Three-equation model

Actually, all methods of closure involve some type of modeling with the introduction of adjustable parameters that must be fixed by comparison with data. The only question is where in the hierarchy of equations the empiricism should be introduced. Many different systems of modeling have been developed. The zero-equation models have already been introduced. In addition there are one-equation and two-equation models, stress-equation models, three-equation models, and large-eddy simulation models. Depending on the complexity of the model and the problem... 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

Three Parameter Models. Most fluids deviate from the predicted corresponding states values. Thus the acentric factor, CO, was introduced to account for asymmetry in molecular stmcture (79). The acentric factor is defined as the deviation of reduced vapor pressure from 0.1, measured at a reduced temperature of 0.7. In equation form this becomes ... 

The drilling model presented here is a simplified and modified model developed by Bourgoyne and Young [137]. The model includes three equations. Instantaneous drilling rate equation... 

Values for rs p and T for the three different models considered above are given in Table III. Once again, the equation for the hydrogen atom stripping mechanism was directly integrated to give I to1al, but for the... 

When the SVE technology is applied in a contaminated site, the NAPL is gradually removed. Towards the end of the remediation and when NAPL is no longer present, a three-phase model should be considered to calculate the phase distribution of contaminants (see Table 14.3). In this case, the vapor concentration in pore air (Ca) is calculating using the Henry s Law equation (Equation 14.5), which describes the equilibrium established between gas and aqueous phases ... 

At a molecular scale, a three-function model can be defined with no direct interaction between the so-called reductant and NO. In this case, the apparent contradiction to the global equation (1) can be ruled out, and the F>cNOx reaction by itself occurs owing to the third function of the model (Figure 5.1). 

In reality, the slip velocity may not be neglected (except perhaps in a microgravity environment). A drift flux model has therefore been introduced (Zuber and Findlay, 1965) which is an improvement of the homogeneous model. In the drift flux model for one-dimensional two-phase flow, equations of continuity, momentum, and energy are written for the mixture (in three equations). In addition, another continuity equation for one phase is also written, usually for the gas phase. To allow a slip velocity to take place between the two phases, a drift velocity, uGJ, or a diffusion velocity, uGM (gas velocity relative to the velocity of center of mass), is defined as... 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

There are seven unknowns but only three equations that relate these quantities. Therefore, four of the unknowns can be chosen arbitrarily. This process is not really arbitrary, however, because we are constrained by certain practical considerations such as a lab model that must be smaller than the field pipeline, and test materials that are convenient, inexpensive, and readily available. For example, the diameter of the pipe to be used in the model could, in principle, be chosen arbitrarily. However, it is related to the field pipe diameter by Eq. (2-11) ... 

By means of Equations 5 and 6, the adsorption process can be described in terms of the parameters and Ks. Since the effective area of solid surface available to polymer adsorption is not generally known in practice, Ng is a third parameter which must be fitted from experiment and Equations 5 and 6 define a three-parameter model for the process. 

The equation representing this curve was introduced in Section 4.4 (equation (4.4)). However, a more contemporary model based on results from Proficiency Testing schemes has shown that the relationship is best represented if three equations are used to cover from high to low concentrations, as shown in... 

The measured variance oj may also be used to estimate Pe, in conjunction with equations 19.4-64 (open-open), 19.4-70 (closed-closed), and 19.4-72 (closed-open, Table 19.7). For large PeL values the results are nearly the same, but fa- small Pe, values, they differ significantly. For these three equations, and also the Gauss solution in equation 19.4-58, as Pe, -> <, cr2 - 0, consistent with PF behavior. These results are illustrated in Table 19.8, winch (conversely) gives values of aj calculated from the four equations for specified values of Pe,. Note that aj becomes ill-behaved fir small values of Pet for the Gauss solution, and for open-open and closed-open conditions. This is most apparent for Pe, = 0 for these cases, in which a - cc. The expected result, cr%= 1, is given by equation 19.4-70 for closed-closed conditions. This is the expected result because Pe, = 0 corresponds to BMF, which in turn corresponds to N = 1 in the TIS model, and for this, cr = 1, firm equation 19.4-26. 

