q q0 in this equation is the total concentration of the adsorption sites on the adsorbents; K is the reciprocal of the equilibrium constant of irreversible adsorption on the sites. For data processing of the Langmuir type of equa tions, a linearization step is often the only thing needed. For example, rearrangement of Equation 3.11 reveals the linear relationship of 1/q vs. 1/y (Equation 3.12):
q q0 q0y
Here, q0 is considered as a constant.
The Freundlich isotherm is defined as the following (Equation 3.13):
q = Kyn where n and K are experimentally determined constants; in many cases, n is less than 1 (but never equal to or larger than 1). Occasionally, for some adsorption curves of Freundlich isotherm type that has an n value close to 1, the linear isotherm is used for approximation purposes. To use the Freundlich isotherm to fit the experimental data, one has to transform the equation into a linear form by taking logs on both sides of Equation 3.13, resulting in the following (Equation 3.14):
It is obvious that plotting of log q versus log y yields a linear line.
The common way of determining the best fit of the isotherms in an adsorption experimental data is to plot data according to three common isotherms until a best-fit model (isotherm) is found. It is possible that none of the isotherms fits precisely to the data; in that case, a best fit (least deviated from the majority of data points—a statistical analysis of data fitting of the linearized isotherm) may be needed.
In adsorption operations, adsorbents often reside in a packed bed. This arrangement, which is akin to a plug flow reactor, has the best mass transfer driving force of adsorption of adsorbates onto the adsorbents in the packed bed. As adsorbates pass through the bed over time, the adsorbents in the bed extract the adsorbates from the solution and, eventually, the adsorbents in the bed become saturated and the concentrations of the adsor-bates from the effluent (called eluent) start to rise as illustrated in Fig. 3.11. The s-shaped curve of adsorbates' concentrations versus eluent volume (expressed often as time) that depicts the starting points of sharp rising and leveling off corresponding to exhaustion of the adsorbents of ad-
sorbates' concentrations leaving the packed bed is called the breakthrough curve. The breakthrough curve is used to compare different adsorbents in a packed bed. In addition to the breakthrough curve obtained from evaluation of adsorbents in a laboratory or provided by the vendor, another issue, called bed efficiency, is also important to practical application of adsorption technology and is related to the breakthrough curve. The packed bed efficiency over time is a measurement of the effect of the bed (the shell/container holding the adsorbents and the packing factor) on adsorption capacity (Equation 3.15):
2t where 9 is the fraction of the packed bed that is loaded with adsorbates, tE is eluent time (corresponding to the starting point of leveling off of the breakthrough curve in Fig. 3.11) and tB is the starting point of the breakthrough curve. This equation, through 9, indicates the actual saturation of the adsorbents in the bed when the breakthrough curve is observed, which
is always lower than 100%, hence the bed efficiency. It should be emphasized that Equation 3.15 is an approximation because its derivation involves the assumption of the breakthrough curve as a step function, a sharp, vertical rise of the eluent concentration in contrast to the more rounded curve shown in Fig. 3.11; however, it is a relatively adequate approximation because of the concentration profile developing quickly in the bed and "sharpening" of the curve. The attempt to make a numerical integration over the breakthrough curve does not lead to more accurate results since the gain from elaborate mathematical treatment of the breakthrough curve will be negated by the errors associated with the numerical methods used as well as the curve itself. The approximation method shown in Equation 3.15 is often recommended by adsorbent manufacturers.
Once 9 is known, the unused bed length l' can be calculated as follows (Equation 3.16):
where l is the bed length. So the used bed length l-l' is therefore saturated as follows (Equation 3.17):
where v is the superficial velocity of the eluent; £ is the void fraction of the bed; and y0 and q0 are the initial concentration of the adsorbate and saturated concentration of the adsorbate on the adsorbent, respectively. For a batch-packed bed, if we know the initial and final concentrations of the adsorbate and the amount of the feed that goes through the bed, we can easily calculate the amount of the adsorbent used based on mass balance on the adsorbate (Equation 3.18):
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