Cycle length

The Chevron process was used in two U.S. plants, although it is no longer used. Cycle lengths tanged from 6—30 d, depending on catalyst age and OX content of the feed. Operating conditions were temperature of 370—470°C and space velocity of about 0.5/h. Addition of 5 wt % steam reduced disproportionation losses. 

Scaling Up Test Results The results of small-scale tests are determined as dry weight of sohds or volume of filtrate per unit of area per cycle. This quantity multiplied by the number of cycles per day permits the calculation of either the filter area reqiiired for a stipulated daily capacity or the daily capacity of a specified plant filter. The scaled-up filtration area should be increased by 25 percent as a factor of uncertainty. In the calculation of cycle length, proper account must be made of the downtime of a batch filter. 

Figures 3,20, 3,21 and 3,22 show how the number of cyclic states (Nc), the average cycle length Crmave) and total fraction of states on cycles fc) changes as...

Fig. 3.21 Cycle length as a function of lattice size for a fejw representative one-dimensional elementary rules see text.

Lemmas 1 and 2 allow us to characterize cycle lengths for all even-sized systems. The following result effectively determines the cyclic structure for the case when N is odd. 

The previous discussion has shown us how to calculate the total number of possible cyclic states. We also know, from Lemma 2, that all cycle lengths must divide the maximal cycle length Hiv obtain the exact number of distinct cycles and their lengths takes a little bit more work. If flw prime, we know that the only possible cyclic lengths are 1 and It can then be shown that only the null configuration is a fixed point unless N is some multiple of 3, it which case there are exactly four distinct cycles of length one. If Hat i ot prime, there can exist as many cycles as there are divisors of Although there is no currently known closed form... 

Cycle Lengths. Suppose we want to find the possible cycle lengths for a size N system. Theorem 2 shows that any configuration yl(x) on a cycle may be written in the form yl(x) = (1 + B[x), where B x) is some polynomial and (we recall)... 

D2 N) is defined as the largest value of 2 that divides N. The cycle length must then be the minimum power n such that... 

To summarize, we must perform the following two basic computations to find the lengths of all possible cycle lengths for a system of size N ... 

The goal, as before, is to find the state graph (= Gl) description of the system, where Gl is a directed graph with q vertices and is defined by G jj = 1 S(j) = L Familiar quantities of interest include cycle lengths, number of... 

Table 5.5 gives all possible cycle lengths supported by (symmetric) C Q (for systems in P[2]), along with the volume of cycle state vol C) = Y iniU. Bracketed cycle lengths are those which appear for all C Q. The coefficients ai are defined by Pi(a ) = l + X ... 

K State Cycle Length Number of State Cycle Attractors Stability with respect to minimal perturbations... 

Table 8.9 State cycle length and the number of state cycles for random boolean nets of size N and connectivity k a = pn — 1/2, where Pk is the mean internal homogeneity of all Boolean functions on K inputs (see text).

Table 8.9 shows that while the number of attractors increases exponentially, the average cycle length increases rather slowly as a function of N. Random Boolean nets with connectivity one are also generally only moderately stable with respect to minimal perturbations. 

Internal Homogeneity Clusters Walker and Ashby [walker66] first showed that increasing the sameness of the entries or the internal homogeneity - of a Boolean function s rule table tends to decrease the cycle length. 

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