Stirred equations

Thus far we have considered systems where stirring ensured homogeneity witliin tire medium. If molecular diffusion is tire only mechanism for mixing tire chemical species tlien one must adopt a local description where time-dependent concentrations, c r,f), are defined at each point r in space and tire evolution of tliese local concentrations is given by a reaction-diffusion equation... 

From this equation we see that increasing k leads to a shorter analysis time. For this reason controlled-potential coulometry is carried out in small-volume electrochemical cells, using electrodes with large surface areas and with high stirring rates. A quantitative electrolysis typically requires approximately 30-60 min, although shorter or longer times are possible. 

In many types of contactors, such as stirred tanks, rotary agitated columns, and pulsed columns, mechanical energy is appHed externally in order to reduce the drop si2e far below the values estimated from equations 36 and 37 and thereby increase the rate of mass transfer. The theory of local isotropic turbulence can be appHed to the breakup of a large drop into smaller ones (66), resulting in an expression of the form... 

In this equation, represents the rate of energy dissipation per unit mass of fluid. In pulsed and reciprocating plate columns the dimensionless proportionahty constant K in equation 38 is on the order of 0.3. In stirred tanks, the proportionaUty constant has been reported as 0.024(1 + 2.5 h) in the holdup range 0 to 0.35 (67). The increase of drop si2e with holdup is attributed to the increasing tendency for coalescence between drops as the concentration of drops increases. A detailed survey of drop si2e correlations is given by the Hterature (65). 

The Rate Law The goal of chemical kinetic measurements for weU-stirred mixtures is to vaUdate a particular functional form of the rate law and determine numerical values for one or more rate constants that appear in the rate law. Frequendy, reactant concentrations appear raised to some power. Equation 5 is a rate law, or rate equation, in differential form. 

For maximum purification, the zone should move as slowly as possible, the sample should be many zone lengths long, the melt should be stirred, and more zone passes should be made than several times the number of zone lengths in the sample. Stirring and slow zone travel lower 5U/Z7, which in turn increases 1 —, as shown by equation 5 and Figure 6. In practice, it is sufficient for bVjD - 0.1. 

Volt mmetiy. Diffusional effects, as embodied in equation 1, can be avoided by simply stirring the solution or rotating the electrode, eg, using the rotating disk electrode (RDE) at high rpm (3,7). The resultant concentration profiles then appear as shown in Figure 5. A time-independent Nernst diffusion layer having a thickness dictated by the laws of hydrodynamics is estabUshed. For the RDE,... 

The alkylthio group is replaceable by nucleophiles. The positions 7 and 4 react under mild conditions in that order the 2-alkylthio functions require more drastic treatment. Conversion of l-methyl-4-methylthiopteridin-2-one (157) into the 4-methylamino derivative (158) can be achieved by stirring with methylamine at room temperature (equation 48). The reactivity of an alkylthio group can often be further enhanced by oxidation to the corresponding sulfoxide and sulfone. Thus, reaction of l,3-dimethyl-7-methylthiolumazine (160) with m-chloroperbenzoic acid yields 7-methylsulfinyl- (161) and 7-methylsulfonyl-l,3-dimethyllumazine (162 equation 49) (82UP21601). 4-Amino-2-methylthio-7-... 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

This equation has two unknowns Xq and Xe), and an empirical relation between them is needed. Many have been tried, and one of the best is to assume that the excess of To over Te expressed as a ratio to Tp (zero for a perfectly stirred chamber) is a constant A [ (Tg — Tg)/ Tp]. Although A should vaiy with burner type, the effects of firing rate and percent excess air are small. In the absence of performance data on the land of furnace under study, assume A = 300/Tp, °R or 170/Tp, K. The left side of Eq. (5-178) then becomes D 1 —Xc + A), and with coefficients of Xc and Xc collected, the equation becomes... 

Integration of the Ergim equation for well stirred flow gives... 

For operation with an inert tracer, the material balances are conveniently handled as Laplace transforms. For a stirred tank, the differential equation... 

Three basic fluid contacting patterns describe the majority of gas-liquid mixing operations. These are (1) mixed gas/mixed liquid - a stirred tank with continuous in and out gas and liquid flow (2) mixed gas/batch mixed liquid - a stirred tank with continuous in and out gas flow only (3) concurrent plug flow of gas and liquid - an inline mixer with continuous in and out flow. For these cases the material balance/rate expressions and resulting performance equations can be formalized as ... 

