Dimensional Equation

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  Dimensional Analysis

  • (Author)
  • 2014(Publication Date)
  • White Word Publications

    (Publisher)

  ____________________ WORLD TECHNOLOGIES ____________________ Chapter 1 Dimensional Analysis In physics and science, dimensional analysis is a tool to find or check relations among physi cal quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions (usually mass, length, time, electric charge, and temperature) which describe it; for example, speed has the dimension length / time, and may be measured in meters per second, miles per hour, or other units. Dimensional analysis is based on the fact that a physical law must be independent of the units used to measure the physical variables. A straightforward practical consequence is that any meaningful equation (and any inequality and inequation) must have the same dimensions in the left and right sides. Checking this is the basic way of performing dimensional analysis. Dimensional analysis is routinely used to check the plausibility of derived equations and computations. It is also used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena, and to categorize types of physical quantities and units based on their relations to or dependence on other units, or their dimensions if any. The basic principle of dimensional analysis was known to Isaac Newton (1686) who referred to it as the Great Principle of Similitude . James Clerk Maxwell played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived. The 19th -century French mathematician Joseph Fourier made important contributions based on the idea that physical laws like F = ma should be independent of the units employed to measure the physical variables.

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  Handbook of Dynamic System Modeling

  • Paul A. Fishwick(Author)
  • 2007(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  5 -4 Handbook of Dynamic System Modeling them. There is a technique called dimensional analysis , which H. L. Langhaar has defined as “a method by which we deduce information about a phenomenon from the single premise that the phenomenon can be described by a dimensionally correct equation among certain variables.” Some of the available tools of dimensional analysis are now described. 5.2.1 Dimensions and Units The physical quantities used to model objects or systems represent concepts , such as time, length, and mass, to which are also attached numerical values or measurements. If the width of a soccer field is said to be 60 m, the concept invoked is length or distance, and the numerical measure is 60 m. A numerical measure implies a comparison with a standard that enables (1) communication about and (2) comparison of objects or phenomena without their being in the same place. In other words, common measures provide a frame of reference for making comparisons. The physical quantities used to describe or model a problem are either fundamental or primary quanti-ties, or they are derived quantities. A quantity is fundamental if it can be assigned a measurement standard independent of that chosen for the other fundamental quantities. In mechanical problems, for example, mass, length, and time are generally taken as the fundamental mechanical variables, while force is derived from Newton’s law of motion. For any given problem, enough fundamental quantities are required to express each derived quantity in terms of these primary quantities. The word dimension is used to relate a derived quantity to the fundamental quantities selected for a particular model. If mass, length, and time are chosen as primary quantities, then the dimensions of area are (length) 2 , of mass density are mass/(length) 3 , and of force are (mass × length)/(time) 2 .

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  Applied Mathematics

  • J. David Logan(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  scaling. The former permits us to understand the dimensional (meaning length, time, mass, etc.) relationships of the quantities in the equations and the resulting implications of dimensional homogeneity. Scaling is a technique that helps us understand the magnitude of the terms that appear in the model equations by comparing the quantities to intrinsic reference quantities that appear naturally in the physical situation. A side benefit in scaling differential equations, for example, is in the great economy it affords; more often than not, the number of independent parameters can be significantly reduced.

  1.1.2 Dimensional Methods

  One of the basic techniques that is useful in the initial, modeling stage of a problem is the analysis of the relevant quantities and how they must relate to each other in a dimensional way. Simply put, apples cannot equal oranges plus bananas; equations must have a consistency to them that precludes every possible relationship among the variables. Stated differently, equations must be dimensionally homogeneous. These simple observations form the basis of the subject known as dimensional analysis. The methods of dimensional analysis have led to important results in determining the nature of physical phenomena, even when the governing equations were not known. This has been especially true in continuum mechanics, out of which the general methods of dimensional analysis evolved.

