Mass Transfer Coefficient Estimation in Cross-Flow
Reference ID: MET-D76D | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The estimation of the mass transfer coefficient is a fundamental task in process engineering, particularly in the design and optimization of separation equipment such as membrane modules, heat exchangers, and packed bed reactors. The mass transfer coefficient, kL, quantifies the rate at which a solute species moves across a phase boundary or through a fluid medium under a concentration gradient. Accurate determination of this parameter is critical for predicting flux, sizing equipment, and ensuring the efficiency of mass transfer operations.
Methodology & Formulas
The calculation follows a dimensionless analysis approach, utilizing the Reynolds number (Re) to characterize the flow regime and the Schmidt number (Sc) to relate momentum and mass diffusivity. The Sherwood number (Sh) is then determined via empirical correlations specific to the system geometry and flow conditions.
The fundamental dimensionless groups are defined as follows:
\[ Re = \frac{\rho \cdot u \cdot L}{\mu} \]
\[ Sc = \frac{\mu}{\rho \cdot D_{AB}} \]
Once the flow regime is identified, the Sherwood number is calculated using the appropriate correlation. For internal turbulent pipe flow, the standard Chilton–Colburn analogy yields:
\[ Sh = 0.023 \cdot Re^{0.8} \cdot Sc^{1/3} \]
Finally, the mass transfer coefficient is derived from the Sherwood number:
\[ k_{L} = \frac{Sh \cdot D_{AB}}{L} \]
Flow Regime
Geometry
Criteria
Internal Turbulent
Circular Pipe
Re ≥ 4000; 0.6 ≤ Sc ≤ 2500
Internal Laminar
Circular Pipe
Re < 2300; Re · Sc · (D/L) > 10
External Laminar
Flat Plate
Re < 5 · 105; Sc > 0.6
External Turbulent
Flat Plate
Re > 5 · 105
To select the correct correlation, you must first characterize your flow regime and geometry. Consider the following factors:
Determine the Reynolds number range to identify if the flow is laminar, transitional, or turbulent.
Assess the Schmidt number to account for the ratio of momentum diffusivity to mass diffusivity.
Verify if the geometry matches standard configurations such as tube banks or packed beds.
Ensure the correlation accounts for the specific orientation of the cross-flow relative to the transfer surface.
The mass transfer coefficient is sensitive to several physical and operational parameters. Key variables include:
Fluid velocity, which dictates the thickness of the boundary layer.
Fluid properties, specifically kinematic viscosity and molecular diffusivity.
Characteristic length of the obstruction or surface in the cross-flow path.
Temperature and pressure, as these affect the physical properties of the fluid stream.
Turbulence promoters significantly enhance mass transfer by disrupting the concentration boundary layer. When estimating the coefficient in these systems, keep the following in mind:
The effective mass transfer coefficient will typically increase compared to smooth surface flow.
You must apply a correction factor or use a specific empirical correlation that accounts for the geometry of the promoter.
Increased pressure drop is a common trade-off that must be balanced against the gain in mass transfer efficiency.
Engineers often encounter errors when applying correlations outside of their validated range. Avoid these common mistakes:
Extrapolating correlations beyond the experimental Reynolds number range provided by the literature.
Neglecting the impact of concentration polarization in high-flux systems.
Using physical property values at bulk conditions when significant temperature gradients exist near the interface.
Failing to account for the entrance effects in short-path cross-flow configurations.
Worked Example: Mass Transfer Coefficient in a Cross-Flow Tubular Membrane
Scenario: A dilute aqueous solution flows inside a tubular membrane module operating in cross-flow. The mass transfer of a solute from the tube wall into the bulk fluid is to be quantified. The flow is turbulent and fully developed. Estimate the liquid-phase mass transfer coefficient kL.
Tube diameter:D = 0.02 m
Average fluid velocity:u = 0.5 m/s
Density:ρ = 1000.0 kg/m3
Dynamic viscosity:μ = 0.001 Pa·s
Solute diffusivity:DAB = 1 × 10-9 m2/s
Compute the Reynolds number (Re):
\[ Re = \frac{\rho \cdot u \cdot D}{\mu} = \frac{1000.0 \times 0.5 \times 0.02}{0.001} = 10000.0 \]
This exceeds 4000, confirming turbulent flow.
Compute the Schmidt number (Sc):
\[ Sc = \frac{\mu}{\rho \cdot D_{AB}} = \frac{0.001}{1000.0 \times 1 \times 10^{-9}} = 1000.0 \]
Sc = 1000 is within the valid range \(0.6 \le Sc \le 2500\).
Select and apply the Sherwood correlation: For turbulent, fully developed internal flow, use the Chilton–Colburn analogy \(Sh = 0.023\,Re^{0.8}\,Sc^{1/3}\).
\[ Sh = 0.023 \times (10000.0)^{0.8} \times (1000.0)^{1/3} = 0.023 \times 1584.9 \times 10.0 = 364.5 \]
The correlation is valid because Re > 4000 and Sc is within the required range.
Solve for the mass transfer coefficient:
\[ k_L = \frac{Sh \cdot D_{AB}}{D} = \frac{364.5 \times 1 \times 10^{-9}}{0.02} = 1.823 \times 10^{-5}\ \text{m/s} \]
Final answer: The estimated liquid-phase mass transfer coefficient is kL = 1.82 × 10-5 m/s. This value lies within the typical range for aqueous liquid systems (10-5 to 10-3 m/s), confirming the result is physically plausible.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
