Calculations errors

The Kohonen Self-Organizing Maps can be used in a. similar manner. Suppose Xj., k = 1,. Nis the set of input (characteristic) vectors, Wy, 1 = 1,. l,j = 1,. J is that of the trained network, for each (i,j) cell of the map N is the number of objects in the training set, and 1 and j are the dimensionalities of the map. Now, we can compare each with the Wy of the particular cell to which the object was allocated. This procedure will enable us to detect the maximal (e max) minimal ( min) errors of fitting. Hence, if the error calculated in the way just mentioned above is beyond the range between e and the object probably does not belong to the training population. 

There is a great difference between various simulators (5) in terms of how easily and how well the hypothetical calculation units can be incorporated in the simulation. The trial-and-error calculations, which ate called iterative calculations, do not always converge for every flow sheet being simulated. Test problems can be devised to be tried with various simulators to see if the simulator will give a converged solution (11). Different simulators could take different numbers of iterations to converge and take different amounts of computet time on the same computet. 

One way of overcoming these disadvantages of the (DCFRR) method is to make estimates of the times required to reach certain values of (DCFRR). For example, how many years will it take to reach (DCFRR)s of 10 percent, 15 percent, 20 percent per year, etc. Although (DCFRR) trial-and-error calculations and (NPV) calculations are tedious if done manually, computer programs which are suitable for programmable pocket calculators can readily be written to make calculations easier. 

In concentrated wstems the change in gas aud liquid flow rates within the tower and the heat effects accompanying the absorption of all the components must be considered. A trial-aud-error calculation from one theoretical stage to the next usually is required if accurate results are to be obtained, aud in such cases calculation procedures similar to those described in Sec. 13 normally are employed. A computer procedure for multicomponent adiabatic absorber design has been described by Feiutnch aud Treybal [Jnd. Eng. Chem. Process Des. Dev., 17, 505 (1978)]. Also see Holland, Fundamentals and Modeling of Separation Processes, Prentice Hall, Englewood Cliffs, N.J., 1975. 

The actual yield may be obtained from algebraic calculations or trial-and-error calculations when the heat effects in the process and any resultant evaporation are used to correc t the initial assumptions on calculated yield. When calculations are made by hand, it is generally preferable to use the trial-and-error system, since it permits easy adjustments for relatively small deviations found in practice, such as the addition of wash water, or instrument and purge water additions. The following calculations are typical of an evaporative ciy/staUizer precipitating a hydrated salt, if SI units are desired, kilograms = pounds X 0.454 K = (°F 459.7)/I.8. 

The second section of the spreadsheet contains the overall flows, the calculated component flows, and the material balance closure of each. The weighted nonclosure can be calculated using the random error calculated above, and a constraint test can be done with each component constraint if desired. Whether the measurement test is done or not, the nonclosure of the material balance for each component gives an indication of the validity of the overall flows and the compositions. If particiilar components are found to have significant constraint error, discussions with laboratory personnel about sampling and analysis and with instrument personnel about flow-measurement errors can take place before any extensive computations begin. 

Another view is given in Figure 3.1.2 (Berty 1979), to understand the inner workings of recycle reactors. Here the recycle reactor is represented as an ideal, isothermal, plug-flow, tubular reactor with external recycle. This view justifies the frequently used name loop reactor. As is customary for the calculation of performance for tubular reactors, the rate equations are integrated from initial to final conditions within the inner balance limit. This calculation represents an implicit problem since the initial conditions depend on the result because of the recycle stream. Therefore, repeated trial and error calculations are needed for recycle... 

Obviously, if over-simplifying assumptions are to be avoided, a tremendous amount of trial-and-error calculation is involved in such a study. 

The cost of utilities is one of the most significant, yet difficult chores encountered in estimating operating costs. As discussed earlier, the amount of utilities required for both the process and the offsite areas must be estimated as accurately as possible. If utilities are generated in the project, the utilities required to operate the utility area must be included. Any increase in the project requires re-estiraating the utilities consumed in the utility area. This can result in a trial and error calculation to get the total cost of utilities. 

Mak used the same Lapple article to develop a method for designing flare headers. His method has the advantage of starting calculations from the flare tip (atmospheric) end, thus avoiding the trial and error calculations of methods starting at the inlet. 

Although Figures 19 and 20 can be used for line sizing, it should be noted that Figure 19 requires more extensive trial and error calculations. 

Therefore, after several trial-and error calculations these results indicate that after flashing, there would be 70% vapor (approximately) of above composition and 30% (mol) liquid. 

A trial-and-error calculation is necessary to solve for W until a value is found from the In Wj/W equation above that matches the xq avg which represents the required overhead distillate composition. By material balance ... 

Using Equation 9-31B trial and error, calculate using Gf, for given Lf/Gf until the desired AP is obtained. Lf must be below 20,000. For higher Lf use chart in the original article not included here. 

To calculate the outside film coefficient, you need to know the difference in temperature of the condensing vapor (T, ) and the pipe wall temperature (L). The pipe wall temperature is determined hy trial-and-error calculations using the following equation/ ... 

