9.2 Dimensionless Numbers in Mass Transfer

Dimensionless numbers are crucial tools in mass transfer analysis. They help simplify complex problems, compare different systems, and identify dominant mechanisms. By understanding these numbers, engineers can better predict and optimize mass transfer processes across various applications.

These numbers, like the Sherwood and Schmidt numbers, relate different physical properties and phenomena. They allow for scaling, generalization of data, and identification of rate-limiting steps in mass transfer processes. This knowledge is essential for designing efficient systems.

Dimensionless Numbers in Mass Transfer

Key Dimensionless Numbers and Their Interpretations

• The Sherwood number (Sh) represents the ratio of convective mass transfer to diffusive mass transport
  • Analogous to the Nusselt number in heat transfer
  • Quantifies the relative importance of convective and diffusive mass transfer mechanisms (high Sh indicates convection-dominated transfer)
  • Defined as: $Sh = \frac{hL}{D}$, where $h$ is the convective mass transfer coefficient, $L$ is a characteristic length, and $D$ is the mass diffusivity
• The Schmidt number (Sc) represents the ratio of momentum diffusivity (kinematic viscosity) to mass diffusivity
  • Characterizes the relative thickness of the hydrodynamic and concentration boundary layers
  • Relates the viscous and mass diffusion rates in a fluid (high Sc indicates a thinner concentration boundary layer relative to the velocity boundary layer)
  • Defined as: $Sc = \frac{\nu}{D}$, where $\nu$ is the kinematic viscosity and $D$ is the mass diffusivity
• The Péclet number for mass transfer (Pe) represents the ratio of advective transport rate to diffusive transport rate
  • Indicates the relative importance of advective and diffusive mass transport mechanisms (high Pe implies advection-dominated transfer)
  • Defined as: $Pe = \frac{uL}{D}$, where $u$ is the fluid velocity, $L$ is a characteristic length, and $D$ is the mass diffusivity
• The Stanton number for mass transfer (St) represents the ratio of the mass transfer coefficient to the product of fluid velocity and density
  • Characterizes the ratio of the actual mass transfer rate to the maximum possible mass transfer rate
  • Quantifies the efficiency of mass transfer in a fluid flow (high St indicates efficient mass transfer)
  • Defined as: $St = \frac{h}{\rho u}$, where $h$ is the convective mass transfer coefficient, $\rho$ is the fluid density, and $u$ is the fluid velocity
• The Lewis number (Le) represents the ratio of thermal diffusivity to mass diffusivity
  • Characterizes the relative rates of heat and mass transfer in a fluid
  • Relates the thickness of the thermal and concentration boundary layers (Le > 1 implies a thinner concentration boundary layer relative to the thermal boundary layer)
  • Defined as: $Le = \frac{\alpha}{D}$, where $\alpha$ is the thermal diffusivity and $D$ is the mass diffusivity

Dimensionless Number Relationships and Analogies

• Dimensionless numbers can be combined to form other dimensionless groups
  • The Péclet number is the product of the Reynolds number (Re) and the Schmidt number: $Pe = Re \cdot Sc$
  • The Sherwood number can be expressed as a function of the Reynolds and Schmidt numbers: $Sh = f(Re, Sc)$
• Analogies between heat and mass transfer can be established using dimensionless numbers
  • The Chilton-Colburn analogy relates the Stanton numbers for heat and mass transfer: $St_h = St_m \cdot (Pr/Sc)^{2/3}$, where $Pr$ is the Prandtl number
  • The heat and mass transfer coefficients can be estimated using the Nusselt and Sherwood numbers: $Nu = \frac{hL}{k}$ and $Sh = \frac{h_mL}{D}$, where $k$ is the thermal conductivity and $h_m$ is the mass transfer coefficient

Applying Dimensionless Numbers to Mass Transfer

Characterizing Mass Transfer Mechanisms

• Dimensionless numbers help identify the dominant mass transfer mechanisms in a given problem
  • High Sherwood numbers (Sh >> 1) indicate convection-dominated mass transfer, while low Sh suggests diffusion-dominated transfer
  • High Péclet numbers (Pe >> 1) imply advection-dominated mass transport, while low Pe indicates diffusion-dominated transport
  • The Schmidt number (Sc) characterizes the relative thickness of the velocity and concentration boundary layers, influencing the mass transfer rate
• Example: In a laminar flow over a flat plate, the Sherwood number can be expressed as a function of the Reynolds and Schmidt numbers: $Sh = 0.664 Re^{1/2} Sc^{1/3}$
  • This correlation helps determine the relative importance of convective and diffusive mass transfer based on the flow conditions (Re) and fluid properties (Sc)

