Mass kinetic parameters estimation

The techniques referred to above (Sects. 1—3) may be operated for a sample heated in a constant temperature environment or under conditions of programmed temperature change. Very similar equipment can often be used differences normally reside in the temperature control of the reactant cell. Non-isothermal measurements of mass loss are termed thermogravimetry (TG), absorption or evolution of heat is differential scanning calorimetry (DSC), and measurement of the temperature difference between the sample and an inert reference substance is termed differential thermal analysis (DTA). These techniques can be used singly [33,76,174] or in combination and may include provision for EGA. Applications of non-isothermal measurements have ranged from the rapid qualitative estimation of reaction temperature to the quantitative determination of kinetic parameters [175—177]. The evaluation of kinetic parameters from non-isothermal data is dealt with in detail in Chap. 3.6. 

Estimation of parameters. Model parameters in the selected model are then estimated. If available, some model parameters (e.g. thermodynamic properties, heat- and mass-transfer coefficient, etc.) are taken from literature. This is usually not possible for kinetic parameters. These should be estimated based on data obtained from laboratory expieriments, if possible carried out isothermal ly and not falsified by heat- and mass-transport phenomena. The methods for parameter estimation, also the kinetic parameters in complex organic systems, and for discrimination between models are discussed in more detail in Section 5.4.4. More information on parameter estimation the reader will find in review papers by Kittrell (1970), or Froment and Hosten (1981) or in the book by Froment and Bischoff (1990). 

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

The results confirm that the adsorption of ammonia is very fast and that ammonia is strongly adsorbed on the catalyst surface. The data were analyzed by a dynamic isothermal plug flow reactor model and estimates of the relevant kinetic parameters were obtained by global nonlinear regression over the entire set of runs. The influences of both intra-particle and external mass transfer limitations were estimated to be negligible, on the basis of theoretical diagnostic criteria. 

To obtain a more realistic estimation of the behavior of an autocatalytic reaction under adiabatic conditions, it is possible to identify the kinetic parameters of the Benito-Perez model from a set of isothermal DSC measurements. In the example shown in Figure 12.11, the effect of neglecting the induction time assumes a zero-order reaction leading to a factor of over 15 during the time to explosion. Since this factor strongly depends on the initial conversion or concentration of catalyst initially present in the reaction mass, this method must be applied with extreme care. The sample must be truly representative of the substance used at industrial scale. For this reason, the method should be only be applied by specialists. 

Those based on strictly empirical descriptions Mathematical models based on physical and cnemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input/output data without any physiochemical analysis of the process. For these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component... 

The substance-specific kinetic constants, kx and k2, and partition coefficient Ksw (see Equations 3.1 and 3.2) can be determined in two ways. In theory, kinetic parameters characterizing the uptake of analytes can be estimated using semiempirical correlations employing mass transfer coefficients, physicochemical properties (mainly diffusivities and permeabilities in various media), and hydro-dynamic parameters.38 39 However, because of the complexity of the flow of water around passive sampling devices (usually nonstreamlined objects) during field exposures, it is difficult to estimate uptake parameters from first principles. In most cases, laboratory experiments are needed for the calibration of both equilibrium and kinetic samplers. 

The parameter estimation is made in a stirred Lewis-type cell (Fig. 10.11) where, in the plateau region, chemical reaction prevails. With increasing stirring speed the initial mass transfer rates will increase as diffusional effects are diminished and a plateau is reached where mass transfer is independent of diffusion, as discussed elsewhere [31, 46]. A further increase in stirrer speed will disrupt the planar interface, leaving the operating area for determining the kinetics parameters. 

Lehtonen et al. (1998) considered polyesterification of maleic acid with propylene glycol in an experimental batch reactive distillation system. There were two side reactions in addition to the main esterification reaction. The equipment consists of a 4000 ml batch reactor with a one theoretical plate distillation column and a condenser. The reactions took place in the liquid phase of the reactor. By removing the water by distillation, the reaction equilibrium was shifted to the production of more esters. The reaction temperatures were 150-190° C and the catalyst concentrations were varied between 0.01 and 0.1 mol%. The kinetic and mass transfer parameters were estimated via the experiments. These were then used to develop a full-scale dynamic process model for the system. 

