Carbon 43 (2005) 2258–2263 www.elsevier.com/locate/carbon
Adsorption equilibrium modeling for water on activated carbons Nan Qi, M. Douglas LeVan
Department of Chemical Engineering, Vanderbilt University, VU Station B #351604, 2301 Vanderbilt Place, Nashville, TN 37235, USA Received 12 May 2004; accepted 30 March 2005 Available online 17 May 2005
Abstract A new equation for describing adsorption equilibria of water on activated carbon is developed based on a mechanism proposed by Dubinin. It is HenryÕs law consistent, mathematically simple, and explicit in pressure. The model describes the full range of the adsorption isotherm with high accuracy using only a small number of parameters. Adsorption equilibrium data for water on several activated carbons quite diﬀerent in surface area, surface chemical properties, and pore structure are used to test the model. A twovariable series expansion is used to extend a single adsorption isotherm to an adsorption isotherm family at multiple temperatures. The model description of pure water adsorption isotherms at diﬀerent temperatures on BPL activated carbon is in good agreement with experimental data. 2005 Elsevier Ltd. All rights reserved. Keywords: Activated carbon; Adsorption; Modeling; Adsorption properties; Surface oxygen complexes; Water
1. Introduction Water vapor adsorption equilibria have been measured on diﬀerent kinds of porous adsorbents [1–6] and on carbons with diﬀerent degrees of surface oxidation [7,8]. Unlike organic adsorption on a microporous carbonaceous adsorbent, for which the isotherm shows type I behavior according to the classiﬁcation of Brunauer et al. , water adsorption isotherms show type IV behavior on carbons with a highly oxidized surface, or type V behavior (S-shaped) on carbons with a strongly hydrophobic surface. Generally, water adsorption exhibits hysteresis on porous media. Diﬀerent theories try to explain and model water vapor adsorption equilibrium. DubininÕs  water adsorption mechanism is a commonly accepted one. Dispersion interactions between water molecules and the carbon surface are negligibly weak while hydrogen *
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bonds play a more signiﬁcant role. Oxygen complexes on the adsorbent surface act as the primary sites for water molecules to be adsorbed by hydrogen bonding. The adsorbed water molecules supply secondary sites for more water vapor molecules to be adsorbed via hydrogen bonds. As more and more water molecules are adsorbed, clusters are formed. Heat released during water adsorption is close to the heat of condensation, which is approximately 45 kJ/mol. Water adsorption hysteresis occurs predominantly because of the porous structure of the adsorbent and can be explained as the coalescence of water clusters on the adsorption branch and evaporation of capillary condensed water on the desorption branch . The hysteresis loop vanishes, giving a closed single S-shape isotherm, for water adsorbed on nonporous carbon black . Based on DubininÕs water adsorption mechanism, four water adsorption isotherm models have been developed from kinetic theory in a way similar to the classical development of the Langmuir isotherm. They are the DS-1 , DS-2 , DS-3 , and DS-4  equations.
N. Qi, M.D. LeVan / Carbon 43 (2005) 2258–2263
n ¼ n0 cpr =ð1 cpr Þ
n ¼ cpr ðn0 þ nÞð1 jnÞ ðDS 2Þ
n ¼ pr ðcjn3 cjn0 n2 þ cnÞ ðDS 3Þ n h io 2 n ¼ pr cn0 þ cn 1 exp j2 ðn nc Þ
ð3Þ ðDS 4Þ ð4Þ
where n, pr, c, n0, j and nc are, respectively, the loading (mol/kg), relative pressure (p/psat), kinetic constant, concentration of primary sites, constant for decrease in number of adsorption sites, and parameter initiating the decline in adsorptive power . The DS-1 equation normally gives only a fair description at low relative pressures (e.g., pr < 0.5 for BPL carbon at 25 C ) and fails at higher relative pressures. Diﬀerent empirical factors have been introduced to account for the decrease of available adsorption sites as (1 jn) for DS-2, (1 jn2) for DS-3 and [1 exp(j2(n nc)2)] for the DS-4 equation. The data correlations are improved such that DS-3 is more accurate than DS-2, and DS-4 is the most accurate among the series of the four equations. Sircar developed a general model for type I, IV, and V isotherms on porous carbons [15,16]. The amount adsorbed is determined through the pore ﬁlling mechanism in micropores and through both physical adsorption and capillary condensation in macropores. A full range pore size distribution is used to account for pore structure heterogeneity. Water adsorption on sugar charcoal are ﬁt nicely by SircarÕs model . Although the model is explicit, it involves complicated procedures to evaluate a gamma function and seven parameters. Talu and Meunier  developed a thermodynamic model for type V behavior by treating cluster formation as a series of chemical reactions for self-associating molecules. Water adsorption on activated carbons was described with a good ﬁt at ambient temperature with 3 parameters and at multiple temperatures with 5 parameters. All parameters are physically signiﬁcant. Do and Do  proposed a model to describe both type IV and type V isotherms for water on diﬀerent carbons. Water molecules form clusters around functional groups. In their model, a cluster of 5 molecules can penetrate into a micropore as adsorbed water. Capillary condensation was evaluated using the Kelvin equation in mesopores as water vapor approaches the saturation pressure. The model gives very good descriptions for some water adsorption isotherms. Mahle  derived an equation for a type V isotherm by describing capillary condensation with the Kelvin equation and integrating over a pore size distribution. The model successfully describes water adsorption equilibria on activated carbon for both adsorption and desorption branches. It can be written explicitly in terms of either pressure or loading.
