6.2 Mass Transfer Coefficients

Mass transfer coefficients are key to understanding how substances move between phases. They quantify the rate of mass transfer and help us design better systems for processes like absorption and extraction.

These coefficients depend on fluid properties, flow conditions, and system geometry. By using correlations and understanding their limitations, we can calculate mass transfer rates and optimize processes for various applications.

Mass transfer coefficients

Definition and interpretation

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• Mass transfer coefficients quantify the rate of mass transfer between a fluid and a surface or between two fluids in convective mass transfer processes
• Defined as the ratio of the mass flux to the driving force (concentration difference) for mass transfer
• The overall mass transfer coefficient (K) combines individual mass transfer coefficients on either side of the interface, analogous to the overall heat transfer coefficient in heat transfer
• Higher values indicate faster mass transfer rates, while lower values suggest slower mass transfer rates ($h_{m,high} > h_{m,low}$)
• Typically expressed in units of length per time (m/s) or molar flux per unit concentration difference (mol/(m^2·s·mol/m^3))

Significance and applications

• Mass transfer coefficients are essential for designing and analyzing processes involving mass transfer, such as absorption, desorption, extraction, and membrane separation
• Used to calculate mass transfer rates and determine the effectiveness of mass transfer equipment (packed towers, falling film absorbers)
• Help optimize process conditions and select appropriate materials and geometries to achieve desired mass transfer performance
• Enable the comparison of different mass transfer systems and the evaluation of mass transfer enhancements (surface modifications, flow disruptions)

Coefficients vs. system properties

Dependence on physical properties

• Mass transfer coefficients depend on the physical properties of the fluids involved, such as density, viscosity, diffusivity, and Schmidt number (Sc)
• The Schmidt number (Sc) relates the momentum diffusivity (kinematic viscosity) to the mass diffusivity, representing the relative importance of these two transport phenomena ($Sc = \frac{\nu}{D}$)
• Higher Schmidt numbers indicate that momentum diffusion dominates over mass diffusion, leading to thinner concentration boundary layers and higher mass transfer coefficients
• Lower Schmidt numbers suggest that mass diffusion is more significant, resulting in thicker concentration boundary layers and lower mass transfer coefficients

Influence of flow conditions

• The flow regime (laminar, transition, or turbulent) significantly influences mass transfer coefficients
• Turbulent flow generally results in higher mass transfer rates due to enhanced mixing and thinner concentration boundary layers (smaller $\delta_c$)
• Increasing the fluid velocity or reducing the characteristic length scale of the system (smaller pipe diameter or shorter plate length) typically enhances mass transfer coefficients
• Higher Reynolds numbers (Re) indicate more turbulent flow and generally lead to higher mass transfer coefficients ($h_m \propto Re^n$, where $n > 0$)

Calculating mass transfer rates

Correlations for mass transfer coefficients

• Mass transfer rates can be calculated using correlations that relate mass transfer coefficients to system parameters such as Reynolds number (Re), Schmidt number (Sc), and geometry-specific dimensionless numbers
• For external flow over flat plates, the Chilton-Colburn analogy relates the mass transfer coefficient to the friction factor and Schmidt number, allowing the use of heat transfer correlations for mass transfer calculations ($j_m = St Sc^{2/3}$)
• For flow in pipes and ducts, correlations such as the Dittus-Boelter equation (turbulent flow) and the Sieder-Tate equation (laminar flow) can be used to estimate mass transfer coefficients based on Re, Sc, and geometry ($Sh = a Re^b Sc^c$)
• For packed beds and other complex geometries, empirical correlations specific to the system configuration and packing characteristics are often employed (Onda correlations, Shulman correlations)

Assumptions and limitations

• When using correlations, it is essential to ensure that the assumptions, limitations, and applicable ranges of the correlations are met for accurate mass transfer rate calculations
• Correlations are often developed for specific geometries, flow regimes, and fluid properties, and their applicability to other systems should be carefully evaluated
• The presence of surface reactions, non-Newtonian fluids, or other complex phenomena may require specialized correlations or numerical simulations to accurately predict mass transfer rates

Influence on mass transfer coefficients

Geometry effects

• The geometry of the system, such as the shape and size of the interface between the fluids or the surface area available for mass transfer, significantly affects mass transfer coefficients
• Larger surface areas and smaller characteristic lengths (smaller pipe diameters or shorter distances between plates) lead to higher mass transfer coefficients due to increased contact between the fluids and reduced resistance to mass transfer
• Complex geometries (packed beds, structured packings) can enhance mass transfer by increasing the interfacial area and promoting turbulence and mixing

Flow regime and surface interactions

• The flow regime (laminar, transition, or turbulent) greatly influences mass transfer coefficients
• Turbulent flow enhances mass transfer by promoting mixing, reducing concentration boundary layer thickness, and increasing the effective surface area for mass transfer
• In laminar flow, mass transfer coefficients are typically lower due to the presence of thicker concentration boundary layers and reduced mixing compared to turbulent flow
• The presence of surface roughness, turbulence promoters, or other flow disruptions can enhance mass transfer coefficients by increasing turbulence and promoting mixing near the interface
• Surface wettability and interfacial phenomena (surface tension, capillary effects) can also influence mass transfer coefficients, particularly in gas-liquid and liquid-liquid systems