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| | There are a number of chemical procedures that require catalysts for the reactions to be completed. In case you loved this short article and you would like to receive more details relating to [http://randolphl4.blog.com/2014/05/29/la-fibra-de-vidrio-un-elemento-provechoso-con-formidable-potencial-en-el-reciclaje/ resina epoxi] i implore you to visit our own site. These are responsible for the proper functioning of the chemical reactions, so that the desired results can be achieved.<br><br>Reactive thinners are one of these catalysts which are widely used in different ways and forms to get the most useful results with the help of the same. The use of epoxy resin in different forms can be done only with the help of adding the right amount of diluent in it.<br><br>They are used to form adhesive bonds that are strong and even unbreakable at times. It's essential you understand the specific characteristics of diluents so that you know mixing them in what proportion is going to get you what sort of results.<br><br>You need to properly understand about the characteristics of these diluents so that you can use them according to your needs and requirements to get the desired reactions in the end.<br><br>The first and most visible quality of a diluent is the fact that it increases the solubility of the agent you mix it with. This means that when you mix the reactive, it shall reduce the viscosity or the thickness of the material.<br><br>This is what makes it really important when you need to use a thinner form of a particular chemical. It basically helps in reducing the molecular weight of the epoxy resins. They tend to make diluted counterparts of the pure form of the chemical which can be used for various applications.<br><br>The next advantage of it is that the vapor they produce resultantly has a low content of pressure. Another great feature of these reactive diluents is that they tend to reduce the substance and the wetting capability of the fillers that are made for using these adhesive joints.<br><br>It helps the agent become water and chemical resistant as well and they cannot tend to have harmful effects on it through their set of reactions.<br><br>Listed below are a few typical uses of these diluents with epoxy:<br><br>Flooring-While creating seamless floors for homes, this strong form of adhesive is used because it one of the most strong and long-staying adhesive that can be counted upon to stay put for a long time.<br><br>Another form of application epoxies with diluents is in the leather industry. Each product is careful coated with an efficient layer of epoxy because it helps to render properties such as water resistance, shine, and a finished and smooth look. |
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| '''Nondimensionalization''' is the partial or full removal of [[Units of measurement|unit]]s from an [[mathematical equation|equation]] involving [[physical quantity|physical quantities]] by a suitable substitution of [[Variable (mathematics)|variables]]. This technique can simplify and [[parametric equation|parameterize]] problems where [[Measurement|measured]] units are involved. It is closely related to [[dimensional analysis]]. In some physical [[system]]s, the term '''scaling''' is used interchangeably with ''nondimensionalization'', in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities [[Wiktionary:intrinsic|intrinsic]] to the system, rather than units such as [[International System of Units|SI]] units. Nondimensionalization is not the same as converting [[intensive and extensive properties|extensive quantities]] in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
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| Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic [[Resonance|resonance frequency]], [[length]], or [[time constant]], nondimensionalization can recover these values. The technique is especially useful for systems that can be described by [[differential equation]]s. One important use is in the analysis of [[control system]]s.
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| One of the simplest characteristic units is the [[doubling time]] of a system experiencing [[exponential growth]], or conversely the [[half-life]] of a system experiencing [[exponential decay]]; a more natural pair of characteristic units is mean age/[[mean lifetime]], which correspond to base ''e'' rather than base 2.
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| Many illustrative examples of nondimensionalization originate from simplifying differential equations. This is because a large body of physical problems can be formulated in terms of differential equations. Consider the following:
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| * [[List of dynamical systems and differential equations topics]]
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| * [[List of partial differential equation topics]]
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| * [[Differential equations of mathematical physics]]
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| Although nondimensionalization is well adapted for these problems, it is not restricted to them. An example of a non-differential-equation application is dimensional analysis; another example is [[Normalization (statistics)|normalization]] in [[statistics]].
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| [[Measuring instrument|Measuring device]]s are practical examples of nondimensionalization occurring in everyday life. Measuring devices are calibrated relative to some known unit. Subsequent measurements are made relative to this standard. Then, the absolute value of the measurement is recovered by scaling with respect to the standard.
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| == Rationale ==
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| Suppose a [[pendulum]] is swinging with a particular [[Frequency|period]] ''T''. For such a system, it is advantageous to perform calculations relating to the swinging relative to ''T''. In some sense, this is normalizing the measurement with respect to the period.
