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9.3 Nusselt's Technique

The Nusselt's method is a bit more labor intensive, in that the governing equations with the boundary and initial conditions are used to determine the dimensionless parameters. In this method, the boundary conditions together with the governing equations are taken into account as opposed to Buckingham's method. A common mistake is to ignore the boundary conditions or initial conditions. The parameters that results from this process are the dimensional parameters which control the problems. An example comparing the Buckingham's method with Nusselt's method is presented. In this method, the governing equations, initial condition and boundary conditions are normalized resulting in a creation of dimensionless parameters which govern the solution. It is recommended, when the reader is out in the real world to simply abandon Buckingham's method all together. This point can be illustrated by example of flow over inclined plane. For comparison reasons Buckingham's method presented and later the results are compared with the results from Nusselt's method.

Example 9.10

Utilize the Buckingham's method to analyze a two dimensional flow in incline plane. Assume that the flow infinitely long and thus flow can be analyzed per width which is a function of several parameters. The potential parameters are the angle of inclination, \(\theta\), liquid viscosity, \(\nu\), gravity, \(g\), the height of the liquid, \(h\), the density, \(\rho\), and liquid velocity, \(U\). Assume that the flow is not affected by the surface tension (liquid), \(\sigma\). You furthermore are to assume that the flow is stable. Develop the relationship between the flow to the other parameters.

Solution 9.10

Under the assumptions in the example presentation leads to following
\[
    \label{dim:eq:buckinghamNusseltCompar}
    \dot{m} = f\left( \theta, \nu, g, \rho, U\right) \tag{1}
\]  
The number of basic units is three while the number of the parameters is six thus the difference is \(6-3=3\). Those groups (or the work on the groups creation) further can be reduced the because angle \(\theta\) is dimensionless. The units of parameters can be obtained in Table 9.3 the following table.

Parameter Units Parameter Units

\(\nu\)

\(L^2t^{-1}\)

\(g\)

\(L^{1}t{-2}\)

\(\dot{m}\)

\(M\,t^{-1}L^{-1}\)

\(\theta\)

none

\(U\)

\(L^{1}t^{-1}\)

\(\rho\)

\(M\,L^{3}\)

The basic units are chosen as for the time, \(U\), for the mass, \(\rho\), and for the length \(g\). Utilizing the building blocks technique provides
\[
    \label{inclidePlane:ini}
    \overbrace{\dfrac{M}{t\,L} }^{\dot{m}}
     = \left( \overbrace{\dfrac{M}{L^3}}^{\rho} \right)^a  
            \left( \overbrace{\dfrac{L}{t^2} }^{g} \right)^b
            \left( \overbrace{\dfrac{L}{t} }^{U} \right)^c \tag{2}
\]
The equations obtained from equation (2) are
\[
    \label{inclidePlane:gov0}
    \left.
    \begin{array}{rrl}
    \text{Mass}, M & a   =& 1 \\
    \text{Length}, L & -3a + b +c    =& -1 \\
    \text{time}, t & -2b -c =& - 1
    \end{array}
    \right\} \Longrightarrow  
    \pi_1 = \dfrac{\dot{m} \,g}{\rho\,\,U^3} \tag{3}
\]
\[
    \label{inclidePlane:ini1}
    \overbrace{\dfrac{L^2}{t} }^{\nu}
     = \left( \overbrace{\dfrac{M}{L^3}}^{\rho} \right)^a  
            \left( \overbrace{\dfrac{L}{t^2} }^{g} \right)^b
            \left( \overbrace{\dfrac{L}{t} }^{U} \right)^c \tag{4}
\]
The equations obtained from equation (2) are
\[
    \label{inclidePlane:gov}
    \left.
    \begin{array}{rrl}
    \text{Mass}, M & a   =& 0 \\
    \text{Length}, L & -3a + b +c    =& 2 \\   
    \text{time}, t & -2b -c =& - 1
    \end{array}
    \right\} \Longrightarrow  
    \pi_2 = \dfrac{\nu \,g}{U^3} \tag{5}
\]
Thus governing equation and adding the angle can be written as
\[
    \label{inclidePlane:}
    0 = f\left(\dfrac{\dot{m} \,g}{\rho\,\,U^3} , \dfrac{\nu \,g}{U^3} ,\theta\right) \tag{6}
\]
The conclusion from this analysis are that the number of controlling parameters totaled in three and that the initial conditions and boundaries are irrelevant.

