Dispersion first order reaction

FIG. 23-15 Chemical conversion by the dispersion model, (a) First-order reaction, volume relative to plug flow against residual concentration ratio, (h) Second-order reaction, residual concentration ratio against kC t. 

Equations 8-148 and 8-149 give the fraction unreacted C /C o for a first order reaction in a closed axial dispersion system. The solution contains the two dimensionless parameters, Np and kf. The Peclet number controls the level of mixing in the system. If Np —> 0 (either small u or large [), diffusion becomes so important that the system acts as a perfect mixer. Therefore,... 

Danckwerts et al. (D6, R4, R5) recently used the absorption of COz in carbonate-bicarbonate buffer solutions containing arsenate as a catalyst in the study of absorption in packed column. The C02 undergoes a pseudo first-order reaction and the reaction rate constant is well defined. Consequently this reaction could prove to be a useful method for determining mass-transfer rates and evaluating the reliability of analytical approaches proposed for the prediction of mass transfer with simultaneous chemical reaction in gas-liquid dispersions. 

FIGURE 9.10 Relative error in the predicted conversion of a first-order reaction due to assuming piston flow rather than axial dispersion, kt versus Pe. 

Example 9.6 Compare the nonisothermal axial dispersion model with piston flow for a first-order reaction in turbulent pipeline flow with Re= 10,000. Pick the reaction parameters so that the reactor is at or near a region of thermal runaway. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

This example models the dynamic behaviour of an non-ideal isothermal tubular reactor in order to predict the variation of concentration, with respect to both axial distance along the reactor and flow time. Non-ideal flow in the reactor is represented by the axial dispersion flow model. The analysis is based on a simple, isothermal first-order reaction. 

The dispersion model of example DISRE is extended for non-isothermal reactions to include the dispersion of heat from a first-order reaction. 

For a first-order reaction, the reaction rate, M = kCsr] where Cs is the concentration at the particle surface. On the basis of unit volume of the three-phase dispersion, the reaction rate becomes kmol/m3s. 

Solve the dispersion equation for first order reaction for several... 

Find the general solution of the equations of first order reaction with dispersion, closed end conditions. 

Fig. 18. General design chart for the dispersion model for first-order reaction A R with no change in volume (e = 0). Ordinate gives the dispersed flow reactor volume divided by the volume of an ideal PFR which achieves the same conversion. - - -, Constant kr ------, constant DIuL.

The remaining portion of the first tank (about 3.1 x 10s kg) rests on the bottom of the river from which it is introduced by a first-order reaction into the flowing water (rate constant kb = 3 x 10 3 Ir1). Dispersion of this cloud can be disregarded, but not air-water exchange. 

Figure 17.4. Dispersion model. Conversion of first-order reaction as function of the Peclet number.

We will consider a dispersed plug-flow reactor in which a homogeneous irreversible first order reaction takes place, the rate equation being 2ft = k, C. The reaction is assumed to be confined to the reaction vessel itself, i.e. it does not occur in the feed and outlet pipes. The temperature, pressure and density of the reaction mixture will be considered uniform throughout. We will also assume that the flow is steady and that sufficient time has elapsed for conditions in the reactor to have reached a steady state. This means that in the general equation for the dispersed plug-flow model (equation 2.13) there is no change in concentration with time i.e. dC/dt = 0. The equation then becomes an ordinary rather than a partial differential equation and, for a reaction of the first order ... 

To incorporate mixing by the dispersed plug flow mechanism into the model for the bubble column, we can make use of the equations developed in Chapter 2 for dispersed plug flow accompanied by a first-order chemical reaction. In the case of the very fast gas-liquid reaction, the reactant A is transferred and thus removed from the gas phase at a rate which is proportional to the concentration of A in the gas, i.e. as in a homogeneous first-order reaction. Applied to the two-phase bubble column for steady-state conditions, equation 2.38 becomes ... 

The reaction term follows from the assumption of a first-order reaction with a rate constant k, defined on the basis of unit volume of particle (see Chapter 3) ef allows for the change in basis to unit volume of dispersion. The effectiveness factor r is also included to take into account any diffusional resistance within the pores of the particle. 

Equation (19-22) indicates that, for a nominal 90 percent conversion, an ideal CSTR will need nearly 4 times the residence time (or volume) of a PFR. This result is also worth bearing in mind when batch reactor experiments are converted to a battery of ideal CSTRs in series in the field. The performance of a completely mixed batch reactor and a steady-state PFR having the same residence time is the same [Eqs. (19-5) and (19-19)]. At a given residence time, if a batch reactor provides a nominal 90 percent conversion for a first-order reaction, a single ideal CSTR will only provide a conversion of 70 percent. The above discussion addresses conversion. Product selectivity in complex reaction networks may be profoundly affected by dispersion. This aspect has been addressed from the standpoint of parallel and consecutive reaction networks in Sec. 7. 

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