  The cornerstone result in dimensional analysis is known as the Pi theorem. The Pi theorem states that if there is a physical law that gives a relation among a certain number of dimensioned physical quantities, then there is an equivalent law that can be expressed as a relation among certain dimensionless quantities (often noted by π1 , π2 ,…, and hence the name). In the early 1900s, E. Buckingham formalized the original method used by Lord Rayleigh and gave a proof of the Pi theorem for special cases; now the theorem often carries his name. Birkhoff (1950) can be consulted for a bibliography and history.

  Example 1.1

  (Atomic explosion) To communicate the flavor and power of this classic result, we consider a calculation made by the British applied mathematician G. I. Taylor in the late 1940s to compute the yield of the first atomic explosion after viewing photographs of the spread of the fireball. In such an explosion a large amount of energy E is released in a short time (essentially instantaneously) in a region small enough to be considered a point. From the center of the explosion a strong shock wave spreads outward; the pressure behind it is on the order of hundreds of thousands of atmospheres, far greater than the ambient air pressure whose magnitude can be accordingly neglected in the early stages of the explosion. It is plausible that there is a relation between the radius of the blast wave front r, time t, the initial air density ρ, and the energy released E. Hence, we assume

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  Introduction to Food Process Engineering

  • Albert Ibarz, Gustavo V. Barbosa-Canovas(Authors)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Dimensional analysis is an analytical method in which, once the variables that intervene in a physical phenomenon are known, an equation to bind them can be established. That is, dimensional analysis provides a general relationship TABLE 2.6 (continued) Conversion Factors 1 kilowatt hour (kW h) 3.6 × 10 6 J 860 kcal 1 atm. liter 0.0242 kcal 10.333 kg m Viscosity 1 poise (P) 0.1 Pa s 1 pound/(ft h) 0.414 mPa s 1 stoke (St) 10 −4 m 2 /s Mass flow 1 lb/h 0.126 g/s 1 ton/h 0.282 kg/s Mass flux 1 lb/(ft 2 h) 1.356 g/s m 2 Thermal magnitudes 1 Btu/(h ft 2 ) 3.155 W/m 2 1 Btu/(h ft 2 °F) 5.678 W/(m 2 K) 1 Btu/lb 2.326 kJ/kg 1 Btu/(lb °F) 4.187 kJ/(kg K) 1 Btu/(h ft °F) 1.731 W/(m K) 15 Unit Systems, Dimensional Analysis, and Similarities among the variables that should be completed with the assistance of experimentation to obtain the final equation that binds all the variables. 2.2.1 B UCKINGHAM ’ S π T HEOREM Every term that has no dimensions is defined as factor π . According to Bridgman, there are three fundamental principles of dimensional analysis: 1. All of the physical magnitudes may be expressed as power functions of a reduced number of fundamental magnitudes. 2. The equations that relate physical magnitudes are dimensionally homogeneous; this means that the dimensions of all their terms must be equal. TABLE 2.7 Dimensionless Modules Modules Expression Equivalence Biot (Bi) hd k Bodenstein (Bo) vd D (Re)(Sc) Euler (Eu) 2 ∆ P v ρ Froude (Fr) d N g P Graetz (Gz) 2 ρ vd kL ˆ C P (Re)(Pr)( d / L ) Grashof (Gr) 3 2 2 g d β ρ η ∆ T Hedstrom (He) d σ ρ η 0 ′ Nusselt (Nu) hd k Peclet (Pe) ρ vdC k P ˆ (Re)(Pr) Power (Po) P d N P 5 ρ Prandtl (Pr) ˆ C k P η Reynolds (Re) ρ η vd Schmidt (Sc) η ρ D Sherwood (Sh) k d D g Stanton (St) h C v P ˆ ρ (Nu)[(Re)(Pr)] −1 Weber (We) ρ σ lv 2 16 Introduction to Food Process Engineering 3. If an equation is dimensionally homogeneous, it may be reduced to a relation among a complete series of dimensionless rates or groups.

