Condensation approximation

Figure 10.5 shows the appearance of numerous failures that had occurred in a short time in the water box inlet end. Cracking of this type was a recurrent problem in this condenser. Approximately 9% of the tubes in the condenser had been plugged. The condenser was in cyclic service, although the failures had occurred while the boiler itself was out of service. 

Figure 11.7 illustrates the internal surface at the inlet end of the condenser. Approximately 2 in. (5 cm) of the surface is marked by mutually intersecting depressions and grooves. Areas of the internal surface downstream of this zone are smooth and covered with a thin layer of deposits. This typical case of inlet-end erosion can be eliminated by the techniques discussed earlier in this chapter under Elimination. ... 

A cold-finger type of condenser approximately 10 X 40 cm. is satisfactory. 

Condenser - either silica cold finger condensers, effective length 140 mm, or soda glass air condensers, approximately 750 mm. 

Recently, the Horvath-Kawazoe (HK) method for slit-like pores [40] and its later modifications for cylindrical pores, such as the Saito-Foley (SF) method [41] have been applied in calculations of the mesopore size distributions. These methods are based on the condensation approximation (CA), that is on the assumption that as pressure is increased, the pores of a given size are completely empty until the condensation pressure corresponding to their size is reached and they become completely filled with the adsorbate. This is a poor approximation even in the micropore range [42], and is even worse for mesoporous solids, since it attributes adsorption on the pore surface to the presence of non-existent pores smaller than those actually present (see Fig. 2a) [43]. It is easy to verify that the area under the HK PSD peak corresponding to actually existing pores does not provide their correct volume, so the HK-based PSD is not only excessively broad, but also provides underestimated volume of the actual pores. This is a fundamental problem with the HK-based methods. An additional problem is that the HK method for slit-like pores provides better estimates of the pore size of MCM-41 with cylindrical pores than the SF method for cylindrical pores. This shows the lack of consistency [32,43]. Since the HK-based methods use CA, one can replace the HK or SF relations between the pore size and pore filling pressure by the properly calibrated ones, which would lead to dramatic improvement of accuracy of the pore size determination [43] (see Fig. 2a). However, this will not eliminate the problem of artificial tailing of PSDs, since the latter results from the very nature of HK-based methods. 

The chlorine stream is interrupted, and the reaction mixture is allowed to stand for a short time, then it is poured into a separatory funnel and washed thoroughly with hydrochloric acid, soda solution, and water. If part of the mixture has not already precipitated as a white crystalline solid, the mixture is transferred to a beaker until part of it solidifies, then it is filtered, cooled to about 10°, and centrifuged. About 95 grams of practically pure m-nitrochlorobenzene is obtained. The filtrate and the liquid from the centrifuge are combined and subjected to hractional distillation in vacuo. For this purpose, a column should be used which is about 80 cm. in length and provided with a partial condenser. Approximately the following fractions are collected ... 

The cycle has many similarities to a Rankine cycle which uses electrons as a working fluid. Unlike the normal Rankine cycle, however, the working fluid s "heat of evaporation", approximately the emitter work function, and its "heat of condensation", approximately the collector work function, can be varied in the thermionic converter. This feature provides the converter with great flexibility in matching the operating constraints of any particular system. 

This material is an anionic fatty amide condensate, approximately 98% active solids and is useful as a synthetic emulsifier and detergent. 

The applicability of the HK method for the pore size analysis of active carbons was questioned on the basis of adsorption isotherms obtained via density functional theory [19,20] as well as computer simulations [21,22]. The crudest assumption in this method is the use of the condensation approximation to represent the micropore filling, which in fact has a... 

It is shown that the position of PSDs calculated by the HK method depend strongly on the A(w)-relationship used. For instance, the pore widths at the maxima of PSDs obtained by the HK method with the Saito-Foley expression for cylindrical pores are underestimated about 1.4 nm. However, the HK method with the relationship between A and w established on the basis of good-quality MCM-41 materials [26] provides an accurate estimation of the pore widths of mesoporous silicas. While the position of PSD may be improved by a proper selection of the A(w)-relation, its unphysical features remain. The height of main peak is significantly reduced in order to compensate the appearance of an artificial small peak and tail in the micropore-mesopore transition range. These artifacts arise from the condensation approximation used in the HK method, which does not provide a good representation for the volume filling of micropores. 

