Lumping defined

Following the same arguments, the ring isomerization deactivation rate expression can be directly extended to include the effects of all kinetic lumps (defined in Table VII) ... 

Traditional hydrocarbon conversion process models have implemented lumped kinetics schemes, where the molecules are aggregated into lumps defined by global properties, such as boiling point or solubility. Molecular information is obscured due to the multi-component nature of each lump. However, increasing environmental concerns and the desire for better control and manipulation of the process chemistry have focused attention on the molecular composition of both the feedstocks and their refined products. Modeling approaches that account for the molecular fundamentals underlying reaction of complex feeds and the subsequent prediction of molecular properties require an unprecedented level of molecular detail. 

Lump sum contracts tend to be favoured by companies awarding work (if the scope of work can be well defined) as they provide a clear incentive for the contractor to complete a project on time and within an agreed price. 

The exponential term appears for the same reason as it does in diffusion it describes the rate at which molecules can slide past each other, permitting flow. The molecules have a lumpy shape (see Fig. 5.9) and the lumps key the molecules together. The activation energy, Q, is the energy it takes to push one lump of a molecule past that of a neighbouring molecule. If we compare the last equation with that defining the viscosity (for the tensile deformation of a viscous material)... 

An analogous situation occurs in the catalytic cracking of mixed feed gas oils, where certain components of the feed are more difficult to crack (less reactive or more refractory) than the others. The heterogeneity in reactivities (in the form of Equations 3 and 5) makes kinetic modelling difficult. However, Kemp and Wojclechowskl (11) describe a technique which lumps the rate constants and concentrations into overall quantities and then, because of the effects of heterogeneity, account for the changes of these quantities with time, or extent of reaction. First a fractional activity is defined as... 

The very basis of the kinetic model is the reaction network, i.e. the stoichiometry of the system. Identification of the reaction network for complex systems may require extensive laboratory investigation. Although complex stoichiometric models, describing elementary steps in detail, are the most appropriate for kinetic modelling, the development of such models is time-consuming and may prove uneconomical. Moreover, in fine chemicals manufacture, very often some components cannot be analysed or not with sufficient accuracy. In most cases, only data for key reactants, major products and some by-products are available. Some components of the reaction mixture must be lumped into pseudocomponents, sometimes with an ill-defined chemical formula. Obviously, methods are needed that allow the development of simple... 

In whole tissue or cell monolayer experiments, transcellular membrane resistance (Rm = Pm1) lumps mucosal to serosal compartment elements in series with aqueous resistance (R = P ). The operational definition of Lm depends on the experimental procedure for solute transport measurement (see Section VII), but its magnitude can be considered relatively constant within any given experimental system. Since the Kp range dwarfs the range of Dm, solute differences in partition coefficient dominate solute differences in transcellular membrane transport. The lumped precellular resistance and lumped membrane resistance add in series to define an effective resistance to solute transport ... 

The lumped material resistance properties of intestinal membranes as defined by permeability can be described macroscopically by the ratio of mass flux (J) through the membrane to the solute concentration difference across the membrane. Permeability (reciprocal resistance) has units of length per unit time. 

In the molten state polymers are viscoelastic that is they exhibit properties that are a combination of viscous and elastic components. The viscoelastic properties of molten polymers are non-Newtonian, i.e., their measured properties change as a function of the rate at which they are probed. (We discussed the non-Newtonian behavior of molten polymers in Chapter 6.) Thus, if we wait long enough, a lump of molten polyethylene will spread out under its own weight, i.e., it behaves as a viscous liquid under conditions of slow flow. However, if we take the same lump of molten polymer and throw it against a solid surface it will bounce, i.e., it behaves as an elastic solid under conditions of high speed deformation. As a molten polymer cools, the thermal agitation of its molecules decreases, which reduces its free volume. The net result is an increase in its viscosity, while the elastic component of its behavior becomes more prominent. At some temperature it ceases to behave primarily as a viscous liquid and takes on the properties of a rubbery amorphous solid. There is no well defined demarcation between a polymer in its molten and rubbery amorphous states. 

Comparison of the Experimental and Simulation Results. The preceding discussion has shown that both the experimental anthracene fluorescence profiles and the simulated anthracene concentration profiles decrease in a manner which closely follows an exponential decay. Therefore, the most convenient way to compare the simulation results to the experimental data is to define an effective overall photosensitization rate constant, kx or k2, as described above. Adoption of this lumped-parameter effective kinetic constant allows us to conveniently and efficiently compare the experimental data to the simulation results by contrasting the rate constant obtained from the steady-state fluorescence decay with the value obtained from the simulated decrease in the anthracene concentration. 

