Chapter 8

8.8 Radial Variations in a Tubular Reactor

In the previous sections we have assumed that there were no radial variations in velocity, concentration, temperature or reaction rate in the tubular and packed bed reactors. As a result the axial profiles could be determined using an ordinary differential equation (ODE) solver. In this section we will consider the case where we have both axial and radial variations in the system variables in which case will require a partial differential (PDE) solver. A PDE solver such as COMSOL Multiphysics, will allow us to solve tubular reactor problems for both the axial and radial profiles.

Figure P8-24a Shell balance.

Molar Flux

In order to derive the governing equations we need to add a couple of definitions. The first is the molar flux of Species A, W_A mol/m²•s. The molar flux has two components, the radial component W_Ar, and the axial component, W_Az. The molar flow rates are just the product of the molar fluxes and the cross sectional areas normal to their direction of flow A_cz_. E.g. for species A

F_Az= W_Az A_cz

where A_cz is the cross sectional area of the tubular reactor.

In chapter 11 we discuss the molar fluxes in some detail, but for now let us just say they consist of a diffusional component, , and a convective flow component,

where D_e is the effective diffusivity m²/s, and U_z is the axial molar average velocity, (m/s). Similarly the flux in the radial direction is

where U_r is the molar average velocity in the radial direction. A mole balance on a cylindrical system volume of length Dz and thickness Dr gives

Figure 8-4 Cylindrical shell of thickness Dr and length Dz

Mole Balance

Dividing by 2prDrDz and taking the limit as Dr and Dz ® 0

Similarly for any species i

(8-X)

Substituting for W_iz and W_ir

Assumption 1. U_r is essentially zero.

Assumption 2. U_z is constant wrt z.*

*1F U_z is not constant then we have another term , with

Energy Flux

When we applied the first law of thermodynamics to a reactor to relate either temperature and conversion or molar flow rates and concentration, we arrived at Equation (8-9). Neglecting the work term we have

The term is the heat added to the system most always includes a conduction component of some form. Neglecting PV work, the total energy of the system

We now define an energy flux, e, (J/m²•s).

where q is given by Fourier's law. For axial conduction Fourier's law is

where k_c is the thermal conductivity (J/m•s•K). The energy transfer (flow) is the flux times the cross sectional area, A_c, normal to the energy flux

Energy flow = e • A_c

Energy Balance

Using the energy flux, e, to carry out an energy balance on our system volume 2prDrDz we have

Dividing by 2prDrDz and taking the limit as Dr and Dz ® 0

Substituting for e

and expanding the convective energy flux, ,

we obtain

Recognizing the term in brackets is related to Eqn (8-X) and is just the rate of disappearance of species i, -r_i. We have

Recalling

and

we have

Some initial approximations

Assumption 3. The diffusive flux, , in the radial direction is greater than the convective flux, i.e. , thus

Assumption 4. The convective flux in the axial direction, UC_i, is greater than the diffusive flux, , consequently the flux of species i in the axial direction is

With these approximations we have

Further approximations

Assumption 5. Neglect the convective energy flux in the radial direction with respect to the conductive flux, . The energy balance now becomes

where and is assumed constant.

Boundary and initial conditions

A. Initial conditions

B. Boundary condition

(1) Radial

(a) At r = 0, we have symmetry

(b) At the tube wall r = R, the flux to the wall on the reaction side equals the convective flux out of the reactor into the shell side of the heat exchanger.

with