Isotherm step response

The step responses (Fig. 1) for the two-site surfaces of catalysts 1 and 2 are compared from a plot of the functions generated by Eq. (4c). These are simulations of the type of data obtained (pressure versus time or moles adsorbed versus time) from adsorption isotherms. It is seen how easily experimental error at short times might make the separation and identification of the two sites difficult a tenfold variation in rate constant gives a barely noticeable change in the initial slope of the step response. 

That there be some previous experimental data on classical isothermal adsorptions (step response data) available for comparison by mathematical transformation techniques. 

The hydrogen-supported nickel system has been reported by Schuit and de Boer 48) as having both fast and slow adsorptions occurring. In addition precise data are available 45) on the isothermal variation of pressure with adsorption time on a similar hydrogen-supported nickel system. This classical adsorption data (step response data) can be mathematically transformed to simulate frequency response data by means of suitable mathematical techniques before any experimentation on the system is begun. The results of this simulation will be much the same as though an actual frequency response of Doerner s hydrogen-supported nickel system had been made. Actually any published adsorption isotherm data can be treated. However, the limitations of the simulation method are threefold (1) very accurate adsorption versus time data are required (2) the accuracy and dependability of the result at very fast times are subject to question (3) the adsorptions are not reproducible in the sense that only one real experiment was made for all adsorptions and the sensitivity of the mathematics could distort the result. 

The step responses should be modelled quantitatively by using a transient plug-flow model. The isothermal plug flow model for the components in the gas phase is written as... 

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

Brace et al. [92] investigated polymer/water interactions using SAW devices coated with either polyimide or cellulose acetate butyrate (CAB). In this study thermodynamic parameters were evaluated from the absorption isotherms, and transient responses to step changes in concentration were monitored. The transient responses observed were not consistent with Fickian diffusion, but could be described using a generalized relaxation equation containing two additive terms. Results under various conditions indicated that relaxation in the polymer system is much slower than diffusion of water. 

The experimental method used in TEOM for diffusion measurements in zeolites is similar to the uptake and chromatographic methods (i.e., a step change or a pulse injection in the feed is made and the response curve is recorded). It is recommended to operate with dilute systems and low zeolite loadings. For an isothermal system when the uptake rate is influenced by intracrystalline diffusion, with only a small concentration gradient in the adsorbed phase (constant diffusivity), solutions of the transient diffusion equation for various geometries have been given (ii). Adsorption and diffusion of o-xylene, / -xylene, and toluene in HZSM-5 were found to be described well by a one-dimensional model for diffusion in a slab geometry, represented by Eq. (7) (72) ... 

Figure 9.38. Schematic representation of the response of the chromatographic column for different adsorption isotherms (top). The column breakthrough of a step concentration change without disperson effects is shown (below) for different isotherms (a) linear and (b) convex. The abscissa tItQ is equivalent to the number of pore volumes eluted. (Adapted from Biirgisser et al., 1993.)...

The results described in this report compare well with data of Van-Den-Begin et al. [15] obtained on silicalite samples with an equivalent radius of 31pm by means of Single-Step Frequency-Response. The authors report a self diffusion coefficient for n-hexane of about 2 10 cmVs at a temperature of444 K. However, it has to be considered that, due to the shape of the sorption isotherm, the self-diffusion coefficient will be somewhat smaller than the transport diffusion coefficient. Caro et al [16] report a transport diffusion coefficient of 1.8 10 cmVs for the system n-hexane/HZSM-5 at 298 K, determined gravimetrically. The crystals used in that study were of prismatic shape, the dimensions being 330 pm (z-axis), 110 pm... 

FIGURE 9.10 HSDSC traces for 1 1 mixtures of aspiring with lactose (top) and magnesium stearate (bottom) following a step-isothermal temperature program. Aspirin/lactose shows no apparent incompatibility whereas aspirin/magnesium stearate shows a considerable heat response. (Wissing, 2000, unpublished data.)... 

The method consists of monitoring and analyzing the response of an adsorption column to a pulse input or a step change in concentration of an adsorbate. The carrier gas is a mixture of an inert gas and the adsorbate of known composition. The retention time of the pulse is related to slope of the equilibrium curve at the carrier gas composition. The slopes of the equilibrium curve at different points on the curve can be determined by carrying out experiments with different carrier gas compositions. The equilibrium curve can, then, be easily obtained by integration of the slopes of the isotherm curve. For binary sorption equilibria, the experiments are similar except the carrier gas is a mixture of the two adsorbates. 

In the left side, gas A flows in the direction as shown and picks up B due to the diffusion of B from the other side of the porous medium. Similarly, B at the other side will pick up A from its diffusion through the porous medium. The flow rates of the two sides can be carefully adjusted to give zero pressure gradient across the media (that is the total pressure is uniform throughout the porous medium). The concentrations of gases A and B are analysed by detectors, such as the thermal conductivity cell, and then the diffusive fluxes of A and B can be calculated. This is the steady state method. Recently, this method was extended to allow for transient operation such as a step change or square pulse in one chamber and the response is monitored in the other chamber. With this transient operation, the contribution from the dead end pore can be studied. This contribution is not seen by the steady state method, but its advantage is the ease of operation under isothermal operations. Detailed analysis of diffusion cell under steady state and transient conditions is provided in Chapter 13. 

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