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Nondimensionalization

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Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term scaling is used interchangeably with nondimensionalization, in order to suggest that certain quantities are better measured relative to some appropriate unit.

Physical quantityPhysical quantityA physical quantity (or simply quantity) is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a value, which is a pair of a numerical value and a unit of measurement. For example, the physical quantity mass, symbol m, can be quantified as m=nkg, where n is the numerical value and kg is the unit symbol (for kilogram). Vector quantities have, besides numerical value and unit, direction or orientation in space.Intensive and extensive propertiesIntensive and extensive propertiesProperties independent of system size, and proportional to system size ThermodynamicsThe classical Carnot heat engine Branches Classical Statistical Chemical Quantum thermodynamics Equilibrium / Non-equilibrium Laws Zeroth First Second Third Systems Closed system Open system Isolated system State Equation of state Ideal gas Real gas State of matter Phase (matter) Equilibrium Control volume Instruments Processes Isobaric Isochoric Isothermal Adiabatic Isentropic Isenthalpic Quasistatic Polytropic Free expansion Reversibility Irreversibility Endoreversibility Cycles Heat engines Heat pumps Thermal efficiency System propertiesNote: Conjugate variables in italics Property diagrams Intensive and extensive properties Process functions Work Heat Functions of state Temperature / Entropy (introduction) Pressure / Volume Chemical potential / Particle number Vapor quality Reduced properties Material properties Property databases Specific heat capacity c = {\displaystyle c=} T {\displaystyle T} ∂ S {\displaystyle \partial S} N {\displaystyle N} ∂ T {\displaystyle \partial T} Compressibility β = − {\displaystyle \beta =-} 1 {\displaystyle 1} ∂ V {\displaystyle \partial V} V {\displaystyle V} ∂ p {\displaystyle \partial p} Thermal expansion α = {\displaystyle \alpha =} 1 {\displaystyle 1} ∂ V {\displaystyle \partial V} V {\displaystyle V} ∂ T {\displaystyle \partial T} Equations Carnot's theorem Clausius theorem Fundamental relation Ideal gas law Maxwell relations Onsager reciprocal relations Bridgman's equations Table of thermodynamic equations Potentials Free energy Free entropy Internal energy U ( S , V ) {\displaystyle U(S,V)} Enthalpy H ( S , p ) = U + p V {\displaystyle H(S,p)=U+pV} Helmholtz free energy A ( T , V ) = U − T S {\displaystyle A(T,V)=U-TS} Gibbs free energy G ( T , p ) = H − T S {\displaystyle G(T,p)=H-TS} HistoryCulture History General Entropy Gas laws "Perpetual motion" machines Philosophy Entropy and time Entropy and life Brownian ratchet Maxwell's demon Heat death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva ("living force") Mechanical equivalent of heat Motive power Key publications An Inquiry Concerning theSource ...Parametric equationParametric equationIn mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or more variables called parameters. In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point. For this case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve.SystemSystemA system is a group of interacting or interrelated elements that act according to a set of rules or set of constraints to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and is expressed in its functioning. Systems are the subjects of study of systems theory and other systems sciences.Characteristic propertyA characteristic property is a chemical or physical property that helps identify and classify substances. The characteristic properties of a substance are always the same whether the sample being observed is large or small. Thus, conversely, if the property of a substance changes as the sample size changes, that property is not a characteristic property. Examples of physical properties that are not characteristic properties are mass and volume.International System of UnitsInternational System of UnitsThe International System of Units, internationally known by the abbreviation SI (from its official French name, Système international d'unités), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce.EquationEquationIn mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign=. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables while an égalité does not require variables, but in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.Change of variablesChange of variablesMathematical technique for simplification In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem. Change of variables is an operation that is related to substitution.Dimensional analysisIn engineering and science, dimensional analysis of different physical quantities is the analysis of their physical dimension or quantity dimension, defined as a mathematical expression identifying the powers of the base quantities involved (such as length, mass, time, etc.), and tracking these dimensions as calculations or comparisons are performed. The concepts of dimensional analysis and quantity dimension were introduced by Joseph Fourier in 1822.LengthLengthLength is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system, the base unit for length is the metre. Length is commonly understood to mean the most extended dimension of a fixed object.MeasurementMeasurementMeasurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to a basic reference quantity of the same kind. The scope and application of measurement are dependent on the context and discipline.ResonanceResonancePhysical characteristic of oscillating systemsintrinsicintrinsicRelated word or term.

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