Draft:Scale Analysis in Convective Mass Transfer

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Scale Analysis in Convective Mass Transfer

Overview

Whenever convective events occur in nature, mass transfer—the movement of materials that function as species or components within a fluid mixture often occurs simultaneously. One illustration is the phenomenon of double-diffusive convection, in which ocean currents propelled by differential heating act as shipping containers for salt in the form of saline water. Combustion processes are also known to exhibit mass transfer. The evaporation of water from a pond to the atmosphere, the purifying of blood in organs like the liver and kidneys, and the distillation of alcohol are common examples of mass transfer. Mass transfer is essential in industrial applications for processes including liquid-liquid extraction systems, scrubbers, activated carbon beds, and distillation columns, which segregate chemical components.

Governing Equations

The mass transfer process is governed by the species conservation equation, which is similar in form to the energy conservation equation used in heat transfer. The concentration C of the species in the fluid is analogous to temperature T, and mass diffusivity D replaces thermal diffusivity α (alpha).

Mass Conservation Equation:

The general form of the mass conservation equation for a component of interest can be written as:

$\operatorname {D} \!C/\operatorname {D} \!t=D\nabla ^{2}\ C+{\dot {m}}'''$

Where:

• C: Concentration of the species,
• v: Velocity vector,
• D: Diffusion coefficient (mass diffusivity),
• t: Time,
• ${\dot {m}}'''$: Volumetric mass generation rate.

This equation resembles the heat transfer equation, where temperature is replaced by concentration, and diffusivity terms are swapped between mass and heat transfer contexts.

Dimensionless Numbers

To simplify and generalize the behavior of mass transfer systems, we introduce dimensionless numbers that characterize the balance between various forces (diffusion, convection, etc.) in the system.

1. Schmidt Number (Sc)

The Schmidt number (Sc) is a dimensionless ratio that compares the rate of momentum diffusivity (viscosity) to the rate of mass diffusivity:

​$Sc={\frac {\nu }{D}}$

Where:

• $\nu$: Kinematic viscosity of the fluid (momentum diffusivity),
• D: Mass diffusivity.

This number provides insight into the relative effectiveness of momentum transport (due to fluid viscosity) compared to mass transport (due to diffusion). A high Schmidt number means that momentum diffuses more easily than mass, while a low Schmidt number indicates that mass diffuses more readily than momentum.

2. Sherwood Number (Sh)

The Sherwood number (Sh) is analogous to the Nusselt number in heat transfer and represents the ratio of convective to diffusive mass transfer:

$Sh={\frac {h_{m}L}{D}}$

Where:

• h_m: Convective mass transfer coefficient,
• L: Characteristic length scale of the system,
• D: Mass diffusivity.

The Sherwood number quantifies how effectively mass is transferred in a system due to convective effects compared to molecular diffusion. In boundary layer theory, the Sherwood number can be derived by analyzing how the concentration profile changes within the layer and relates to the mass transfer rate.

3. Mass Transfer Rayleigh Number ($Ra_{m}$)

The mass transfer Rayleigh number ($Ra_{m}$​) characterizes buoyancy-driven flow caused by concentration differences within the fluid:

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

Where:

• g: Gravitational acceleration,
• β_c​: Concentration expansion coefficient (the rate of change of density with concentration),
• C₀​ and C_∞​: Concentrations at the wall and in the bulk fluid, respectively,
• L: Characteristic length scale,
• $\nu$: Kinematic viscosity,
• D: Mass diffusivity.

A higher Rayleigh number indicates stronger buoyancy forces due to the concentration gradient, which enhances convective flow.

4. Lewis Number (Le)

The Lewis number (Le) compares thermal diffusivity to mass diffusivity:

$Le={\frac {\alpha }{D}}$

Where α (alpha) is the thermal diffusivity. A Lewis number greater than one means that heat diffuses faster than mass, while a Lewis number less than one implies that mass diffuses more rapidly than heat. The Lewis number helps in understanding whether heat or mass transfer dominates in coupled heat and mass transfer processes.

Scale Analysis in Mass Transfer

Scale analysis simplifies the governing equations by focusing on the dominant terms and neglecting those that are small. This method provides an approximate but often highly accurate solution by estimating the magnitude of each term and parameter in the system.

• 
  Figure 1. Combined mass and heat transfer effected by a buoyant boundary layer flow.

