Conservative Forces and Potential Energy

Chapter 8 Potential Energy and Conservation of Energy In this chapter we will introduce the following concepts: Potential Energy Conservative and non-conservative forces Mechanical Energy Conservation of Mechanical Energy The conservation of energy theorem will be used to solve a variety of problems As was done in Chapter 7 we use scalars such as work ,kinetic energy, and mechanical energy rather than vectors. Therefore the approach is mathematically simpler. (8-1) A B g v o h v o Work and Potential Energy: Consider the tomato of mass m shown in the figure. The tomato is taken together with the earth as the system we wish to study. The tomato is thrown upwards with initial speed v o at point A. Under the action of the gravitational force it slows down and stops completely at point B. Then the tomato falls back and by the time it reaches point A its speed has reached the original value v o . Below we analyze in detail what happens to the tomato-earth system. During the trip from A to B the gravitational force F g does negative work W 1 = -mgh. Energy is transferred by F g from the kinetic energy of the tomato to the gravitational potential energy U of the tomato-earth system. During the trip from B to A the transfer is reversed. The work W 2 done by F g is positive ( W 2 = mgh ). The gravitational force transfers energy from the gravitational potential energy U of the tomato-earth system to the kinetic energy of the tomato. The change in the potential energy U is defined as: U W ∆ = − (8-2) A A B B k m Consider the mass m attached to a spring of spring constant k as shown in the figure. The mass is taken together with the spring as the system we wish to study. The mass is given an initial speed v o at point A. Under the action of the spring force it slows down and stops completely at point B which corresponds to a spring compression x. Then the mass reverses the direction of its motion and by the time it reaches point A its speed has reached the original value v o . As in the previous example we analyze in detail what happens to the mass- spring system . During the trip from A to B the spring force F s does negative work W 1 = -kx 2 /2 . Energy is transferred by F s from the kinetic energy of the mass to the potential energy U of the mass-spring system. During the trip from B to A the transfer is reversed. The work W 2 done by F s is positive ( W 2 = kx 2 /2 ). The spring force transfers energy from the potential energy U of the mass-spring system to the kinetic energy of the mass. The change in the potential energy U is defined as: U W ∆ = − (8-3) m m A B v o f k f k x d Conservative and non-conservative forces. The gravitational force as the spring force are called “conservative” because the can transfer energy from the kinetic energy of part of the system to potential energy and vice versa. Frictional and drag forces on the other hand are called “non-conservative” for reasons that are explained below. Consider a system that consists of a block of mass m and the floor on which it rests. The block starts to move on a horizontal floor with initial speed v o at point A. The coefficient of kinetic friction between the floor and the block is μ k . The block will slow down by the kinetic friction f k and will stop at point B after it has traveled a distance d. During the trip from point A to point B the frictional force has done work W f = - μ k mgd. The frictional force transfers energy from the Conservative Forces and Potential Energy Conservative Forces and Potential Energy Bởi: OpenStaxCollege Potential Energy and Conservative Forces Work is done by a force, and some forces, such as weight, have special characteristics A conservative force is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken We can define a potential energy (PE) for any conservative force, just as we did for the gravitational force For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you work against its spring and store energy in it (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring Indeed, the reason that the spring has this characteristic is that its force is conservative That is, a conservative force results in stored or potential energy Gravitational potential energy is one example, as is the energy stored in a spring We will also see how conservative forces are related to the conservation of energy Potential Energy and Conservative Forces Potential energy is the energy a system has due to position, shape, or configuration It is stored energy that is completely recoverable A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken We can define a potential energy (PE) for any conservative force The