Analogies and differences in different transport phenomena
From ThermalFluidsPedia
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{{EquationRef(1)}}  {{EquationRef(1)}}  
}  }  
  where  +  where <math>{\dot{m}}'''_{0}</math> is the rate of oxidant consumption (kg/m3*s), hc,0 is the heat released by combustion per unit mass consumption of the oxidant (J/kg), which is different from the combustion heat, and<math>\omega _{0}</math> is the mass fraction of the oxidant in the gaseous mixture. 
  <math>{\dot{m}}'''_{0}</math>  +  
  +  
The corresponding boundary conditions of eqs. (4.366) to (4.369) are  The corresponding boundary conditions of eqs. (4.366) to (4.369) are  
  +  { class="wikitable" border="0"  
+    
+   width="100%" <center>  
<math>u\to U_{\infty }\begin{matrix}  <math>u\to U_{\infty }\begin{matrix}  
, & T\to T_{\infty }\begin{matrix}  , & T\to T_{\infty }\begin{matrix}  
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\end{matrix} \\  \end{matrix} \\  
\end{matrix}\_at\_y\to \infty </math>  \end{matrix}\_at\_y\to \infty </math>  
  +  </center>  
  +   {{EquationRef(1)}}  
+  }  
+  { class="wikitable" border="0"  
+    
+   width="100%" <center>  
<math>u=0\begin{matrix}  <math>u=0\begin{matrix}  
, & v=\frac{{\dot{m}}''_{f}}{\rho }\begin{matrix}  , & v=\frac{{\dot{m}}''_{f}}{\rho }\begin{matrix}  
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\end{matrix} \\  \end{matrix} \\  
\end{matrix}\_at\_y=0</math>  \end{matrix}\_at\_y=0</math>  
  +  </center>  
  +   {{EquationRef(1)}}  
  <math>{\dot{m}}''_{f}</math>  +  } 
  +  
+  where <math>{\dot{m}}''_{f}</math> is the rate of solid fuel sublimation per unit area (kg/m2?s) and ρ is the density of the mixture.  
The shear stress at the solid fuel surface is  The shear stress at the solid fuel surface is  
  +  { class="wikitable" border="0"  
+    
+   width="100%" <center>  
<math>\tau _{w}=\mu \frac{\partial u}{\partial y}\begin{matrix}  <math>\tau _{w}=\mu \frac{\partial u}{\partial y}\begin{matrix}  
, & y=0 \\  , & y=0 \\  
\end{matrix}</math>  \end{matrix}</math>  
  +  </center>  
+   {{EquationRef(1)}}  
+  }  
+  
The heat flux at the solid fuel surface is  The heat flux at the solid fuel surface is  
  +  { class="wikitable" border="0"  
+    
+   width="100%" <center>  
<math>{q}''_{w}=k\frac{\partial T}{\partial y}\begin{matrix}  <math>{q}''_{w}=k\frac{\partial T}{\partial y}\begin{matrix}  
, & y=0 \\  , & y=0 \\  
\end{matrix}</math>  \end{matrix}</math>  
  +  </center>  
+   {{EquationRef(1)}}  
+  }  
+  
The exact solution of the heat and mass problem described by eqs. (4.366) to (4.369) can be obtained using numerical simulation. It is useful here to introduce the results obtained by Kaviany (2001) using an analogy between momentum, heat, and mass transfer. Multiplying eq. (4.369) by hc,0 and adding the result to eq. (4.368), one obtains  The exact solution of the heat and mass problem described by eqs. (4.366) to (4.369) can be obtained using numerical simulation. It is useful here to introduce the results obtained by Kaviany (2001) using an analogy between momentum, heat, and mass transfer. Multiplying eq. (4.369) by hc,0 and adding the result to eq. (4.368), one obtains  
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}  }  
Considering the assumption that the Lewis number is unity, i.e. <math>Le=\nu /D=1</math>, eq. (4.374) can be rewritten as  Considering the assumption that the Lewis number is unity, i.e. <math>Le=\nu /D=1</math>, eq. (4.374) can be rewritten as  
  +  { class="wikitable" border="0"  
+    
+   width="100%" <center>  
<math>\begin{align}  <math>\begin{align}  
& \frac{\partial }{\partial x}\left[ \rho u(c_{p}T+\omega _{o}h_{c,o}) \right]+\frac{\partial }{\partial y}\left[ \rho v(c_{p}T+\omega _{o}h_{c,o}) \right] \\  & \frac{\partial }{\partial x}\left[ \rho u(c_{p}T+\omega _{o}h_{c,o}) \right]+\frac{\partial }{\partial y}\left[ \rho v(c_{p}T+\omega _{o}h_{c,o}) \right] \\  
& =\frac{\partial }{\partial y}\left[ \rho \alpha \frac{\partial }{\partial y}(c_{p}T+\omega _{o}h_{c,o}) \right] \\  & =\frac{\partial }{\partial y}\left[ \rho \alpha \frac{\partial }{\partial y}(c_{p}T+\omega _{o}h_{c,o}) \right] \\  
\end{align}</math>  \end{align}</math>  
  +  </center>  
  which can be viewed as an energy equation with quantity ({ class="wikitable" border="0"  +   {{EquationRef(1)}} 
+  }  
+  
+  which can be viewed as an energy equation with quantity <math>c_{p}T+\omega _{0}h_{c,0}</math>) as a dependent variable.  
+  Since <math>\partial \omega _{o}/\partial y=0</math> at y = 0, i.e., the solid fuel surface is not permeable for the oxidant, eq(4.373) can be rewritten as  
+  { class="wikitable" border="0"  
    
