Lumping rules

Glinos K, Malone MF. Minimum reflux, product distribution, and lumping rules for multicomponent distillation. Ind Eng Chem Process Des Dev 1984 23 764. 

Glinos, K., Malone, M. F. (1984). Minimum Reflux, Product Distribution and Lumping Rules for Multicomponent Distillation. Ind. Eng. Chem. Process Des. Dev., 23,764. 

In the case of complex reactions, usually also a simplification of the reaction model is needed to arrive at workable rate terms with a limited number of regression parameters. This is all the more, necessary, as kinetic parameters like the preexponential factor and activation energy of a reaction step are strongly correlated to each other /II/. Wei /12/ and many other authors /I3,I4,15/ have developed lumping rules for complex reactions and shown the consequences on the overall kinetics, resulting from the so-called exact, approximate or proper lumping /16/. Unfortunately, these rules are limited mainly to first order steps, whereas in many cases of practical importance, rate laws of different orders prevail. 

In the life insurance annuity a person contributes equal amounts over a number of years, and then at a given age (assuming he has not died previously) he receives a lump sum of money or some other form of payment. To determine how this compares with other forms of investment, the investor must determine at what interest rate his money would need to be invested in order to earn that lump sum in the same period of time. The first payment would earn compound interest for n periods. The second payment, which is made at the end of the first period, would earn interest for (n - 1) periods. The general rule is that each payment earns interest for one less period than the proceeding one. This can be expressed as... 

It is not easy to determine whether lumping in a process model is a valid technique for representing the process. A good rule of thumb is that if the response is... 

If the number of components is very large, a mixture can be regarded as continuous and sharp distinctions between individual components are not made. Methods for dealing with stoichiometry, thermodynamics and kinetics for continuous mixtures are discussed by Aris and Gavalas [33]. An indication is given that rules for grouping in such mixtures depend on the nature of the reaction scheme. Wei and Kuo [34] considered ways in which species in a multicomponent reaction mixture could be lumped when the reaction network was composed of first-... 

If the pathway or segment of a portion producing a non-trace intermediate is irreversible, no subsequent portion of the overall network feeds back into it. As a rule, this allows the subsequent portion or portions to be studied independently by using the separately synthesized non-trace intermediate as starting material. It also allows the portion yielding the non-trace intermediate to be studied independently For this purpose, all subsequent intermediates and products are lumped with the intermediate produced by the portion (i.e., the concentrations are added) to obtain the total production of the portion. Alternatively, before analysis, all intermediates are converted to end products, and only these then need to be analyzed for and lumped. 

It is obvious from the data of Tables ll-VIII to I l-XII that the emanation power varies widely. As a rule, the finer the particle size the greater the emanation power. However, when samples are severely weathered or damaged, as in the case of radioactive ores, further comminution of a specimen has little effect on the emanation power. In both cases it is believed that Ra has migrated from the original site of its parent in the crystal lattice and has adsorbed on the surfaces of microfractures facilitating Rn escape. When certain samples are pulverised in the laboratory they may even emit less Rn than the whole lump. This can happen when U and its decay products are associated with minerals that do not pulverise as easily, i.e., are less friable. Starik and Melikova (1957) found that a whole lump of ore had an emanation power of 46%... 

The two Eqs. 6.57a and 6.57b are classical relationships of the most critical importance in linear chromatography. Combined, they constitute the famous Van Deemter equation, which shows that the effects of the axial dispersion and of the mass transfer resistances are additive. This is the basic tenet of the equilibrium-dispersive model of linear chromatography. We will assume that this rule of additivity and Eqs. 6.57a remain valid when we apply the equilibrium-dispersive model to nonlinear chromatography. In this case, however, it is only an approximation because the retention factor, k = dq/dC, is concentration dependent. These equations have been derived from the lumped kinetic model. Thus, they show that the kinetic model and the equilibrium-dispersive model are equivalent as long as the rate of the equilibrium kinetics in the chromatographic system is not very slow. 

Once again using vertical lumping, only one reference component every five C-atoms can be selected in the range between 20 and 35 carbon atoms, for each family. These components are schematically reported in Fig. 8. Primary distribution products for all these species are generated by the MAMA program, using the same rules and procedures previously discussed. 

The grouping or lumping of components does not refer to unique rules, but it is only dependent on the successive uses of the kinetic scheme. Consequently,... 

Lastly, a post-processor lumps the end products according with pre-assigned rules specified by the user. 

On the other hand, diffusion of a substance into an emulsion droplet of a few fm in radius would take far less than a second. Diffusion thus proceeds very fast at very small distances and takes a very long time at long distances it cannot be expressed as a linear rate, in m s 1. If a substance has to diffuse into a lump of material, say salt in a loaf of cheese by immersing it in brine, similar rules hold. In the beginning of the process, the quantity of salt taken up is proportional to the square root of time, according to Eq. (5.19) if it takes one day to obtain a total salt content of 1% in the cheese, it will take four days to obtain 2%. 

This example also demonstrates another point. The denominator of a rate equation obtained from the general formula (eqns 6.4 to 6.6) may contain several terms that involve the same combination of concentrations of co-reactants. Such terms can, of course, be lumped. Thus, in the original rate equation 6.10, the first two denominator terms both contained the co-factor pco, the third and fourth both contained pHr In eqn 6.11, obtained by lumping the terms of these pairs, the number of denominator terms had been reduced from four to two. The one-plus form 6.12 was then obtained by division by the second of these two, actually the sum of the third and fourth terms in the original equation. [Rule 24 in Section 7.3.1 allows such lumping even before the rate equation is derived.]... 

Storage containers handling granular or lump materials are often fitted with feeder tables to control the discharge rate. As a general rule, the discharge aperture should be not less than 6 times the top size of the material being handled. 

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