Subintervals

In applying Simpson s rule, over the interval [a, i>] of the independent variable, the interval is partitioned into an even number of subintervals and three consecutive points are used to determine the unique parabola that covers the area of the first... 

Figure 1-3 Areas Under a Parabolic Arc Covering Two Subintervals of a Simpson s Rule Integration.

Monte Carlo simulation uses computer programs called random number generators. A random number may be defined as a nmnber selected from tlie interval (0, 1) in such a way tliat tlie probabilities that the number comes from any two subintervals of equal lengtli are equal. For example, the probability tliat tlie number is in tlie subinter al (0.1, 0.3) is the same as the probability tliat tlie nmnber is in tlie subinterval (0.5, 0.7). Random numbers thus defined are observations on a random variable X having a uniform distribution on tlie interval (0, 1). Tliis means tliat tlie pdf of X is specified by... 

This pdf assigns equal probability to subintervals of equal lengtli in tlie interval... 

Subinterval defined by N (as listed) 50 counts, except that lowest and highest subintervals include all extreme values. 

Targeting is performed by cascading water from one concentration interval to the next, until the last concentration interval. Within each concentration interval, water is cascaded from one time subinterval to the next without degradation. This appears to be the mass transfer version of the targeting procedure presented for heat exchangers. 

Secondly, the network layout showed in Fig. 12.7 shows that 12.5 t of water should be supplied to Process 3, instead of 25 t stipulated in the problem specification. This can only be true if this process does not have flowrate constraints, but has a fixed mass load. The assumption of fixed mass loads was never mentioned in the analysis. This variation of flowrate is contrary to the assumption made in targeting. During targeting it was implicitly assumed that the flowrates were fixed as shown by the calculation of water demand in each of the time subintervals. 

It is worthy of note that the problem solution is such that in each time subinterval, the concentration constraints is met. This implies that, as long as water is available at the right time, it can safely be reused in any of the time subintervals within the concentration interval, i.e. the secondary constraints (concentration) is met in every step of the analysis, and the primary constraints (time) guides the formulation of the final solution (target design network). 

Figure 12.11 represents targeting in interval (0.25-0.51 kg salt/kg water). This interval, as shown in Fig. 12.8, has the B and the C reactions with an overall water demand of 560 kg. Since both these reactions start before the completion of the washing operation of product A, no reusable water is available in the reaction time subintervals. This implies that fresh water will have to be used. The accumulated fresh water demand is, therefore, 1560 kg. As this is the last concentration interval, this quantity presents itself as the target for the optimal design. This is equivalent to a 34% reduction in freshwater demand compared to the base case. 

Figure 12.13 represents targeting interval (0.25-0.51 kg salt/kg water) with water from the B wash reused in the C wash. Note that there is no longer any available water in the (4-5.5 h) subinterval, since it was transferred to the (6-7.5 h) subinterval, where it was used in the C wash. The amount of water remaining from the A wash is now 600 kg, since only 400 kg was reused for the washes. Nonetheless, there is still a need for 560 kg of fresh water due to the time constraints as mentioned previously. 

Instead of splitting the problem into concentration intervals and time subintervals, the problem is split into time intervals and concentration subintervals, with water demand plotted on the horizontal axis. The boundaries for time intervals and concentration subintervals are set by the process end-points. However, unlike in a case where time is taken as a primary constraints, the streams that are required or available for reuse in each concentration subinterval are plotted separately. This approach has proven to ease the analysis as will be shown later in this section. 

Available water from one concentration subinterval in a specific time interval is cascaded throughout the subsequent concentration subintervals, either within the same or subsequent time intervals without degeneration. This simply means that water which is available in any concentration subinterval can never be reused within the same concentration subinterval, irrespective of the time interval. If possible any surplus is transferred to higher concentration subintervals in the same time interval for reuse, or stored for reuse in later time intervals. However, this water cannot be reused in lower concentration subintervals or previous time intervals. Any shortfall within any concentration subinterval can be made up from lower concentration subintervals from previous time intervals, or from fresh water. As in the previous case, the eventual surplus becomes the system effluent and the accumulated fresh water make up constitutes the system intake. 

Parabolic Rule (Simpsonys Rule) This procedure consists of subdividing the interval a[Pg.47]

This method approximates//) by a parabola on each subinterval. This rule is generally more accurate than the trapezoidal rule. It is the most widely used integration formula. 

Fig. 4.1. Stratification consists of splitting the interval of interest into subintervals, thereby reducing the free energy barriers inside each window. The umbrella sampling method can bias the sampling and attempt to make it more uniform...

The interval of the variable c where the integral kernel is nonnegligible is divided into n subintervals, and the integral is approximated by rectangle or trapeze sums. 

This procedure is repeated for all subintervals. Because the free energy is a continuous function of z, the different sections of the free energy can be mafched to produce fhe final free energy profile over the whole range of... 

We can break up the total interval (0 to t,) into a number of unequal subintervals of length At. Then the FIT can be written, with no loss of rigor, as a sum of integrals ... 

A = is the dimensionless thickness parameter, p is the number of time subintervals, and / is the series number of time subintervals (see Appendix). 

The applicability of the foregoing procednre has been tested by modeling simple reaction under semi-infinite diffusion conditions (reaction 1.1) and EC mechanism coupled to adsorption of the redox couple (reaction (2.177)) [2]. The solutions derived by the original and modified step-function method have been compared in order to evaluate the error involved by the proposed modification. As expected, the precision of the modified step-function method depends solely on the value of p, i.e., the number of time subintervals. For instance, for the complex EC mechanism, the error was less than 2% for p>20. This slight modification of the mathematical procedure has opened the gate toward modeling of very complex electrode mechanisms such as those coupled to adsorption equilibria and regenerative catalytic reactions [2] and various mechanisms in thin-film voltammetry [5-7]. 

To obtain higher-accuracy solutions for Problem 15.2, divide the interval into more subintervals. Increase n from 4 as in Problem 15.2 to 8, and decrease h from 1 to. You will generally obtain a tridiagonal system of n — 1 linear equations up to 7 in the present case. For large n, solving the system of equations with paper and pencil becomes impractical, and it is necessary to find algorithms suitable for computation by computers. 

It turns out that for the second and third rows of elements the variations of the functions presented in Fig. (6a) are roughly similar, whereas some exceptions appear for the fourth row. One of these occurs in the subinterval Z = 20-23, and is... 

Compounding interest means that you split up the rate of interest into a designated number of subintervals (every three months, twice a year, daily, and so on), figure the interest earned during that subinterval, add the interest to the principal, and then figure the next interval s interest on the sum of the original principal plus the interest you ve added. As you may expect, you ll have more money in the end if you deposit it where you can earn compound interest rather than just a flat amount. The formula for compound interest... 

---