Mathematical model reduced

The few investigations into the properties and characteristics of coupled NCLs include Salazar et al. (1988) for idealized coupled loops with specified energy input and extraction over the primary and secondary loops, respectively. Wu (2011) has analyzed the case of a general number of coupled loops following the idealized approach of Welander. The mathematical model reduces to a system of coupled ordinary differential equations (ODEs) analogous to a coupled system of the equations developed by Lorenz (1963). 

Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

Both the need to reduce experimental costs and increasing reHabiHty of mathematical modeling have led to growing acceptance of computer-aided process analysis and simulation, although modeling should not be considered a substitute for either practical experience or reHable experimental data. 

Parameter estimation is a procedure for taking the unit measurements and reducing them to a set of parameters for a physical (or, in some cases, relational) mathematical model of the unit. Statistical interpretation tempered with engineering judgment is required to arrive at realistic parameter estimates. Parameter estimation can be an integral part of fault detection and model discrimination. 

The accuracy of absolute risk results depends on (1) whether all the significant contributors to risk have been analyzed, (2) the realism of the mathematical models used to predict failure characteristics and accident phenomena, and (3) the statistical uncertainty associated with the various input data. The achievable accuracy of absolute risk results is very dependent on the type of hazard being analyzed. In studies where the dominant risk contributors can be calibrated with ample historical data (e.g., the risk of an engine failure causing an airplane crash), the uncertainty can be reduced to a few percent. However, many authors of published studies and other expert practitioners have recognized that uncertainties can be greater than 1 to 2 orders of magnitude in studies whose major contributors are rare, catastrophic events. 

Over the years there have been many attempts to simulate the behaviour of viscoelastic materials. This has been aimed at (i) facilitating analysis of the behaviour of plastic products, (ii) assisting with extrapolation and interpolation of experimental data and (iii) reducing the need for extensive, time-consuming creep tests. The most successful of the mathematical models have been based on spring and dashpot elements to represent, respectively, the elastic and viscous responses of plastic materials. Although there are no discrete molecular structures which behave like the individual elements of the models, nevertheless... 

Mao and White developed a mathematical model for discharge of an Li / TiS2 cell [39]. Their model predicts that increasing the thickness of the separator from 25 to 100 pm decreases discharge capacity from 95 percent to about 90 percent further increasing separator thickness to 200 pm reduced discharge capacity to 75 percent. These theoretical results indicate that conventional separators (25-37 pm thick) do not significantly limit mass transfer of lithium. 

Before the advent of modem computer-aided mathematics, most mathematical models of real chemical processes were so idealized that they had severely limited utility— being reduced to one dimerrsion and a few variables, or Unearized, or limited to simplified variability of parameters. The increased availability of supercomputers along with progress in computational mathematics and numerical functional analysis is revolutionizing the way in which chemical engineers approach the theory and engineering of chemical processes. The means are at hand to model process physics and chenustry from the... 

Mathematical models have also predicted a low volatility for methyl parathion (Jury et al. 1983 McLean et al. 1988). One study using a laboratory model designed to mimic conditions at soil pit and evaporation pond disposal sites (Sanders and Seiber 1983) did find a high volatility from the soil pit model (75% of the deposited material), but a low volatility for the evaporation pond model (3. 7% of the deposited material). A study of methyl parathion and the structurally similar compound ethyl parathion, which have similar vapor pressures, foimd that methyl parathion underwent less volatilization than ethyl parathion in a review of the data, the reduced level of volatilization for methyl parathion was determined to be due to its adsorption to the soil phase (Alvarez-Benedi et al. 1999). 

A mathematical model for reservoir souring caused by the growth of sulfate-reducing bacteria is available. The model is a one-dimensional numerical transport model based on conservation equations and includes bacterial growth rates and the effect of nutrients, water mixing, transport, and adsorption of H2S in the reservoir formation. The adsorption of H2S by the roek was considered. 

E. Sunde, T. Thorstenson, T. Torsvik, J. E. Vaag, and M. S. Espedal. Field-related mathematical model to predict and reduce reservoir souring. In Proceedings Volume, pages 449-456. SPE Oilfield Chem Int Symp (New Orleans, LA, 3/2-3Z5), 1993. 

The mathematical model for a hydrocarbon reservoir consists of a number of partial differential equations (PDEs) as well as algebraic equations. The number of equations depends on the scope/capabilities of the model. The set of PDEs is often reduced to a set of ODES by grid discretization. The estimation of the reservoir parameters of each grid cell is the essence ofhistory matching. 

The basic mechanism for surfactants to enhance solubility and dissolution is the ability of surface-active molecules to aggregate and form micelles [35], While the mathematical models used to describe surfactant-enhanced dissolution may differ, they all incorporate micellar transport. The basic assumption underlying micelle-facilitated transport is that no enhanced dissolution takes place below the critical micelle concentration of the surfactant solution. This assumption is debatable, since surfactant molecules below the critical micelle concentration may improve the wetting of solids by reducing the surface energy. 

In the production of formic acid, a slimy of calcium formate in 50% aqueous formic acid containing urea is acidified with strong nitric acid to convert the calcium salt to free acid, and interaction of formic acid (reducant) with nitric acid (oxidant) is inhibited by the urea. When only 10% of the required amount of urea had been added (unwittingly, because of a blocked hopper), addition of the nitric acid caused a thermal runaway (redox) reaction to occur which burst the (vented) vessel. A small-scale repeat indicated that a pressure of 150-200 bar may have been attained. A mathematical model was developed which closely matched experimental data. 

Reaction (52) occurs at the gradient interface of the bolus addition until local Hb(02) concentrations have been reduced, at which point additional NO reduces the iron(III) to iron(II) which can further react with free NO to form Hb(NO). The validity of this mechanism was verified by the observation that addition of CN- ion, which binds irreversibly to metHb to form metHb(CN), significantly attenuated the formation of Hb(NO) in both cell-free Hb and RBC. Mathematical models used to simulate bolus addition of NO to cell-free Hb and RBC were compatible with the experimental results (147). In the above experiments, SNO-Hb was a minor reaction product and was formed even in the presence of 10 mM CN, suggesting that RSNO formation does not occur as a result of (hydrolyzed) NO+ formation during metHb reduction. However, formation of SNO-Hb was not detectable when NO was added as a bolus injection to RBC or through thermal decomposition of DEA/NO in cell free Hb (DEA/NO = 2-(A/ A/ diethylamino)diazenolate). SNO-Hb was observed... 

Simplified mathematical models These models typically begin with the basic conservation equations of the first principle models but make simplifying assumptions (typically related to similarity theory) to reduce the problem to the solution of (simultaneous) ordinary differential equations. In the verification process, such models must also address the relevant physical phenomenon as well as be validated for the application being considered. Such models are typically easily solved on a computer with typically less user interaction than required for the solution of PDEs. Simplified mathematical models may also be used as screening tools to identify the most important release scenarios however, other modeling approaches should be considered only if they address and have been validated for the important aspects of the scenario under consideration. 

In specifying the insulation thickness for a cylindrical vessel or pipe, it is necessary to consider both the costs of the insulation and the value of the energy saved by adding the insulation. In this example we determine the optimum thickness of insulation for a large pipe that contains a hot liquid. The insulation is added to reduce heat losses from the pipe. Next we develop an analytical expression for insulation thickness based on a mathematical model. 

Many of these assumptions are made to reduce the complexity of the mathematical model for the distillation process. Some may have negligible adverse effects in a specific process, whereas others could prove to be too restrictive. 

The results of application of different mathematical models can be reduced by allocation of some additives to the Young-Laplace equation... 

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