Asce Standardized Reference Evapotranspiration Equation
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Chapter 11. Three Dimensional Analytic Geometry and Vectors
Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1. -
Evapotranspiration in the Urban Heat Island Students Will Be Able To: Explain Transpiration in Plants
Objectives: Evapotranspiration in the Urban Heat Island Students will be able to: explain transpiration in plants. Explain evapotranspiration in the environment Background: Objectives: Living in the desert has always been a challenge for people and other living organisms. observe transpiration by collecting water vapor Students will be able to: There is too little water and, in most cases, too much heat. As Phoenix has grown, in plastic bags, which re-condenses into liquid • observe and explain tran- the natural environment has been transformed from the native desert vegetation into water as it cools. spiration. a diverse assemblage of built materials, from buildings, to parking lots, to roadways. measure and compare changes in air tempera- • explain evapotranspira- Concrete and asphalt increase mass density and heat-storage capacity. This in turn ture due to evaporation from a wet surface vs. tion in the environment means that heat collected during the day is slowly radiated back into the environment a dry one. • measure and compare at night. The average nighttime low temperature in Phoenix has increased by 8ºF over changes in air tempera- the last 30 years. For the months of May through September, the average number of understand that evapotranspiration cools the ture due to evaporation hours per day with temperatures over 100ºF has doubled since 1948. air around plants. from a wet surface vs. a Some researchers have found that the density and diversity of plants moderate tem- relate evapotranspiration to desert landscap- dry one. peratures in neighborhoods (Stabler et al., 2005). Landscaping appears to be one way ing choices in an urban heat island. -
A Quick Algebra Review
A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation. -
Writing the Equation of a Line
Name______________________________ Writing the Equation of a Line When you find the equation of a line it will be because you are trying to draw scientific information from it. In math, you write equations like y = 5x + 2 This equation is useless to us. You will never graph y vs. x. You will be graphing actual data like velocity vs. time. Your equation should therefore be written as v = (5 m/s2) t + 2 m/s. See the difference? You need to use proper variables and units in order to compare it to theory or make actual conclusions about physical principles. The second equation tells me that when the data collection began (t = 0), the velocity of the object was 2 m/s. It also tells me that the velocity was changing at 5 m/s every second (m/s/s = m/s2). Let’s practice this a little, shall we? Force vs. mass F (N) y = 6.4x + 0.3 m (kg) You’ve just done a lab to see how much force was necessary to get a mass moving along a rough surface. Excel spat out the graph above. You labeled each axis with the variable and units (well done!). You titled the graph starting with the variable on the y-axis (nice job!). Now we turn to the equation. First we replace x and y with the variables we actually graphed: F = 6.4m + 0.3 Then we add units to our slope and intercept. The slope is found by dividing the rise by the run so the units will be the units from the y-axis divided by the units from the x-axis. -
The Geometry of the Phase Diffusion Equation
J. Nonlinear Sci. Vol. 10: pp. 223–274 (2000) DOI: 10.1007/s003329910010 © 2000 Springer-Verlag New York Inc. The Geometry of the Phase Diffusion Equation N. M. Ercolani,1 R. Indik,1 A. C. Newell,1,2 and T. Passot3 1 Department of Mathematics, University of Arizona, Tucson, AZ 85719, USA 2 Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK 3 CNRS UMR 6529, Observatoire de la Cˆote d’Azur, 06304 Nice Cedex 4, France Received on October 30, 1998; final revision received July 6, 1999 Communicated by Robert Kohn E E Summary. The Cross-Newell phase diffusion equation, (|k|)2T =∇(B(|k|) kE), kE =∇2, and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order “self-dual” equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. 1. Introduction The mathematical models discussed in this paper are motivated by physical systems, far from equilibrium, which spontaneously form patterns. -
Numerical Algebraic Geometry and Algebraic Kinematics
Numerical Algebraic Geometry and Algebraic Kinematics Charles W. Wampler∗ Andrew J. Sommese† January 14, 2011 Abstract In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work. Contents 1 Introduction 4 2 Notation 5 I Fundamentals of Algebraic Kinematics 6 3 Some Motivating Examples 6 3.1 Serial-LinkRobots ............................... ..... 6 3.1.1 Planar3Rrobot ................................. 6 3.1.2 Spatial6Rrobot ................................ 11 3.2 Four-BarLinkages ................................ 13 3.3 PlatformRobots .................................. 18 ∗General Motors Research and Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090- 9055, U.S.A. Email: [email protected] URL: www.nd.edu/˜cwample1. This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and by General Motors Research and Development. †Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, U.S.A. Email: [email protected] URL: www.nd.edu/˜sommese. This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and the Duncan Chair of the University of Notre Dame. -