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| Measurements made relative to an intrinsic property of a system will apply to other systems which also have the same intrinsic property. It also allows one to compare a common property of different implementations of the same system. Nondimensionalization determines in a systematic manner the '''characteristic units''' of a system to use, without relying heavily on prior knowledge of the system's intrinsic properties
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| (one should not confuse characteristic units of a ''system'' with [[natural units]] of ''nature''). In fact, nondimensionalization can suggest the parameters which should be used for analyzing a system. However, it is necessary to start with an equation that describes the system appropriately.
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| == Nondimensionalization steps ==
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| To nondimensionalize a system of equations, one must do the following:
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| #Identify all the independent and dependent variables;
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| #Replace each of them with a quantity scaled relative to a characteristic unit of measure to be determined;
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| #Divide through by the coefficient of the highest order polynomial or derivative term;
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| #Choose judiciously the definition of the characteristic unit for each variable so that the coefficients of as many terms as possible become 1;
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| #Rewrite the system of equations in terms of their new dimensionless quantities.
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| The last three steps are usually specific to the problem where nondimensionalization is applied. However, almost all systems require the first two steps to be performed.
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| As an illustrative example, consider a first order differential equation with [[constant coefficients]]:
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| ::<math>a\frac{dx}{dt} + bx = Af(t).</math>
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| # In this equation the independent variable here is ''t'', and the dependent variable is ''x''.
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| # Set <math>x = \chi x_c, \ t = \tau t_c</math>. This results in the equation
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| #:<math>a \frac{x_c}{t_c} \frac{d \chi}{d \tau} + b x_c \chi = A f(\tau t_c) \ \stackrel{\mathrm{def}}{=}\ A F(\tau).</math>
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| # The coefficient of the highest ordered term is in front of the first derivative term. Dividing by this gives
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| #:<math>\frac{d \chi}{d \tau} + \frac{b t_c}{a} \chi = \frac{A t_c}{a x_c} F(\tau).</math>
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| # The coefficient in front of χ only contains one characteristic variable ''t<sub>c</sub>'', hence it is easiest to choose to set this to unity first:
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| #:<math>\frac{b t_c}{a} = 1 \Rightarrow t_c = \frac{a}{b}.</math> Subsequently, <math>\frac{A t_c}{a x_c} = \frac{A}{b x_c} = 1 \Rightarrow x_c = \frac{A}{b}.</math>
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| # The final dimensionless equation in this case becomes completely independent of any parameters with units:
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| #:<math>\frac{d \chi}{d \tau} + \chi = F(\tau).</math>
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| === Substitutions ===
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| Suppose for simplicity that a certain system is characterized by two variables - a dependent variable ''x'' and an independent variable ''t'', where ''x'' is a [[function (mathematics)|function]] of ''t''. Both ''x'' and ''t'' represent quantities with units. To scale these two variables, assume there are two intrinsic units of measurement ''x''<sub>c</sub> and ''t''<sub>c</sub> with the same units as ''x'' and ''t'' respectively, such that these conditions hold:
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| :<math>\tau = \frac{t}{t_c} \Rightarrow t = \tau t_c </math>
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| :<math> \chi = \frac{x}{x_c} \Rightarrow x = \chi x_c.</math>
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| These equations are used to replace ''x'' and ''t'' when nondimensionalizing. If differential operators are needed to describe the original system, their scaled counterparts become dimensionless differential operators.
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| ==== Conventions ====
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| There are no restrictions on the variable names used to replace "''x''" and "''t''". However, they are generally chosen so that it is convenient and intuitive to use for the problem at hand. For example, if "''x''" represented mass, the letter "''m''" might be an appropriate symbol to represent the dimensionless mass quantity.
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| In this article, the following conventions have been used:
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| * ''t'' - represents the independent variable - usually a time quantity. Its nondimensionalized counterpart is ''τ''.
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| * ''x'' - represents the dependent variable - can be mass, voltage, or any measurable quantity. Its nondimensionalized counterpart is ''χ''.
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| A subscripted ''c'' added to a quantity's variable-name is used to denote the characteristic unit used to scale that quantity. For example, if ''x'' is a quantity, then ''x<sub>c</sub>'' is the characteristic unit used to scale it.
