Mixed-effects model

In this section we will develop the techniques for modeling mixing effects on reactor performance. As pointed out above, three approaches have received the most attention (1) direct use of RTD information, (2) mixing-cell approximations, and (3) dispersion-plug-flow models. 

With these goals in mind, several investigators have undertaken to set down quantitative expressions which will predict propellant burning rates in terms of the chemical and physical properties of the individual propellant constituents and the characteristics of the ingredient interactions. As in the case of ignition, the basic approach taken in these studies must consider the different types of propellants currently in use and must make allowances for their differences. In the initial combustion studies, the effort was primarily concerned with the development of combustion models for double-base propellants. With the advent of the heterogeneous composite propellants, these studies were redirected to the consideration of the additional mixing effects. 

Pinheiro JC, Bates DM. Approximations to the loglikelihood function in the nonlinear mixed effects model. / Comput Graphical Stat, 1995 4 12-35. 

Lindstrom MJ, Bates DM. Nonlinear mixed effects models for repeated measures data. Biometrics, 1990 46 673-87. 

Examination of Table 25.3 and the comments above clearly demonstrate that both HYDRUS and UNSAT-H are likely to produce very good estimates of water movement within the soil profile. However, they do not estimate snowmelt, model mixed plant communities, directly estimate surface runoff, or consider the effect of soil density on root growth and water use.14... 

For example, (A) and (B) can be computed using Eqs. (49) and (50), respectively. Note that instead of Eq. (55), we could use the simpler expression for given by Eq. (33), which avoids the need for numerical quadrature. In both cases, the mean and variance of the mixture fraction are identical (and thus both models account for finite-rate mixing effects.) In practical applications, the differences in the predicted values of () can often be small (Wang and Fox, 2004). 

PF, to model the effect of earliness or lateness of mixing, depending on the sequence, on the performance of a single-vessel reactor. The following two examples explore the consequences of such series arrangements-first, for the RTD of an equivalent single vessel, and second, for the fractional conversion. The results are obtained by methods already described, and are not presented in detail. 

Shkedy, Z. et al., Modeling anti-KLH ELISA data using two-stage and mixed effects models in support of immunotoxicological studies, J. Biopharm. Stat., 15, 205, 2005. 

In CRE textbooks (Hill 1977 Levenspiel 1998 Fogler 1999), the types of reactors considered in this book are referred to as non-ideal. The flow models must take into account fluid-mixing effects on product yields. 

Tsai, K. and R. O. Fox (1993). PDF modeling of free-radical polymerization in an axisymmetric reactor. EES Report 254, Kansas State University, Manhattan, Kansas. (1994a). Modeling the effect of turbulent mixing on a series-parallel reaction in a tubular reactor. ICRES Report 9403, Kansas State University, Manhattan, Kansas. 

At any level in the transition region, there will be a balance between the mixing effects attributable to (a) axial dispersion and to (b) the segregating effect which will depend on the difference between the interstitial velocity of the liquid and that interstitial velocity which would be required to produce a bed of the same voidage for particles of that size on their own. On this basis a model may be set up to give the vertical concentration profile of each component in terms of the axial mixing coefficients for the large and the small particles. 

Munoz MJ, Merino-Sanjuan M, Lledo-Garcia R, Casabo VG, Manez-Castillejo F and Nacher A (2005) Use of Nonlinear Mixed Effect Modeling for the Intestinal Absorption Data Application to Ritonavir in the Rat. Eur J Pharm Biopharm 61 ... 

There are two common methods for obtaining estimates of the fixed effects (the mean) and the variability the two-stage approach and the nonlinear, mixed-effects modeling approach. The two-stage approach involves multiple measurements on each subject. The nonlinear, mixed-effects model can be used in situations where extensive measurements cannot or will not be made on all or any of the subjects. 

Pharmacokinetic/pharmacodynamic model using nonlinear, mixed-effects model in two compartment, best described time course of concentration strong correlation with creatinine clearance predicted concentration at the efi ect site and in reduction of heart rate during atrial fibrillation using population kinetic approach... 

Modeling the Effects of Oxidant Stress on a Western Mixed-Conifer Forest Ecoqrstem... 

A series of works by Matsuda et al. composed perhaps the first systematic study to explore the physical foundation for such a mixing effect. Using PC/DME as a model system, they investigated the dependence of vapor pressure, dielectric constant, and viscosity on solvent composition, and they correlated these variations with ion conductions. It was found that the dielectric constant varied with solvent composition by following an almost linear relation, with slight positive deviations, while viscosity always showed a pronounced negative deviation from what a linear relation would predict (Figure 6b). For such binary solvent systems, approximate quantifications... 

A lake mixes on some time scale so it might be modeled as a stirred reactor for times longer than the mixing time, and it is a tubular reactor for shorter times. Most lakes turn over seasonally because of the density rrtirtimum of water at 4°C so they are mixed on at least a 1 year time scale. Small lakes are mixed effectively by wind on shorter time scales. Deep lakes exhibit an upper layer that turns over and mixes and a lower layer that does not. The inlets to a lake are aU the rivers that flow into it and also water coming from rainfall, and the outlets are aU the rivers that flow out of it and evaporatiorr... 

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