Similar equations can be written for Pb and Pb using their appropriate radioactive parents and decay constants. If t = 0 is taken to represent the time of the formation of the Earth s crust, then these three equations describe the trajectory of the isotopic composition of terrestrial lead from that time. If T is the time elapsed since the formation of the Earth, (i.e., the age of the Earth), and tm is the time before present at which the lead minerals were formed, then, using the assumptions of the Holmes Houtermans model given above, the isotopic composition of a common lead deposit formed tm years ago is given as follows ... 

Table 5.3. The terms in the transport equations for a three-environment model.

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

The moments of the solutions thus obtained are then related to the individual mass transport diffusion mechanisms, dispersion mechanisms and the capacity of the adsorbent. The equation that results from this process is the model widely referred to as the three resistance model. It is written specifically for a gas phase driving force. Haynes and Sarma included axial diffusion, hence they were solving the equivalent of Eq. (9.10) with an axial diffusion term. Their results cast in the consistent nomenclature of Ruthven first for the actual coefficient responsible for sorption kinetics as ... 

A. Jones It does not distinguish between the lattice models. I don t think we are going to easily distinguish between the lattice models. The same basic equations are employed in each case. The three lattice models I mentioned have spectral densi-... 

While the chemical mass balance receptor model is easily derivable from the source model and the elements of its solution system are fairly easy to present, this is not the case for multivariate receptor models. Watson (9) has carried through the calculations of the source-receptor model relationship for the correlation and principal components models in forty-three equation-laden pages. 

Mason (58JCS674) used all three equations (6), (7) and (8), so that the effect of Me in the two model compounds cancelled out to some extent. Albert (56JCS1294) employed modified equations (9) or (10), which were subsequently criticized by Mason because the effect of Me no longer cancels. 

Three-parameter models such as the VFT equation (4.14) can be lit to data by first placing the equation in linear form ... 

In the following, the one-dimensional model will be presented. The basic ideal models assume that concentration and temperature gradients occur in the axial direction (Froment and Bischoff, 1990). The model for a fixed-bed reactor consists of three equations, which will be presented in the following sections and are... 

X2° = X30 = 0 assumed to be known exactly. The only observed variable is = x. Jennrich and Bright (ref. 31) used the indirect approach to parameter estimation and solved the equations (5.72) numerically in each iteration of a Gauss-Newton type procedure exploiting the linearity of (5.72) only in the sensitivity calculation. They used relative weighting. Although a similar procedure is too time consuming on most personal computers, this does not mean that we are not able to solve the problem. In fact, linear differential equations can be solved by analytical methods, and solutions of most important linear compartmental models are listed in pharmacokinetics textbooks (see e.g., ref. 33). For the three compartment model of Fig. 5.7 the solution is of the form... 

The interfacial diffusion model of Scott, Tung, and Drickamer is somewhat open to criticism in that it does not take into account the finite thickness of the interface. This objection led Auer and Murbach (A4) to consider a three-region model for the diffusion between two immiscible phases, the third region being an interface of finite thickness. These authors have solved the diffusion equations for their model for several special cases their solutions should be of interest in future analysis of interphase mass transfer experiments. 

Inhomogeneous systems. If Eq. 21-46 is an inhomogeneous system, that is, if at least one Ja is different from zero, then usually all eigenvalues are different from zero and negative, at least if the equations are built from mass balance considerations. Again, the eigenvalue with the smallest absolute size determines time to steady-state for the overall system, but some of the variables may reach steady-state earlier. In Illustrative Example 21.6 we continue the discussion on the behavior of tetrachloroethene (PCE) in a stratified lake (see also Illustrative Example 21.5). Problem 21.8 deals with a three-box model for which time to steady-state is different for each box. 

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