Simplified equations technique

ISA TR 84.00.02-2 Illustrates a simplified equation technique for calculating the probabilities of failure for safety instrumented functions designed in accordance with IEC 61511-1 ANSI/ISA- 

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Numerical simulations are designed to solve, for the material body in question, the system of equations expressing the fundamental laws of physics to which the dynamic response of the body must conform. The detail provided by such first-principles solutions can often be used to develop simplified methods for predicting the outcome of physical processes. These simplified analytic techniques have the virtue of calculational efficiency and are, therefore, preferable to numerical simulations for parameter sensitivity studies. Typically, rather restrictive assumptions are made on the bounds of material response in order to simplify the problem and make it tractable to analytic methods of solution. Thus, analytic methods lack the generality of numerical simulations and care must be taken to apply them only to problems where the assumptions on which they are based will be valid. 

Bacterial leaching of copper by Thiobacillus ferrooxidans has been practiced for years. Yearly, 250000 tons of copper are recovered by microbial leaching techniques in the USA. This process can be represented by the following simplified equations ... 

The calculations may be done with simplified equations, fault trees, Markov models or other techniques depending on the complexity of the model and the demand mode of operation. 

The PFDavg may be calculated using one of several different techniques as described in Chapter 5. The most common approaches involve simplified equations, fault trees or Markov models. The resultant PFDavg value must be compared with a table from ANSl/ISA-84.00.01-2004 (lEC 61511 Mod)... 

When this model is solved used numerical techniques and the time dependent PFD values averaged, the result is a PFDavg = 0.013067. The difference between this answer and the previous answer of 0.01318 represents the approximation of the simplified equations. 

It should be noted that the simplified equations derived by approximation techniques are only valid for low failure rates. 

ISA-TR84 00.02-2002-Part 2 Safety Instrumented Functions (SIF)-Safety Integrity Level (SIL) Evaluation Techniques Part 2 Determining the SIL of a SIP via Simplified Equations Introduction, ISA, June 2002. 

In Chapter 2, we saw that the configuration integral is the key quantity to be calculated if one seeks to compute thermal properties of classical (confined) fluids. However, it is immediately apparent that this is a formidable task because it n (iuire.s a calculation of Z, which turns out to involve a 3/V-dimcnsional integration of a horrendously complex intcgiand, namely the Boltzmann factor exp [-U (r ) /kuT] [see Eq. (2.112)]. To evaluate Z we either need additional simplifying assumptions (such as, for example, mean-field approximations to be introduced in Chapter 4) or numerical approaches [such as, for instance, Monte Carlo computer simulations (see Chapters 5 and 6), or integral-equation techniques (see Chapter 7)]. 

Despite obvious deficiencies, this simplified equation has been tested on the Sn2 barrier data and other data as well, and gave reasonable quantitative results in comparison with equation (3). Perhaps more importantly, the expression in equation (7) reproduces the qualitative trends of equation (3). Specifically, the equation shows that the barrier arises from an interplay of the curvature-scaled promotion energy, the reaction driving force, and the TS resonance energy. Other simplifications are possible, but the general outcome is virtually identical. A useful feature of equation (3) and its simplified versions is their interface quality between experiment and theory. Thus, without calculating the whole SCD, the key factors can be either computed by means of any ab initio technique or extracted from thermochemical or spectroscopic data, and used to rationalize computational and experimental data. 

Dyna.micPerforma.nce, Most models do not attempt to separate the equiUbrium behavior from the mass-transfer behavior. Rather they treat adsorption as one dynamic process with an overall dynamic response of the adsorbent bed to the feed stream. Although numerical solutions can be attempted for the rigorous partial differential equations, simplifying assumptions are often made to yield more manageable calculating techniques. 

Dual solvent fractional extraction (Fig. 7b) makes use of the selectivity of two solvents (A and B) with respect to consolute components C and D, as defined in equation 7. The two solvents enter the extractor at opposite ends of the cascade and the two consolute components enter at some point within the cascade. Solvent recovery is usually an important feature of dual solvent fractional extraction and provision may also be made for reflux of part of the product streams containing C or D. Simplified graphical and analytical procedures for calculation of stages for dual solvent extraction are available (5) for the cases where is constant and the two solvents A and B are not significantly miscible. In general, the accurate calculation of stages is time-consuming (28) but a computer technique has been developed (56). 

For ideal solutions (7 = 1) of a binary mixture, the equation simplifies to the following, which appHes whether the separation is by distillation or by any other technique. 

If V is not a function of time, the Schrodinger equation can be simplified using the mathematical technique known as separation of variables. If we write the wavefunction as the product of a spatial function and a time function ... 

In computer operations with other kinetic systems, Equation 8 may be used, and all the unique features of the kinetic system may be incorporated into the value of Q which may of course be a very complex expression. This technique is of interest only in that it simplifies the work necessary to analyze data using any specific kinetics for a chemical reaction. The technique requires sectioning the catalyst bed in most cases with normal space velocities, 50-100 sections which require 2-3 min of time on a small computer, appear to be sufficient even when very complex equations are used. 

An approximation technique can greatly simplify calculations when the change in composition (x) is less than about 5% of the initial value. To use it, assume that x is negligible when added to or subtracted from a number. Thus, we can replace all expressions like A + x or A - 2x, by A. When x occurs on its own (not added to or subtracted from another number), it is left unchanged. So, an expression such as (0.1 — 2x)2x simplifies to (0.1 )2x, provided that 2x 0.1 (specifically, if x < 0.005). At the end of the calculation, it is important to verify that the calculated value of x is indeed smaller than 5% of the initial values. If it is not, then we must solve the equation without making an approximation. 

We need to be able to write balanced chemical equations to describe redox reactions. It might seem that this task ought to he simple. However, some redox reactions can be tricky to balance, and special techniques, which we describe in Sections 12.1 and 12.2, have been developed to simplify the procedure. 

Let us now specialize the above equations for the special case when only the population Ur of the r-th MO changes, and the reference scheme is a simple a>-technique [10] applied to an extended-Hiickel method, which is a highly simplified form of the BMV procedure. 

The technique allows immediate interpretation of the regression equation by including the linear and interaction (cross-product) terms in the constant term (To or stationary point), thus simplifying the subsequent evaluation of the canonical form of the regression equation. The first report of canonical analysis in the statistical literature was by Box and Wilson [37] for determining optimal conditions in chemical reactions. Canonical analysis, or canonical reduction, was described as an efficient method to explore an empirical response surface to suggest areas for further experimentation. In canonical analysis or canonical reduction, second-order regression equations... 

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