Model equilibrium transport dispersive

A common modeling approach for chromatographic batch reactors is the equilibrium-transport dispersive model (Chapter 6.2.4.1). Therefore, only the equations for this approach are discussed here. The differential mass balance (Fig. 8.6) takes into account axial dispersion as well as mass transfer between fluid and both solid phases. 

The linear driving force (LDF) model can be classified in the group of equilibrium transport dispersive models (Fig. 9.5). For this model it is no longer assumed that the mobile and the stationary phases are permanently in equilibrium state, so that an additional mass-balance equation for the stationary phase is required. Assuming a linear concentration gradient an effective mass-transfer coefficient keff is implemented, where all mass-transfer resistances and the diffusion into the pores of the particle are lumped together. In this model a constant local equilibrium between the solid and the liquid in the pores is assumed. 

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 other words, the simplification of using the number of stages for process optimization is best applied if either mass transfer or dispersion dominates the peak broadening. Therefore, the optimization strategies discussed later in this chapter apply a validated transport dispersive model, which can flexibly consider mass transfer and/ or dispersion effect. Here, the number of stages is used as independent variable for the optimization criteria like productivity or eluent consumption. Another possible approach would be the use of simplified simulation model like equilibrium dispersive model (Seidel-Morgenstern, 1995). 

SMBR, TMBR-equilibrium transport dispersive A— B + C A B + C Linear Model-based design Fridce and Schmidt-Traub (2003)... 

Depending on the main cause of sluggishness in reaching equilibrium in the column, we can distinguish several kinetic models. If the kinetics of the retention mechanism (e.g., the kinetics of adsorption-desorption) is slower than the other steps of the chromatographic process, we use the reaction-dispersive model. If the slowest step in the chromatographic process is the mass transfer kinetics, we have the transport-dispersive model. 

Figure 16.13 Comparison of the experimental (symbols) overloaded elution band profiles of the racemic mixture of the R and S enantiomers of 3-chloro-l-phenyl-l-propanol on a Chiracel OB-H 250 x 4.6 mm column eluted with n-hexane/ethyl acetate, 95/5 v/v and the profiles calculated with the equilibrium-dispersive (dotted lines) and the transport-dispersive models (solid lines). Sample volume, 1 ml, loading factor Lj = 5%. Reproduced with permission from D. Cher-rak, S. Khattahi, G. Guiochon, J. Chromatogr. 877 (2000) 109, (Figure 7d).

Figure 7.36a and b proves the applicability of shortcut calculations based on the ideal equilibrium model for the estimation of process conditions. The results of rigorous process simulation based on the transport-dispersive model are in very good agreement with the shortcut calculation for isocratic (a) as well as nonisocratic (b) SMB processes. Expectedly safety margins have to be taken into account when the process conditions of an SMB process are estimated by shortcut calculation. The scattering of the numerical data results from an increased grid size for the numerical calculations that has been chosen in order to reduce computer time. The model parameters coincide with the data for the protein separation presented in Section 6.6.2.2.3 the separation quality of the SMB process was set to 99.9% purity. 

Thomas Model Equilibrium Dispersive Model Equilibrium Transport Model... 

Sorbed pesticides are not available for transport, but if water having lower pesticide concentration moves through the soil layer, pesticide is desorbed from the soil surface until a new equihbrium is reached. Thus, the kinetics of sorption and desorption relative to the water conductivity rates determine the actual rate of pesticide transport. At high rates of water flow, chances are greater that sorption and desorption reactions may not reach equilibrium (64). Nonequihbrium models may describe sorption and desorption better under these circumstances. The prediction of herbicide concentration in the soil solution is further compHcated by hysteresis in the sorption—desorption isotherms. Both sorption and dispersion contribute to the substantial retention of herbicide found behind the initial front in typical breakthrough curves and to the depth distribution of residues. 

In the dispersive model it is assumed - like in the ideal model - that the mobile and the stationary phases are in equilibrium state. In addition to the convective transport a reverse-directed diffusive transport is included, which is responsible for the change of the concentration profile along the longitudinal coordinate z by backmixing. By means of Pick s 1st law a dispersion term can be added to Eq. (9.2), which leads to ... 

Cameron, D. R., and Klute, A. (1977). Convective-dispersive solute transport with a combined equilibrium and kinetic adsorption model. Water Res. 13, 183-188. 

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]. 

In the RT3D simulation, advective/dispersive transport of each contaminant is assumed. Sorption is modeled as a linear equilibrium process and biodegradation is modeled as a first-order process. Due to the assumed degradation reaction pathways (Fig. 2) transport of the different compounds is coupled. In the study, four reaction zones were delineated, based on observed geochemistry data. Each zone (two anaerobic zones, one transition zone, and one aerobic zone) has a different value for the biodegradation first-order rate constant for each contaminant. For example, since PCE is assumed to degrade only under... 

Mathematical models for mass transfer at the NAPL-water interface often adopt the assumption that thermodynamic equilibrium is instantaneously approached when mass transfer rates at the NAPL-water interface are much faster than the advective-dispersive transport of the dissolved NAPLs away from the interface [28,36]. Therefore, the solubility concentration is often employed as an appropriate concentration boundary condition specified at the interface. Several experimental column and field studies at typical groundwater velocities in homogeneous porous media justified the above equilibrium assumption for residual NAPL dissolution [9,37-39]. 

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