Balance Model

The first model to be discussed is based upon consideration of the mass balance in the column. A small slice across the column is considered. The mass balance equation for a single component takes the form [8]  

2 The Practical Application of Theory in Preparative Liquid Chromatography 

The first difficulty arises in that the kinetics of mass transfer between the mobile and stationary phases are not always rapid. If one tries to introduce a finite rate of mass transfer, however, except for the simplest of cases (which conditions are not at all relevant to preparative LC) the set of equations becomes intractable and no solution can be found. Hence we are forced to assume instantaneous mass transfer in order to proceed. 

A second difficulty arises from the diffusion term. This, too, renders the equations intractable even to current numerical methods of solution. The problem is avoided by the assumption of infinite column efficiency. This instantly introduces a difficulty in that the model is then far from representing real chromatographic systems. The equations, however, may now be solved numerically. In fact, for a single solute, there is an analytical solution of the equations resulting from these assumptions [3]. When two solutes are of interest, similar equations are written for both and the set of equations is solved numerically. The above assumptions reduce the differential Eq. (A 2.1) to a simpler equation  

The solution of the differential equations is possible by use of the right computer program. The calculation is performed by setting up a two dimensional grid of points, with time in one direction and distance travelled along the column in the other Fig. 2-A2.1). In fact, the differential equation is replaced by a finite difference equation [9]. In order to calculate the mobile phase concentration at a distance, time point k,l) on the grid, the equation takes the form [10]  

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The energy laws of Bond, Kick, and Rittinger relate to grinding from some average feed size to some product size but do not take into account the behavior of different sizes of particles in the mill. Computer simulation, based on population-balance models [Bass, Z. Angew. Math. Phys., 5(4), 283 (1954)], traces the breakage of each size of particle as a function of grinding time. Furthermore, the simu-... 

Capture efficiency can also be measured by first estimating workspace emission rates and local exhaust emissions. The local exhaust emission rate equals the duct concentration (mass/volume) multiplied by the duct flow rate (volume/time). The workspace emission rates can be calculated using appropriate mass balance models and measured ventilation rates and workspace concentrations. Capture efficiency is the ratio of duct emission rate to total emission rate (duct plus workspace). ... 

Gertlauer, A., Mitrovic, A., Motz, S. and Gilles, E.-D., 2001. A population balance model for crystallization processes using two independent particles properties. Chemical Engineering Science, 56(7), 2553-2565. 

Inexpensive balances - Models SC 340 (500g) SC 330 (2 kg) Solex International, 44 Main Street, Broughton Astley Leicester LE9 6RD... 

Salter A D Balances - Models FX-3000 and FY 3000 Scale Services, Hillcrest Way, Gerrards Cross Bucks SL2 SDN... 

In this model, energy balances are set up for the reactor and the separator tube separately, and two equations are obtained. The gas holdup can then be obtained from combining these two equations. Details can be found in Zhang et al. [7]. The comparison between the measured and calculated cross-sectional mean gas holdups is shown in Fig. 5. It can be seen that there is a satisfactory agreement between the experimental and calculated gas holdup in the different operating conditions. Therefore, it is reasonable to conclude that the energy balance model used in this work can describe the circulation flow behavior in the novle internal-loop airlift reactor proposed in this work. 

IX vcloping Dynamic Balance Models, Simulation Approach to Model Solving, Dynamic Mass and Energy Balances... 

B. Griffiths and D. Robin.son, Root-induced nitrogen mineralisation a nitrogen balance model. Plant Soil 759 253 (1992). 

Khire, M.V., Benson, C.H., and Bosscher, P.J., Water balance modeling of earthen final covers, Journal of Geotechnical and Geoenvironmental Engineering, 123, 744-754, 1997. 

While we laud the virtue of dynamic modeling, we will not duphcate the introduction of basic conservation equations. It is important to recognize that all of the processes that we want to control, e.g. bioieactor, distillation column, flow rate in a pipe, a drag delivery system, etc., are what we have learned in other engineering classes. The so-called model equations are conservation equations in heat, mass, and momentum. We need force balance in mechanical devices, and in electrical engineering, we consider circuits analysis. The difference between what we now use in control and what we are more accustomed to is that control problems are transient in nature. Accordingly, we include the time derivative (also called accumulation) term in our balance (model) equations. 

