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Engineering LibreTexts

9.2.3.2: Building Blocks Method: Constructing Dimensional Parameters

Note, as opposed to the previous method, this technique allows one to find a single or several dimensionless parameters without going for the whole calculations of the dimensionless parameters.

Example 9.8

Assume that the parameters that effects the centrifugal pumps are

\(Q\) Pump Flow Rate \(rpm\) or \(N\) angular rotation speed
\(D\) Rotor Diameter \(\rho\) liquid density (assuming liquid phase)
\(B_{T}\) Liquid Bulk Modulus \(\mu\) liquid viscosity
\(\epsilon\) Typical Roughness of pump surface \(g\) gravity force (body force)
\(\Delta P\) Pressure created by the pump    

Construct the functional relationship between the variables. Discuss the physical meaning of these numbers. Discuss which of these dimensionless parameters can be neglected as it is known reasonably.

Solution 9.8

The functionality can be written as
\[
    \label{pumpScaling:fun}
    0 = f \left( D,\, N,\,\rho,\,Q,\,B_T,\, \mu ,\, \epsilon,\,g,\,\Delta P \right)  \tag{31}
\]
The three basic parameters to be used are \(D\) [L], \(\rho\) [M], and \(N\) [t]. There are nine (9) parameters thus the number of dimensionless parameters is \(9-3=6\). For simplicity the $RPM$ will be denoted as \(N\). The first set is to be worked on is \(Q,\,D,\,\rho,\,N\) as  
\[
    \label{pumpScaling:Q:ini}
    \overbrace{\dfrac{L^3}{t} }^Q =   
    \left(\overbrace{ {L} }^{D}\right)^a
    \left(\overbrace{\dfrac{M}{L^3} }^{\rho}\right)^b
    \left(\overbrace{\dfrac{1}{t} }^{N}\right)^c  \tag{32}
\]
\[
    \label{pumpScaling:Q:gov}
    \left.
    \begin{array}{rrl}
    \text{Length}, L & a - 3b    =& 3 \\   
    \text{Mass}, M & b   =& 0  \\
    \text{time}, t & -c =& - 1    
    \end{array}
    \right\} \Longrightarrow  
    \pi_1 = \dfrac{Q}{N\,D^3}  \tag{33}
\]
For the second term \(B_T\) it follows
\[
    \label{pumpScaling:BT:ini}
    \overbrace{\dfrac{M}{L\,t^2} }^{B_T} =   
    \left(\overbrace{ {L} }^{D}\right)^a
    \left(\overbrace{\dfrac{M}{L^3} }^{\rho}\right)^b
    \left(\overbrace{\dfrac{1}{t} }^{N}\right)^c \tag{34}
\]
\[
    \label{pumpScaling:BT:gov}
    \left.
    \begin{array}{rrl}
    \text{Mass}, M & b   =& 1  \\
    \text{Length}, L & a - 3b    =& -1 \\   
    \text{time}, t & -c =& - 2    
    \end{array}
    \right\} \Longrightarrow  
    \pi_2 = \dfrac{B_T}{\rho\,N^2\,D^2} \tag{35}
\]
The next term, \(\mu\),
\[
    \label{pumpScaling:mu:ini0}
    \overbrace{\dfrac{M}{L\,t} }^{\mu} =   
    \left(\overbrace{ {L} }^{D}\right)^a
    \left(\overbrace{\dfrac{M}{L^3} }^{\rho}\right)^b
    \left(\overbrace{\dfrac{1}{t} }^{N}\right)^c \tag{36}
\]
\[
    \label{pumpScaling:mu:gov1}
    \left.
    \begin{array}{rrl}
    \text{Mass}, M & b   =& 1  \\
    \text{Length}, L & a - 3b    =& -1 \\   
    \text{time}, t & -c =& - 1    
    \end{array}
    \right\} \Longrightarrow  
    \pi_3 = \dfrac{\rho\,N^2\,D^2}{\mu} \tag{37}
\]
The next term, \(\epsilon\),
\[
    \label{pumpScaling:mu:ini1}
    \overbrace{L}^{\epsilon} =   
    \left(\overbrace{ {L} }^{D}\right)^a
    \left(\overbrace{\dfrac{M}{L^3} }^{\rho}\right)^b
    \left(\overbrace{\dfrac{1}{t} }^{N}\right)^c \tag{38}
