Technology & Engineering
Dimensional Analysis
Dimensional analysis is a mathematical technique used to check the consistency of equations and to convert units from one system to another. It involves examining the dimensions of physical quantities and using them to derive relationships between different variables. By analyzing the dimensions of various terms in an equation, engineers can ensure that their calculations are accurate and that the units of measurement are consistent.
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10 Key excerpts on "Dimensional Analysis"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
____________________ WORLD TECHNOLOGIES ____________________ Chapter- 4 Dimensional Analysis In mathematics and science, Dimensional Analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature, (denoted M , L , T , Q , and Θ (theta), respectively). For example, speed, which may be measured in meters per second (m/s) or miles per hour (mi/h), has the dimension L / T , or alternatively LT -1 . 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 . 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. This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually formalized in the Buckingham π theorem . This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.
- Ahlam I. Shalaby(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
7 Dimensional Analysis 7.1IntroductionDimensional Analysis is an essential and invaluable tool in the deterministic (as opposed to stochastic) mathematical modeling of fluid flow problems (or any problems other than fluid flow). Deterministic mathematical models are used in order to predict the performance of physical fluid flow situations. As such, in the deterministic mathematical modeling of physical fluid flow situations (both internal and external flow), while some fluid flow situations may be modeled by the application of pure theory (e.g., creeping flow, laminar flow, sonic flow, or critical flow), some (actually, most, due to the assumption of real turbulent flow, which is stochastic in nature) fluid flow situations are too complex for theoretical modeling—for instance, turbulent flow, subsonic (or supersonic or hypersonic) flow, subcritical (or supercritical) flow. However, regardless of the complexity of the fluid flow situation, the analysis phase of mathematical modeling involves the formulation, calibration, and verification of the mathematical model, while the subsequent synthesis or design phase of mathematical modeling involves application of the mathematical model in order to predict the performance of the fluid flow situation. Analysis, by definition, is “to break apart” or “to separate into its fundamental constituents,” where synthesis, by definition, is “to put together” or “to combine separate elements to form a whole.” As such, Dimensional Analysis is an essential and invaluable tool used in both the analysis and synthesis phases of the deterministic mathematical modeling of fluid flow problems.Regardless of the complexity of the fluid flow situation, Dimensional Analysis plays an important role in both the analysis and the synthesis phases of the mathematical modeling of a fluid flow problem (or any problem other than fluid flow). In the deterministic mathematical modeling of a fluid flow (or any) process, the model formulation step begins by applying the theories/principles (conservation of momentum, conservation of mass, conservation of energy, etc.) (see Chapters 3 through 5 ) that govern the fluid flow situation. Then, the initial form of the model is determined by identifying and defining the physical forces and physical quantities that play an important role in the given flow situation. In the case where each physical force and physical quantity can be theoretically modeled, one may easily analytically/theoretically derive/formulate the mathematical model and apply the model without the need to calibrate or verify the model (e.g., see the definitions of Stokes Law, Poiseuille’s law, sonic velocity, critical velocity, etc.). Additionally, if one wishes to experimentally formulate, calibrate, and verify the theoretical mathematical model, one may use the Dimensional Analysis procedure itself (Buckingham π theorem) and the results of Dimensional Analysis (laws of dynamic similarity/similitude) to do so. However, in the case where there is at least one physical force or one physical quantity that cannot be theoretically modeled, one may not analytically/theoretically derive/formulate the mathematical model (as in the case of empirically modeling the flow resistance in real turbulent flow: major head loss, minor head loss, actual discharge, drag force, and the efficiency of a pump or turbine), and, furthermore, one needs to actually calibrate and verify the model before applying the model. In such a case, the Dimensional Analysis procedure itself (Buckingham π theorem) (see Chapter 7 ) is actually necessary in order to formulate (derive) the mathematical model while reducing/minimizing the number of variables required to be calibrated. Furthermore, the results of Dimensional Analysis (laws of dynamic similarity/similitude) (see Chapter 11
eBook - PDF- 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 .
eBook - PDF- Albert Ibarz, Gustavo V. Barbosa-Canovas(Authors)
- 2002(Publication Date)
- CRC Press(Publisher)
The objective of Dimensional Analysis is to relate the different variables involved in the physical processes. For this reason, the variables are grouped in dimensionless groups or rates, allowing discovery of a relationship among the different variables. Table 2.7 presents the dimensionless modules usually found in engineering problems. Dimensional Analysis is an analytical method in which, once the variables that intervene in a physical phenome-non are known, an equation to bind them can be established. That is, dimen-sional analysis provides a general relationship among the variables that should be completed with the assistance of experimentation to obtain the final equation binding all the variables. 2.2.1 Buckingham’s Theorem Every term that has no dimensions is defined as factor . According to Bridgman, there are three fundamental principles of the Dimensional Analysis: 1. All 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. 3. If an equation is dimensionally homogeneous, it may be reduced to a relation among a complete series of dimensionless rates or groups. These induce all the physical variables that influence the phenomenon, the dimensional constants that may correspond to the selected unit system, and the universal constants related to the phenomenon treated. This principle is denoted as Buckingham’s π theorem. A series of dimen-sionless groups is complete if all the groups among them are independent; any other dimensionless group that can be formed will be a combination of two or more groups from the complete series. Because of Buckingham’s π theorem, if the series q 1 , q 2 , …, q n is the set of n independent variables that define a problem or a physical phenomenon, then there will always exist an explicit function of the type:
eBook - PDF- 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.
