Lumped definition

Also in relumped form, single-event microkinetics account for all reactions at molecular level [2,3,13], This requires a molecular composition of the lumps considered. The definition of the lumps in hydrocracking is such that thermodynamic equilibrium can be assumed within the lumps. Per carbon number 12 lumps are considered, i.e., normal, mono-, di- and tribranched alkanes, mono-, di-, tri- and tetracycloalkanes and mono-, di-, tri- and tetra-aromatic components. 

This observation is the first part of the cancellation puzzle [20, 21, 27, 29]. We know from Section lll.B that we should be able to solve it directly by applying Eq. (19), which will separate out the contributions to the DCS made by the 1-TS and 2-TS reaction paths. That this is true is shown by Fig. 9(b). It is apparent that the main backward concentration of the scattering comes entirely from the 1-TS paths. This is not a surprise, since, by definition, the direct abstraction mechanism mentioned only involves one TS. What is perhaps surprising is that the small lumps in the forward direction, which might have been mistaken for numerical noise, are in fact the products of the 2-TS paths. Since the 1-TS and 2-TS paths scatter their products into completely different regions of space, there is no interference between the amplitudes f (0) and hence no GP effects. 

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 ... 

If you mix sulfur and iron filings in a certain proportion and then heat the mixture, you can see a red glow spread through the mixture. After it cools, the black solid lump which has been produced, even if crushed into a powder, does not dissolve in carbon disulfide and is not attracted by a magnet. The material has a new set of properties it is a compound, called iron(II) sulfide. It has a definite composition, and if, for example, you had mixed more iron with the sulfur originally, some iron(II) sulfide and some leftover iron would have resulted. The extra iron would not have become part of the compound. 

Molecular diffusion (or self-diffusion) is the process by which molecules show a net migration, most commonly from areas of high to low concentration, as a result of their thermal vibration, or Brownian motion. The majority of reactive transport models are designed to simulate the distribution of reactions in groundwater flows and, as such, the accounting for molecular diffusion is lumped with hydrodynamic dispersion, in the definition of the dispersivity. 

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.)... 

Here the summation index i covers both traps and recombination centers. We suppose initially that there is only one trap and that the rest of the impurities or defects act as recombination centers. A trap, of course, is by definition more likely to release its charge to the appropriate band than to hold it until annihilation by subsequent capture of the opposite charge. The reverse is true for recombination centers. We can often lump the sum effect of all the recombination centers into a single lifetime t ... 

The term phase defines any homogeneous and physically distinct part of a system which is separated from other parts of the system by definite bounding surfaces. For example, ice, liquid water, and water vapor are three phases. Each is physically distinct and homogeneous, and there are definite boundaries between ice and water, between ice and water vapor, and between liquid water and water vapor. Thus, we say that we have a three-phase system solid, liquid, and gas. One particular phase need not be continuous. For instance, the ice may exist as several lumps in the water. 

Consider first the series junction of N waveguides containing transverse force and velocity waves. At a series junction, there is a common velocity while the forces sum. For definiteness, we may think of A ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance Rj (s), as depicted in Fig. 10.11 for TV= 4. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., V(s) = C v for velocity waves and F(s) = C / for force waves, where jC denotes the Laplace transform. In the discrete-time case, we use the z transform instead, but otherwise the story is identical. 

All the works just cited have an important and common assumption the well-known, three-lump Weekman model with three kinetic constants. This three-lump model was a significant achievement, but it lumps together gas and coke, which are clearly different. Therefore, we consider it absolutely necessary to expand this model by using a five-lump model with eight cracking constants and the following definitions A = feedstock (bp > 350 °C), O = gas oil (bp 221-350 °C), E = gasoline (bp 36-221 °C), G = gas (bp < 36 ° C), and C = coke (by combustion). 

We have already likened the macroscopic transport of heat and momentum in turbulent flow to their molecular counterparts in laminar flow, so the definition in Eq. (5-60) is a natural consequence of this analogy. To analyze molecular-transport problems (see, for example. Ref. 7, p. 369) one normally introduces the concept of mean free path, or the average distance a particle travels between collisions. Prandtl introduced a similar concept for describing turbulent-flow phenomena. The Prandtl mixing length is the distance traveled, on the average, by the turbulent lumps of fluid in a direction normal to the mean flow. 

A pure substance has a definite composition, which stays the same in response to physical changes. A lump of copper is a pure substance. Water (with nothing dissolved in it) is also a pure substance. Diamond, carbon dioxide, gold, oxygen, and aluminum are pure substances, too. 

You would never express the mass of a lump of gold, like the one in Figure 5.11, in atomic mass units. You would express its mass in grams. How does the mole relate the number of atoms to measurable quantities of a substance The definition of the mole pertains to relative atomic mass, as you learned in section 5.1. One atom of carbon-12 has a mass of exactly 12 u. Also, by definition, one mole of carbon-12 atoms (6.02 x 1023 carbon-12 atoms) has a mass of exactly 12 g. 

Commercial ammonium carbonate is amixture of ammonium hydrogen carbonate and ammonium carbamate, NH2-CO-ONH4, and is probably a definite compound of the two salts.7 It is formed by distillation of a mixture of ammonium chloride with carbonate of potassium, sodium, or calcium, and was formerly manufactured by the dry distillation of animal excrement, horn, and other substances. It is now obtained as a by-product of the gas-manufacture, and after sublimation condenses in hard lumps. 

So far, lumping has been defined, but nothing has been said concerning the dynamic behavior of the" system. Now we come to the definition of exact lumping A system is said to be exactly lumpable by the matrix L if there exists an N X N matrix K, enjoying the same properties as K does (i.e., off-diagonal elements of K are nonpositive, = 0, and there exists at least an m" =... 

The dimensionless quantities defined above for a plane wall can also be used for a cylinder or sphere by replacing the space variable x by r and the half-thickness L by the outer radius r. Note that the characieristic length in the definition of the Biot number is taken to be the half-thickness L for the plane wall, and the radius for the long cylinder and sphere instead of 1//A used in lumped system analysis. 

Discussion Note that the Biot number in lumped system analysis was defined differenfjy.as Bi = hLJk= h rJ3)lk. However, either definition can be used in determijiifig the applicability of the lumped system analysis unless Bi = 0.1. 

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