Basic procedures

The elimination method basically involves the elimination of variables in such a way that the final equation will involve only one variable. The procedure for a set of N equations is as follows. First, from one equation solve for x, as a function of other variables, X2, X3. Substitute this x, into the remaining N - I equations to obtain a new set of N - 1 equations with N - I unknowns, X2, X3. x,s[- Next, using one of the equations in the new set, solve for X2 as a function of other variables, X3, X4. and then substitute this X2 into the remaining N - 2 equations to obtain a new set of N - 2 equations in terms of N - 2 unknown variables. Repeat the procedure until you end up with only one equation with one unknown, x, from which we can readily solve for x, . Knowing x, we can use it in the last equation in which X j was written in terms of x y. Repeat the same procedure to find x,. The process of going backward to find solutions is called back substitution. 

Let us demonstrate this elimination method with this set of three linear equations. 

Substitute this Xj into Eqs. B.31 and B.32 to eliminate Xj from the remaining two equations, and we have 

We see that the patterns of Eqs. B.36 and B.39 are exactly the same and this pattern is independent of the number of equations. This serial feature can be exploited in computer programming. 

the elimination process finally yields one equation in terms of the variable X3, from which it can be solved as 

According to the concept of the rate-determining step, any one of these steps can be controlling. 

Focusing now on the case where surface reaction is controlling, the basic procedure in developing an LHHW model is to write the rate equation in terms of the surface coverage of reactant A rather than its concentration [A], Sometimes, as in reactions requiring a second (vacant) site for adsorbing a product (e.g., A — P + / ), the rate will also directly depend on the fraction of surface covered by vacant sites 6 and when there is dissociation of a reactant, a pair of adjacent vacant sites should be available, so that the rate of adsorption would now be proportional to 6F rather than 9. One of the characteristics of the surface reactions is that in most of the situations the adsorption is much faster than the rest of the steps. In such a situation, we can easily assume that the adsorption step is at pseudoequilibrium, indicating that the rate of adsorption is equal to the rate of desorption  

For this simple case, the coverage of A is simply what can be predicted from Langmuir adsorption isotherm (Equation 5.20). Another very frequently encountered situation is the dissociative adsorption, or adsorption requiring more than one site. 

Let us now turn to the case where adsorption of one of the components is controlling. Thus, consider the reaction 

Bodenstein approximation At some point during the reaction, the rate of formation and the rate of disappearance of an intermediate are identical causing a steady-state concentration, or surface coverage, of fhe reactive intermediate. 

---

The basic procedure for the derivation of a least squares finite element scheme is described in Chapter 2, Section 2.4. Using this procedure the working equations of the least-squares finite element scheme for an incompressible flow are derived as follows ... 

When the catalyst is triethylamine, the yield is nearly 100% cycHc oligomers but if pyridine is used, the polymer is nearly 100% linear. A basic catalyst in the second step, such as lithium stearate or an organic titanate [bis-(acetylacetonato)diisopropoxytitanium], produces a polycarbonate with a molecular weight of 250,000—300,000 when polymerized at 300°C for 30 min. A fiber glass composite has been prepared using this basic procedure (39). 

Painting of all metallic surfaces of a switchgear or a controlgear assembly is an essential requirement to provide it with an aesthetic appearance, on the one hand, and to prevent it from rust and corrosion, on the other. Painting serves these purposes by providing the machine with a hard and longer-lasting metallic surface. We describe briefly, the basic procedure to paint and test painted surfaces. In the discussion, we have laid more emphasis on MS sheet-metal surfaces as these are more typical. 

Fig. 2.41. Basic procedure for EELS quantification exercised for BN (a) background extrapolation and subtraction ...

The same basic procedures are common to the CRBR PSA (1977) Apostolokis and Kazarians (1980) and Gallucci (1980). They consist of the following steps ... 

Using the procedure described above, the target levels can be determined. Relevant calculation methods and expertise are needed in all the phases. Although the calculation of target levels takes place in different ways for particular cases, the basic procedure remains the same. The target level calculation also varies for different outside temperatures and different process parameters. 

The same basic procedures as described above can be used if the source and hood are rectangular rather than circular. The two dimensions of the plume and hood must be calculated independently by substituting them for d, in the equations developed above. 

Predictive human error analysis (PHEA) is the process via which specific errors associated with tasks or task steps are predicted. The process also considers how these predicted errors might be recovered before they have negative consequences. The inputs to the process are the task structure and plans, as defined by the task analysis, and the results of the PIF analysis. The basic procedure of the PHEA is as follows ... 

The final step in the process of standardizing our columns was to try and maintain the high quality of columns from batch to batch of gel from the manufacturer. This was done by following the basic procedures outlined earlier for the initial column evaluation with two exceptions. First, we did not continue to use the valley-to-peak ratios or the peak separation parameters. We decided that the D20 values told us enough information. The second modification that we made was to address the issue of discontinuities in the gel pore sizes (18,19). To do this, we selected six different polyethylenes made via five different production processes. These samples are run every time we do an evaluation to look for breaks or discontinuities that might indicate the presence of a gel mismatch. Because the resins were made by several different processes, the presence of a discontinuity in several of these samples would be a strong indication of a problem. Table 21.5 shows the results for several column evaluations that have been performed on different batches of gel over a 10-year period. Table 21.5 shows how the columns made by Polymer Laboratories have improved continuously over this time period. Figure 21.2 shows an example of a discontinuity that was identified in one particular evaluation. These were not accepted and the manufacturer quickly fixed the problem. 

The design procedure will vary with the type of flue, but the basic procedure will not change. It is necessary to consider the relevant parameters discussed above and decide on flue dimensions and materials of construction to satisfy them. The parameters to consider are ... 

Commissioning procedures will be specific to particular plant, but the basic procedure is general to all. There are four consecutive stages ... 

