Design Project Analysis

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Adele Stafford
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1 Design Project Analysis You are tasked with building a water turbine and generator that will convert the kinetic energy of falling water into electrical energy. The water starts (and remains) at a given height, shoots out of a small hole, and drops another given height before it hits the turbine blades. Using your foundation in fluid mechanics, you will be required to calculate the power in the water. Initially, you will derive all equations using variables and necessary constants. To complete the project calculations, you will ultimately need to measure all necessary variables and refer to notes/lectures for given constants. The analysis of this project requires a separate analysis of the turbine and generator. To simplify, you will first analyze the mechanical properties of the turbine and compute the turbine s potential power withdrawal from a jet of water with a fixed power. In the process of calculating the turbine s efficiency, you will find an input torque and various resistive torques. Ultimately, the input torque from the jet of water must equal the sum of the resistive torques and the torque from the generators electromotive force (e.m.f.). T input T resistive = T electric Equating these torques will allow you to solve for the power and rotational velocity derived from your design project. With all the potential power withdrawals from the turbine and the generator, you can obtain the total efficiency of your design project. Power in the Water V 3 Define the variables that are required to determine the final velocity, v 3, of the water. Hint: You can use the image above as a guideline. 1

2 List the necessary assumptions that need to be made in order to determine the final velocity based on the parameters in the problem. Hint: One assumption comes as a result from the observation that the water is recycled from the outflow hole back into the bucket. It maintains that the elevation component is constant in Bernoulli s equation. Calculate the output volumetric flow, Q 2, of the water leaving the hole. Remember that the volumetric flow is equal to the area of the outlet hole times the velocity, v 2. Hint: Bernoulli s equation could be helpful in finding the velocity. The turbine blades need to be placed in the parabolic path of the falling water. Assume the turbine blade will be placed a vertical height, H, from the midpoint of the hole to the point at which the water hits the turbine blades. Calculate the horizontal distance from the hole that the turbine will be placed in terms of your previous variables and H. 2

3 Find the velocity v 3 at which the water hits your turbines blades. In all of the lectures and handouts, the equation to calculate the power of water was never provided. Intuitively, however, think of all the variables previously used that would influence the power in the jet of water. List these variables. Hint: There are between 2 and 4 depending on which variables you choose. Units are a good way to check your answer. Knowing that the SI unit of power for a jet of water is also watts, derive a relationship for the power of the water in terms of all the previous variables. Hint: Bernoulli s equation is the fluid mechanics equivalent of the conservation of energy and 1 Watt = 1 Joule/second. If you re stuck, the kinetic energy equation (SI units = Joules) might help. 3

4 Turbine Equations Let us begin by assuming that the contact areas of the blades are moving at an absolute velocity, rω. From the reference frame of the blades, the input water jet is moving slower than v 3 (because the blades are moving in the same direction). Find the relative speed of the input jet in terms of v 3 and rω. rω r The exit speed after the jet hits the blades is defined by v f. From the reference frame of the rotating blades, find the relative exit velocity v f, in terms of v 3 and rω (just put into scalar terms for now without worrying about the direction of the sign). Hint: Can the mass or volumetric flow change? y x How does the angle of the blade affect the output of water leaving the blades, and thus the force felt by the turbine blade? Ultimately, the force felt by the turbine blade is only that felt in the x direction. Include an angle β that solves for only the x component of the relative exit velocity based on the shape of the blades. Note: This angle can be measured by finding the width and depth of the spoon, and assuming a straight line connecting the two. 4

