Parabolic Trough Solar Collector Analysis


 Charlotte Higgins
 7 months ago
 Views:
Transcription
1 Parabolic Trough Solar Collector Analysis Derek Lipp AME 4043 Capstone Design Project 28 February 2008 Abstract The purpose of this trade study is to examine a parabolic trough and determine the ideal width and focal length given constraints on the amount of light available, the length of the trough, and reflector material properties. It will show for a 48 inch heat collection element and Southwall s SilverBSR, the ideal width is 0 inches and the ideal focal length is.7 inches given South Bend, IN solar radiation. 1 Developing the Trough Characterization The problem presented is to determine the ideal width and focal length of a parabolic trough. In order to do this, we must first develop the mathematical characterization of the trough and rays incident upon it, shown in Figure 1 provided the parabolic equation y = x2 1, where 4f f is the focal length of the parabola. The rays incident upon the trough are assumed to be parallel due to the sun s approximately infinite distance, and at perfect alignment, these rays are parallel to the yaxis. The slope of the trough is tan θ such that, tan θ = dy dx = x ( ) x 2f θ = tan 1. 2f The normal at an arbitrary value of x bisects the incident ray and its reflection, and the incident ray is shown to be perpendicular to the xaxis. This observation coupled with the 1 Price, et al., 2002, Advances in Parabolic Trough Solar Power Technology, ASME Journal of Solar Energy Engineering, 124(2), pp
2 y y φ Focus n^ Incident light ray n^ ψ f y* y Reflection β θ θ θ θ t^ Parabola x y = 2 4f x f * y* y β θ t^ x Figure 1: The geometric definition of the simple case. Figure 2: The geometric definition including misalignment definition of y (shown in Figure 1) yields, y = f y = x tan β (1) f = x tan β + y (2) In Equations 1 and 2, f is the y coordinate where the incident light is reflected to, and mathematically β = π 2θ. In Figure 1, f and f are equal, while in Figure 2, f > f. 2 Introduce a misalignment, φ, and variable ψ and rederive y and f according to Figure 2. Algebraically, this is represented by, ψ = θ φ β = π 2 ψ θ = π 2 2θ + φ ( π ) [ ( ) ] π x y = x tan 2 2θ + φ = x tan 2 2 tan 1 + φ 2f (3) With the expanded definition of Equation 1 seen in 3, Equation 2 becomes Equation 4. [ ( ) ] π x f (x, f, φ) = x tan 2 2 tan 1 + φ + x2 2f 4f. (4) 2
3 As seen in Equation 4, f, is a function of the x coordinate of the trough, the focal length, f, and any misalignment of the trough from perfect alignment, φ. 2 Considerations for Variables of the Trade Study The design of the parabolic trough is defined by four main characteristics: the width, the length, the focal length, and the reflective material. For the present study, the length was constrained to 48 inches by the choice of an evacuated glass solar tube. Due to time constraints, a reflective material was chosen based on a response to an inquiry concerning Southwall Technologies SilverBSR (data sheet provided in Appendix) material 2. Design requirements state that the system must run given solar radiation typical to South Bend, IN. Using NOAA data for the month of December, this was found to be 20 W. m 2 The design of the parabolic trough produces a number of state variables including, the power output, the incident area of light, the amount of material needed, the cost, the motion envelope, and the losses from misalignment. By a simple energy efficiency calculation involving approximate generator, engine, and heat transfer efficiencies, power output was determined to be W. The incident area of light is the projection of light hitting the trough onto a plane (A = cos φ L W ), and when φ is small, cos φ 1. φ was limited to less than four degrees, because preliminary results revealed losses from greater values of φ were near 100%. The total cost of the material is dependent on the amount needed. The final