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9.2.3.3 Mathematical Method: Constructing Dimensional Parameters

Advance Material

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|}

ParseError: EOF expected (click for details)
Callstack:
    at (Core/Chemical_Engineering/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
\ \hline \endfoot \hline \hline \endlastfoot grains amount & q & M/L & total grains per unit length \ cross section area & A & L2 & pipe cross section \ grains per volume & gr & grains/L3 & count of grain per V \ grain weight & e & M/grain & count of grain per V \ \end{longtable} \end{center} Notice that grains and grain are the same units for this discussion. Accordingly, the dimensional matrix can be constructed as

\begin{center} 
\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 
\multicolumn{5}{c} 
{\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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Callstack:
    at (Core/Chemical_Engineering/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
 \ \hline 
\endfoot 
\endlastfoot 
  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

  • 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.