DOCSLIB.ORG

Scaling Laws and Similarity Models for the Preliminary Design of Multirotor Drones Marc Budinger, Aurélien Reysset, Aitor Ochotorena, Scott Delbecq

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Scaling Laws and Similarity Models for the Preliminary Design of Multirotor Drones Marc Budinger, Aurélien Reysset, Aitor Ochotorena, Scott Delbecq Scaling laws and similarity models for the preliminary design of multirotor drones Marc Budinger, Aurélien Reysset, Aitor Ochotorena, Scott Delbecq To cite this version: Marc Budinger, Aurélien Reysset, Aitor Ochotorena, Scott Delbecq. Scaling laws and similarity models for the preliminary design of multirotor drones. Aerospace Science and Technology, Elsevier, 2020, 98, pp.1-15. 10.1016/j.ast.2019.105658. hal-02997598 HAL Id: hal-02997598 https://hal.archives-ouvertes.fr/hal-02997598 Submitted on 10 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Open Archive Toulouse Archive Ouverte (OATAO ) OATAO is an open access repository that collects the wor of some Toulouse researchers and ma es it freely available over the web where possible. This is an author's version published in: https://oatao.univ-toulouse.fr/26688 Official URL : https://doi.org/10.1016/j.ast.2019.105658 To cite this version : Budinger, Marc and Reysset, Aurélien and Ochotorena, Aitor and Delbecq, Scott Scaling laws and similarity models for the preliminary design of multirotor drones. (2020) Aerospace Science and Technology, 98. 1-15. ISSN 1270-9638 Any correspondence concerning this service should be sent to the repository administrator: [email protected] Scaling laws and similarity models for the preliminary design of multirotor drones ∗ M. Budinger a, , A. Reysset a, A. Ochotorena a, S. Delbecq b a ICA, Université de Toulouse, UPS, INSA, ISAE-SUPAERO, MINES-ALBI, CNRS, 3 rue Caroline Aigle, 31400 Toulouse, France b ISAE-SUPAERO, Université de Toulouse, 10 Avenue Edouard Belin, 31400 Toulouse, France a b s t r a c t Multirotor drones modelling and parameter estimation have gained great interest because of their vast application for civil, industrial, military and agricultural purposes. At the preliminary design level the challenge is to develop lightweight models which remain representative of the physical laws and the system interdependencies. Based on the dimensional analysis, this paper presents a variety of modelling approaches for the estimation of the functional parameters and characteristics of the key components Keywords: of the system. Through this work a solid framework is presented for helping bridge the gaps between Multirotor drones optimizing idealized models and selecting existing components from a database. Special interest is given Scaling laws to the models in terms of reliability and error. The results are compared for various existing drone Dimensional analysis platforms with different requirements and their differences discussed. Surrogate models Propulsion system Landing gear 1. Context which can be integrated into any optimization tool for rapid pre- liminary design of multirotor drones [8,9]. During the last decade, technological innovations [1–3]have Section 2 places emphasis on the sizing scenarios that influ- significantly contributed to the development of smart and power- ence the design of the components and introduces the concept of ful multirotor UAVs: miniaturization and microelectronic with the dimensional analysis for the creation of estimation models. Mod- integration of inertial sensors, increased computational power of els based on similarities, also called scaling laws, are described in control processors, new battery technologies with higher energy section 3 and applied in section 4 for the electrical components: density, permanent magnet motors with higher torque density and motor, battery, ESC and cables. high power densities electric-converters. Regression models are described in section 5 and can adopt The expansion of drones market has led to the decrease of different forms: polynomial forms exemplified in section 6 for pro- drone sizes and the increase of the availability of drone compo- pellers and variable power law models in section 7 for structural nents at very competitive prices. This has facilitated the exper- components. imentation and optimization of drone designs based on succes- Finally, in section 8 several examples of existing drone plat- sive physical tests. Most of recent designs address very specific forms are compared with the results obtained using the models. application-cases with particular performances/needs for which a design optimization process is needed. The design optimization 2. Needs for prediction models during preliminary design process becomes mandatory when the payload scale factor dif- fers from market trend. Particularly in research projects of large Within the whole product development process, the purpose of multirotor UAVs designed for the transport of commercial loads or preliminary design phase is the evaluation of architecture feasibil- passengers [4–7] where expensive prototypes are used, it is sug- ity, technology selection and components high-level specifications gested to perform advanced design studies before manufacturing, definition based on product requirements and operational scenar- integration and testing. To accelerate the design process, a general ios. Yet, to do so, the critical scenarios should be identified and the trend is to extend the role of modelling in design and specification. associated main components characteristics identified. The present paper proposes a set of efficient prediction