The type of optimum reaetor that will proeess 200 m /hr is a eon-tinuous flow stirred tank reaetor (CFSTR). This eonfiguration operates at the maximum reaetion rate. The volume V[ of the reaetor ean be determined from the design equation ... 

In Chapter 3, the analytieal method of solving kinetie sehemes in a bateh system was eonsidered. Generally, industrial realistie sehemes are eomplex and obtaining analytieal solutions ean be very diffieult. Beeause this is often the ease for sueh systems as isothermal, eonstant volume bateh reaetors and semibateh systems, the designer must review an alternative to the analytieal teehnique, namely a numerieal method, to obtain a solution. Eor systems sueh as the bateh, semibateh, and plug flow reaetors, sets of simultaneous, first order ordinary differential equations are often neeessary to obtain die required solutions. Transient situations often arise in die ease of eontinuous flow stirred tank reaetors, and die use of numerieal teehniques is die most eonvenient and appropriate mediod. 

In a single eontinuous flow stirred tank reaetor, a portion of the fresh feed eould exit immediately in the produet stream as soon as the reaetants enter the reaetor. To reduee this bypassing effeet, a numher of stirred tanks in series is frequently used. This reduees the prohahility that a reaetant moleeule entering the reaetor will immediately find its way to the exiting produet stream. The exit stream from the first stirred tank serves as the feed to the seeond, the exit stream from the seeond reaetor serves as the feed to the third, and so on. For eonstant density, the exit eoneentration or eonversion ean he solved hy eonseeutively applying Equation 5-158 to eaeh reaetor. The following derived equations are for a series of tliree stii+ed tanks (Figure 5-23) with eonstant volume Vr. 

There are eight nonlinear equations involving the material balanees of speeies A and B in the four stirred tank reaetors. Rearranging these equations yields the following ... 

The experimental study of solid eatalyzed gaseous reaetions ean be performed in bateh, eontinuous flow stirred tank, or tubular flow reaetors. This involves a stirred tank reaetor with a reeyele system flowing through a eatalyzed bed (Figure 5-31). For integral analysis, a rate equation is seleeted for testing and the bateh reaetor performanee equation is integrated. An example is the rate on a eatalyst mass basis in Equation 5-322. 

Consider an exothermie iiTeversible reaetion with first order kineties in an adiabatie eontinuous flow stirred tank reaetor. It is possible to determine the stable operating temperatures and eonversions by eom-bining bodi die mass and energy balanee equadons. For die mass balanee equation at eonstant density and steady state eondition. 

The various types of reaetors employed in the proeessing of fluids in the ehemieal proeess industries (CPI) were reviewed in Chapter 4. Design equations were also derived (Chapters 5 and 6) for ideal reaetors, namely the eontinuous flow stirred tank reaetor (CFSTR), bateh, and plug flow under isothermal and non-isothermal eonditions, whieh established equilibrium eonversions for reversible reaetions and optimum temperature progressions of industrial reaetions. 

In these model equations it is assumed that turbulence is isotropic, i.e. it has no favoured direction. The k-e model frequently offers a good compromise between computational economy and accuracy of the solution. It has been used successfully to model stirred tanks under turbulent conditions (Ranade, 1997). Manninen and Syrjanen (1998) modelled turbulent flow in stirred tanks and tested and compared different turbulence models. They found that the standard k-e model predicted the experimentally measured flow pattern best. 

The simplest method to measure gas solubilities is what we will call the stoichiometric technique. It can be done either at constant pressure or with a constant volume of gas. For the constant pressure technique, a given mass of IL is brought into contact with the gas at a fixed pressure. The liquid is stirred vigorously to enhance mass transfer and to allow approach to equilibrium. The total volume of gas delivered to the system (minus the vapor space) is used to determine the solubility. If the experiments are performed at pressures sufficiently high that the ideal gas law does not apply, then accurate equations of state can be employed to convert the volume of gas into moles. For the constant volume technique, a loiown volume of gas is brought into contact with the stirred ionic liquid sample. Once equilibrium is reached, the pressure is noted, and the solubility is determined as before. The effect of temperature (and thus enthalpies and entropies) can be determined by repetition of the experiment at multiple temperatures. 

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