Lumping

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- 

The mechanisms considered above are all composed of steps in which chemical transformation occurs. In many important industrial reactions, chemical rate processes and physical rate processes occur simultaneously. The most important physical rate processes are concerned with heat and mass transfer. The effects of these processes are discussed in detail elsewhere within this book. However, the occurrence of a diffusion process in a reaction mechanism will be mentioned briefly because it can lead to kinetic complexities, particularly when a two-phase system is involved. Consider a reaction scheme in which a reactant A migrates through a non-reacting fluid to reach the interface between two phases. At the interface, where the concentration of A is Caj, species A is consumed in a first-order chemical rate process. In effect, consecutive rate processes are occurring. If a steady state is achieved, then 

Equation (55) describes a situation in which the rate is said to be chemically controlled whereas eqn. (56) describes mass transfer or diffusion control. Such descriptions become possible because the two rate processes are occurring in sequence. 

Consider, now, a reaction scheme in which A diffuses from a phase interface into a stagnant reaction fluid in which a first-order chemical reaction occurs. Here, the chemical rate process and the physical rate process (diffusion) occur in parallel, so that 

/ is the distance from the interface. If Ca is the concentration of A at the surface of the reaction fluid and Ca attains a very low value as I increases, then from eqn. (58), we can obtain an expression for the rate of consumption of A. 

Symmetry boundary conditions reduce the computational effort due to the requirement that all solution variables have zero normal gradient at the symmetry plane. As such, only one symmetric solution is obtained, but in reality several solutions may exist. 

Periodic boundary conditions are very useftil when a system has periodic behavior. The inflow at one boundary is set as the outflow from another surface, e.g. only 1/6 of a tank reactor with six impeller blades and no baffles is calculated by using the radial outflow on one side as the inflow on the other side. 

Rgure Simplilied geometry. When only the outer part of a complex body is active, the body may be simplified to a slab. 

In many cases, irregular-shaped particles or a size distribution of particles can be treated as spheres with an equivalent diameter that gives the same sinface/volmne ratio as the irregular-shaped particle. The Saunter mean diameter, usually denoted t/j2, defined as the total volume divided by the total surface area, is a very common method used to characterize a size distribution of bubbles or drops. With this definition, the total volume can be calculated from the munber of particles multiphed by the volume of a sphere with diameter and the total sinface area can be calculated by multiplying the number of particles by the surface of the sphere. 

The use of hydrauhc diameter in calculating pressure drop is a commonly used concept for simplifying geometry. The hydraulic diameter for an opening is calculated by multiplying the open surface area by 4, then dividing by the wetted periphery, e.g. for a cylindrical opening d = Mnd IAt)l(nd), and for a square duct d = / Ad). 

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Lump Sum contract contractor manages and executes specified work to an agreed delivery date for a fixed price. Penalties may be due for late completion of the work, and this provides an incentive for timely completion. Payment may be staged when agreed milestones are reached. 

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. 

If the reference date is set at the beginning of the year (e.g. 1.1.98) then full year discount factors imply that t is a whole number and that cashflows occur in lump sums at the end of each year. If the cashflow occurs uniformly throughout the year and the reference date is the beginning of the year then mid-year discount factors are more appropriate, in which case the discounting equation would be ... 

The total number of bits set on A is a + c. and the total number of bits set on B is b + c. These totals form the basis of an alternative notation that uses a instead of a + c, and b instead oib + c [16]. This notation, however, lumps together similarity and dissimilarity components" - a disadvantage when interpreting a similarity measure. 

Carry out this preparation precisely as described for the a-compound, but instead of zinc chloride add 2 5 g. of anhydrous powdered sodium acetate (preparation, p. 116) to the acetic anhydride. When this mixture has been heated on the water-bath for 5 minutes, and the greater part of the acetate has dissolved, add the 5 g. of powdered glucose. After heating for I hour, pour into cold water as before. The viscous oil crystallises more readily than that obtained in the preparation of the a-compound. Filter the solid material at the pump, breaking up any lumps as before, wash thoroughly with water and drain. (Yield of crude product, io o-io 5 g.). Recrystallise from rectified spirit until the pure -pentacetylglucose is obtained as colourless crystals, m.p- 130-131° again two recrystallisations are usually sufficient for this purpose. 

CAUTION. The vapour of selenium dioxide is poisonous, and all operations involving the hot material, alone or in solution, should be performed in a fume-cupboard. If lumps of selenium dioxide have to be powdered in a mortar, the latter should also be in a fume-cupboard, with the window lowered as far as possible, to avoid inhaling the fine dust. (cf. p. 191)... 

In order to prepare the free base, place the remaining half of the crude hydrochloride in a 200 ml. beaker, add 20 ml. of water, and then stir the mixture with a glass rod until a thin paste of uniform consistency (quite free from lumps) is obtained. Now... 

Required Ethyl acetoacetate, 32 g. (32 ml.) acetaldehyde-ammonia, lO g. Note. The aldehyde-ammonia should preferably be fresh material the quantity should be increased to 15 g. if an old sample, which has formed brown sticky lumps, is employed.)... 

