Thomas model

ANSPIPE Calculates pipe break probability using the Thomas Model BETA Calculates and draws event trees using word processor and other input BNLDATA Failure rate data... 

As above except that the condition of the second kind assumed the Boundary for Bi -> 0, the Thomas model becomes the Semenov Model for Bi oo, the Thomas model becomes the Frank-Kamanetskii Model critical conditions for various geometries given. 

The more recent Thomas model [209] comprises elements of both the Semenov and Frank-Kamenetskii models in that there is a nonuniform temperature distribution in the liquid and a steep temperature gradient at the wall. Case C in Figure 3.20 shows a temperature distribution curve from self-heating for the Thomas model. The appropriate model (Semenov, Frank-Kamenetskii, or Thomas) is determined by the ratio of the heat removal from the vessel and the thermal conductivity in the vessel. This ratio is determined by the Biot number (Nm) which has been described previously as hx/X, in which h is the film heat transfer coefficient to the surroundings (air, cooling mantle, etc.), x is the distance such as the radius of the vessel, and X is the effective thermal conductivity. 

FIGURE 3.20. Typical Temperature Distributions during Self-Heating in a Vessel. A = The Semenov Model B = The Frank-Kamenetskii Model C = The Thomas Model... 

Thomas model is usually appropriate somewhere between these two. These criteria are guidelines only and must be carefully applied. All three models should be tested for borderline cases. In practice, the models are valid only if no mass flows to or from the vessels, negligible reactants are consumed, and heat is generated only by reactions. 

Thomas Model Equilibrium Dispersive Model Equilibrium Transport Model... 

The solution of the simplest kinetic model for nonlinear chromatography the Thomas model [9] can be calculated analytically. The Thomas model entirely ignores the axial dispersion, i.e., 0 =0 in the mass balance equation (Equation 10.8). For the finite rate of adsorption/desorption, the following second-order Langmuir kinetics is assumed... 

For the case of a Dirac impulse injection, the solution of the Thomas model was given by Wade et al. [10] ... 

The Thomas model is also applicable to the design of ion-exchange columns (Kapoor and Viraraghavan, 1998). The Thomas equation constants qmm and values can be obtained from the column data and can be used in the design of a full-scale adsorption bed. This equation is simple since it can be used in its linear form ... 

The Thomas model has been used for the sorption of heavy metals using fungal biomass, bone char, chitin, and goethite (Kapoor and Viraraghavan, 1998 Lehmann et al.,... 

Wheeler-Jonas model for VOCs adsorption The Wheeler-Jonas equation is used for adsorption of VOCs using carbons. This equation is of the same form of the Thomas model with some modifications ... 

In the special case of ion exchange and unfavorable equilibrium, i.e. aA B < 1, with A originally in the solution, under the condition of sufficiently long bed, Walter s solution could be used. Walter s equation is a special case of the Thomas model for arbitrary isotherm and the kinetic law equivalent to a reversible second-order chemical reaction (Helfferich, 1962) ... 

In the fifth step, heat exchange with the surroundings is also considered the ambient temperature is different from the initial temperature of the reactive mass. This assessment also requires the heat transfer coefficient from the wall to the surroundings and uses the Thomas model. If the situation is assessed to be critical under these conditions, real kinetics can be used in order to give a more precise assessment. 

Step 5 The heat exchange at the wall with the overall heat transfer coefficient of 50 Wm 2 K can also be considered, following the Thomas model. Thus, the Frank-Kamenetskii criterion (8) is to be compared with the Thomas criteria... 

As a fifth attempt, an increase of the heat transfer at the wall in the Thomas model is not practicable and would not be efficient, since the major part of the resistance to heat transfer is the conductivity in the product itself, as shown by the high value of the Biot criterion, 300, which is closer to Frank-Kamenetskii conditions than to Semenov conditions. 

Figure 14.6 Evolution of the KL23L23 satellite intensities in (a) Cu and (b) Ni metals, related to the intensity of the ]D2 main Auger line (Sat 1 final-state shake up Sat 2 initial-state shake up), as a function of the excess photon energy above the K-absorption threshold [17]. For comparison, the predictions of the Thomas model [21] and the generic model [17] are also indicated.

