Uptake model equation

The shape and length of the wave that propagates in the bed is related to the mathematical form of the uptake rate expression that is substituted into the last term in Eq. (9.10). Equation (9.15) is the pde describing diffusion of an adsorbate from the gas phase into the adsorbent This is but one of many diffusion based uptake models that might be substituted into the uptake rate term of Eq. (9.9) or (9.10). 

It is well known that swelling of resins increases the particle radius. When resin beads are immersed in the solution, water uptake takes place almost immediately, resulting in a new swollen radius. The swollen radius is inversely proportional to the initial solution concentration (Hellferich, 1962). This effect has not been taken into account in the original model of Levenspiel (1972). However, the actual swollen radius of the resin should be used in the model equations, and so measurements should be performed in order to estimate this radius. 

Thomann (1989) and Gobas (1993) have developed physiologically based kinetic models, including rate constants for chemical uptake and elimination based on physiological parameters such as gill ventilation rates, feeding rates and chemical assimilation rates as well as empirical correlations. Tables 9.6 and 9.7 list the model equations and parameters. 

The Davidson and Peppas model [Equation (3)] was applied to these data to study the mechanism and the rate of water uptake. 

Mass balance within an arbitrarily chosen biofilm section, or slice, taken parallel to the surface of attachment, is described by the one-dimensional, advection-dispersion-reaction equation, Eq. [1-5], with steady-state conditions and no advection. The sink term is microbial uptake, modeled using the parameters discussed in Section 2.6.3 see Eqs. [2-71 a] and [2-72],... 

The values in Table 5.1 are valid for water uptake at 30°C. Other data have shown that water uptake decreases with increasing temperature [4,16,42,46, 47], According to these various data, the water uptake of the membrane exposed to a saturated-vapor reservoir decreases from a value of 3, = 14 to a value of around X = 9 or 10 at 60°C with perhaps a slight upswing afterwards. To account for the temperature dependence rigorously, the temperature dependence of all the parameters listed in Table 5.1 must be known. This leads to a very complicated fitting procedure with more unknowns than data points. Furthermore, the effects may counteract each other. To simplify the analysis, all the parameters are assumed independent of temperature except for Ki, where the temperature appears explicitly as seen in Eq. (5.13). Fitting the model equations to data yields an expression for K2... 

Very often in the literature short time solutions are needed to investigate the behaviour of adsorption during the initial stage of adsorption. For linear problems, this can be achieved analytically by taking Laplace transform of model equations and then considering the behaviour of the solution when the Laplace variable s approaches infmity. Applying this to the case of linear isotherm (eq. 9.2-3), we obtain the following solution for the fractional uptake at short times ... 

In the calculation of the predicted response curves the axial dispersion coefficient and the external mass transfer coefficient were estimated from standard correlations and the effective pore diffusivily was determined from batch uptake rate measurements with the same adsorbent particles. The model equations were solved by orthogonal collocation and the computation time required for the collocation solution ( 20 s) was shown to be substantially shorter than the time required to obtain solutions of comparable accuracy by various other standard numerical methods. It is evident that the fit of the experimental breakthrough curves is good. Since all parameters were determined independently this provides good evidence that the model is essentially correct and demonstrates the feasibility of modeling the behavior of fairly complex multicomponent dynamic systems. 

The non-linear regression program supplied with the Hewlett-Packard 9845 minicomputer was used to solve the kinetic uptake phase equation for a one-compartment open model operating under first order kinetics ... 

Many models have been suggested to describe anomalous (non-Fickian) uptake and a number of the more relevant to structural adhesives will be discussed. Diffusion-relaxation models are concerned with moisture transport when both Case I and Case II mechanisms are present. Berens and Hopfenberg (1978) assumed that the net penetrant uptake could be empirically separated into two parts, a Fickian diffusion-controlled uptake and a polymer relaxation-controlled uptake. The equation for mass uptake using Berens and Hopfenbergs model is shown below. 

In calculations of pore size from the Type IV isotherm by use of the Kelvin equation, the region of the isotherm involved is the hysteresis loop, since it is here that capillary condensation is occurring. Consequently there are two values of relative pressure for a given uptake, and the question presents itself as to what is the significance of each of the two values of r which would result from insertion of the two different values of relative pressure into Equation (3.20). Any answer to this question calls for a discussion of the origin of hysteresis, and this must be based on actual models of pore shape, since a purely thermodynamic approach cannot account for two positions of apparent equilibrium. 