A small note, it is well established that the combination of angle gravity or effective body force is significant to the results. Hence, this analysis misses, at the very least, the issue of the combination of the angle gravity. Nusselt's analysis requires that the governing equations along with the boundary and initial conditions to be written. While the analytical solution for this situation exist, the parameters that effect the problem are the focus of this discussion. In Chapter 8, the Navier–Stokes equations were developed. These equations along with the energy, mass or the chemical species of the system, and second laws governed almost all cases in thermo–fluid mechanics. This author is not aware of a compelling reason that this fact should be used in this chapter. The two dimensional NS equation can obtained from equation (??) as

\[
    \label{dim:eq:twoDNSx}
       \begin{array} {rll}
      \rho \left(\dfrac{\partial U_x}{\partial t} +
             \right. & U_x\dfrac{\partial U_x}{\partial x} +
       U_y \dfrac{\partial U_x}{\partial y} + \left. U_z
                \dfrac{\partial U_x}{\partial z}\right)  =  &\\
      &-\dfrac{\partial P}{\partial x} + \mu \left(\dfrac{\partial^2 U_x}{\partial x^2} +
      \dfrac{\partial^2 U_x}{\partial y^2} + \dfrac{\partial^2 U_x}{\partial z^2}\right) +
         \rho g\,\sin\theta
   \end{array} \tag{7}
\]
and
\[
    \label{dim:eq:twoDNSy}
       \begin{array} {rll}
      \rho \left(\dfrac{\partial U_y}{\partial t} +
             \right. & U_x\dfrac{\partial U_y}{\partial x} +
       U_y \dfrac{\partial U_y}{\partial y} + \left. U_z
                \dfrac{\partial U_y}{\partial z}\right)  =  &\\
      &-\dfrac{\partial P}{\partial x} + \mu \left(\dfrac{\partial^2 U_y}{\partial x^2} +
      \dfrac{\partial^2 U_y}{\partial y^2} + \dfrac{\partial^2 U_y}{\partial z^2}\right) +
         \rho g\,\sin\theta
   \end{array} \tag{8}
\]
With boundary conditions  
\[
    \label{dim:eq:twoDNSbcBotton}
    \begin{array}{l}
    U_x (y=0) = U_{0x} f(x) \\
    \dfrac{\partial U_x}{\partial x} (y=h) = \tau_0 f(x)   
    \end{array} \tag{9}
\]
The value \(U_0x\) and \(\tau_0\) are the characteristic and maximum values of the velocity or the shear stress, respectively. and the initial condition of  
\[
    \label{dim:eq:twoDNSic}
    U_x (x=0) = U_{0y}\, f(y)     \tag{10}
\]
where \( U_{0y}\) is characteristic initial velocity. These sets of equations (7-??) need to be converted to dimensionless equations. It can be noticed that the boundary and initial conditions are provided in a special form were the representative velocity multiply a function. Any function can be presented by this form.
    In the process of transforming the equations into a dimensionless form associated with  
some intelligent guess work. However, no assumption is made or required about whether or not the velocity, in the \(y\) direction. The only exception is that the \(y\) component of the velocity vanished on the boundary. No assumption is required about the acceleration or the pressure gradient etc.
    The boundary conditions have typical velocities which can be used. The velocity is selected according to the situation or the needed velocity. For example, if the effect of the initial condition is under investigation than the characteristic of that velocity should be used. Otherwise the velocity at the bottom should be used. In that case, the boundary conditions are  
\[
    \label{dim:eq:twoDNScbles}
    \begin{array}{l}
    \dfrac{ U_x (y=0)}{ U_{0x}} = f(x) \\
    \mu \dfrac{\partial U_x}{\partial x} (y=h) = \tau_0 \,g(x)   
    \end{array} \tag{11}
\]
Now it is very convenient to define several new variables:
\[
    \label{dim:eq:twoDNSdefVer1}
    \begin{array}{rlcrl}
    \overline{U} = & \dfrac{ U_x (\overline{x} ) }{ U_{0x}}\\
    where:\
    \overline{x} = & \dfrac{x}{h} &\qquad& \overline{y} &= \dfrac{y}{h} \\
    \end{array} \tag{12}
\]
The length \(h\) is chosen as the characteristic length since no other length is provided. It can be noticed that because the units consistency, the characteristic length can be used for "normalization'' (see Example 9.11). Using these definitions the boundary and initial conditions becomes
\[
    \label{dim:eq:twoDNScbles1}
    \qquad
    \begin{array}{l}
    \dfrac{ \overline{U_x} (\overline{y}=0)}{ U_{0x}} = f^{'}(\overline{x}) \\
    \dfrac{h \, \mu}{U_{0x}}\,  
        \dfrac{\partial \overline{U_x}}{\partial \overline{x}} (\overline{y}=1) = \tau_0\, g^{'}(\overline{x})   
    \end{array} \tag{13}
\]
It commonly suggested to arrange the second part of equation (13) as
\[
    \label{dim:eq:twoDNScbles2}
    \dfrac{\partial \overline{U_x}}{\partial \overline{x}} (\overline{y}=1) =  
    \dfrac{\tau_0\,U_{0x}}{h \, \mu}\,  
         g^{'}(\overline{x})     \tag{14}
\]
Where new dimensionless parameter, the shear stress number is defined as  
\[
    \label{dim:eq:tauDef}
    \overline{\tau_0} = \dfrac{\tau_0\,U_{0x}}{h \, \mu} \tag{15}
\]
With the new definition equation (14) transformed into
\[
    \label{dim:eq:twoDNScbles3}
    \dfrac{\partial \overline{U_x}}{\partial \overline{x}} (\overline{y}=1) =  
    \overline{\tau_0} \, g^{'}(\overline{x})     \tag{16}
\]  

