Five-lump model

Assume that this distributed system can be adequately modeled by a five-lump model of equal lengths. Inside each lump the gas temperature and the composition vary with time, as does the packing temperature. 

All the works just cited have an important and common assumption the well-known, three-lump Weekman model with three kinetic constants. This three-lump model was a significant achievement, but it lumps together gas and coke, which are clearly different. Therefore, we consider it absolutely necessary to expand this model by using a five-lump model with eight cracking constants and the following definitions A = feedstock (bp > 350 °C), O = gas oil (bp 221-350 °C), E = gasoline (bp 36-221 °C), G = gas (bp < 36 ° C), and C = coke (by combustion). 

Table 1. Definition of lumps used in the five lump model...

The cracking experiments have been modeled according to the five lump model with different deactivation orders. The parameters of the model have been calculated using a Levenberg-Marquard minimization routine written in Fortran. Discrimination of models was based on ... 

For the models evaluated in this work, the best model to describe all experiments was the five lump model with a first order deactivation, although it did not describe the first part of the reactor correctly, obviously due to an incorrect description of the initial effects. When the initial effects were excluded, a model with a constant activity described the data satisfactory. Therefore, coke deposition and catalyst deactivation have to be divided in an initial process (<0.15 s) and a process on a longer time scale. 

The method is applied for the catalytic cracking reactions and the kinetic parameters of a five-lump model are determined by using successively NLRA with various 3- and 4-lump kinetic models. 

Since five-lump model is unable to capture feed quality and hence impact of recycle slurry flow rate on the performance of unit. It was necessary to develop a model, which is based upon detailed characterisation of feed. In present study. 

It should be noted that using five lump model (Dave and Saraf, 2002), maximisation of conversion is equivalent to maximisation of gasoline conversion. 

It should be noted that again five lump model (Dave and Saraf, 2002) is used. It was assumed that composition of feed is maintained constant by fixing recycle ratio. Appropriate bounds on the decision variables are specified and the constraints for this problem are same as that are listed in problem 1. 

Figure 3 Pareto optimal solutions obtained for problem 1 using five lump model.

Li and Rabitz" used the above approach to further contract the well-known 10-lump FCC model to a five-lump model with essentially the same predictive power for product slate changes. Moreover, if gasoline is the only unlumped species, then three lumps suffice. For nonane reforming, this approach reduces the number of liunps from 14 to 5 without significant errors if the total aromatics is kept imliunped. ... 

Ancheyta, J., Sotelo, R. 2007. Estimation of kinetic constants of a five-lump model for fluid catalytic cracking process using simpler sub-models. Energy Fuels 14 1226-1231. 

Petroleum Cut Temperature Range (°Q Experimental Five-Lump Model Continuous Model... 

A model for the riser reactor of commercial fluid catalytic cracking units (FCCU) and pilot plants is developed This model is for real reactors and feedstocks and for commercial FCC catalysts. It is based on hydrodynamic considerations and on the kinetics of cracking and deactivation. The microkinetic model used has five lumps with eight kinetic constants for cracking and two for the catalyst deactivation. These 10 kinetic constants have to be previously determined in laboratory tests for the feedstock-catalyst considered. The model predicts quite well the product distribution at the riser exit. It allows the study of the effect of several operational parameters and of riser revampings. 

Selection of an appropriate lumping scheme was one of the most important issues in this modelling exercise. Ten lump kinetic scheme developed by Jacob et al. (1976) and five lump kinetic model proposed by Ancheyta et al. (1999) were examined closely. The virtue of more detailed lumping scheme over other less detailed models is that rate constants is that rate constants are independent of feed composition. But utilisation of these models are limited by two problems i.e. detailed characterisation of streams is not available on a regular basis and elaborate kinetic information is scarcely available. Thus, a balance between kinetic description required and cost of laboratory analysis often decides selection of lumping strategy. 

Dave and Saraf (2002) modified the original scheme of Ancheyta et al. (1999) by assuming gasoline and LPG also convert to coke. Figure 1 presents the modified kinetic study, used in the present work. Since rate constants for this model are dependent upon feed quality, they are obtained by tuning industrial data available. In the present work, five lump kinetic model developed by Dave and Saraf (2002) is used for multi objective analysis for objective functions gasoline production maximisation versus minimisation of CO emission from regenerator. 

Mosby et al. reported a seven-lump residue hydroconversion model. Lumped models for steam cracking of naphtha and gas oils can be found in Dente and Ranzi s review. The literature aboimds with FCC kinetic models, with the number of lumps being three four °, five, six , eight, ten °. 

The catalytic hydrocracking of heavy oil has been well represented by the five-lump kinetic model shown in Figure 6.15 (Sdnchez et al., 2005). Although catalytic and thermal reactions follow different mechanisms, the same kinetic model was used to represent the NHDC. Hydrocracking of vacuum residue was assumed to follow second order as demonstrated earlier, while first order was considered for the other reactions. The reaction rate (r for each lump as a function of the product composition (y and the corresponding kinetic constant k is as follows ... 