In applying the model, some mineral parameters, such as numbers, n, and mean radii, Rq of various mineral particles may be estimated by mineralogical techniques. For physical properties such as phase equilibrium constants, K, published ternary and binary data may be used on an approximate basis. Kinetic parameters such as reaction rate constants, k, or mass transfer coefficients can be very roughly estimated based on laboratory experiments. Their values may then be varied in a series of computer runs until the results match pilot plant data. A reasonably good match will, at the same time, confirm the remaining variables, rate equations and other assumptions. 

The main variable of design for a CSTR is the hydraulic retention time (HRT), which represents the ratio between volume and flow rate, and it is a measure of the average length of time that a soluble compound remains in the reactor. Capital costs are related to HRT, as this variable directly influences reactor volume [83]. HRT can be calculated by means of a mass balance of the system in that case, kinetic parameters are required. Some authors obtained kinetic models from batch assays operating at the same reaction conditions, and applied them to obtain the HRT in continuous operation [10, 83, 84]. When no kinetic parameters are available, HRT can be estimated from the time required to complete the reaction in a discontinuous process. One must take into account that the reaction rate in a continuous operation is slower than in batch systems, due to the low substrate concentration in the reactor. Therefore, HRT is usually longer than the total time needed in batch operation [76]. 

In this chapter the aspects of model selection/discrimination and parameter estimation and the experimental acquisition of kinetic data are not dealt with, since they fall fell outside its scope. Moreover, in interpreting the observed temperature dependency of the rate coefficients in this chapter it was assumed that we are dealing with intrinsic kinetic data. As will be shown in a Chapter 7, other, parasitic, phenomena of mass and heat transfer may interfere, disguising the intrinsic kinetics. Criteria will be presented there, however, to avoid this experimental problem. 

Figure 11.2 Kinetic plots (initial rate versus sucrose concentration) of fructosyltransferase from A aculeatus. Transfer activity ( ) and hydrolytic activity (o). Reactions were carried out in 0.2 M sodium acetate buffer (pH 5.5) at 60°C. Kinetic parameters were calculated estimating a molecular mass of 135 kDa for the active enzyme. Adapted from Ref [33].

Jantti s approach was originally developed enabling short adsorption measurements when a simple molecular adsorption model is applicable and the number of sites available for adsorption is infinite. In the present paper we show that with also in the case of a restricted number of sites combined with the possibility of more than one layer of adsorbed molecules Jantti s approach can lead to the saving of time. An estimate of the equilibrium value of the adsorbed mass as well as of other kinetic parameters in an early stage of the measurements is... 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

Conceptual models of percutaneous absorption which are rigidly adherent to general solutions of Pick s equation are not always applicable to in vivo conditions, primarily because such models may not always be physiologically relevant. Linear kinetic models describing percutaneous absorption in terms of mathematical compartments that have approximate physical or anatomical correlates have been proposed. In these models, the various relevant events, including cutaneous metabolism, considered to be important in the overall process of skin absorption are characterized by first-order rate constants. The rate constants associated with diffusional events in the skin are assumed to be proportional to mass transfer parameters. Constants associated with the systemic distribution and elimination processes are estimated from pharmacokinetic parameters derived from plasma concentration-time profiles obtained following intravenous administration of the penetrant. 

Hence there is no simple or direct relation between the observable gas outlet concentration and the kinetic parameters of the gas/solid reaction. The conversion also depends on the rate of mass transfer between the two phases, which varies in a complicated manner with bubble size and rise velocity, particle size distribution, diffusivity, bed geometry, the presence of bed inserts, etc. A number of empirical correlations are available to estimate these parameters. In most modelling studies however, adjustable parameters such as the bubble size are used to fit the experimental data. 

---