 width="100%" <center>   width="100%" <center>  
  
  
  
  
  
  
<math>{q}''_{w}=\rho \alpha \frac{\partial }{\partial y}(c_{p}T+\omega _{o}h_{c,o})\begin{matrix}  <math>{q}''_{w}=\rho \alpha \frac{\partial }{\partial y}(c_{p}T+\omega _{o}h_{c,o})\begin{matrix}  
, & y=0 \\  , & y=0 \\  
\end{matrix}</math>  \end{matrix}</math>  
  +  </center>  
+   {{EquationRef(1)}}  
+  }  
+  
Analogy between surface shear stress and the surface energy flux yields  Analogy between surface shear stress and the surface energy flux yields  
  +  { class="wikitable" border="0"  
+    
+   width="100%" <center>  
<math>\begin{align}  <math>\begin{align}  
& {q}''_{w}=\frac{\tau _{w}}{U_{\infty }}\left[ (c_{p}T+\omega _{o}h_{c,o})_{w}(c_{p}T+\omega _{o}h_{c,o})_{\infty } \right] \\  & {q}''_{w}=\frac{\tau _{w}}{U_{\infty }}\left[ (c_{p}T+\omega _{o}h_{c,o})_{w}(c_{p}T+\omega _{o}h_{c,o})_{\infty } \right] \\  
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\end{matrix} \\  \end{matrix} \\  
\end{align}</math>  \end{align}</math>  
  +  </center>  
+   {{EquationRef(1)}}  
+  }  
+  
The energy balance at the surface of the solid fuel is  The energy balance at the surface of the solid fuel is  
Revision as of 06:13, 7 April 2010
Most early work in theoretically predicting the heat and/or mass transfer in both laminar and turbulent flow cases was done using the analogy between momentum, heat, and mass and by predicting the approximate results for the heat and/or mass transfer coefficient from the momentum transfer or skin friction coefficient. Clearly there are severe limitations in using this simple approach; however, it is beneficial to understand the advantages and similarities for physical and mathematical modeling as well as the constraints involving this approach. We present this analogy for the classic problem of heat and mass transfer over a flat plate in this section. Its application to more complicated geometries and boundary conditions as well as turbulent flow is not proven and caution should be taken in applying this approach to other cases.
As presented before, a flat plate at constant wall temperature Tw is exposed to a free stream of constant velocity U∞, temperature T∞ and mass fraction ω1,∞. Due to binary diffusion it can be presented with the following dimensionless conservation boundary layer equations and boundary conditions corresponding to Fig. 1. Continuity

Momentum

Energy

Species




where the dimensionless variables are defined as:

Let’s first consider the analogy between momentum and heat transfer. Equations (4.349) and (4.350) and appropriate boundary conditions are the same if Pr = 1. The solutions for u+ and θ are, therefore, exactly the same if Pr = 1 and one expects to have a relationship between the skin friction coefficient, cf, and heat transfer coefficient, h.


Since θ = u+ for Pr = 1, one also concludes that

Therefore, combining eqs. (4.356) and (4.357) and using eq. (4.358) gives

This relationship between the friction coefficient and Nu is referred to as Reynolds analogy and is appropriate when Pr = 1. If Pr ≠ 1, we already concluded that from the similarity solution presented in Section 4.6. Using this information, one can generalize the result of the Reynolds analogy to Pr ≠ 1 by

Now, we focus on similarities between the heat and mass transfer for comparison of eqs. (4.350) and (4.351) with their appropriate boundary conditions. It is clear that the solution for differential equations (4.350) and (4.351) for θ and are the same if Sc and Pr are interchanged appropriately. We already know the solution for eq. (4.350) for v_{w} = 0 from the similarity solution for 0.5 ≤ Pr ≤ 10

Therefore, it can also be assumed that the solution of eq. (4.351) for v_{w} = 0 is

Combining eqs. (4.361) and (4.362) gives

It should be noted that the convective effect due to vertical velocity at the surface was neglected in predicting h and hm in the above analysis and therefore the analogy presented in (4.363) is for very low mass transfer at the wall. The effect is that the contribution of v_{w} in heat and mass flux was primarily due to diffusion.


