Stage Models

With a larger N, the width of the elution curve becomes narrower at a given VR, and a better resolution will be attained. The dependence of the value of N on the 

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Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... 

Naplitali-Sandholm SC Method This method employs the equilibrium-stage model of Figs. 13-48 and 13-49 but reduces the number of vari les by 2N so that only N(2C + 1) equations in a hke number of unknowns must be solved. In place of Vj, Lj, Xij, and iji j, component flow rates are used according to their definitions ... 

The iC values (vapor-liquid equihbrium ratios) in Equation (13-123) are estimated from the same equation-of-state or aclivity-coefficient models that are used with equilibrium-stage models. Tray or packed-section pressure drops are estimated from suitable correlations of the type discussed by Kister (op. cit.). 

From the above list of rate-based model equations, it is seen that they total 5C -t- 6 for each tray, compared to 2C -t-1 or 2C -t- 3 (depending on whether mole fractious or component flow rates are used for composition variables) for each stage in the equihbrium-stage model. Therefore, more computer time is required to solve the rate-based model, which is generally converged by an SC approach of the Newton type. 

Dozov I., Temkin S., Kirov N. Two-stage model for molecular orientational relaxation in liquid crystals, Liq. Crystak 8, 727-38 (1990). 

The simplest experimental model of carcinogenesis is the three-stage model (Pitot and Dragan, 1994) consisting of ... 

FIVE STAGE COUNTERCURRENT EXTRACTION CASCADE EQUILIBRIUM STAGE MODEL... 

Five stage countercurrent extraction cascade with backmixing Equilibrium stage model... 

The most widely-accepted dose response model at the present time is the multi-stage model, which has great flexibility in curve-fitting, and also has a strong physiological justification. Although it is difficult to implement, there are already computer codes in existence that estimate the model parameters (13). The two most widely-used models, until recently, were the one-hit model and the log-probit model. They are both easy to implement, and represent opposite extremes in terms of shape - the former represents the linear non-threshold assumption, whereas the latter has a steep threshold-like curvature. In numerous applications with different substances it has been found that these three... 

Ponnamperuma, R.M., Shimizu, Y., Kirchhof, S.M., and De Luca, L.M. 2000. Beta-carotene fails to act as a tumor promoter, induces RAR expression, and prevents carcinoma formation in a two-stage model of skin carcinogenesis in male Senear mice. Nutr Cancer 37 82-88. 

Mathematical optimization models that explicitly consider such a multi-stage structure belong to the class of multi-stage stochastic programs. A deterministic optimization model with uncertain parameters is extended to a multi-stage model by three measures ... 

In this fashion, we extend our deterministic model with a prediction horizon of H = 2 to a multi-stage model. The multi-stage tree of the possible outcomes of the demand within this horizon (starting from period i = 1) with four scenarios is shown in Figure 9.5. Each scenario represents the combination fc out of the set of all combinations of the demand outcomes within the horizon. The production decision x has to be taken under uncertainty in all future demands. The decision xj can react to each of the two outcomes of d i, but has to be taken under uncertainty in the demand di. The corrective decisions are explicitly modeled by replacing xj by two variables 2,1 and 2.2 ... 

However, the description of the tree structure of a multi-stage model leads to complicated constraints. To simplify the original multi-stage model, it is approximated by a model with two stages. It consists of only one sequence of decisions-observation-decisions. The two-stage structure leads to considerably simpler optimization problems. It is also adequate from a practical point of view in the moving horizon scheme, only the first decision x is applied to the plant while all the remaining variables are used to compute the estimated performance only. 

The moving horizon scheme using the two-stage model is shown in Figure 9.6. In contrast to the deterministic scheduler which uses the expected value of the demands dls and ds i (see Section 9.2.2), the stochastic scheduler updates the demand in form of the distribution given in Table 9.1 d, df, d]+1, and dj+1. 

Stacey, J.S. and Kramers, J.D. (1975). Approximation of terrestrial lead isotope evolution by a two-stage model. Earth and Planetary Science Letters 26 207-221. 

Run the model for the 5-stage model and ascertain if equilibrium is attained for each stage. Reduce KLa by a factor of 10 and check again. Reduce L and G by the same factor. Explain the effects in terms of the process time constants. 

Mehendale HM. 1991. Role of hepatocellular regeneration and hepatolobular healing in the final outcome of liver injury A two-stage model of toxicity. Biochem Pharmacol 42(6) 1155-1162. 

Sirica AE, Wilkerson CS, Wu LL, et al. 1989. Evaluation of chlordecone in a two-stage model of hepatocarcinogenesis A significant sex difference in the hepatocellular carcinoma incidence. Carcinogenesis 10(6) 1047-1054. 

Figure 1. Three isotope plot of O isotopes in Allende inclusions. Deviations are plotted in 6 units which are %o deviations relative to the terrestrial SMOW standard. In a two stage model, normal inclusions had initially a composition close to point A and exchanged with a reservoir poorer in O in the region of point D (Clayton et al. 1973). FUN inclusions underwent an intermediate step along a fractionation line between point A and point C. Then each inclusion exchanged with the same O poor reservoir D (Clayton and Mayeda 1977).

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