Ordinary differential equation regime

In the stationary regime and for a non-reactive solid, the volumetric solid flow rate is constant and equal to the inlet flow rate given by the vibrating feeder (Qv ). From Eq. (1), the following ordinary differential equation is directly obtained and can be solved with a boundary condition (exit height fixed) in order to compute the bed profile along the kiln ... 

Equation (2) (as an ordinary differential equation) and Eq. (3) apply now with Eq. (4). As already implied, a laboratory well-mixed reactor for heterogeneous catalysis is more difficult to realize than a PFR. Many versions have been used 12), and Froment and Bischoff 13) illustrate reactors with external recycle, with internal recycle 1,14), and with an internal spinning basket 15). When using these reactors for experiments in the transient regime, it is important to keep to a minimum the volume outside the bed of catalyst. Internal recycle reactors involve bearings exposed to hot reactive gases and require a magnetic drive system for leak-proof operation. Exter-... 

The equations governing the steady-state current as a function of tq can be obtained by solving the ordinary differential equations for spherical diffusion governing the appropriate kinetic scheme or by using the reaction layer approximation (7, 91-94). The relevant behavior at microspherical electrodes in any time regime can also be obtained through digital simulation (17). 

A series of differential equations describes the evolution of liquid phase products with time in a semi-batch reactor with continuous feed of the gas. If the formations of the by-products E and F are second order in the reacting gas A and all other reactions are first order, and moreover if A absorbs and reacts significantly only in the bulk liquid phase, so that film reaction is negligible and the slow regime applies, then a series of ordinary differential equations describe the concentration trajectories of liquid phase products with time. 

The advantage of Eq. (160) (representing a parabolic partial differential equation from a mathematical viewpoint) is that it can be solved exactly by standard numerical techniques e.g., by the finite-difference Crank-Nicholson scheme under transient (nonstationary) conditions [37,64]. These calculations showed that the duration of the transient regimes is of the order of seconds, as previously estimated. Under the stationary conditions, Eq. (160) is simplified to the ordinary one-dimensional differential equation which can be solved by standard numerical techniques [18,76,118,119]. 

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