state, and merit of the project, generated is the loss due to misalignment, with minimal losses being desirable. At this time, note the limitations of the study. This study is meant to examine trends in data produced from the derivation and not intended as final design criteria. The final numerical results for this study apply solely to the material stated above and are theoretical based on the aforementioned and derived analysis. Results are generated assuming perfect alignment can be achieved. Furthermore, validation of this study through experimentation 2 Information provided by Mr. Matthew Coda, Sr. Staff Product Engineer, Southwall Technologies 3
4 should be considered to account for scattering affects of the reflective material, manufacturing imperfections and misalignments greater than the studied range (0 < φ < 4 ). 3 Results The following is the development and results of the trade study for the Southwall SilverBSR material. Given a full spectrum (above 0 nm) reflectance of 94%, the length of 48 inches, available power of 20 W, and required power output of W yields a necessary width m 2 from the following balance: ( ) m in 20 W 1 width W = in 2 in m 48 in 2 Now recall Equation 4, and from above the x range was 0 < x < 2 inches and the range of misalignment was 0 < φ < 4. The remaining design variable, focal length, was initially tested from 1 < f < 19 inches in increments of two inches. Loss was calculated as the percentage of rays falling outside f ± 0.7 inches (the radius of inner, absorbent tube). This method can be seen graphically in Figure 3 as the reflected light position (blue line) plotted within the range of working values (red lines). After examining data from the initial run, minimal losses were seen φ ( ) losses (%) to lie within a focal length range of three to thirteen inches (see MATLAB terminal output in the Appendix). The search parameters were refined to reflect this new range and discretized to increments of 0.01 inches and degrees for the focal length and misalignment respectively. Finally, values for the focal length with minimal loss were weighted to calculate the ideal focal length. Running this new simulation code (in the Appendix) revealed the ideal focal length to be inches. This focal length produces the accompanying table of losses (seen 4
5 f =.7443, φ = 0 f =.7443, φ = 1 Incident light (inches).. Incident light (inches) x coordinate (inches) f =.7443, φ = x coordinate (inches) f =.7443, φ = 3.. Incident light (inches). Incident light (inches) x coordinate (inches) f =.7443, φ = x coordinate (inches). Incident light (inches) x coordinate (inches) Figure 3: The incident light vs. x position results for varying φs
6 graphically in Figure 3). 4 Impact of Results As expected from Equation 4 and confirmed by examining output over ranges of x, the ideal focal length varies sinusoidally. This was noted early in the development process and thus the values were simply evenly weighted to give a single output value. A more complex weighting method could be developed and used to account for larger retained incidence from small x values where losses are minimal. This may also be used to account for the trend that a focal length greater than the ideal is preferred over a focal length shorter than the ideal, since the latter produces higher losses. With this in mind, the general trend in focal length is to increase as the overall width of the trough increases. This could be used to redimension the trough if a longer or shorter heat collection element is used, since the necessary width is linearly proportional with the amount of energy required of the unit. Results indicate a larger trough than initially conceived but at 1.48 square meters remains under the two square meters limit. The ability to easily solve for an ideal focal length given a necessary width (determined by material reflectivity) as demonstrated by the MATLAB script, is encouraging for future material changes that the project may encounter. A negative result is the essentially four degree limit of misalignment revealed in the results. This means an accurate alignment system with tight tolerances will be required to maximize the benefits of the trough. Additional State Information Two additional pieces of state information were generated with the information from the study. To find the surface area of reflective material needed for the trough, recall the line
