Hand calculations

A simplified calculation of the TIP is possible on the basis of crystal field theory. The diagonal components of the /1-tensor, restricted to only the first excitation, are 

For a hexacoordinate Cu2+ complex adopting the geometry of an elongated bipyramid (D symmetry) the individual term functions can be replaced by the corresponding /-orbitals. Hence, according to Table 5.6, we have 

For an octahedral [Co(NH3)6]3+ complex the first excitation energy - TX is AE/hc = 21000cm-1 and the matrix element is 

These numbers yield Xtip = +2.50 x 10-9m3mol-1. On the other hand, the diamagnetic contribution is calculated according to Table 5.8. The final value of Xdia = —1-21 x 10 9m3morI is only half of the temperature-independent paramagnetism. 

Normally the TIP contribution (the van Vleck term) is only a minor correction to the underlying diamagnetism (the Langevin term). In complex systems like metalloproteins the diamagnetic contribution may also dominate the temperature-dependent paramagnetism. 

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Procedure. One approach to the problem is to select a value for a that is obviously too small and to increment it iteratively until the equation is satisfied. This is the method of program WIEN, where the initial value of x is taken as 1 (clearly, -E i < 1 as you can show with a hand calculator). 

Write a program in BASIC to carry out the multiplications in Problem 1. Crosscheck yoru program results with your hand calculations from Problem 1. 

Using a hand calculator, find the slope of the linear regression line that passes through the origin and best satisfies the points... 

Let us follow the first few iterations for the allyl system by hand calculations. We subtract the matrix xl from the HMO matrix to obtain the matrix we wish to diagonalize, just as we did with ethylene. With the rotation block in the upper left conrer of the R matrix (we are attacking an and aai), we wish to find... 

Since it is possible to obtain logarithms by using hand calculators, this Handbook contains no logarithm tables. 

A piping system can be evaluated for its displacement and stress either by manual methods (charts, tables, hand calculation) or computerized solution. The latter has become the standard approach desktop and laptop computers can handle all but the most compHcated problems. Manual methods are used only for rough estimates on very simple systems. 

Noncomputerized Calculation. Design charts or tables are used for hand calculations or simplified formulas such as the guided cantilever equation given ia Reference 31, described briefly below. 

This approach attempts to permit the designer to evaluate and optimize alternatives quickly, using shortcut hand calculations. However, if it is to be adopted by practicing design engineers it must be complemented by software for at least the analysis, evaluation, and optimization stages. 

There are many simple two-parameter equations for Hquid mixture constituents, including the Wilson (25), Margules (2,3,18), van Laar (3,26), nonrandom two-Hquid (NRTI.v) (27), and universal quasichemical (UNIQUAC) (28) equations. In the case of the NRTL model, one of the three adjustable parameters has been found to be relatively constant within some homologous series, so NRTL is essentially a two-parameter equation. The third parameter is usually treated as a constant which is set according to the type of chemical system (27). A third parameter for Wilson s equation has also been suggested for use with partially miscible systems (29,30,31). These equations all require experimental data to fit the adjustable constants. Simple equations of this type have the additional attraction of being useful for hand calculations. 

When estimates are being made by hand calculations, Eq. (19-2) is frequently applied until cr = 0.8L. This will cause an inflection point in a plot of concentrations with distance. 

If data gathering is also computer-linked, then the problem is to check out the correctness of the program. Hand calculation of one rated point for How head, efficiency, and horsepower will serve as verification. 

In compensator design, hand calculation is cumbersome, and a suitable computer package, such as MATLAB is generally used. 

A hand calculation method that can be used to take into account two-phase relief when the materials m the vessel are natural" surface active foamers. To account for disengagement, the vessel void fraction at disengagement should be evaluated (i.e.. the point at which the vent flow ceases to be two-phase and starts to be vapor only). ... 