Leblanc and Fogler developed a population balance model for the dissolution of polydisperse solids that included both reaction controlled and diffusion-controlled dissolution. This model allows for the handling of continuous particle size distributions. The following population balance was used to develop this model. 

Johnson and Swindell [77] developed a method for evaluating the complete particle distribution and its effect on dissolution. This method divided the distribution into discrete, noncontinuous partitions, from which Johnson and Swindell determined the dissolution of each partition under sink conditions. The dissolution results from each partition value were then summed to give the total dissolution. Oh et al. [82] and Crison and Amidon [83] performed similar calculations using an expression for non-sink conditions based on a macroscopic mass balance model for predicting oral absorption. The dissolution results from this approach could then be tied to the mass balance of the solution phase to predict oral absorption. 

SE LeBlanc, HS Fogler. Population balance modeling of the dissolution of polydis-perse solids Rate limiting regimes. AIChE J 33 54-63, 1987. 

I mass balance model at the farm level Calculation of inputs and outputs. 

Wania F, Persson J, Di Guardo A, McLachlan MS (2000) CoZMo-POP. A fugacity-based multi-comparlmental mass balance model of the fate of persistent organic pollutants in the coastal zone. WECC report 1/2000. Toronto (April)... 

Mackay D (1998) Multimedia mass balance models of chemical distribution and fate. In Schuurmann G, Markert B (eds) Ecotoxicology. Wiley, New York, pp 237-257... 

Symbols G = general polymer system, PB = population balance model, ADA = age... 

Figure 35. Granule growth mechanisms for population balance modeling. (From Litster and Ennis, 1994.)...

The authors would like to acknowledge that the section dealing with population balance modeling of granulation processes is an abbreviated version of material prepared by Dr. J. D. Litster, University of Queensland, for a joint short course given by Dr. B. J. Ennis and Dr. J. D. Litster. 

Adetayo, A. A., Litster, J. D., Pratsinis, S. E., and Ennis, B. J., Population Balance Modelling of Dram Granulation of Materials with Wide Size Distributions, Powder Tech, 82 37-49 (1995)... 

In this section we describe the key physical-chemical properties and discuss how they may be used to calculate partition coefficients for inclusion in mass balance models. Situations in which data require careful evaluation and use are discussed. 

MASS BALANCE MODELS OF CHEMICAL FATE 1.5.1 Evaluative Environmental Calculations... 

Paterson and Mackay (1985), Mackay and Paterson (1990, 1991), and a recent text (Mackay 2001). Only the salient features are presented here. Three evaluations are completed for each chemical, namely the Level I, II and III fugacity calculations. These calculations can also be done in concentration format instead of fugacity, but for this type of evaluation the fugacity approach is simpler and more instructive. The mass balance models of the types described below can be downloaded for the web site www.trentu.ca/cemc... 

North, G. R., J. G. Mengel, and D. A. Short. 1983. Simple energy balance model resolving the seasons and the continents Application to the astronomical theory of the ice ages. J. Geophys. Res. 88, 6576-86. 

As originally derived, however, the mass balance model has an important (and well acknowledged) limitation implicit in its formulation is the assumption that fluid and minerals in the modeled system remain in isotopic equilibrium over the reaction path. This assumption is equivalent to assuming that isotope exchange between fluid and minerals occurs rapidly enough to maintain equilibrium compositions. 

In this chapter, we develop a mass balance model of the fractionation in reacting systems of the stable isotopes of hydrogen, carbon, oxygen, and sulfur. We then demonstrate application of the model by simulating the isotopic effects of the dolomitization reaction of calcite. 

For the process area described in Problem 7-24, determine the concentration of propane in the area as a function of time if at t = 0 a 3/4-in propane line breaks (the propane main header is at 100 psig). The temperature is 80°F. See chapter 4 for the appropriate source model and chapter 3 for material balance models. 

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