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  Qualitative Analysis of Physical Problems

  • M Gitterman(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Chapter 2 Dimensional Analysis 2.1 Introduction Dimensional analysis is based on two simple features of physical formulas. First, such formulas are not just mathematical equations that relate numbers; they are equations that relate physical quantities. Every physical quantity can be described by the product of a number and a unit, and these units have dimensions. Second, physical formulas and equations must be dimensionally homogeneous, i.e., every term must be of the same dimensions. While these statements are fairly obvious, it is surprising how much information can be derived from the dimensional analysis of problems. Fundamental and Derived Units An essential preliminary step before any analysis of dimensions is to assign the appropriate dimensions to all physical quantities. We denote the dimension of a quantity by enclosing the symbol for it in square brackets, i.e., [x] denotes the dimensions of the quantity χ with no regard to its nu-merical value. Physical quantities are generally divided into two groups, according to whether their dimensions are fundamental (or primary, or basic) ones or derived (or secondary) ones. The dimensions of the derived quantities are expressed in terms of the dimensions of the fundamental quantities with the aid of the appropriate physical formulas in which the dimensions of all quantities except the one under consideration are known. The division into fundamental and derived quantities is to some extent arbitrary and a matter of convenience. For instance, while mass M, length 36 2.1 Introduction 37 L, and time Τ are generally treated as fundamental units, quantities such as force F and temperature θ can be chosen to be fundamental or derived ones. An examination of these two examples will clarify the advantages and dis-advantages of increasing the number of fundamental dimensions. Since acceleration a is of the form d 2 x/dt 2 , its dimensions are LT 2 .

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  Introductory Elements of Analysis and Design in Chemical Engineering

  • Bruce C. Gates, Robert L. Powell(Authors)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  4 Units, Dimensions, and Dimensional Analysis

  DOI: 10.1201/9781003429944-4

  Roadmap

  The dimensions that are characteristic of all physical quantities provide insight into physical processes. We review the basic ideas of dimensions and units, showing the usefulness of the fact that each term in an equation must have the same dimensions. The essential point of this chapter is that, solely through analysis of the dimensions, it is possible to predict the forms of equations representing physical phenomena; the process is called dimensional analysis. Terms that are combined to form dimensionless groups appear frequently in the analysis of physical phenomena and sometimes provide simple criteria for such phenomena, illustrated by laminar and turbulent flow predicted by the dimensionless group called the Reynolds number. Dimensional analysis is illustrated for the draining of water from a tank through an orifice in its base, confirming the results developed in Chapter 2 .

  Dimensions

  We associate the word “dimensions” with figures and objects; examples of dimensions are lengths, depths, and radii. In geometry, two points are separated by a distance—the length of the straight line connecting them—this is a dimension. Dimensions have units; we commonly measure lengths in meters (m) or feet (ft). Periods of time have dimensions that we usually express in the units of seconds (s), minutes (min), or hours (h). Fundamental physical quantities such as the velocity of light and the velocity of sound in air have dimensions measured in units such as meters per second (m×s−1 [m s−1 ]). All physical properties of gases, liquids, and solids, such as density, thermal conductivity, and viscosity, have dimensions. Density is often measured in units of kg m−3 .

  Fundamental dimensions include length (L), time (T), mass (M), and force (F). Other fundamental dimensions, such as those encountered in electricity and magnetism, are left out of our discussion because we do not consider such phenomena in this book.

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  Principles of Mathematical Modeling

  • Clive Dym(Author)
  • 2004(Publication Date)
  • Academic Press

    (Publisher)

