Transport-dispersive model

According to the Scher-MontroU model, the dispersive current transient (Fig. 5b) can be analyzed in a double-log plot of log(i) vs log(/). The slope should be —(1 — ct) for t < and —(1 + a) for t > with a sum of the two slopes equal to 2, as shown in Figure 5c. For many years the Scher-MontroU model has been the standard model to use in analyzing dispersive charge transport in polymers. 

Thomas Model Equilibrium Dispersive Model Equilibrium Transport Model... 

The transport-dispersive model consists of one differential mass balance equation for each component, i, in the mobile phase ... 

The results of theoretical calculation using both general rate and transport-dispersive models were in good agreement with the overloaded band profiles determined experimentally, therefore, the method has been found to be suitable for the prediction of band profiles [88], Natural pigments were generally used as a complicated mixture of various compounds with chromophore substructure. Their separation by preparative RP-HPLC is not necessary, and the application of preparative RP-HPLC for the purification of one or more pigment fractions is not expected in the near future. 

Egan, B. A., and J. R. Mahoney. Applications of a numerical air pollution transport model to dispersion in the atmospheric boundary layer. J. Appl. Meteorol. 11 1023-1039, 1972. 

In the lumped kinetic model, various kinetic equations may describe the relationship between the mobile phase and stationary phase concentrations. The transport-dispersive model, for instance, is a linear film driving force model in which a first-order kinetics is assumed in the following form ... 

Fig. 2 Structure of an off-line air quality modelling system, showing the inputs and outputs of a meteorological and a chemical-transport dispersion model...

As contaminant transport occurs over times much greater than the times over which groundwater flow fluctuates, steady flow is frequently assumed. For steady groundwater flow in three dimensions, the following vector equation, developed based on mass conservation principles, is typically used to model advective/ dispersive transport of a dissolved reactive contaminant (after [53]) ... 

BIOPLUME III is a public domain transport code that is based on the MOC (and, therefore, is 2-D). The code was developed to simulate the natural attenuation of a hydrocarbon contaminant under both aerobic and anaerobic conditions. Hydrocarbon degradation is assumed due to biologically mediated redox reactions, with the hydrocarbon as the electron donor, and oxygen, nitrate, ferric iron, sulfate, and carbon dioxide, sequentially, as the electron acceptors. Biodegradation kinetics can be modeled as either a first-order, instantaneous, or Monod process. Like the MOC upon which it is based, BIOPLUME III also models advection, dispersion, and linear equilibrium sorption [67]. 

Contaminants in the soil compartment are associated with the soil, water, air, and biota phases present. Transport of the contaminant, therefore, can occur within the water and air phases by advection, diffusion, or dispersion, as previously described. In addition to these processes, chemicals dissolved in soil water are transported by wicking and percolation in the unsaturated zone.26 Chemicals can be transported in soil air by a process known as barometric pumping that is caused by sporadic changes in atmospheric pressure and soil-water displacement. Relevant physical properties of the soil matrix that are useful in modeling transport of a chemical include its hydraulic conductivity and tortuosity. The dif-fusivities of the chemicals in air and water are also used for this purpose. 

In the next example the PHREEQC job is presented that simulates the experiment. To adjust the model to the data observed, the exchange capacity (X under EXCHANGE, here 0.0015 mol per kg water), the selectivity coefficients in the data set WATEQ4F.dat and the chosen dispersivity (TRANSPORT, dispersivity, here 0.1 m) are decisive besides the spatial discretisation (number of cells, here 40). If one sets the dispersivity to a very small value (e.g. T10"6) in the input file Exchange and rerun the job, one will see that no numerical dispersion occurs showing that numerical stability criteria are maintained properly. 

A useful description of mixing in bubble columns is provided by the dispersion model. The global mixing effects are generally characterized by the dispersion coefficients El and Eq of the two phases which are defined in analogy to Fick s law for diffusive transport. Dispersion in bubble columns has been the subject of many investigations which have recently been reviewed by Shah et al. (45). Particularly, plenty of data are available for liquid-phase dispersion. 

In order to further substantiate this conclusion, it is of interest to compare it with the prediction obtained from a simple theoretical model. Glueckauf s well-known transport model (19, p. 449-453), supplemented by the more modern concept of hydro-dynamic dispersion, is well suited for this purpose. The model simulates dispersion-affected solute transport with ion exchange for which diffusion processes are rate limiting. In his development, Glueckauf assumes 1) exchange takes place in porous... 

The purpose of this study is to illustrate some important differences between online and offline model systems and to evaluate transport, dispersion and deposition of the online coupled meteorological and chemical transport and dispersion model Enviro-HIRLAM (High Resolution Limited Area Model), which is developed at the Danish Meteorological Institute (DM1). 

Within the completely online Enviro-HIRLAM (Baklanov et al. 2008a Korsholm et al. 2008b) the transport of chemical species is achieved in the same way like for other variables in HIRLAM (actually it was performed via the Tracer subroutine inside HIRLAM like for other scalars) on the same time steps and with the same grid. There is no need for any interface in Enviro-HIRLAM, because ACT is inside the HIRLAM model, so all the HIRLAM parameters are available for ACT. The model is designed to be used for operational as well as research purposes and comprises aerosol and gas transport, dispersion and deposition, aerosol physics and chemistry, as well as gas-phase chemistry. A Climate version Enviro-HIRHAM is also planned and will be developed in the near future. 

