Mass transfer rate exposure time

Given that, from the penetration theory for mass transfer across an interface, the instantaneous rale ol mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure lime in the Higbie mode and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be deri ved from first principles. 

Equation (35) predicts that the mass transfer coefficient increases with increases in the screw speed and the number of parallel channels on the screw. The explanation for this is rather simple and is related to the fact that each time the film on the barrel wall is regenerated and the surface of the nip is renewed, a uniform concentration profile is reestablished, which means that the driving force for mass transfer is maximized. Since the instantaneous mass transfer rate decreases with time, mass transfer rates can be maximized by keeping the exposure time as short as possible, and... 

Mechanical forces can disturb the elaborate structure of the enzyme molecules to such a degree that de-activation can occur. The forces associated with flowing fluids, liquid films and interfaces can all cause de-activation. The rate of denaturation is a function both of intensity and of exposure time to the flow regime. Some enzymes show an ability to recover from such treatment. It should be noted that other enzymes are sensitive to shear stress and not to shear rate. This characteristic mechanical fragility of enzymes may impose limits on the fluid forces which can be tolerated in enzyme reactors. This applies when stirring is used to increase mass transfer rates of substrate, or in membrane filtration systems where increasing flux through a membrane can be accompanied by increased fluid shear at the surface of the membrane and within membrane pores. Another mechanical force, surface... 

Given that the instantaneous mass transfer rate = Kt 1/ 2, then for the Higbie model, the average mass transfer rate for an exposure time te is given by ... 

The instantaneous mass transfer rate is expressed as a function of time. In order to calculate an average mass transfer coefEcient we need to average the instantaneous coefEcient over the total exposure time period. To do this we need to know the age distribution function, which represents the fraction of elements having ages between t and t + dt t the surface. In the penetration theory, it is assumed that all the elements reside at the interface for a time period of the same length. As a consequence of this assumption the age distribution function is [6] ... 

Thus, the shorter the time of exposure the greater is the rate of mass transfer. No precise value can be assigned to te in any industrial equipment, although its value will clearly become less as the degree of agitation of the fluid is increased. 

At 60 minutes only, dc potentiodynamic curves were determined from which the corrosion current was obtained by extrapolation of the anodic Tafel slope to the corrosion potential. The anodic Tafel slope b was generally between 70 to 80 mV whereas the cathodic curve continuously increased to a limiting diffusion current. The curves supported impedance data in indicating the presence of charge transfer and mass transfer control processes. The measurements at 60 minutes indicated a linear relationship between and 0 of slope 21mV. This confirmed that charge transfer impedance could be used to provide a measure of the corrosion rate at intermediate exposure times and these values are summarised in Table 1. 

High solubility (and / or short exposure times), no reaction. In this case, the rate of mass transfer due to diffusion and solubility is large and Eq. (PPP) becomes... 

As an alternative to investigating the kinetics of a gas-liquid reaction on a laboratory scale, the mass transfer resistance may be minimised or eliminated so that the measured rate corresponds to the rate of the homogeneous liquid-phase reaction. This method of approach will be considered after first describing those reactors giving rise to controlled surface exposure times. 

With kinetic or TWA sampling, it is assumed that the rate of mass transfer to the sorption phase is linearly proportional to the difference between the chemical activity of the contaminant in the water phase and that in the sorption phase. During the initial phase of sampler exposure, the rate of desorption of analyte from the sorption phase to water is negligible and the sampler works in the linear uptake mode. The amount of analyte accumulated is therefore linearly proportional to its TWA concentration in water, even for situations where aqueous concentrations fluctuate over time (Figure 3.2). In this case Equation 3.1 reduces to... 

Apart from this, each model contains an empirical parameter effective film thickness effective exposure time 0, and effective rate of surface renewals. Consequently, when predicting the average rate of purely physical absorption for large-scale conditions, one cannot expect better results from refined models than from the empirical mass-transfer coefficient first defined by Whitman (W5). 

The calculation of k using Eqs. 9.2.11 and 9.2.12 requires a priori estimation of the exposure time or the surface renewal rate s. In some cases this is possible. For bubbles rising in a liquid the exposure time is the time the bubble takes to rise its own diameter. In other words, the jacket of the bubble is renewed every time it moves a diameter. If we consider the flow of a liquid over a packing, when the liquid film is mixed at the junction between the packing elements, then is the time for the liquid to flow over a packing element. For flow of liquid in laminar jets and in thin films, the exposure time is known but in these cases it may be important to take into account the distribution of velocities along the interface. In the penetration model, this velocity profile is assumed to be flat (i.e., plug flow). For gas-liquid mass transfer in stirred vessels, the renewal frequency in the Danckwerts model s may be related to the speed of rotation (see Sherwood et al. 1975). 

While the treatment of adsorption at the DME generally follows that for linear sweep voltammetry at a stationary electrode, it is complicated by the growth of the drop with time and the continuous exposure of fresh surface. In this case, the rate of mass transfer of reactant and product (see Section 13.5.3) and the rate of adsorption can affect the height of the adsorption wave. Although the first explanation of adsorption in voltam-metric methods and the explanation of prewaves and postwaves arose from the classic studies by Brdicka (48, 49), dc polarography is not the method of choice in the study of adsorption. Only a brief discussion will be given here more detailed treatments have appeared (33, 44, 50). 

Fnel when the exposure time is greater than 10 5 sec, and the gas phase is saturated. From this discussion it can be concluded that the net rate of phase change, that is, the rate of mass transfer, can be calculated from either Eq. (65) or Eq. (66) with an equal degree of accuracy. However, the absolute rates of vaporization and condensation can only be calculated from the rate expressions based on the molecular interchange process. It is shown later in this chapter that the absolute rates must be determined to describe accurately the energy transfer that accompanies phase change. 

The average rate during the exposure of the surface is obtained by integrating R(t)dt over the time interval 0 to tp and dividing the result by tp. The mass-transfer coefhcient becomes... 

The film model according to the Hinterland concept assumes a stagnant film independent of the liquid bulk [38]. The penetration model [39] assumes that elements of liquid film are mixed up with the liquid bulk after a certain superficial exposure time. Therefore, all the species present inside the liquid film are considered to be part of the liquid phase. This results in a larger overall liquid phase concentration. For processes controlled by intrinsic reaction kinetics (Ha <0.3), the two approaches yield similar results, because most species are present in the bulk liquid and the excess of species present in the liquid film can be neglected. For processes controlled completely or partly by the rate of the mass transfer (Ha >0.3), the two models give different results, because the excess of species present in the liquid film can no longer be neglected when compared with the total amount of species present in the liquid phase (see Chapter 11). 

It should be noted that the semicircle part of Nyquist plots develops with the exposure time t (Figure 9.26). Such plots are typical of the processes the rate of which Is controlled by charge transfer and diffusive mass transport simultaneously. Impedance spectra obtained for Sn(II) solutions can be described with a frequency error less than 2% by means of equivalent circuit, the description code of which is. RnG- ctQdlQdi) Figure 9.26). Here, elements in series are given in square brackets and elements in parallel are enclosed in parentheses. The ohmic resistance of the solutions was found to be actually independent of tetraethy-lene glycol (TEG) concentration and equal to 0.24 0.01 2 cm. Diffusion and double-layer impedances are presented by the constant phase elements (CPEs)... 

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