7 integral for the parabola, A reflector = x 2 dx 48 = in Provided a cost of $ per square inch courtesy of Mr. Coda, the final state is a reflector cost of $1.7. 7
8 Appendix.1 MATLAB code %Derek Lipp %Solar Power Parabolic Trough Trade Study %12 February 2008 %Code to examine the optical properties of parabolic troughs as %characterized by the accompanying development and geometric derivations %This specialized version developed from a generic version close all clear all clc %dimensions given per inch %specify target focal lengths and deviation from incidence focal = [3:.01:13]; phid = [ ]; phi = phid.*pi/180; widths = [2]; for z = 1:1 %specify xvalues width = widths(z); x = [0:.01:width]; parameter = width*100+1; %%%%% Loop for testing multiple values of widths and focal lengths %%%%%% % for i = 1:1001 % for k = 1:41 % count = 0; % for j = 1:parameter % y_star(j) = x(j)*tan(pi/22*atan(x(j)/2/focal(i)) + phi(k)) % + x(j)^2/4/focal(i); % if abs(focal(i)y_star(j)) >.7 % count = count + 1; % loss(i,k) = count/parameter; 8
9 % % ideal = 0; % k = 2; % for i = 2:41 % for j = 2:1001 % if loss(j,i) < loss(j1,i) % k = j; % ideal = focal(k)/40 + ideal; % best(z) = ideal; % % widths % best % code to plot results of the ideal case focal(1) =.7443; for i = 1:1 for k = 1: count = 0; for j = 1:parameter y_star(j) = x(j)*tan(pi/22*atan(x(j)/2/focal(i)) + phi(k)) + x(j)^2/4/focal(i); if abs(focal(i)y_star(j)) >.7 count = count + 1; end end loss(i,k) = count/parameter; y_min = focal(i) .7; y_max = focal(i) +.7; one = ones(1,parameter); figure plot(x,y_star) hold on plot(x,y_min.*one, r ) plot(x,y_max.*one, r ) end end end 9
10 .2 MATLAB losses Printout This sample printout is loss values in terms of percentage divided by 100. Values of φ increase across the columns from left to right and values of focal length increase down the rows. loss = Columns 1 through
11 .3 Provided Reflective Material Data Sheet Figure 4: The Southwall SilverBSR data sheet 11
Reflection and Refraction
Equipment Reflection and Refraction Acrylic block set, planeconcaveconvex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationGeometric Optics Converging Lenses and Mirrors Physics Lab IV
Objective Geometric Optics Converging Lenses and Mirrors Physics Lab IV In this set of lab exercises, the basic properties geometric optics concerning converging lenses and mirrors will be explored. The
More informationGraphical Presentation of Data
Graphical Presentation of Data Guidelines for Making Graphs Titles should tell the reader exactly what is graphed Remove stray lines, legends, points, and any other unintended additions by the computer
More informationFRICTION, WORK, AND THE INCLINED PLANE
FRICTION, WORK, AND THE INCLINED PLANE Objective: To measure the coefficient of static and inetic friction between a bloc and an inclined plane and to examine the relationship between the plane s angle
More information11.1 Parabolas Name: 1
Algebra 2 Write your questions and thoughts here! 11.1 Parabolas Name: 1 Distance Formula The distance between two points, and, is Midpoint Formula The midpoint between two points, and, is,, RECALL: Standard
More informationwaves rays Consider rays of light from an object being reflected by a plane mirror (the rays are diverging): mirror object
PHYS1000 Optics 1 Optics Light and its interaction with lenses and mirrors. We assume that we can ignore the wave properties of light. waves rays We represent the light as rays, and ignore diffraction.