It is Lewis complementarity, on the other hand, that is operative when the mating of the molecules is determined by acid-base interactions, one that is described and predicted by the complementary mating of the lumps with the holes in the two associated Laplacian distributions. A molecule s reactive surface is defined by the zero envelope of the Laplacian distribution, the envelope that separates the shells of charge concentration from those of charge depletion. The reactive surfaces make immediately clear the locations of the lumps, the nucleophilic sites, and the holes, the electrophilic sites, that are brought into juxtaposi-... 

It is assumed that at time zero a defined quantity of Dursban enters a lake and is distributed subsequently between the fish, the soil and the plants. The soil and plants are lumped together as one compartment. The quantities in each compartment are expressed as a percentage of the initial contamination. 

Let us lump together the m observations y (i = 1,..., m) into a vector y, the polynomial coefficients otj (/=0,...,n — 1) into a vector jc of unknowns, and define the (/—l)th power of the ith observable (uj> 1 as the current term atJ of the matrix Am, . We now apply the usual method. Polynomials of high degrees tend to generate nearly singular matrices A which result in excessive fluctuations. 

We now lump the n Lagrange multipliers into the vector X, and the g3(x) into a vector g(x). We further define the m x n matrix F of partial derivatives by its current term fp which is the derivative of the y th constraint with respect to the ith parameter, as... 

Smog-chamber studies are needed for validating both the detailed chemical models and the lumped models. Many of the past chamber studies have not used sufficiently well-defined initial conditions. Measurements of more products and of the reactive intermediates will provide more stringent tests for models. 

As described in Chapter 1, the first term on the left-hand side describes the kinetic energy of the electron, V is the potential energy of an electron interacting with the nuclei, VH is the Flartree electron-electron repulsion potential, and Vxc is the exchange-correlation potential. This approach divides electron-electron interactions into a classical part, defined by the Flartree term, and everything else, which is lumped into the exchange-correlation term. The Flartree potential describes the Coulomb repulsion between the electron and the system s total electron density ... 

In this rate expression we have lumped C/js into the effective surface rate coefficient by defining k" — CC s- AU sohd reactions have reaction steps similar to those in catalytic reactions, and the rate expressions we need to consider are basically Langmuir-Hinshelwood kinetics, which were considered in Chapter 7. Our use of a first-order irreversible rate expression is obviously a simplification of the more complex rate expressions that can arise from these situations. 

Service Facilities For a process plant, utihty services such as steam, water, electric power, fuel, compressed air, shop facilities, and a cafeteria require capital ejq)enditures. The cost of these facihties lumped together may be 10 to 20 percent of the fixed capital investment for a preliminary estimate. Note Buildings, yards, and service facihties must oe well defined to obtain a definitive or detailed estimate.)... 

Clearly, Bohr s atom is a long way from Dalton s. No longer is it an indivisible lump it is made from subatomic particles - the electrons and the nucleus - and is mostly just space. The size of the atom is defined not by hard boundaries but by how far the electron orbits reach. 

The model includes fundamental hydrocarbon conversion kinetics developed on fresh catalysts (referred to as start-of-cycle kinetics) and also the fundamental relationships that modify the fresh-catalyst kinetics to account for the complex effects of catalyst aging (deactivation kinetics). The successful development of this model was accomplished by reducing the problem complexity. The key was to properly define lumped chemical species and a minimum number of chemical reaction pathways between these lumps. A thorough understanding of the chemistry, thermodynamics, and catalyst... 

An important part of the model development was first defining a set of lumped chemical species from the 300 identified species and then defining... 

The hydrocarbon lumps and reaction network for both the start-of-cycle and the deactivation kinetics were defined. 

Start-of-cycle kinetic lumps in KINPTR are summarized in Table V. A C5-light gas lump is required for mass balance. Thirteen hydrocarbon lumps are defined. The reforming kinetic behavior can be modeled without splitting the lumps into their individual isomers (e.g., isohexane and n-hexane). Also, the component distribution within the C5- lump can be described by simple correlations, as discussed later. The start-of-cycle reaction network that defines the interconversions between the 13 kinetic lumps is shown in Fig. 9. This reaction network results from kinetic studies on pure components and narrow boiling fractions of naphthas. It includes the basic reforming reactions... 

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