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Laminar Boundary Layer Flow: Heat and Mass Transfer

Fig 1.1 Concentration boundary layer near a flat plate

In laminar boundary layer flow, the comparison between heat transfer and mass transfer becomes evident, as depicted in Fig. 1.1. Consider a solid surface covered with or made of a material that is soluble in the flowing stream. This material is exposed to a uniform U-shaped stream. As an example, we can consider the forced convection drying of a water-saturated porous solid wall. The vapor content in the humid air stream, denoted by $U_{\infty }$, is at a distance from the wall, and the free-stream concentration is represented as $C_{\infty }$. Using the boundary layer method, we anticipate a concentration boundary layer at the wall. This boundary layer balances the difference between the relatively moist flow near the wall, with concentration $C_{0}$, and the drier free stream, represented by $C_{\infty }$.

Due to the concentration gradient between the wall and the free stream, the soluble material diffuses from the wall. The purpose of this section is to predict the rate of mass transfer. Here, $C_{0}$ represents the concentration of the fluid batch adjacent to the wall, not the concentration of the soluble material within the wall. The concentration on the fluid side of the wall-stream interface, at $y=0^{+}$, is denoted by $C_{0}$. For example, in the drying wall case, the water content can vary considerably between $y=0^{-}$ (within the wall) and $y=0^{+}$ (outside the wall).

Assuming that ${\frac {dP_{\infty }}{dx}}=0$ and that there is no temperature gradient between the wall and stream, the concentration at the fluid side of the wall-stream interface is determined by the equilibrium vapor pressure at the free-stream pressure and temperature. According to Fick’s law, the mass flux from the wall into the stream is given by:

$$j_{0}=-D\left({\frac {\partial C}{\partial y}}\right)_{y=0}$$ (Eq. 1.36)

The concentration field, $C(x,y)$, is obtained by solving the boundary layer concentration equation::

$$u{\frac {\partial C}{\partial x}}+v{\frac {\partial C}{\partial y}}=D{\frac {\partial ^{2}C}{\partial y^{2}}}$$ (Eq. 1.37)

This equation is subject to the following boundary conditions:

$$C=C_{0}{\text{ at }}y=0,\quad C\rightarrow C_{\infty }{\text{ as }}y\rightarrow \infty$$ (Eq. 1.38)

The constants $C_{0}$ and $C_{\infty }$ are both present. From the Blasius solution for laminar boundary layer flow, we can derive the concentration gradient at the wall using the transformation $T\rightarrow C$ and $\alpha \rightarrow D$ since this mass transfer problem is similar to Pohlhausen’s solution for heat transfer

$$\left({\frac {\partial C}{\partial y}}\right){y=0}=\left(C\infty -C_{0}\right)\left({\frac {U_{\infty }}{\nu x}}\right)^{1/2}\left\{\int _{0}^{\infty }\exp \left[-{\frac {\nu }{2D}}\int _{0}^{\gamma }f(\beta )\,d\beta \right]d\gamma \right\}^{-1}$$ (Eq. 1.39)

This result introduces the Schmidt number, $Sc$, analogous to the Prandtl number in heat transfer:

$$Sc={\frac {\nu }{D}}$$ (Eq. 1.40)

From the two asymptotes of the nested integral in Eq. (1.39), we conclude:

$${\frac {\left({\frac {\partial C}{\partial y}}\right){y=0}}{\left(C\infty -C_{0}\right)\left({\frac {U_{\infty }}{\nu x}}\right)^{1/2}}}={\begin{cases}0.332\,Sc^{1/3}&(Sc>1)\\0.564\,Sc^{1/2}&(Sc<1)\end{cases}}$$ (Eq. 1.41)

This result is expressed dimensionlessly using the local Sherwood number, analogous to the Nusselt number for temperature gradient (or wall heat flux) in convective heat transfer:

$$Sh=\left({\frac {\partial C}{\partial y}}\right){y=0}\left({\frac {x}{C\infty -C_{0}}}\right)=j_{0}\left({\frac {x}{D(C_{0}-C_{\infty })}}\right)$$ (Eq. 1.42)

Thus, we have:

$$Sh={\begin{cases}0.332\,Sc^{1/3}\,Re_{x}^{1/2}&(Sc\geq 0.5)\\0.564\,Sc^{1/2}\,Re_{x}^{1/2}&(Sc\leq 0.5)\end{cases}}$$ (Eq. 1.43, Eq. 1.44)

Taking the analogy further, the Sherwood number can also be defined as:

$$Sh={\frac {h_{m}x}{D}}$$ (Eq. 1.46)

where $h_{m}={\frac {j_{0}}{C_{0}-C_{\infty }}}$ is the mass transfer coefficient:

$$h_{m}={\frac {j_{0}}{C_{0}-C_{\infty }}}$$ (Eq. 1.45)

The local mass transfer coefficient, $h_{m}$, can be computed using Eqs. (1.43) and (1.44), with $h_{m}={\frac {Sh\,D}{x}}$. This coefficient can also be averaged over the entire surface of length $L$:

$${\bar {h}}_{m}={\frac {1}{L}}\int _{0}^{L}h_{m}\,dx$$ (Eq. 1.48)

Thus, the overall Sherwood number based on this average mass transfer coefficient is:

$${\bar {Sh}}={\frac {{\bar {h}}_{m}L}{D}}$$ (Eq. 1.49)

Using Eqs. (1.43) and (1.44), we can calculate:

$${\bar {Sh}}={\begin{cases}0.664\,Sc^{1/3}\,Re_{L}^{1/2}&(Sc\geq 0.5)\\1.128\,Sc^{1/2}\,Re_{L}^{1/2}&(Sc\leq 0.5)\end{cases}}$$ (Eq. 1.50, Eq. 1.51)

To determine the total mass transfer rate, we use the following equation:

$${\dot {m}}={\bar {h}}mL(C_{0}-C\infty )$$ (Eq. 1.52)

For a general case, the total mass transfer rate over a surface area $A$ is:

$${\dot {m}}={\bar {h}}mA(C_{0}-C\infty )$$ (Eq. 1.53)

Impermeable Surface Model

The mass transfer and heat transfer analogy holds when the transversal velocity of the flow, $v$, at the wall is zero. This approximation is valid when the concentration of the species of interest is sufficiently low. When the concentration is low, the mass flux through the surface is negligible, and the wall behaves as impermeable.

