Accuracy of Simulations

Feldberg has suggested that fi = 55x that is, the reaction layer should be at least 5 times the grid space size. In terms of the number of time increments, we 

The number of time increments needed for a simulation then becomes nt = time of experiment x 50 x kchem- 

However, our experience in using the program CVSIM is that the constraint  

Simulations of several mechanisms are shown that illustrate the difference in simulation time and the difference in the result when the two contrasting constraints are used. 

Generally, highly accurate results for all mechanisms have been obtained ( 0.5 /o error). In any case, it is important to remember that the number of time increments, which determines the accuracy, can be set by the user. For any new mechanism, a test simulation can be done using a more rigorous constraint (use 10 times more time increments than the default) to check for accuracy. 

---

The numerical accuracy of simulations performed using this model is affected by several factors. These include a) the degree of triangulation, b) the number of marching steps taken along the flow direction and c) the order of the polynomial basis function. Numerical accuracy improves as a, b and c increase, however the computational time can become excessive. Therefore, it was necessary to quantitatively determine the effects of these variables on numerical accuracy. 

One major difficulty in assessing the accuracy of any transport simulation method is the inaccuracy in runoff estimations. The basin selected for the study described was of minimum size for application of the Stanford watershed model. This is reflected in the corresponding uncertainties in all computations. However, it is worth noting that the measured and simulated concentrations of Sr and 137Cs seldom differed by more than a factor of two. This observation suggests that increasing the accuracy of simulated runoff processes will result in an increased accuracy in radioaerosol transport estimates as well. 

Except the kinetic equations, now various numerical techniques are used to study the dynamics of surfaces and gas-solid interface processes. The cellular automata and MC techniques are briefly discussed. Both techniques can be directly connected with the lattice-gas model, as they operate with discrete distribution of the molecules. Using the distribution functions in a kinetic theory a priori assumes the existence of the total distribution function for molecules of the whole system, while all numerical methods have to generate this function during computations. A success of such generation defines an accuracy of simulations. Also, the well-known molecular dynamics technique is used for interface study. Nevertheless this topic is omitted from our consideration as it requires an analysis of a physical background for construction of the transition probabilities. This analysis is connected with an oscillation dynamics of all species in the system that is absent in the discussed kinetic equations (Section 3). 

As most chemical and virtually all biochemical processes occur in liquid state, solvation of the reaction partners is one of the most prominent topics for the determination of chemical reactivity and reaction mechanisms and for the control of reaction conditions and resulting materials. Besides an exhaustive investigation by various experimental methods [1,2,3], theoretical approaches have gained an increasing importance in the treatment of solvation effects [4,5,6,7,8], The reason for this is not only the need for sufficiently accurate models for a physically correct interpretation of the experimental data (Theory determines, what we observe ), but also the limitation of experimental methods in dealing with ultrafast reaction dynamics in the pico- or even subpicosecond regime. These processes have become more and more the domain of computational simulations and a critical evaluation of the accuracy of simulation methods covering experimentally inaccessible systems is of utmost importance, therefore. 

As the increase of the computer power, both size and accuracy of simulations have incredibly grown, and also many indicators to probe ergodicity and mixing have been proposed. One should first note, nevertheless, that the situation is still far from being clear and appropriately understood, even after 50 years have passed since the first attempt by FPU. 

It must be noted here that even for Eulerian-Lagrangian simulations, although there is no complexity of averaging over trajectories, the accuracy of simulations of individual bubble trajectories depends on lumped interphase interaction parameters such as drag force, virtual mass force and lift force coefficients. All of these interphase interaction parameters will be functions of bubble size and shape, presence of other bubbles or walls, surrounding pressure field and so on. Unfortunately, adequate information is not available on these aspects. To enhance our understanding of basic... 

In Section V.A we will provide arguments for the joint kinetic description of oxidative transformations of methane and C2 hydrocarbons. Regarding molecules containing more than two carbon atoms, their influence on the overall kinetics and on the formation of many important products is below the anticipated accuracy of simulations (Arutyunov et al., 2005). This is why their formation and transformations can be not accounted for in methane and ethane oxidation models for many applications. At least it would not compensate the excessive complication of the model accounting for reactions of C3+ species. 

A number of experimental studies have established that both microbial and chemical degradation can be approximately described by first-order kinetics (24). Most pesticide models employ such an approach. As with linear sorption, this relatively naive representation of a fundamentally more complicated process is a simplifying assumption to make mathematical solutions possible and data requirements reasonable. Implicit in the assumption is the belief that the accuracy of simulation of pesticide fate is more dependent upon other factors than a very precise representation of the degradation process. These factors include spatial and temporal variability of the degradation process itself as affected by water, temperature, substrate, and pH, and variability in the transport of pesticide through the soil profile. There is little information to substantiate this assumption, although some field experiments on water and solute movement (discussed below) indicate it to be reasonable at this point in model development. 

Improvements in the accuracy of simulations of structure will also undoubtedly stem from the continued evolution of efficient collection, database construction, and analysis of crystal structural information. The Cambridge Crystallographic Database (Allen et al., 1991), for example, presently contains more than 100 000 crystal structures. There is a wealth of information contained within such a compendium and a steady stream of publications appear based on such analyses. 

The accuracy of simulated pmf s for Systems I and III has also been examined. Table 5 shows that Eq. 18 provides a reasonable estimation of the... 

J. Pape, H. Potente, and C. Obermann, Influence of Model Simplifications on the Accuracy of Simulation Results in Single Screw Extruders, 15 Annual Meeting of the Polymer Processing Society, Den Bosch, the Netherlands (1999)... 

Interphase mass transfer. Of all the aspects of the model which influence predictions and the accuracy of simulations, interphase mass transfer is the most important. While there have been about a dozen studies of transfer for single bubbles, there are very few in which bubbles have been allowed to interact and coalesce. Interactions have been shown to lead to enhancement in the rate of interphase transfer. Based on their own results and those of other workers, Sit and Grace (28) proposed the following expression for k 2 freely bubbling three-dimensional beds ... 

Since the slip of particle appears during the impact to the wall, a negative distribution function / i is introduced to eliminate the effect of sUp. The accuracy of simulation by this method is better than that by bounce-back. 

For testing the accuracy of simulation, Chen also performed the experiment under the simulated condition. The experimental setup is shown in Fig. 9.27. 

Water is essential to the very existenee of life, playing its important role in a myriad of physieal, ehemical, and biological processes. Despite having a simple moleeular strueture, it forms one of the most eomplex substances. Specific interactions amongn water molecules in the condensed phase are responsible for its anomalous behavior. Extensive atomistic simulations have been performed to connect the microscopic structure of water to its macroscopic properties. The accuracy of simulation results strongly depends on the quality of the applied intermolecular potentials. To this end, more than 50 empirical water potentials have emerged in the literature, broadly differing in the... 

The result is showed in Fig. 2 and found to be in good quantitative agreement with related investigations [10]. This result supports the accuracy of simulation models and methods used in this work. 

---