work done against a conservative force to reach a final configuration depends on the configuration, not the path followed, and is the potential energy added 1/8 Conservative Forces and Potential Energy Potential Energy of a Spring First, let us obtain an expression for the potential energy stored in a spring (PEs ) We calculate the work done to stretch or compress a spring that obeys Hooke’s law (Hooke’s law was examined in Elasticity: Stress and Strain, and states that the magnitude of force F on the spring and the resulting deformation ΔL are proportional, F = kΔL.) (See [link].) For our spring, we will replace ΔL (the amount of deformation produced by a force F) by the distance x that the spring is stretched or compressed along its length So the force needed to stretch the spring has magnitude F = kx, where k is the spring’s force constant The force increases linearly from at the start to kx in the fully stretched position The average force is kx / Thus the work done in stretching kx or compressing the spring is Ws = Fd = x = kx2 Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of F vs x is the work done by the force In [link](c) we see that this area is also kx2 We therefore define the potential energy of a spring, PEs, to be ( ) PEs = kx2, where k is the spring’s force constant and x is the displacement from its undeformed position The potential energy represents the work done on the spring and the energy stored in it as a result of stretching or compressing it a distance x The potential energy of the spring PEs does not depend on the path taken; it depends only on the stretch or squeeze x in the final configuration (a) An undeformed spring has no PEs stored in it (b) The force needed to stretch (or compress) the spring a distance x has a magnitude F = kx , and the work done to stretch (or compress) it is 2 kx Because the force is conservative, this work is stored as potential energy (PEs) in the spring, and it can be fully recovered (c) A graph of F vs x has a slope of k, and the area under 1 the graph is kx2 Thus the work done or potential energy stored is kx2 The equation PEs = kx2 has general validity beyond the special case for which it was derived Potential energy can be stored in any elastic medium by deforming it Indeed, the general definition of potential energy is energy due to position, shape, or 2/8 Conservative Forces and Potential Energy configuration For shape or position deformations, stored energy is PEs = kx2, where k is the force constant of the particular system and x is its deformation Another example is seen in [link] for a guitar string Work is done to deform the guitar string, giving it potential energy When released, the potential energy is converted to kinetic energy and back to potential as the string oscillates back and forth A very small fraction is dissipated as sound energy, slowly removing energy from the string Conservation of Mechanical Energy Let us now consider what form the work-energy theorem takes when only conservative forces are involved This will lead us to the conservation of energy principle The workenergy theorem states that the net work done by all forces acting on a system equals its change in kinetic energy In equation form, this is 1 Wnet = mv2 − mv02 = ΔKE If only conservative forces act, then Wnet = Wc, where Wc is the total work ...1|Page  Washington State Pulp and Paper Mill Boilers : Current and Potential Renewable Energy Production Final Report September 2009 Richard Gustafson 1 and Natalia Raffaeli University of Washington School of Forest Resources Department of Ecology Publication No. 09-07-048  1 InquiriesshouldbeaddressedtoRichardGustafson.Email:pulp@u.washington.edu 2|Page  This report is available on the Department of Ecology home page on the World Wide Web at http://www.ecy.wa.gov/biblio/0907048.html For a printed copy of this report, contact: Department of Ecology Address: P.O. Box 47600, Olympia, WA 98504-7600 E-mail: mdav461@ecy.wa.gov Phone: (360) 407-6129 Refer to Publication Number #09-07-048 Any use of product or firm names in this publication is for descriptive purposes only and does not imply endorsement by the authors or the Department of Ecology. The Department of Ecology is an equal-opportunity agency and does not discriminate on the basis of race, creed, color, disability, age, religion, national origin, sex, marital status, disabled veteran’s status, Vietnam-era veteran’s status, or sexual orientation. If you have special accommodation needs or require this document in alternative format please contact Kathy Vermillion at (360) 407-6916 or call 711 or 877-833-6341 (TYY). 