Example 9.11

Non–dimensionalize the following boundary condition. What are the units of the coefficient in front of the variables, \(x\). What are relationship of the typical velocity, \(U_0\) to \(U_{max}\)?   
\[
    \label{twoDNSbc:bc}
    U_x (y = h) = U_0 \left( a\,x^2 + b\,xp(x) \right) \tag{17}
\]

Solution 9.11

The coefficients \(a\) and \(b\) multiply different terms and therefore must have different units. The results must be unitless thus \(a\)
\[
    \label{twoDNSbc:aDef1}
    L^0 = a \, \overbrace{ {L^2} }^{x^2}  
    \Longrightarrow a = \left[ \dfrac{1}{L^2} \right] \tag{18}
\]
From equation (18) it clear the conversion of the first term is \(U_x = a \, h^2 \overline{x}\). The exponent appears a bit more complicated as
\[
    \label{twoDNSbc:bDef1}
    {L}^{0} = b \, xp\left( h\,\dfrac{x}{h}\right) =
        b \, xp\left( h \right)
        \, xp\left( \dfrac{x}{h}\right) =  
        b \, xp\left( h \right)
        \, xp\left( \overline{x}\right)   \tag{19}
\]
Hence defining  
\[
    \label{twoDNSbc:bDef12}
    \overline{b} = \dfrac{1}{xp{h}} \tag{20}
\]
With the new coefficients for both terms and noticing that \(y=h\longrightarrow \overline{y} =1\) now can be written as  
\[
    \label{twoDNSbc:}
    \dfrac{ U_x (\overline{y} =1)}{U_{0}} =  \overbrace{a\,h^2}^{\overline{a}}\,x^2 +  
        \overbrace{b \, xp\left( h \right) }^{\overline{b}}
        \, xp\left( \overline{x}\right)  
        = \overline{a}\,\overline{x}^2 + \overline{b} xp{\overline{x}} \tag{21}
\]
Where \(\overline{a}\) and \(\overline{b}\) are the transformed coefficients in the dimensionless presentation.

