Transient actual boundary concentrations

In Example 10.1 the case where the aerosol concentration does not change with time was considered. In many practical situations, however, the aerosol concentration does change with time, possibly as a result of diffusion and subsequent loss of particles to a wall or other surface. In this event, Fick s second law, Eq. 9.2, must be used. Solution of this equation is possible in many cases, depending on the initial and boundary conditions chosen, although the solutions generally take on very complex forms and the actual mechanics involved to find these solutions can be quite tedious. Fortunately, there are several excellent books available which contain large numbers of solutions to the transient diffusion equation (Barrer, 1941 Jost, 1952). Thus, in most cases it is possible to fit initial and boundary conditions of an aerosol problem to one of the published solutions. Several commonly occurring examples follow. 

The measurement of induction times (see second of Eqs. 7.31) or the measurements of Ihe temporal change of concentration profiles for transient diffusion, i.e. diffusion before reaching the steady-state condition or with time-dependent concentration boundary conditions, only provide values for experimental quantities in which both De and R are included. Usually, only the so-called apparent or retarded diffusion coefficient Da = DJR is determined. Due to the t)q)ical ranges of values for 0, /"and R the values for the three diffusion coefficients De, De and Da differ by up to 2 orders of magnitude. Since the terminology is sometimes ambiguous, literature data about diffusion coefficients must be scrutinised very carefully to see which coefficient has actually been determined or used. 

Coupling may also occur via boundary conditions, e.g. the reaction rate in a catalyst pellet depends on the concentration and the temperature of the fluid surrounding the pellet. At steady state, when coupling between equations occnrs throngh boundary conditions, an exact or approximate analytical solution can be calculated with boundary conditions as variables, e.g. the effectiveness factor for a catalyst particle can be formulated as an algebraic fimction of surface concentrations and temperature. The reaction rate in the catalyst can then be calculated using the effectiveness factor when solving the reactor model. However, this is not possible for transient problems. The transport in and out of the catalyst also depends on the accumulation within the catalyst, and the actual reaction rate depends on the previous history of the particle. 

---