More informationMeasurement of ChargetoMass (e/m) Ratio for the Electron
Measurement of ChargetoMass (e/m) Ratio for the Electron Experiment objectives: measure the ratio of the electron chargetomass ratio e/m by studying the electron trajectories in a uniform magnetic
More informationA Guide to AcoustoOptic Modulators
A Guide to AcoustoOptic Modulators D. J. McCarron December 7, 2007 1 Introduction Acoustooptic modulators (AOMs) are useful devices which allow the frequency, intensity and direction of a laser beam
More informationEXPERIMENT 6 OPTICS: FOCAL LENGTH OF A LENS
EXPERIMENT 6 OPTICS: FOCAL LENGTH OF A LENS The following website should be accessed before coming to class. Text reference: pp189196 Optics Bench a) For convenience of discussion we assume that the light
More informationTorsion Testing. Objectives
Laboratory 4 Torsion Testing Objectives Students are required to understand the principles of torsion testing, practice their testing skills and interpreting the experimental results of the provided materials
More informationHow to add sine functions of different amplitude and phase
Physics 5B Winter 2009 How to add sine functions of different amplitude and phase In these notes I will show you how to add two sinusoidal waves each of different amplitude and phase to get a third sinusoidal
More informationReview Sheet for Third Midterm Mathematics 1300, Calculus 1
Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,
More informationIllumination Models and Shading. Foley & Van Dam, Chapter 16
Illumination Models and Shading Foley & Van Dam, Chapter 16 Illumination Models and Shading Light Source Models Ambient Illumination Diffuse Reflection Specular Reflection Polygon Rendering Methods Flat
More information1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x and yintercepts of graphs of equations. Use symmetry to sketch graphs
More informationSection 1.8 Coordinate Geometry
Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationExample 1. Example 1 Plot the points whose polar coordinates are given by
Polar Coordinates A polar coordinate system, gives the coordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points
More informationScientific Programming
1 The wave equation Scientific Programming Wave Equation The wave equation describes how waves propagate: light waves, sound waves, oscillating strings, wave in a pond,... Suppose that the function h(x,t)
More informationE/M Experiment: Electrons in a Magnetic Field.
E/M Experiment: Electrons in a Magnetic Field. PRELAB You will be doing this experiment before we cover the relevant material in class. But there are only two fundamental concepts that you need to understand.
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More information1051232 Imaging Systems Laboratory II. Laboratory 4: Basic Lens Design in OSLO April 2 & 4, 2002
05232 Imaging Systems Laboratory II Laboratory 4: Basic Lens Design in OSLO April 2 & 4, 2002 Abstract: For designing the optics of an imaging system, one of the main types of tools used today is optical
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationWAVELENGTH OF LIGHT  DIFFRACTION GRATING
PURPOSE In this experiment we will use the diffraction grating and the spectrometer to measure wavelengths in the mercury spectrum. THEORY A diffraction grating is essentially a series of parallel equidistant
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationInterference. Physics 102 Workshop #3. General Instructions
Interference Physics 102 Workshop #3 Name: Lab Partner(s): Instructor: Time of Workshop: General Instructions Workshop exercises are to be carried out in groups of three. One report per group is due by
More informationProcedure: Geometrical Optics. Theory Refer to your Lab Manual, pages 291 294. Equipment Needed
Theory Refer to your Lab Manual, pages 291 294. Geometrical Optics Equipment Needed Light Source Ray Table and Base Threesurface Mirror Convex Lens Ruler Optics Bench Cylindrical Lens Concave Lens Rhombus
More informationMachine Learning and Data Mining. Regression Problem. (adapted from) Prof. Alexander Ihler
Machine Learning and Data Mining Regression Problem (adapted from) Prof. Alexander Ihler Overview Regression Problem Definition and define parameters ϴ. Prediction using ϴ as parameters Measure the error
More informationO6: The Diffraction Grating Spectrometer
2B30: PRACTICAL ASTROPHYSICS FORMAL REPORT: O6: The Diffraction Grating Spectrometer Adam Hill Lab partner: G. Evans Tutor: Dr. Peter Storey 1 Abstract The calibration of a diffraction grating spectrometer
More informationHALFWAVE PARABOLIC REFLECTOR ANTENNA OPTIMIZATION
HALFWAVE PARABOLIC REFLECTOR ANTENNA OPTIMIZATION Parker Singletary, Carson Smith Advisor: Dr. Gregory J. Mazzaro Department of Electrical & Computer Engineering The Citadel, The Military College of South
More informationThe impact of high latitudes on the optical design of solar systems
The impact of high latitudes on the optical design of solar systems Mats Rönnelid 1, Björn Karlsson 2 and J M Gordon 3 1 Solar Energy Research Center, Dalarna University, S781 88 Borlänge, Sweden 2 Vattenfall
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationThe Nature of Electromagnetic Radiation
II The Nature of Electromagnetic Radiation The Sun s energy has traveled across space as electromagnetic radiation, and that is the form in which it arrives on Earth. It is this radiation that determines
More informationPOLAR COORDINATES DEFINITION OF POLAR COORDINATES
POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationwith functions, expressions and equations which follow in units 3 and 4.
Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model
More informationPhysics 9 Fall 2009 Homework 2  Solutions
Physics 9 Fall 009 Homework  s 1. Chapter 7  Exercise 5. An electric dipole is formed from ±1.0 nc charges spread.0 mm apart. The dipole is at the origin, oriented along the y axis. What is the electric
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationAgilent AEDB9140 Series Three Channel Optical Incremental Encoder Modules with Codewheel, 100 CPR to 500 CPR Data Sheet
Agilent AEDB9140 Series Three Channel Optical Incremental Encoder Modules with Codewheel, 100 CPR to 500 CPR Data Sheet Description The AEDB9140 series are three channel optical incremental encoder modules
More informationEquations, Lenses and Fractions
46 Equations, Lenses and Fractions The study of lenses offers a good real world example of a relation with fractions we just can t avoid! Different uses of a simple lens that you may be familiar with are
More informationTheremino System Theremino Spectrometer Technology
Theremino System Theremino Spectrometer Technology theremino System  Theremino Spectrometer Technology  August 15, 2014  Page 1 Operation principles By placing a digital camera with a diffraction grating
More informationComputing Euler angles from a rotation matrix
Computing Euler angles from a rotation matrix Gregory G. Slabaugh Abstract This document discusses a simple technique to find all possible Euler angles from a rotation matrix. Determination of Euler angles
More informationMathematics PreTest Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics PreTest Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {1, 1} III. {1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More informationFactoring Patterns in the Gaussian Plane
Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood
More informationCalculus with Analytic Geometry I Exam 10 Take Home part
Calculus with Analytic Geometry I Exam 10 Take Home part Textbook, Section 47, Exercises #22, 30, 32, 38, 48, 56, 70, 76 1 # 22) Find, correct to two decimal places, the coordinates of the point on the
More informationTutorial 2: Using Excel in Data Analysis
Tutorial 2: Using Excel in Data Analysis This tutorial guide addresses several issues particularly relevant in the context of the level 1 Physics lab sessions at Durham: organising your work sheet neatly,
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationAlgebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2
1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationIntroduction to Cassegrain antenna
Introduction to Cassegrain antenna Srinivasan Ashwyn from June 01th 2009 to June 07th 2009 1 Introduction A number of microwave antennas have been developed which employ doublereflector systems. Each
More informationAntennas. Antenna Design Kit. M. Kesteven ATNF 25/September/2001. The primary elements of a synthesis array. The Antenna Structure
Antennas The Antenna Structure The primary elements of a synthesis array M. Kesteven ATNF 25/September/2001 * Backup structure * Reflector surface(s) shape accuracy construction * Two axis Mount Antenna
More informationChapter 2 Laser Diode Beam Propagation Basics
Chapter 2 Laser Diode Beam Propagation Basics Abstract Laser diode beam propagation characteristics, the collimating and focusing behaviors and the M 2 factor are discussed using equations and graphs.
More informationStudy Guide and Review for Electricity and Light Lab Final
Study Guide and Review for Electricity and Light Lab Final This study guide is provided to help you prepare for the lab final. The lab final consists of multiplechoice questions, usually two for each unit,
More informationk u (t) = k b + k s t
Chapter 2 Stripe domains in thin films 2.1 Films with perpendicular anisotropy In the first part of this chapter, we discuss the magnetization of films with perpendicular uniaxial anisotropy. The easy
More informationVector Algebra II: Scalar and Vector Products
Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define
More informationBinary Stars. Kepler s Laws of Orbital Motion
Binary Stars Kepler s Laws of Orbital Motion Kepler s Three Laws of orbital motion result from the solution to the equation of motion for bodies moving under the influence of a central 1/r 2 force gravity.