The condensation approximation (CA) was introduced in the study of equilibrium adsorption on heterogeneous surfaces by Roginsky in the forties [40, 41]. However, it was only after Harris s rediscovery [42] and Cerofolini s systematic application to realistic model isotherms [43] that this method has become of wide use in the analysis of real adsorption systems. 

Figure 2 shows the distribution functions calculated for second-order kinetics in the frame of the condensation approximation for Elovich equations with different values of l — it is immediately seen that the larger is the ratio (i.e., the more accurate is the... 

In addition to the reduction in steam usage, there is also a reduction in cooling water required to operate the last effect condenser. Approximately 30 poimds of cooling water must be provided for each pound of steam... 

Solution. The normal boiling points of the six species are listed in Table 1.5. Because both trans- and cis-butene-2 are contained in the butenes product and are adjacent when species are ordered by relative volatility, they need not be separated. Using the method of Section 12.1, we find that all ordinary distillation columns could be operated above atmospheric pressure and with cooling-water condensers. Approximate relative volatilities assuming ideal solutions at 150°F (65.6°C) are as follows for all... 

The most popular solutions of Eq. (73) are limited to the range of the multilayer adsorption and capillary condensation [13]. By replacing the kernel 9x(A,x) by the condensation isotherm, one can express the function J(x) as the derivative of the amount adsorbed with respect to the pore width (the condensation approximation method). In order to carry out this differentiation one needs to express a(A) as a function of the pore size. This can be done by using a simplest form of the Kelvin equation, which is valid for the mesopore range [7] ... 

In 1983 Horvath and Kawazoe [143] proposed a method to derive analytical equations for the average potential in a micropore of a given geometry, which in fact relate the adsorption potential with the pore size x. These equations are used to express the amount adsorbed in micropores as a function of the pore width and subsequently to calculate the micropore volume distributioa Thus, the Horvath-Kawazoe (HK) procedure is a logical extension of the metliod based on the Kelvin equation to the micropore range, and can be considered as an extension of the condensation approximation method to the region of fine pores [4]. Further improvements and modifications of this method are reported elsewhere [144, 153-157]. 

Another important conclusion concerns the geometrical heterogeneity of nanoporous carbons, which is characterized by the micropore and mesopore volume distributions. The current work demonstrates that in terms of the condensation approximation both these dishibutions are directly related to the adsorption potential distribution. As shown the pore volume distribution can be obtained by multiplication of the adsorption potential distribution... 

The replacement of the kernel function q(p, Uq) with the condensation isotherm has been used by several authors, invoking the so-called condensation approximation at experimental temperatures far from absolute zero. These efforts generally use Eq. (18) without observing the temperature limitation. The condensation approximation can also be applied in Eq. (9) directly ... 

Applications of the condensation approximation have been well reviewed by Rudzinski and Everett [15], In general we can say that ignoring the equilibrium thermal distribution of the adsorbate among the surface patches of different values of Vq can result in a seriously distorted picture of the adsorptive potential distribution. 

B, The Condensation Approximation Approach to Adsorption Equiiibria and Kinetics... 

There is one approach in the theories of adsorption that is extremely useful in considering the effects of solid surface energetic heterogeneity. It has been used mainly to study adsorption equilibria, but it can also be applied to the study of adsorption kinetics, as we will soon show. For that reason we repeat briefly the principles of that approach, called the CA (condensation approximation) approach. 

Calculating 0, in this way has been known for a long time in theories of adsorption equilibria and is called the condensation approximation. 

The first term on the right-hand side of Eq. (20) is just the result obtained by applying the condensation approximation, i.e., assuming the hypothetical limit r 0. Because of the symmetry of the function 90/9s, the second term on the right disappears, and the first nonvanishing correction to —X(Zc) is the second correction term. For the Langmuir model of adsorption, to a good approximation [371,... 

However, the Elovich equation can be obtained in another way by using the ART and the condensation approximation. That method was first used in the theoretical work of Roginski [12] and reviewed later by Aharoni and Tompkins [9]. 

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