Triangle method

The alternative triangle method. Making a. substantiated choice of the operator B will be justified in more detail a little latter. Recall that any product B of economical operators is also an economical operator. This is certainly true for triangle operators Bi and B2 for which the operator B — B B2 would be economical . 

DRAVNIEKS, A. (1976). Development of a dynamic forced choice triangle method for the measurement of emissions and ambient odours. Final report IITRI project C8217. IITRI, Chicago. 

To have a quantitative idea of the problem of intraparticle diffusion, effectiveness factors for the two catalysts were calculated from the observed second order rate constants (based on surface area) using the "triangle method" suggested by Saterfield (4). The effectiveness factors for Monolith and Nalcomo 474 catalysts on Synthoil liquid at 371°C (700 F) were calculated to be 0.94 and 0.216, respectively. In applying the relationship between the "Thiele Modulus," 4>> and the "effectiveness factor," n> the following simplifying assumptions were made ... 

Die Dreiecksmethode ( triangle method") 160>. Auch hier werden 2 Reaktionsgeschwindigkeitsmessungen und ein Diagramm rj = ri([Pg.58]

Techniques without a preceding separation step such as direct inlet probe MS [163] or H-NMR [69] allow one to estimate the sum concentration of toxaphene. In connection with GC/ECD, Walter and Ballschmiter used the so-called triangle method for this purpose. It estimates the area under the raised baseline caused by not resolved toxaphene peaks [174]. Although this method has to be... 

Operating point S is the theoretical optimal operating condition derived from triangle method [S]. The purity of the extract and the raf te at this experimental point is... 

The "triangle method" of activity evaluation. When Dw. is not independently known, other methods must be used (Hougen and Watson, 10). One of these is the "size method, or "triangle method. In this method the rate of reaction per unit volume of catalyst, dn /dt and dn /dt, is measured for two different particle sizes, Ri and R2, under otherwise identical conditions. It follows from equation (28) that the ratio of measured rates gives the ratio of the diffusion factors... 

Fig. 10. Triangle" method for determination of diffusion and activity parameters from measurements on two particle sizes.

For this evaluation it is a very fortunate fact that the shape of the 77 versus

reaction order (except when approaching zero order), nor to effects of molar volume change, nor to particle geometry. This makes it possible to determine 771 and 772 by the triangle method and, consequently, with considerable independence of the variables mentioned. 

Another method applicable to resolving the case in which two experimental particle sizes of the catalyst are both in the region of strong diffusion effects is to reduce the temperature and thus (and consequently y) until the triangle method can be applied. While thus 7c is found for the lower temperature only. Aft may also be found and, since De(f is essentially constant for all practical changes in temperature encountered in this procedure, it can now be used to solve the original case of higher temperature by the use of equation (29). 

Example of application of triangle method. A study was made by Goodwin and Weisz (unpublished) of the intrinsic activity of ZnO when sintered at various temperatures to achieve various amounts of defect structure. In such a series of catalyst samples the surface area and pore structure, and therefore diffusivity, changes over a wide range. Measured activities have to be scrutinized for diffusion effects. Methanol decomposition rates at 270°C. were measured in a Schwab reactor, on particles of two different sizes, Ri = 0.13 cm., and = 0.4 cm. Therefore, = 3. For the various samples the measured activities per gram of... 

The sample sintered at the highest temperature presents an example of the limitation of the triangle method in that the triangle fit becomes indeterminate. However, this result places a lower limit to the activity of this sample which was sufficient for examining the relative activity of these samples. 

Comparison of the Safety Margin and the Triangle Method for Linear Isotherms 815... 

Equations 17.51 and 17.52 define a sufficient number of criteria to allow the correct choice of the operating conditions in an SMB operating rmder linear conditions. This set of conditions is equivalent to the one derived by Storti et al. [16] (see later. Subsection 17.6.5), the so called Triangle Method. However, both sets of conditions are based on the assumption that all columns have identical characteristics and an infinite efficiency. In practice, the different columns of an SMB separator cannot be identical. Their individual average porosity, permeability, retention factors, and efficiency are more or less different, however slightly. The influence of the possible differences between the colunms of an SMB imit on its performance is discussed later (Subsection 17.7.1.5)... 

Optimization of the SMB Process with a Linear Isotherm Using the Triangle Method... 

The two main methods used to select the experimental conditions under which there is complete separation in linear SMB are the sa/ety margin approach [27,46] and the triangle method [16,28,43], It is easy to show that both approaches are equivalent in the linear case [47]. Equations. 17.50a to 17.50c result from the Safety Margin approach. Rewriting them and using the classical relationship Q =... 

The left hand sides of these four equations are identical to the corresponding ratios of the net fluid flow rates and the solid phase flow rate, nij. These ratios are used as the basis for the separation conditions derived in the triangle method (see Eqs. 17.59a through 17.59d of the previous section). In these equations, m2 must be larger than m3 because the liquid phase flow rate must be larger in section III than in section II. Substituting m2 and m3 in this condition (m2 > m3) (Eq. 17.60) with the equivalent expressions m2 = and m3 = 02/ 111, respectively, gives directly > 2/ nir i-s., Eq. 17.51 < ot). Similarly, if my in Eqs. 17.59a... 

A second approach is the triangle method which was developed based on the equilibrium theory model which assumes that the adsorption equilibrium is established everywhere at any time in the column. The equivalent TMB configuration with a four-section emit will be considered here. The model equations consist in four sets of mass balance equations, one for each section j j = 1,- , 4), with the relevant boundary conditions and the integral material balances at the column ends and at the nodes of the unit [16,28]. These equations were given earlier, in Section 17.2 (Eqs. 17.4 to 17.6). 

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