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| ==== Differential operators ====
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| Consider the relationship
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| <!-- The \,\! is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| :<math>\,\! t = \tau t_c \Rightarrow dt = t_c d\tau \Rightarrow \frac{d\tau}{dt} = \frac{1}{t_c}.</math>
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| The dimensionless differential operators with respect to the independent variable becomes
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| :<math>\frac{d}{dt} = \frac{d \tau}{dt} \frac{d}{d \tau} = \frac{1}{t_c} \frac{d}{d \tau} \Rightarrow \frac{d^n}{dt^n} = \left( \frac{d}{dt} \right)^n = \left( \frac{1}{t_c} \frac{d}{d \tau} \right)^n = \frac{1}{t_c^n} \frac{d^n}{d \tau^n}.</math>
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| ==== Forcing function ====
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| If a system has a [[forcing function (differential equations)|forcing function]] ''f''(''t''), then
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| <!-- The \,\! is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| :<math>\,\! f(t) = f(\tau t_c) = f(t(\tau)) = F(\tau).</math>
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| Hence, the new forcing function ''F'' is made to be dependent on the dimensionless quantity ''τ''.
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| == Linear differential equations with constant coefficients ==
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| === First order system ===
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| Let us consider the differential equation for a first order system:
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| :<math>a\frac{dx}{dt} + bx = Af(t).</math>
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| The [[#Nondimensionalization steps|derivation]] of the characteristic units for this system gives
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| :<math>t_c = \frac{a}{b}, \ x_c = \frac{A}{b}.</math>
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| === Second order system ===
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| A second order system has the form
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| :<math>a \frac{d^2 x}{dt^2} + b\frac{dx}{dt} + cx = A f(t).</math>
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| ==== Substitution step ====
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| Replace the variables ''x'' and ''t'' with their scaled quantities. The equation becomes
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| :<math>a \frac{x_c}{t_c^2} \frac{ d^2 \chi}{d \tau^2} + b \frac{x_c}{t_c} \frac{d \chi}{d \tau} + c x_c \chi = A f(\tau t_c) = A F(\tau) .</math>
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| This new equation is not dimensionless, although all the variables with units are isolated in the coefficients. Dividing by the coefficient of the highest ordered term, the equation becomes
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| :<math> \frac{d^2 \chi}{d \tau^2} + t_c \frac{b}{a} \frac{d \chi}{d \tau} + t_c^2 \frac{c}{a} \chi = \frac{A t_c^2}{a x_c} F(\tau).</math>
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| Now it is necessary to determine the quantities of ''x''<sub>''c''</sub> and ''t''<sub>''c''</sub> so that the coefficients become normalized. Since there are two free parameters, at most only two coefficients can be made to equal unity.
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| ==== Determination of characteristic units ====
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| Consider the variable ''t''<sub>''c''</sub>:
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| #If <math> t_c = \frac{a}{b} </math> the first order term is normalized.
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| #If <math> t_c = \sqrt{\frac{a}{c}} </math> the zeroth order term is normalized.
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| Both substitutions are valid. However, for pedagogical reasons, the latter substitution is used for second order systems. Choosing this substitution allows ''x''<sub>''c''</sub> to be determined by normalizing the coefficient of the forcing function:
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| :<math>1 = \frac{A t_c^2}{a x_c} = \frac{A}{c x_c} \Rightarrow x_c = \frac{A}{c}.</math>
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| The differential equation becomes
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| :<math>\frac{d^2 \chi}{d \tau^2} + \frac{b}{\sqrt{ac}} \frac{d \chi}{d\tau} + \chi = F(\tau). </math>
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| The coefficient of the first order term is unitless. Define
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| :<math>2 \zeta \ \stackrel{\mathrm{def}}{=}\ \frac{b}{\sqrt{ac}}. </math>
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| The factor 2 is present so that the solutions can be parameterized in terms of ζ. In the context of mechanical or electrical systems, ζ is known as the [[damping ratio]], and is an important parameter required in the analysis of [[control system]]s. 2ζ is also known as the [[linewidth]] of the system. The result of the definition is the [[Harmonic oscillator#Universal oscillator equation|universal oscillator equation]].
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| :<math>\frac{d^2 \chi}{d \tau^2} + 2 \zeta \frac{d \chi}{d\tau} + \chi = F(\tau) .</math>
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| === Higher order systems ===
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| The general n-th order linear differential equation with constant coefficients has the form:
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| :<math>a_n \frac{d^n x(t)}{dt^n} + a_{n-1} \frac{d^{n-1} x(t)}{dt^{n-1}} + \ldots + a_1 \frac{dx(t)}{dt} + a_0 x(t) = \sum_{k = 0}^n a_k \frac{d^k x(t)}{dt^k} = Af(t). </math>
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| The function ''f''(''t'') is known as the [[forcing function (differential equations)|forcing function]].