From the experimental evidence, a five-lump kinetic model with six kinetic parameters was found to represent in an adequate manner the NHDC of heavy oils. 

In this section, a five-lump kinetic model and a time-dependent nonselective catalyst deactivation expression are used to represent the reactions occurring during hydrocracking of heavy oils. 

The kinetic model used is shown in Figure 6.15 (Chapter 6). It consists of five lumps and was previously reported in the literature (Sfinchez et al., 2005). The lumps are gases, naphtha (IBP— 204°C), middle distillates (204°C-343°C), VGO (343 C-538 C), and unconverted vacuum residue (538°C+). As mentioned earlier, experiments were performed under a disguised kinetic regime due to the presence of internal diffusion limitations this made necessary the introduction of the effectiveness factor (n) since internal gradients are indeed present. 

A sequential method for determining the parameter values was employed, which is reported elsewhere (Ancheyta and Sotelo, 2000). The original kinetic model was divided into two-, three- and four-lump kinetic models to simplify parameters estimation, as presented in Figure 10.18. From this approach, it is possible to obtain some kinetic parameters that become constant when solving the five-lump kinetic model, thus reducing the number of parameters to be estimated simultaneously. 

The continuous kinetic lumping model, in contrast to the discrete lump model, is able to predict the whole distillation curve of hydrocracked product. For instance. Table 11.2 shows predicted product compositions for two cases, five lumps and nine lumps, obtained with Equation 11.9 with the optimized parameter values. In the case of the continuous kinetic lumping model, it allows for predicting the entire boiling point, and the yield of any fraction can be defined at convenience without any other recalculation of parameters that is why results for five and nine lumps are easily generated, while in the case of the discrete lumping model, only results for five lumps are reported, and to obtain information for nine lumps apart from the new parameter estimation, more experiments are indeed necessary. The comparison between the two models can then be only made for five-lumps results. The difference of the predicted product composition with both models compared with the experimental values is small. 

Comparison between Experimental and Predicted Product Compositions using a Discrete Five-Lump Kinetic Model and the Continuous Kinetic Lumping Model (Temperature = 400°C, LHSV = 0.5 h )... 

Preliminary work showed that first order reaction models are adequate for the description of these phenomena even though the actual reaction mechanisms are extremely complex and hence difficult to determine. This simplification is a desired feature of the models since such simple models are to be used in numerical simulators of in situ combustion processes. The bitumen is divided into five major pseudo-components coke (COK), asphaltene (ASP), heavy oil (HO), light oil (LO) and gas (GAS). These pseudo-components were lumped together as needed to produce two, three and four component models. Two, three and four-component models were considered to describe these complicated reactions (Hanson and Ka-logerakis, 1984). 

In this class of models, the five bitumen pseudo-components are lumped into two in an effort to describe the following reaction... 

By lumping pseudo-components, we can formulate five three-component models of interest. Pseudo-components shown together in a circle are treated as one pseudo-component for the corresponding kinetic model. 

Similar to Eq. (67), the first reaction (incorporating the enzyme phosphofructo-kinase) exhibits a Hill-type inhibition by its substrate ATP [126]. The overall ATP utilization v3 (ATP) is modeled by a saturable Michaelis Menten function. The system is specified by five kinetic parameters (with Gx lumped into Vm ), the Hill coefficient n, and the total concentration, 4 / = [ATP] + [ADP]. Note that the model is not intended to capture biological realism, rather it serves as a paradigmatic example to identify dynamic behavior in metabolic pathways. 

Another point illustrated by Table XXIII is the need to carefully consider the effect of lumping isomers for convenience, when mechanistic models are generated. Thus, in Example 4, isomerization of 1-butene is neglected in selecting the elementary steps for butadiene production. In effect, it is assumed that all intermediates are indistinguishable whether 1-butene or a mixture of n-butenes reacts. If that scheme were used in the present case, we could consider a model in which only s, s2, s3, s4, s6, and s j 3 were retained, which would correspond to only a single direct mechanism. However, if instead we chose to retain all the elementary steps as possibilities except Sj j and s12, we would obtain five direct mechanisms for a system producing only 1-butene (in which p — a = 0). 

We are going to use as an example a five-organ compartment model for the metabolism of ethanol in humans. We will apply the CRE algorithm to the tissue water volume in each organ. The TWVs are lumped according to their perfusion rates and... 

Saidel et al. (1975) developed a lumped parameter, deterministic model to describe these five steps in the metastatic process. This model describes their data on pulmonary metastatic formation from a fibrosarcoma implanted in mice and simulates the effects of various perturbations (e.g., external tumor massage, primary tumor amputations) on metastatic process. These authors also developed a stochastic model of metastasis... 

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