3|Page  Acknowledgements The authors wish to thank several individuals for their help in this study. We thank all the engineers in the mills who took the time to complete the survey. Dave Krawchuk of Harris Group was invaluable in helping us design the survey, providing data on modern boilers, and in providing cost data for the economic analysis. Llewellyn Mathews, Kathryn VanNatta, and company representatives of the Northwest Pulp and Paper Association helped get the project funded and encouraged mills to participate in the survey. All their efforts are appreciated. This work was made possible by funding provided by and under the mandate of the Washington State Legislature through the Washington Department of Ecology. The funding by the Legislature for us to conduct this important and fascinating study is greatly appreciated. 4|Page  Table of Contents EXECUTIVE SUMMARY 5 INTRODUCTION 7 Steam and Electricity Generation 9 Biopower Technologies 11 Recovery Boilers 17 Lime Kilns 19 Modern pulp mills 20 MOTIVATION AND OBJECTIVES FOR THIS STUDY 21 RESULTS AND DISCUSSION 23 Boiler Survey 23 Survey Results 23 Fossil fuel boilers 23 Biomass boilers 24 Recovery boilers 26 Steam turbines 27 Lime kilns 27 Survey conclusions 28 Energy production capability 28 CONCLUSIONS AND RECOMMENDATIONS 31 REFERENCES 32 APPENDIX I 33  5|Page  Executive Summary At the request of the Legislature, we have conducted a thorough investigation on the state of boilers in Washington State pulp and paper mills and the potential for these boilers to provide additional renewable energy and renewable fuels. The specific objective of the project is to assess the current energy profile of the Washington pulp and paper industry and to determine the renewable energy production of the industry with implementation of state-of-the-art technologies. There are two phases to this investigation: Phase 1. Assess the current energy production in Washington State pulp and paper mills. In this phase of the study, we determined the energy (steam and J. Sci. Dev. 2009, 7 (Eng.Iss.1): 70 - 78 HA NOI UNIVERSITY OF AGRICULTURE 70 Energy recovery potential from landfill and environmental evaluation of landfill gas power generation system at nam son landfill, Vietnam Tiềm năng thu hồi năng lượng từ bãi rác và đánh giá lợi ích môi trường của hệ thống phát điện sử dụng khí từ bãi rác tại bãi rác Nam Sơn, Việt Nam Pham Chau Thuy 1 , Sohei Shimada 2 1 Department of Environmental Technology, Faculty of Natural Resource and Environment, Hanoi Agricultural University, Trau Quy, Gia Lam, Hanoi 2 Graduate School of Frontier Sciences, Institute of Environmental Studies, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN TÓM TẮT Khí từ bãi rác là nguồn năng lượng xanh, sạch, có thể tái tạo được và có thể sử dụng để tạo ra điện, hay sử dụng trong công nghiệp năng lượng. Bài báo này đánh giá tiềm năng thu hồi năng lượng từ khí bãi chôn lấp chất thải rắn đô thị, mục đích làm giảm lượng phát thải methan nói riêng và giảm phát thải khí nhà kính nói chung. Ngoài ra, bài báo cung cấp cách sử dụng mô hình đánh giá lượng khí methan tạo ra từ bãi chôn lấp chất thải rắn đô thị và tiềm năng tạo ra năng lượng từ khí đã thu hồi. Đặc biệt, bài báo sử dụng phương pháp đánh giá vòng đời để đánh giá việc giảm phát thải khí nhà kính của hệ thống phát điện sử dụng khí từ bãi rác. Kết quả nghiên cứu chỉ ra rằng, bãi rác Nam Sơn là một bãi rác có tiềm năng lương lượng lớn cần thu hồi và sử dụng, góp phần đáng kể vào việc làm giảm phát thải khí nhà kính, hướng tới sự phát triển bền vững. Bài báo cung cấp một cách nhìn mới về công nghệ năng lượng sử dụng khí từ bãi rác cho Viêt Nam: hệ thống phát điên sử dụng động cơ khí và tuabin khí. Kết quả còn chỉ ra rằng, hệ thống phát điện bằng động cơ khí tỏ ra hiệu quả hơn về lợi ích về môi trường so với hệ thống phát điện bằng tuabin khí. Hệ thống này có thể ứng dụng cho bãi rác Nam Sơn và ứng dụng cho các bãi rác khác của Việt Nam trong tương lai. Từ khóa: Đánh giá vòng đời, giảm phát thải khí nhà kính, khí từ bãi rác, mô hình phát thải khí bãi rác. SUMMARY Landfill gas (LFG), a green, clean, and renewable energy source can be used for electricity generation or fuel industries. This research presents an attempt to assess the energy recovery potential from the Municipal Solid Waste (MSW) landfill, targeting at gas recovery and gas utilization, in mitigating methane emission in particular and green house gas (GHG) emission in general. Our research provides the using of landfill gas emission model (LFGEM) to quantify the methane generation volume for MSW landfill. We then evaluate of energy generation potential from recovered gas. Especially, this research conducted the Life Cycle Inventory to evaluate GHG emission mitigation of power generation