Homework No. 4 - Solution
Homework No. 4 - Solution 1. For Lake Nasser, Egypt, in July, the average net radiation is 170 W m-2; the mean daily air temperature is 28.5 ºC, the relative humidity is 55 percent and the wind speed is 2.7 m s-1 at z2=2 m. D At the above temperature is equal to 3.6. Assume zo=0.0003 m and p=101300 Pa. g a. Obtain the open water evaporation rate in millimeters per day using the Thornthwaite- Holzmann aerodynamic or mass transfer method. b. Obtain the open water evaporation rate in millimeters per day using the Penman equation. Use both the aerodynamic method and the empirical Penman equation for the evaluation of the drying power of the air, and compare your results. For each of these conditions, what is the Bowen Ratio? c. Obtain the equilibrium evaporation rate, Ee. d. Obtain the Bowen Ratio implied by the equilibrium evaporation rate and compare with the Bowen Ratio obtained in part b. e. Obtain estimates of the sensible heat flux for these conditions. f. Obtain estimates of the evaporation rate under conditions of minimal advection, Epe. Useae = 1.28. Obtain the Bowen Ratio implied by this rate of evaporation under conditions of minimal advection. g. Obtain the evapotranspiration rate if the above conditions apply to a (well-watered) vegetated surface for which the stomatal resistance equals 200 s/m. For the aerodynamic resistance use the following expression: 2 z2 ln zo r := av k 2 u O restart;with(student); D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint, changevar, completesquare, distance, equate, integrand, intercept, intparts, leftbox, leftsum, makeproc, middlebox, middlesum, midpoint, powsubs, rightbox, rightsum, showtangent, simpson, slope, summand, trapezoid Define evaporation functions needed for this solution: Saturation vapor pressure function (Clausius-Clapeyron eqn.) in Pascals. -
1 Supplementary Material the Blue Water Footprint of Electricity from Hydropower B Corresponding Author
Supplementary material The blue water footprint of electricity from hydropower Mesfin M. Mekonnena,b and Arjen Y. Hoekstraa aDepartment of Water Engineering and Management, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands b corresponding author: [email protected] Method The water footprint of electricity (WF, m3/GJ) generated from hydropower is calculated by dividing the amount of water evaporated from the reservoir annually (WE, m3/yr) by the amount of energy generated (EG, GJ/yr): WE WF (1) EG The total volume of evaporated water (WE, m3/yr) from the hydropower reservoir over the year is: 365 WE 10 E A t1 (2) where E is the daily evaporation (mm/day) and A the area of the reservoir (ha). There are a number of methods for the measurement or estimation of evaporation. These methods can be grouped into several categories including (Singh and Xu, 1997): (i) empirical, (ii) water budget, (iii) energy budget, (iv) mass transfer and (v) a combination of the previous methods. Empirical methods relate pan evaporation, actual lake evaporation or lysimeter measurements to meteorological factors using regression analyses. The weakness of these empirical methods is that they have a limited range of applicability. The water budget methods are simple and can potentially provide a more reliable estimate of evaporation, as long as each water budget component is accurately measured. However, owing to difficulties in measuring some of the variables such as the seepage rate in a water system the water budget methods rarely produce reliable results in practice (Lenters et al., 2005, Singh and Xu, 1997). -
Calibration of Crop Water Use with the Penman Equation Under United Arab Emirates Conditions
AN ABSTRACT OF THE THESIS OF Fareed H. A. N. Mohammed for the degree of Master of Sciencein Soil Science presented on March 9, 1988. Title: Calibration of Crop Water Use With the Penman Equation Under United Arab Emirates Conditions Redacted for Privacy Abstract approved: r BenriO.