Using mass flux $j_{0}$, we derive the velocity scale associated with the addition of the mass flux from the wall side:

$$v_{0}={\frac {j_{0}}{\rho }}$$ (Eq. 1.55)

The transverse velocity in the boundary layer is given by:

$$v_{\infty }\sim U_{\infty }Re_{x}^{-1/2}$$ (Eq. 1.56)

For low Schmidt numbers, the impermeability condition can be written as:

$${\frac {C_{0}-C_{\infty }}{\rho }}<Sc^{1/2}<1$$ (Eq. 1.57)

For high Schmidt numbers, where the concentration boundary layer thickness $\delta _{c}$ is smaller than the velocity boundary layer thickness $\delta$, the condition becomes:

$${\frac {C_{0}-C_{\infty }}{\rho }}<1$$ (Eq. 1.62)

Natural Convection

In natural convection: over a vertical wall, as shown in Fig. 1.2, the boundary layer flow is driven by the density difference due to temperature and concentration gradients. The boundary layer momentum equation is:

$u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}=\nu {\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {g}{\rho }}\left(\rho _{\infty }-\rho \right)$

The density difference $\rho _{\infty }-\rho$ can be approximated using the Boussinesq approximation. This approximation assumes that the density difference arises from small temperature and concentration variations, leading to the following expression for the density:

Fig 1.2 Combined mass and heat transfer effected by a buoyant boundary layer flow

$\rho \approx \rho _{\infty }+\left({\frac {\partial \rho }{\partial T}}\right)P(T-T\infty )+\left({\frac {\partial \rho }{\partial C}}\right)P(C-C\infty )+\cdots$

Using the definition of the thermal expansion coefficient:

$\beta =-{\frac {1}{\rho }}\left({\frac {\partial \rho }{\partial T}}\right)_{P}$

we introduce the concentration expansion coefficient

$\beta _{c}=-{\frac {1}{\rho }}\left({\frac {\partial \rho }{\partial C}}\right)_{P}$

Substituting these coefficients into the boundary layer momentum equation, we find:

$u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}=\nu {\frac {\partial ^{2}v}{\partial x^{2}}}+g\beta (T-T_{\infty })+g\beta _{c}(C-C_{\infty })$

This equation shows that the flow field is influenced by both temperature and concentration variations, where the terms represent inertial effects, frictional forces, and body forces due to non-uniform temperature and concentration.

To determine the temperature and concentration fields, we solve the boundary layer equations for energy and mass transfer:

$u{\frac {\partial T}{\partial x}}+v{\frac {\partial T}{\partial y}}=\alpha {\frac {\partial ^{2}T}{\partial x^{2}}}$

$u{\frac {\partial C}{\partial x}}+v{\frac {\partial C}{\partial y}}=D{\frac {\partial ^{2}C}{\partial x^{2}}}$

Mass-Transfer-Driven Flow

To calculate the mass transfer rate between the wall and the fluid reservoir, we analyze two limiting cases. First, we consider the scenario where heat transfer is negligible, meaning the flow is driven entirely by the concentration gradient. This simplifies our problem to solving equations (1.82) and (1.83) with the velocity and concentration boundary conditions illustrated in Fig. 1.2.

This mass transfer problem can be solved analytically, producing results identical to those derived for heat transfer. By applying the transformations $Nu\to Sh$, $\alpha \to D$, $Pr\to Sc$, and $\beta (T_{0}-T_{\infty })\to \beta _{c}(C_{0}-C_{\infty })$, we derive the mass transfer results:

$Sh=0.503Ra_{m,y}^{1/4}\quad (Sc>1)$

$Sh=0.6\left(Ra_{m,y}Sc\right)^{1/4}\quad (Sc<1)$

where $Ra_{m,y}$ is the local mass transfer Rayleigh number defined as:

$Ra_{m,y}=g\beta _{c}(C_{0}-C_{\infty }){\frac {y^{3}}{\nu D}}$

This conclusion highlights the interplay between the velocity, temperature, and concentration fields in determining the mass transfer characteristics in natural convection scenarios.

Conclusion

This overview of laminar boundary layer flow in the context of heat and mass transfer emphasizes the critical role of both temperature and concentration gradients in influencing flow behavior. The mathematical framework established here not only lays the foundation for understanding complex heat and mass transfer interactions but also serves as a guide for practical applications in engineering and environmental contexts.

See also

• Boundary layer separation
• Boundary-layer thickness
• Thermal boundary layer thickness and shape
• Boundary layer suction
• Boundary layer control
• Boundary microphone
• Blasius boundary layer
• Logarithmic law of the wall
• Shape factor (boundary layer flow)

References

• Bejan, Adrian (2013). Convection Heat Transfer. John Wiley & Sons, Inc. doi:10.1002/9781118671627. ISBN 9780470900376.