Macroscopic mixing time

Copolymerizations. The uniform chemical environment of a CSTR makes it ideally suited for the production of copolymers. If the assumption of perfect mixing is justified, there will be no macroscopic composition distribution due to monomer drift, but the mixing time must remain short upon scaleup. See Sections 1.5 and 4.4. A real stirred tank or loop reactor will more closely... 

Im a 5(<.. The mixing time can be shortened somewhat by setting the turbine at one-third the liquid depth, in which case the trough of a concentration fluctuation transported back to the impeller by the lower (upper) circulation loop arrives with the peak from the upper (lower) loop. Depending somewhat on the method used to assess macroscopic homogeneity, (m will then be roughly... 

As mentioned previously, fast liquid-phase polymerisation processes are characterised by the reaction time, which may be equal to or shorter than the mixing time of the initial components. One of the most important results of the investigation of these processes is evidence of several macroscopic process modes. Each of the modes is characterised by certain temperature field structures and reactant concentrations and therefore, different reaction fronts and rates [1-3]. 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

The macroscopic mass action rate law, which holds for a well-mixed system on sufficiently long time scales, may be written... 

Although there is a distribution of residence times, the complete mixing of fluid at the microscopic and macroscopic levels leads to an averaging of properties across all fluid elements. Thus, the exit stream has a concentration (average) equivalent to that obtained as if the fluid existed as a single, large fluid element with a residence time of t = V/q (equation 2.3-1). 

One of the simplest methods to generalize formal kinetics is to treat reactant concentrations as continuous stochastic functions of time, which results in a transformation of deterministic equations (2.1.1), (2.1.40) into stochastic differential equations. In a system with completely mixed particles the macroscopic concentration n (t) turns out to be the average of the stochastic function Cj(<)... 

The term macroscopic diffusion control has been used to describe processes in which the rate of reaction is determined essentially by the rate of mixing of the reactant solutions. The nitration of toluene in sulpholane by the addition of a solution of nitronium fluoroborate in sulpholane appears to fall into this class (Ridd, 1971a). Obviously, if a reaction is subject to microscopic diffusion control when the reactants meet in a homogeneous solution, it must also be subject to macroscopic diffusion control when preformed solutions of the same reactants are mixed. However, the converse is not true. The difficulty of obtaining complete mixing of solutions in very short time intervals implies that a reaction may still be subject to macroscopic diffusion control when the rate coefficient is considerably below that for reaction on encounter. The mathematical treatment and macroscopic diffusion control has been discussed by Rys (Ott and Rys, 1975 Rys, 1976), and has been further developed recently (Rys, 1977 Nabholtz et al, 1977 Nabholtz and Rys, 1977 Bourne et al., 1977). It will not be considered further in this chapter. 

Consider a set of 10 coins that forms the system. The most ordered macroscopic states are 10 heads up or 10 tails up. In either case, there is one possible configuration, and hence the entropy according to Eq. 7.4-13 is zero. The most disordered state consists of 5 heads up and 5 tails up, which allows for 252 different configurations. This would be the case if we reflip each coin sequentially for a long time. What we have done is mix the system. 

System with random fluxes is defined as the nonequilibrium system where the fluxes of substance, heat, etc. change randomly. One can cite numerous examples of such systems turbulent gas-liquid systems with intensive heat/mass transfer, turbulent fluids containing dispersed solids, etc. In the case of pore formation, such situation is realized when the heat fluxes change randomly because of air fluidization or mechanical mixing. All macroscopic measured parameters of stationary turbulent flows, like their pressure, temperature, excess (free) energy, entropy, etc. do not change with time, while their values and directions in different spots of the flows can vary significantly. 

The macroscopic approach, in which it is not taken into account what happens inside the cell in detail, but only an overall view of the system is described. In fact, the system is considered as a black box from the fluid dynamic point of view and then, it is assumed that the cell behaves a mixed tank reactor (the values of the variables only depend on time and not on the position since only one value of every variable describes all positions). This assumption allows simplifying directly all the set of partial differential equations to an easier set of differential equations, one for each model species. For the case of a continuous-operation electrochemical cell, the mass balances take the form shown in (4.5), where [.S, ]... 

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