system using LFG. The results show that the methane gas flow at Nam Son landfill can provide a considerable energy potential. LFG recovery and utilization could contribute remarkable to GHG emission mitigation, toward to sustainability. The research supplies a new vision of energy technology from LFG for Viet Nam: Gas Engine and Gas Turbine. The research found that Gas Engine is more attractive in term of environmental benefit, which can be applied primarily for Nam Son landfill and continue applied for other landfill in Vietnam for the future. Journal of Science and Development 2009: Tập VI, No 6: 69-77 HA NOI UNIVERSITY OF AGRICULTURE 71 Key words: Green House Gas emission mitigation, landfill gas, landfill gas emission model, life cycle Inventory. 1. INTRODUCTION Climbing LFG is considered as the largest anthropogenic emission source in the developed countries and also as a considerable emission source in developing countries up to now. Landfill gas (LFG) is produced from anaerobic biodegradable decomposition of organic content of landfilled waste. Release of LFG is one of the dangerous contaminations due to high methane content contributing to GHG J. Sci. Dev. 2009, 7 (Eng.Iss.1): 70 - 78 HA NOI UNIVERSITY OF AGRICULTURE 70 Energy recovery potential from landfill and environmental evaluation of landfill gas power generation system at nam son landfill, Vietnam Tiềm năng thu hồi năng lượng từ bãi rác và đánh giá lợi ích môi trường của hệ thống phát điện sử dụng khí từ bãi rác tại bãi rác Nam Sơn, Việt Nam Pham Chau Thuy 1 , Sohei Shimada 2 1 Department of Environmental Technology, Faculty of Natural Resource and Environment, Hanoi Agricultural University, Trau Quy, Gia Lam, Hanoi 2 Graduate School of Frontier Sciences, Institute of Environmental Studies, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN TÓM TẮT Khí từ bãi rác là nguồn năng lượng xanh, sạch, có thể tái tạo được và có thể sử dụng để tạo ra điện, hay sử dụng trong công nghiệp năng lượng. Bài báo này đánh giá tiềm năng thu hồi năng lượng từ khí bãi chôn lấp chất thải rắn đô thị, mục đích làm giảm lượng phát thải methan nói riêng và giảm phát thải khí nhà kính nói chung. Ngoài ra, bài báo cung cấp cách sử dụng mô hình đánh giá lượng khí methan tạo ra từ bãi chôn lấp chất thải rắn đô thị và tiềm năng tạo ra năng lượng từ khí đã thu hồi. Đặc biệt, bài báo sử dụng phương pháp đánh giá vòng đời để đánh giá việc giảm phát thải khí nhà kính của hệ thống phát điện sử dụng khí từ bãi rác. Kết quả nghiên cứu chỉ ra rằng, bãi rác Nam Sơn là một bãi rác có tiềm năng lương lượng lớn cần thu hồi và sử dụng, góp phần đáng kể vào việc làm giảm phát thải khí nhà kính, hướng tới sự phát triển bền vững. Bài báo cung cấp một cách nhìn mới về công nghệ năng lượng sử dụng khí từ bãi rác cho Viêt Nam: hệ thống phát điên sử dụng động cơ khí và tuabin khí. Kết quả còn chỉ ra rằng, hệ thống phát điện bằng động cơ khí tỏ ra hiệu quả hơn về lợi ích về môi trường so với hệ thống phát điện bằng tuabin khí. Hệ thống này có thể ứng dụng cho bãi rác Nam Sơn và ứng dụng cho các bãi rác khác của Việt Nam trong tương lai. Từ khóa: Đánh giá vòng đời, giảm phát thải khí nhà kính, khí từ bãi rác, mô hình phát thải khí bãi rác. SUMMARY Landfill gas (LFG), a green, clean, and renewable energy source can be used for electricity generation or fuel industries. This research presents an attempt to assess the energy recovery potential from the Municipal Solid Waste (MSW) landfill, targeting at gas recovery and gas utilization, in mitigating methane emission in particular and green house gas (GHG) emission in general. Our research provides the using of landfill gas emission model (LFGEM) to quantify the methane generation volume for MSW landfill. We then evaluate of energy generation potential from recovered gas. Especially, this research conducted the Life Cycle Inventory to evaluate GHG emission mitigation of power generation system using LFG. The results show that the methane gas flow at Nam Son landfill can provide a considerable energy potential. LFG recovery and utilization could contribute remarkable to GHG emission mitigation, toward to sustainability. The research supplies a new vision of energy technology from LFG for Viet Nam: Gas Engine and Gas Turbine. The research found that Gas Engine is more attractive in term of environmental benefit, which can be applied primarily for Nam Son landfill and continue applied for other landfill in Vietnam for the future. Journal of Science and Development 2009: Tập VI, No 6: 69-77 HA NOI UNIVERSITY OF AGRICULTURE 71 Key words: Green House Gas emission mitigation, landfill gas, landfill gas emission model, life cycle Inventory. 