Linear mass transfer coefficient

In vivo experiments can also be utilized to estimate mass transfer coefficients. Under the condition of steady-state blood concentrations of drug, a linear mass transfer coefficient can be estimated from [10]... 

A moment method to estimate linear mass transfer coefficients for noneliminating organs was derived by Gallo et al. [52], hi is estimated as... 

Mass transfer coefficients can also be estimated from in vivo experiments. Under the condition of steady-state drug concentrations in blood, a method was derived to estimate a linear mass transfer coefficient. Another method, referred to as the moment method, was derived by Gallo et al. to estimate linear mass transfer coefficients for non-eliminating organs. By using Monte Carlo methods it was demonstrated that the method was both accurate and precise. 

Even though the linear mass transfer coefficient (h ) is more frequently used in heterogeneous chemical kinetics, in enzyme heterogeneous kineties it is customary to... 

Equations 11 and 12 caimot be used to predict the mass transfer coefficients directly, because is usually not known. The theory, however, predicts a linear dependence of the mass transfer coefficient on diffusivity. 

The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

The unknown intermediate concentration C, has been mathematically ehminated from the last term. In this case, r can be solved for explicitly, but that is not always possible with surface rate equations of greater complexity. The mass transfer coefficient /ci is usually obtainable from correlations. When the experimental data are of (C, r) the other constants can be found by linear plotting. 

Whenever die rich and the lean phases are not in equilibrium, an interphase concentration gradient and a mass-transfer driving force develop leading to a net transfer of the solute from the rich phase to the lean phase. A common method of describing the rates of interphase mass transfer involves the use of overall mass-transfer coefficients which are based on the difference between the bulk concentration of the solute in one phase and its equilibrium concentration in the other phase. Suppose that the bulk concentradons of a pollutant in the rich and the lean phases are yi and Xj, respectively. For die case of linear equilibrium, the pollutant concnetration in the lean phase which is in equilibrium with y is given by... 

In model equations, Uf denotes the linear velocity in the positive direction of z, z is the distance in flow direction with total length zr, C is concentration of fuel, s represents the void volume per unit volume of canister, and t is time. In addition to that, A, is the overall mass transfer coefficient, a, denotes the interfacial area for mass transfer ifom the fluid to the solid phase, ah denotes the interfacial area for heat transfer, p is density of each phase, Cp is heat capacity for a unit mass, hs is heat transfer coefficient, T is temperature, P is pressure, and AHi represents heat of adsorption. The subscript d refers bulk phase, s is solid phase of adsorbent, i is the component index. The superscript represents the equilibrium concentration. 

For a non-linear equilibrium relationship, in which the slope of the equilibrium curve varies with concentration, the magnitudes of the overall mass transfer coefficients will also vary with concentration, even when the film coefficients themselves remain constant. The use of overall mass transfer coefficients in mass transfer rate equations should therefore be limited to the case of linear equilibria or to situations in which the mass transfer coefficient is known to be... 

For a linear equilibrium curve with constant film coefficients, Icl and Icq, the overall coefficient, Kl, will also be constant, but for the case of a non-linear equilibrium relationship, the value of m, which is the local slope of the equilibrium curve, will vary with solute concentration. The result is that the overall coefficient, Kl, will also vary with concentration, and therefore in modelling the case of a non-linear equilibrium extraction, further functional relationships relating the mass transfer coefficient to concentration will be required, such that... 

Overall mass transfer coefficients are only constant when both liquid film coefficients are constant and also when the slope of the equilibrium line is constant. Thus, for a non-linear equilibrium relationship, the overall mass transfer coefficient will vary with concentration. How would you implement this effect into the program ... 

Eqn. 3.86 shows the linear relationship between ip and the original analyte concentration C, and also the influence of the mass transfer coefficient, i.e. its type of motion (stirring or rotation). However, basically eqn. 3.85 indicates that ip is directly proportional to ALCa = VCa, i.e. the mass of metal dissolved in the ultra-thin mercury this relationship was also confirmed by Roe and Toni105 in their approximate equation... 

