Reactor design parabolic

Boundary layer similarity solution treatments have been used extensively to develop analytical models for CVD processes (2fl.). These have been useful In correlating experimental observations (e.g. fi.). However, because of the oversimplified fiow description they cannot be used to extrapolate to new process conditions or for reactor design. Moreover, they cannot predict transverse variations In film thickness which may occur even In the absence of secondary fiows because of the presence of side walls. Two-dimensional fully parabolized transport equations have been used to predict velocity, concentration and temperature profiles along the length of horizontal reactors for SI CVD (17,30- 32). Although these models are detailed, they can neither capture the effect of buoyancy driven secondary fiows or transverse thickness variations caused by the side walls. Thus, large scale simulation of 3D models are needed to obtain a realistic picture of horizontal reactor performance. 

The sixth reactor design criterion requires that the pressure drop at the minimum residence time be less than 100 psi. For a small diameter channel, the flow through that channel wiU be laminar for all flow rates of interest for this particular applicahon. Neglecting end effects, the solutions to the equations of continuity and of motion for steady-state laminar flow of an incompressible Newtonian fluid are well-known, yielding a parabolic velocity distribution and the Hagen-Poiseuille equahon for pressure drop, as given in Eqs. (9) and (10) ... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Within the general concept of the ESVE models, Alfano et al. conceived a model for the radiant power profile of a tubular light source located in the focal axis of a parabolic reflector in order to analyze the design of a cylindrical photochemical reactor irradiated from the bottom [118]. Differences between experimental and calculated (ESVE) results were always less than 15%. 

Non-isothermal and non-adiabatic conditions. A useful approach to the preliminary design of a non-isothermal fixed bed reactor is to assume that all the resistance to heat transfer is in a thin layer near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the approximate design of reactors. Neglecting diffusion and conduction in the direction of flow, the mass and energy balances for a single component of the reacting mixture are ... 

It has been shown that under some geometric restrictions that involve conditions in distances and dimensions of the complete experimental device that is, lamps, reflectors, and reactors, the radiation field produced by the tubular lamp, and the parabolic reflector can be modeled by a onedimensional representation (Alfano et al., 1986). These limitations were imposed on the equipment design of this work. Since is a function of the radiation-absorption species concentration, in this case. Equation (34) is coupled with Equation (32). 

Design of a turbulent reactor requires consideration of V and ijf since both will affect reaction yields. Eor turbulent flow in long, empty pipes, the time-average velocities in the radial and tangential directions are zero since there is no net flow in these directions. The axial velocity component will have a nonzero time-average profile y (r). This profile is considerably flatter than the parabolic profile of laminar flow, but a profile nevertheless exists. The zero-slip boundary condition still applies and forces V (R) = 0. The time average-velocity changes very rapidly near the tube wall. 

For turbulent flow in pipes the velocity profile can be calculated from the empirical power law design formula (1.360). Similar balance equations with purely molecular diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.359) might suffice. It is important to note that the difference between the cross section averaged ID axial dispersion model equations (discussed in the previous section) and the simplified 2D model equations (presented above) is that the latter is valid locally at each point within the reactor, whereas the averaged one simply gives a cross sectional average description of the axial composition and temperature profiles. 

Compound parabolic collectors (CPCs) belong to the most promising photocatalytic solar reactors which combine the advantages of parabolic trough concentrator and non-concentrating system [178]. CPCs are low-concentration static collectors with reflective surface and can be designed for any given reactor shape (see Fig. 7.3a) [182]. The CPC reflectors are usually made from polished aluminum... 

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