First-order dissipation model

The k,E-model is based on a first order turbulence model closure according to Boussinesq. In analogy to laminar flows, the Reynolds stresses are assumed to be proportional to the gradients of the mean velocities. Transport equations for the turbulent kinetic energy and the turbulent dissipation are developed from the Navier-Stokes equations assuming an isotropic turbulence. The implementation of this model and the parameters used can be found in [10],... 

For instance, Einstein s model, which only includes the linear term with ai = 2.5, was derived based on a viscous dissipation balance. The quadratic term in the equation represents the first-order effects of interaction between the filler particles. Geisbiisch suggested a model with a yield stress and, where the strain rate of the melt increases by a factor k as... 

Model Ecosystem Studies. Dissipation of aminocarb and fenitrothion in stream water Measurements of the concentrations (ppb) of fortified aminocarb and fenitrothion in the stream water as a function of time (t [h]), and graphing of the data (Figure 1 and 2) showed that the concentration of these two insecticides decreased exponentially with time and followed the first-order rate... 

The classic extrusion model gives insight into the screw extrusion mechanism and first-order estimates. For more accurate design equations, it is necessary to eliminate a long series of simplifying assumptions. These, in the order of significance are (a) the shear rate-dependent non-Newtonian viscosity (b) nonisothermal effects from both conduction and viscous dissipation and (c) geometrical factors such as curvature effects. Each of these... 

Soil dissipation rate will be modeled as first order ... 

The presented form was developed heuristicaUy, and it shows that the resistive torques depend on both the joint angle and its angular velocity. The two first terms are the contributions of passive tissues crossing the joints (dissipative properties of joints) reduced to first-order functions. The other terms are the nonlinear components of the resistive torques around the terminal positions, and are modeled as double exponential curve [73],... 

We can also turn the question around. In chemical kinetics, we need a model to fit the data. This model can be simple, as in first-order reactions where the decay is exponential, or more complicated depending on a complex mechanism. If we do not have a model, our data are just that, data. We could try to fit to a variety of functions, but as there is an infinite number of different functions, that is a pointless exercise. As we have seen in the classical part of this chapter, even for a simple reaction a variety of models are possible, based on dissipative classical dynamics, and we can use these models to try to understand our data. This often involves varying the external parameters, temperature, pH, viscosity, and polarizabihty, but our model should tell us what to expect for such variations for instance, how the rate constant for a reaction depends on those parameters. If our models are quantum mechanical in nature, it is mandatory that we also provide a mechanism for decay, and show how the decay constant or constants depend on external parameters. 

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