1 = 0.90,
• Model diameter, D_1 = 260\,\text{mm} = 0.26\,\text{m},
• Full-scale diameter, D_2 = 1.2\,\text{m},
• Exponent, n = 0.25.
First, compute the ratio
\frac{D_1}{D_2}=\frac{0.26}{1.2}\approx 0.2167\,.
Then raising this ratio to the 0.25 power:
\left(\frac{D_1}{D_2}\right)^{0.25}\approx (0.2167)^{0.25}\approx 0.682\,.
Now, compute the full-scale efficiency:
\eta_2=1 - (1-0.90)\times 0.682 = 1 - 0.10\times 0.682 \approx 1 - 0.0682 \approx 0.9318\,.
Thus, the turbine scaled up to a 1.2 m impeller is predicted to have an efficiency of about 93.2%.
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 2.5: Combining and Resizing Turbomachines Part 2 (HIGHLY RELEVANT)
- Chapter 2.2: Nondimensional Turbomachine Parameters Part 2 (HIGHLY RELEVANT)
- Chapter 2.4: Selecting a Turbine Class (HIGHLY RELEVANT)
- Chapter 2.1: Turbomachines Part 3 (HIGHLY RELEVANT)
- Chapter 2.6: Turbomachine Inefficiencies (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.