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Mass transfer coefficient

The mass transfer coefficient is a fundamental parameter in transport phenomena, particularly in chemical and biological engineering, that quantifies the rate at which a species diffuses or convects across a phase boundary or within a fluid due to a concentration gradient.^[1] It serves as a proportionality constant linking the molar flux of the species to the driving force provided by the difference in concentrations between the interface and the bulk fluid, simplifying the analysis of complex diffusive and convective processes.^[2] Analogous to the heat transfer coefficient in thermal systems, it accounts for the resistance to mass transfer near interfaces, such as in gas-liquid absorption or solid-liquid dissolution, and is influenced by factors like fluid velocity, geometry, and physical properties.^[2]^[1] Mathematically, the local mass transfer coefficient k_C (with units of length per time, such as m/s) is defined by the equation N_A = k_C (c_{A,s} - c_{A,\infty}), where N_A is the molar flux of species A (moles per area per time), c_{A,s} is the concentration at the surface or interface, and c_{A,\infty} is the bulk concentration far from the boundary.^[2] This linear driving force model assumes a thin "film" layer where diffusion dominates, though actual profiles may vary with flow conditions.^[2] In convective scenarios, the coefficient depends on hydrodynamic boundary layers and is often determined experimentally or via dimensionless correlations like the Sherwood number (Sh = k_C L / D_{AB}, where L is a characteristic length and D_{AB} is the diffusivity).^[1] For interphase mass transfer, such as in gas-liquid contactors, overall mass transfer coefficients (e.g., K_L or K_G) are employed to incorporate resistances from both phases, expressed as N_A = K_L (c_{A,L}^* - c_{A,L}) for the liquid phase or N_A = K_G (p_{A,G} - p_{A,G}^*) for the gas phase, where starred terms denote equilibrium concentrations or partial pressures.^[3] These overall coefficients relate to individual phase coefficients through resistances in series, for instance, \frac{1}{K_L} = \frac{1}{k_L} + \frac{1}{H k_G} (with H as Henry's law constant), enabling design of unit operations like distillation columns or bioreactors.^[3] Applications span environmental engineering (e.g., pollutant removal from air or water), biomedical processes (e.g., oxygen delivery in tissues), and industrial separations, where accurate prediction of coefficients enhances efficiency and safety.^[1]^[3]

Fundamentals

Definition

The mass transfer coefficient, often denoted as k, is defined as the proportionality constant relating the total molar flux of a species N_A to the driving force given by the concentration difference across an interface, expressed as

N_A = k (c_{A,s} - c_{A,b}),

where c_{A,s} is the concentration of species A at the surface and c_{A,b} is the bulk concentration.^[2] This formulation quantifies the rate of convective mass transfer in processes where species move from regions of higher to lower concentration due to bulk fluid motion.^[1] The concept draws a direct analogy to the heat transfer coefficient in Newton's law of cooling, where the heat flux is proportional to the temperature difference, and similarly to the friction factor in momentum transfer, enabling unified approaches across transport phenomena.^[1] Unlike pure molecular diffusion governed solely by Fick's law, the mass transfer coefficient k incorporates both diffusive transport and convective effects from fluid velocity gradients and turbulence. The mass transfer coefficient was first introduced by W.G. Whitman in 1923 to describe absorption at gas-liquid interfaces, laying the groundwork for modeling interphase transport.^[4] In dimensionless terms, it is often expressed through the Sherwood number, which scales k by a characteristic length and diffusivity.

