Two- and one-dimensional model

The results of calculations of the Nusselt number are presented in Fig. 10.19. Here also the data of the calculated heat transfer by the quasi-one-dimensional model by Khrustalev and Faghri (1996) is shown. The comparison of the results related to one and two-dimensional model shows that for relatively small values of wall superheat the agreement between the one and two-dimensional model is good enough (difference about 3%), whereas at large At the difference achieves 30%. 

The boundary conditions that govern both the one- and two-dimensional models are usually stated in the following manner. 

Georg RB, Reynolds BC, Frank M, Halhday AN (2006) Mechanisms controlhng the silicon isotopic compositions of river water. Earth Planetary Sci Lett 249 290-306 Gerdes ML, Baumgartner LP, Person M, Rumble D (1995) One- and two-dimensional models of fluid flow and stable isotope exchange at an outcrop in the Adamello contact aureole, Southern Alps, Italy. Am Miner 80 1004-1019... 

Lin et al. [44] simulated the enhancement using the two-dimensional model, as a function of all important parameters. It is clear from this work that the role of lateral diffusion depends mainly on the particle capacity, that is, on the value of partition coefficient, H. This conclusion was drawn by Brilman et al. [42,54], as well. The one-dimensional model does not contain the lateral effect and, as a consequence, the absorption rate would be underestimated by this model and the difference of the results obtained by the one- and two-dimensional models would be increased with increasing partition coefficient. The question is in which parameter range this error can be neglected. 

Comparison between One and Two-Dimensional Models. In a recent article ( 2), the differences between the responses of the one and two-dimensional versions of the same type of model were compared. It was found that they depend to a greater extent on kinetic parameters (j3 and y) than on physical parameters (Re and dp/dt). In order to explain this behaviour an analysis of the error in the evaluation of the reaction rate for a one-dimensional model is carried out. Since the reaction rate at radial mean concentration and temperature generally differs from the radial mean reaction rate, we can define ... 

An extensive analysis of the behaviour of different types of non-adiabatic fixed bed reactor models is carried out and the importance of the heterogeneous one and two-dimensional models III-0 and III-T is stressed. Although in these models the heat and mass transfer phenomena are correctly taken into account, they... 

One-and two dimensional models are used to describe deactivation of nonadiabatic packed bed reactors. 

For the deactivation process studied,the time step At=30 min utes has been found to be satisfactory for both the one-and two-dimensional models. The axial step size Az=0.02m has been used for both models and 10 grid points in the radial direction have been adapted for the two-dimensional descriptions. To improve the order of approximation of the explicit integration process,the nonlinear reaction rate term has been evaluated at the (i+1/2)level. The values of concentration and temperature at (i+1/2) have been determined by extrapolation from the profiles i-1 and i. 

A comparison between axial temperature profiles of temperature, activity, poison,and benzene concentrations calculated from one-and two-dimensional models is presented in Figures 1-4, respectively. The values calculated from a two-dimensional model are drawn in Figures 1-4 at the radial position r=0.707Ifo. The theory reveals (12) that the axial profile taken for a two-dimensional model at r = 0.707Ro should well agree with the one-dimensional approximation. The calculated results prove that there is a very good agreement between the one- and two-dimensional models. 

The radial activity profile is a simple parabola-like function with a minimum in the center of the tube. As a result, for a 2.54 cm tube, the deactivation process can be simulated very accurately from the one-dimensional approximation. We can also notice that both the one-and two-dimensional models predict correctly the growing transient hot spot temperature. This effect was predicted by Blaum (3) for extreme reaction conditions and was experimentally observed by Mikus et al. (7). in a quasiadiabatic laboratory reaction. Evidently,this phenomenon can be observed also for a rather mild condition in a deactivating bed of full size. After 25 hours of deactivation,the hot spot moved from z=0.75m to z=1.65m and the temperature increased by 15°C. 

Frequently,for higher values of or RQ,the one-dimensional model can qualitatively predict a rather complex interaction between the temperature and concentration fields. Such a situation is presented in Figures 5-7. For high values of E the activity 0 is very sensitive to temperature fields and the activity calculated from one- and two-dimensional models can be different. For higher values of R (e.g. R lM),the activity profile can be affected by both the temperature and concentration fields. With higher temperatures, the consumption of benzene and poison and the rate of deactivation is higher however, the concentration of poison is lower. This complex interaction may result in radial profiles of activity with minima outside the reactor axis (c.f., Figure 8). Of course, the one-dimensional model cannot correctly describe such a behavior. 

