Effective swept volume rate

The collision tube concept is familiar from the kinetic theory of gases. Consider a particle in snch a cube, moving with a relative speed with respect to the other particles which are fixed. The particle in the tnbe sweeps a volume per unit time (m s ). Veimeker [118] named the rate of volume swept by the particle for the effective swept volume rate. Henceforth this name is nsed referring to this qnantity. 

As for the collision density in the macroscopic model formulation, the average collision frequency of fluid particles is usually described assuming that the mechanisms of collision is analogous to collisions between molecules as in the kinetic theory of gases. The volume average coalescence frequency, ac d d, Y), can thus be defined as the product of an effective swept volume rate hc d d, Y) and the coalescence probability, pc d d, Y) (e.g., [16, 92, 114, 39, 46, 118]) ... 

The coalescence terms in the average microscopic population balance can then be expressed in terms of the local effective swept volume rate and the coalescence probability variables ... 

Lame coefficients, metric coefficients, or scale factors effective swept volume rate m /s)... 

The volume integrated coalescence frequency, sometimes denoted the effective swept volume rate, hc(d) is given by ... 

Since the collision radius for two particles of equal size is two times the particle radius, the effective volume swept out will be four times that given by Eq. 9-33. Since both particles are diffusing, the effective diffusion constant will be twice that used in obtaining Eq. 9-28. Thus, the effective volume swept out by the particle in a second will be eight times that given by Eq. 9-33. The volume swept out by one mole of particles is equal to /cD (recall that the second-order rate constant has dimensions of liter mol-1 s 1). Thus, when converted to a moles per liter basis and multiplied by 8, Eq. 9-33 should (and does) become identical with the Smolu-chowski equation (Eq. 9-30). 

If the viscosity varies during flow for some reason (decreases with rising temperature or increases as a result of a chemical reaction such as polymerization), the linear Poiseuille P-vs-Q relation is violated and the pressure drop - flow rate curve may become nonmonotonic. This effect in polymerizing reactors can be explained by the fact that the most viscous products of a reaction are swept out of the reactor with increasing flow rate and are replaced. Instead, a reactor is refilled with a fresh reactive mixture of low viscosity. This leads to a decrease of the volume-averaged integral viscosity and therefore the pressure drop decreases. This can be illustrated by the following relationship ... 

Refrigerating capacity Qe is the product of mass flow rate of refrigerant m and refrigerating effect R which is (for isobaric evaporation) R = /ievapomior outlet ievapoiator inlet- Powcr P required for the compressiou, necessary for the motor selection, is the product of mass flow rate m and work of compression W. The latter is, for the isentropic compression, W = /idischarge suction- Both of these characteristics could be c culated for the ideal (without losses) and for the actual compressor. Ideally, the mass flow rate is equal to the product of the compressor displacement Vi per unit time and the gas density p m = Vj p. The compressor displacement rate is volume swept through by the pistons (product of the cylinder number n, and volume of cylinder V = stroke d T 4) per second. In reality, the actual compressor delivers less refrigerant. 

If accurate efficiency data are required, the variance of the injection profile should be measured as a function of the flow rate, with a correction applied by subtracting the first moment of the injection profile from the peak retention time and the variance of the injection profile from the band variance. In actual analytical practice it is often sufficient to minimize this contribution by making sure that the volume of the injection device is much smaller than the volume of the column, that the device is properly swept by the mobile phase, and that actual injection is rapid. The use of a bypass at the column inlet permits a higher flow rate through the injection device than through the column, effectively reducing the band-broadening contribution of the injection system. 

On the other hand, when a reactant B Is frequently scattered by the solvent, its trajectory is entangled (case (b)) and It sweeps the same place repeatedly. Consequently the effective volume swept per unit time becomes smaller than ra2v. This is the reason the rate of reaction or the rate constant decreases with decreasing mean free path of reactant. Case (a) corresponds to low-pressure gas phase reactions, while case (b) corresponds to (narrowly defined) diffusion-controlled reactions. 

Assume that the trajectory of a reactant B is engangled (case (b) in Fig. 1). Assume also that the reactant B carries a charge. Reactants A need not carry any charge. If an external electric field E is applied on the system, the entangled trajectory of reactant B will become more or less straight along the electric field (case (c)). Consequently the effective volume swept per unit time should Increase, and thus the rate of reaction should increase. 

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