Units and dimensions

The mass transfer coefficient possesses dimensions of velocity, = [L]/, arising from its definition in the flux equation where the molar flux (mol m^{-2} s^{-1}) is proportional to the concentration driving force difference (mol m^{-3}), yielding = (mol m^{-2} s^{-1}) / (mol m^{-3}) = m s^{-1}.^[5]^[6] For liquid-phase or concentration-based coefficients such as k_c or k_L, the standard units are meters per second (m/s), reflecting the velocity-like nature of the coefficient in convective mass transfer contexts.^[6]^[5] In gas-phase applications driven by partial pressure differences, the coefficient k_G adopts units of mol m^{-2} s^{-1} Pa^{-1} or equivalently mol m^{-2} s^{-1} atm^{-1}, accounting for the pressure-based driving force (Pa or atm).^[5]^[3] Variations occur depending on the chosen driving force and phase; for instance, mole fraction-based coefficients like k_x or k_y also carry units of m/s when multiplied by total concentration, maintaining dimensional consistency.^[5] In interphase mass transfer under the two-film theory, overall coefficients such as K_L (liquid-based, m/s) or K_G (gas-based, mol m^{-2} s^{-1} atm^{-1}) describe combined resistances, while volumetric forms like k_L a or K_G a (where a is the interfacial area per unit volume, m^{-1}) yield units of s^{-1} for applications in absorbers or reactors.^[3] Conversions between these forms in gas-liquid systems rely on Henry's law constant H, which relates equilibrium partial pressure and concentration (p = H c). The liquid-phase coefficient k_L connects to the gas-phase k_G through such relations, for example, in overall resistance expressions like \frac{1}{K_G} = \frac{1}{k_G} + \frac{H}{k_L}, allowing transformation based on phase-specific resistances and H's value (typically in atm m³ mol^{-1}).^[3]^[5]

Theoretical models

Film theory

The film theory of mass transfer was developed by Lewis and Whitman in 1924 to describe gas absorption processes, proposing a model where mass transfer across a phase interface occurs through a hypothetical stagnant boundary layer. This classical approach conceptualizes a thin, unmoving film adjacent to the interface, with the bulk fluid assumed to be well-mixed and at uniform concentration. Within this film, solute transport is governed exclusively by molecular diffusion, leading to a linear concentration profile. The key relation derived from the model equates the liquid-side mass transfer coefficient k_L to the solute diffusivity D divided by the film thickness \delta:

k_L = \frac{D}{\delta}

This expression arises from a steady-state diffusion analysis. The molar flux N_A of species A across the film, based on Fick's law, is given by

N_A = \frac{D (c_{A,i} - c_{A,b})}{\delta},

where c_{A,i} is the interfacial concentration and c_{A,b} is the bulk concentration. By the definition of the mass transfer coefficient, N_A = k_L (c_{A,i} - c_{A,b}), it follows directly that k_L = D / \delta. The film thickness \delta is treated as a characteristic parameter influenced by hydrodynamics, though not explicitly derived in the original formulation. The model rests on several assumptions: mass transfer is steady-state with no accumulation within the film; the film thickness \delta remains constant; and convective effects are absent inside the film, confining transport to pure diffusion. These simplifications, while enabling analytical tractability, introduce limitations. Notably, the theory predicts mass transfer rates independent of contact time between phases, which contradicts experimental observations showing time dependence in unsteady systems. Additionally, it tends to overpredict coefficients at low fluid velocities where the steady-state film assumption inadequately captures boundary layer dynamics, and it implies a linear dependence of k_L on D, whereas data often indicate a weaker proportionality like D^{0.5} to D^{0.75}. Despite these shortcomings, the film theory provides a foundational framework for understanding interfacial resistance and underpins extensions like the two-film model for overall transfer coefficients.

Penetration and surface renewal theories

Penetration theory, proposed by Higbie in 1935, models mass transfer as an unsteady-state diffusion process where fluid elements at the gas-liquid interface are exposed to the solute for a fixed contact time t_e.^[7] This approach assumes that fresh fluid elements arrive at the interface, absorb solute via molecular diffusion perpendicular to the surface, and then depart after t_e, without mixing during exposure. The theory solves the one-dimensional unsteady diffusion equation

