Mass transfer equilibrium distribution coefficient

Note that volatility of the liquid reactant is accounted for by the parameter y which reflects the magnitude of the equilibrium distribution coefficient relative to that of the liquid-side mass transfer coefficient. 

Water containing urea at a bulk phase concentration of c, = 0.008 kmol/m is flowing on one side of a membrane. The bulk phase concentration of urea on the other side of the membrane is Cj = 0.002 kmol/m The diffusivity of urea in the membrane is D = 2.5 x 10" m /s and the membrane thickness is L = 4 x 10" m. The equilibrium distribution coefficient on either side of the membrane is K= 1.4. The mass transfer coefficients are k, = 3 x lO m/s and kj = 2 x 10 m/s. Calculate the flux of urea through the membrane, the permeance and permeability, and the concentrations of urea at the film interface with the membrane and on the membrane surface on both sides. 

Generally, the solids are not structurally homogeneous, but the solid and liquid nevertheless will be called phases and leaching will be treated as a two-phase, mass transfer process- The solid consists of a matrix of insoluble solids, the mare, and the occluded solution. It may also contain undissolved solute and a nonextractable secondary phase, for example, coffee oil in water-soaked coffee grounds. This secondary phase is treated as part of the mate. Dimensionless parameters that can affect solnta transfer include the solute equilibrium distribution coefficients, m and M tha Pick number, v the strippirg factor, a the Biot number, Bi and the Peclet number, Be. These parameters are defined more precisely in the Notation section. 

Various parameters such as adsorption and desorption isotherms, diffusion coefficients, liquid/gas, gas/solid and liquid/solid equilibrium distribution coefficients, as well as mass transfer coefficients and many other physicochemical property values have to be used in the models proposed for supercritical fluid extractions. These parameter values are either obtained from existing correlations, or from independent data sources using parameter estimation. However, in those cases where the above stated means are not sufficient to estimate the values of all parameters used in the model, the researcher(s) may be forced to use the model and the associated data to evaluate best fit or optimal values for the missing parameters. The fact is that, the number of reliable correlation s and methods for the SFE are still quite scarce. 

Equilibrium relations. Even when mass transfer is occurring equilibrium relations are important to determine concentration profiles for predicting rates of mass transfer. In Section 10.2 the equilibrium relation in a gas-liquid system and Henry s law were discussed. In Section 7.1C a discussion covered equilibrium distribution coefficients between two phases. These equilibrium relations will be used in discussion of mass transfer between phases in this section. 

Local flow rate inside the membrane separator G General driving force property H Henry s constant K Equilibrium distribution coefficient k Mass transfer coefficient L Membrane thickness M Molecular weight M Factor defined by Equation 18.26 N Flux P Total pressure Pm Permeability p Partial pressure Pm Permeance... 

In fluid-fluid separation processes, mass transfer rate frequently limits the efficiency of the operation. The fundamental physical problem is transport of a chemical species from a dispersed phase in the form of a bubble or drop to a continuous phase. Once one has selected a chemical system, the equilibrium distribution coefficient is established and one must turn to physical means to improve the transport efficiency. 

Study the effect of the extraction stage on reactor performance by varying the magnitudes of the the mass transfer coefficient Ka, the equilibrium distribution ratio m, the recycle ratio R, the relative reactor and extraction volumes and solvent flowrate. 

Due to the ionic nature of cephalosporin molecules, the interfacial chemical reaction may in general be assumed to be much faster than the mass transfer rate in the carrier facilitated transport process. Furthermore, the rate controlling mass transfer steps can be assumed to be the transfer of cephalosporin anion or its complex, but not that of the carrier. The distribution of the solute anion at the F/M and M/R interfaces can provide the equilibrium relationship [36, 69]. The equilibrium may be presumably expressed by the distribution coefficients, mf and m at the F/M and M/R interfaces, respectively and these are defined as... 

The experimental determination of the film coefficients kL and kc is very difficult. When the equilibrium distribution between the two phases is linear, over-all coefficients, which are more easily determined by experiment, can be used. Over-all coefficients can be defined from the standpoint of either the liquid phase or gas phase. Each coefficient is based on a calculated over-all driving force Ac, defined as the difference between the bulk concentration of one phase (cL or cc) and the equilibrium concentration (cL or cc ) corresponding to the bulk concentration of the other phase. When the controlling resistance is in the liquid phase, the over-all mass transfer coefficient KLa is generally used ... 

Complexity in multiphase processes arises predominantly from the coupling of chemical reaction rates to mass transfer rates. Only in special circumstances does the overall reaction rate bear a simple relationship to the limiting chemical reaction rate. Thus, for studies of the chemical reaction mechanism, for which true chemical rates are required allied to known reactant concentrations at the reaction site, the study technique must properly differentiate the mass transfer and chemical reaction components of the overall rate. The coupling can be influenced by several physical factors, and may differently affect the desired process and undesired competing processes. Process selectivities, which are determined by relative chemical reaction rates (see Chapter 2), can thenbe modulated by the physical characteristics of the reaction system. These physical characteristics can be equilibrium related, in particular to reactant and product solubilities or distribution coefficients, or maybe related to the mass transfer properties imposed on the reaction by the flow properties of the system. 

