K.R. Westerterp

Notes on wave theory in heat conduction: a new boundary condition

The physical basis behind the simple hyperbolic heat transport model is discussed. The model equations are interpreted as energy equations for a system in a state of local nonequilibrium. On this basis a new boundary condition for the hyperbolic model is proposed when the temperature of a surface is given. It is based on a nonequilibrium situation and it reduces to the conventional boundary condition under the same assumptions as needed to justify Fouriers law of heat conduction and reduces to the known temperature-jump boundary condition at steady-state conditions. The significance of the new boundary condition is demonstrated through its application to one-dimensional heat transfer problems via hyperbolic equations. The difference between solutions of hyperbolic and Fourier models is found to be much larger than reported before and it does not vanish in the steady-state.

Experimental study of vaporization effect on steady state and dynamic behavior of catalytic pellets

The impact of the combined evaporation of the liquid phase and reaction on single catalyst pellet performance has been studied experimentally. The exothermic, catalyzed hydrogenation of α-methylstyrene (AMS) to cumene has been employed as a model reaction. Steady state and dynamic experiments have been performed in a single catalytic pellet reactor using five catalytic pellets of different porous structures, thermal conductivity, apparent catalytic activity and distribution of catalyst in the pellet. Gas-phase temperature, concentration of AMS in the gas phase and the liquid flow rates have been varied. The measured center and surface temperatures of each pellet reveal the existence of two significantly different steady states in the range of liquid flow rate. The range of the liquid flow rate over which the two steady states were observed, the pellet temperature and the pellet dynamics depend strongly on the amount of AMS vapor in the gas phase and the catalyst properties. The obtained experimental data are helpful to elucidate the mechanism of hot-spot formation and runaway in multiphase fixed-bed reactors.

Nonequilibrium effects in fixed-bed interstitial fluid dispersion

Continuum models for the role of the interstitial fluid with respect to mass and heat dispersion in a fixed bed are discussed. It is argued that the departures from local equilibrium and not the concentration and temperature gradients as such should be considered as the driving forces for mass and heat dispersion fluxes. This general principle results in a new two-dimensional, continuum model for dispersion in the fluid flowing through a packed bed. An essential feature of the model is that mass and heat dispersion flux vectors are taken to be state variables additional to concentration and temperature; their values characterize the departures from local equilibrium. The differential equations for the dispersion model are of the hyperbolic type and they require fundamentally different boundary conditions compared to the usual conditions for the conventionally used diffusion type models. The new model avoids the physical drawbacks inherent to the diffusion models. It provides a new physical interpretation of the scatter in point temperature measurements and of the temperature drop observed near the reactor wall under conditions of heat transfer through the wall. A part of the temperature drop and the temperature scatter are direct consequences of the macroscopic local thermal nonequilibrium state of the fluid. The results predicted by the new model and diffusion model are essentially equivalent in case of slow processes in reactors with large tube-to-particle diameter ratios. Otherwise the diffusion model does not have a solid physical basis and may not be valid for predictive purposes because of the influence of chemical reaction on the transport parameters.

How to Prevent Runaways in Trickle-Bed Reactors for Pygas Hydrogenation

In the past, several runaways have occurred in Trickle-Bed Reactors (TBR) used for the hydrogenation of pyrolysis gasoline as produced in ethylene cracking installations. This phenomenon has been studied in the framework of a special program in the Netherlands, which is administered by the National Science Foundation and which aims to keep scientific groups in the former Soviet Union working in their current difficult times. To this end a working party was set up at the Boreskov Institute of Catalysis (BIC) in Novosibirsk, Siberia in Russia. The day-to-day management of the working party was conducted locally by Professor Kirillov and the authors Westerterp and Kronberg scientifically and administratively managed the project. This paper reports the results of the study that took place in the years from 1997-2000. The study has led to certain recommendations for the design and operation of TBRs for the hydrogenation of pygas, which will be elucidated and explained here. To this end the phenomenon of particle runaway will first be discussed, and then the major results of the study at the BIC will be described, and eventually conclusions will be reached for the running of pygas hydrogenation TBRs. Later, in depth reports will be published on all work done, but now only the most important results are discussed, without going into detail.

Wave Concept in the Theory of Hydrodynamical Dispersion—a Maxwellian Type Approach

A new approach to the modelling of chemical reactors and contactors is discussed. This approach argues that the dispersion should, under most circumstances, be based on Maxwells, rather than Ficks diffusion law. As a pair of first-order partial differential equations of the hyperbolic type and requiring only inlet conditions, the wave model is more realistic physically, has a much wider range of validity and in many practical cases is simpler mathematically. Only mass transfer problems are considered, but the results apply equally well to the hydrodynamic dispersion of heat. It is explained why the standard dispersion model fails in many practical applications and why the new wave model gives much better results.

Solution of the hyperbolic model for heat and mass transfer in packed bed reactors

Publisher Summary The importance of packed bed reactors in chemical industry has prompted extensive research on mathematical modeling of such systems. One of the key problems involved is the formulation of conservation equations for the smoothed concentration and temperature profiles in the flowing fluid. to use these equations to determine concentration and temperature distributions additional relations for the mass and energy dispersion fluxes j mx, jmr , jhx, and jhr are needed. The only approach used for practical calculations is to represent the mixing in terms of modified Ficks law of diffusion and Fouriers law of heat conduction. For the numerical approximation of hyperbolic equations, the method of characteristics is extensively used. The method maximally exploits the properties of differential equations, converting them to ordinary differential equations. It is particularly convenient if the slopes of characteristics are constant. This chapter shows that Wendroffs algorithm adopted to a problem is unconditionally stable, does not depart much from the characteristic method, requires less computer memory, and is faster.