Simulated isotherm

Chrzanowski FA, Ulissi LA, Fegely BJ, Newman AC. Preformulation excipient compatibility testing application of a differential scanning calorimetric method versus a wet granulation simulating, isothermal stress method. Drug Devel Ind Pharm 1986 12(6) 783-800. 

Figure 20. Isotherms and interface shapes for the time t = 1.0 for batchwise simulations of CZ growth. Results are shown for (a) uncontrolled, (b-d) integral control, and (e) proportional-integral control simulations. Isotherms are spaced as described for Figure 19. The figure is taken from Atherton et al. (153).

The chapter ends with a case study. Four different reduced kinetic models are derived from the detailed kinetic model of the phenol-formaldehyde reaction presented in the previous chapter, by lumping the components and the reactions. The best estimates of the relevant kinetic parameters (preexponential factors, activation energies, and heats of reaction) are computed by comparing those models with a wide set of simulated isothermal experimental data, obtained via the detailed model. Finally, the reduced models are validated and compared by using a different set of simulated nonisothermal data. 

First, the detailed model is used to simulate the behavior of the real system, and a set of simulated isothermal experimental data is generated including the total heat released by reaction. Then, these data are used to estimate the kinetic parameters of the reduced models and the heats of reaction of the lumped reactions. Finally, the reduced kinetic models are tested in a validation procedure which simulates the operation of a batch reactor and allows one to identify the best reduced model. 

Figure 12.4. Simulated isotherms for the adsorption of argon and nitrogen at 77 K by a 4.4 nm bucky-tube (Maddox and Gubbins, 1995).

As we can see from Figure 4, the simulated isotherm (solid line) is qualitatively different from the experimental isotherm (dotted line). The bimodal character of the PSD is reflected in the simulated isotherm, producing two "steps". On the other hand, the experimental isotherm has a typical S shape. Furthermore, the simulation results lie almost entirely at lower pressures than the experiment, which suggests that the concentration of surface sites is too high. As an attempt to improve the model, we have performed simulations... 

N2 adsorption isotherms in internal and ineterstitial nanopores of single wall carbon nanohorn (SWNH) were calculated by GCMC simulation and is compared with experimental one. Fitting of the GCMC-simulated isotherm to experimental one in internal nanopores gave the average pore width w = 2.9 nm. The N2 adsorption isotherm in the interstitial nanopores of the bundled SWNH particles well coincided with the observed one. 

In this study, N2 adsorption in the internal pore of single SWNH particle and on external pores of bundled SWNH particles is simulated with grand canonical Monte Carlo (GCMC) method and the simulated isotherms are compared with the experimental results. 

Simulated adsorption isotherms on the bundled SWNH assembly of the hexagonal symmetry for D = 3.2 nm (w = 2.9 nm) is shown in Figure 5. Here it is assumed that the internal pores are available for adsorption and the interparticle spacing is 0.4 nm. The simulated isotherm in the external pores well expresses the experimental isotherm, indicating the presence of the bundled structure. Figure 6 shows N2 adsorption behaviors in external pores at different P/Pq values using the snapshot. Molecules are adsorbed at the tube and neck sites from an extremely low pressure region. 

Fig. 3. Experimental adsorption Isotherm ofN2 on SWNH at 77 K (o) and simulated isotherms of Nj on three models at 77 K triangular array (+ symbols) square array ( ) adsorption on the external surface of an isolated model (A). 

Figure 2. Simulated isotherms for methane and ethane adsorption in single-walled carbon nanotubes of diiierent diameters.

With regards to the repulsive case, the model adsorption isotherm reproduces the MC data fairly well for w/keT. The discrepancy in the case w/kal links to the order-disorder phase transitions that dimers develop on a square lattice at w/kBTe w3, 6c =l/2 and w/kaTc 5, 0c =2/3 (see ref. [5]). These transitions can not be reproduced by the approximation (mean field) used in eq. (8). As it is clear from Figure 2, the simulated isotherm for w/keT=4 display a plateau (characteristic of an ordered phase) at 6=>l/2 since w/kBTc <4. However, only a knee around 0=2/3 is visible at this temperature because w/ksTc 4. A deeper discussion of the referred phase transitions can be found in ref. [5]. 

Figure 2. Simulated isotherms for methane-in carbon slit pores of varying widths are shown here. The number of adsorbed molecules per unit area of pore wall is plotted as a function of the pressure times the Henry s constant, which gives the single straight line shown for the limiting low pressme parts of the isotherms. Pore widths in A are shown on the Figure. From Ref. [22], Sep. Sci. and Tech. 27 (1992), 1837-1856.

Computer simulation is an appealing way to approach this sort of problem, because the results of simulated isotherm analyses can be compared against geometric analyses of the exactly know simulated pore structures. If the computer model of the adsoi-bent under study is realistic, qualitative (and perhaps quantitiative) results from this approach should be relevant for the... 

Figure 1. Simulated isotherms for models of G3. adsorption x desorption (uniform porosity) o desorption (randomised pore size, uniform porosity) + desorption (spin density image porosity). Macroscopic model pore size allocated from T, image except where stated.

Simulated isotherms of nitrogen at 77 K for closed square SWCNT arrays are shown in Figure 11. At small tube separations, the amount adsorbed increases with increasing tube diameter. However, at larger tube separations, the amount adsorbed decreases with increasing tube diameter. The isotherms are mainly type I or type IV, depending on tube diameters and separations. If the tubes are open, the amount of nitrogen adsorbed will increase markedly as... 

In what follows we describe the simulation model for N2 adsorption in activated carbons for slit-like and triangular section pores and the characterization method to find the MSD from adsorption data by fitting simulated isotherms to experimental ones... 

In order to characterize the micropore structure by obtaining the MSD from adsorption isotherm data, it is necessary to obtain a collection of simulated isotherms for different size d, (/ = 1,. . ., ), in the form of adsorbed volume of gas at STP as a function of p i] d, p). Then, assuming the hypothesis of independent pores as valid, as commonly done, the global theoretical isotherm for a microporous material having a size distribution/ (d,) can be written as ... 

With a slight modification, the flowchart in Fig. 4.7 can serve to simulate isothermal chromatography of short-lived nuclides when we detect the nuclei which survived at the column exit see Fig. 1.7. 

Fig. 2 Comparison of experimental and simulated isotherms of benzene in NaX zeolite vs. pressure for three temperatures indicated. (View this art in color at www.dekker.com.)...

Some systematic deviations of simulated isotherms from experiment in the BET-region may be explained on assumption that real surfaces of amorphous oxide are characterized not just by the atomic roughness of the irregular amorphous atomic structure, but may also have another kind of roughness with a much larger scale of length. 

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