Recycle Models

These data do not provide strong support for the heat water stress/urea recycling model (Ambrose 1991). This model may be incorrect or inapplicable to rats. Were the experimental conditions inappropriate, with temperatures too low and/or protein levels too low or too high In the heat stress experiments that inspired this research animals were kept at a temperature of 40°C for 12 hours each day rather than 36°C in this study. In our heat and water stress experiments the protein content of the diets were set at 20% and 70%. These are relatively high levels compared to those in herbivore diets. It would be necessary to repeat the experiments with ruminant herbivores or lower protein diets to conclusively determine if rats are an inappropriate model. 

Scenario 4 Formulation for fixed water quantity with reusable water storage Constraints (4.18), (4.19), (4.3), (4.20), (4.16), (4.17), (4.21), (4.22), (4.23), (4.24), (4.25) and (4.26) together constitute a complete water reuse/recycle model for a situation in which the quantity of water in each water using operation is fixed. This is also a nonconvex MINLP for which exact linearization is not possible. 

Gibilaro [49] has considered a recycle model of the form of eqn. (60) where Gj (s) and G2(s) are general series combinations of PFR and equal size CSTR reactors and he gives sixteen references to published work involving more restricted forms of Gj (s) and G2 is). With an infinite choice over the forms of G (s) and G2(s) and the magnitude of R, the recycle model is seen to be the most flexible of all flow-mixing models. The performance of each specific form of Gj (s) as a potential reactor must be investigated individually in practice, the model is often reduced to a pure PFR element... 

Imagine a first-order reaction taking place in such a system under conditions where rk, i.e. VkjQ, is 10 and R is 5. Using the technique previously adopted in Sect. 5.1 and outlined in Appendix 2, we can readily calculate that this system would achieve 96.3% conversion of reactant. Under these conditions, the recycle reactor volume turns out to be 3.03 times that of an ideal PFR required for the same duty. This type of calculation allows Fig. 14 to be constructed this is similar in form to Fig. 12, but lines of constant for the tanks-in-series model have been replaced by lines of constant recycle ratio for the recycle model. From a size consideration alone, the choice of a PFR recycle reactor is not particularly... 

Fig. 14. General design chart for the recycle model described by eqn. (61), first-order reaction A R with no change in volume (e = 0). Ordinate gives the recycle system volume divided by the volume of an ideal PFR which achieves the same conversion. —, Constant fer ----, constant recycle ratio, R.

One might intuitively expect that infinite recycle rates associated with a system as described by eqn. (61) would produce a completely well-mixed volume with concentration independent of location. This is indeed so and under these conditions, the performance tends to that of an equal sized CSTR. At the other extreme, when R is zero, PFR performance pertains. Fractional conversions at intermediate values of R may be determined from Fig. 14. The specific form of recycle model considered is thus seen to be continuously flexible in describing flow mixing between the PFR and CSTR extremes just as was the tanks-in-series model. The mean and variance of this model are given by eqns. (62) and (63) and these may be used for moments matching purposes of the type illustrated in Example 6. 

If Fig. 12 and Fig. 14 were laid on top of each other, then conditions of equivalence could be determined under which the performance of the tanks-in-series model with specified N would be the same as that of the recycle model, that is the value of R could be found which would result in the same conversion and V/Vpp ratio. Levenspiel [17] gives these values for a variety of conditions for both first- and second-order reactions. His data are reproduced in Table 8. 

The general recycle model has been used by Weinstein and co-workers [55, 56] as a device for describing variable levels of micromixing. Dudukovic s work in this area has shown that approximations involved in Weinstein s work can introduce greater errors than were originally estimated [57] and he advocates the use of the more conventional micromixing models referred to in Sect. 4.3. 

Many correlations for mixing time (see above) have been proposed in the literature (142). One of the most comprehensive treatments of this problem was published by Khang and Levenspiel (143), on the basis of a recycle model 9m is defined as the time constant for the exponential decrease of pseudo-periodic oscillations after a pulse injection of tracer in a batch stirred reactor. 

Reactor Selection Ideal CSTR and PFR models are extreme cases of complete axial dispersion (De = oo) and no axial dispersion (De = 0), respectively. As discussed earlier, staged ideal CSTRs may be used to represent intermediate axial dispersion. Alternatively, within the context of a PFR, the dispersion (or a PFR with recycle) model may be used to represent increased dispersion. Real reactors inevitably have a level of dispersion in between that for a PFR or an ideal CSTR. The level of dispersion may depend on fluid properties (e.g., is the fluid newtonian),... 

We note that our model does not include removal of (wet) solids from the reactor and other secondary reactions to nonregenerable byproducts. Therefore, the model must predict no loss of FeE complexes and the condition Cp add = 0 can be used to check the correctness of the combined absorber-bioreactor-recycle model. 

It is chosen here to illustrate the following characteristic features of limiter tokamak recycling models ... 

The CFM can be completed with a recycling model (the trajectory of which can be considered as a CFM, such as a PF with PM, series of PM, etc.), or with models with slip flows and models with multiple closed currents. 

Calculated bulk rock trace-element systematics of eclogites have wider implications for mantle recycling models and bulk silicate earth mass balance. The subchondritic Nb/Ta, Nb/La, and Ti/Zr of both continental cmst and depleted mantle require the existence of an additional reservoir with superchondritic ratios to complete the terrestrial mass balance. Rudnick et al. (2000) have shown that rutile-bearing eclogites from cratonic mantle have suitably superchondritic Nb/Ta, Nb/La, and Ti/Zr such that if this component formed 1 -6% by weight of the bulk silicate earth, this would resolve the mass imbalance. This mass fraction far exceeds the likely mass of eclogite in the continental lithosphere and so the material is proposed to reside in the lower mantle, possibly at the core-mantle boundary. 

Carberry has investigated the effect of mixing using his tubular reactor with recycle model and has been particularly interested in yields with simultaneous reactions. There are again situations for which a degree of mixing somewhere between the two extremes of the plug flow reactor and the stirred tank is optimal. 

APPENDIX 13.6 ENTEROHEPATIC RECYCLING MODEL NONMEM Input... 

Example 123Jb-2 Recycle Model for Large-ScaU Mixing Effects... 

As ky - 0, the ordinary recycle model for reactors, with delay in both streams, is obtained. In addition, if VijF - 0, the mode used by Carberry [88] to study reaction selectivity behavior is obtained and if V F 0, there is... 

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