IMPORTANCE must be run to find the probability of the fault tree, even if the importances are not of interest. Go to FTAPSUIT and type 7. It asks for the input file name type "pvn.ii." The output is pvn.io which after editing (for format) is presented as Figure 7.4-5, It has calculated the Fussel-Vesely importances because of the "FV" in Figure 7.4-3. The probability of the top event is 4.5538E-3 whereas 4.56E-3 was calculated by the previous hand calculation. It says that the cutset con ting of the relief valve-switch-timer is the most important. 

It can be seen from the previous description that the design of both a cold-feed stabilizer and a stabilizer with reflux is a rather complex and involved procedure. Distillation computer simulations are available that can be used to optimize the design of any stabilizer if the properties of the feed stream and desired vapor pressure of the bottoms product are known. Cases should be run of both a cold-feed stabilizer and one with reflux before a selection is made. Because of the large number of calculations required, it is not advisable to use hand calculation techniques to design a distillation process. There is too much opportunity for computational eiToi. 

A computer program was developed by Grandhidsan et al., to estimate the number of trays and the circulation rate of lean TEG needed to dry nat-ual gas. It was found that more accurate predictions of the rate could be achieved using this program than using hand calculation. ... 

Wills, M.J.N. and JOHNSTON, D. 22nd Nat. Heat Transfer Conf, HTD., 36. (ASME, New York, 1984) A new and accurate hand calculation method for shell side pressure drop and flow distribution. 

Kaplan and Thornton (1967) used three different sets of vibrational frequencies to estimate the zero-point energies of the reactants and products of the equilibrium, which provided three different isotope exchange equilibrium constants 1-163, 1-311 and 1-050. The value 1-311 is considered to be most reasonable, whereas the others are rejected as unrealistic for the case in hand. Calculations using the complete theory led to values that varied from 1-086 to 1-774 for different sets of valence-force constants for the compounds involved. 

On the theoretical hand, calculations have been performed as soon as in the 50ies [56,63] since formaldehyde represents the smallest member of the carbonyl series. References to early works are avalaible in the compilation by Davidson and McMurchie [64] and in references [56-58,63]. Of particular interest for a comprehensive assignment of the experimental transitions are the very fine and accurate calculations by Harding and Goddard using their GVB-CI method [60,65]. 

The Engineering Sciences Data Unit has also published a method for estimating shell-side the pressure drop and heat transfer coefficient, EDSU Design Guide 83038 (1984). The method is based on a simplification of Tinker s work. It can be used for hand calculations, but as iterative procedures are involved it is best programmed for use with personal computers. 

Each time one of these changes is made, there is a chance for an error to occur. So after each change the program must be checked to see that it responds correctly under as many varied situations as possible. Most programs have special input card decks, or tapes, that have been prepared to perform these tests. These are merely a set of specific input data for which the responses have been hand-calculated. If the computer and hand-calculated results agree, the program is considered correct. Of course, when the data bank is altered the answers are altered. This means a corrected set of hand calculations must be made whenever the data bank is changed. 

These expressions apply to constant volume systems in which vA = — 1. The sums are taken from i = 1 to i = n. Although the use of these equations is somewhat laborious for hand calculations, they are easily handled by even the simplest types of computers. 

The data are worked up in Table 10.1.1 in a form suitable for hand calculations, although a computerized solution would be more accurate, particularly with regard to the integration. 

Inspection of this equation indicates that the pressure drop will be quite sensitive to the pipe diameter employed. To minimize the pressure drop, large tubes are desired, but the larger the pipe, the lower the heat transfer area per unit volume. The three specified pipe sizes have been chosen to illustrate the optimization problem. We are now prepared to carry out the calculations for a single length of each of these pipe sizes. For hand calculations we have the necessary relations in equations 13.1.24, 13.1.29, and... 

McCaffrey, Quintiere and Harkleroad (26) used a simple conservation of energy expression and a correlation of a relatively wide range of data to develop a hand- calculation formula for the hot layer temperature in a naturally ventilated compartment. 

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