St is the total sorbed concentration (M/M), a, is the first-order mass-transfer rate coefficient for compartment i (1/T), / is the mass fraction of the solute sorbed in each site at equilibrium (assumed to be equal for all compartments), Kp is the distribution coefficient (L3 /M), C is the aqueous solute concentration (M/L3), St is the mass sorbed in compartment i with respect to the total mass of the sorbent (M/M), 0 is the volumetric flow rate through the reactor (L3/T), C, is the influent concentration of solute (M/L3), M, is the mass of sorbent in the reactor (M), and V is the aqueous reactor volume (L3). Using the T-PDF, discrete values for the mass-transfer rate coefficients were generated for the NK compartments. The median value of the mass-transfer rate coefficient within each compartment was chosen as the representative value. The resulting system of ordinary differential equations was solved numerically using a 4th-order Runge-Kutta integration technique. 

The model I is very simple, and it is not very sensitive to the physical properties of the bed, but the values of the overall mass transfer coefficients optimised are strongly dependent from the equilibrium relation assumed and it only is able to describe the initial part of the extraction. Ke was determined by mass balance assuming a uniform distribution in solid bed. Ke values of 0.5, 0.2, and 0.6 were obtained for 7, 10 and 15 MPa. Assuming a = 3000 nAn", the mass transfer coefficients calculated with model I are of some orders of magnitude lower than those for external mass transfer coefficients. This type of models have being applied with success to the extraction of edible oils from seeds were the solute is in a... 

Liquid extraction is a separation process in which a liquid feed solution is combined with a second solvent that is immiscible or nearly immiscible with the feed solvent, causing some (and ideally most) of the solute to transfer to the phase containing the second solvent. The distribution coefficient is the ratio of the solute mass fractions in the two phases at equilibrium. Its value determines how much solvent must be added to the feed solution to achieve a specified solute transfer. When the two solvents are partially miscible, a triangular phase diagram like that in Figure 6.6-1 simplifies balance calculations on extraction processes,... 

Under equilibrium or near-equilibrium conditions, the distribution of volatile species between gas and water phases can be described in terms of Henry s law. The rate of transfer of a compound across the water-gas phase boundary can be characterized by a mass-transfer coefficient and the activity gradient at the air—water interface. In addition, these substance-specific coefficients depend on the turbulence, interfacial area, and other conditions of the aquatic systems. They may be related to the exchange constant of oxygen as a reference substance for a system-independent parameter reaeration coefficients are often known for individual rivers and lakes. 

Dj are distribution coefficients at equilibrium at feed-membrane and membrane-strip interfaces Relation for the overall mass transfer resistance can be derived [11,25,91] ... 

Henry constant for absorption of gas in liquid Free energy change Heat of reaction Initiator for polymerization, modified Bessel functions, electric current Electric current density Adsorption constant Chemical equilibrium constant Specific rate constant of reaction, mass-transfer coefficient Length of path in reactor Lack of fit sum of squares Average molecular weight in polymers, dead polymer species, monomer Number of moles in electrochemical reaction Molar flow rate, molar flux Number chain length distribution Number molecular weight distribution... 

The resistance to mass transfer according to (1.221) and (1.223) is made up of the individual resistances of the gas and liquid phases. Both equations show how the resistance is distributed among the phases. This can be used to decide whether one of the resistances in comparison to the others can be neglected, so that it is only necessary to investigate mass transfer in one of the phases. Overall mass transfer coefficients can only be developed from the mass transfer coefficients if the phase equilibrium can be described by a linear function of the type shown in eq. (1.217). This is normally only relevant to processes of absorption of gases by liquids, because the solubility of gases in liquids is generally low and can be described by Henry s law (1.217). So called ideal liquid mixtures can also be described by the linear expression, known as Raoult s law. However these seldom appear in practice. As a result of all this, the calculation of overall mass transfer coefficients in mass transfer play a far smaller role than their equivalent overall heat transfer coefficients in the study of heat transfer. 

Mass transfer capacity coefficient Equilibrium distribution constant Volume... 

Overall mass-transfer coefficients on the feed and strip sides are calculated using Eqs (8) and (9). Ep and Ep, or IBM + LMF potentials are determined experimentally through distribution coefficients at membrane-based equilibrium forward and backward extraction. 

SPME is a multiphase equilibrium technique and, therefore, the analytes are not completely extracted from the matrix. Nevertheless, the method is useful for quantitative work and excellent precision and Unearity have been demonstrated. An extraction is complete when the concentration of analytes has reached distribution equilibrium between the sample and coating. This means that once the equihbrium is achieved, the amount extracted is independent of further increase in extraction time. If extraction is terminated before the equihbrium is reached, good precision and reproducibihty is still obtained if incubation temperature, sample agitation, sample pH and ionic strength, sample and headspace volume, extraction and desorption time are kept constant. The theory of the thermodynamic, kinetic and mass transfer processes underlying direct immersion and HS-SPME has been extensively discussed by Pawhszyn [2]. The sensitivity and time required to reach adsorption equilibriiun depends on the partition coefficients between the fiber and the analytes, and the thickness of the phase. Limits of detection and quantitation are often below 1 ppb. 

As mentioned earher, the plate theory has played a role in the development of chromatography. The concept of "plate" was originally proposed as a measmement of the performance of distillation processes. It is based upon the assumption that the column is divided into a number of zones called theoretical plates, that are treated as if there exists a perfect equilibrium between the gas and the Hquid phases within each plate. This assumption imphes that the distribution coefficient remains the same fi-om one plate to another plate, and is not affected by other sample components, and that the distribution isotherm is hnear. However, experimental evidences show that this is not true. Plate theory disregards that chromatography is a dynamic process of mass transfer, and it reveals httle about the factors affecting the values of the theoretical plate number. In principle, once a sample has been introduced, it enters the GC column as a narrow-width "band" or "zone" of its composite molecules. On the column, the band is further broadened by interaction of components with the stationary phase which retains some components more than others. Increasing... 

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