T Trban airshed models are mathematical representations of atmospheric transport, dispersion, and chemical reaction processes which when combined with a source emissions model and inventory and pertinent meteorological data may be used to predict pollutant concentrations at any point in the airshed. Models capable of accurate prediction will be important aids in urban and regional planning. These models will be used for ... 

In the following the most relevant models for liquid chromatography are derived in a bottom-up procedure related to Fig. 6.2. To illustrate the difference between these models their specific assumptions are discussed and the level of accuracy and their field of application are pointed out. The mass balances are completed by their boundary conditions (Section 6.2.7). For the favored transport dispersive model a dimensionless representation will also be presented. 

The transport dispersive model (TDM) is an extension of the transport model and summarizes the internal and external mass transfer resistance in one lumped film (= effective) transfer coefficient, keii (compare Eq. 6.30) ... 

As mentioned in Section 6.2.2, the mass transfer term in Eq. 6.3 is defined by the linear driving force approach. Therefore, the transport dispersive model consists of the balance equations in the mobile phase (Eq. 6.71) written with the pore concentration... 

Notably, concerning the overall peak shape, as with the equilibrium dispersive model (Section 6.2.4.1), the analytical solution of the transport dispersive model is always an asymmetric peak, and the asymmetry is enhanced by increasing Dax as well as decreasing keff (Lapidus and Amundson 1952). 

In this book,. .transport dispersive model" always refers to the liquid film linear driving force model (Eqs. 6.71, 6.74 and 6.37). 

The other subgroup of the lumped rate approach consists of the reaction dispersive model where the adsorption kinetic is the rate-limiting step. It is an extension of the reaction model (Section 6.2.4.3). Like the mass transfer coefficient in the transport dispersive model, the adsorption and desorption rate constants are considered as effective lumped parameters, kads,eff and kdes.eff- Since no film transfer resistance exists (Cpi = q), the model can be described by Eq. 6.79 ... 

The transport dispersive model (TDM, Section 6.2.5.1) is thus appropriate to simulate systems with considerable band broadening (Section 6.6), using only two different parameters to characterize packing properties (Dax) and mass transfer (keff). From theoretical viewpoint, Kaczmarski and Antos et al. (1996) provide rules, in which case both TDM and GRM give identical results. 

As increasing computational power and sophisticated numerical solvers are now available it is no longer necessary to use even more simplified models to reduce computing time. For these reasons the rest of this chapter is restricted to the transport dispersive model. 

Transport dispersive model Adsorption chromatography for products with low molecular weights Generally high accuracy Chiral separation... 

Introducing Eqs. 6.101-6.105 into the equations of the transport dispersive model leads to the following dimensionless mass balances ... 

The above list refers to the transport dispersive model and, if other models are selected, the model equations and hence the number of parameters are expanded or reduced. 

The method of consistent parameter determination is depicted in Fig. 6.9. It was first published by Altenhoner et al. (1997) for the transport dispersive model. The basic idea is to start with the simplest parameters and to use them subsequently to determine more complex ones. The procedure is structured into the following steps ... 

The corresponding equations for the first and second moments for the transport dispersive model (TDM) (Lapidus and Amundson, 1952 and van Deemter et al., 1956) are ... 

Table 6.5 Determination of parameters of the transport dispersive model based on moment analysis only.

The retention time tRjin and the second moment for the Gaussian profile (Eq. 6.61) have been replaced by variables indexed with g . These parameters tg and og must be optimized by curve fitting. Equation 6.143 is only suitable for symmetric peaks. Analytical solutions of, for example, the transport dispersive model (which describes asymmetric band broadening only for a very low number of stages) are not suited to describing the asymmetry often encountered in practical chromatograms. Thus, many different, mostly empirical functions have been developed for peak modeling. A recent extensive review by Marco and Bombi (2001) lists over 90 of them. 

The following examples illustrate a few effects encountered in model validation based on our research (Epping, 2005 and Jupke, 2004). All process simulations are based on the transport dispersive model. Model equations were solved by the gPROMS Software (PSenterprise, UK) using the OCFE method (Section 6.4). 

The validity of the transport dispersive model was further confirmed by experiments with other chromatographic systems such as Troger s base (Mihlbachler et al., 2001 and Jupke, 2004), the WMK-Keton (Epping, 2005) and fructose-glucose (Jupke, 2004). 

Even for the fructose-glucose isomer system, with liquid phase concentrations that are an order of magnitude higher than those of the enantiomer system, the transport dispersive model is valid. Figure 6.32 shows that experimental and theoretical concentration profiles for a pulse experiment match very well using the isotherm data of Eq. 6.182. Eight semi-preparative columns were connected in series to... 

Because of the analogy between simulated and true counter-current flow, TMB models are also used to design SMB processes. As an example, the transport dispersive model for batch columns can be extended to a TM B model by adding an adsorbent volume flow Vad (Fig. 6.38), which results in a convection term in the mass balance with the velocity uads. Dispersion in the adsorbent phase is neglected because the goal here is to describe a fictitious process and transfer the results to SMB operation. For the same reason, the mass transfer coefficient feeff as well as the fluid dispersion Dax are set equal to values that are valid for fixed beds. 

---