This matrix is usually diagonalized using a simple mass lumping technique (Pittman and Nakazawa, 1984) to minimize the computational cost of pressure calculations in this method. 

STRESS. Applies the variational recovery method to calculate nodal values of pressure and, components of the stress. A mass lumping routine is called by STRESS to diagonalize the coefficient matrix in the equations to eliminate the... 

A force field that can produce vibrational spectra has a second advantage in that the Ay// calculations can be put on a much more satisfactory theoretical base by calculating an enthalpy of formation at 0 K as in ab initio procedures and then adding various thermal energies by more r igorous means than simply lumping them in with empirical bond enthalpy contributions to Ay//-. The stronger the theoretical base, the less likely is an unwelcome surprise in the output. 

By adding a few lumps of solid carbon dioxide this produces a number of cold spots here and there, and assists the formation of crystals. 

Hydrogen chloride. Method 1 from concentrated sulphuric acid and fused ammonium chloride). The most convenient procedure is to allow concentrated sulphuric acid to react with lumps of fused ammonium chloride in a Kipp s apparatus. The gas may be dried by passage through a wash bottle containing concentrated sulphuric acid the latter should be followed by an empty wash bottle or flask as a precaution against sucking back of the contents of the reaction vessel. 

The hard lump of sulphur remaining in the flask is best removed by boiling with concentrated nitric acid in the fume cupboard. 

The surface layer on lumps of sodium is removed with a large knife, the clean sodium is rapidly weighed out on a few large Alter papers and immediately trans. ferred to a beaker containing sodium-dried ether. The sodium may then be removed at leisure, cut into small pieces and transferred to the flask. 

Place 50 g. of o-chloronitrobenzene and 75 g. of clean dry sand in a 250 ml. flask equipped with a mechanical stirrer. Heat the mixture in an oil or fusible metal bath to 215-225° and add, during 40 minutes, 50 g. of copper bronze or, better, of activated copper bronze (Section 11,50, 4) (1), Maintain the temperature at 215-225° for a further 90 minutes and stir continuously. Pour the hot mixture into a Pyrex beaker containing 125 g. of sand and stir until small lumps are formed if the reaction mixture is allowed to cool in the flask, it will set to a hard mass, which can only be removed by breaking the flask. Break up the small lumps by powdering in a mortar, and boil them for 10 minutes with two 400 ml. 

Equip a 500 ml. three-necked flask with an efficient stirrer (e.g., a Hershberg stirrer. Fig. II, 7, 8) and a reflux condenser stopper the third neck. Place a solution of 30 g. of sodium hydroxide in 100 ml. of water, and also 20-5 g. (17-1 ml.) of pure nitrobenzene in the flask, immerse it in a water bath maintained at 55-60°, and add 21 g. of anhydrous dextrose in small portions, with continuous stirring, during 1 hour. Then heat on a boiUng water bath for 2 hours. Pour the hot mixture into a 1 litre round-bottomed flask and steam distil (Fig. II, 40, 1) to remove aniline and nitrobenzene. When the distillate is clear (i.e., after about 1 htre has been collected), pour the residue into a beaker cooled in an ice bath. The azoxybenzene soon sohdifies. Filter with suction, grind the lumps of azoxybenzene in a mortar, wash with water, and dry upon filter paper or upon a porous plate. The yield of material, m.p. 35-35-5°, is 13 g. Recrystallise from 7 ml. of rectified spirit or of methyl alcohol the m.p. is raised to 36°. ... 

Beckmann rearrangement of benzophenone oxime to benz-anilide. Dissolve 2 g. of benzophenone oxime in 20 ml. of anhydrous ether in a small conical flask and add 3 g. of powdered phosphorus pentachloride (or 3 ml. of pure tbionyl chloride). Distil off the solvent and other volatile products on a water bath CAUTION ether), add 25 ml. of water, boil for several minutes and break up any lumps which may be formed. Decant the supernatant liquid, and recrystallise, in the same vessel, from boiling alcohol. The product is benzanilide, m.p. 163° confirm this by a mixed m.p. determination with an authentic specimen. 

Dissolve 5 g. of phenol in 75 ml. of 10 per cent, sodium hydroxide solution contained in a wide-mouthed reagent bottle or conical flask of about 200 ml. capacity. Add 11 g. (9 ml.) of redistilled benzoyl chloride, cork the vessel securely, and shake the mixture vigorously for 15-20 minutes. At the end of this period the reaction is usually practically complete and a sohd product is obtained. Filter oflf the soUd ester with suction, break up any lumps on the filter, wash thoroughly with water and drain well. RecrystaUise the crude ester from rectified (or methylated) spirit use a quantity of hot solvent approximately twice the minimum volume required for complete solution in order to ensure that the ester does not separate until the temperature of the solution has fallen below the melting point of phenyl benzoate. Filter the hot solution, if necessary, through a hot water funnel or through a Buchner funnel preheated by the filtration of some boiling solvent. Colourless crystals of phenyl benzoate, m.p. 69°, are thus obtained. The yield is 8 g. 

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