Simplified Solution

If the distribution coefficient is constant, and if there is essentially no mutual solubility, the fraction not extracted, P, can be calculated directly as a function of the extraction factor, E, and the nmnber of stages, n. 

TreybalPl discusses the derivation of these equations and presents a graphical solution reproduced here as Fig, 11. 

Even when the two limitations of immiscibility and constant distribution coefficient do not quite hold. Fig. 11 does allow a quick estimate of the trade-offs between solvent/feed ratio and number of stages required to obtain a desired degree of extraction (raffinate purity). 

The above solutions are all based on ideal or theoretical stages. Even in discrete stage systems, like mixer-settlers, equilibrium may not be attained because of insufficient time for diffusion of solute across the phase boundary or insufficient time for complete clarification of each stage. 

In continuous differential extractors (columns) it has been convenient to think in terms of a height equivalent to a theoretical stage (HETS), and to correlate HETS as a function of system and equipment variables. Alternately, correlations may be obtained on the basis of the height of a transfer unit (HTU), which is more amenable to calculations which separately include the effects of backmixing.l H  

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For steady-state design scenarios, the required vent rate, once determined, provides the capacity information needed to properly size the relief device and associated piping. For situations that are transient (e.g., two-phase venting of a runaway reactor), the required vent rate would require the simultaneous solution of the applicable material and energy balances on the equipment together with the in-vessel hydrodynamic model. Special cases yielding simplified solutions are given below. For clarity, nonreactive systems and reactive systems are presented separately. 

The use of isotopic models in the literature—practical limits of usage As mentioned above, simplified solutions are employed in ion exchange for the estimation of diffusion coefficients. For example, the equations of Vermeulen and Patterson, derived from isotopic exchange systems, have been successfully used, even in processes that are not isotopic. Inglezakis and Grigoropoulou (2001) conducted an extended review of the literature on the use of isotopic models for ion-exchange systems. 

Emulsion phase gas in plug flow Solutions for bubble phase free of solids In the following, a simplified solution is presented under the following assumptions first-order reactions, gas flow only through the bubble phase (fh = 1), and absence of solids in the bubble phase (yb = 0). Under these conditions, the material balances (3.519) and (3.520) become the following. 

This problem is a good example of the importance of formulating a complex diffusion problem in terms of the equations of change. Hence the simplified treatment given here is discussed in terms of the simplified solutions to the three basic equations. 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

To avoid the complex form of the error function, simplified solutions have been proposed in the literature [10]. To solve for the ignition delay time (tP fig), a first-order Taylor series expansion of Equation 3.19 is conducted. The range of validity of this expansion is limited, and thus, cannot be used over a large range of incident heat fluxes. Therefore, the domain has to be divided at least into two. 

Solving Equations 3.20 and 3.21 for the pyrolysis time t will yield a theoretical value for the time at which the solid fuel sample begins to pyrolyze and produce fuel vapors. The use of the appropriate simplified solution will allow the evaluation of the pyrolysis time t over the entire domain of the imposed incident heat fluxes. 

While numerical methods come into question for solutions involving variable D, D can be assumed to be constant or practically constant for most cases of practical interest. In addition, simplified solutions for diffusion along the x-axis can be used instead of the general solution, except for some particular cases which will be pointed out later. This greatly simplifies presentation of the problem and the resulting equation for diffusion is ... 

A few quick observations are in order at this point. The conducted energy is clearly reduced for large values of the heat of ablation. Similarly, the rate of material removal pVa is dependent on the heat of ablation and decreases for increased values of Hab. In order for this very simplified solution to apply, the overall slab thickness 8 must be large compared with the depth of penetration of melting. In terms of the above parameters this means that... 

In the important case in which the two drops are volatile and the same liquid, the interior temperatures are uniform. This leads to the simplified solutions ... 

Helfferich [2,3,30] states that in addition to the mutual interference of substances i and j, characterized by the phenomenological cross coefficients of the type L,j, one should take into account the presence of a coion in the ion exchanger as well. As a result, the simplified solution is inappropriate, even to the problem of ordinary IE. By use of only one diffusion mass-transfer equation, as in this case, account for the presence of co-ion has been neglected. It is, as a consequence, necessary to consider the Nemst-Planck relation for the co-ion also. 

The dependence of vapor pressure on temperature follows from a simplified solution to the Clapeyron equation (Schwarzenbach et al., 1993) ... 

The temperature dependence of the amorphization dose can be easily obtained by solving for ) in Equation (13). The simplified solution is ... 

Challenge Problem. In example 10-5, we neglected the contribution of nitrous acid to the ionic strength. We also used the simplified solution for the hydronium ion concentration,... 

A simplified solution to the Fokker-Planck equation is used in the updating of particle positions. 

Examples 13-3 to 13-5 illustrate the simplified design method for different cases. The first is for the endothermic styrene reaction, where the temperature decreases continually with catalyst-bed depth. Example 13-4 is for an exothermic reaction carried out under conditions where radial temperature gradients are not large. Example 13-5 is also for an exothermic case, but here the gradients are severe, and the simplified solution is not satisfactory. 

Two classes of simplified solutions are useful. The first is relevant to a phytoplankton bloom, starting from high ambient nutrient concentration St=0 and ending when this external nutrient has been exhausted and cellular concentration of limiting nutrient has fallen to the susbsistence quota. There is supposed to be no losses. In the case of a monospecific bloom, the final biomass is given by... 

Reverse osmosis is simply the application of pressure on a solution in excess of the osmotic pressure to create a driving force that reverses the direction of osmotic transfer of the solvent, usually water. The transport behavior can be analyzed elegantly by using general theories of irreversible thermodynamics however, a simplified solution-diffusion model accounts quite well for the actual details and mechanism in most reverse osmosis systems. Most successful membranes for this purpose sorb approximately 5 to 15% water at equilibrium. A thermodynamic analysis shows that the application of a pressure difference, Ap, to the water on the two sides of the membrane induces a differential concentration of water within the membrane at its two faces in accordance with the following (31) ... 

Cohen [C6] made the following assumptions to simplify solution ... 

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