Tubular reactors with axial temperature gradients

Tubular reactors with axial temperature gradients 

In many processes exothermic reactions are carried out in tubular reactors in a practically adiabatic manner. This means that the reaction temperature increases with the length of the reactor, and the reaction rate increases at first exponentially. The conversion can he calculated from mass and heat balances, that are written in the form of two coupled differential equations. For simple first order processes the mass balance was formulated in eq. (7.18). However, the reaction rate constant varies with temperature, therefore eq. (8.1) has to be substituted in eq. (7.18). The boundary conditions were given by eqs. (7.19a) and (7.19b). The heat balance is analogous to the mass balance  

Note that gas flow rate u varies with T. When the number of moles remains constant during the reaction, u is proportional to T for ideal gases. When the number of moles does change, both Boyle s law and eq. (3.9) have to be taken into account. Analogous boundary conditions are 

For reactors with practically complete conversion, the temperature at jc = / will be the same as the exit temperature, and the last equation becomes dTIdz = 0. The two coupled differential equations, eq. (7.18) and eq. (8.12), combined with eq. (8.1), together with the boundary conditions, can be solv using numerical methods. The solution will give the reactant concentration (or the degree of conversion)and the temperature as functions of the length coordinate. This is shown in qualitatively figure 8.3. 

In many processes it is practically impossible to reach complete conversion in one adiabatic tubular reactor, because the temperature range would be too large. The entrance temperature has to be such that the reaction will start at the beginning of the reactor, but a certain maximum temperature may not be surpassed. A simple solution is offered by a series of tubular reactors with intermediate cooling. The 

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Tubular reactors with axial temperature gradients... 

Tubular Reactors. The simplest model of a tubular reactor, the plug-flow reactor at steady state is kinetically identical to a batch reactor. The time variable in the batch reactor is transformed into the distance variable by the velocity. An axial temperature gradient can be imposed on the tubular reactor as indicated by Gilles and Schuchmann (22) to obtain the same effects as a temperature program with time in a batch reactor. Even recycle with a plug flow reactor, treated by Kilkson (35) for stepwise addition without termination and condensation, could be duplicated in a batch reactor with holdback between batches. 

Tubular reactors with both axial and radial temperature gradients In many exothermic processes the reactor temperature has to be controlled within much narrower limits. This is particularly true for many catalytic gas phase reactions. The reasons are usu y that undesired side reactions have to be avoided, or that the catalyst has to be protected against sintering. There are two reactor types for solid/gas-reactions that make good temperature control possible ... 

The flow patterns, composition profiles, and temperature profiles in a real tubular reactor can often be quite complex. Temperature and composition gradients can exist in both the axial and radial dimensions. Flow can be laminar or turbulent. Axial diffusion and conduction can occur. All of these potential complexities are eliminated when the plug flow assumption is made. A plug flow tubular reactor (PFR) assumes that the process fluid moves with a uniform velocity profile over the entire cross-sectional area of the reactor and no radial gradients exist. This assumption is fairly reasonable for adiabatic reactors. But for nonadiabatic reactors, radial temperature gradients are inherent features. If tube diameters are kept small, the plug flow assumption in more correct. Nevertheless the PFR can be used for many systems, and this idealized tubular reactor will be assumed in the examples considered in this book. We also assume that there is no axial conduction or diffusion. 

FEMLAB is a partial differential equation solver (PDE) available commercially from COMSOL. Inc, Included with this text is a special version of FEMLAB that has been prepared to solve problems intolving tubular reactors. Specifically, one can solve CRE problems with heat elTccts involving both axial and radial gradients in concentration and temperature simply by loading the FEMLAB CD on one s computer and running the program. One can also use it to solve isothermal CRE problem,s with reaction and diffusion. 

Tubular reactors, which may be open or packed with catalyst, are considered ideal if there is plug flow of fluid and there are no radial gradients of temperature, concentration, or velocity. In plug-flow reactors, or PFRs, there are axial gradients of concentration and perhaps also axial gradients of temperature and pressure, but in the ideal PFR there is no axial diffusion or conduction. 

To carry out an exothermic reaction in a tubular reactor under nearly isothermal conditions, a small diameter is needed to give a high ratio of surface area to volume. The reactor could be made from sections of jacketed pipe or from a long coil immersed in a cooling bath. The following analysis is for a constant jacket temperature, and the liquid is assumed to be in plug flow, with no radial gradients of temperature or concentration and no axial conduction or diffusion. 

The assumption of plug flow is not always correct. The plug flow assumes that the convective flow (flow by velocity q/A, = v, caused by a compressor or pump) is dominating over any other transport mode. In fact, this is not always correct, and it is sometimes important to include the dispersion of mass and heat driven by concentration and temperature gradients. However, the plug flow assumption is valid for most industrial units because of the high Peclet number. We will discuss this model in some detail, not only because of its importance but also because the techniques used to handle these two-point boundary-value differential equations are similar to that used for other diffusion-reaction problems (e.g., catalyst pellets) as well as countercurrent processes and processes with recycle. The analytical analysis as well as the numerical techniques for these systems are very similar to this axial dispersion model for tubular reactors. 

The model of a non-isothermal wall-cooled ideal tubular reactor discussed in the following neglects radial gradients of temperature and concentration, and only considers axial gradients. The control space is a slab with the differential length dz in the direction of flow. For steady-state operation, the mass balance for a differential element dVa (= zdAt.im) and a constant volume reaction of reactant A is given by... 

A polymerization reaction carried out in a tubular reactor under conditions of laminar flow is associated with complex radial profiles of vdodty, and axial and radial temperature and concentration gradients exist. This is only beginning to be explored in terms of the fundamental... 

Additional complications peculiar to the analysis of copolymerization data obtained in continuous tubular processes are associated with the fact that such reaches never operate isothermally and often with a substantial axial pressure drop. In addition, there often exists a substantial reridence time distribution, accompanied by a radial concentration gradient, as already pointed out. Data obtained in such reactors are induded in Table 10, but it can be seen readily how the copol3mer compositions reported — and hence the reactivity ratios calculated — could depend not only on feed composition, pressure, entrance and exit temperature and average residence time, but also on residence time distribution, pipe diameter and flow rate. Again, such S3 tems are usually very inadequately defined, but data obtained in tubular reactors must not be rejected a priori for the reasons already cited. 

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