ADDING A COOLING JACKET- NON-ADIABATIC PLUG FLOW

Mass and energy balance

Adding a cooling jacket to the reactor also means adding radial energy effects, and the energy balance will no longer take on a one-dimensional form. The radial temperature effects will affect the rate of reaction and thereby result in radial concentration gradients. Consequently the problem is shifted into a two-dimensional problem. The equations used to solve this problem thereby take these forms of the general equations derived above.

                         (5)

                (11)

Boundary conditions

To solve the energy balance we need two more boundary conditions in the radial dimension. Since this is a two-dimensional problem, COMSOL Multiphysics is solving the two equations for a two-dimensional slice of the reactor. To get the entire solution this slice is rotated. Thus, the boundaries of this slice are by the reactor wall and in the center of the reactor.

At the center boundary, we assume symmetry or insulation, meaning that there is no net flux leaving or entering the center boundary in the radial direction. This boundary condition comes from the fact that there are no radial concentration or temperature gradients in the infinitely small volume element in the center of the reactor.

No mass can leave the reactor through the reactor wall, i.e. the boundary is insulated. The wall boundary condition will therefore be the same as for the center; there are no radial concentration gradients over the volume element close to the reactor wall. However, there is a net energy flux leaving the reactor due to the cooling jacket. The boundary condition is simply set to the energy flux.

The boundary conditions can be expressed as:

                         Mass/Mole condition       Energy condition

at z = 0           

  at r = 0            

         at r = R            

T_a is the outer wall temperature and U is the overall heat transfer coefficient.

Assuming that the cooling jacket results in a constant wall temperature, can you foresee how the Temperature Surface plot and the Radial Profiles will look like? Take some time to think about it!