models 2.1. Multirotor main components * Corresponding author. Fig. 1 shows a typical mass distribution for the different compo- E-mail address: [email protected] (M. Budinger). nents of a drone according to several examples [10–12]. This paper https://doi.org/10.1016/j.ast.2019.105658 2 Nomenclature N ABS Acrylonitrile Butadiene Styrene k Equivalent stiffness................................. eq m2 DC Direct Current Nm K T Torque constant.................................... A DoE Design of Experiments rpm K V Velocity constant.................................. V EMF ElectroMotive Force L Characteristic length................................. m ESC Electronic Speed Controller larm Arm length........................................... m FEM Finite Element Method M Mach number IGBT Insulated Gate Bipolar Transistor M Mass................................................. kg LiPo Lithium-Polymer np Parallel branches LG Landing Gear ns Series branches MTOW Maximum Takeoff Weight nT Coil turns MOSFET Metal Oxide Semiconductor Field Effect Transistor P iron Iron losses........................................... W PBS Product Breakdown Structure Pon Conduction losses................................... W RSM Response Surface Model P switch Commutation losses................................. W UAV Unmanned Aerial Vehicle r Diameter ratio....................................... m VPLM Variable Power Law Metamodel R Resistance............................................ Latin formula symbols U Voltage............................................... V U DC DC voltage............................................ V B Flux density.......................................... T T f Friction torque..................................... Nm Br Magnetic remanence................................. T Tnom Nominal torque.................................... Nm C Capacity............................................. Ah V Flight speed......................................... m −1 s Crate C-rate.............................................. h m V impact Impact speed........................................ s C P Power coefficient C T Thrust coefficient Greek formula symbols D Blade diameter....................................... m β Ratio pitch-to-diameter E No-load voltage...................................... V rad δ Rotational speed.................................... W s h Convection coefficient............................. m2 K Average relative error................................ % Harm Arm height........................................... m rad ω Rotational speed.................................... s I O No-load current...................................... A π Dimensionless number J Advance ratio ρ Resistivity.......................................... m A J Winding current density........................... m2 σ Standard deviation................................... % N K Air compressibility................................. θ Maximum admissible temperature................... K m2 edge of the technology or physical-domain. The so-called compo- nents design drivers are summarized in Table 1 with the corre- sponding scenarios for which they should be assessed as well as literature references. 2.3. Dimensional analysis and Buckingham’s theorem Some references [19]or tools [20]directly use databases to evaluate the performance resulting from the association of defined components. The explosion of combinatorial solutions which in- crease with the databases size may lead to very high computation time. This is why, most of the design methodologies, despite the wide variety of implementations [9,16,8], are based on analytical Fig. 1. Multirotor drone average mass distribution based on [10–12]. expressions or catalogue data regressions and benefit, as most con- tinuous problems, of numerical
Load more

Recommended publications
  • Dimensional Analysis and the Theory of Natural Units
    LIBRARY TECHNICAL REPORT SECTION SCHOOL NAVAL POSTGRADUATE MONTEREY, CALIFORNIA 93940 NPS-57Gn71101A NAVAL POSTGRADUATE SCHOOL Monterey, California DIMENSIONAL ANALYSIS AND THE THEORY OF NATURAL UNITS "by T. H. Gawain, D.Sc. October 1971 lllp FEDDOCS public This document has been approved for D 208.14/2:NPS-57GN71101A unlimited release and sale; i^ distribution is NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral A. S. Goodfellow, Jr., USN M. U. Clauser Superintendent Provost ABSTRACT: This monograph has been prepared as a text on dimensional analysis for students of Aeronautics at this School. It develops the subject from a viewpoint which is inadequately treated in most standard texts hut which the author's experience has shown to be valuable to students and professionals alike. The analysis treats two types of consistent units, namely, fixed units and natural units. Fixed units include those encountered in the various familiar English and metric systems. Natural units are not fixed in magnitude once and for all but depend on certain physical reference parameters which change with the problem under consideration. Detailed rules are given for the orderly choice of such dimensional reference parameters and for their use in various applications. It is shown that when transformed into natural units, all physical quantities are reduced to dimensionless form. The dimension- less parameters of the well known Pi Theorem are shown to be in this category. An important corollary is proved, namely that any valid physical equation remains valid if all dimensional quantities in the equation be replaced by their dimensionless counterparts in any consistent system of natural units.
