More Rigorous Modeling

The diffusion model is not very satisfactory from a physical point of view since liquid transport is known to be convective. Therefore, Jomaa et al. made an effort to develop a more fundamental description by volume averaging of the heat and mass balances in the deforming saturated gel (Caceres et al., 2007). For the evolution of the volume-average liquid density, or equivalently the local porosity ip = q /q, they obtained 

mechanical equflibrium is assumed, that is, a divergence-free total stress tensor. The total stress tensor decomposes into soUd network stress and liquid stress (with identity matrix I)  

Network stress is then related to the nite strain tensor with an elastic material law, and normal stress is set to zero at the gel surface. This closes the mathematical problem for the four independent variables, ip, Vj and T. 

The previous drying experiment ona gel rod is used to assess the model. Moisture and temperature evolution are fairly well reproduced by the simulation and maximal tensile stresses are obtained that are similar to those computed with the diffusion model However, the numerical Uquid pressure attains unrealistically low values, which is a result of the non-penetration assumption for gas. Here, the missing link of the macroscopic model to microstructural properties, that is, pore size, becomes obvious. 

Altogether, this model formulation provides a complete and physicaUy correct description of fluid transfer and seems to be suited for extension to the partiaUy saturated gel of the second drying period. A major challenge remains the definition of an air-penetration criterion that initiates the second drying period. 

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The EVB approach described in this chapter provides a convenient way for estimating the energetics of chemical reactions in various solvents. However, the approximation involved in eq. (2.21) cannot be justified without detailed studies by more rigorous models. Such models will be described in Chapter 3. 

The simpler models diseussed above (sueh as eases A and B) are usually good enough for continuoits-flow systems where the charrges in liquid arrd vapor holdups and temperatures are not very large. Bateh systems may require the more rigorous models (cases C and D) because of the big variations of most... 

Staby and Mollerup105 published a more rigorous model describing the retention of a protein and the salt concentration in the mobile phase. The capacity factor k or the relative retention can be further defined as... 

Another limitation of the regular solution theory is the assumption diat is negligible. While this assumption may be valid for solutions in which all components (solute and solvent) are of similar sizes, it breaks down when the molar voinmes of the components are significantly different, i.e., in the case of high molecular weight (polymeric) solvents and low molecular weight solutes. For such cases, more rigorous models that include entropic considerations, such as... 

Using an entirely different approach to the modeling of multicomponent mass transfer in distillation (an approach that we describe in Chapter 14), Krishnamurthy and Taylor (1985c) found significant differences in design calculations involving nonideal systems. For an almost ideal system (a hydrocarbon mixture), pseudobinary methods were found to be essentially equivalent to a more rigorous model that accounted for diffusional interaction effects. 

In this section we just keep this relation in mind and present a survey of the rather complicated mathematical derivation involved. A more rigorous model derivation is given in App. B. 

The simplifying assumptions that are necessary for the Y-X solution method are then relaxed to include enthalpy balances. A graphical solution of this more rigorous model, which requires enthalpy-composition data, is presented. 

In this article we report some experimental studies on spreading and penetration of an aqueous IJ ink on a variety of commercial printing papers. The data are tested against existing simple theoretical models for liquid penetration in porous media. No attempt has been made to develop or employ more rigorous models. 

The extraction of water by TBP requires a more rigorous model, but the extraction of U and HNO3 by TBP can be modeled fairly accurately using simplified models based on equilibrium constants and the law of mass action. At least one reaction must be assumed for each species that extracts. In the case of uranium and nitric acid, the following simplified reactions [5] can be assumed ... 

P7-2Sf (Open-ended problem) You may have to look up/guess/vary some of the coo-s(ants. if methanol is ingested, it can be metabolized to formaldehyde, which can cause blindness if the formaldehyde reaches a conceniralion of 0,16 tt/dm of fluid in the body. A concentration of 0.75 g/dm will be lethal. After ail the methanol ha,s been removed from the stomach, the primary [neaimeni is to inject ethanol inirasenously to tie up (competitive inhibition) the enzyme alcohol dehydrogenase (ADH so that methanol is not converted lo formaldehyde and is eliminated from the body through the kidney and bladder (iy). We will assume as a lirsi approximation that the body is a well-mixed C.5TR of 40 dm [total body fluid i. In Section 7.5, we applied a more rigorous model-... 

Mitchell (7), Blauer, and co-workers (7, 40) modeled foams as Bingham plastic. Foams were modeled as power law fluids by Patton et al. (41). A more rigorous model is the Herschel—Bulkley model, which combines both the Bingham plastic and Power law models (2,11, 42, 43). 

The development of more rigorous models may proceed along three different paths ... 

In the above simplified explanation, the effects of chemical gradients, principally diffusion, have been neglected. A more rigorous model has been given by Narasimhan and Sri Ranjan (2000). 

Figure 4 Comparison of demetallization performance of nine hydrotreating catalysts with model predictions using more rigorous model(from Spry and Sawyer [140]). 

Another approach, based on the theory of the transition state (by Wigner,t Eyring, Polanyi, and Evans in 1930 see Laidler and King 1998), provides a more rigorous model based on quantum considerations on the transition freqnencies of a reaction intermediate, which is the following ... 

For the esterification with n-butanol, Liu, Zhang, and Chen (2001) used a cross-linked PVA-ceramic composite membrane (in the temperature range of 60—90 °C). This paper is a rare case of use of the Zr(S04)2 -4H20 catalyst. On the basis of a simple Fickian model, the influence of several esterification process variables, such as process temperature, initial mole ratio of acetic acid to n-butanol, the ratio of the effective membrane area to the volume of reacting mixture, and catalyst content, were discussed. A more rigorous model was defined by Liu and Chen (using a kinetic approach) comparing model outcomes with experimental outcomes (Inoue et al., 2007). 

Simple models may be used to study specific mass transfer, transformation, or transport processes, or more rigorous models may be used to study macroscale performance. Unfortunately, because air sparging creates dynamic, nonequilibrium processes, models based on simplifying assumptions may provide misleading results. Additionally, many models have yet to be validated with controlled laboratory test data and field data. 

Compared with a more rigorous model, this model gives better approximation when intraparticle diffusion plays a dominant role in mass transfer. When intraparticle diffusion is the only rate determining step of mass transfer, Kjty can be replaced by particle phase mass transfer coefficient, k,fiy, which is related to the intraparticle diffusion parameters as... 

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