Thus, as pointed out by Cohan who first suggested this model, condensation and evaporation occur at difi erent relative pressures and there is hysteresis. The value of r calculated by the standard Kelvin equation (3.20) for a given uptake, will be equal to the core radius r,. if the desorption branch of the hysteresis loop is used, but equal to twice the core radius if the adsorption branch is used. The two values of should, of course, be the same in practice this is rarely found to be so. 

Any interpretation of the Type I isotherm must account for the fact that the uptake does not increase continuously as in the Type II isotherm, but comes to a limiting value manifested in the plateau BC (Fig. 4.1). According to the earlier, classical view, this limit exists because the pores are so narrow that they cannot accommodate more than a single molecular layer on their walls the plateau thus corresponds to the completion of the monolayer. The shape of the isotherm was explained in terms of the Langmuir model, even though this had initially been set up for an open surface, i.e. a non-porous solid. The Type I isotherm was therefore assumed to conform to the Langmuir equation already referred to, viz. 

For cellular models, a more compUcated form of the above equation is needed, to factor in paracellular, facilitated uptake and effiux transport, etc. [22].)... 

While most authors have used the finite-difference method, the finite element method has also been used—e.g., a two-dimensional finite element model incorporating shrinkable subdomains was used to de.scribe interroot competition to simulate the uptake of N from the rhizosphere (36). It included a nitrification submodel and found good agreement between ob.served and predicted uptake by onion on a range of soil types. However, while a different method of solution was used, the assumptions and the equations solved were still based on the Barber-Cushman model. 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

The initial conditions are CD = CD(0) at t = 0 and CR = 0 at t = 0. Efforts to obtain analytical solutions are tedious and unnecessary. By applying the change in concentrations (or mass) in the donor and receiver solutions with time to the Laplace transforms of Eqs. (140) and (141), the inverse of the simultaneous transformed equations can be numerically calculated with appropriate software for best estimates of a, (3, and y. It is implicit here that P Pap, Pbh and Ke are functions of protein binding. Upon application of the transmonolayer flux model to the PNU-78,517 data in Figure 32, the effective permeability coefficients from the disappearance and appearance kinetics points of view are in good quantitative agreement with the permeability coefficients determined from independent studies involving uptake kinetics by MDCK cell monolayers cultured on a flat dish... 

The solutions for moisture uptake presented in this section are based on the experimental condition of a pure water vapor atmosphere. In the next section a derivation of moisture uptake equations is based on both heat and mass transport that are characteristic of moisture uptake in air. The final section of this chapter presents the results of studies where heat transport is unimportant and mass transport dominates the process. Thus, we will have a collection of solutions covering models that are (1) heat transport limited, (2) mass transport limited, (3) heat and mass transport limited, and (4) mass transport limited with a moving boundary for the uptake of water by water-soluble substances. 

EPA. 1994b. Technical support document Parameters and equations used in integrated exposure uptake biokinetic model for lead in children (v0.99d). EPA/540/R-94/040, PB94-963505. 

Table 18.2 lists 30 of the molecules used in this study that are known to be substrates for active transport or active efflux. The mechanistic ACAT model was modified to accommodate saturable uptake and saturable efflux using standard Michaelis-Menten equations. It was assumed that enzymes responsible for active uptake of drug molecules from the lumen and active efflux from the enterocytes to the lumen were homogeneously dispersed within each luminal compartment and each corresponding enterocyte compartment, respectively. Equation (5) is the overall mass balance for drug in the enterocyte compartment lining the intestinal wall. 

Figure 4 also shows the excellent fit to the GAB equation (Eq. 10) of the sorption and desorption isotherms for microcrystalline cellulose. In this regard, this equation offers considerable practical utility in fitting isotherms for these types of materials over the entire relative humidity range, especially in contrast to the BET equation, which usually only fits uptake data up to about 40% relative humidity. As we have mentioned, however, this does not in itself confirm the validity of the GAB model for describing moisture sorption data on these materials. Rather, independent confirmation of the physical meaning is necessary. 

Modelling biouptake processes helps in the understanding of the key factors involved and their interconnection [1]. In this chapter, uptake is considered in a general sense, without distinction between nutrition or toxicity, in which several elementary processes come together, and among which we highlight diffusion, adsorption and internalisation [2-4], We show how the combination of the equations corresponding with a few elementary physical laws leads to a complex behaviour which can be physically relevant. Some reviews on the subject, from different perspectives, are available in the literature [2,5-7]. 

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