\]
\[
    \label{pumpScaling:mu:gov2}
    \left.
    \begin{array}{rrl}
    \text{Mass}, M & b   =& 0  \\
    \text{Length}, L & a - 3b    =& 1 \\   
    \text{time}, t & -c =& 0
    \end{array}
    \right\} \Longrightarrow  
    \pi_4 = \dfrac{\epsilon}{D} \tag{39}
\]
The next term, \(g\),
\[
    \label{pumpScaling:mu:ini2}
    \overbrace{\dfrac{L}{t^2}}^{g} =   
    \left(\overbrace{ {L} }^{D}\right)^a
    \left(\overbrace{\dfrac{M}{L^3} }^{\rho}\right)^b
    \left(\overbrace{\dfrac{1}{t} }^{N}\right)^c \tag{40}
\]
\[
    \label{pumpScaling:mu:gov3}
    \left.
    \begin{array}{rrl}
    \text{Mass}, M & b   =& 0  \\
    \text{Length}, L & a - 3b    =& 1 \\   
    \text{time}, t & -c =& -2  
    \end{array}
    \right\} \Longrightarrow  
    \pi_5 = \dfrac{g}{D\,N^2} \tag{41}
\]
The next term, \(\Delta P\), (similar to \(B_T\))
\[
    \label{pumpScaling:mu:ini}
    \overbrace{\dfrac{L}{t^2}}^{\Delta P} =   
    \left(\overbrace{ {L} }^{D}\right)^a
    \left(\overbrace{\dfrac{M}{L^3} }^{\rho}\right)^b
    \left(\overbrace{\dfrac{1}{t} }^{N}\right)^c \tag{42}
\]
\[
    \label{pumpScaling:mu:gov}
    \left.
    \begin{array}{rrl}
    \text{Mass}, M & b   =& 1  \\
    \text{Length}, L & a - 3b    =& -1 \\   
    \text{time}, t & -c =& -2  
    \end{array}
    \right\} \Longrightarrow  
    \pi_6 = \dfrac{\Delta P}{\rho \,N^2\,D^2} \tag{43}
\]
The first dimensionless parameter \(\pi_1\) represents the dimensionless flow rate. The second number represents the importance of the compressibility of the liquid in the pump. Some argue that this parameter is similar to Mach number (speed of disturbance to speed of sound. The third parameter is similar to Reynolds number since the combination \(N\,D\) can be interpreted as velocity. The fourth number represents the production quality (mostly mode by some casting process The fifth dimensionless parameter is related to the ratio of the body forces to gravity forces. The last number represent the "effectiveness'' of pump or can be viewed as dimensionless pressure obtained from the pump.
    In practice, the roughness is similar to similar size pump and can be neglected.
However, if completely different size of pumps are compared then this number must be considered. In cases where the compressibility of the liquid can be neglected or the pressure increase is relatively insignificant, the second dimensionless parameter can be neglected.
    A pump is a device that intends to increase the pressure. The increase of the pressure involves energy inserted to to system. This energy is divided to a useful energy (pressure increase) and to overcome the losses in the system. These losses has several components which includes the friction in the system, change order of the flow and "ideal flow'' loss. The most dominate loss in pump is loss of order, also know as turbulence (not covered yet this book.). If this physical phenomenon is accepted than the resistance is neglected and the fourth parameter is removed. In that case the functional relationship can be written as
\[
    \label{pumpScaling:finalFun}
    \dfrac{\Delta P }{N^2, D^2} = f \left( \dfrac{Q}{N\,D^3}\right)  \tag{44}
\]

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.