After the boundary conditions the initial condition can undergo the non–dimensional process. The initial condition (10) utilizing the previous definitions transformed into
\[
    \label{dim:eq:twoDNSicDless}
    \dfrac{U_x(\overline{x}=0)}{U_{0x}} = \dfrac{U_{0y}}{U_{0x}}  
    f(\overline{y})   \tag{22}
\]
Notice the new dimensionless group of the velocity ratio as results of the boundary condition. This dimensionless number was and cannot be obtained using the Buckingham's technique. The physical significance of this number is an indication to the "penetration'' of the initial (condition) velocity. The main part of the analysis if conversion of the governing equation into a dimensionless form uses previous definition with additional definitions. The dimensionless time is defined as \(\overline{t} = t\,U_{0x}/h\). This definition based on the characteristic time of \(h/U_{0x}\). Thus, the derivative with respect to time is  
\[
    \label{dim:eq:twoDNSdefDevT}
    \dfrac{\partial U_x}{\partial t} =  
        \dfrac{\partial \overbrace{\overline{U_x}}^{\dfrac{U_x}{U_{0x}} } U_{0x}}
             {\partial \underbrace{\overline{t}}_{ \dfrac{t\,U_{0x}}{ h } }\dfrac{h}{U_{0x}} }
        =  
        \dfrac

