Dispersed plug-flow model with first-order chemical reaction

Dispersed Plug-Flow Model with First-Order Chemical Reaction 

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  

The solution to this equation (unlike the partial differential equation 2.14) has been shown not to depend on the precise formulation of the inlet and outlet conditions, i.e. whether they are open or closed 051. In the following derivation, however, the reaction vessel is considered to be closed , i.e. it is connected at the inlet and outlet by piping in which plug flow occurs and, in general, there is a flow discontinuity at both inlet and outlet. The boundary conditions to be used will be those which properly apply to a closed vessel. (See Section 2.3.5 regarding the significance of the boundary conditions for open and closed systems.) 

In order to understand these boundary conditions, let us consider that the inlet pipe in which ideal plug flow occurs has the same diameter (shown by broken lines) as the reactor itself (Fig. 2.21). Inside the reactor, across any section perpendicular to the z-direction, the flux of the reactant, i.e. the rate of transfer is made up of two contributions, the convective flow uC and the diffusion-like dispersive flow 

Because there is no dispersion in the inlet pipe, upstream of the reactor, there is only the convective contribution. 

---

FIG. 19-2 Chemical conversion by the dispersion model, (a) Volume relative to plug flow against residual concentration ratio for a first-order reaction. (b) Residual concentration ratio against kCQt for a second-order reaction, (c) Concentration profile at the inlet of a closed-ends vessel with dispersion for a second-order reaction with kC01 = 5. 

---