Estimating Mass Transfer Coefficients

• Dimensionless number correlations can be used to estimate mass transfer coefficients in various geometries and flow conditions
  • Empirical correlations relate the Sherwood number to the Reynolds and Schmidt numbers for different geometries (e.g., flat plates, cylinders, spheres)
  • These correlations are typically of the form: $Sh = a Re^b Sc^c$, where $a$, $b$, and $c$ are constants that depend on the geometry and flow regime
• Example: For forced convection mass transfer over a sphere, the Frössling correlation can be used: $Sh = 2 + 0.552 Re^{1/2} Sc^{1/3}$
  • This correlation allows the estimation of the mass transfer coefficient for a sphere in a flowing fluid, given the fluid properties (Re, Sc) and the sphere diameter

Simplifying Governing Equations

• Dimensionless numbers can be used to simplify the governing equations for mass transfer problems
  • By introducing dimensionless variables based on the relevant dimensionless numbers, the number of variables and parameters in the equations can be reduced
  • The simplified dimensionless equations are easier to solve and provide insights into the fundamental behavior of the system
• Example: The convection-diffusion equation for mass transfer can be non-dimensionalized using the Péclet number:
  • Original equation: $\frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x} = D \frac{\partial^2 C}{\partial x^2}$
  • Non-dimensionalized equation: $\frac{\partial C^*}{\partial t^*} + \frac{\partial C^*}{\partial x^*} = \frac{1}{Pe} \frac{\partial^2 C^*}{\partial x^{*2}}$, where $C^*$, $t^*$, and $x^*$ are dimensionless concentration, time, and position, respectively

Significance of Dimensionless Numbers for Mass Transfer Data

Generalizing Mass Transfer Data

• Dimensionless numbers allow for the generalization of mass transfer data obtained from experiments or simulations
  • By expressing the data in terms of dimensionless numbers, it becomes applicable to a wide range of systems with similar dimensionless parameters
  • This generalization enables the comparison and analysis of mass transfer processes across different scales and geometries
• Example: The Sherwood number can be used to generalize mass transfer data for different fluids and geometries
  • Experimental data for mass transfer in a specific fluid and geometry can be expressed as a Sherwood number correlation: $Sh = f(Re, Sc)$
  • This correlation can then be applied to predict mass transfer coefficients in other systems with similar Re and Sc values, even if the fluid and geometry are different

Scaling Up Mass Transfer Processes

• Dimensionless correlations can be used to scale up mass transfer processes from laboratory to industrial scales
  • By ensuring that the key dimensionless parameters (e.g., Re, Sc, Sh) remain constant during the scale-up process, the mass transfer performance can be maintained
  • This approach allows for the design and optimization of industrial-scale mass transfer equipment based on laboratory-scale experiments or simulations
• Example: In the design of a packed bed reactor for a gas-liquid mass transfer process, the Sherwood number correlation obtained from laboratory experiments can be used to determine the required packing size and fluid flow rates for the industrial-scale reactor
  • The Sherwood number correlation, $Sh = f(Re, Sc)$, is used to calculate the mass transfer coefficient for the industrial-scale reactor
  • The reactor dimensions and operating conditions are then adjusted to maintain the same Re and Sc values as in the laboratory experiments, ensuring similar mass transfer performance

Identifying Rate-Limiting Steps

• The analysis of dimensionless numbers in mass transfer problems can provide insights into the underlying physical mechanisms and help identify the rate-limiting steps
  • By comparing the values of different dimensionless numbers, the relative importance of various mass transfer mechanisms can be assessed
  • This analysis helps in identifying the factors that control the overall mass transfer rate and guides the optimization of the process
• Example: In a gas-liquid mass transfer process, the Sherwood number (Sh) and the Péclet number (Pe) can be compared to determine the rate-limiting step
  • If Sh >> Pe, the mass transfer is likely limited by the diffusion through the liquid-side boundary layer, and efforts should focus on increasing the liquid-side mass transfer coefficient
  • If Pe >> Sh, the mass transfer is likely limited by the convective transport in the bulk fluid, and efforts should focus on increasing the fluid mixing or velocity to enhance convective mass transfer