A second study by McKeen, Liu, and Kiang [7] supports the conclusion that SO2 oxidation does not consume HO. They modelled the oxidation of SO2 in the stratosphere from the 1982 eruption of the El Chichon volcano. Using one and two dimensional models, they examined the photochemical effects on the injection of several megatons of sulfur into the stratosphere and compared the results with the chemical lifetimes of sulfur obtained from observations of stratospheric SO2 and sulfate particles. The SO2 to sulfate conversion scheme they tested assumed that odd hydrogen radicals were consumed (as in Reaction (5)). Under this condition their model predicted that the HO species were significantly suppressed and that the chemical lifetime of SO2 was greater than 100 days. This can be compared with the observations which imply a lifetime... 

One- and two-dimensional models of fixed-bed reactors are compared for a numerical case in G. F. Froment, Current Design Status, Fixed-bed Catalytic Reactors, Ind. Eng. Chem., 59, 18 (1967). 

When more than one reaction occurs the calculation procedures are similar to those illustrated in Example 13-6. A difference equation is written for each component, and these equations are solved simultaneously with the difference equation for the conservation of energy. Froment has used one- and two-dimensional models to predict conversion and temperatures in a fixed-bed reactor for the oxidation of o-xylene to phthalic anhydride, CO, and COj, with a V2O5 catalyst. The reaction scheme is... 

Fig. 19. Comparison of the one- and two-dimensional models for Mn distribution in the top 0-18 cm of sediment at NWC. The production rate in both cases is that found for core NWC-4. The anoxic precipitation rate is assumed to be zero. The effective cylinder geometry used in the two-dimensional model is that determined for NH4 in Part I r, = 0.14 cm, rj = 4.5 cm. The basal gradient is constrained to be zero. The diffusion geometry created by irrigated burrows results in a vertical pore-water solute profile exhibiting apparent precipitation.

As mentioned earlier, the one- and two-dimensional models of reaction profiles discussed above can be generalized to any number of dimensions. The only prerequisite is that sufficient information on potential constants, especially anharmonic ones, is available. 

A physical interpretation of the perturbation constants a and b, in the one- and two-dimensional models, respectively, can be inferred from a description of the spontaneous hydrolysis of acetals in terms of the n,a interaction hypothesis. According to this hypothesis, the process is driven by the double-bond-no-bond resonance that provides stabilization energy SE proportional to the overlap of the 0(2) n- and C(l)-0(1) o orbitals and inversely proportional to the separation of their energy levels... 

In Equation 5.220, P and Pq denote the total pressure at the reactor outlet and inlet, respectively. For an exact calculation, the pressure drop expression, Equation 5.216, needs to be solved simultaneously with the mass and energy balances of one- and two-dimensional models. 

Figure 2 shows a comparison of the results obtained with the one and two dimensional models when diffusional limitations in the solid pellets (heterogeneous models) were considered. These curves correspond to the operating condition of = 10 kg/s and for three different inlet methanol partial pressures - 1 atm,... 

It was shown [163 that there are two stable stationary solutions for a proper choice of the control parameter values. With Xq = 20, Bq = 62.5, Dq = 750, the system is bistable for 0 < kc < 1.68. These values are used in the rest of the work. One and two dimensional model-reactors were used. In a one dimensional reactor of length , the convection velocity is all along equal to If the space is dis-... 

Srinivasan, Hurwitz, and Bockris [57] derived expressions for the current-distribution and the /x=o x = o relation in one- and two-dimensional models for a single pore with a flat meniscus. The influence of electrolytic resistance, diffusion, and an inhibited discharge step with a simplified kinetic equation were considered. It was shown that the one-dimensional treatment may be used for current densities lower by a factor of 4 than the limiting current density. The limiting current density was obtained by the two-dimensional treatment. Special cases of predominance of one or two processes were discussed. As expected, the conclusions for a special case are the same as those for the corresponding case of the flooded pore (see sections 1 to 3) in the one-dimensional model. 

All of the previous one- and two-dimensional models assumed that oxygen concentration in the gas channel, or, more specifically, at the gas diffusion layer-gas channel interface, is known. However, oxygen concentration along the channel changes as oxygen is being depleted in the electrochemical reaction. At the same time, water content along the channel increases. 

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