  2.4 How Do We Do Dimensional Analysis? Dimensional analysis is the process by which we ensure dimensional con-sistency. It ensures that we are using the proper dimensions to describe the problem being modeled, whether expressed in terms of the correct number of properly dimensioned variables and parameters or whether written in terms of appropriate dimensional groups. Remember, too, that we need consistent dimensions for logical consistency, and we need consistent units for arithmetic consistency. 20 Chapter 2 Dimensional Analysis How do we ensure dimensional consistency? First, we check the dimen-sions of all derived quantities to see that they are properly represented in terms of the chosen primary quantities and their dimensions. Then we identify the proper dimensionless groups of variables, that is, ratios and products of problem variables and parameters that are themselves dimensionless. We will explain two different techniques for identifying such dimensionless groups, the basic method and the Buckingham Pi theorem . 2.4.1 The Basic Method of Dimensional Analysis The basic method of dimensional analysis is a rather informal, unstructured approach for determining dimensional groups. It depends on being able to construct a functional equation that contains all of the relevant variables, for which we know the dimensions. The proper dimensionless groups are then identified by the thoughtful elimination of dimensions. For example, consider one of the classic problems of elementary mecha-nics, the free fall of a body in a vacuum. We recall that the speed, V , of such a falling body is related to the gravitational acceleration, g , and the height, h , from which the body was released. Thus, the functional expression of this knowledge is: V = V ( g , h ) . (2.8) Note that the precise form of this functional equation is, at this point, entirely unknown—and we don’t need to know that form for what we’re doing now.

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  Engineering Fundamentals

  An Introduction to Engineering, SI Edition

  • Saeed Moaveni(Author)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  192 CHAPTER 6 Fundamental Dimensions and Systems of Units concepts using fundamental dimensions. Moreover, we will explain them in a way that could be easily grasped at a freshman level. As we explained previously, from the observation of our surroundings, we have learned that we need only a few physical quantities (fundamental dimen- sions) to describe events and our surroundings. With the help of these funda- mental dimensions, we can then define or derive engineering variables that are commonly used in analysis and design. As you will see in the following chapters, there are many engineering design variables that are related to these fundamen- tal dimensions (quantities). As we also discussed and emphasized previously, we need not only physical dimensions to describe our surroundings, but we also need some way to scale or divide these physical dimensions. For example, time is considered to be a physical dimension, but it can be divided into both small and large portions (such as seconds, minutes, hours, and so on). To become a successful engineer, you must first fully understand these fundamentals. Then it is important for you to know how these variables are measured, approximated, calculated, or used in engineering analysis and design. A summary of funda- mental dimensions and their relationship to engineering variables is given in Table 6.7. After you understand these concepts, we will explain the concepts of energy and power in Chapter 13.

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  Similarity and Dimensional Methods in Mechanics

  • L. I. Sedov(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  1 General Dimensional Theory for Different Quantities 1. INTRODUCTION

  In the course of studying mechanical phenomena a number of concepts are introduced, for example, energy, velocity, stress, etc., which characterize the considered phenomenon and can be specified and determined with the help of numbers.

  All the questions concerning motion and equilibrium can be formulated as problems for determining certain functions and numerical values for the quantities characterizing the phenomenon. By solving such problems, we present laws of nature and various geometrical relations in the form of functional equations, which are usually differential.

  In purely theoretical investigations, these equations are used to establish general qualitative properties of motions and for the actual calculation of unknown functional relations with the aid of various mathematical operations. However, it is not always possible to perform mechanical investigation by mathematical considerations and calculations. In a number of cases, in solving mechanical problems one encounters certain insuperable mathematical difficulties. Very often we have no mathematical statement of a problem because the studied mechanical phenomenon is so complicated that there is not yet a satisfactory scheme and equations of motion. We encounter such problems while solving a variety of important problems in the field of aeromechanics, hydromechanics, and the theory of structures. In these cases, experimental methods of investigation, which make it possible to define the simplest experimental facts, are the most important. In general, any investigation of natural phenomena starts with the determination of the simplest experimental facts, on the basis of which one can formulate the laws governing the considered phenomenon and write these laws in the form of certain mathematical relations.

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  Scale-up in Chemical Engineering

  • Marko Zlokarnik(Author)
  • 2006(Publication Date)
  • Wiley-VCH

    (Publisher)

