Chapter 14 Models for Nonideal ReactorspNot all tank reactors are perfectly mixed nor do all tubular reactor exhibit plug-flow behavior.pIn these situations, some means must be used to allow for deviations from ideal behavior.pChapter 13 showed how the RTD was sufficient if the reaction was first-order or if the fluid was either in state of complete segregation or maximum mixedness. But the other situations compose a great majority of reactor analysis problem and cannot be ignored.14.1 One Parameter ModelpUse a single parameter to account for the nonideality of a real reactor, such as a nonideal tubular reactor.pFor nonideal CSTR, the one-parameter models include the reactor dead volume Vd or the fraction f of fluid bypassing the reactor.14.2 Tanks-in-Series ModelpA real reactor can be modeled as a number of tanks-in- series. pThe number of tanks necessary, n, is determined from the E(t) curve. For a first order reactionpKnowing the value of n, we can determine the conversion and/or effluent concentrations for the reactor.For reactions other than first order and for multiple reactionspThe sequential equations must be solved: 14.3 Dispersion modelpThis model is also used to describe nonideal tubular reactors.pIn this model,there is an axial dispersion of the materials, which is governed by an analogy to Ficks law of diffusion, superimposed on the flow.pTo illustrate how dispersion affects the concentration profile in a tubular reactor, we consider the injection of a perfect tracer pulse.pThe following figure shows how dispersion causes the pulse to broaden as it moves down the reactor and becomes less concentrated.The Concentration ProfilepThe molar flow rate of tracer( FT) by both convection and dispersion is :pA mole balance on the inert tracer T givesAxial Dispersion ModelpSubstituting for FT and dividing by the cross-sectional area AC , we obtain the axial dispersion model as follows:Dispersion in a Tubular Reactor with Laminar FlowpIn a laminar flow reactor we know that the axial velocity varies in the radial direction according to the Hagen- Poiseuille equation:pwhere U is the average velocity.RTD FunctionpFor laminar flow we saw that the RTD function E(t) was given by :pIn arriving at this distribution E(t) it was assumed that there was no transfer of molecules in the radial direction between streamlines.Residence Time CalculationpWith the aid of equation:pWe know that the molecules on the center streamline (r=0) exited the reactor at a time t = / 2, and molecules traveling on the streamline at r = 3R/4 exited the reactor at time t = 8/ 7 The Question?pWhat would happen if some of the molecules traveling on the streamline at r = 3R/4 jumped (i.e., diffused) to the streamline at r = 0? pThe answer is that they would exit sooner than if they had stayed on the streamline at r = 3R/4. Analogously, if some of the molecules from the faster streamline at r = 0 jumped (i.e., diffused) to the streamline at r = 3R/4, they would take a longer time to exit. Radial Diffusion in Laminar Flow Radial diffusion in laminar flowAxial Dispersion Coefficient DapTo answer this question we will derive an equation for the axial dispersion coefficient, Da, that accounts for the axial and radial diffusion mechanisms.pIn deriving Da, which is referred to as the Aris-Taylor dispersion coefficient, we closely follow the development given by Brenner and Edwards.Axial DispersionpIn addition to the molecules diffusing between streamlines, they can also move forward or backward relative to the average fluid velocity by molecular diffusion (Ficks law). pWith both axial and radial diffusion occurring, the question arises as to what will be the distribution of residence times when molecules are transported between and along streamlines by diffusion. The Convective-Diffusion EquationpThe convective-diffusion equation for solute (e.g., tracer) transport in both the axial and radial direction is:where c is the solute concentration at a particular r, z and t.pWe are going to change the variable in the axial direction z to z*, which corresponds to an observer moving with the fluid:pA value of z* = 0 corresponds to an observer moving with the fluid on the center streamline. Using the chain rule, we obtain:Average Axial ConcentrationpBecause we want to know the concentrations and conversions at the exit to the reactor, we are really only interested in the average axial concentration , which is given by Equation Solutionpwe are going to solve Equation (14-16) for the solution concentration as a function of r and then substitute the solution c(r, z, t) into Equation (14-17) to find (z, t). AssumptionspTo solve the equations above to determine the Aris-Taylor dispersion coefficient, we make the following four assumptions: pWe now apply the approximations above to Equation (4-17) to arrive at the following equation: pFinally, the equation describing the variation of the averag