Logarithmic concentration difference

In larger and especially higher reactors, such as bubble columns, a constant value of cL can normally not be assumed. A differential balance over the height of the column is necessary. An approximation for this case for oxygen transfer was recommended by Zlokamik (1980), the so-called logarithmic concentration difference to be used for (cL -cL) and defined as ... 

Since part of the gas mixture (e.g. air) is absorbed as it bubbles through the liquid column, the composition of the gas mixture changes. Furthermore, the pressure in the liquid is higher at the gas inlet than in the head space above it. This difference in partial pressure is taken into account by the mean logarithmic concentration difference, Acm ... 

NOTE For Nfe < 3 convective contributions which are not included may become [important. Use with logarithmic concentration difference (integrated form) or with arithmetic concentration difference (differential form). 

The average logarithmic concentration difference is calculated with the oxygen gas concentrations in the bulk liquid phase and saturation conditions at the inlet (1) and outlet (2). In this application the value of Cl is commonly assumed to be 0 (cl < c ). 

For volume gassing the average logarithmic concentration difference Acm (similar to ATm in heat exchangers) applies, if it can be assumed that the gas phase is not backmixed (which is the case with high aspect ratio bubble columns and fermenters) ... 

Typical plots of the variation of the surface tension with the logarithmic concentration difference of a surfactant, a saturated hydrocarbon alcohol and an inorganic salt in an aqueous solution are shown in Figure 5.4. Surfactant molecules find a lower free-energy environment at the interface of an aqueous solution than they have in the bulk solution and prefer to concentrate at the surface. The adsorption isotherm of most surfactants shows a sharp decrease initially with the increase in the solute concentration until a plateau appears, due to the formation of micelles following the critical micelle concentration (CMC), where all the solution surface is covered with surfactant molecules. Any further addition of surfactant does not decrease the surface tension of the solution because it is directly consumed in the micelle formation process (see also Section 5.7). Saturated hydrocarbon alcohols also show a decrease in surface tension with increases in their aqueous solution concentration because they do not like to stay in the bulk solution and prefer to enrich on... 

The overall length-averaged mass-transfer coefficient Kieo is defined via a logarithmic concentration difference... 

This is the equation of a straight line. One constructs a plot of the incremental concentration differences against the logarithm of their ratios. The line gives k = -slope and k = intercept/r. Figure 2-11 presents this plot for the same data set points to 90 percent completion were used, with 375 s chosen for t. 

A Cta concentration difference Logarithmic mean value of C kmol/m3 NL-3... 

For isotope effects on equilibrium constants in both gas and condensed phase the take-home lesson is there is no direct proportionality between measured isotope effects on logarithmic concentration or pressure ratios and isotopic differences in... 

In a bioreactor, one is interested in the transfer per unit of volume of reactor, called Kia or the volumetric mass-transfer coefficient, a is the interfacial surface area per unit of volume of liquid. In a perfectly mixed tank, C has identical values at any point and C depends on the conditions in the gas phase at the outlet of the reactor. Several authors [60] consider that a better estimate of the driving force is given by the logarithmic mean concentration difference between the entry and the exit of gas. 

The data can be evaluated using any commonly available non-linear regression program or with a linear regression, in which k,a is the slope from the plot of the natural log of the concentration difference versus time. Linearity of the logarithmic values over one decade is required for the validity of the measurement. Of course the assumptions inherent in the model must apply to the experimental system, especially in respect to completely mixed gas as well as liquid phases and reactions are negligible. Two common problems are discussed below. Other common pitfalls and problems are summarized in Table 3-3. 

Now let us discuss the sensitivity of the taste sensor. The sensor had detection errors ( in the unit of logarithmic concentration) 0.73% for saltiness, 0.65% for sourness and 3.4% for bitterness in the mixed aqueous solution [27]. Humans usually cannot distinguish two tastes with a concentration difference below 20% [4], Here, 20% means the error of 7.9% (=log 1.2). Therefore, ability of detection of the sensor is superior to that of... 

The set of data in Table 6.4 was generated so that the concentration of dissolved ions increases from data set 1 to set 5, but the relative abundance of the ions is preserved (check it, for example, by comparing the Mg Ca ratio in data sets 1-5, Table 6.4). This imitates dilution of a saline water by different amounts of a fresh (ideally, distilled) water, a common occurrence in nature. The data of Table 6.4 have been plotted once with a linear concentration axis (Fig. 6.2) and once with a logarithmic concentration axis (Fig. 6.3). The outcome is striking the same data plotted with a different concentration axes reveal intrinsically different patterns. On the linear plot the lines of the individual water samples differ in their gradients, whereas on the semilogarithmic plot the lines have the same gradients. The latter well reflects the dilution of saline water with fresh water, whereas the former... 

O. Shell side of microporous hollow fiber module for solvent extraction Na, = V[dha-)/L]N%N°s M Nsh- D Nlt = K = overall mass-transfer coefficient (3 = 5.8 for hydrophobic membrane. (3 = 6.1 for hydrophilic membrane. [E] Use with logarithmic mean concentration difference. dh = hydraulic diameter 4 x cross-sectional area of flow wetted perimeter (p = packing fraction of shell side. L = module length. Based on area of contact according to inside or outside diameter of tubes depending on location of interface between aqueous and organic phases. Can also be applied to gas-liquid systems with liquid on shell side. [118]... 

The mass transfer rate (or the evaporation rate) in this case can be determined by defining the logarithmic mean concentration difference in an analogous manner to the logarithmic mean temperature difference. 

The ratio in the logarithm term gets larger as the concentration difference increases. However, when the mobilities of the cation and anion in the concentrated solution are the same, the sum in front of the logarithm is zero so that j A2< ) is zero. As stated earlier, the electrolyte most commonly used for which this condition is approximately met is KCl. Another suitable system is concentrated sodium formate. 

Figure 5.4 Variation of surface tension with the logarithm of the concentration difference of a typical surfactant, saturated hydrocarbon alcohol and an inorganic salt in an aqueous solution.

FIG U RE 2.23 The effect of micelle formation. Surface tension as a function of logarithmic concentration for solutions of n-dodecyl hexaoxyethylene in water at different temperatures ( ) 15°C ( ) 25°C (o) 35°C. (From Corkill, Goodman, and Ottewill, 1961, Micellization of homogeneous non-ionic detergents. Transactions of the Faraday Society, 57, 1627-1635. Reproduced hy permission of The Royal Society of Chemistry.)... 

FIGURE 4.26 Plots of the Fowler-Guggenheim isotherm for different values of the interaction parameter BpQ in (a) linear and (b) logarithmic concentration scale. 

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