    [Show full text]
  • Chapter 5 Dimensional Analysis and Similarity
    Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences.
    [Show full text]
  • Three-Dimensional Analysis of the Hot-Spot Fuel Temperature in Pebble Bed and Prismatic Modular Reactors
    Transactions of the Korean Nuclear Society Spring Meeting Chuncheon, Korea, May 25-26 2006 Three-Dimensional Analysis of the Hot-Spot Fuel Temperature in Pebble Bed and Prismatic Modular Reactors W. K. In,a S. W. Lee,a H. S. Lim,a W. J. Lee a a Korea Atomic Energy Research Institute, P. O. Box 105, Yuseong, Daejeon, Korea, 305-600, [email protected] 1. Introduction thousands of fuel particles. The pebble diameter is 60mm High temperature gas-cooled reactors(HTGR) have with a 5mm unfueled layer. Unstaggered and 2-D been reviewed as potential sources for future energy needs, staggered arrays of 3x3 pebbles as shown in Fig. 1 are particularly for a hydrogen production. Among the simulated to predict the temperature distribution in a hot- HTGRs, the pebble bed reactor(PBR) and a prismatic spot pebble core. Total number of nodes is 1,220,000 and modular reactor(PMR) are considered as the nuclear heat 1,060,000 for the unstaggered and staggered models, source in Korea’s nuclear hydrogen development and respectively. demonstration project. PBR uses coated fuel particles The GT-MHR(PMR) reactor core is loaded with an embedded in spherical graphite fuel pebbles. The fuel annular stack of hexahedral prismatic fuel assemblies, pebbles flow down through the core during an operation. which form 102 columns consisting of 10 fuel blocks PMR uses graphite fuel blocks which contain cylindrical stacked axially in each column. Each fuel block is a fuel compacts consisting of the fuel particles. The fuel triangular array of a fuel compact channel, a coolant blocks also contain coolant passages and locations for channel and a channel for a control rod.
    [Show full text]
  • Units of Measurement and Dimensional Analysis
    Measurements and Dimensional Analysis POGIL ACTIVITY.2 Name ________________________________________ POGIL ACTIVITY 2 Units of Measurement and Dimensional Analysis A. Units of Measurement- The SI System and Metric System here are myriad units for measurement. For example, length is reported in miles T or kilometers; mass is measured in pounds or kilograms and volume can be given in gallons or liters. To avoid confusion, scientists have adopted an international system of units commonly known as the SI System. Standard units are called base units. Table A1. SI System (Systéme Internationale d’Unités) Measurement Base Unit Symbol mass gram g length meter m volume liter L temperature Kelvin K time second s energy joule j pressure atmosphere atm 7 Measurements and Dimensional Analysis POGIL. ACTIVITY.2 Name ________________________________________ The metric system combines the powers of ten and the base units from the SI System. Powers of ten are used to derive larger and smaller units, multiples of the base unit. Multiples of the base units are defined by a prefix. When metric units are attached to a number, the letter symbol is used to abbreviate the prefix and the unit. For example, 2.2 kilograms would be reported as 2.2 kg. Plural units, i.e., (kgs) are incorrect. Table A2. Common Metric Units Power Decimal Prefix Name of metric unit (and symbol) Of Ten equivalent (symbol) length volume mass 103 1000 kilo (k) kilometer (km) B kilogram (kg) Base 100 1 meter (m) Liter (L) gram (g) Unit 10-1 0.1 deci (d) A deciliter (dL) D 10-2 0.01 centi (c) centimeter (cm) C E 10-3 0.001 milli (m) millimeter (mm) milliliter (mL) milligram (mg) 10-6 0.000 001 micro () micrometer (m) microliter (L) microgram (g) Critical Thinking Questions CTQ 1 Consult Table A2.
