EQUATIONS

Energy balance

 The equation below formulates the general energy balance that takes into consideration radial variations in a tubular reactor. In Chapter 8, section 8.9 you can see the derivation of the energy balance.

  Where  Cp_i is the heat capacity of species i,

            T the temperature,

            k_e the thermal conductivity of the reaction mixture,

            W_ir and W_iz are the radial and axial fluxes of species i respectively,

  (see Chapter 11 of CRE)

the heat of reaction,

            and -r_A the reaction rate based on the limiting species A. For example, the reaction rate for an elementary first-order reaction can be expressed as: -r_A = kC_A

Similar to the Eqn (1) for the mass balance, the term on the left hand side represents accumulation of energy. The first and the second term on the right hand side represents the conductive fluxes in the radial and axial directions respectively. The two following terms represents heat flux generated by a flux in mass. This flux consists as always of two parts; Flux by diffusion and convection. The last term represents the heat production through the heat of reaction. So, a difference in energy flux over a volume element is either due to accumulation of heat or the production or consumption of it.

Assumptions

In the same way that the expression for the mass balance can be simplified for steady-state conditions, so can the energy expression:

                    (7) 

The convective flux in the radial direction is smaller than the diffusive flux and can therefore be neglected, i.e. U_r» 0. Therefore the radial flux will only consist of the diffusive term.

            (8)

Furthermore, we assume that U_z is constant throughout the length of the reactor and equal to the inlet velocity and that the convective flux is greater in the axial direction than the diffusive flux. Therefore the axial flux will only consist of the convective term.

        (9)

The convective energy flux in the radial direction can be neglected with respect to the conductive flux, i.e.

Then Eqn (9) becomes

                                        (10)

Finally, we expand the radial conduction term and change the axial convection term slightly. C_iCp_i is equivalent to ρ_iCp_i if Cp_i is  mass based instead of molar based. Assuming that ρC_p is constant, the energy balance becomes:

                                              (11)

Eqn (5) and (11) represents the simplified general forms of Eqn (1) and (6) that has been used by COMSOL Multiphysics to solve the problem stated above.