The analogy in momentum, heat, and mass transfer can also be applied to complex and/or coupled transport phenomena problems including phase change and chemical reactions. Obviously, in these circumstances, the simple relationship developed in eq. (4.363) is not applicable. To show the usefulness of this analogy to more complex transport phenomena problems we will apply it to sublimation with a chemical reaction for forced convective flow over a flat plate. During combustion involving a solid fuel, the solid fuel may either burn directly or be sublimated before combustion. In the latter case – which will be discussed in this subsection – gaseous fuel diffuses away from the solidvapor interface. Meanwhile, the gaseous oxidant diffuses toward the solidvapor interface. Under the right conditions, the mass flux of vapor fuel and the gaseous oxidant meet and the chemical reaction occurs at a certain zone known as the flame. The flame is usually a very thin region with a color dictated by the temperature of combustion. Figure 4.32 shows the physical model of the problem under consideration. The concentration of the fuel is highest at the solid fuel surface, and decreases as the location of the flame is approached. The gaseous fuel diffuses away from the solid fuel surface and meets the oxidant as it flows parallel to the solid fuel surface. Combustion occurs in a thin reaction zone where the temperature is the highest, and the latent heat of sublimation is supplied by combustion. The combustion of solid fuel through sublimation can be modeled as a steadystate boundary layer type flow with sublimation and chemical reaction. To model the problem, the following assumptions are made: 1. The fuel is supplied by sublimation at a steady rate. 2. The Lewis number is unity, so the thermal and concentration boundary layers have the same thickness. 3. The buoyancy force is negligible.
The conservation equations for mass, momentum, energy, and species in the boundary layer are




where is the rate of oxidant consumption (kg/m3*s), hc,0 is the heat released by combustion per unit mass consumption of the oxidant (J/kg), which is different from the combustion heat, andω_{0} is the mass fraction of the oxidant in the gaseous mixture. The corresponding boundary conditions of eqs. (4.366) to (4.369) are


where is the rate of solid fuel sublimation per unit area (kg/m2?s) and ρ is the density of the mixture. The shear stress at the solid fuel surface is

The heat flux at the solid fuel surface is

The exact solution of the heat and mass problem described by eqs. (4.366) to (4.369) can be obtained using numerical simulation. It is useful here to introduce the results obtained by Kaviany (2001) using an analogy between momentum, heat, and mass transfer. Multiplying eq. (4.369) by hc,0 and adding the result to eq. (4.368), one obtains

Considering the assumption that the Lewis number is unity, i.e. Le = ν / D = 1, eq. (4.374) can be rewritten as

which can be viewed as an energy equation with quantity c_{p}T + ω_{0}h_{c,0}) as a dependent variable. Since at y = 0, i.e., the solid fuel surface is not permeable for the oxidant, eq(4.373) can be rewritten as

Analogy between surface shear stress and the surface energy flux yields
Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): \begin{align} & {q}''_{w}=\frac{\tau _{w}}{U_{\infty }}\left[ (c_{p}T+\omega _{o}h_{c,o})_{w}(c_{p}T+\omega _{o}h_{c,o})_{\infty } \right] \\ & \begin{matrix} {} & =\frac{\tau _{w}}{U_{\infty }}\left[ c_{p}(T_{w}T_{\infty })+h_{c,o}(\omega _{o,}_{w}\omega _{o}_{,\infty }) \right] \\ \end{matrix} \\ \end{align} 
The energy balance at the surface of the solid fuel is

where the two terms on the righthand side of eq. (4.378) represent the latent heat of sublimation, and the sensible heat required to raise the surface temperature of the solid fuel to sublimation temperature and heat loss to the solid fuel. Combining eqs. (4.377) and (4.378) yields the rate of sublimation on the solid fuel surface

where Z is the transfer driving force or transfer number defined as

Using the friction coefficient gives

eq. (4.379) therefore becomes

The surface blowing velocity of the gaseous fuel is then

where the friction coefficient, cf, can be obtained from the solution of the boundary layer flow over a flat plate with blowing on the surface. The similarity solution of the boundary layer flow problem exists only if the blowing velocity satisfies . In this case, one can define a blowing parameter as

Combining eqs. (4.383) and (4.384) yields

Glassman (1987) recommended an empirical form of eq. (4.385) based on numerical and experimental results:
