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Scale Analysis of boundary layer in combined mass and heat transfer about the adjacent vertical heated wall and its similarity with double-diffusive convection

Abstract

This study seeks to examine the boundary layer adjacent to a vertical heated wall that undergoes both heat and mass transfer simultaneously. We aim to compare this situation with double-diffusive convection to gain insight into the similar characteristics and interactions between heat and mass transport in this intricate system.

Introduction

Double diffusive convection is a complex fluid dynamics phenomenon characterized by the presence of two or more distinct density gradients within the fluid. These gradients often arise from variations in temperature or concentration, which can give rise to buoyancy-driven instabilities. The differing diffusion rates of these gradients play a crucial role in determining the nature and intensity of the convective flow.

Convective fingers Temperature: white(hot)black(cold)green(moderate)

Applications

Double diffusive convection plays a significant role in various natural and engineered systems including:

• Oceanography: Influencing Ocean Circulation, nutrient Distribution and marine ecosystems
• Atmosphere: Contributing to cloud formation, weather patterns, and climate variability.
• Geophysics: Driving processes in the Earth's mantle and contributing to the formation of geological structures.

• Industrial Processes: Impacting crystal growth, chemical engineering, and material science.

Governing Equations

To analyze this system, we employ the following governing equations:

• Continuity Equation: $\partial \rho /\partial t+\nabla \ \rho v=0$
• Momentum Equation: $\rho (\partial v/\partial t+v\nabla \ v)=-\nabla \ p+\mu \nabla ^{2}\ v+\rho g\beta (T-T\infty )+\rho g(\beta *)(C-C\infty )$
• Energy Equation: $\rho Cp(\partial T/\partial t+v\nabla \ T)=k\nabla ^{2}\ T$
• Species Conservation Equation: $\rho (\partial C/\partial t+v\nabla \ C)=D\nabla ^{2}\ C$

where

• $\rho$-density of the fluid
• v-velocity of fluid
• p-pressure applied on fluid
• $\mu$-dynamic viscosity
• g-gravitational acceleration
• $\beta$-thermal expansion coefficient
• $\beta *$-concentration expansion coefficient
• Cp-specific heat at constant pressure
• k-thermal conductivity
• D-Diffusion coefficient
• T-Surface temperature
• $T\infty$-Ambient Temperature
• C-Surface Concentration
• $C\infty$-Ambient Concentration

Scale Analysis of Vertical Heated Wall

We consider a vertical wall in contact with a fluid reservoir. The wall is maintained at a constant temperature($T\circ$) and concentration($C\circ$), while the reservoir fluid is at a constant temperature($T\infty$) and concentration($C\infty$). The temperature and concentration difference creates a density gradient, driving a buoyancy-induced vertical boundary layer flow.

Fig 1:Combined Heat and mass transfer boundary layer

The boundary layer momentum equation for this flow is

$u\partial v/\partial x+v\partial v/\partial y=\nu \partial ^{2}v/\partial ^{2}x+(1/\rho )(\rho \infty -\rho )g$

In the case of heat transfer, we saw that the density difference($\rho \infty -\rho$) is approximately proportional to the temperature difference ($T-T\infty$),in accordance with the Boussinesq approximation. In the presence of mass transfer, the driving density difference ($\rho \infty -\rho$) may also be due to the concentration difference ;a vertical buoyant layer forms if the wall releases a substance less dense than the reservoir fluid mixture. The thermodynamic state of the fluid mixture depends on pressure, temperature and composition.

In the limit of small density variations at constant pressure, we can write

$\rho \cong \rho \infty +(\partial \rho /\partial T)(T-T\infty )+(\partial \rho /\partial C)(C-C\infty )$

The definition of thermal expansion coefficient

$\beta =-(1/\rho )(\partial \rho /\partial T)$

and the concentration expansion coefficient

$\beta *=-(1/\rho )(\partial \rho /\partial C)$

Coefficient $\beta$ and $\beta *$ can be positive or negative; hence, in the scale analysis of $\beta$($T\circ -T\infty$) means the absolute value of $\beta$($T\circ -T\infty$). Based on the approximation, the boundary layer momentum equation becomes

$u\partial v/\partial x+v\partial v/\partial y=\nu \partial ^{2}v/\partial ^{2}x+g\beta (T-T\infty )+g(\beta *)(C-C\infty )$

The boundary layer energy, concentration, and continuity equations will be

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

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

$\partial u/\partial x+\partial v/\partial y=0$

respectively.

The wall-reservoir temperature difference drives the mass transfer to the vertical boundary layer as thermal diffusivity is much larger than the concentration diffusivity ($\alpha >>D$).

Let $\delta _{c},\delta _{t},\delta _{v}$ be the boundary layer thickness of the concentration profile, temperature profile, and velocity profile respectively.

In the flow region of thickness $\delta c$ and height H, from boundary layer continuity and concentration equations

$\partial u/\partial x+\partial v/\partial y=0$

and

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

we can convert it into

$u/\delta _{c}\backsim v/H$ , $u\ \bigtriangleup C/\delta c,v\bigtriangleup C/H\thicksim D\bigtriangleup C/\delta _{c}^{2}$

$v\bigtriangleup C/H\thicksim D\bigtriangleup C/\delta _{c}^{2}$

from scale analysis.

we finally get

$v\thicksim DH/\delta _{c}^{2}$

Note that v is the vertical velocity scale in the region of thickness $\delta _{c}$; Naturally, v will depend on the relative size of $\delta _{c}$ and the other two length scales of the $\bigtriangleup T$ -driven boundary layer flow.

Four possibilities are exist, as shown in Fig 2

Fig 2 :Relative size of boundary layer thickness in natural convection mass and heat transfer

1. $Pr>1,\delta _{c}<\delta _{t}$
2. $Pr>1,\delta _{c}>\delta _{t}$
3. $Pr<1,\delta _{c}<\delta _{v}$
4. $Pr<1,\delta _{c}>\delta _{v}$

We examine in some detail the first possibility ( $Pr>1,\delta _{c}<\delta _{t}$ ) , from Fig 3

Fig 3:Boundary Layer for $Pr>1,\delta _{c}<\delta _{t}$

As boundary layers are generally very small in size, ranging from $\mu m$ to mm. So, their velocity ratios will be linearly related, i.e.,

$v_{c}/v_{t}=\delta _{c}/\delta _{t}$

$v_{c}=v_{t}\delta _{c}/\delta _{t}$;

In a heat-transfer-driven boundary layer, the vertical velocity reaches the order of magnitude ($\alpha /H$) *$Ra_{H}^{1/2}$ at a distance of order $H*Ra_{H}^{-1/4}$ away from the wall (if $Pr>1$).

So, $v_{t}=(\alpha /H)*Ra_{H}^{1/2}$ and $\delta _{t}=H*Ra_{H}^{-1/4}$.

Velocity scale in the $\delta _{c}$-thin layer

$v\thicksim (\delta _{c}/\delta _{t})*(\alpha /H)*Ra_{H}^{1/2}$ where $\delta _{c}<\delta _{t}$

So, by replacing value of $v,\delta _{t}$ from above equation to get $\delta _{c}$,

$\delta _{c}\thicksim H*(D/\alpha )^{1/3}*Ra_{H}^{-1/4}$.

$L_{e}=(\alpha /D),S_{c}=(\nu /D)$ and $Pr=S_{c}/L_{e}$ are Lewis number, Schmidt number, and Prandlt number, respectively.

Similarly, we examine in some detail the third possibility ($Pr<1,\delta _{c}<\delta _{v}$) , from Fig 4

Fig 4:Boundary Layer for$Pr<1,\delta _{c}<\delta _{v}$

As boundary layers are generally very small in size, ranging from $\mu m$ to mm. So, their velocity ratios will be linearly related, i.e.,

$v_{c}/v_{v}=\delta _{c}/\delta _{v}$

$v_{c}=v_{v}\delta _{c}/\delta _{v}$;

In a heat-transfer-driven boundary layer, the vertical velocity reaches the order of magnitude ($\alpha /H$) *$Pr*Ra_{H}^{1/2}$ at a distance of order $H*(Ra_{H}/Pr)^{-1/4}$ away from the wall (if $Pr<1$ and $S_{c}>1$).

So, $v_{v}=(\alpha /H)*Pr*Ra_{H}^{1/2}$ and $\delta _{v}=H*(Ra_{H}/Pr)^{-1/4}$

So, by replacing value of $v,\delta _{v}$ from above equation to get $\delta _{c}$,

$\delta _{c}\thicksim H*(L_{e})^{-1/3}*Pr^{-1/12}*Ra_{H}^{-1/4}$.

Similarly, we found the boundary layer thickness of the concentration and peak velocity and predicted Sherwood Number(just like Nusselt Number in thermal boundary layer conditions).

Table 1: Mass transfer Rate scales for a vertical boundary layer driven by heat

Fluid Condition	Overall Sherwood Number $H/\delta _{c}$	Concentration Boundary Layer Thickness $\delta _{c}$	Peak Velocity obtain in $\delta _{c}*H$ profile $v_{c}$
$Pr>1,\delta _{c}<\delta _{t}(orL_{e}>1)$	$L_{e}^{1/3}*Ra_{H}^{1/4}$	$H*L_{e}^{-1/3}*Ra_{H}^{-1/4}$	$D*H/(H*L_{e}^{-1/3}*Ra_{H}^{-1/4})^{2}$
$Pr>1,\delta _{c}>\delta _{t}(orL_{e}<1),S_{c}>1$	$L_{e}^{1/2}*Ra_{H}^{1/4}$	$H*L_{e}^{-1/2}*Ra_{H}^{-1/4}$	$D*H/(H*L_{e}^{-1/2}*Ra_{H}^{-1/4})^{2}$
$Pr>1,\delta _{c}>\delta _{t}(orL_{e}<1),S_{c}<1$	$L_{e}*Pr^{1/2}*Ra_{H}^{1/4}$	$H*L_{e}^{-1}*Pr^{-1/2}*Ra_{H}^{-1/4}$	$D*H/(H*L_{e}^{-1}*Pr^{-1/2}*Ra_{H}^{-1/4})^{2}$
$Pr<1,\delta _{c}<\delta _{v}(orS_{c}>1)$	$L_{e}^{1/3}*Pr^{1/12}*Ra_{H}^{1/4}$	$H*L_{e}^{-1/3}*Pr^{-1/12}*Ra_{H}^{-1/4}$	$D*H/(H*L_{e}^{-1/3}*Pr^{-1/12}*Ra_{H}^{-1/4})^{2}$
$Pr<1,\delta _{c}>\delta _{v}(orS_{c}<1),L_{e}>1$	$L_{e}^{1/2}*Pr^{1/4}*Ra_{H}^{1/4}$	$H*L_{e}^{-1/2}*Pr^{-1/4}*Ra_{H}^{-1/4}$	$D*H/(H*L_{e}^{-1/2}*Pr^{-1/4}*Ra_{H}^{-1/4})^{2}$
$Pr<1,\delta _{c}>\delta _{v}(orS_{c}<1),L_{e}<1$	$L_{e}*Pr^{1/4}*Ra_{H}^{1/4}$	$H*L_{e}^{-1}*Pr^{-1/4}*Ra_{H}^{-1/4}$	$D*H/(H*L_{e}^{-1}*Pr^{-1/4}*Ra_{H}^{-1/4})^{2}$

For $Pr=7,H=1$;

Fig 5: Lewis Number Relation for Pr=7(water),where $x\longrightarrow L_{e}$

Result

This study examines the similarities between combined mass and heat transfer in a boundary layer adjacent to a vertical heated wall and double-diffusive convection. While these two phenomena exhibit distinct characteristics, their underlying mechanisms share commonalities.

By leveraging the scale analysis techniques developed for combined mass and heat transfer, we can gain valuable insights into double-diffusive convection. This approach allows for the estimation of key parameters, such as boundary layer thicknesses and velocities without resorting to complex numerical simulations.

The study demonstrates that the concepts and methodologies employed in combined mass and heat transfer can be effectively applied to double-diffusive convection, simplifying the analysis of this complex phenomenon. This finding offers a promising avenue for future research and applications in fields such as oceanography, atmospheric sciences, and materials science.

Fig 6: Double diffusive convection in oceanographyis where the vertical plate changes to the horizontal surface of seawater, which is heated by sunlight and is concentrated by saline water over the surface. The ambient water is cold freshwater(downside).$T\circ \rightarrow Temperature_{SalineWater}$ $C\circ \rightarrow Concentration_{SalineWater}$$T\infty \rightarrow Temperature_{FreshWater}$$C\infty \rightarrow Concentration_{FreshWater}$

References

• https://core.ac.uk/reader/36728010
• https://www.researchgate.net/publication/358386004_Dynamics_of_fingering_convection_a_numerical_study
• Double diffusive convection
• https://books.google.co.in/books/about/Convection_Heat_Transfer.html?id=9yC91-gpU8sC&redir_esc=y

Article prepared by

• Piyush Kumar (Roll No:-21135096), IIT BHU Varanasi