8.4 Correlations for convective mass transfer

Convective mass transfer correlations are key tools for engineers to estimate mass transfer rates in various systems. These empirical relationships link mass transfer coefficients to system parameters and fluid properties, helping predict performance in industrial processes.

Choosing the right correlation depends on factors like geometry and flow regime. Common ones include the Sherwood-Reynolds-Schmidt correlation and the Ranz-Marshall correlation. Understanding these helps optimize mass transfer operations in real-world applications.

Selecting Correlations for Convective Mass Transfer

Factors Influencing Correlation Choice

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• Convective mass transfer correlations are empirical relationships that relate the mass transfer coefficient to system parameters and fluid properties
• The choice of an appropriate correlation depends on factors such as the geometry of the system (flat plate, cylindrical, spherical), the flow regime (laminar or turbulent), and the presence of forced or natural convection
• Common correlations for convective mass transfer include the Sherwood-Reynolds-Schmidt correlation for forced convection over a flat plate, the Ranz-Marshall correlation for mass transfer to a single sphere or droplet, and the Chilton-Colburn analogy for relating heat and mass transfer coefficients
• The Sherwood number (Sh) represents the ratio of convective mass transfer to diffusive mass transfer, analogous to the Nusselt number in heat transfer

Dimensionless Numbers in Mass Transfer

• The Reynolds number (Re) characterizes the flow regime (laminar or turbulent), while the Schmidt number (Sc) relates the viscous diffusion rate to the molecular diffusion rate
• The Sherwood number (Sh) is defined as $Sh = (hm × L) / DAB$, where $hm$ is the mass transfer coefficient, $L$ is a characteristic length, and $DAB$ is the binary diffusion coefficient of species A in species B
• The Reynolds number (Re) is defined as $Re = (ρ × u × L) / μ$, where $ρ$ is the fluid density, $u$ is the fluid velocity, $L$ is a characteristic length, and $μ$ is the fluid viscosity
• The Schmidt number (Sc) is defined as $Sc = μ / (ρ × DAB)$, where $μ$ is the fluid viscosity, $ρ$ is the fluid density, and $DAB$ is the binary diffusion coefficient

Estimating Mass Transfer Coefficients

Using Correlations for Specific Geometries

• For forced convection over a flat plate, the Sherwood-Reynolds-Schmidt correlation is given by $Sh = a × Re^b × Sc^c$, where $a$, $b$, and $c$ are empirical constants that depend on the flow regime and geometry
  • Example: In laminar flow over a flat plate, the correlation is $Sh = 0.664 × Re^{0.5} × Sc^{0.33}$
• For mass transfer to a single sphere or droplet, the Ranz-Marshall correlation is given by $Sh = 2 + 0.6 × Re^{1/2} × Sc^{1/3}$
  • This correlation is applicable for Reynolds numbers up to 200,000 and Schmidt numbers between 0.6 and 400
• The Chilton-Colburn analogy relates the mass transfer coefficient to the heat transfer coefficient using the Chilton-Colburn j-factors, where $jm = Sh / (Re × Sc^{1/3})$ and $jh = Nu / (Re × Pr^{1/3})$
  • This analogy allows for the estimation of mass transfer coefficients from known heat transfer coefficients or vice versa

Calculating Mass Transfer Rates

• The mass transfer coefficient ($hm$) obtained from correlations can be used to calculate the mass transfer rate ($NA$) using the equation $NA = hm × A × (CA,s - CA,∞)$, where $A$ is the surface area, $CA,s$ is the concentration of species A at the surface, and $CA,∞$ is the concentration of species A in the bulk fluid
  • Example: In a gas absorption process, the mass transfer rate of a solute from the gas phase to the liquid phase can be determined using the mass transfer coefficient and the concentration difference between the gas-liquid interface and the bulk liquid

Applying Convective Mass Transfer Results

Design and Optimization

• In design problems, the mass transfer coefficient can be used to determine the required surface area for a given mass transfer rate or to estimate the time required for a certain amount of mass transfer to occur
  • Example: In the design of a packed bed absorber, the mass transfer coefficient can be used to calculate the required packing height or surface area for a desired removal efficiency
• Understanding the factors that influence the mass transfer coefficient, such as fluid velocity, temperature, and concentration gradients, can help optimize process conditions and improve the efficiency of mass transfer operations
  • Example: Increasing the fluid velocity or temperature can enhance the mass transfer coefficient, leading to faster mass transfer rates and smaller equipment sizes

Industrial Applications

• The results from convective mass transfer correlations can be applied to various industrial processes, such as drying, absorption, adsorption, distillation, and extraction, where mass transfer plays a crucial role
  • Example: In a drying process, the mass transfer coefficient determines the rate at which moisture is removed from the material being dried
• Mass transfer coefficients are essential in the design and analysis of heat and mass exchangers, such as cooling towers, humidifiers, and dehumidifiers
  • Example: In a cooling tower, the mass transfer coefficient governs the rate of evaporation and the overall cooling performance

Limitations of Convective Mass Transfer Correlations

Assumptions and Simplifications

• Convective mass transfer correlations are empirical relationships based on experimental data and are valid only within the range of conditions for which they were developed
• The correlations assume steady-state conditions, meaning that the fluid properties and flow conditions do not change with time
• Most correlations are derived for simple geometries, such as flat plates, cylinders, or spheres, and may not accurately represent more complex shapes or flow patterns encountered in real-world applications
• The presence of surface roughness, non-uniform fluid properties, or non-Newtonian fluids can affect the accuracy of the correlations

Factors Not Accounted For

• The correlations do not account for the presence of chemical reactions or phase changes, which can significantly impact the mass transfer process
  • Example: In a catalytic reactor, the mass transfer coefficient may be affected by the presence of chemical reactions on the catalyst surface
• In some cases, the assumptions of constant fluid properties (density, viscosity, diffusivity) may not hold, especially when there are significant temperature or concentration gradients in the system
  • Example: In high-temperature mass transfer processes, the variation of fluid properties with temperature can lead to deviations from the predicted mass transfer coefficients
• When applying convective mass transfer correlations to real-world problems, it is essential to consider the limitations and assumptions of the correlations and to use appropriate safety factors or experimental validation when necessary
  • Example: In the design of a mass transfer equipment, a safety factor may be applied to the calculated mass transfer coefficient to account for uncertainties and ensure adequate performance