Note Using the above obtained values for K, A and S, one may attempt to optimize drilling parameters from Equations 4-278, 4-279 and 4-280 however, in the case considered, the bit life is limited by bearings wear. Consequently Equations 4-278, 4-279 and 4-280 are not applicable. Nevertheless a simple trial-and-error calculation can be used to find the desired parameters. 

Extensive comparisons of experimental frequencies with HF, MP2 and DFT results have been reported [7-10]. Calculated harmonic vibrational frequencies generally overestimate the wavenumbers of the fundamental vibrations. Given the systematic nature of the errors, calculated raw frequencies are usually scaled uniformly by a scaling factor for comparison with the experimental data. 

Interproton distances of 0-ceIIobiose (see Ref. 49) error 0.01 A. Interproton distances of 1,6-anhydro- -D-glucopyranose (see Ref. 49) error 0.01 A. Interproton distances of -cellobiose octaacetate (see Ref. 49) error 0.05 A. Interproton distances of 2,3,4-tri-0-acetyl-l,6-anhydro- -D-glucopyranose (see Ref. 49) error 0.05 A. Error calculations based on the errors of the measured quantities in Eqs. 18 and 21. Interproton distances calculated from the relaxation parameters of the methylene protons. 

Error calculations based on a 5% error in the measured R, values. Calculated from Eq. 5/ and the py values given in Ref. 85. Estimated error of 0.01 A. Only the average value could be determined for thqse protons, because their resonances were not separately resolved for the isotopomers 44c and 44d. Calculations based on the assumption that = p2t = Ps.sc = Pi,3 - From Refs. 78 and 85. 

Figure 2. Histograms of Monte Carlo simulations for two synthetic analyses (Table 1) of a 330 ka sample. The lower precision analysis (A) has a distinctly asymmetric, non-Gaussian distribution of age errors and a misleading first-order error calculation. The higher precision analysis (B) yields a nearly symmetric, Gaussian age distribution with confidence limits almost identical those of the first-order error expansion.

Th/U age and error calculation, with or without decay-constant errors, via first-... 

With simple reactions it is usually possible to balance the stoichiometric equation by inspection, or by trial and error calculations. If difficulty is experienced in balancing complex equations, the problem can always be solved by writing a balance for each element present. The procedure is illustrated in Example 2.3. 

The value of r is found by trial-and-error calculations. Finding the discount rate that just pays off the project investment over the project s life is analogous to paying off a mortgage. The more profitable the project, the higher the DCFRR that it can afford to pay. 

This is found by trial-and-error calculations. The present worth has been calculated at discount rates of 25, 35 and 37 per cent. From the results shown in Table 6.8 it will be seen that the rate to give zero present worth will be around 36 per cent. This is the discounted cash-flow rate of return for the project. 

Efficient techniques for the solution of the trial and error calculations necessary in multicomponent flash calculations are given by several authors Hengstebeck (1976) and King (1980). 

Usually an approximate estimate of the wall temperature is sufficient, but trial-and-error calculations can be made to obtain a better estimate if the correction is large. 

The disadvantage of this method is that if three or more alternatives are being compared the process is time-consuming unless a digital computer is used. First any two projects are compared, then the best is compared with one of those remaining, and this process is continued until all have been considered and only the best remains. Each comparison involves trial-and-error calculations. On the other hand, the Net Present Value method requires only one calculation for each project. 

The obvious time to use computers is when some calculation is repeated over and over again. This can be in a trial-and-error calculation such as the calculation of the rate of return. As noted in Chapter 10, the best way to do this is to assume an interest rate, perform the calculations, and determine whether the net present value is zero. If it is not, another choice is made, and the net present value for this choice is calculated. This procedure is repeated until the desired answer is obtained. 

Suppose one wanted to compare the behavior of two polymers and their blends. Let us define the signal as the difference between the logarithims of the viscoelastic quantities and the noise as the error calculated for a specific set of viscoelastic properties associated with a specific composition. The signal to noise ratio would have the appearance of the three curves shown in Figure 2 for a PMMA/Hytrel blend >3/1. Selection of the optimum conditions for comparison is apparent in that figure. Emphasis should be placed at those temperatures with high signal/noise ratios. 

As there are few data on the ratio of thoron concentration to radon, we have assumed that the value is constant and is equal to 0.14, Error(4) and Error(5) are assumed to be negligible for the error calculation. The error due to ambiguities in classification and discrimination of tracks according to the shape or the size was assessed by experiment and the relative error (1 S.D.) was less than 10%. The relative error due to the variations of F, f, and (V a) is less than 13% as described in the previous section. The combined error due to these factors is estimated to be 16%. The total error can be obtained by combining this error and Error(l). If we use the definition that the lowest detectable limit is the radon concentration at 50 % relative S.D., the lowest detectable... 

Equation (e) is merely a definition of the mass flow rate. Equation (/) is a standard correlation for the friction factor for turbulent flow. (Note that the correlation between /and the Reynold s number (Re) is also available as a graph, but use of data from a graph requires trial-and-error calculations and rules out an analytical solution.)... 

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