1. INTRODUCTION Climbing LFG is considered as the largest anthropogenic emission source in the developed countries and also as a considerable emission source in developing countries up to now. Landfill gas (LFG) is produced from anaerobic biodegradable decomposition of organic content of landfilled waste. Release of LFG is one of the dangerous contaminations due to high methane content contributing to GHG CHAPTER 4 ENERGY AND POTENTIAL In the previous two chapters we became acquainted with Coulomb's law and its use in finding the electric field about several simple distributions of charge, and also with Gauss's law and its application in determining the field about some symmetrical charge arrangements. The use of Gauss's law was invariably easier for these highly symmetrical distributions, because the problem of integration always disappeared when the proper closed surface was chosen. However, if we had attempted to find a slightly more complicated field, such as that of two unlike point charges separated by a small distance, we would have found it impossible to choose a suitable gaussian surface and obtain an answer. Coulomb's law, however, is more powerful and enables us to solve problems for which Gauss's law is not applicable. The application of Coulomb's law is laborious, detailed, and often quite complex, the reason for this being precisely the fact that the electric field intensity, a vector field, must be found directly from the charge distribution. Three different integrations are needed in general, one for each component, and the resolution of the vector into components usually adds to the complexity of the integrals. Certainly it would be desirable if we could find some as yet undefined scalar function with a single integration and then determine the electric field from this scalar by some simple straightforward procedure, such as differentiation. 83 | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents This scalar function does exist and is known as the potential or potential field. We shall find that it has a very real physical interpretation and is more familiar to most of us than is the electric field which it will be used to find. We should expect, then, to be equipped soon with a third method of finding electric fieldsÐa single scalar integration, although not always as simple as we might wish, followed by a pleasant differentiation. The remaining difficult portion of the task, the integration, we intend to remove in Chap. 7. 4.1 ENERGY EXPENDED IN MOVING A POINT CHARGE IN AN ELECTRIC FIELD The electric field intensity was defined as the force on a unit test charge at that point at which we wish to find the value of this vector field. If we attempt to move the test charge against the electric field, we have to exert a force equal and opposite to that exerted by the field, and this requires us to expend energy, or do work. If we wish to move the charge in the direction of the field, our energy expenditure turns out to be negative; we do not do the work, the field does. Suppose we wish to move a charge Q a distance dL in an electric field E. The force on Q due to the electric field is F E  QE 1 where the subscript reminds us that this force is due to the field. The component of this force in the direction dL which we must overcome is F EL  F Á a L  QE Á a L where a L  a unit vector in the direction of dL: The force which we must apply is equal and opposite to the force due to the field, F appl ÀQE Á a L and our expenditure of energy is the product of the force and distance. That is, Differential work done by external source moving Q ÀQE Á a L dL ÀQE Á dL dW ÀQE Á dL 2 where we have replaced a L dL by the simpler expression dL: This differential amount of work required may be zero under several con- ditions determined easily from (2). There are the trivial conditions for which E, Q,ordL is zero, and a much more important case in which E and dL are 84 ENGINEERING ELECTROMAGNETICS or | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents perpendicular. Here the charge is moved always in a direction at right angles to the electric field. We can draw on a good analogy between the ... ΔKE If only conservative forces act, then Wnet = Wc, where Wc is the total work done by all conservative forces Thus, Wc = ΔKE 3/8 Conservative Forces and Potential Energy Now, if the conservative. .. Potential energy can be stored in any elastic medium by deforming it Indeed, the general definition of potential energy is energy due to position, shape, or 2/8 Conservative Forces and Potential Energy. . .Conservative Forces and Potential Energy Potential Energy of a Spring First, let us obtain an expression for the potential energy stored in a spring (PEs ) We