The mass transfer coefficients may also be expressed in units of time-1 by multiplying by the appropriate compartmental volume term. Irreversible drug elimination from the tissue requires the addition of an expression to the differential equation that represents the subcompartment in which elimination occurs. For instance, hepatic drug elimination would be described by a linear or nonlinear expression added to the intracellular liver compartment mass balance equation since this compartment represents the hepatocytes. Formal elimination terms are given below for the simplified tissue models. 

Due to the difficulties in having rigorous analytical expressions for the flux at any given geometry and flow conditions, in many instances it is convenient to include all the characteristics of the supply in the mass transfer coefficient ms and use expression (50). It must be pointed out that expression (50), stating a linearity between the flux and the difference between bulk and surface concentrations, cannot be - in general valid for nonlinear processes, such as coupled complexation of the species i with any other species (see Chapters 4 and 10 for a more detailed discussion). 

In the particular case dealt with now (fully labile complexation), due to the linearity of a combined diffusion equation for DmCm + DmlL ml, the flux in equation (65) can still be seen as the sum of the independent diffusional fluxes of metal and complex, each contribution depending on the difference between the surface and bulk concentration value of each species. But equation (66) warns against using just a rescaling factor for the total metal or for the free metal alone. In general, if the diffusion is coupled with some nonlinear process, the resulting flux is not proportional to bulk-to-surface differences, and this complicates the use of mass transfer coefficients (see ref. [II] or Chapter 3 in this volume). 

We will now describe the application of the two principal methods for considering mass transport, namely mass-transfer models and diffusion models, to PET polycondensation. Mass-transfer models group the mass-transfer resistances into one mass-transfer coefficient ktj, with a linear concentration term being added to the material balance of the volatile species. Diffusion models employ Fick s concept for molecular diffusion, i.e. J = — D,v ()c,/rdx, with J being the molar flux and D, j being the mutual diffusion coefficient. In this case, the second derivative of the concentration to x, DiFETd2Ci/dx2, is added to the material balance of the volatile species. 

A mass transfer coefficient determined from some subsurface soil column elution studies was inverse log-linearly related to the octanol-water partition coefficient (i.e., K0Vf) for closely related compounds, and polarity in the molecule caused an additional decline in the mass transfer coefficient. 

The rotating disc electrode is constructed from a solid material, usually glassy carbon, platinum or gold. It is rotated at constant speed to maintain the hydrodynamic characteristics of the electrode-solution interface. The counter electrode and reference electrode are both stationary. A slow linear potential sweep is applied and the current response registered. Both oxidation and reduction processes can be examined. The curve of current response versus electrode potential is equivalent to a polarographic wave. The plateau current is proportional to substrate concentration and also depends on the rotation speed, which governs the substrate mass transport coefficient. The current-voltage response for a reversible process follows Equation 1.17. For an irreversible process this follows Equation 1.18 where the mass transfer coefficient is proportional to the square root of the disc rotation speed. 

For the turbulent motion in a tube, the mass transfer coefficient k is proportional to the diffusion coefficient at the power of 2/3. It is easy to realize by inspection that this value of the exponent is a result of the linear dependence of the tangential velocity component on the distance y from the wall. For the turbulent motion in a tube, the shear stress t r0 = const near the wall, whereas for turbulent separated flows, the shear stress is small at the wall near the separation point (becoming zero at this point) and depends on the distance to the wall. Thus, the tangential velocity component has, in the latter case, no longer a linear dependence on y and a different exponent for the diffusion coefficient is expected. For separated flows, it is possible to write under certain conditions that [90]... 

The statement cA = c0/ (1 + K) in Eqs. (157a and b) above is tantamount to saying that cA + cB = Co, where c0 is the total concentration of both species of the dissolved solute. If the diffusivities SDA8 and DBs are assumed to be equal, then cB can be eliminated from Eqs. (155) and (156) and a fourth-order, linear partial-differential equation is obtained. The solution of this equation consistent with the conditions in Eq. (157) is obtainable by Laplace transform techniques (S9). Sherwood and Pigford discuss the results in terms of the behavior of the liquid-film mass transfer coefficient. 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

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 ... 

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