\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2},

with boundary conditions of constant surface concentration and initial bulk concentration, yielding an instantaneous mass flux described by the error function solution.^[7] The average mass transfer coefficient over the exposure time is then

k_L = 2 \sqrt{\frac{D}{\pi t_e}},

which highlights the square-root dependence on diffusivity D, reflecting the transient nature of penetration.^[7] This model is particularly applicable to systems with uniform contact times, such as gas absorption in bubble columns or droplet flows. Surface renewal theory, developed by Danckwerts in 1951, extends Higbie's framework to account for random fluid element replacement at the interface in turbulent flows.^[8] Here, fluid elements are renewed at a constant average rate s, with exposure times following an exponential probability distribution f(t) = s e^{-s t} for t \geq 0. Integrating the penetration flux over this distribution yields the average mass transfer coefficient

k_L = \sqrt{D s},

again showing proportionality to \sqrt{D}, but independent of specific exposure times due to the statistical averaging.^[8] This theory better suits highly turbulent conditions, such as in packed columns or stirred tanks, where surface elements are irregularly swept away and replaced.^[9] The primary difference between penetration and surface renewal theories lies in their treatment of exposure: Higbie's assumes uniform t_e for orderly flows like rising bubbles, while Danckwerts' incorporates stochastic renewal via s for chaotic turbulence. Both models predict k_L \propto \sqrt{D}, contrasting with the linear dependence in steady-state film models. Despite their advantages in capturing unsteadiness, these theories have limitations: they overpredict transfer rates for prolonged contact times, as the models idealize short exposures and neglect eventual steady-state gradients; additionally, t_e or s must be estimated from hydrodynamics, introducing uncertainty without direct measurement.^[9]

Dimensionless formulation

Sherwood number

The Sherwood number (Sh) is a dimensionless parameter that characterizes the mass transfer coefficient in convective processes, defined as Sh = \frac{k L}{D}, where k is the mass transfer coefficient, L is a characteristic length scale (such as the pipe diameter in flow systems or the particle diameter in packed beds), and D is the molecular diffusivity of the transferring species.^[10] This formulation renders the mass transfer coefficient independent of specific units, facilitating scaling and comparison across different systems. The Sherwood number is analogous to the Nusselt number in heat transfer, replacing thermal conductivity with diffusivity and heat transfer coefficient with mass transfer coefficient. Physically, the Sherwood number represents the ratio of the rate of convective mass transport to the rate of diffusive mass transport across the boundary layer, quantifying the enhancement of mass transfer due to fluid motion over pure diffusion.^[11] The minimum value of Sh for a given geometry (e.g., Sh = 2 for diffusion from an isolated sphere in stagnant fluid) indicates mass transfer dominated solely by diffusion, without convection.^[10] Higher values reflect increasing influence of convection, which thins the concentration boundary layer and accelerates transfer.^[10] The Sherwood number can be expressed locally or as an average over a surface. The local Sherwood number, Sh_x = \frac{k_x L}{D}, varies with position x along the surface due to developing flow and concentration profiles. The average Sherwood number over a length L is then given by Sh_m = \frac{1}{L} \int_0^L Sh_x \, dx, providing an overall measure for engineering calculations. In scenarios involving dissolving solids or evaporating liquids under constant surface concentration conditions, the Sherwood number approximates the inverse of the dimensionless concentration boundary layer thickness, Sh \approx \frac{L}{\delta_c}, where \delta_c is the thickness of the layer over which the concentration gradient occurs.^[11] This relation highlights how convection reduces \delta_c, thereby increasing the mass transfer rate.

Mass transfer correlations

Mass transfer correlations provide empirical and semi-empirical relations to predict the Sherwood number (Sh) as a function of the Reynolds number (Re), which quantifies the ratio of inertial to viscous forces, and the Schmidt number (Sc), which quantifies the ratio of momentum diffusivity to mass diffusivity, typically in the form Sh = f(Re, Sc). These correlations are derived from experimental data and theoretical analogies, enabling the estimation of mass transfer coefficients in various flow regimes without direct measurement.^[5] A classic example for laminar boundary layer flow over a flat plate is the local Sherwood number, expressed as:

\text{Sh}_x = 0.332 \text{Re}_x^{1/2} \text{Sc}^{1/3}

This relation arises from the similarity solution to the convective diffusion equation and applies to dilute systems with constant properties. For the average Sherwood number over the plate length, it doubles to 0.664 Re_L^{1/2} Sc^{1/3}.^[5] In turbulent pipe flow, a widely used correlation is:

\text{Sh} = 0.023 \text{Re}^{0.8} \text{Sc}^{1/3}

valid for 0.6 < Sc < 3000 and Re > 10,000, assuming fully developed flow and smooth walls. This form stems from the Chilton-Colburn analogy, which links mass transfer to momentum and heat transfer by defining the mass transfer j-factor as j_D = Sh / (Re Sc^{1/3}) ≈ f/2, where f is the Fanning friction factor; the analogy holds well for both gases (Sc ≈ 1) and liquids (Sc >> 1) in turbulent regimes.^[5] Correlations are also influenced by surface roughness, which enhances transfer in turbulent flows by promoting eddy mixing, and by geometry; for instance, in low-Re external flow around spheres, the Frössling correlation gives:

\text{Sh} = 2 + 0.6 \text{Re}^{1/2} \text{Sc}^{1/3}

for 2 < Re < 800 and 0.6 < Sc < 2.7, where the 2 accounts for pure diffusion at Re = 0.^[6]

Determination and applications

Experimental measurement

Experimental determination of the mass transfer coefficient typically involves controlled laboratory setups where the flux of a species is measured under well-defined conditions, allowing inference of the coefficient from fundamental relations like the convective flux equation N_A = k \Delta c, where N_A is the molar flux, k is the mass transfer coefficient, and \Delta c is the concentration driving force.^[12] These methods prioritize isolating mass transfer from other processes, such as reaction kinetics, to ensure accuracy.^[13] In solid-liquid systems, the dissolution method employs sparingly soluble solids, such as benzoic acid, immersed in a flowing liquid; the rate of mass loss is monitored gravimetrically or volumetrically to compute k from N_A = k (c_s - c_b), where c_s is the saturation concentration at the surface and c_b is the bulk concentration.^[14] This technique is widely used for its simplicity and applicability to pipeline or stirred tank hydrodynamics, with experiments often conducted at controlled Reynolds numbers to mimic flow conditions.^[15] Electrochemical methods complement this by measuring the limiting current density i_L = n F k c_b for a reversible reduction or oxidation reaction, where n is the number of electrons, F is Faraday's constant, and c_b is the bulk concentration of the electroactive species; ferricyanide/ferrocyanide couples are common due to their fast kinetics.^[16] These approaches enable precise quantification of k in the range of $10^{-5} to $10^{-4} m/s for typical aqueous systems.^[17] For controlled hydrodynamics in flow systems, the rotating disk electrode (RDE) provides a benchmark; a disk electrode rotates in an electrolyte, generating a well-defined boundary layer, and k is derived from the Levich equation:

k = 0.62 D^{2/3} \omega^{1/2} \nu^{-1/6}

where D is the diffusion coefficient, \omega is the angular rotation speed, and \nu is the kinematic viscosity.^[18] Voltammetric measurements at varying \omega yield i_L, from which k is extracted, offering reproducibility for validating transport models in laminar and turbulent regimes.^[19] In gas-liquid systems, wetted-wall columns facilitate measurement of gas-phase coefficients k_G by absorbing a soluble gas, such as ammonia or CO₂, into a thin liquid film flowing down a vertical tube; the absorption rate is determined from inlet/outlet gas compositions, and the average k_G is calculated using the log-mean driving force \Delta p_{LM}, via N_A = k_G \Delta p_{LM}. This setup minimizes interfacial area uncertainties and is effective for Reynolds numbers up to 2000, yielding k_G values on the order of $10^{-6} to $10^{-5} mol/(m²·s·Pa).^[20] Experimental measurements are subject to uncertainties from end effects, such as non-uniform flow at column entrances or electrode edges, which can inflate k by 10-20%; corrections involve extrapolating data or using guard rings.^[21] Natural convection interference, particularly in low-flow dissolution or absorption setups, introduces buoyancy-driven transport that overestimates k if not suppressed by sufficient forced flow (e.g., Re > 1000).^[14] Validation against established correlations, like those for pipe flow, confirms reliability, with typical uncertainties in k ranging from 5-15% after accounting for these factors.^[17]