    [Show full text]
  • Planck Dimensional Analysis of Big G
    Planck Dimensional Analysis of Big G Espen Gaarder Haug⇤ Norwegian University of Life Sciences July 3, 2016 Abstract This is a short note to show how big G can be derived from dimensional analysis by assuming that the Planck length [1] is much more fundamental than Newton’s gravitational constant. Key words: Big G, Newton, Planck units, Dimensional analysis, Fundamental constants, Atomism. 1 Dimensional Analysis Haug [2, 3, 4] has suggested that Newton’s gravitational constant (Big G) [5] can be written as l2c3 G = p ¯h Writing the gravitational constant in this way helps us to simplify and quantify a long series of equations in gravitational theory without changing the value of G, see [3]. This also enables us simplify the Planck units. We can find this G by solving for the Planck mass or the Planck length with respect to G, as has already been done by Haug. We will claim that G is not anything physical or tangible and that the Planck length may be much more fundamental than the Newton’s gravitational constant. If we assume the Planck length is a more fundamental constant than G,thenwecanalsofindG through “traditional” dimensional analysis. Here we will assume that the speed of light c, the Planck length lp, and the reduced Planck constanth ¯ are the three fundamental constants. The dimensions of G and the three fundamental constants are L3 [G]= MT2 L2 [¯h]=M T L [c]= T [lp]=L Based on this, we have ↵ β γ G = lp c ¯h L3 L β L2 γ = L↵ M (1) MT2 T T ✓ ◆ ✓ ◆ Based on this, we obtain the following three equations Lenght : 3 = ↵ + β +2γ (2) Mass : 1=γ (3) − Time : 2= β γ (4) − − − ⇤e-mail [email protected].
    [Show full text]
  • Department of Physics Notes on Dimensional Analysis and Units
    1 CSUC Department of Physics 301A Mechanics: Notes on Dimensional Analysis and Units I. INTRODUCTION Physics has a particular interest in those attributes of nature which allow comparison processes. Such qualities include e.g. length, time, mass, force, energy...., etc. By \comparison process" we mean that although we recognize that, say, \length" is an abstract quality, nonetheless, any two \lengths" may successfully be compared in a generally agreed upon manner yielding a real number which we sometimes, by custom, call their \ratio". It's crucial to realize that this outcome it isn't really a mathematical ratio since the participants aren't numbers at all, but rather completely abstract entities . viz. \Physical Extent in Space". Thus we undertake to represent physical qualities (i.e. certain aspects) of our system with real numbers. In general, we recognize that only `lengths' can be compared to `other lengths' etc - i.e. that the world divides into exclusive classes (e.g. `all lengths') of entities which can be compared to each other. We say that all the entities in any one class are `mutually comparable' with each other. These physical numbers will be our basic tools for expressing the relationships we observe in nature. As we proceed in our understanding of the physical world we will come to understand that any (and ultimately every !) properly expressed relationship (e.g. our theories) has several universal aspects of structure which we will study using what we may call \dimensional analysis". These structural aspects are of enormous importance and will be, as you progress, among the very first things you examine in any physical problem you encounter.