Two primary cases are considered in the study of mass transfer:

1. Mass Transfer Driven Solely by Concentration Gradients

In this case, there is no temperature gradient, so the transfer is purely driven by differences in concentration. The analysis is analogous to heat transfer by replacing the Nusselt number Nu with the Sherwood number Sh and thermal diffusivity α (alpha) with mass diffusivity D.

The boundary layer equations lead to the following scaling for the Sherwood number:

$Sh={\begin{cases}0.503\,Ra_{m}^{1/4}&{\text{for }}Sc>1\\0.6\,(Ra_{m,y}\,Sc)^{1/4}&{\text{for }}Sc<1\end{cases}}$

This result shows that as the Rayleigh number increases (indicating stronger buoyancy forces), the mass transfer rate also increases, although at a reduced rate (proportional to the quarter power of the Rayleigh number).

2. Mass Transfer by Temperature Difference

In this case, the concentration and thermal boundary layers are distinct, and the scales for mass transfer and hydrodynamics must be separately derived.

• 
  Table 1 Length scale where temperature difference dominates

Heat-Transfer Driven Flow  

When heat transfer drives the mass transfer, the boundary layer's behavior is governed by both thermal and concentration profiles. The resulting vertical velocity in the boundary layer is proportional to the thermal boundary layer thickness. Scale analysis is used to estimate the mass transfer rate (Sherwood number), providing results that are within 25% accuracy of more detailed analyses.

• 
  Figure 2 Relative size of the boundary layer thicknesses in natural convection mass and heat transfer.
• 
  Table 2. Mass transfer rate scales for a vertical boundary layer driven by heat transfer

Relative Size of Boundary Layers

The boundary layers for concentration ($\delta _{c}$) and temperature ($\delta _{T}$) interact in different ways depending on fluid properties like the Prandtl (Pr) and Lewis (Le) numbers. In cases where Pr > 1 and Le > 1, the concentration boundary layer is thinner than the thermal boundary layer, influencing the overall mass transfer rate.

How to decide a layer is driven by mass transfer or heat transfer?

By comparing the thickness of the boundary layers connected to each process, we can determine whether heat transfer or mass transfer is responsible for a layer's movement during natural convection. Mass transfer predominates if the mass transfer boundary layer is thinner than the heat transfer boundary layer.

$(\delta _{c})_{HT}>(\delta _{c})_{MT}$

This is because the mass flux will follow the easier path, which corresponds to the thinner layer. The criterion demonstrates that heat transfer is the main mechanism when temperature-driven density changes outweigh concentration differences for fluids with Prandtl (Pr) > 1 and Lewis (Le) > 1.

${\frac {\beta (T_{0}-T_{\infty })}{\beta _{c}(C_{0}-C_{\infty })}}>{\text{Le}}^{-1/3}$

Practical Applications and Importance of Scale Analysis

1. Industrial Processes:

Scale analysis is used in designing separation equipment such as distillation columns, absorbers, and membrane systems, where precise control over mass transfer rates is essential.

2. Natural Convection in Engineering Systems:

In cooling systems, heat exchangers, and environmental systems like atmospheric or oceanic flows, understanding the balance between heat and mass transfer helps in optimizing performance and energy efficiency.

3. Environmental Systems:

Natural processes like ocean currents driven by temperature and salinity differences can be studied using scale analysis. The transport of pollutants, nutrients, or other species in the environment depends on both convective and diffusive mechanisms.

Conclusion

Scale analysis in convective mass transfer provides a powerful method to simplify complex transport phenomena and predict key behaviors in a variety of systems. By drawing analogies between heat and mass transfer, and using dimensionless numbers such as the Sherwood, Schmidt, and Rayleigh numbers, engineers can estimate mass transfer rates and optimize the design of systems involving fluid flows. The interplay between heat and mass transfer, particularly in systems where both gradients are present, requires careful analysis of boundary layer behavior to determine which process dominates.

Article Prepared By

• Astha Singh (Roll No.: 21135036), IIT (BHU) Varanasi
• Pooja Kumari (Roll No.: 21135097), IIT (BHU) Varanasi
• Tejal Arora (Roll No.: 21135153), IIT (BHU) Varanasi
• Pranjali Yadav (Roll No.: 21134034), IIT (BHU) Varanasi
  

References

2. Scale analysis (mathematics)#:~:text=Rules of scale analysis,-Scale analysis is&text=The object of scale analysis,results produced by exact analyses.