The development of objective methods is not difficult, and the basic procedures are available. The difficulty resides in establishing the proper relationships between analytical results and quality and condition in terms of consumer acceptance and utility, and in terms of the results to be expected in good, economical, commercial practice. 

There are many protocols and different types of platforms available, e.g. GeneChips from Affymetrix, Illumina Bead Arrays, Arrays from Agilent, Applied Biosystems, GE Healthcare, customized spotted cDNA microarrays etc. the basic procedure for a large-scale measurement of gene expression involves the preparation of total or mRNA from the biological sample(s) under investigation (e.g. candidate tissue) and the hybridization of copied labelled RNA or cDNA to the DNA elements on the array surface (Fig. 1). 

There are two basic procedures that have been successfully used for the separation of isomers. The first is to add a chiral agent to the mobile phase such that it is adsorbed, for example, on the surface of a reverse phase, producing a chirally active surface. This approach has been discussed on page (38) in chapter 2. The alternative is to employ a stationary phase that has been produced with chiral groups bonded to the surface. 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation (5.28). The same basic procedure can be used for Equation (5.25). The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from Hout after each time step. 

After metabolically stable cultures have been obtained, pure cultures of the relevant organisms may then be obtained by any of three basic procedures ... 

If the tu s and/or the Si/s are functions of Vj (this is often the case) then the basic procedure may be repeated until convergence is obtained on Vj. 

The basic procedure of the VILM model is to send an initial distribution of drops through a specified number of strong and weak zones. With each pass through the strong and weak zones, the evolution of the drop distribution is determined based on the fundamentals of breakup and coalescence. 

The basic procedures for calculating the Mond indices are similar to those used for the Dow index. 

The basic procedure used in the Thiele-Geddes method, with examples, is described in books by Smith (1963) and Deshpande (1985). The application of the method to computers is covered in a series of articles by Lyster el al. (1959) and Holland (1963). 

Preliminary assessments have been conducted at more than 31,000 sites reported as possible sources of contamination. In 1990 there were over 1100 sites (presenting the greatest health risk and hence eligible for Superfund reimbursement) on the NPL.8-9 The NCP has outlined the level of cleanup necessary at Superfund sites and established the basic procedures that have to be followed for the discovery, notification, response, and remediation of the hazardous waste sites.10... 

The basic theory of mass transfer to a RHSE is similar to that of a RDE. In laminar flow, the limiting current densities on both electrodes are proportional to the square-root of rotational speed they differ only in the numerical values of a proportional constant in the mass transfer equations. Thus, the methods of application of a RHSE for electrochemical studies are identical to those of the RDE. The basic procedure involves a potential sweep measurement to determine a series of current density vs. electrode potential curves at various rotational speeds. The portion of the curves in the limiting current regime where the current is independent of the potential, may be used to determine the diffusivity or concentration of a diffusing ion in the electrolyte. The current-potential curves below the limiting current potentials are used for evaluating kinetic information of the electrode reaction. 

The basic procedure for the power law fluid is the same as above for the Newtonian fluid. We get a first estimate for the Reynolds number by ignoring fittings and assuming turbulent flow. This is used to estimate the value of / (hence Kpipe) using Eq. (6-44) and the Knt values from the equivalent 3-K equation. Inserting these into Eq. (7-50) then gives a first estimate for the diameter, which is then used to revise the Reynolds number. The iteration continues until successive values agree, as follows ... 

In 1972, we reported a general procedure for the preparation of highly reactive metal powders. The basic procedure involved the reduction of a metal salt in a hydrocarbon or ethereal solvent. The reductions are most generally carried out with alkali metals such as potassium, sodium, or lithium. A wide range of methods have been developed to carry out the reductions. The reactivities of these resulting black powders exceed other reports in the literature for metal powders. This high reactivity has resulted in the development of several new synthetic techniques and vast improvements in many older, well established reactions. This review concentrates on the metals Mg, Ni, Zn, Cd, Co, Cu, Fe, and U. 

Biochemical EPR samples are almost always collections of randomly oriented molecules (frozen) aqueous solutions in which each paramagnetic molecule points in a different direction. In order to generate simulations of these powder EPR spectra we have to calculate the individual spectrum for many different orientations and then add these all up to obtain the powder pattern. Numerical procedures that generate sufficient spectra to approximate a powder pattern are collectively known as walking the unit sphere algorithms. Here is the basic procedure ... 

Basic procedure (ACW kit) Mix 1500 pL of ACW reagent 1 (diluter) with 1000 pL of ACW reagent 2 (buffer) and 25 pL of photosensitizer reagent (lumi-nol based). Start measurement after brief vortexing. Assayed solution (or control) is added before addition of photosensitizer reagent. Volume of ACW reagent 1 is reduced by the volume of assayed plasma sample. Standard substance ascorbic acid. Duration of measurement 2-3 min. Measured parameter effective lag phase = lag-phase sample - lag-phase blank. Assayed amount of human blood plasma 2 pL. 

Basic procedure LDL is isolated from fresh EDTA(Na2) plasma by sequential ultracentrifugation in KBr step gradient at 120,000 rpm (microultracentrifuge HITACHI CS 120) and a temperature of 10°C for 2 X 2 h. The LDL was covered with nitrogen in a screw-cap tube and stored in the refrigerator at 4°C until further use. 

The functionalization of a carbon-carbon double bond is a basic procedure in organic chemistry. Oxidation is one of the many different ways in which such a double bond may be transformed. Although the term oxidation is rather a general one, we will mostly concern ourselves in this review with oxygenations, i.e. reactions in which oxygen is added to the substrate with or without cleavage of the carbon-carbon bond. Only some mention will be made of other formally oxidative procedures such as dehydrogenation. 

---