Comments on the behaviour of a fixed-bed reactor

Dear Sirs, In a recent series co-workers (see e.g. of papers by Morbidelli and Varma and Morbidelli and Varma 1986a; Morbidelli et al., 1986h Lee and Varma, 1987, 1988; Bauman and Varma 1990), the multiplicity behaviour of a fixed bed reactor is discussed. The basis for this analysis was given by Morbidelli and Varma (1986a) and Morbid& et al. (1986b), where the authors devote many columns to mathematically prove, by means of “a rigorous application of the Implicit Function Theorem”, that the multiplicity of a heterogeneous plug Row packed bed reactor is fully determined by the multiplicity of the inlet particle. Thus, according to Morbidelli et al., this model even in its basic form, does not exhibit an infinite number of steady states, a property referred to as infinite multiplicity (Liu and Amundson, 1962; Eigenberger, 1972). Eigenbcrger (1972) showed quantitatively that infinite multiplicity will not occur if axial dispersion of heat is present. Morbidelli et al. claim that they have proven that, in order to exclude infinite multiplicity, it is not necessary to take into account axial dispersion. We will show that their proof is incorrect and that the modelling equations of the heterogeneous plug flow reactor without thermal axial dispersion do have an infinite number of steady-state solutions. The notation and the equations can be found in Morbidelli and Varma (1986a). The solution of the heterogeneous plug flow model involves the integration of an ordinary differential equation, eq. (7), where at each step in the integration, the non-linear constraint eq. (l’lt_representing the state of the particles needs to be solved to yield OP. As is well known, there can be either one or two stable solutions for this constraint. If the particles operate in the multiplicity region, then there are two possible solutions for the constraint eq. (17). In each step of the integration we must choose between these two solutions. In other words, we must choose the particle to be either in the lower or in the higher state and during the integration we must make this choice for each individual particle over and over again. Now, according to Morbidelli et al. we cannot choose arbitrarily between these two states. They concluded this from the following statement: “from the implicit function theorem it can be seen that the obtained solution of eq. (17) is unique and continuous as long as the following condition is satisfied: (Received 30 August 1990; accepted 23 April 1991)