9.2.3.3 Mathematical Method: Constructing Dimensional Parameters
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- 764
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que the new evolution is the mathematical method. It can be noticed that in the previous technique the same matrix was constructed with different vector solution (the right hand side of the equation). This fact is the source to improve the previous method. However, it has to be cautioned that this technique is overkill in most cases. Actually, this author is not aware for any case this technique has any advantage over the ``building block'' technique. In the following hypothetical example demonstrates the reason for the reduction of variables. Assume that water is used to transport uniform grains of gold. The total amount grains of gold is to be determined per unit length. For this analysis it is assumed that grains of gold grains are uniformly distributed. The following parameters and their dimensions are considered: \begin{center} \begin{longtable}{|l|c|c|l|} \caption[Units of the Pendulum] {Units and Parameters of gold grains \label{dim:tab:gold}} \ \hline \multicolumn{1}{|c|}{ {|c|}{ { Parameters } & \multicolumn{1}{c|}{ {c|}{ { Units } & \multicolumn{1}{c|}{ {c|}{ { Dimension } & \endfirsthead \multicolumn{4}{c} {\bfseries ablename\ \thetable{} – continued from previous page} \ \hline \endhead \hline \multicolumn{4}{|r|}
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at (Bookshelves/Civil_Engineering/Book:_Fluid_Mechanics_(Bar-Meir)/09:_Dimensional_Analysis/9.2:_Buckingham–π–Theorem/9.2.3:_Implementation_of_Construction_of_Dimensionless_Parameters/9.2.3.3_Mathematical_Method:_Constructing_Dimensional_Parameters), /content/body/p[2]/span, line 1, column 11
\begin{longtable}{c|cccc}
\caption[gold grain dimensional matrix] { gold grain dimensional matrix } \
\multicolumn{1}{c|}{} &
\multicolumn{1}{p{0.5in}}{\centering{\bf q}} &
\multicolumn{1}{p{0.5in}}{\centering{\bf A}} &
\multicolumn{1}{p{0.5in}}{\centering{\bf gr}} &
\multicolumn{1}{p{0.5in}}{\centering{\bf e}} \ \hline
\endfirsthead
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{\bfseries ablename\ \thetable{} – continued from previous page} \
\multicolumn{1}{c|}{} &
\multicolumn{1}{p{0.5in}}{\centering{\bf q}} &
\multicolumn{1}{p{0.5in}}{\centering{\bf A}} &
\multicolumn{1}{p{0.5in}}{\centering{\bf gr}} &
\multicolumn{1}{p{0.5in}}{\centering{\bf e}} \ \hline
\endhead
\multicolumn{5}{|r|}
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at (Bookshelves/Civil_Engineering/Book:_Fluid_Mechanics_(Bar-Meir)/09:_Dimensional_Analysis/9.2:_Buckingham–π–Theorem/9.2.3:_Implementation_of_Construction_of_Dimensionless_Parameters/9.2.3.3_Mathematical_Method:_Constructing_Dimensional_Parameters), /content/body/div[1]/span/span/span/span, line 1, column 11
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M & 1 & 0& 0& 1 \
L & 1 & 2 & 3 & 0 \
grain & 0 & 0 & 1 & -1 \
\end{longtable}
\end{center}
In this case the total number variables are 4 and number basic units are 3. Thus, the total of one dimensional parameter. End ignore section
End Advance Material
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.