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Efficiency of a Water Turbine (3D Printing Lab)

Developed by Deva O'Neil, Benjamin Hancock, and Benjamin Hanks - Published June 7, 2021

This Exercise Set describes one way to incorporate 3D printing into lab sessions in Physics I: Students design and print a water-wheel, and measure its efficiency in lifting a load. An optional exercise at the end uses video analysis to verify that the system is approximately in equilibrium for almost all of the lift process. Concepts applied include power, energy, and efficiency.
Subject Area Mechanics First Year Students who complete this set of exercises will be able to: * Apply the concept of density to calculate the potential energy of a water reservoir **(Exercise 1)** * Calculate work done in lifting a load vertically and relate it to power **(Exercise 2)** * Calculate efficiency and identify sources of energy loss **(Exercise 3)** * Practice design thinking and develop CAD skills in designing and printing a water turbine **(Exercise 4)** * Use video analysis to verify that the system is in equilibrium **(Exercise 5)** 200 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

**Exercise 1: Finding Potential Energy of Water Flowing Through a Turbine** ![](images/waterturbine/diag1.JPG "") - Not shown to scale In this lab, we will be placing a water wheel under a water reservoir. The amount of energy available to the wheel depends on the potential energy of the water in the reservoir. The equation for potential energy of the water must incorporate the measurable dimensions $V$ and $h$, and will also depend on $g$. Using the numbers in the figure above, calculate the potential energy of the water; you'll have to look up the density that water would have in your lab room environment. **** **Exercise 2: Finding the Power Exhibited by a Water Turbine** Your 3D-printed water wheel will be lifting a small weight. In order to find the power of the wheel, it must perform work on the object by moving it through a distance $d$. ![](images/waterturbine/diag2.JPG "") The time the wheel is exerting a force on the mass is also needed, since $P_{out} = W/t$, where $W$ is the work being done on the block by the string. To get started, we will assume that in the course of 2 minutes, the falling water induced the wheel to lift a 10 gram block 70 cm into the air. Assume the entirety of the water was used for this action. We need to calculate $W$, the work being done on the block by the string, so it is necessary to find the force on the block. Use a free-body diagram to determine the force on the block due to the string. -- What condition must be true to make the tension on the string be equal to $mg$? Using this information, calculate the power of the water turbine, listing all assumptions that must be made. ___ **Exercise 3: Finding the Efficiency of the Water Turbine** As with all machines, efficiency ($\eta$) is an important factor in determining a water wheel's viability in a real world scenario. It compares the amount of useful energy (work) generated compared to the amount of energy that went into the machine. $$\eta=W_{block} / E_{water}.$$ - Efficiency can also be expressed as a power ratio: $$\eta = P_{out} / P_{in} .$$ Using the potential energy of the water and the wheel's work output, calculate the efficiency of the water wheel. If not all of the water contributes to the lifting process, then one should not count all of it as an energy input. In this scenario, we will assume that all of the water contributes. According to the Law of Energy Conservation, final and initial energy in the universe is the same. Since the efficiency of the wheel is not 1, energy must have left the system. In what ways could it have left the system? **Exercise 4: Efficiency of a 3D-Printed Water Wheel** In this exercise, you'll need to design and 3D print a water wheel that will lift a load. Before starting your drawing in a CAD program, discuss with your instructor options for mounting the wheel so that it can rotate freely. In designing your wheel, your goal is to optimize the efficiency. As you create the wheel, size it appropriately in the CAD program so you can predict what its dimensions will be when printed. If there are holes in your shape, keep in mind that the filament may expand during printing so that holes generally end up 1 mm smaller in diameter than the design dimension predicts. Keep in mind also the following constraints when designing the axle: * a string will be attached to the axle; it must wind around the axle as the wheel rotates. * the wheel must be able to be mounted in a stable way Maximize the lifted mass so that it will maintain a constant velocity of the wheel after motion has been established. Using the process indicated by exercises 1-3, calculate the efficiency of your 3D-printed water wheel. You'll need to decide which efficiency equation (power ratio or energy ratio) best applies to your experiment, based on whether you can assume that all the potential energy of the water contributed to the lift. If the lift occurred over 30 seconds but it took 50 seconds for the water to drain, using power would be a better choice. Thinking back to the calculation in Exercise 2, answer the following: - Why is it desirable to maintain a constant velocity? **Exercise 5: Analysis of Acceleration by Tracker** One of the primary assumptions made to justify the calculations used was that the angular acceleration of the wheel is approximately zero once motion is established. Using video analysis, measure the acceleration as a function of time and determine to what degree the system is in equilibrium. The instructions in the Experiment Tab guide you through this process in detail, if you have not done video analysis before. - What average acceleration do you obtain? Does the answer vary depending on what part of the motion you capture? - Relate your observations to the assumptions made earlier in this lab. - Discuss what factors could have resulted in error. The software can be downloaded at [Tracker](https://physlets.org/tracker/). ___

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### Credits and Licensing

Deva O'Neil, Benjamin Hancock, and Benjamin Hanks, "Efficiency of a Water Turbine (3D Printing Lab)," Published in the PICUP Collection, June 2021.

The instructor materials are ©2021 Deva O'Neil, Benjamin Hancock, and Benjamin Hanks. 