For all these reasons, the mathematical aspects of the theory become much more complex. The mathematics of nonlinear chromatography are so complex that even for a single solute, there is no analytical, closed-form solution available, except with two simplified models, the ideal model and the Thomas model [120]. The ideal model is based upon the assumption of an infinite column efficiency. Its solutions are discussed in detail in Chapters 7 to 9. The Thomas model is based upon the assumptions that there is a slow Langmuir adsorption-desorption kinetics and that there are no other nvass transfer resistances, nor any axial dispersion. The system of equations of this model has been solved by Goldstein [121], and this general solution has been simplified for pulse injection by Wade et al. [122]. In aU other cases, the problem must be solved numerically. The Thomas model is discussed with other kinetic models in Chapter 14 and 16. 

The reaction-kinetic model is the only kinetic model for which an analytical solution can be derived for a pulse injection. The solution of Thomas model has been derived by Goldstein [42] in the case of a rectangular pulse injection of width tp and concentration Cq. It is... 

Equations 14.58 and 14.65 are the solutions of the Thomas model, depending on the width of the rectangular injection profile. We must imderline the fact that these solutions are exact. No simplifications have been made in the derivation of the solution beyond the assumption made in the model itself, that D is 0. 

Wade et al. [46] have compared the experimental band profiles of p-nitrophenyl-ff-D-mannopyranoside on silica-bonded Concanavalin A, obtained in affinity chromatography, and the best fit parameters to their model. This model i.e., Thomas model) uses a Langmuir kinetic and neglects the axial dispersion. The best values of the parameters are calculated using a Simplex program to minimize the sum of the residuals of the predided and experimental band profiles. Figure 14.10 illustrates the results obtained and shows excellent agreement. 

The Thomas model [23] is the only kinetic model that has an analytical solution in the single-component case, in nonlinear chromatography. In all other cases, the... 

The reaction-dispersive model is an extension of the Thomas model with a Langmuir kinetics... 

This result means that even if the kinetics follows the solid film linear driving force model the breakthrough curve can be fitted successfully to the Thomas model, provided that the apparent desorption rate corrstant, k, given by Eq. 14.82b is used. 

The band profiles for overloaded elution calculated with the reaction-dispersive, transport, and transport-dispersive models can be fitted to Thomas model [1]. 

This last result has been verified by calculating numerical solutions of these models, fitting them to the equation of Wade et al. [46] for the Thomas model, and determining the best values of the three parameters Aq, Lf, and N a- A modified Simplex algorithm [1,59-61] was designed for the successive estimation of these three parameters. Typical results are shown in Figures 14.14 and 14.15, and in Table 14.1. 

If the effect of dispersion is not taken into accoimt in the apparent number of reaction units (N ), there will be a large difference between the solutions of the Thomas model and the transport-dispersive or the reaction-dispersive models. This is illustrated in Figure 14.14, which compares the analytical solution of the Thomas model and the numerical solution of the reaction-dispersive model. The front of the latter solution is less steep than that of the former because the Thomas model does not take into accoimt axial dispersion, but only the kinetics of adsorption-desorption. 

Figure 14.14 Comparison of the numerical solution of the reaction-dispersive model (solid line) and the analytical solution of the Thomas model (dotted line). Both models kg = 5 Nrea = fcgfcdL/M = 2000. Thomas model Noisp = co. Reaction-dispersive model 2NDisp[fc()/(l + = 2000. (Left Lf =1%. (Right) Lf = 1, 1% 2, 5% 3, 10% 4, 20%.

The values of the parameters of the Thomas model (see Table 14.1) were obtained by fitting the profiles obtained as numerical solutions of the reaction-dispersive model (solid lines) to Eq. 14.65, for the different values of the loading factor. Reproduced with permission from S. Golshan-Shirazi and G. Guiockon,. Chromatogr., 603 (1992) 1 (Figs. 1 and 2). 

The best values of the parameters obtained when fitting the numerical solution of the reaction-dispersive model to Eq. 14.65 for the Thomas model are given in Table 14.1. The results in Table 14.1 show a very small error, of the order of 0.1% for the retention factor. For the loading factor, the error is approximately 1%. Both errors are independent of the loading factor. On the other hand, the value obtained for is not constant. It is significantly lower than 1000 and it decreases... 

Table 14.1 Best Values of the Parameters Obtained by Fitting the Band Profiles Calculated with the Reaction-Dispersive Model to the Thomas Model "...

Table 14.2 Best Values of the Parameters Calculated by Fitting the Transport and the Transport Dispersive " Models to the Thomas Model...

---