Axial collocation mass balances

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

For these simulations, the discretization of the mass balance equations used first order backward finite differences over a uniform grid of 100 intervals in the axial direction and a third order orthogonal collocation over 50 finite elements in the radial direction [16]. 

For nonlinear reaction kinetics, a numerical solution of the balance Equation 4.121 is carried out. For example, for second-order kinetics, R = kcACB, with an arbitrary stoichiometry, the generation rate expressions, ta = —va CaCb and tb = —vb caCb, are inserted into the mass balance expression, which is solved numerically using, for example, a polynomial approximation (orthogonal collocation method). The performances of the normal dispersion model and its segregated or maximum-mixed variants are compared in Figure 4.34. The symbols are explained in the figure. The comparison reveals that the differences between the segregated, maximum-mixed, and normal axial dispersion models are notable at moderate Damkohler numbers R = Damkohler number). 

Comparative simulations for a dynamic bubble column reactor are presented in Figure 7.23. The time-dependent mass balances (axial dispersion model) and the partial differential equations (PDEs) were discretized with respect to the reactor length coordinate. Finite differences were used for the discretization of the reactor length coordinate, while global collocation was used for film equations. The original system of PDEs was... 

A model for transient simulation of radial and axial composition and temperature profiles In pressurized dry ash and slagging moving bed gasifiers Is described. The model Is based on mass and energy balances, thermodynamics, and kinetic and transport rate processes. Particle and gas temperatures are taken to be equal. Computation Is done using orthogonal collocation In the radial variable and exponential collocation In time, with numerical Integration In the axial direction. 

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