Introduction & Context
Dimensionless groups are fundamental to process engineering, providing a framework to scale experimental data from laboratory models to industrial-scale equipment. In mass transfer operations, these groups allow engineers to predict the mass transfer coefficient (kc), which dictates the rate at which species move across phase boundaries. This methodology is critical for designing absorption towers, catalytic reactors, and drying equipment, where the interaction between fluid dynamics and molecular diffusion determines overall process efficiency.
Methodology & Formulas
The calculation of the mass transfer coefficient relies on the determination of the Reynolds (Re), Schmidt (Sc), and Sherwood (Sh) numbers. The following algebraic expressions define the relationship between these parameters:
The Reynolds number, representing the ratio of inertial forces to viscous forces, is defined as:
\[ Re = \frac{V \cdot L}{\nu} \]
The Schmidt number, representing the ratio of momentum diffusivity to mass diffusivity, is defined as:
\[ Sc = \frac{\nu}{D_{AB}} \]
The Sherwood number is calculated using the Whitaker correlation for flow over a sphere, assuming low mass flux conditions:
\[ Sh = 2 + \left( 0.4 \cdot Re^{0.5} + 0.06 \cdot Re^{2/3} \right) \cdot Sc^{0.4} \cdot \left( \frac{\mu}{\mu_s} \right)^{0.25} \]
Finally, the mass transfer coefficient (kc) is extracted from the Sherwood number definition:
\[ k_c = \frac{Sh \cdot D_{AB}}{L} \]
| Parameter |
Condition / Threshold |
Validity Range |
| Reynolds Number (Re) |
Whitaker Correlation Limit |
3.5 < Re < 80,000 |
| Schmidt Number (Sc) |
Whitaker Correlation Limit |
0.6 ≤ Sc ≤ 400 |
| Mass Flux |
Assumption of Dilute Concentration |
Low mass flux (μ/μs ≈ 1) |
Dimensionless groups allow process engineers to maintain geometric, kinematic, and dynamic similarity between laboratory-scale experiments and full-scale industrial reactors. By using these ratios, you can predict performance without needing to replicate the exact physical size of the equipment. Key benefits include:
- Reducing the number of variables required for experimental correlation.
- Enabling the extrapolation of data across different fluid properties and flow regimes.
- Providing a standardized framework to compare mass transfer efficiency across diverse unit operations.
Worked Example: Mass Transfer Coefficient for Air Flow Over a Sphere
In a process engineering application, such as the drying of a spherical catalyst pellet in an air stream, it is essential to determine the mass transfer coefficient to design efficient mass transfer equipment. Consider air flowing over a solid sphere under the following conditions:
- Fluid temperature: \( T = 25.0 \, ^\circ\mathrm{C} \) ( \( = 298.15 \, \mathrm{K} \) )
- Pressure: \( P = 1.0 \, \mathrm{atm} \)
- Air velocity: \( V = 2.0 \, \mathrm{m/s} \)
- Sphere diameter: \( D = 0.05 \, \mathrm{m} \)
- Diffusivity of species in air: \( D_{AB} = 2.6 \times 10^{-5} \, \mathrm{m}^2/\mathrm{s} \)
- Kinematic viscosity of air: \( \nu = 1.56 \times 10^{-5} \, \mathrm{m}^2/\mathrm{s} \)
- Viscosity ratio: \( \mu / \mu_s \approx 1.0 \) (assumed for low mass flux)
- Calculate the Reynolds number (\( Re \)):
The Reynolds number is defined as \( Re = \frac{V D}{\nu} \).
Using the known values: \( V = 2.0 \, \mathrm{m/s} \), \( D = 0.05 \, \mathrm{m} \), and \( \nu = 1.56 \times 10^{-5} \, \mathrm{m}^2/\mathrm{s} \), the calculated Reynolds number is \( Re = 6410.256 \).
- Calculate the Schmidt number (\( Sc \)):
The Schmidt number is given by \( Sc = \frac{\nu}{D_{AB}} \).
With \( \nu = 1.56 \times 10^{-5} \, \mathrm{m}^2/\mathrm{s} \) and \( D_{AB} = 2.6 \times 10^{-5} \, \mathrm{m}^2/\mathrm{s} \), the Schmidt number is \( Sc = 0.6 \).
- Calculate the Sherwood number (\( Sh \)) using the Whitaker correlation:
For flow over a sphere, the Whitaker correlation is \( Sh = 2 + \left[0.4 Re^{1/2} + 0.06 Re^{2/3}\right] Sc^{0.4} (\mu/\mu_s)^{1/4} \).
Given \( Re = 6410.256 \), \( Sc = 0.6 \), and \( \mu/\mu_s = 1.0 \), the term \( 0.4 Re^{1/2} + 0.06 Re^{2/3} \) is approximately \( 52.730 \), and the calculated Sherwood number is \( Sh = 44.985 \).
- Calculate the mass transfer coefficient (\( k_c \)):
The mass transfer coefficient is derived from \( Sh = \frac{k_c L}{D_{AB}} \), with characteristic length \( L = D \). Thus, \( k_c = \frac{Sh \cdot D_{AB}}{D} \).
Plugging in \( Sh = 44.985 \), \( D_{AB} = 2.6 \times 10^{-5} \, \mathrm{m}^2/\mathrm{s} \), and \( D = 0.05 \, \mathrm{m} \), the mass transfer coefficient is \( k_c = 0.023 \, \mathrm{m/s} \).
Final Answer: The mass transfer coefficient for the system is \( k_c = 0.023 \, \mathrm{m/s} \).
