Diffusion radial temperature profiles

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 ... 

Figure 4. Radial temperature profiles in a laminar propane diffusion flame (24)...

The backward differencing method requires the solution of 7+1 simultaneous equations to find the radial temperature profile. It is semi-implicit since the solution is still marched-ahead in the axial direction. Fully implicit schemes exist where (7-I- l)(7-l-1) equations are solved simultaneously, one for each grid point in the total system. Fully implicit schemes may be used for problems where axial diffusion or conduction is important so that second derivatives in the axial direction, or 9 r/9z, must be retained in the partial differential equa-... 

The flow patterns, composition profiles, and temperature profiles in a real tubular reactor can often be quite complex. Temperature and composition gradients can exist in both the axial and radial dimensions. Flow can be laminar or turbulent. Axial diffusion and conduction can occur. All of these potential complexities are eliminated when the plug flow assumption is made. A plug flow tubular reactor (PFR) assumes that the process fluid moves with a uniform velocity profile over the entire cross-sectional area of the reactor and no radial gradients exist. This assumption is fairly reasonable for adiabatic reactors. But for nonadiabatic reactors, radial temperature gradients are inherent features. If tube diameters are kept small, the plug flow assumption in more correct. Nevertheless the PFR can be used for many systems, and this idealized tubular reactor will be assumed in the examples considered in this book. We also assume that there is no axial conduction or diffusion. 

For a case that one stable steady state exists transient temperature profiles calculated agree satisfactorily with the measurements. For a case of three steady states the situation is quite complicated. The model used describes propagation of the fronts however, apparently cannot describe front multiplicity. A detailed calculation of the two-dimensional steady state equations including also the radial dispersion terms indicates that the onedimensional model is a very rough approximation for the diffusion" regime. We expect that dynamic calculations with the one-phase two-dimensional model could explain multiplicity of the fronts. 

When gum formation proceeds, the minimum temperature in the catalyst bed decreases with time. This could be explained by a shift in the reaction mechanism so more endothermic reaction steps are prevailing. The decrease in the bed temperature speeds up the deactivation by gum formation. This aspect of gum formation is also seen on the temperature profiles in Figure 9. Calculations with a heterogenous reactor model have shown that the decreasing minimum catalyst bed temperature could also be explained by a change of the effectiveness factors for the reactions. The radial poisoning profiles in the catalyst pellets influence the complex interaction between pore diffusion and reaction rates and this results in a shift in the overall balance between endothermic and exothermic reactions. 

The DE (3-95) is identical in form to the familiar heat equation for radial conduction of heat in a circular cylindrical geometry. Thus we see that the evolution in time of the steady Poiseuille velocity profile is completely analogous to the conduction of heat starting with an initial parabolic temperature profile -(1 - r2)/4. In our problem, the final steady velocity profile is established by diffusion of momentum from the wall of the tube so that the initial profile for w eventually evolves to the asymptotic value uT - 0 as 1 oo. The characteristic time scale for any diffusion process (whether it is molecular diffusion, heat conduction, or the present process) is (f y cli fl iisivity ), where tc is the characteristic distance over which diffusion occurs. In the present process, tc = R and the kinematic viscosity v plays the role of the diffusivity so that... 

Two-dimensional models have been developed for non adiabatic reactors with pronounced heat effects and when radial diffusion cannot be considered infinite. In one-dimensional models the radial heat and mass transfer rates are considered infinite and thus the radial concentration and temperature profiles are flat. It therefore predicts uniform temperature and concentrations in the radial direction at every axial position along the length of the reactor. On the other hand two-dimensional models predict the detailed temperature and concentration patterns in the reactor (Froment, 1972 Karanth and Hughes, 1974 Froment and Bischoff, 1979). 

After extracting the kinetic parameters, selected results for CO oxidation over were used to analyze the effect of non-uniform temperature and velocity distributions on the conversion of CO. In order to determine the optimum number of multiple CSTR s to capture the behavior of a PFR, the rate law of Oh and Carpenter (14) for the NO+CO reaction was used to model a monolith channel as a CSTR in series. The results indicated that it was sufficient to use 5 reactors in series to capture the performance of the PFR behavior in the NO+CO reaction The cells of a monolith reactor were taken as independent parallel reactors ignoring the mass transfer and diffusion through the ceramic pores. The axial and radial temperature and velocity profiles collected from the literature(4,5) are used to calculate the... 

B.2, Here, we demonstrate once more how Brownian dynamics relates to diffusive behavior, by simulating spherical particles of radius 1 mm in water at room temperature. At time f = 0, a particle is released at the origin and undergoes 3-D Brownian motion. Write a program that repeats this simulation many times and plots the radial concentration profile of particles as a function of time. It is easier to do the data analysis if you do the simulations concurrently. Then, solve the corresponding time-dependent diffusion equation in spherical coordinates and compare the results to that obtained fi om Brownian dynamics. 

FIGURE 8.21 Comparison of the radial profiles for scattering cross-section gvv and temperature (uncorrected) as a function of radial position for a coannular ethane diffusion flame (from Santoro et al. [85]). 

A practical difficulty, encountered in this study was performing the AUC runs at temperatures higher than 40 °C at such high temperatures, the oil vapors from the diffusion pump interfere with the UV absorption optics. They circumvented this problem by using a different type of optical scanning known as the Schlieren optics, which generates the data as profiles of radial derivative of concentration distributions as a function of radius (as opposed to the concentration versus radius scans obtained from UV optics). 

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