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U_{0x}
         \dfrac{\partial \overbrace{\overline{U_x} }^{\dfrac{U_x}{U_{0x}}} U_{0x}}
            {\partial \underbrace{\overline{x} }_{\dfrac{x}{h}} h }     
        =
        \dfrac
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{\partial \overline{x} }   \tag{25}
\]
The second derivative of velocity looks like  
\[
    \label{dim:eq:twoDNSdef2Dir}
    \dfrac{\partial^2 U_x}{\partial x^2} =  
    \dfrac{\partial}{\partial \left( \overline{x} h \right) }
    \dfrac{\partial \left( \overline{U_x} U_{0x} \right) }{\partial \left( \overline{x} h \right) } =  
    \dfrac{U_{0x}}{h^2}
    \dfrac{\partial^2 \overline{U_x} }{\partial \overline{x}^2}    \tag{26}
\]
The last term is the gravity \(g\) which is left for the later stage. Substituting all terms and dividing by density, \(\rho\) result in  
\[
    \label{dim:eq:twoDNSxx}
       \begin{array} {rll}
       \dfrac
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+
       \overline{U_y} \dfrac{\partial \overline{U_x}}{\partial \overline{y}} +  
            \left. \overline{U_z} \dfrac{\partial \overline{U_x}}{\partial \overline{z}}\right) = &\\
      &- \dfrac{P_0-P_\infty}{h\,\rho}
        \dfrac{\partial \overline{P}}{\partial \overline{x}} +
             \dfrac
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\,\sin\theta
   \end{array} \tag{27}
\]
Dividing equation (27) by \({U_{0x}}^2/ {h}\) yields
\[
    \label{dim:eq:twoDNSx1i}
       \begin{array} {rll}
       \left(\dfrac{\partial \overline{U_x} }{\partial \overline{t} } +
             \right. & \overline{U_x}\dfrac{\partial \overline{U_x} }{\partial \overline{x}} +
       \overline{U_y} \dfrac{\partial \overline{U_x}}{\partial \overline{y}} +  
            \left. \overline{U_z} \dfrac{\partial \overline{U_x}}{\partial \overline{z}}\right) = &\\
      &- \dfrac{P_0-P_\infty}
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{\partial \overline{x}} +
             \dfrac{\mu}{U_{0x}\,h\,\rho} \left(\dfrac{\partial^2 U_x}{\partial x^2} +
      \dfrac{\partial^2 U_x}{\partial y^2} + \dfrac{\partial^2 U_x}{\partial z^2}\right) +
         \dfrac{ g \,h }{{U_{0x}}^2}\,\sin\theta
   \end{array} \tag{28}
\]
Defining "initial'' dimensionless parameters as
\[
    \label{dim:eq:iniDimlessPara}
    \begin{array}{lcccr}
    Re = \dfrac{U_{0x}\,h\,\rho}{\mu}  &&  
        Fr = \dfrac{{U_{0x}}}{\sqrt{g \,h}} &&
        Eu = \dfrac{P_0-P_\infty}{{U_{0x}}^2\,\rho}    
    \end{array} \tag{29}
\]  
Substituting definition of equation (29) into equation (28) yields
\[
    \label{dim:eq:twoDNSx1}
       \begin{array} {rll}
       \left(\dfrac{\partial \overline{U_x} }{\partial \overline{t} } +
             \right. & \overline{U_x}\dfrac{\partial \overline{U_x} }{\partial \overline{x}} +
       \overline{U_y} \dfrac{\partial \overline{U_x}}{\partial \overline{y}} +  
            \left. \overline{U_z} \dfrac{\partial \overline{U_x}}{\partial \overline{z}}\right) = &\\
      &- Eu\,  
        \dfrac{\partial \overline{P}}{\partial \overline{x}} +
             \dfrac{1}{Re} \left(\dfrac{\partial^2 U_x}{\partial x^2} +
      \dfrac{\partial^2 U_x}{\partial y^2} + \dfrac{\partial^2 U_x}{\partial z^2}\right) +
         \dfrac{ 1 }{Fr^2}\,\sin\theta
   \end{array} \tag{30}
\]
Equation (30) show one common possibility of a dimensionless presentation of governing equation. The significance of the large and small value of the dimensionless parameters will be discuss later in the book. Without actually solving the problem, Nusselt's method provides several more parameters that were not obtained by the block method. The solution of the governing equation is a function of all the parameters present in that equation and boundaries condition as well the initial condition. Thus, the solution is
\[
    \label{dim:eq:twoDNSx2}
    U_x = f \left( \overline{x}, \overline{y}, Eu, Re, Fr, \theta, \overline{\tau_0}, f_u, f_{\tau},
        \dfrac{U_{0y}}{U_{0x}} \right) \tag{31}
\]
The values of \( \overline{x},\, \overline{y}\) depend on \(h\) and hence the value of \(h\) is an important parameter. It can be noticed with Buckingham's method, the number of parameters obtained was only three (3) while Nusselt's method yields 12 dimensionless parameters. This is a very significant difference between the two methods. In fact, there are numerous examples in the literature that showing people doing experiments based on Buckingham's methods. In these experiments, major parameters are ignored rendering these experiments useless in many cases and deceiving.

Common Transformations

Fluid mechanics in particular and Thermo–Fluid field in general have several common transformations that appear in boundary conditions, initial conditions and equations. It recognized that not all the possibilities can presented in the example shown above. Several common boundary conditions which were not discussed in the above example are presented below. As an initial matter, the results of the non dimensional transformation depends on the selection of what and how is nondimensionalization carried. This section of these parameters depends on what is investigated. Thus, one of the general nondimensionalization of the Navier–Stokes and energy equations will be discussed at end of this chapter. Boundary conditions are divided into several categories such as a given given derivative (Neumann b.c.), mixed condition, and complex conditions. The first and second categories were discussed to some degree earlier and will be expanded later. The third and fourth categories were not discussed previously. The non–dimensionalization of the boundary conditions of the first category requires finding and diving the boundary conditions by a typical or a characteristic value. The second category involves the nondimensionalization of the derivative. In general, this process involve dividing the function by a typical value and the same for length variable (e.g. \(x\)) as

