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Mass transfer
Mass transfer
Mass transfer
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Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction, or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtration, and distillation. Mass transfer is used by different scientific disciplines for different processes and mechanisms. The phrase is commonly used in engineering for physical processes that involve diffusive and convective transport of chemical species within physical systems.

Some common examples of mass transfer processes are the evaporation of water from a pond to the atmosphere, the purification of blood in the kidneys and liver, and the distillation of alcohol. In industrial processes, mass transfer operations include separation of chemical components in distillation columns, absorbers such as scrubbers or stripping, adsorbers such as activated carbon beds, and liquid-liquid extraction. Mass transfer is often coupled to additional transport processes, for instance in industrial cooling towers. These towers couple heat transfer to mass transfer by allowing hot water to flow in contact with air. The water is cooled by expelling some of its content in the form of water vapour.

Astrophysics

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In astrophysics, mass transfer is the process by which matter gravitationally bound to a body, usually a star, fills its Roche lobe and becomes gravitationally bound to a second body, usually a compact object (white dwarf, neutron star or black hole), and is eventually accreted onto it. It is a common phenomenon in binary systems, and may play an important role in some types of supernovae and pulsars.

Chemical engineering

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Mass transfer finds extensive application in chemical engineering problems. It is used in reaction engineering, separations engineering, heat transfer engineering, and many other sub-disciplines of chemical engineering like electrochemical engineering.[1]

The driving force for mass transfer is usually a difference in chemical potential, when it can be defined, though other thermodynamic gradients may couple to the flow of mass and drive it as well. A chemical species moves from areas of high chemical potential to areas of low chemical potential. Thus, the maximum theoretical extent of a given mass transfer is typically determined by the point at which the chemical potential is uniform. For single phase-systems, this usually translates to uniform concentration throughout the phase, while for multiphase systems chemical species will often prefer one phase over the others and reach a uniform chemical potential only when most of the chemical species has been absorbed into the preferred phase, as in liquid-liquid extraction.

While thermodynamic equilibrium determines the theoretical extent of a given mass transfer operation, the actual rate of mass transfer will depend on additional factors including the flow patterns within the system and the diffusivities of the species in each phase. This rate can be quantified through the calculation and application of mass transfer coefficients for an overall process. These mass transfer coefficients are typically published in terms of dimensionless numbers, often including Péclet numbers, Reynolds numbers, Sherwood numbers, and Schmidt numbers, among others.[2][3][4]

Analogies between heat, mass, and momentum transfer

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Main article: Transport phenomena

There are notable similarities in the commonly used approximate differential equations for momentum, heat, and mass transfer.[2] The molecular transfer equations of Newton's law for fluid momentum at low Reynolds number (Stokes flow), Fourier's law for heat, and Fick's law for mass are very similar, since they are all linear approximations to transport of conserved quantities in a flow field. At higher Reynolds number, the analogy between mass and heat transfer and momentum transfer becomes less useful due to the nonlinearity of the Navier–Stokes equation (or more fundamentally, the general momentum conservation equation), but the analogy between heat and mass transfer remains good. A great deal of effort has been devoted to developing analogies among these three transport processes so as to allow prediction of one from any of the others.

References

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See also

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Mass transfer

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from Grokipedia
Mass transfer is the net transport of mass from one location to another within a system, primarily driven by a of a in a . This phenomenon arises in multicomponent systems, such as those involving chemical reactions, separations, or phase changes, where differences in species abundance create a driving force for movement. At its core, mass transfer occurs through two main mechanisms: , which is the molecular-scale random motion resulting in net flux from high to low concentration regions as described by Fick's first law, and , which involves bulk fluid motion carrying species along with it, often quantified using mass transfer coefficients analogous to . These processes can occur within a single phase (intraphase) or across phase boundaries (interphase), such as gas-liquid or liquid-solid interfaces. In , mass transfer is a foundational that governs the of numerous unit operations essential for . Key applications include for separating liquid mixtures by and , absorption for capturing solutes from gases into liquids (e.g., removing CO₂ from flue gases), liquid-liquid extraction for partitioning components between immiscible solvents, to remove moisture from solids, and humidification or in gas-liquid contactors. These operations rely on controlled mass transfer rates to achieve desired separations, often modeled using , penetration theory, or surface renewal models to predict coefficients and optimize equipment design like packed columns or tray towers. Beyond traditional separations, mass transfer principles extend to biological systems (e.g., oxygen in tissues) and emerging technologies like separations and . The study of mass transfer draws strong analogies to heat and momentum transfer, sharing similar mathematical frameworks—such as the diffusion equation and boundary layer concepts—allowing unified transport phenomena analysis via dimensionless numbers like the Sherwood, Schmidt, and Reynolds numbers. Accurate prediction of mass transfer rates is critical for scaling laboratory processes to industrial levels, ensuring energy efficiency and product purity in sectors ranging from petrochemicals to pharmaceuticals.

