Separation equation for

This integrated version of Equation (3.15) requires viscosity to be constant as well as density, but this assumption is not strictly necessary. See Problem 3.15. Write separate equations for the pressure drop in the large and small reactors and take their ratio. The physical properties cancel to give the following, general relationship ... 

Equations 11 and 12 are not written for constant molality, and can not be easily used with the Gibbs-Duhem equation to obtain an analytical expression for the activity of water in the ternary solution. However, it is possible to propose a separate equation for the activity coefficient of water that is consistent with the proposed model of concentrated solutions. 

One of the most successful methods for calculating the number of plates necessary for a given separation is due to Lewis and Matheson(38). This is based on the Lewis-Sorel method, described previously for binary mixtures. If the composition of the liquid on any plate is known, then the composition of the vapour in equilibrium is calculated from a knowledge of the vapour pressures or relative volatilities of the individual components. The composition of the liquid on the plate above is then found by using an operating equation, as for binary mixtures, although in this case there will be a separate equation for each component. 

Experiments carried out with accelerating horizontal bubble flow, that is, with variable quality, were correlated by similar but separate equations for f-in., -in., and -in. diameter pipe. For the -in. pipe,... 

The critical gas velocity, Vocb, at the start of entrainment is expressed by three separate equations, for low, medium and high Reynolds numbers. 

We place no upper bound on j since we assume that the chain is long enough to neglect the end effect, which manifests itself in separate equations for dPNjdt and dPs i dt. 

Of particular interest is the case of anisotropic tensorial conductivity, which couples together the various field and current components. It is then, of course, impossible to obtain a separate equation for Hz and, in principle, a dynamo is possible with an increment determined by the off-diagonal (in x, y, z coordinates) components of the conductivity tensor. Usually the direction of anisotropy is related to the magnetic field, but then the whole problem becomes nonlinear. 

Computational fluid dynamics were used to describe the flow which undergoes a fast transition from laminar (at the fluid outlets) to turbulent (in the large mixing chamber) [41]. Using the commercial tool FLUENT, the following different turbulence models were applied a ke model, an RNC-ki model and a Reynolds-stress model. For the last model, each stream is solved by a separate equation for the two first models, two-equation models are applied. To have the simulations at... 

The mechanisms and resulting kinetic equations are shown in Figure 4. Other mechanisms are possible as well as modifications of these—e.g., disproportion termination of chain reactions, and condensation between unlike monomers. The left sides of the equations represent the reactor operator (note that all resulting differential equations are nonlinear because of the second-order propagation and termination reactions). To this is added the complexity of considering separate equations for the thousands of separate species frequently required to define completely commercially useful polymers. Solution by direct application of classical techniques is impractical or impossible in most cases even direct numerical solution is often difficult. Simplifying assumptions or special mathematical techniques must be used (described below in the calculations of MWD). 

They are discrete transforms and can therefore operate directly on the separate equations for each species, reducing them to one expression. Nonlinear terms arising from condensation polymerization can be handled and, with some difficulty, so can realistic terminations in free radical polymerization. They are a special case of the generating functions and can be used readily to calculate directly the moments of the distribution, and thus, average molecular weights and dispersion index, etc. Abraham (2) provided a short table of Z-transforms and showed their use with stepwise addition. 

In Bloch s original treatment of NMR,23 he postulated a set of phenomenological equations that accounted successfully for the behavior of the macroscopic magnetization M in the presence of an rf field. These relations are based on Eq. 2.41, where M replaces X, and B is any magnetic field—static (B0) or rotating (B,). By expanding the vector cross product, we can write a separate equation for the time derivative of each component of AT ... 

The table below shows what the symbols represent in the separate equations for conversion and temperature. 

From (11.70) we can obtain the separate equation for the migrated anomalous electric field ... 

Substituting the excess electric current j for the extraneous electric current j , and the background conductivity Ub for the total conductivity a in equation (12.1), we obtain the separate equations for the electric and magnetic anomalous fields ... 

In calculating titration curves, separate equations for different regions of the curve ("before the equivalence point", "at the equivalence point", "after the equivalence point", etc.) are often employed. This section illustrates how to use a single "master" equation to calculate points on a titration curve. Instead of calculating pH as a function of the independent variable V, it is convenient to use pH as the independent variable and V as the dependent variable. The species distribution at a particular pH value is calculated from the [H ], and the volume of titrant required to produce that amount of each species is calculated. For example, in the titration of a weak monoprotic acid HA, we can calculate the concentration of A at a particular pH and then calculate the number of moles of base required to produce that amount of A . In general (J -j) moles of base per mole of acid are required to produce the species HjA from the original acid species HjA. 

These are equivalent to ( - 1) separate equations for each component. The number of conditions arising in this way is thus c ( - 1) for the whole system. 

Since this is a total quantity, it has the advantage that it is independent of temperature and pressure, unlike the concentrations of its constituent species. Experimentally, DIG is determined by acidifying the sample, so that all the HCO3 and CO3 react with H+ to become CO2 and H2O, and then measuring the amount of CO2 gas evolved. To create a plot of the concentrations of the three dissolved carbonate species as a function of pH we assign the DIG its average value in seawater (Table 4.1). Gombining Eq. (4.15) with Eqs. (4.9) and (4.11) yields separate equations for the carbonate species as a function of equilibrium constants, DIG and pH ... 

Allen et al. derived separate equations for CO and CO2 formation based on a mechanism involving reaction between CH4 gas and adsorbed H2O, where the overall rate was controlled by the desorption of the two products. The CO shift reaction took place between adsorbed CO and H2O in the gas phase. The rate expression for CO formation was that given in (15). The numerator relates... 

Separate equations for eight groups of monomers such as styrenes, chlorinated olefins, acrylates and methacrylates were developed and the correlation coefficients for most of the equations were greater than 0.99. The method has been extended to the calculation of the monomer reactivity ratios from the 13C NMR spectra of the monomers for which no Q and e values are known.362... 

Write separate equations for the oxidation and reduction haif-reactions. 

The fact that the individual electrons distributions are different suggests that there should be a separate equation for each orbital. 

The single equation which determines all the orbitals (electron distributions) of these models must contain an operator which generate. each separate equation for each orbital when actually applied to that orbital. 

This shows that any C—H bond in any alkane (or cycloalkane) can be converted to a C-halogen bond by reaction with molecular halogen. We learn the generalization to avoid having to learn separate equations for each of the thousands and thousands of alkanes and for the different halogens. 

The effect of high catalyst concentrations near the inlet and exit of the reactor could be allowed for by using separate equations for each section of the reactor, where a and K might be different. 

Rolls. Consider the solution A1 = i/T/Sexpi i, A2 = As = 0. This solution exists only for T > 0. Linearizing the system (41)-(43) around this solution, we find that the system splits into two sub-systems a separate equation for Ai, and a coupled system of equations for A2 and A3. The equation for Ai is... 

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