    [Show full text]
  • 3.1 Dimensional Analysis
    3.1 DIMENSIONAL ANALYSIS Introduction (For the Teacher).............................................................................................1 Answers to Activities 3.1.1-3.1.7........................................................................................8 3.1.1 Using appropriate unit measures..................................................................................9 3.1.2 Understanding Fundamental Quantities....................................................................14 3.1.3 Understanding unit definitions (SI vs Non-sI Units)................................................17 3.1.4 Discovering Key Relationships Using Fundamental Units (Equations)...................20 3.1.5 Using Dimensional Analysis for Standardizing Units...............................................24 3.1.6 Simplifying Calculations: The Line Method.............................................................26 3.1.7 Solving Problems with Dimensional Analysis ..........................................................29 INTRODUCTION (FOR THE TEACHER) Many teachers tell their students to solve “word problems” by “thinking logically” and checking their answers to see if they “look reasonable”. Although such advice sounds good, it doesn’t translate into good problem solving. What does it mean to “think logically?” How many people ever get an intuitive feel for a coluomb , joule, volt or ampere? How can any student confidently solve problems and evaluate solutions intuitively when the problems involve abstract concepts such as moles, calories,
    [Show full text]
  • [PDF] Dimensional Analysis
    Printed from file Manuscripts/dimensional3.tex on October 12, 2004 Dimensional Analysis Robert Gilmore Physics Department, Drexel University, Philadelphia, PA 19104 We will learn the methods of dimensional analysis through a number of simple examples of slightly increasing interest (read: complexity). These examples will involve the use of the linalg package of Maple, which we will use to invert matrices, compute their null subspaces, and reduce matrices to row echelon (Gauss Jordan) canonical form. Dimensional analysis, powerful though it is, is only the tip of the iceberg. It extends in a natural way to ideas of symmetry and general covariance (‘what’s on one side of an equation is the same as what’s on the other’). Example 1a - Period of a Pendulum: Suppose we need to compute the period of a pendulum. The length of the pendulum is l, the mass at the end of the pendulum bob is m, and the earth’s gravitational constant is g. The length l is measured in cm, the mass m is measured in gm, and the gravitational constant g is measured in cm/sec2. The period τ is measured in sec. The only way to get something whose dimensions are sec from m, g, and l is to divide l by g, and take the square root: [τ] = [pl/g] = [cm/(cm/sec2)]1/2 = [sec2]1/2 = sec (1) In this equation, the symbol [x] is read ‘the dimensions of x are ··· ’. As a result of this simple (some would say ‘sleazy’) computation, we realize that the period τ is proportional to pl/g up to some numerical factor which is usually O(1): in plain English, of order 1.
    [Show full text]
  • Dimensional Analysis
    Dimensional Analysis Taylor Dupuy November 14, 2013 Abstract Hannes Schenck explained this to me. 1 Units and Buckingham's Theorem Given a physical system there are basic units such as mass M, time T and length L and then there are \derived units" like speed S = LT −1, force F = MLT −2 and energy E = FL. Really a unit is derived from a collection of others if there is an algebraic dependence relation. For a given physical system (an experiment or thought experiment) one can take all the units you can measure (or more that will vary while you are considering them) and form an abelian group. For example with the units above if we chose the bijection M = (1; 0; 0), T = (0; 1; 0) and L = (0; 0; 1) then speed is just LT −1 = (1; −1; 0), force is (1; −2; 1) and energy is (1; 2; −1). Really the choice of which units are \derived" is quite arbitrary. All we really know is that we have a collection of units (for example M; T; L; S; F; E) and that they satisfy certain relations. There is a physical map from these units to the physical world for which one has prescribed dimensions. A dimensionless unit is one which is in the kernel of this map. In our example, −1 −1 −2 π1 = SL T is such an element (there are acually two more π2 = F MLT −1 2 −2 and π3 = E ML T ). Note that the free abelian group on our parameters is generated by the fundamental units together with the dimensionless ones.