\[
    \label{dim:eq:nondimDirative1}
    \dfrac{\partial U}{\partial x} =  
    \dfrac{\ell}{U_0}
    \dfrac{\partial \left( \dfrac{U}{U_0} \right) }{ \partial \left( \dfrac{x}{\ell} \right)} =
    \dfrac{\ell}{U_0}
    \dfrac{\partial \overline{U}}{\partial \overline{x}}   \tag{32}
\]
In the Thermo–Fluid field and others, the governing equation can be of higher order than second order. It can be noticed that the degree of the derivative boundary condition cannot exceed the derivative degree of the governing equation (e.g. second order equation has at most the second order differential boundary condition.). In general "nth'' order differential equation leads to
\[
    \label{dim:eq:nondimDirative}
    \dfrac{\partial^n U}{\partial x^n} =  
    \dfrac{U_0}{\ell^n}
    \dfrac{\partial^n \left( \dfrac{U}{U_0} \right) }
            { \partial  \left(\dfrac{x}{\ell}\right)^n } =
    \dfrac{U_0}{\ell^n}
    \dfrac{\partial^n \overline{U}}{\partial \overline{x}^n}   \tag{33}
\]
 The third kind of boundary condition is the mix condition. This category includes combination of the function with its derivative. For example a typical heat balance at liquid solid interface reads
\[
    \label{dim:eq:mixBCex}
    h(T_0 - T) = - k \dfrac{\partial T}{ \partial x} \tag{34}
\]
This kind of boundary condition, since derivative of constant is zero, translated to  
\[
    \label{dim:eq:mixBCexDim1}
    h\, \cancel{(T_0 -T_{max})}\, \left( \dfrac{T_0 - T }{ T_0 -T_{max}} \right) =  
        - \dfrac{ k \,\cancel{\left( T_0 -T_{max} \right)} }{ \ell} \,  
            \dfrac{ - \partial \left( \dfrac{ T - T_0}   { T_0 -T_{max} } \right) }  
            { \partial \left(\dfrac{x}{\ell} \right) } \tag{35}
\]
or
\[
    \label{dim:eq:mixBCexDim}
    \left( \dfrac{T_0 - T }{ T_0 -T_{max}} \right) =   
         \dfrac{ k }{h\, \ell}  
            \dfrac{\partial \left( \dfrac{ T - T_0}   { T_0 -T_{max} } \right) }
          { \partial \left(\dfrac{x}{\ell} \right) }
        \Longrightarrow
    \Theta = \dfrac{1}{Nu} \dfrac{\partial \Theta}{\partial \overline{x}}   \tag{36}
\]
temperature are defined as  
\[
    \label{dim:eq:NuDef}
    Nu = \dfrac{h\,\ell}{k} && \Theta = \dfrac{ T - T_0}   { T_0 -T_{max} }    \tag{37}
\]
and \(T_{max}\) is the maximum or reference temperature of the system. The last category is dealing with some non–linear conditions of the function with its derivative. For example,
\[
    \label{dim:eq:surfaceTensionBCi}
    \Delta P \approx {\sigma}\left({ \dfrac{1}{r_1} + \dfrac{1}{r_2} } \right) =  
    \dfrac{\sigma}{ r_1}\, \dfrac{r_1+ r_2}{r_2}       \tag{38}
\]
Where \(r_1\) and \(r_2\) are the typical principal radii of the free surface curvature, and, \(\sigma\), is the surface tension between the gas (or liquid) and the other phase. The surface geometry (or the radii) is determined by several factors which include the liquid movement instabilities etc chapters of the problem at hand. This boundary condition (38) can be rearranged to be
\[
    \label{dim:eq:surfaceTensionBC}
    \dfrac{\Delta P \, r_1}{\sigma} \approx \dfrac{r_1+ r_2}{r_2} \Longrightarrow
    Av \approx \dfrac{r_1+ r_2}{r_2}   \tag{39}
\]
The Avi number represents the geometrical characteristics combined with the material properties. The boundary condition (39) can be transferred into
\[
    \label{dim:eq:surfaceTensionBCdim}
    \dfrac{ \Delta P\,r_1}{\sigma} = Av \tag{40}
\]
Where \(\Delta P\) is the pressure difference between the two phases (normally between the liquid and gas phase). One of advantage of Nusselt's method is the object-Oriented nature which allows one to add additional dimensionless parameters for addition "degree of freedom.'' It is common assumption, to initially assume, that liquid is incompressible. If greater accuracy is needed than this assumption is removed. In that case, a new dimensionless parameters is introduced as the ratio of the density to a reference density as
\[
    \label{dim:eq:refereceRho}
    \overline{\rho} = \dfrac{\rho}{\rho_0} \tag{41}
\]
In case of ideal gas model with isentropic flow this assumption becomes  
\[
    \label{dim:eq:gasDensity}
    \bar{\rho} = {\rho \over \rho_0} = \left(\dfrac{ P_{0}}{ P } \right)^{\dfrac{1}{n}} \tag{42}
\]
The power \(n\) depends on the gas properties.