  It can be simply explained by the engineer’s preference for depicting test results in dou- ble-logarithmic plots. Curve sections which can be approxi- mated as straight lines are then analytically expressed as power products. Where this proves less than easy, the engi- neer will often be satisfied with the curves alone, cf. Fig 1. 2. The “benefits” of dimensional analysis are often discussed. The above example provides a welcome opportunity to make the following comments. The five-parametrical dimensional relationship {Dp/l; d; q, m, q } can be represented by means of dimensional analysis as f(Re) and plotted as a single curve (Fig. 1). If we wanted to represent this relationship in a dimensional way and avoid creating a “galaxy” at the same time [24], we would need 25 diagrams with 5 curves in each! If we had assumed that only 5 measurements per curve were sufficient, the graphic repre- sentation of this problem would still have required 625 mea- surements. The enormous savings in time and energy made possible by the application of dimensional analysis are con- sequently easy to appreciate. These significant advantages have already been pointed out by Langhaar [74]. Finally a critical remark must be made concerning a frequent, but completely wrong denotation of diagrams which can often be found in physical and chemical publications. Instead of denoting an axis of a diagram by, e.g., d [m] or d in m d/m is used. This suggests that a dimensionless expression (a numerical value) is obtained by dividing a physical quantity by its unit of measurement. 23 3 Generation of Pi-sets by Matrix Transformation This is real nonsense. Such a “pseudo-dimensionless” representation only means that, e.g., at a value of d/m = 0.35 the diameter d has this value only for the chosen measuring unit (here m).

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  Scaling

  • Grigory Isaakovich Barenblatt(Author)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  Therefore, it must be possible to express them using relationships between quantities that do not depend on this arbitrary choice, i.e., dimensionless combinations of the variables. This was realized long ago, and concepts from dimensional analysis were in use long before the -theorem had been explicitly recognized, formu-lated and proved formally. The outstanding names that should be mentioned here are Galilei, Newton, Fourier, Maxwell, Reynolds and Rayleigh. Dimensional analysis may be successfully applied (see below) in theoretical studies where a mathematical model of the problem is available, in the pro-cessing of experimental data and also in the preliminary analysis of physical phenomena preceding the construction of each model. The point that we are trying to make here is the following. In order to determine the functional dependence of some quantity a , (1.19), on each of the governing parameters, it is necessary to either measure or cal-culate the function f for, let us say, 10 values of each governing parameter. Of course, the number 10 is somewhat arbitrary; a smaller number of mea-surements or calculations may suffice for some smooth functions, while even 100 measurements are insufficient for other functions. Thus, it is necessary to carry out a total of 10 k + m measurements or calculations to determine a . After applying dimensional analysis, the problem is reduced to one of determining a function of m dimensionless arguments 1 ,..., m , and only 10 m (i.e. a factor of 10 k fewer) experiments or calculations are required to determine this function. As a result, we reach the following important conclusion: the amount of work required to determine the desired function is reduced by as many orders of magnitude as there are governing parameters with independent dimensions.

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  Dimensional Analysis of Food Processes

  • Guillaume Delaplace, Karine Loubière, Fabrice Ducept, Romain Jeantet(Authors)
  • 2015(Publication Date)
  • ISTE Press - Elsevier

    (Publisher)

  Chapter 6 . This chapter addresses the semantics of the dimensional analysis process and provides several keys for understanding the rules (coherence of unit systems and independence of variables) and choices made (unit systems, repeated variables and rearrangements of dimensionless numbers) with which the user is faced in constructing the dimensionless numbers. Knowledge of all these aspects will allow the readers to feel confident in carrying out their own dimensional analysis modeling and/or in analyzing other works using this approach.

  This chapter is divided into four sections:

  –  

  in section 2.1 , basic concepts are defined and the fundamental rules for constructing the set of dimensionless numbers are provided;

  –  

  section 2.2 examines the elements which allow the readers to choose (1) a set of dimensionless numbers suitable for the experimental program implemented, and (2) the mathematical form of the process relationship correlating the dimensionless numbers with each other;

  –  

  section 2.3 focuses on the definition of notions regarding the configuration of the system and operating points, which are basic elements for addressing problems of process scale-up or scale-down (Chapter 5 );

  –  

  section 2.4 concludes with a guided example, illustrating all the notions and rules previously described.

  2.1 Terminology and theoretical elements

  2.1.1 Physical quantities and measures. Dimensions and unit systems

  2.1.1.1 Physical quantities and dimensions

  Science uses physical quantities (or entities)1

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