More informationPage 1 Class 10 th Physics LIGHT REFLECTION AND REFRACTION
Page 1 LIGHT Light is a form of energy, which induces the sensation of vision in our eyes and makes us able to see various things present in our surrounding. UNITS OF LIGHT Any object which has an ability
More informationExperiment #1, Analyze Data using Excel, Calculator and Graphs.
Physics 182  Fall 2014  Experiment #1 1 Experiment #1, Analyze Data using Excel, Calculator and Graphs. 1 Purpose (5 Points, Including Title. Points apply to your lab report.) Before we start measuring
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Thursday, January 29, 2004 9:15 a.m. to 12:15 p.m.
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
More informationExperimental study of a parabolic solar concentrator
Revue des Energies Renouvelables CICME 08 Sousse (008) 193199 Experimental study of a parabolic solar concentrator A.R. El Ouederni 1*, A.W. Dahmani, F. Askri 3, M. Ben Salah 3 and S. Ben Nasrallah 4
More informationMathematics (Project Maths Phase 1)
2012. M128 S Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination, 2012 Sample Paper Mathematics (Project Maths Phase 1) Paper 2 Ordinary Level Time: 2 hours, 30
More informationGeometric Optics Physics 118/198/212. Geometric Optics
Background Geometric Optics This experiment deals with image formation with lenses. We will use what are referred to as thin lenses. Thin lenses are ordinary lenses like eyeglasses and magnifiers, but
More informationPROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving
More informationPhysics 41, Winter 1998 Lab 1  The Current Balance. Theory
Physics 41, Winter 1998 Lab 1  The Current Balance Theory Consider a point at a perpendicular distance d from a long straight wire carrying a current I as shown in figure 1. If the wire is very long compared
More informationLecture Notes for Chapter 34: Images
Lecture Notes for hapter 4: Images Disclaimer: These notes are not meant to replace the textbook. Please report any inaccuracies to the professor.. Spherical Reflecting Surfaces Bad News: This subject
More informationPHYS 39a Lab 3: Microscope Optics
PHYS 39a Lab 3: Microscope Optics Trevor Kafka December 15, 2014 Abstract In this lab task, we sought to use critical illumination and Köhler illumination techniques to view the image of a 1000 linesperinch
More informationC) D) As object AB is moved from its present position toward the left, the size of the image produced A) decreases B) increases C) remains the same
1. For a plane mirror, compared to the object distance, the image distance is always A) less B) greater C) the same 2. Which graph best represents the relationship between image distance (di) and object
More informationCORNU S SPIRAL. Such diffraction is called Fraunhofer diffraction.
CORNU S SPIRAL If a parallel beam of light from a distant source encounters an obstacle, the shadow of the obstacle is not a simple geometric shadow but is, rather, a diffraction pattern. For example,
More informationUseful Mathematical Symbols
32 Useful Mathematical Symbols Symbol What it is How it is read How it is used Sample expression + * ddition sign OR Multiplication sign ND plus or times and x Multiplication sign times Sum of a few disjunction
More informationParametric Curves. EXAMPLE: Sketch and identify the curve defined by the parametric equations
Section 9. Parametric Curves 00 Kiryl Tsishchanka Parametric Curves Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t), y = g(t) (called
More informationChapter 23. The Reflection of Light: Mirrors
Chapter 23 The Reflection of Light: Mirrors Wave Fronts and Rays Defining wave fronts and rays. Consider a sound wave since it is easier to visualize. Shown is a hemispherical view of a sound wave emitted
More informationPlane Stress Transformations
6 Plane Stress Transformations ASEN 311  Structures ASEN 311 Lecture 6 Slide 1 Plane Stress State ASEN 311  Structures Recall that in a bod in plane stress, the general 3D stress state with 9 components
More informationSpectrophotometry and the BeerLambert Law: An Important Analytical Technique in Chemistry
Spectrophotometry and the BeerLambert Law: An Important Analytical Technique in Chemistry Jon H. Hardesty, PhD and Bassam Attili, PhD Collin College Department of Chemistry Introduction: In the last lab