Fundamentals

Definition and Scope

Mass transfer refers to the net transport of mass—encompassing molecules, ions, or particles—from regions of higher concentration to regions of lower concentration, primarily driven by concentration gradients within a . This process is fundamental to understanding how substances move in physical, chemical, and biological systems, distinguishing the relative motion of from the overall movement of the medium. The scope of mass transfer spans microscopic scales, where individual molecular interactions dominate, to macroscopic scales involving bulk flows that enhance transport across larger distances. It explicitly focuses on the migration of specific components rather than the bulk fluid motion, which represents the collective velocity of the mixture; the latter contributes to total transport but is separated analytically from species-specific . Mass transfer is closely related to , serving as its broader . Historically, mass transfer concepts trace their origins to 19th-century advancements in and the study of diffusive processes, with Adolf Fick's 1855 formulation providing the initial mathematical foundation for quantifying rates. Subsequent developments in the expanded this into a unified theory of , integrating mass transfer with heat and momentum transport. Quantitatively, mass transfer is measured through , defined as the rate of mass crossing a unit area per unit time, with common units of mol/m²·s for molar flux or kg/m²·s for . , representing the amount of a per unit , is typically expressed in mol/m³ or kg/m³, serving as the key variable in rate equations that link to driving forces like concentration differences. These metrics enable the prediction and analysis of without requiring detailed molecular derivations.

Driving Forces

The primary driving force for mass transfer is the gradient in between regions of differing composition, which drives from areas of higher potential to lower potential until equilibrium is achieved. This fundamental thermodynamic quantity, denoted as μ, quantifies the free energy change associated with the addition or removal of a and is particularly pronounced in processes involving concentration differences, as higher concentrations correspond to elevated chemical potentials in dilute systems. In practical terms, this gradient often appears as a concentration disparity, prompting diffusive or convective movement to homogenize the system. The chemical potential for a species in a solution is expressed as μ=μ0+RTln⁡a,\mu = \mu_0 + RT \ln a, where μ0\mu_0 is the standard chemical potential at a reference state, RR is the universal gas constant, TT is the absolute temperature, and aa is the activity, which incorporates deviations from ideal behavior through the activity coefficient and concentration or mole fraction. This formulation links mass transfer directly to thermodynamic non-equilibrium, as activity gradients reflect interactions in non-ideal solutions, such as electrolyte mixtures where ionic strength alters species availability. Mass transfer persists under this driving force until the chemical potentials equalize across phases or regions, yielding zero net flux and establishing equilibrium, as seen in dissolution or phase separation processes. Secondary driving forces supplement the chemical potential gradient in specialized contexts. Pressure gradients can induce barodiffusion, where species migrate from high- to low-pressure regions due to partial molar volume differences, though this effect is minor except under extreme conditions like high-pressure gas separations. Temperature variations influence mass transfer indirectly by altering solubility—gases typically exhibit decreased solubility with rising temperature, thereby steepening concentration gradients at interfaces—and directly via thermal diffusion (Soret effect), where species separate along temperature gradients in multicomponent mixtures. Electrostatic fields serve as a driving force in electrophoresis, propelling charged species toward oppositely charged electrodes, as utilized in biomolecular separations where electric potential differences overcome diffusive resistances. Additionally, gravitational forces drive sedimentation, causing denser particles or droplets to settle against buoyant fluids, facilitating phase separation in suspensions or emulsions. These secondary mechanisms often couple with the primary chemical potential gradient to enhance overall transfer rates in engineered systems.