Characteristics Values

Normally, the characteristics values are determined by physical values e.g. The diameter of cylinder as a typical length. There are several situations where the characteristic length, velocity, for example, are determined by the physical properties of the fluid(s). The characteristic velocity can determined from \(U_0 =\sqrt{2P_{0} / \rho}\). The characteristic length can be determined from ratio of \(\ell = \Delta P/\sigma\).

Example 9.12

One idea of renewable energy is to use and to utilize the high concentration of brine water such as in the Salt Lake and the Salt Sea (in Israel). This process requires analysis the mass transfer process. The governing equation is non-linear and this example provides opportunity to study nondimensionalizing of this kind of equation. The conversion of the species yields a governing nonlinear equation (??) for such process is  
\[
    \label{highMass:gov}
    U_0 \dfrac{\partial C_A}{\partial x} =  
    \dfrac{\partial }{ \partial y }
        \dfrac {D_{AB}}{ \left( 1 - X_A\right) } \dfrac{\partial C_A}{ \partial y } \tag{43}
\]
Where the concentration, \(C_A\) is defended as the molar density i.e. the number of moles per volume. The molar fraction, \(X_A\) is defined as the molar fraction of species \(A\) divide by the total amount of material (in moles). The diffusivity coefficient, \(D_{AB}\) is defined as penetration of species \(A\) into the material. What are the units of the diffusivity coefficient? The boundary conditions of this partial differential equation are given by
\[
    \label{highMass:BCy}
    \dfrac{\partial C_A}{\partial y} \left(y=\infty\right) = 0   \tag{44}
\]
\[
    \label{highMass:BCy2i}
    C_A (y=0) = C_e \tag{45}
\]
Where \(C_e\) is the equilibrium concentration. The initial condition is
\[
    \label{highMass:BCy2}
    C_A (x=0) = C_0 \tag{46}
\]
Select dimensionless parameters so that the governing equation and boundary and initial condition can be presented in a dimensionless form. There is no need to discuss the physical significance of the problem.

