Multiplying proportions

Let us put the signal from the external reflux flow transmitter into the B connection and the signal from the summer into the C connection. Let us further assume that the output of the multiplier, proportional to the internal reflux, has the same span as that of the external reflux flow transmitter (assumed linear). We now need to find the/and S factors. 

In a canonical ensemble the probability f canon(T E) of visiting a point in phase space wi an energy E is proportional to the Boltzmarm factor, = exp(—E/lcgT), multiplied by tl density of states, (E), where the number of states between E and E + dE is given 1 n E)6E. Thus ... 

It would seem that the proeess of evaluating all of the <(l)i(l)j g (l)k(l)i>, eaeh of whieh requires additions and multiplieations, would require eomputer time proportional to N. However, it is possible to perform the full transformation of the two-eleetron integral list in a time that seales as. This is done by first performing a transformation of the to an intermediate array labeled as follows ... 

If this is multiplied by 100, it becomes atom%, a percentage that relates the proportion of heavy isotope to the sum of the heavy and light isotopes. 

Aside from the side chains, the movement of the backbone along the main reptation tube is still given by Eq. (2.67). With the side chains taken into account, the diffusion velocity must be decreased by multiplying by the probability of the side-chain relocation. Since the diffusion velocity is inversely proportional to r, Eq. (2.67) must be divided by Eq. (2.69) to give the relaxation time for a chain of degree of polymerization n carrying side chains of degree of polymerization n ... 

It is the net intensity, not the electric field, which concerns us. We previously used the fact that intensity is proportional to E to evaluate i. Using complex numbers to represent E requires one slight modification of this procedure. In the present case we must multiply E by its complex conjugate -obtained by replacing / 1 by to evaluate intensity ... 

The velocity head JT in a pipe flow is related to Hquid velocity hy H = I Qc The Hquid velocity in a mixing tank is proportional to impeller tip speed 7zND. Therefore, JTin a mixing tank is proportional to The power consumed by a mixer can be obtained by multiplying and H and is given... 

Fig. 2. Stress—strain curve for standard polycarbonate resin at 23°C where the points A, B, and C correspond to the proportional limit (27.6 MPa), the yield point (62 MPa), and the ultimate strength (65.5 MPa), respectively. To convert MPa to psi, multiply by 145.

R-Values are directly proportional to thickness shown. To convert R-value in m K/W to h-ft F/Btu, multiply by 5.7. 

Velocity meters with density compensation. The signal from the velocity meter (e.g., turbine meter, electromagnetic meter, or sonic velocity meter) is multiplied by the signal from a densitometer to give a signal proportional to the mass flow rate. 

Tlocciilant addition rate can be regulated in proportion to the thickener oliirnetric feed rate or solids mass How in a feed-fonvard mode, or in a feed-back mode on either rake torque, iindertlow density, settling solids (sludge) bed le el, or solids settling rate. Of these, feed-fonvard on mass How or feed-back on bed le el are probably the most common. In some feed-fonvard schemes, the ratio multiplier is trimmed by one of the other parameters. 

From dimensional analysis, the expander blade speed, u, is directly proportional to the wheel diameter, D, of the expander, multiplied by the rotational speed, N, of the expander, both of which are dependent on the volume flow of gas and mechanical stresses. The equivalent velocity energy, C, is dependent on the inlet gas conditions to the expander and can be directly translated into available energy by the following equation ... 

The equations have been expressed as proportionals however, they can be used by simply ratioing an old to a new value. To add credibility to fan law adaptation, recall the flow coefficient, Equation 5.19, The term Qj/N is used which shows a direct proportion between volume Qj and speed N. Equation 5.12 indicates the head, Hp, to be a function of the tip speed, squared. The tip speed is, in turn, a direct function of speed making head proportional to speed. Finally, the power, Wp, is a function of head multiplied by flow, from which the deduction of power, proper tional to the speed cubed, may be made. 

Since dependency analysis is not needed, we can go on to the BUILD program. Go to FTAPSUIT and select 5 "Run Build." It asks you for the input file name including extender. Type "pv.pch," It asks you for name and extender of the input file for IMPORTANCE. Type, for examle, "pv.ii . It next asks for the input option. Type "5" for ba.sic event failure probabilities. This means that any failure rates must be multiplied by their mission times as shown in Table 7.4-1. (FTAPlus was written only for option 5 which uses probabilities and error factors. Other options will require hand editing of the pvn.ii file. The switch 1 is for failure rate and repair time, switch 2 is failure rate, 0 repair time, switch 3 is proportional hazard rate and 0 repair time, and switch 4 is mean time to failure and repair time.)... 

Such MO integrals are required for all electron correlation methods. The two-electron AO mtegrals are the most numerous and the above equation appears to involve a computational effect proportional to M AO integrals each multiplied by four sets of M MO coefficients). However, by performing the transformation one index at a time, the computational effort can be reduced to. ... 

The above method is applicable directly to all oils containing above 50 per cent, of cineol in oils containing lower proportions of cineol the precipitate is not solid enough to permit convenient handling and if the cineol-content drops below 25 per cent, the separation of cineol arsenate is not quantitative. It was found that the addition of an equal volume of eucalyptol to such oils (i.e. mixing 5 c.c. of the oil with 5 c.c. of eucalyptol) successfully overcomes this difficulty it then only becomes necessary to subtract from the volume of cineol, as observed in the neck of the flask, 5 c.c., and to multiply the difference by 10, in order to obtain the percentage of cineol in the oil. 

This calculation of the proportions by partial pressure can be converted to proportions by weight, by multiplying each pressure by the molecular mass (Avogadro s hypothesis), to give ... 

Notice from the rate expressions just written that the rate of an elementary step is equal to a rate constant k multiplied by the concentration of each reactant molecule. This rule is readily explained. Consider, for example, a step in which two molecules, A and B, collide effectively with each other to form C and D. As pointed out earlier, the rate of collision and hence the rate of reaction will be directly proportional to the concentration of each reactant. 

Coulometric analysis is an application of Faraday s First Law of Electrolysis which may be expressed in the form that the extent of chemical reaction at an electrode is directly proportional to the quantity of electricity passing through the electrode. For each mole of chemical change at an electrode (96487 x n) coulombs are required i.e. the Faraday constant multiplied by the number of electrons involved in the electrode reaction. The weight of substance produced or consumed in an electrolysis involving Q coulombs is therefore given by the expression... 

In the phosphor-photoelectric detector used as just described, the x-ray quanta strike the phosphor at a rate so great that the quanta of visible light are never resolved they are integrated into a beam of visible light the intensity of which is measured by the multiplier phototube. In the scintillation counters usual in analytical chemistry, on the other hand, individual x-ray quanta can be absorbed by a single crystal highly transparent to light (for example, an alkali halide crystal with thallium as activator), and the resultant visible scintillations can produce an output pulse of electrons from the multiplier phototube. The pulses can be counted as were the pulses-from the proportional counter. 

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