Industrial applications

In gas absorption processes, packed towers are commonly employed to facilitate the transfer of soluble gases into a liquid absorbent, where the overall mass transfer coefficient K_G a is determined using the height equivalent to a theoretical plate (HETP) for column design and performance evaluation.^[22] Turbulence induced by the packing material enhances the gas-phase mass transfer coefficient k_G by promoting better mixing and reducing boundary layer thickness at the gas-liquid interface.^[23] This approach is critical in industrial operations such as CO₂ capture and flue gas desulfurization, allowing for efficient scaling of absorber height based on required separation efficiency.^[24] In distillation and liquid-liquid extraction, the liquid-phase mass transfer coefficient k_L governs solute transfer rates in countercurrent column operations, particularly for systems with significant liquid-side resistance.^[23] Column design relies on the height of a transfer unit (HTU), expressed as \text{HTU} = G / (k_G a), where G is the gas molar flow rate, to predict the required packing height for achieving desired separations. These principles are applied in petrochemical refining for hydrocarbon fractionation and in pharmaceutical extraction to isolate active compounds from aqueous feeds, optimizing throughput and energy use. Convective drying and evaporation processes utilize the mass transfer coefficient to model moisture removal rates, given by m = k (\rho_s - \rho_\infty), where m is the drying rate, k is the coefficient, \rho_s is the vapor density at the surface, and \rho_\infty is the ambient vapor density. This formulation influences dryer sizing in industries such as food processing and pulp production, where airflow velocity and temperature gradients dictate the transition from constant-rate to falling-rate drying periods. In biological and environmental applications, the mass transfer coefficient is essential for oxygen supply in membrane bioreactors (MBRs), where k quantifies oxygen transfer across membranes to support microbial growth in wastewater treatment.^[25] For wastewater aeration, typical volumetric mass transfer coefficients k_L a range from 0.01 to 0.1 s⁻¹, influenced by biomass concentration and aeration intensity, enabling efficient pollutant degradation in activated sludge systems.^[25] Enhancements to the mass transfer coefficient are achieved through structured packings and agitators, which can increase k by 10- to 100-fold by maximizing interfacial area and inducing turbulence in gas-liquid and liquid-liquid contactors.^[22] In packed towers, random or structured packings like Raschig rings or Mellapak elevate effective area and k_G via improved wetting and flow distribution, while agitators in mixer-settlers boost k_L through intensified dispersion and reduced drop size in extraction columns.^[26] These modifications are pivotal for scaling industrial processes, reducing equipment volume while maintaining high transfer rates.^[23]

