1068 / JOURNAL OF ENVIRONMENTAL ENGINEERING / NOVEMBER 2000 EQUILIBRIUM GASEOUS ADSORPTION AT DIFFERENT TEMPERATURES a Discussion by Sukanta Basu, 4 Associate Member, ASCE, Paul F. Henshaw, 5 and Nihar Biswas 6 The authors are to be commended for their investigation on the effect of temperature on equilibrium adsorption for pure gases. The aforementioned paper compared different types of correlations reported in the literature with a proposed equation for pure component gaseous adsorption data at different tem- peratures. The purpose of this discussion is to point out some conceptual shortcomings in their work. LIMITING ADSORPTION VALUE The term q 0 in the Dubinin-Astakhov equation [(3b)] is not a constant over a temperature range, as reported in Table 3. The dependence of q 0 on temperature is determined by the thermal coefficient of limiting adsorption  (Dubinin 1975): 0 0 d ln q 1 dq  = - = - (7) 0 dT q dT  is practically constant over a wide range of temperature. If the limiting adsorption value is determined experimentally 0 q 0 for a certain temperature T 0 , then according to (7), the limiting adsorption values q 0 for other temperatures can be expressed as 0 0 q = q exp[-(T - T )] (8) 0 0 Nikolayev and Dubinin (1958) (Dubinin 1975) proposed a method for calculating the density of a substance in the ad- sorbed state (adsorbate) for the range of temperatures from the normal boiling point (or triple point) T b to the critical temper- ature T cr from the physical constants of the substance being adsorbed (adsorptive). This method can be used to calculate  (Dubinin 1975):  b log  * cr  = (9) 0.434(T - T ) cr b where  b = density of the bulk liquid at T b ; and = adsorbate * cr density at T cr . Effective limiting values of adsorption for T > T cr are obtained by extrapolation of (8). Thus, the Dubinin- Astakhov equation can then be written as  A 0 q = q exp - + (T - T ) (10) e 0 0     E HETEROGENEITY PARAMETER The heterogeneity parameter  in the Dubinin-Astakhov equation is expressed as a small integer (Dubinin et al. 1970) rather than the nonintegral values, as reported in Table 3. The parameter  requires only a tentative estimation, since it is a June 1999, Vol. 125, No. 6, by A. R. Khan, R. Ataullah, and A. Al- Haddad (Paper 16201). 4 Grad. Student, Civ. and Envir. Engrg., Univ. of Windsor, ON N9B 3P4, Canada. 5 Asst. Prof., Civ. and Envir. Engrg., Univ. of Windsor, ON N9B 3P4, Canada. 6 Chair, Civ. and Envir. Engrg., Univ. of Windsor, ON N9B 3P4, Canada. expressed by an integer. This allows more accurate determi- nation of q 0 and E by nonlinear regression analysis, because of fewer unknown parameters. In general,  = 2 refers to the microporous adsorbents such as activated carbon (first kind of adsorbents), and  = 1 refers to the wide pore carbons (second kind of adsorbents) (Tien 1994). For molecular sieves the value is 2, while very fine pore carbons and zeolites may re- quire values up to 5 or 6 (Rudzinski et al. 1992). SATURATION PRESSURE TERM The discussers agree with the authors that physically mean- ingful pressures, P*, are not possible above the critical tem- perature. But, experimental data on adsorption of vapors by microporous adsorbents indicate that there are no discontinu- ities in adsorption characteristics during transition to the su- percritical region, i.e., to gas adsorption. So, by adopting a standard state in the supercritical region, the concept of vol- ume filling of micropores can be extended to supercritical tem- peratures (Dubinin 1975). Effective values of P* can be de- termined using the following expression (Dubinin 1975): N log P*= M - (11)  T whose constants M and N are calculated from the critical pres- sure P cr for the critical temperature T cr and the normal boiling point T b when P* = 1 atm. This equation is in satisfactory agreement with experimental data in the range of P* from 1 atm to P cr . Effective values of P* for T > T cr can be calculated from (11). Usually, in such cases P* is used to find fugacities, f s , which are then used for calculating differential molar works of adsorption (A). MODIFIED OBJECTIVE FUNCTION The authors have defined the objective function for nonlin- ear regression as the sum of the squares of the percentage errors to avoid poor fitting at low values of pressure. This modification undoubtedly leads to the assignment of equal weight to each value throughout the experimental range, but it also introduces greater mean relative error in the case of the Dubinin-Astakhov equation. The lower boundary of approximate applicability of the Du- binin-Astakhov equation is usually restricted to fillings (q e /q 0 ) of 0.1–0.2 (Dubinin 1975). This equation is applicable only to medium and high fillings. Since there are no physical grounds for considering the applicability of the Dubinin-As- takhov equation in the range of low fillings, it is expected to have poor fit at low pressure values corresponding to low fill- ings. Thus, it is not logical to have equal weight in low, me- dium, and high fillings. The discussers used the sum of the squares of the numerical errors as the objective function and found an excellent fit of values in the medium and high fill- ings. The mean relative errors were also lower than the values reported by the authors in column 8 of Table 3. NONLINEAR REGRESSION ANALYSIS The Dubinin-Astakhov equation is based on rational phys- ical and chemical concepts rather than on the empirical fit of experimental data. Since the theory is rational, parameters that affect the adsorption process can be identified and estimated. The generalized equation by Khan et al. [(5)] has five param- eters (q 0 , b, , , n), in comparison to only two parameters ( , E ) in the Dubinin-Astakhov equation. The heterogeneity 0 q 0 parameter () in the Dubinin-Astakhov equation is not con- sidered as unknown because of its known range of integral values for different adsorbent systems. The discussers used the Marquardt-Levenberg method for nonlinear regression analy- J. Environ. Eng. 2000.126:1068-1069. Downloaded from ascelibrary.org by North Carolina State University on 08/02/15. Copyright ASCE. For personal use only; all rights reserved.