About the Mass Transfer Coefficient

The mass transfer coefficient (often denoted as k, k_c, or k_L) is a crucial parameter in chemical engineering that quantifies the rate at which a substance moves from one phase to another (e.g., from a gas bubble into a liquid). This Mass Transfer Coefficient calculator is designed to provide estimations based on various theoretical models and established empirical correlations.

What This Calculator Does

This tool estimates the mass transfer coefficient by implementing several widely recognized models and correlations. Depending on the physical situation you are analyzing, you can select the most appropriate model:

• Fundamental Theories: Includes Film Theory, Penetration Theory (Higbie), and Surface Renewal Theory (Danckwerts) to provide a theoretical basis for mass transfer.
• Geometric Correlations: Provides calculations for common engineering geometries, including flow over a flat plate (both laminar and turbulent), turbulent flow inside a pipe, flow in packed beds, and around single spheres or bubbles.
• System-Specific Models: Includes correlations for agitated or stirred tanks and the Two-Film Theory for calculating overall coefficients when resistance exists in both gas and liquid phases.

When to Use It

The mass transfer coefficient is essential in the design, analysis, and optimization of processes involving interphase mass transfer. Common applications include:

• Chemical Reactors: Designing gas-liquid reactors where a gaseous reactant must dissolve into a liquid to react.
• Separation Processes: Sizing equipment for distillation, absorption (like gas scrubbers), and extraction.
• Environmental Engineering: Modeling the aeration of water bodies or the volatilization of pollutants.
• Bioprocessing: Ensuring adequate oxygen supply to microorganisms in fermenters.

Inputs Explained

• Diffusivity (D_AB): The rate at which molecules of substance A diffuse through substance B. It is a property of the specific chemical pair and conditions (e.g., O₂ in water). Units: m²/s.
• Fluid Density (ρ): The mass per unit volume of the primary fluid phase. Units: kg/m³.
• Fluid Viscosity (μ): The fluid's resistance to flow (dynamic viscosity). Units: Pa·s.
• Fluid Velocity (v): The speed of the fluid relative to the surface or in the system. Units: m/s.
• Characteristic Length (L or D): A representative dimension of the system, such as the plate length or pipe diameter. Units: m.
• Film Thickness (δ): A hypothetical stagnant layer thickness used in Film Theory. Units: m.
• Exposure Time (t_c): The average time a fluid element is at the interface in Penetration Theory. Units: s.
• Surface Renewal Rate (s): The rate at which the interface surface is replaced with fresh fluid in Surface Renewal Theory. Units: s⁻¹.
• Henry's Constant (H): A dimensionless constant relating the concentration of a species in the gas phase to its concentration in the liquid phase at equilibrium.

Results Explained

• Mass Transfer Coefficient (k, k_c, k_L, k_G): The primary output, representing the velocity of mass transfer. Its units are typically m/s. The subscript indicates the phase or basis (c for concentration, L for liquid, G for gas).
• Overall Coefficients (K_L, K_G): Calculated using the Two-Film Theory, these combine the resistances of both the liquid and gas phases into a single overall coefficient.
• Dimensionless Numbers (Re, Sc, Sh):
  • Reynolds Number (Re): Indicates the flow regime (laminar or turbulent).
  • Schmidt Number (Sc): Relates momentum diffusivity (viscosity) to mass diffusivity.
  • Sherwood Number (Sh): A dimensionless mass transfer coefficient, representing the ratio of convective to diffusive mass transport.

Formula / Method

The calculator uses a library of established formulas. The choice of formula depends on the selected model:

• Film Theory: k = D_AB / δ
• Penetration Theory: k = 2 * √(D_AB / (π * t_c))
• Surface Renewal Theory: k = √(D_AB * s)
• Flow Over Flat Plate (Laminar): Sh = 0.664 * Re^0.5 * Sc^(1/3)
• Flow Over Flat Plate (Turbulent): Sh = 0.037 * Re^0.8 * Sc^(1/3)
• Flow in Pipe (Turbulent): Sh = 0.023 * Re^0.83 * Sc^0.33
• Two-Film Theory (Overall Liquid): 1/K_L = 1/k_L + 1/(H * k_G)

In geometry-based correlations, the final mass transfer coefficient k_c is found by rearranging the Sherwood number definition: k_c = (Sh * D_AB) / L, where L is the characteristic length.

Step-by-Step Example

Let's calculate the mass transfer coefficient for air flowing over a 0.5 m long wet surface (e.g., water evaporating into air).