More informationDevelopment of Optical Wave Microphone Measuring Sound Waves with No Diaphragm
Progress In Electromagnetics Research Symposium Proceedings, Taipei, March 5 8, 3 359 Development of Optical Wave Microphone Measuring Sound Waves with No Diaphragm Yoshito Sonoda, Takashi Samatsu, and
More information2Digital tablets or computer scanners can
Appendix A Measuring Lake Surface Area Lake surface area can be measured with a bathymetric map using any of the following techniques: 1One of the most accurate methods is to use a planimeter to trace
More informationAP Physics B Ch. 23 and Ch. 24 Geometric Optics and Wave Nature of Light
AP Physics B Ch. 23 and Ch. 24 Geometric Optics and Wave Nature of Light Name: Period: Date: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Reflection,
More informationChapter 22: Electric Flux and Gauss s Law
22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we
More informationAP1 Oscillations. 1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationSection 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates
Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in
More informationTesting and Performance of the Convex Lens Concentrating Solar Power Panel Prototype
Testing and Performance of the Convex Lens Concentrating Solar Power Panel Prototype Ankit S. Gujrathi 1, Prof. Dilip Gehlot 2 1 M.tech (2 nd Year), 2 Assistant Professor, Department of Mechanical Engg.,
More informationAppendix E: Graphing Data
You will often make scatter diagrams and line graphs to illustrate the data that you collect. Scatter diagrams are often used to show the relationship between two variables. For example, in an absorbance
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More information2 Absorbing Solar Energy
2 Absorbing Solar Energy 2.1 Air Mass and the Solar Spectrum Now that we have introduced the solar cell, it is time to introduce the source of the energy the sun. The sun has many properties that could
More informationUsing the Spectrophotometer
Using the Spectrophotometer Introduction In this exercise, you will learn the basic principals of spectrophotometry and and serial dilution and their practical application. You will need these skills to
More informationPhysical Science Study Guide Unit 7 Wave properties and behaviors, electromagnetic spectrum, Doppler Effect
Objectives: PS7.1 Physical Science Study Guide Unit 7 Wave properties and behaviors, electromagnetic spectrum, Doppler Effect Illustrate ways that the energy of waves is transferred by interaction with
More informationAstromechanics TwoBody Problem (Cont)
5. Orbit Characteristics Astromechanics TwoBody Problem (Cont) We have shown that the in the twobody problem, the orbit of the satellite about the primary (or viceversa) is a conic section, with the
More information2.4 GHz YagiUda Antenna
P a g e 1 EE 172 Extra Credit Project 2.4 GHz YagiUda Antenna Created by Mario Delgadillo Maringan Pardamean Panggabean P a g e 2 Abstract This report will define antenna theory and design as it relates
More informationThis function is symmetric with respect to the yaxis, so I will let  /2 /2 and multiply the area by 2.
INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,
More informationPrecalculus. What s My Locus? ID: 8255
What s My Locus? ID: 855 By Lewis Lum Time required 45 minutes Activity Overview In this activity, students will eplore the focus/directri and reflection properties of parabolas. They are led to conjecture
More informationSolution Guide for Chapter 6: The Geometry of Right Triangles
Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab
More informationWaves Sound and Light
Waves Sound and Light r2 c:\files\courses\1710\spr12\wavetrans.doc Ron Robertson The Nature of Waves Waves are a type of energy transmission that results from a periodic disturbance (vibration). They are
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationIV. ALGEBRAIC CONCEPTS
IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other
More information10 Polar Coordinates, Parametric Equations
Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationUpon completion of this lab, the student will be able to:
1 Learning Outcomes EXPERIMENT B4: CHEMICAL EQUILIBRIUM Upon completion of this lab, the student will be able to: 1) Analyze the absorbance spectrum of a sample. 2) Calculate the equilibrium constant for
More information