- Teri Junge(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Dimensional Analysis 15 Dimensional Analysis The term Dimensional Analysis is used to describe the process of understanding and applying the relationships between the qualities of various items (such as drugs and systems for identifying and measuring drug dos-ages) based on the physical characteristics (dimensions) of each item. Dimensional Analysis that often occurs in the surgical setting includes comparisons of weight, size (length, width, height, or volume), distance, temperature, and time between various systems (such as the house-hold system versus the metric system, Roman versus Arabic numbering systems, military versus standard time). BASIC MATH As a vital surgical team member, the surgical technolo-gist is responsible for all of the medications, solutions, tissue dyes and stains , contrast media , hemostatic agents , and any other type of chemical within the sterile field. This responsibility is shared with other surgical team members and includes procuring, preparing, handling, and managing all substances according to federal law, state law, and facility policy and procedure. Fundamen-tal knowledge of the commonly used numbering systems and application of accurate basic mathematical skills ap-plied to calculating drug dosages and concentrations in the surgical setting is critical. A calculation error could produce a devastating result for the patient. Symbols and terminology important to understand-ing basic mathematical concepts include: • 2 Symbol that represents the term subtraction • 2 Symbol that indicates division; separates the numerator and denominator of a fraction • % Symbol that represents the term percent • .
eBook - ePub- Guillaume Delaplace, Karine Loubière, Fabrice Ducept, Romain Jeantet(Authors)
- 2015(Publication Date)
- ISTE Press - Elsevier(Publisher)
2 Dimensional Analysis: Principles and Methodology Abstract This chapter examines the principles and methodology involved in Dimensional Analysis, which should be regarded in this context as a tool for establishing the dimensionless numbers linking the causes of the phenomena studied to its effects, using the homogeneity of dimensions. Keywords Agro-food processes Dimensional Analysis Dimension splitting Elimination-substitution method Fluid mechanics Impeller rotational speeds Mathematical relationship Newtonian fluids Power consumption for a Newtonian fluid Vaschy–Buckingham theorem This chapter examines the principles and methodology involved in Dimensional Analysis, which should be regarded in this context as a tool for establishing the dimensionless numbers linking the causes of the phenomena studied to its effects, using the homogeneity of dimensions. The objective is to provide definitions, prerequisites, tools and essential mathematical demonstrations in order to both construct an unbiased space of dimensionless numbers associated with the phenomena studied and to illustrate the rigor and the potential of applicability of the Dimensional Analysis. Dimensional Analysis is a generic and multi-disciplinary approach: it will be applied in this book in order to understand the physics of, and thereby optimize, agro-food processes. We strongly advise readers, whether trained or not, to take the time to consider these theoretical aspects before turning to the examples given. in 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
eBook - PDF- G. Astarita, R. Ocone(Authors)
- 2001(Publication Date)
- Elsevier Science(Publisher)
Chapter 3 Dimensional Analysis, Scaling, and Orders of Magnitude 3.1 INTRODUCTION The title of this chapter may come as a bit of a surprise to the reader. Dimensional Analysis — that has to do with Buckingham’s theorem and the like, and it is perhaps related to scaleup, but the word scaling doesn’t mean the same thing as scaleup, does it? And what have orders of magnitude to do with all this? Well, let’s begin by agreeing on what the terms mean, with perhaps the exception of ‘Dimensional Analysis’ which needs no clarification. Dimensional Analysis is a very powerful tool for the chemical engineer, but one which should be used with care and intelligence. In its classical formulation, it has to do with the identification of the dimensionless groups which will appear in the solution of any given problem. It is discussed (one way or another) in practi-cally every book on chemical engineering, and yet none of the presentations is entirely convincing. For any given problem, Dimensional Analysis is applied, and the results turn out to be both useful and correct. However, the student is left with the uncomfortable feeling that for every new problem there is a new twist in the technique of applying Dimensional Analysis, and one worries that, given a new problem for which there are no guidelines in the textbooks, one would not feel confident about using Dimensional Analysis. There is, however, a book which is a jewel in this regard, so much so that we don’t feel up to improving upon it, in spite of the fact that it was published more than 70 years ago: “Dimensional Analysis” by P.W. Bridgman, Yale University Press, New Haven, 1922. We urge our readers to study this book — it’s not a long one and it is easy to read, and once one has reached its end lots of things one had learned by heart but didn’t really understand have become crystal clear. So we 171
- Norman W. Loney(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
369 8 Dimensional Analysis and Scaling of Boundary Value Problems 8.1 INTRODUCTION In the practice of chemical engineering, Dimensional Analysis and scaling are techniques that can be employed to emphasize similarities between a prototype and its model� Dimensionless quantities are developed, which can serve as new variables� Usually, the number of new variables is much smaller than the original number of physical variables of the system under consideration� For example, suppose one is interested in conduct-ing an investigation to determine the power required to drive an ordinary house fan [1]� More specifically, suppose one wants to relate the size and shape of the fan to the rota-tional speed and torque, where Newton’s second law relates the torque ( t ) to the forces generated when the fan accelerates the air that passes through it� Suppose the torque is chosen as the dependent variable and the physical variables are the following: • Fan diameter ( d ) • Fan design or shape of fan ( R ) • Medium characteristics (air viscosity, density, sound velocity, and ratio of specific heats) • Rotative speed ( n ) Since the house fan is to be uncomplicated, one can neglect the effects of the air viscosity, sound velocity, and the ratio of specific heats� Then, using the mass ( M ), length ( L ), and time ( T ) system, Table 8�1 can be generated� If the torque is divided by the density, Table 8�2 results� If t / ρ is divided by the product D 5 n 2 , there results the quantity ρ = 1 5 2 t D n Therefore, the final analysis yields ρ = , 0 5 2 f t D n R which means that the torque for a given design R is proportional to the dimensionless product ρ 5 2 t D n
eBook - PDF- 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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