Mechanisms

Molecular Diffusion

Molecular diffusion is the primary mechanism of mass transfer in fluids at rest or with negligible bulk motion, arising from the random thermal agitation of molecules that results in a net flux from regions of higher to lower concentration. This process is driven by concentration gradients and occurs at the molecular scale, without reliance on macroscopic flows. In stagnant media, it governs the mixing of species over time, such as the slow interpenetration of gases or solutes in liquids. Key types of molecular diffusion include self-diffusion, mutual diffusion, and . Self-diffusion describes the random displacement of identical molecules within a homogeneous medium, typically quantified using isotopically labeled tracers to reveal intrinsic molecular mobility without net concentration change. Mutual diffusion, or interdiffusion, pertains to binary or multicomponent systems where dissimilar exchange positions due to a concentration , leading to relative motion between components. emerges in confined geometries like narrow pores, where the of molecules exceeds the pore diameter, causing molecules to collide more frequently with solid walls than with each other, as first described by in his 1909 study on gas through small orifices. The foundational description of is provided by Fick's , which states that the diffusive flux J\mathbf{J} (moles per unit area per unit time) is proportional to the negative of concentration cc: J=−D∇c\mathbf{J} = -D \nabla c Here, DD is the diffusion coefficient, a material-specific property with units of squared per time. This law, originally formulated by Fick in 1855 based on analogies to heat conduction, assumes steady-state conditions and isotropic media. A microscopic derivation of Fick's first law can be obtained from the random walk model, which idealizes molecular motion as uncorrelated jumps. In three dimensions, consider atoms in a crystal lattice jumping to nearest-neighbor sites with jump distance λ\lambda and frequency Γ\Gamma (attempts per unit time). The probability of jumping in any of the six directions is equal, but a concentration gradient ∂c/∂x\partial c / \partial x over distance λ\lambda creates an imbalance: more atoms jump right-to-left than left-to-right across a plane at xx. The flux from left to right is JL→R=(Γ/6)cλJ_{L \to R} = (\Gamma / 6) c \lambda, and from right to left is JR→L=(Γ/6)(c+λ∂c/∂x)λJ_{R \to L} = (\Gamma / 6) (c + \lambda \partial c / \partial x) \lambda. The net flux is then J=JL→R−JR→L=−(Γλ2/6)∂c/∂xJ = J_{L \to R} - J_{R \to L} = -(\Gamma \lambda^2 / 6) \partial c / \partial x, yielding D=Γλ2/6D = \Gamma \lambda^2 / 6. In one dimension, the relation simplifies to D=l2/(2τ)D = l^2 / (2 \tau), where ll is step length and τ\tau is time between steps, linking microscopic statistics to macroscopic transport. The diffusion coefficient DD varies with environmental and molecular factors. In gases, DD is inversely proportional to PP due to increased at higher densities, and scales with as D∝T3/2D \propto T^{3/2} from kinetic theory, reflecting enhanced molecular speeds and mean free paths. In liquids, DD depends more strongly on viscosity η\eta and solute size via the Stokes-Einstein relation, D=kBT/(6πηr)D = k_B T / (6 \pi \eta r), where kBk_B is Boltzmann's constant and rr is the of the diffusing species; this relation, derived by in 1905 from theory, assumes spherical particles in a continuum . Pressure effects in liquids are weaker, often negligible except at extremes. Experimental measurement of DD employs techniques like the diaphragm cell and (NMR). The diaphragm cell method, introduced by Northrop and Anson in 1928, involves monitoring the concentration change over time as solute diffuses through a porous barrier separating two compartments of known volumes, allowing DD to be calculated from the logarithmic decay of the concentration difference. NMR, particularly pulsed-field gradient NMR, quantifies self-diffusion by applying gradients to encode molecular positions and measure displacement distributions over short timescales, enabling studies of molecular motion in complex media.