Solution 9.12

This governing equation requires to work with dimension associated with mass transfer and chemical reactions, the "mole.'' However, the units should not cause confusion or fear since it appear on both sides of the governing equation. Hence, this unit will be canceled. Now the units are compared to make sure that diffusion coefficient is kept the units on both sides the same. From units point of view, equation (43) can be written (when the concentration is simply ignored) as
\[
    \label{highMass:govDim}
    \overbrace{\dfrac{L}{t}} ^{U}  
    \overbrace{\dfrac{\cancel{C}}{L} }^{\dfrac{\partial C}{\partial x}} =
    \overbrace{\dfrac{1}{L}}^{\dfrac{\partial }{\partial y}}  
    \overbrace{\dfrac{D_{AB}}{1}}^{\dfrac{D_{AB} }{ \left(1 - X \right)}}  
    \overbrace{\dfrac{\cancel{C}}{L} }^{\dfrac{\partial C}{\partial y}} \tag{47}
\]
It can be noticed that \(X\) is unitless parameter because two same quantities are divided.
\[
    \label{highMass:govGov}
    \dfrac{1}{t} = \dfrac{1}{L^2} D_{AB} \Longrightarrow D_{AB} = \dfrac{L^2}{t} \tag{48}
\]
Hence the units of diffusion coefficient are typically given by \(\left[m^2/sec\right]\) (it also can be observed that based on Fick's laws of diffusion it has the same units). The potential of possibilities of dimensionless parameter is large. Typically, dimensionless parameters are presented as ratio of two quantities. In addition to that, in heat and mass transfer (also in pressure driven flow etc.) the relative or reference to certain point has to accounted for. The boundary and initial conditions here provides the potential of the "driving force'' for the mass flow or mass transfer. Hence, the potential definition is
\[
    \label{highMass:phi}
    \Phi = \dfrac{C_A - C_0 }{C_e - C_0} \tag{49}
\]
With almost "standard'' transformation  
\[
    \label{highMass:coordinates}
    \overline{x} = \dfrac{x}{\ell} &\qquad & \overline{y} = \dfrac{y}{\ell}   \tag{50}
\]
Hence the derivative of \(\Phi\) with respect to time is
\[
    \label{highMass:PhiD}
    \dfrac{\partial \Phi}{\partial \overline{x}} =
    \dfrac{\partial \dfrac{C_A - C_0 }{C_e - C_0} }{\partial \dfrac{x}{\ell} }  
    = \dfrac{\ell}{C_e - C_0}
    \dfrac{\partial \left(C_A - \cancelto{0}{C_0} \,\,\,\,\,\right)} {\partial {x} }  
    = \dfrac{\ell}{C_e - C_0}
    \dfrac{\partial C_A } {\partial {x} }   \tag{51}
\]  
In general a derivative with respect to \(\overline{x}\) or \(\overline{y}\) leave yields multiplication of \(\ell\). Hence, equation (43) transformed into  
\[
    \label{highMass:totalDim}
    \begin{array}{rl}
    U_0\dfrac{\cancel{\left(C_e - C_0\right)}} {\ell}\dfrac{\partial \Phi}{\partial \overline{x}} &=
    \dfrac{1}{\ell}\, \dfrac{\partial }{ \partial \overline{y} }
        \dfrac {D_{AB}}{ \left( 1 - X_A\right) }  
    \dfrac{\cancel{\left(C_e - C_0\right)}} {\ell}\dfrac{\partial \Phi}{\partial \overline{y}} \  
    \mbox{\Huge $\displaystyle\leadsto$}    
    \dfrac{U_0} {\ell}\dfrac{\partial \Phi}{\partial \overline{x}} &=
    \dfrac{1}{\ell^2}\, \dfrac{\partial }{ \partial \overline{y} }
        \dfrac {D_{AB}}{ \left( 1 - X_A\right) }  
    \dfrac{\partial \Phi}{\partial \overline{y}}  
    \end{array} \tag{52}
\]
Equation (52) like non–dimensionalized and proper version. However, the term \(X_A\), while is dimensionless, is not proper. Yet, \(X_A\) is a function of \(\Phi\) because it contains \(C_A\). Hence, this term, \(X_A\) has to be converted or presented by \(\Phi\). Using the definition of \(X_A\) it can be written as
\[
    \label{highMass:X}
    X_A = \dfrac{C_A} {C} = \left( C_e- C_0 \right)\dfrac{C_A - C_0}{ C_e- C_0} \dfrac{1}{C} \tag{53}
\]
Thus the transformation in equation (53) another unexpected dimensionless parameter as
\[
    \label{highMass:Xf}
    X_A = \Phi \, \dfrac{C_e- C_0}{C}   \tag{54}
\]
Thus number, \(\dfrac{C_e- C_0}{C}\) was not expected and it represent ratio of the driving force to the height of the concentration which was not possible to attend by Buckingham's method. 

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.