9.2: Buckingham–π–Theorem
- Page ID
- 758
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)All the physical phenomena that is under the investigation have \(n\) physical effecting parameters such that
\[ \label{dim:eq:bt:initial} F_1(q_1, q_2, q_3, \cdots, q_n) = 0 \]
where \(q_i\) is the "\(i\)'' parameter effecting the problem. For example, study of the pressure difference created due to a flow in a pipe is a function of several parameters such
\[ \label{dim:eq:bt:exPipe} \Delta P = f(L, \, D,\, \mu,\, \rho,\, U) \] In this example, the chosen parameters are not necessarily the most important parameters. For example, the viscosity, \(\mu\) can be replaced by dynamic viscosity, \(\nu\). The choice is made normally as the result of experience and it can be observed that \(\nu\) is a function of \(\mu\) and \(\rho\). Finding the important parameters is based on "good fortune'' or perhaps intuition. In that case, a new function can be defined as
\[ \label{dim:eq:bt:exPipeG} F(\Delta P ,L, D, \mu, \rho, U) = 0 \] Again as stated before, the study of every individual parameter will create incredible amount of data. However, Buckingham's methods suggested to reduce the number of parameters. If independent parameters of same physical situation is \(m\) thus in general it can be written as
\[ \label{dim:eq:bt:initialDimless} F_2(\pi_1, \pi_2, \pi_3, \cdots, \pi_m) = 0 \] If there are \(n\) variables in a problem and these variables contain \(m\) primary dimensions (for example M, L, T), then the equation relating all the variables will have (n-m) dimensionless groups. There are 2 conditions on the dimensionless parameters: beginNormalEnumerate change startEnumerate=1 1. Each of the fundamental dimensions must appear in at least one of the m variables 2. It must not be possible to form a dimensionless group from one of the variables within a recurring set. A recurring set is a group of variables forming a dimensionless group. In the case of the pressure difference in the pipe (Equation (3)) there are 6 variables or \(n = 6\). The number of the fundamental dimensions is 3 that is \(m = 3\) ([M], [L], [t]) The choice of fundamental or basic units is arbitrary in that any construction of these units is possible. For example, another combination of the basic units is time, force, mass is a proper choice. According to Buckingham's theorem the number of dimensionless groups is \(n -m = 6-3 = 3\). It can be written that one dimensionless parameter is a function of two other parameters such as
\[ \label{dim:eq:bt:pipeDim} \pi_1 = f \left(\pi_2, \pi_3\right) \] If indeed such a relationship exists, then, the number of parameters that control the problem is reduced and the number of experiments that need to be carried is considerably smaller. Note, the \(\pi\)-theorem does not specify how the parameters should be selected nor what combination is preferred.Construction of the Dimensionless Parameters
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Contributors and Attributions
Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.