Convective Mass Transfer

Convective mass transfer refers to the transport of species within a fluid due to the bulk motion of the fluid, where advection dominates and couples velocity gradients with concentration gradients to significantly enhance mass transfer rates compared to molecular diffusion alone. This process is characterized by the molar flux NA=xA(NA+NB)−cDAB∇xA\mathbf{N}_A = x_A (\mathbf{N}_A + \mathbf{N}_B) - c D_{AB} \nabla x_A, where the first term represents convective contribution and the second diffusive, as derived in boundary layer analyses. Convective mass transfer is classified into two primary types: natural convection and . Natural convection arises from -driven flows caused by variations, often due to concentration or gradients, with the Grm=gβmL3Δcν2Gr_m = \frac{g \beta_m L^3 \Delta c}{\nu^2} quantifying the ratio of to viscous forces, where βm\beta_m is the concentration expansion coefficient, LL is a , Δc\Delta c is the concentration difference, and ν\nu is kinematic . For instance, in vertical plate configurations, correlations like Sh=0.59(GrmSc)1/4Sh = 0.59 (Gr_m Sc)^{1/4} for laminar regimes illustrate how enhances transfer. , in contrast, is induced by external means such as pumps or fans, imposing a specified field, as seen in pipe flows where governs the flow regime. In convective flows, a concentration boundary layer develops adjacent to surfaces where species gradients are confined, analogous to the velocity but influenced by the Sc=νDABSc = \frac{\nu}{D_{AB}}, the ratio of momentum to mass . For high s typical in liquids (Sc≈103Sc \approx 10^3), the concentration δc\delta_c is much thinner than the hydrodynamic δ\delta, approximated as δc≈δSc1/3\delta_c \approx \frac{\delta}{Sc^{1/3}}, leading to steeper concentration gradients and higher transfer rates. This scaling emerges from similarity solutions to the convection-diffusion equation, where the Sh=kcLDABSh = \frac{k_c L}{D_{AB}} correlates with Re1/2Sc1/3Re^{1/2} Sc^{1/3} for laminar over flat plates. The film theory provides a simplified model for convective mass transfer by assuming a stagnant liquid film of thickness δ\delta adjacent to the interface, where transfer occurs solely by molecular diffusion across a linear concentration profile. The resulting molar flux is given by NA=kc(cA,i−cA,b)N_A = k_c (c_{A,i} - c_{A,b}), with the mass transfer coefficient kc=DABδk_c = \frac{D_{AB}}{\delta}, where cA,ic_{A,i} and cA,bc_{A,b} are interface and bulk concentrations, respectively; this approach, originally proposed for gas absorption, yields overall coefficients via two-film resistances. While basic, it effectively estimates rates in well-mixed bulk fluids with thin films, as validated in early absorption studies. Turbulence in convective mass transfer introduces chaotic fluctuations that augment through , modeled by an effective Deff=DAB+DTD_{eff} = D_{AB} + D_T, where DTD_T is the turbulent , often much larger than DABD_{AB}. In pipe flows, this leads to enhanced Sherwood numbers, such as Sh=0.023Re0.8Sc1/3Sh = 0.023 Re^{0.8} Sc^{1/3} for turbulent regimes, reflecting how eddies transport across the boundary layer more efficiently than . The turbulent flux can be expressed as jA=−ρ(D+ϵm)∇wA\mathbf{j}_A = -\rho (D + \epsilon_m) \nabla w_A, with ϵm\epsilon_m as for mass, emphasizing the role of velocity fluctuations in scaling transfer rates with flow intensity.

Interfacial Phenomena

Interfacial phenomena in mass transfer refer to the processes occurring at the boundary regions where two or more phases come into contact, such as gas-liquid or solid-liquid interfaces, where mass transfer resistance arises due to concentration gradients and limited across the phase boundary. These regions are characterized by potential barriers to solute transport, influenced by the physical properties of the phases and the nature of the contact, leading to non-equilibrium conditions that drive the net of species from one phase to another. The two-film theory, proposed by Whitman in 1923, models mass transfer across gas-liquid interfaces by assuming stagnant liquid and gas films adjacent to the interface, where resistance to transport is concentrated. In this model, the overall KK is determined by the resistances in both films, expressed as 1K=1kg+1Hkl\frac{1}{K} = \frac{1}{k_g} + \frac{1}{H k_l}, where kgk_g and klk_l are the gas- and liquid-phase s, respectively, and HH is Henry's constant relating the equilibrium concentrations between phases. This approach simplifies the analysis of interphase transfer by treating the films as regions of pure , with convective effects contributing to film renewal but not detailed here. Equilibrium relations at the interface govern the maximum possible mass transfer rate and are crucial for understanding resistance distribution. For dilute solutions, describes the linear relationship between the partial pressure pp of a gas and its xx in the phase at equilibrium: p=Hxp = H x, where HH is the Henry's law constant, applicable to sparingly soluble gases under low concentrations and moderate pressures. In non-ideal or concentrated systems, phase diagrams or more complex equilibrium models, such as activity coefficient-based approaches, are used to represent deviations from linearity, capturing interactions that affect and transfer rates. The distribution of mass transfer resistance between phases is often asymmetric, particularly in gas-liquid systems where liquid-side resistance dominates for sparingly soluble gases. To account for varying concentrations along the interface, the log-mean driving force is employed as an effective concentration difference: Δclog⁡=Δc1−Δc2ln⁡(Δc1/Δc2)\Delta c_{\log} = \frac{\Delta c_1 - \Delta c_2}{\ln(\Delta c_1 / \Delta c_2)}, where Δc1\Delta c_1 and Δc2\Delta c_2 are the concentration differences at the ends of the contact region, providing a more accurate representation of the average driving force than arithmetic means. This logarithmic averaging arises from integrating the local flux over the interface, highlighting how resistance allocation influences overall transfer efficiency. Unsteady effects at interfaces become prominent in dynamic systems with short contact times, such as bubble absorption or droplet dissolution. The penetration theory, developed by Higbie in 1935, models this by considering transient into a semi-infinite medium exposed to the interface for a finite time tt, yielding an mass flux proportional to D/t\sqrt{D/t}
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