Modeling deviation profiles

FIGURE 2.11 a Deviation profiles for the different values of Af at fixed Ax = 0.l b deviation profiles of approximation at Ar = 0.005 for three values Ar and c the compromise deviation profile for the single measured task at Ar = 0.1 and At = 0.0025. (Reprinted from Zhuiykov, S., Mathematical modelling of YSZ-based potentiometric gas sensors with oxide sensing electrodes part II Complete and numerical models for analysis of sensor characteristics. Sensors and Actuators B, Chetn. 120 (2007) 645-656, with permission from Elsevier Science.)... 

The effect of steam to methane ratios on the performance of reformer U was investigated by considering three different steam to methane ratios S/M- 2.5, 5/M= 5.0, 5/A/= 8.0). At the entrance of reformer II an increase of steam to methane ratio decreases the effectiveness factors of methane while in the rest of the reformer the increase in steam to methane ratios increase the effectiveness factor of methane as shown in Figure 6.33a. The increase of the steam to methane ratios have a slight effect on the effectiveness factors of carbon dioxide (Figure 6.33b), but near the exit of the reformer, the profiles fluctuate between positive and negative values. The differences between the effectiveness factors of methane obtained by the dusty gas model and simplified models I and II are large, while for the effectiveness factors of carbon dioxide the simplified models deviate from the dusty gas model in the second half of the catalyst tube. 

The validity of Eq. (70) is restricted to the case of small deviations from equilibrium and the quiescent surface (d = 0). A quantitative theory, which is not subjected to the latter two restrictions, can be developed by generalizing the approach of the model concentration profiles from Ref. 35 this is done in Ref. 123. In view of Eq. (68), the generalization of Eqs. (24) and (25) to micellar solutions reads... 

Computer Models, The actual residence time for waste destmction can be quite different from the superficial value calculated by dividing the chamber volume by the volumetric flow rate. The large activation energies for chemical reaction, and the sensitivity of reaction rates to oxidant concentration, mean that the presence of cold spots or oxidant deficient zones render such subvolumes ineffective. Poor flow patterns, ie, dead zones and bypassing, can also contribute to loss of effective volume. The tools of computational fluid dynamics (qv) are useful in assessing the extent to which the actual profiles of velocity, temperature, and oxidant concentration deviate from the ideal (40). 

First we run the model so as to study the influenee of sereen resistanee on the overall flow patterns and on the maldistribution. The resulting profiles of inward radial veloeity at the inner sereen aeross the eatalyst bed appear in Figure 10-13 for different sereen resistanees. It ean be seen that higher sereen resistanee leads to more-uniform flow, as one would expeet. The existing sereens (with resistanee eoeffieients C2 of 2 X 10 /m) appear to be satisfaetory, sinee the deviations experieneed are less than 10%. 

Model is relatively insensitive to BODyu load variations. A doubling of loads (from each point source) results in deviations of 5-9 percent DO saturation from the standard profile. 

Model is relatively insensitive to changes in k over a three-fold range of 0.02-0.06 day Predicted DO concentrations deviate no more than 6 percent saturation from standard profile. 

The reasons for the deviation of the constant-composition model from the full model are apparent when the concentrations of Red] and Red2 are examined. Due to the axi-symmetrical SECM geometry, the eoncentration profiles of Red] and Red2 are best shown as cross-sections over the domain / > 0, Z > 0, as illustrated sehematically in Fig. 6. Note that in this figure the tip position has been inverted eompared to that in Fig. 4. Figure 7... 

Many materials are conveyed within a process facility by means of pumping and flow in a circular pipe. From a conceptual standpoint, such a flow offers an excellent opportunity for rheological measurement. In pipe flow, the velocity profile for a fluid that shows shear thinning behavior deviates dramatically from that found for a Newtonian fluid, which is characterized by a single shear viscosity. This is easily illustrated for a power-law fluid, which is a simple model for shear thinning [1]. The relationship between the shear stress, a, and the shear rate, y, of such a fluid is characterized by two parameters, a power-law exponent, n, and a constant, m, through... 

The black body radiation model for the continuum radiation from stars works well but it is not quite right. Careful consideration of the radiation profile shows deviations from the curves shown in Figure 2.1 due to the structure of the star itself. These deviations form the basis of a more detailed analysis including the effects of circulation within the star and will be left to others to explain we shall use black body radiation as our model for stars. 

The data from the WS model in some cases deviated slightly from the full-bed models. This could be explained by the slightly different layout of the WS model. Some spheres had to be relocated in the WS model to create a two-layer periodicity from the six-layer periodicity in the full-bed models. The differences in velocity magnitudes were mainly found in the transition area between the wall layers and the center layers. The effect of slightly larger gaps between spheres from the nine-sphere wall layers and the three-sphere central layers, due to the sphere relocations, had a noticeable effect on the velocity profile. Differences were also found in the central layer area where the sphere positions were not identical. 

The toxic effects model uses concentration-time profiles from the respiratory and skin protection models as input to estimate casualty probabilities. Two approaches are available a simple linear dose-effect model as incorporated in RAP and a more elaborate non-linear response model, based on the Toxic Load approach. The latter provides a better description of toxic effects for agents that show significant deviations of simple Haber s law behaviour (i.e. toxic responses only depend on the concentration-time product and not on each quantity separately). 

Fig. 12.7 Profiles of means (a,b) and standard deviations (c,d) of the bromacil concentrations at four different time points. Solid curves denote simulated profiles obtained from the advection-dispersion equation (a,c) and the mobile-immobile model (b,d). The different symbols denote measured profiles at different times. Reprinted from Russo D, Toiber-Yasur I, Laufer A, Yaron B (1998) Numerical analysis of field scale transport of bromacil. Adv Water Resour 21 637-647. Copyright 1998 with permission of Elsevier...

For laminar flow in short tubes or laminar flow of viscous materials these models may not apply, and it may be that the parabolic velocity profile is the main cause of deviation from plug flow. We treat this situation, called the pure convection model, in Chapter 15. 

We will now find the RDT for several models of tubular reactors. We noted previously that the perfect PFTR cannot in fact exist because, if flow in a tube is sufficiently fast for turbulence (Rco > 2100), then turbulent eddies cause considerable axial dispersion, while if flow is slow enough for laminar flow, then the parabolic flow profile causes considerable deviation from plug flow. We stated previously that we would ignore this contradiction, but now we will see how these effects alter the conversion from the plug-flow approximation. 

The analysis was completed for 12 compounds for which protein binding, renal and hepatic clearances and microsomal data were available. Plasma concentration versus time profiles in the rat were also available for these compounds. The approach taken was to simulate the individual processes (metabolic clearance, renal clearance, distribution, pharmacological activity). The ability of the PBPK model to simulate the in vivo behavior of the compound was verified in the rat. Thus, the metabolic clearance of the compounds could be reasonably well simulated, based on microsomal data and assuming no binding to microsomes less than twofold deviation between the observed and predicted clearance was achieved for about eight of the... 

Note that in the following analyses, we will drop the prime symbol. It should still be clear that deviation variables are being used. Then this linear representation can easily be separated into the standard state-space form of Eq. (72) for any particular control configuration. Numerical simulation of the behavior of the reactor using this linearized model is significantly simpler than using the full nonlinear model. The first step in the solution is to solve the full, nonlinear model for the steady-state profiles. The steady-state profiles are then used to calculate the matrices A and W. Due to the linearity of the system, an analytical solution of the differential equations is possible ... 

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