1. Select Model: Choose "Geometry: Flow Over Flat Plate (Turbulent)" assuming high air speed.
2. Gather Inputs (for air at 25°C, 1 atm):
   • Diffusivity (D_AB of water in air): 2.6e-5 m²/s
   • Density (ρ of air): 1.184 kg/m³
   • Viscosity (μ of air): 1.85e-5 Pa·s
   • Velocity (v): 5 m/s
   • Length (L): 0.5 m
3. Intermediate Calculations:
   • Reynolds Number (Re): (1.184 * 5 * 0.5) / 1.85e-5 = 1.6e5.

     Wait, this Reynolds number is less than 5e5. We must use the Laminar model instead.

4. Recalculate with Laminar Model:
   • Select Model: Choose "Geometry: Flow Over Flat Plate (Laminar)".
   • Re-calculate Re: Re is still 1.6e5, which is valid for laminar flow.
   • Schmidt Number (Sc): 1.85e-5 / (1.184 * 2.6e-5) = 0.60
   • Sherwood Number (Sh): 0.664 * (1.6e5)^0.5 * (0.60)^(1/3) = 0.664 * 400 * 0.843 = 224
5. Final Result:
   • Mass Transfer Coeff. (k_c): (224 * 2.6e-5) / 0.5 = 0.0116 m/s

Tips + Common Errors

• Unit Consistency: Ensure all inputs are in standard SI units (meters, seconds, kilograms, etc.) as shown in the input fields. The most common error is mixing units (e.g., using centimeters for length).
• Select the Correct Model: The choice of model is critical. Use fundamental theories for idealized cases and geometric correlations for real-world engineering systems.
• Check the Flow Regime: For flat plate and pipe flow, the Reynolds number dictates whether the flow is laminar or turbulent. The calculator will warn you if you select a model inconsistent with the calculated Re, as seen in the example.
• Physical Properties: Use accurate values for density, viscosity, and diffusivity at the system's temperature and pressure. These properties can vary significantly with conditions.
• Assumptions Matter: Be aware of the assumptions behind each model. For example, the Frössling correlation is for rigid spheres and may be less accurate for oscillating liquid drops.

Frequently Asked Questions (FAQs)

What is the difference between k_c, k_L, and k_G?

They are all mass transfer coefficients. The subscript denotes the basis for the driving force. k_c is a general coefficient based on concentration differences. k_L specifically refers to the liquid-side coefficient, and k_G to the gas-side coefficient, often used together in the Two-Film Theory.

How do I choose between Film, Penetration, and Surface Renewal theories?

Film theory is the simplest, assuming a stagnant layer, best for systems with low turbulence. Penetration and Surface Renewal theories are more dynamic, accounting for fluid elements being exposed to the interface for a period of time, making them better for turbulent or agitated systems.

What is the Sherwood number (Sh) and why is it important?

The Sherwood number is the dimensionless form of the mass transfer coefficient. It represents the ratio of convective mass transfer to diffusive mass transfer. It's important because correlations for complex geometries are almost always expressed in terms of dimensionless numbers (Sh, Re, Sc), making them universally applicable regardless of scale.

What if my Reynolds number is in the transitional range (e.g., 2100 < Re < 4000 for pipes)?

The calculator uses correlations strictly for laminar or fully turbulent flow. The transitional regime is complex and less predictable. For calculations in this range, more advanced correlations or a conservative choice (often using the turbulent correlation) is required, but results should be treated with caution.

Why does the calculator use dimensionless numbers?

Dimensionless numbers like Re, Sc, and Sh allow engineers to apply experimental results from a small-scale model to a large-scale industrial process. They simplify complex problems by grouping variables into meaningful ratios.

Can I use this tool for mass transfer between two liquids?

Yes. The geometric correlations (e.g., for stirred tanks or single drops) can be applied to liquid-liquid systems. You would need to use the properties (density, viscosity, diffusivity) relevant to the phase where the main resistance to mass transfer occurs.

What is the "controlling resistance" in the Two-Film model?

In a gas-liquid system, mass transfer faces resistance in both the gas film and the liquid film. One of these resistances is usually much larger than the other and is called the "controlling resistance." For highly soluble gases (low H), the gas film often controls. For poorly soluble gases (high H, like oxygen in water), the liquid film controls.

How do I find the molecular diffusivity (D_AB) for my system?

Diffusivity values can be found in engineering handbooks, scientific literature, or estimated using correlations like the Wilke-Chang equation (for liquids) or the Chapman-Enskog theory (for gases).

References

• Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2006). Transport Phenomena (2nd ed.). John Wiley & Sons.
• Welty, J. R., Rorrer, G. L., & Foster, D. G. (2014). Fundamentals of Momentum, Heat, and Mass Transfer (6th ed.). John Wiley & Sons.
• Geankoplis, C. J. (2003). Transport Processes and Separation Process Principles (4th ed.). Prentice Hall.
• Perry, R. H., & Green, D. W. (2008). Perry's Chemical Engineers' Handbook (8th ed.). McGraw-Hill.