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| If the differential equation only contains real (not complex) coefficients, then the properties of such a system behaves as a mixture of first and second order systems only. This is because the [[root of a function|roots]] of its [[characteristic polynomial]] are either [[Real number|real]], or [[complex conjugate]] pairs. Therefore, understanding how nondimensionalization applies to first and second ordered systems allows the properties of higher order systems to be determined through [[Superposition principle|superposition]].
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| The number of free parameters in a nondimensionalized form of a system increases with its order. For this reason, nondimensionalization is rarely used for higher order differential equations. The need for this procedure has also been reduced with the advent of [[symbolic computation]].
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| === Examples of recovering characteristic units ===
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| A variety of systems can be approximated as either first or second order systems. These include mechanical, electrical, fluidic, caloric, and torsional systems. This is because the fundamental physical quantities involved within each of these examples are related through first and second order derivatives.
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| ==== Mechanical oscillations ====
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| [[Image:Mass-Spring-Damper.png|thumb|300px|A mass attached to a spring and a dashpot.]]
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| Suppose we have a mass attached to a spring and a damper, which in turn are attached to a wall, and a force acting on the mass along the same line.
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| Define
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| : ''x'' = displacement from equilibrium [m]
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| : ''t'' = time [s]
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| : ''f'' = external force or "disturbance" applied to system [kg m s<sup>-2</sup>]
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| : ''m'' = mass of the block [kg]
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| : ''B'' = damping constant of dashpot [kg s<sup>-1</sup>]
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| : ''k'' = force constant of spring [kg s<sup>-2</sup>]
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| Suppose the applied force is a sinusoid ''F'' = ''F''<sub>0</sub> cos(ω''t''), the differential equation that describes the motion of the block is
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| :<math>m \frac{d^2 x}{d t^2} + B \frac{d x}{d t} + kx = F_0 \cos(\omega t)</math>
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| Nondimensionalizing this equation the same way as described under [[#Second order system|second order system]] yields several characteristics of the system.
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| The intrinsic unit ''x<sub>c</sub>'' corresponds to the distance the block moves per unit force
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| :<math>x_c = \frac{F_0}{k}.</math>
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| The characteristic variable ''t<sub>c</sub>'' is equal to the period of the oscillations | |
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| :<math>t_c = \sqrt{\frac{m}{k}}</math>
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| and the dimensionless variable 2''ζ'' corresponds to the linewidth of the system. ''ζ'' itself is the [[damping ratio]].
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| :<math>2 \zeta = \frac{B}{\sqrt{mk}}</math>
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| ==== Electrical oscillations ====
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| ===== First-order series RC circuit =====
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| For a series [[RC circuit|RC]] attached to a [[power supply|voltage source]]
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| :<math>R \frac{dQ}{dt} + \frac{Q}{C} = V(t) \Rightarrow \frac{d \chi}{d \tau} + \chi = F(\tau)</math>
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| with substitutions
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| :<math>Q = \chi x_c, \ t = \tau t_c, \ x_c = C V_0, \ t_c = RC, \ F = V.</math>
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| The first characteristic unit corresponds to the total [[electric charge|charge]] in the circuit. The second characteristic unit corresponds to the [[time constant]] for the system.
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| ===== Second-order series RLC circuit =====
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| For a series configuration of ''R'',''C'',''L'' components where ''Q'' is the charge in the system
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| :<math> L \frac{d^2 Q}{dt^2} + R \frac{d Q}{d t} + \frac{Q}{C} = V_0 \cos(\omega t) \Rightarrow \frac{d^2 \chi}{d \tau^2} + 2 \zeta \frac{d \chi}{d\tau} + \chi = \cos(\Omega \tau) </math>
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| with the substitutions
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| :<math>Q = \chi x_c, \ t = \tau t_c, \ \ x_c = C V_0, \ t_c = \sqrt{LC}, \ 2 \zeta = R \sqrt{\frac{C}{L}}, \ \Omega = t_c \omega.</math>
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| The first variable corresponds to the maximum charge stored in the circuit. The resonance frequency is given by the reciprocal of the characteristic time. The last expression is the linewidth of the system. The Ω can be considered as a normalized forcing function frequency.
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| == Nonlinear differential equation example ==
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| Since there are no general methods of solving nonlinear differential equations, each case has to be considered on an individual basis when nondimensionalizing.
