Various problems regarding the behaviour of fixed bed catalytic reactors involving highly exothermic reactions have been studied in relation to optimal design and control.
Steady and unsteady state mathematical models of various degrees of complexity have been used from those considering axial and radial diffusion to the simple one dimensional representation neglecting both mechanisms. Since these models would be used repeatedly, they must be relatively simply solved by a computer in a reasonable time and without loss of detail necessary to take full advantage of control or optimization processes.
Orthogonal collocation has proved a very efficient method of solution for solution of the radial diffusion and axial diffusion models. It has been shown that in the former case, an optimal distribution of the collocation points in the radial direction requires the minimum number
of points. Double collocation, under certain conditions, is an efficient integration procedure both for steady and unsteady state models. In the case of the axial diffusion model, some orthogonal polynomials converge faster than others depending on the profiles to be approximated.
It has been recognised that further reduction in computing time is usually coupled with a reduction in model dimensionality. A model reduction technique has been used to lump the radial profiles in the unsteady state radial diffusion model. This lumped model has the ability
to regenerate the radial profiles from simple algebraic expressions with reasonable accuracy compared with the distributed parameter system.
Studies on the transient behaviour of the reactor have indicated that the major dynamic factor is the solid heat capacitance and that the inlet temperature and concentration may be manipulated to effectively
control the reactor in a multi-variable mode. Consideration has also been given to the response of the reactor to sinusoidal and damped sinusoidal perturbations at the inlet. It has been found that for certain frequencies severe hot spots may be formed over a part of the
radial profiles before a safe quasi-stationary state is reached. A detailed examination of this behaviour has shown that the differences in the speeds of propagation of the concentration and temperature waves along the reactor were significant factors in determining the resulting
behaviour.
A steady state axial diffusion model in which the radial variation in temperature is approximated by a parabolic radial temperature profile has been considered. The limitations of this approximation have been identified and treated by the model reduction technique. Thus the model
developed gives adequate representation of axial and radial dispersion processes. Axial dispersion becomes important if the axial temperature and concentration gradients increase beyond a certain value. This value may be calculated from the axial profiles of the one dimensional model which neglects axial diffusion.
Consideration has been given to a dynamic model based on the above and the collocation and the reduction technique used to solve the model. The solution time is reduced to reasonable levels making, it suitable for detailed studies. Including the axial dispersion in the dynamic model
did not alter the qualitative behaviour of the reactor.
The exceptional cases are those related to parametric sensitivity or temperature runaway studies. Instability arising from parametric sensitivity or multiple states in either the radial or axial diffusion models has been considered. The criteria developed indicate that if
instability is to occur in the reactor, it is likely to originate from the solid phase regardless of the mechanisms considered in the fluid. Thus, conditions of catalyst particle stability are essential in establishing local stability of the reactor.