5 The turbine blades used are not the most efficient. The angle, β, can influence efficiency dramatically assuming everything else is 100% efficient. What would be the optimal β? In practice, why might this not actually be the most optimal β? Without having to convert the two previously found velocities into absolute quantities (as opposed to relative), equate the force felt by the blades with mass flow, input water velocity (hitting the blades), and output water velocity (leaving the blades). Assume all water collides with the middle of the individual blades. Hint: Force is Newtons or kg*m/s 2. The change in water velocity determines the force on the blade, with a magnitude determined by how much water is flowing into the blades. Using your understanding of volumetric flow, standard fluid mechanic variables, and conservation of energy, think about how the velocities of the jet and exit velocity can relate to create a force. Check your answer to see if it physically makes sense (i.e. if β < 90, would the input force be less than if β > 90 ). In practice, the ideal efficiency factor due to β with the optimal blade is around η β = 0.97 when β = 160. This efficiency equation is derived based on the force equation with β isolated from the rest of the variables. Try and derive the efficiency factor due to β. Hint: Efficiency equations generally have an actual value measured or calculated in the numerator and the most ideal value in the denominator. In the denominator, the ideal value will use the theoretically optimal β and in the numerator, the one variable in the equation will be the actual β. Test your result by plugging in the β and η β previously given. 5

6 By now you should have clearly written down the final input force (F input ). With this equation, calculate the input torque (T input ), and output shaft power,, that the turbine extracts from the water. Hint: You just need to add one variable to the force to get torque and add one more variable to the torque to get output shaft power. Even without the generator, the turbine will experience some resistive forces. Find the two most prominent of these, and find the final equation that describes the torque on the shaft without the generator (including the equations for the two resistive forces). Hint: Assume that the jet of water experiences no resistance (but the turbine will!). Even without a generator connected, the turbine must eventually reach a maximum rotational speed, which means that torques must balance at a certain rotational speed. What does this tell you about at least one of the resistive forces? How does the input force of water affect the resistive forces? Rewrite T input T resistive as the total input torque after including the non electric resistive torques. Rewrite for. to describe the power produced from the turbine with the resistive forces accounted 6

7 Generator Equations The magnets used are very strong rare earth permanent magnets, called neodymium iron boron (NIB) magnets. They are rather brittle so please refrain from dropping them or letting two of them slap together. The magnets will need to be placed in a N S N S alternating sequence in order to generate an alternating current. Each pair of N S magnets creates one period of the output alternating current. As a result, the disc will have some combination of pairs on one side (generally, 4, 6, or 8). For now, the number of magnets will be considered as n magnets or as n magnets /2 pairs of magnets. S The shaft rotation will cause either the magnets or coils to rotate while the other remains stationary. The generator can be thought of as loops of wire moving across an infinite, N N stationary set of evenly spaced magnets (or a belt of evenly spaced magnets moving past a stationary loop of wire). S S Your goal here is to find the electromotive force, e.m.f., induced in one loop of wire. This requires knowing the total magnetic flux, Φ, passing through the loops, which means we need to find a way to describe the magnetic flux density, B, anywhere at any given time. N S N To analyze this system, you need to find a spatial and time relationship for the magnets passing over the loops of wire. Each of these can be combined in the same function to describe the total electromotive force in relation to time and position. Time dependence You should start by trying to find a time dependent function to describe the B field. To do so, we can think in terms of a single reference point moving in a direct line across the middle of every magnet in the belt. For a given height above the belt of magnets, there will be a maximum magnetic flux density directly above a single magnet. Call this B max,z. The B field experienced by our reference point must be alternating between B max,z and B max,z as it passes across the N S sequence of magnets, and it must be somewhere in between those two values when the wire is between two magnets. Call the distance between the middle of one magnet to the other, d. For our model s disc of radius, r with equally spaced magnets, describe d in terms of r and n magnets. 7