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| === Quantum harmonic oscillator ===
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| The [[Schrödinger equation]] for the one dimensional time independent [[quantum harmonic oscillator]] is
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| :<math>\left(-\frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{1}{2}m \omega^2 x^2\right) \psi(x) = E \psi(x).</math>
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| The modulus square of the [[wavefunction]] ''|ψ|^2'' represents probability, which is in a sense already dimensionless and normalized. Therefore, there is no need to nondimensionalize the wavefunction. However, it should be rewritten as a function of a dimensionless variable. Furthermore, the variable ''x'' has units of length. Hence substitute
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| :<math>\chi = \frac{x}{x_c}, \ \psi(x) = \psi(x(\chi)) = \psi(\chi).</math>
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| The differential equation becomes
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| :<math> \left(- \frac{\hbar^2}{2m x_c^2} \frac{d^2}{d \chi^2} + \frac{m \omega^2 x_c^2}{2} \chi^2 \right) \psi(\chi) = E \psi(\chi) \Rightarrow \left( -\frac{d^2}{d \chi^2} + \frac{m^2 \omega^2 x_c^4}{\hbar^2} \chi^2 \right ) \psi(\chi) = \frac{2 E m x_c^2}{\hbar^2} \psi(\chi).</math>
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| To make the term in front of ''χ''² unitless, set
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| :<math> \frac{m^2 \omega^2 x_c^4}{\hbar^2} = 1 \Rightarrow x_c = \sqrt{\frac{\hbar}{m \omega}} . </math> | |
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| Hence, the fully nondimensionalized equation is
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| :<math>\left( -\frac{d^2}{d \chi^2} + \chi^2 \right) \psi(\chi) = \frac{2E}{\hbar \omega} \psi(\chi) \ \stackrel{\mathrm{def}}{=}\ \mathcal{E} \psi(\chi) . </math>
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| The nondimensionalization factor for the energy is the same as the [[ground state]] of the harmonic oscillator. Usually, the energy term is not made dimensionless because a primary emphasis of quantum mechanics is determining the energies of the [[Quantum state|states]] of a system. Rearranging the first equation, the familiar equation for the harmonic oscillator is
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| :<math> \frac{\hbar \omega}{2} \left( -\frac{d^2}{d \chi^2} + \chi^2 \right) \psi(\chi) = E \psi(\chi) .</math>
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| <!--=== Fluid dynamics ===-->
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| <!-- Show for example, how the nondimensional version of the Kortewed-de Vries Equation comes about. See the link http://mathworld.wolfram.com/Korteweg-deVriesEquation.html for details. This example is interesting since it has a 3rd order derivative in it-->
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| <!--
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| === Statistical mechanics ===-->
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| <!-- Second virial coefficients, Law of Corresponding States. Lots of examples from here-->
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| <!--=== Chemical kinetics ===-->
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| <!-- collision theory - Hard ball sphere, potential wells approximations...-->
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| == Statistical analogs ==
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| {{main|Normalization (statistics)}}
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| In [[statistics]], the analogous process is usually dividing a difference (a distance) by a scale factor (a measure of [[statistical dispersion]]), which yields a dimensionless number, which is called ''[[Normalization (statistics)|normalization]].'' Most often, this is dividing [[errors and residuals in statistics|errors or residuals]] by the [[standard deviation]] or sample standard deviation, respectively, yielding [[standard score]]s and [[studentized residual]]s.
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| == See also ==
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| {{colbegin}}
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| * [[Buckingham π theorem]]
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| * [[Dimensional analysis]]
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| * [[Dimensionless number]]
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| * [[Natural units]]
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| * [[List of dynamical systems and differential equations topics]]
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| * [[List of partial differential equation topics]]
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| * [[Differential equations of mathematical physics]]
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| * [[System equivalence]]
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| * [[RLC circuit]]
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| * [[RL circuit]]
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| * [[RC circuit]]
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| * [[Logistic equation]]
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| {{colend}}
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| * [[Normalization (statistics)]]
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| == External links==
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| *[http://www.royalsociety.org.nz/publications/journals/nzja/1998/059/ Analysis of differential equation models in biology: a case study for clover meristem populations] (Application of nondimensionalization to a problem in biology).
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| *[http://www.maths.bath.ac.uk/~masjde/MSc/CourseNotes/MA50176.pdf Course notes for Mathematical Modelling and Industrial Mathematics] ''Jonathan Evans, Department of Mathematical Sciences, [[University of Bath]].'' (see Chapter 3).
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| [[Category:Dimensional analysis]]
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