    [Show full text]
  • Framework for the Strategic Management of Dimensional Variability of Structures in Modular Construction
    Framework for the Strategic Management of Dimensional Variability of Structures in Modular Construction by Christopher Rausch A thesis presented to the University of Waterloo in the fulfillment of the thesis requirement for the degree of Master of Applied Science in Civil Engineering Waterloo, Ontario, Canada, 2016 © Christopher Rausch 2016 Author’s Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Challenges in construction related to dimensional variability exist because producing components and assemblies that have perfect compliance to dimensions and geometry specified in a design is simply not feasible. The construction industry has traditionally adopted tolerances as a way of mitigating these challenges. But what happens when tolerances are not appropriate for managing dimensional variability? In applications requiring very precise dimensional coordination, such as in modular construction, the use of conventional tolerances is frequently insufficient for managing the impacts of dimensional variability. This is evident from the literature and numerous industry examples. Often, there is a lack of properly understanding the rationale behind tolerances and about how to derive case specific allowances. Literature surrounding the use of tolerances in construction indicates that dimensional variability is often approached in a trial and error manner, waiting for conflicts and challenges to first arise, before developing appropriate solutions. While this is time consuming, non-risk averse, prone to extensive rework and very costly in conventional construction, these issues only intensify in modular construction due to the accumulation of dimensional variability, the geometric complexity of modules, and discrepancy between module production precision and project site dimensional precision.
    [Show full text]
  • QUANTUM PHYSICS I Using Simple Dimensional Analysis We Can Derive
    QUANTUM PHYSICS I PROBLEM SET 1 due September 15, 2006 A. Dimensional analysis and the blackbody radiation Using simple dimensional analysis we can derive many results of the (flawed) classical theory of blackbody radiation. In particular we can deduce it is bounded to fail ! Follow the steps below. i) As we showed in class the energy density inside the blackbody cavity ρT (ν) is a function only of the temperature and the frequency. In addition it can depend on the fundamental constants c (speed of light, appearing on the Maxwell equations) and k (Boltzman constant). Convince yourself that the dimensions of these quantities are [T ] = temperature,[ν] = 1/time,[c] = length/time and [k] = energy/temperature = mass length2/time2 temperature ([..] denotes the dimension of ...) . Then prove that no dimensionless quantity can be created using these quantities (hint: write T ανβcγ kδ and prove that no choice of α, ...δ can produce a dimensionless quantity). This result shows that non-polynomial functions like exp(x) = 1 + x + x2/2 + ··· cannot arise since there is no dimensionless x to serve as an argument for exp. ii) Use the same kind of argument show that ρT (ν), which has dimensions of energy/volume frequency = mass/length time, is proportional to ν2. This result was “derived” in class through a more complicated calcula- tion. Show that the total energy, integrated over all frequencies, diverges. iii) The introduction of another fundamental constant h with dimensions of energy time invalidate the argument in i). Find a dimensionless combination of h, ν, k and T . This quantity can appear inside non-polynomial functions and invalidate the result in ii).
    [Show full text]
  • Metric System, Dimensional Analysis and Introduction to Solution Chemistry Version 04-06-12
    College of the Canyons: “Introduction to Biotechnology” Custom Lab Exercises Metric System, Dimensional Analysis and Introduction to Solution Chemistry Version 04-06-12 • The metric system is a decimal system of measurement of such features as: length, mass, area, and other physical conditions of objects. • Dimensional analysis is the conversion of one unit of measurement to a different unit of measurement (i.e. inches to centimeters.) • The conversion is accomplished using equalities and proportions. • Metric familiarity is less common in the U.S., where the British System is still in use. As a result, students of science should make an effort to make themselves more aware of metric units by assessing their surrounding with respect to “metric awareness”. • Proper dimensional analysis uses conversions factors that are equal to one, the units cancel accordingly. • Solution preparation involves conversion factors, volumetric transfer and familiarity with molar equations. • These topics are further explored in supplementary reading and assignments. • The Molarity and Dilutions and Serial Dilutions exercises (at the back of this handout) should be completed after doing the Concentration Calculations and Expressions lab (due February 3/see syllabi) For more information on College of the Canyons’ Introduction Biotechnology course, contact Jim Wolf, Professor of Biology/Biotechnology at (661) 362-3092 or email: [email protected]. Online versions available @ www.canyons.edu/users/wolfj These lab protocols can be reproduced for educational purposes only. They have been developed by Jim Wolf, and/or those individuals or agencies mentioned in the references. I. Objectives: 1. To become familiar with key metric units and relate their values to common British equivalents.
    [Show full text]