References

[1]
Mass Transfer — Introduction to Chemical and Biological Engineering
Mass transfer is mass in transit due to a species concentration gradient in a mixture. By concentration gradient, we mean a spatial difference in the abundance ...
[2]
[PDF] Mass Transfer: Definitions and Fundamental Equations - Research
Then one way to define a mass transfer coefficient kC (units: length/time) is as follows,. NA = kC (cAS - cA∞). (57). NA is the molar mass transfer of A ...
[3]
[PDF] Overall mass transfer coefficients - Michigan Technological University
May 1, 2021 · The defining equations for the mass-transfer coefficients: Bulk convection present- Linear-driving-force model. 30. © Faith A. Morrison ...
[4]
[PDF] The Two-Film Theory of Gas Absorption
WHITMAN. Assistant Professor of Chemical Enいneering. Massachusetts Institute of Technology various theories for the mechanism of gas he general form for.
[5]
None
### Summary of Convective Mass Transfer Sections
[6]
[PDF] All about mass transfer coefficients
Mass transfer coefficients for various solute concentration units. Correlations generally given for , which has the dimensions of a velocity (units e.g. m/s) ...
[7]
The Rate of Absorption of a Pure Gas into a Still Liquid during Short ...
The Rate of Absorption of a Pure Gas into a Still Liquid during Short Periods of Exposure · R. Higbie · Published 1935 · Environmental Science, Physics.
[8]
Significance of Liquid-Film Coefficients in Gas Absorption
https://doi.org/10.1021/ie50498a055. Published June 1, 1951. Publication History. Published. online 1 May 2002. Published. in issue 1 June 1951. research- ...
[9]
Mass Transfer in Multiphase Systems | IntechOpen
Oct 22, 2015 · The surface renewal theory, developed by Danckwerts [5], applies mathematics of the penetration theory to a more plausible situation, where the ...
[10]
Sherwood (Sh) Number in Chemical Engineering Applications—A ...
Aug 30, 2024 · This paper reviews a series of cases for which the correct determination of the mass transfer coefficient is decisive for an appropriate design of the system.
[11]
Sherwood Number - an overview | ScienceDirect Topics
Under flow conditions, Sherwood number ( S h ) is usually used to describe the dimensionless length of diffusion boundary layer, which is defined by S h = L ∕ l ...
[12]
Mass transfer at solid dissolution - ScienceDirect.com
An experimental method is presented for the determination of the individual mass transfer coefficient at solid dissolution and of the solid dissolution rate.
[13]
Calculation of the Mass Transfer Coefficient for the Dissolution of ...
Dec 13, 2019 · In this study, the mass transfer coefficient was derived experimentally while changing the process variables that affect mass transfer.
[14]
[PDF] Drawbacks of the Dissolution Method for Measurement of the Liquid ...
Apr 10, 2023 · The dissolution method was used predominantly at atmospheric conditions to measure the liquid-solid mass-transfer coefficient in two-phase flow ...
[15]
A Laboratory Experiment for Measuring Solid-Liquid Mass Transfer ...
A mass transfer laboratory experiment based on candy dissolution is described. The dissolution of 1–4 spherical pieces of candy was followed ...Missing: methods | Show results with:methods
[16]
Measurement of mass transfer coefficients with an electrochemical ...
Oxidation reduction potential measurements are used to determine the limiting current density. A mathematical model taking into account the diffusion and ...
[17]
Mass-Transfer Measurements by the Limiting-Current Technique
The present paper discusses the underlying principles of electrochemical microfabrication processes. The important role of mass transport and current ...
[18]
Rotating Disk Electrodes beyond the Levich Approximation: Physics ...
Aug 17, 2023 · The Levich approach uses an approximate equation describing the solution velocity as a function of the distance away from the electrode under ...Introduction · Theory · Results and Discussion · Conclusions
[19]
[PDF] Determination of Mass Transfer Coefficient Using a RDE - vscht.cz
This equation is called the Levich equation. Note that the limiting current value depends on the square root of the RDE's rotation speed. [ ]. Ac.<|separator|>
[20]
Mass transfer in wetted-wall columns: Correlations at high Reynolds ...
The rate of gas- and liquid-phase mass transport in a pilot-scale wetted-wall column with an inner diameter of 3.26 cm and a length of 5 m was investigated.
[21]
A Review on Gas-Liquid Mass Transfer Coefficients in Packed-Bed ...
This review provides a thorough analysis of the most famous mass transfer models for random and structured packed-bed columns used in absorption/stripping ...
[22]
[PDF] Mass-Transfer-Correlations-da-Perrys-8th-ed.pdf
Includes solid-liquid mass-transfer data to find s coefficient on NSc. May use NRe. 0.83. Use for liquids. See also Table 5-19. [E] ε . = dilation ...
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None
### Summary of Mass Transfer Coefficients for Oxygen in Membrane Bioreactors and Wastewater Aeration
[25]
https://dspace.lib.cranfield.ac.uk/server/api/core/bitstreams/8b818765-403c-4f26-9dc8-1829062308f9/content