Beta parameter values

Similar analyses were carried out for other equipment in order to define the cumulative number of failures, as well as to evaluate the effect of maintenance and operational environment by assessing the beta parameter value. 

Figure 5.138. Program COMPSEG Dimensionless concentration of desired product YR versus dimensionless time THETA obtained by varying THETAMIX (0.03, 0.1,0.3, and 0.6, curves D, B, A and C). Parameter values used ETA=0.3, BETA=2000, DA=1.0.

BETA cols 11-20 oscillation control parameter default value is set equal to 0.25. To help prevent oscillations (thus slowing convergence) we not only require that the sum of squares, SSQ, decreases... 

Toluene alkylation with isopropyl alcohol was chosen as the test reaction as we can follow in a detail the effect of zeolite structural parameters on the toluene conversion, selectivity to cymenes, selectivity to para-cymene, and isopropyl/n-propyl ratio. It should be stressed that toluene/isopropyl alcohol molar ratio used in the feed was 9.6, which indicates the theoretical toluene conversion around 10.4 %. As you can see from Fig. 2 conversion of toluene over SSZ-33 after 15 min of T-O-S is 21 %, which is almost two times higher than the theoretical toluene conversion for alkylation reaction. The value of toluene conversion over SSZ-33 is influenced by a high rate of toluene disproportionation. About 50 % of toluene converted is transformed into benzene and xylenes. Toluene conversion over zeolites Beta and SSZ-35 is around 12 %, which is due to a much smaller contribution of toluene disproportionation to the overall toluene conversion. A slight increase in toluene conversion over ZSM-5 zeolite is connected with the fact that desorption and transport of products in toluene alkylation with isopropyl alcohol is the rate controlling step of this reaction [9]... 

Supposing that one has decided on bounds for a variable, one can fit a distribution that has a bounded support, such as the beta distribution or Johnson SB distribution. Alternatively, in a Monte Carlo implementation, one may sample the unbounded distribution and discard values that fall beyond the bounds. However, then a source of some discomfort is that the parameters of the distribution truncated in this way may deviate from the specification of the distribution (e.g., the mean and variance will be modified by truncation). It seems reasonable for Monte Carlo software to report the percentage discarded, and report means and variances of the distributions as truncated, for comparison to means and variances specified. 

It is a common experience, however, that if we repeat the calibration experiment, the estimated values of the intercept, b, and the slope, hj, will not be identical. If we then repeat the calibration experiment several times, we will find that the estimated values of the parameters will be distributed around an average value, hj, sometimes denoted (beta hat). The situation is depicted in Fig. 3.10. 

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