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9.4.1: The Significance of these Dimensionless Numbers

Reynolds number, named in the honor of Reynolds, represents the ratio of the momentum forces Historically, this number was one of the first numbers to be introduced to fluid mechanics. This number determines, in many cases, the flow regime.

Example 9.13

Eckert number determines whether the role of the momentum energy is transferred to thermal energy is significant to affect the flow. This effect is important in situations where high speed is involved. This fact suggests that Eckert number is related to Mach number. Determine this relationship and under what circumstances this relationship is true.

This example is based on Bird, Lightfoot and Stuart "Transport Phenomena''. 

Solution 9.13

In Table 9.8 Mach and Eckert numbers are defined as

\[
    \label{eckert:definition}
    Ec = \dfrac{U^2}{C_p\,\Delta T} &\qquad&\qquad&\qquad& M = \dfrac{U}{\sqrt{\dfrac{P}{\rho}} } \tag{55}
\]
The material which obeys the ideal flow model (\(P/\rho = R\,T\) and \(P = C_1\,\rho^k\)) can be written that  
\[
    \label{eckert:MachDefIdealGas}
    M =  U \left/ \sqrt{\dfrac{P}{\rho}} \right.= \dfrac{U}{\sqrt{k\,R\,T}} \tag{56}
\]
For the comparison, the reference temperature used to be equal to zero. Thus Eckert number can be written as
\[
    \label{eckert:expendated}
    \sqrt{Ec} = \dfrac{U}{\sqrt{C_p\,T} } =  
     \dfrac{U}{\sqrt{\underbrace{\left(\dfrac{R \, k}{k-1} \right)}_{C_p}\,T} } =  
     \dfrac{\sqrt{k-1}\,U}{\sqrt{k\,R\,T} } = \sqrt{k -1}\, M   \tag{57}
\]
The Eckert number and Mach number are related under ideal gas model and isentropic relationship.

Brinkman number measures of the importance of the viscous heating relative the conductive heat transfer. This number is important in cases when a large velocity change occurs over short distances such as lubricant, supersonic flow in rocket mechanics creating large heat effect in the head due to large velocity (in many place it is a combination of Eckert number with Brinkman number. The Mach number is based on different equations depending on the property of the medium in which pressure disturbance moves through. Cauchy number and Mach number are related as well and see Example 9.15 for explanation.

Example 9.14

For historical reason some fields prefer to use certain numbers and not other ones. For example in Mechanical engineers prefer to use the combination \(Re\) and \(We\) number while Chemical engineers prefers to use the combination of \(Re\) and the Capillary number. While in some instances this combination is justified, other cases it is arbitrary. Show what the relationship between these dimensionless numbers.

Solution 9.14

The definitions of these number in Table 9.8
\[
    \label{CaWeRe:we}
    We = \dfrac{\rho\, U^2\, \ell}{\sigma} &\qquad &
    Re = \dfrac{\rho\,U\,\ell}{\mu} & \qquad &  
    Ca = \dfrac{\mu\,U}{\sigma} = \dfrac{U}{\dfrac{\sigma}{\mu} } \tag{58}
\]
Dividing Weber number by Reynolds number yields
\[
    \label{CaWeRe:We-d-Re}
    \dfrac{We}{Re} = \dfrac{\dfrac{\rho\, U^2\, \ell}{\sigma} } {\dfrac{\rho\,U\,\ell}{\mu} } =
     \dfrac{U}{\dfrac{\sigma}{\mu} } = Ca \tag{59}
\]

Physicist who pioneered so many fields that it is hard to say what and where are his greatest contributions. Euler’s number and Cavitation number are essentially the same with the exception that these numbers represent different driving pressure differences. This difference from dimensional analysis is minimal. Furthermore, Euler number is referred to as the pressure coefficient, \(C_p\). This confusion arises in dimensional analysis because historical reasons and the main focus area. The cavitation number is used in the study of cavitation phenomena while Euler number is mainly used in calculation of resistances.

Example 9.15

Explained under what conditions and what are relationship between the Mach number and Cauchy number?

Solution 9.15

Cauchy number is defined as  
\[
    \label{M2Ca:CaDef}
    Cau = \dfrac{\rho\,\pmb{U}^2 } {E}   \tag{60}
\]
The square root of Cauchy number is  
\[
    \label{M2Ca:Cau12}
    \sqrt{Cau}  = \dfrac{U}{\sqrt{\dfrac{E}{\rho}}} \tag{61}
\]
In the liquid phase the speed of sound is approximated as  
\[
    \label{M2Ca:liquidSound}
    c = \dfrac{E}{\rho} \tag{62}
\]
Using equation (61) transforms equation (60) into
\[
    \label{M2Ca:eq:Cau12f}
    \sqrt{Cau}  = \dfrac{U}{c} = M \tag{63}
\]
Thus the square root of \(Ca\) is equal to Mach number in the liquid phase. In the solid phase equation (62) is less accurate and speed of sound depends on the direction of the grains. However, as first approximation, this analysis can be applied also to the solid phase. 

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.