# Analogies and differences in different transport phenomena

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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. 4.31. Continuity

 $\frac{\partial u^{+}}{\partial x^{+}}+\frac{\partial v^{+}}{\partial y^{+}}=0$ (1)

Momentum

 Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): u^{+}\frac{\partial u^{+}}{\partial x^{+}}+v^{+}\frac{\partial u^{+}}{\partial y^{+}}=\frac{1}{\operatorname{Re}}\frac{\partial ^{2}u^{+}}{\partial y^{+}^{2}} (1)

Energy

 $u^{+}\frac{\partial \theta }{\partial x^{+}}+v^{+}\frac{\partial \theta }{\partial y^{+}}=\frac{1}{\Pr \operatorname{Re}}\frac{\partial ^{2}\theta }{\partial y^{+2}}$ (1)

Species

 $u^{+}\frac{\partial \varphi }{\partial x^{+}}+v^{+}\frac{\partial \varphi }{\partial y^{+}}=\frac{1}{\text{Sc}\operatorname{Re}}\frac{\partial ^{2}\varphi }{\partial y^{+2}}$ (1)
 $\text{at }y^{+}=0,\text{ u}^{\text{+}}=0,\text{ }\theta \text{=}\varphi \text{=0}$ (1)
 $y^{+}=0,\text{ }v^{+}=v_{w}^{+}$ (1)
 $y^{+}=\infty ,\text{ }u^{+}=1,\text{ }\theta =\varphi =1$ (1)

where the dimensionless variables are defined as:

 \begin{align} & u^{+}=\frac{u}{U_{\infty }},\text{ }\theta =\frac{T-T_{w}}{T_{\infty }-T_{w}}\text{, }\varphi =\frac{\omega _{1}-\omega _{1,w}}{\omega _{1,\infty }-\omega _{1,w}}, \\ & v^{+}=\frac{v}{U_{\infty }},\text{ }y^{+}=\frac{y}{L},\text{ }x^{+}=\frac{x}{L},\text{ }\operatorname{Re}=\frac{U_{\infty }L}{\nu } \\ \end{align} (1)

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.

 $\frac{c_{f}}{2}=\frac{\tau _{w}}{\rho U_{\infty }^{2}}=\frac{\mu \left. \frac{\partial u}{\partial y} \right|_{y=0}}{\rho U_{\infty }^{2}}=\frac{\nu \left. \frac{\partial u^{+}}{\partial y^{+}} \right|_{y^{+}=0}}{U_{\infty }L}=\frac{\left. \frac{\partial u^{+}}{\partial y^{+}} \right|_{y^{+}=0}}{\operatorname{Re}}$ (1)
 $Nu=\frac{hL}{k}=\frac{-k\left. \frac{\partial T}{\partial y} \right|_{y=0}L}{k\left( T_{\omega }-T_{\infty } \right)}=\left. \frac{\partial \theta }{\partial y^{+}} \right|_{y^{+}=0}$ (1)

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

 $\left. \frac{\partial \theta }{\partial y^{+}} \right|_{y^{+}=0}=\left. \frac{\partial u^{+}}{\partial y^{+}} \right|_{y^{+}=0}$ (1)

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

 $\frac{c_{f}}{2}=\frac{\text{Nu}}{\operatorname{Re}}$ (1)

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

$\frac{\delta _{T}}{\delta }=\Pr ^{-1/3}\text{ for }0.5\le \Pr \le 10$

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

 $\frac{c_{f}}{2}=\frac{\text{Nu}}{\operatorname{Re}}\Pr ^{-1/3}$ (1)

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 $\varphi$ are the same if Sc and Pr are interchanged appropriately. We already know the solution for eq. (4.350) for vw = 0 from the similarity solution for 0.5 ≤ Pr ≤ 10

 $\text{Nu}_{x}=0.332\operatorname{Re}_{x}^{1/2}\Pr ^{1/3}$ (1)

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

 $\text{Sh}_{x}=0.332\operatorname{Re}_{x}^{1/2}\text{Sc}^{1/3}$ (1)

Combining eqs. (4.361) and (4.362) gives

 $\frac{\text{Nu}}{\text{Sh}}\text{=}\frac{\text{Pr}^{\text{1/3}}}{\text{Sc}^{\text{1/3}}}$ (1)

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 vw in heat and mass flux was primarily due to diffusion.

 ${q}''_{w}=\rho c_{p}v_{w}\left( T_{w}-T_{\infty } \right)-k\left. \frac{\partial T}{\partial y} \right|_{y=0}$ (1)
 ${\dot{m}}''_{w}=\rho \left( \omega _{1,\omega }v_{w}-D_{12}\left. \frac{\partial \omega _{1}}{\partial y} \right|_{y=0} \right)$ (1)

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 solid-vapor interface. Meanwhile, the gaseous oxidant diffuses toward the solid-vapor 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 steady-state 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.