8 For our reference point moving at a velocity of rω, where ω is our model s rotational speed, describe d in terms of the reference point s velocity and the time variable, t. If we start in between two magnets, the flux density must be zero. As it passes a N pole it peaks at B max,z, slowly drops back to zero as it approaches the midpoint between an N and S pole, then peaks at B max,z as it crosses a S pole. We want to multiply the maximum amplitude B max,z with a function that will oscillate. What type of function do we want, which describes this? Hint: Draw a diagram of the magnetic flux density (B field) with relation to time. If we start in between two magnets, the flux density must be zero. For every multiple of d, we arrive again at a flux density of zero. In the previously found function, what number will achieve this criterion? We previously described d in two different ways. Rearrange to isolate the previous answer (the number ) and plug the other side into the function we want. Hint: Does this make sense? We have n magnets /2 pairs of magnets, and if our model s disc makes one revolution, how many times will our reference point experience peaks in B? Check that at the time needed to travel d, our function will be equal to zero. Also remember that angular velocity and angular frequency are the same: ω. Hint #2: Every multiple of d is half of a period, T. Spatial dependence Because the loop of wire has an area that it encloses, we need a way to describe the B field in an area, or its spatial dependence. Once again, we should look at the circular arrangement of magnets as an infinite, straight line belt of magnets, and we will start exactly between two magnets at a fixed position 8

9 along the z axis. In the y direction, the B field dies exponentially. We will use the function e ky, which makes sense because at y=0, e ky = 1. Now we need to express B field in terms of a function of x. In many ways, this exercise will be similar to the time dependence one for the same reasons, we will now multiply both B max,z and e ky by the same type of function as we did in the timedependence exercise. Find the period, T, of the oscillating function in terms of n magnets and r. Hint: We are working in terms of spatial dependence, which means that T will have units of m Hint#2: A singleton of the triple divided, duodecupled semi d Frequency is the inverse of period and has the units revolutions/m. Rewrite the period as an angular frequency with the units radians/m. As before, we multiply angular frequency with our variable, x, and place it within our function. Time and spatial dependence Our goal is to fully describing the B field at any time and any place, and we can do this by combining both of our previous answers. We will again fix ourselves in the z direction, and we will assume that the function of y is unaffected by time because any distance moved along y will only lower the magnetic flux density (as taken into account by e ky ). We know that the function of x changes with time because that is the direction we are moving in. Both functions of t and x are described by the same type of function and should be very similar. At time t 1, we will have moved a certain distance along the belt, but at this new position, the belt can still be spatially described in the same way. We merely have a new starting point. Find a way to include time dependence in the spatial dependence equation. Hint: We still want our type of function to oscillate between 1 and 1, and time only moves our starting point forward. We only need one type of function that describes the general oscillatory pattern to do this. Review some common identities and properties of this function if you need help with conceptual clarification. Hint #2: The simplest mathematical operator. 9

10 Power generated Now that we know the magnetic flux density, find the flux enclosed in the loop. Assume that the loop is a rectangle (of known height and width). Hint: Integrate twice once in the x direction, and once in the y direction. One side of the loop will have current induced in one direction, but the other will have a counter flow of current (right hand rule). Why is there still an overall induced current? Answer briefly in words. Find the e.m.f. induced from the flux enclosed. What would be the total e.m.f. induced if you have n loops number of loops? Notice that the e.m.f. induced still carries the variable t. To find the average power output, the rootmean square of the e.m.f must first be found. Calculate using computer software such as MAPLE. Find the average power output. Answer briefly in words: Concept Quiz 10

11 1) Where is the electrical power obtained from (since conservation of energy cannot be broken)? 2) When will the maximum angular velocity be reached? 3) What happens when more loops are added? 4) What values change when the length of the blades are changed? 11

Physics 54 Lecture March 1, Micro-quiz problems (magnetic fields and forces) Magnetic dipoles and their interaction with magnetic fields

Physics 54 Lecture March 1, Micro-quiz problems (magnetic fields and forces) Magnetic dipoles and their interaction with magnetic fields Physics 54 Lecture March 1, 2012 OUTLINE Micro-quiz problems (magnetic fields and forces) Magnetic dipoles and their interaction with magnetic fields Electromagnetic induction Introduction to electromagnetic