-P. Warkentin Water shortage is one of the biggest problemsfacing agricultural development in the United Arab Emirates, becauseof the extremely arid climate and the large amount ofirrigation water that is used for crops. Prediction of crop water require- ments is essential to solve water managementproblems in this country. Two versions of the Penman equation, FAO and Wright, were used to estimate reference crop evapotranspirationfrom meteoro- logical data for three regions in the U. A. E.; Northern, Central, and Eastern. Lysimeter measurements at the Northern region wereprovided for five crops; tomato for one season; cabbage, cucumber,and squash each for two seasons; and watermelon for three seasons. Crop evapotranspiration was estimated forthese crops and related to measured crop evapotranspiration. It was found that there is great variation in measured cropevapotranspiration between seasons for individual crops, due totechnical problems with the lysimeter. Combined seasons were used to calibrate the Penman equation, and the resultsindicated that the calibration improved the estimated crop evapotranspiration. Calibration coefficients for individual crops and forindividual methods were calculated. Reference crop evapotranspiration(Etr) value for the three regions were correlated. A high correlation of estimated Etr among the three regions wasfound. Coefficients were developed to predict Etr at the Centraland Eastern regions from the Etr at the Northern region. CALIBRATION OF CROP WATER USE WITH THE PENMAN EQUATION UNDER UNITED ARAB EMIRATES CONDITIONS by Fareed H. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Reference-Evapotranspiration-Report
BUREAU OF METEOROLOGY REFERENCE EVAPOTRANSPIRATION CALCULATIONS C.P. Webb FEBRUARY 2010 ABBREVIATIONS ADAM Australian Data Archive for Meteorology ASCE American Society of Civil Engineers AWS Automatic Weather Station BoM Bureau of Meteorology CAHMDA Catchment-scale Hydrological Modelling and Data Assimilation CRCIF Cooperative Research Centre for Irrigation Futures FAO56-PM equation United Nations Food and Agriculture Organisation’s adapted Penman-Monteith equation recommended in Irrigation and Drainage Paper No. 56 (Allen et al. 1998) ETo Reference Evapotranspiration QLDCSC Queensland Climate Services Centre of the BoM SACSC South Australian Climate Services Centre of the BoM VICCSC Victorian Climate Services Centre of the BoM ii CONTENTS Page Abbreviations ii Contents iii Tables iv Abstract 1 Introduction 1 The FAO56-PM equation 2 Input Data 6 Missing Data 10 Pan Evaporation Data 10 References 14 Glossary 16 iii TABLES I. Accuracies of BoM weather station sensors. II. Input data required to compute parameters of the FAO56-PM equation. III. Correlation between daily evaporation data and daily ETo data. iv BUREAU OF METEOROLOGY REFERENCE EVAPOTRANSPIRATION CALCULATIONS C. P. Webb Climate Services Centre, Queensland Regional Office, Bureau of Meteorology ABSTRACT Reference evapotranspiration (ETo) data is valuable for a range of users, including farmers, hydrologists, agronomists, meteorologists, irrigation engineers, project managers, consultants and students. Daily ETo data for 399 locations in Australia will become publicly available on the Bureau of Meteorology’s (BoM’s) website (www.bom.gov.au) in 2010. A computer program developed in the South Australian Climate Services Centre of the BoM (SACSC) is used to calculate these figures daily. Calculations are made using the adapted Penman-Monteith equation recommended by the United Nations Food and Agriculture Organisation (FAO56-PM equation). -
Transpiration Evaporation Evapotranspiration
FAO WaPOR SUB-NATIONAL LEVEL MAPS (30M) ASSESSING THE WATER CONSUMPTION OF CROPS BEKAA, LEBANON 30M 1000 m This map shows the amount of water consumed through evapotranspiration, or the amount of Legend (water consumed, mm/day) Here, land cover classification is used to identify the spatial distribution of various crops. Legend (crop type) water released back into the air through soil evaporation and plant transpiration, per day, in Wetland Tree cover (dense) Irrigated & rainfed wheat millimetres. Timely information on water consumption represents a critical tool for improving 0.1 mm This map shows the land cover classification of the same area as the one represented in water management in agriculture and irrigation. For example, it provides an objective and the evapotranspiration map on the left. This allows for the identification of the most Grassland Orchard (dense) Other crop common information base for discussing consumption-related water quotas, or for monitoring 2.5 mm common crops grown in any area. Bare Other perennial Grapes the impact of irrigation on water resources. All data are made publicly accessible, thereby Artificial Irrigated maize Irrigated orchard (sparse) allowing for participatory planning. Further distinction between evaporation and transpiration, 5.0 mm If information from both maps, land cover and evapotranspiration, are combined, it can Fallow Irrigated potatoes as allowed by WaPOR, provides key information for reducing non-beneficial water help with setting policies to target specific problem areas, and providing farmers with Irrigated other crops 7.5 mm consumption. recommendations on which agronomic practices best suit their cropping patterns. Maize Irrigated vegetables Irrigated other perennial Potato 10 mm Irrigated orchard (dense) Orchard (sparse) Vegetables Irrigated grapes Evapotranspiration is a key component of the water cycle in agriculture and is a combination of evaporation and plant transpiration.