Stoeckli  expressed water adsortion equilibria by combining a type I and a type V isotherm, both described by the Dubinin–Astakhov (D–A) equation. The method has had success in correlating some data. However, a thermodynamic concern with this approach is that the D–A equation does not follow HenryÕs law at low loadings. Cooperative multi-molecular sorption theory was used by Rutherford  to describe water adsorption isotherms. The model works well below 80% relative pressure. It is not valid above 80% relative pressure where capillary condensation and hysteresis occur. Describing water adsorption equilibrium with high accuracy is important, especially in modeling multi-component adsorption equilibrium, in which accurate pure component adsorption isotherms are required. A simple and explicit adsorption isotherm model is highly desirable in order to increase computation speed in ﬁxedbed simulation and other applications where extensive iterations and calculations are performed. In this paper, we develop an equation to describe water adsorption equilibrium on activated carbon. It obeys HenryÕs law at low loadings and describes the full range of the adsorption isotherm with high accuracy using a small number of parameters. The model is mathematically simple and explicit in pressure. Adsorption equilibrium data for water on BPL, PVDC, UO3-1, NC100, polymeric type E, and ACF activated carbons are used to test the model. We also extend a single adsorption isotherm at one temperature to an adsorption isotherm family at multiple temperatures with very few parameters by using a two-variable Taylor series expansion. An example is given for developing water adsorption isotherms over a wide range of temperatures on BPL activated carbon.
2. Theory 2.1. Adsorption equilibrium of water on activated carbons Similar to the DS equation series [10,12–14], water adsorption on the microscale is depicted as a set of chemical reactions following the mechanism of Dubinin , i.e. k1
SþR ! SþA k2
AþR ! AþA k3
ð7Þ A ! R where S symbolizes a primary adsorption site given by a surface oxygen complex; R symbolizes a water vapor molecule; A symbolizes an adsorbed water molecule and the secondary site it supplies; and k1, k2 and k3 are rate constants. Note that according to this mechanism,
N. Qi, M.D. LeVan / Carbon 43 (2005) 2258–2263
when a water molecule is adsorbed, a site is not removed, but another (secondary) site is supplied. Adsorption equilibrium on the macroscale is taken to be analogous to reaction equilibrium, where the total water adsorption rate on both primary and secondary sites is equal to the water desorption rate from the secondary sites. Thus, we have k 1 n0 pG1 þ k 2 npG2 ¼ k 3 n
where n is the water loading; p is the water pressure in the vapor phase; and n0 is the concentration of primary sites. G1 and G2 are factors to express the macroscopic decreases in primary adsorption sites and secondary adsorption sites, respectively, as the loading increases and water ﬁlls the pore structure of the carbon. Rearranging Eq. (8) gives n p¼ ð9Þ ðk 1 =k 3 Þn0 G1 þ ðk 2 =k 3 ÞnG2 or p¼
n b0 G1 þ b1 nG2
with b0 ¼ k 1 n0 =k 3
b1 ¼ k 2 =k 3
Two empirical and ﬂexible functional relations proposed here to describe how G1 and G2 change with water loading are G 1 ¼ 1 b1 n b 2 n 2 b 3 n 3
G 2 ¼ 1 c 1 n c 2 n2
where b and c are constants related to oxygen complexation and the pore size distribution, respectively. Substituting Eqs. (13) and (14) into Eq. (10) gives p¼
n b0 þ ðb0 b1 þ b1 Þn þ ðb0 b2 b1 c1 Þn2 þ ðb0 b3 b1 c2 Þn3 þ
or, with further simpliﬁcation, n p¼ n0 þ n1 n þ n2 n2 þ n3 n3 þ
where n0 ¼ b0
n1 ¼ b0 b1 þ b1
ni ¼ b0 bi b1 ci1
ði P 2Þ
Eq. (16) is the equation used here to describe water adsorption isotherms on activated carbon. Values of n are only functions of system characteristic values (e.g., b0, b1, b, and c) and are constants for a speciﬁc adsorbate–adsorbent pair.
If molecules are independently exploring a surface without interacting with one another, then the system will be in the HenryÕs law region, irrespective of whether the surface is homogeneous or heterogeneous. The fraction of the time each molecule is associated with the surface will be proportional to the adsorbed-phase concentration, and the fraction of the time the molecule is not associate with the surface is proportional to the ﬂuid-phase concentration. Eq. (16) obeys HenryÕs law at low loadings. n0 is the HenryÕs law constant and signiﬁes the adsorption aﬃnity of water molecules for the primary sites. The model reduces to mathematical formulas equivalent to the DS-1 and DS-2 models as the polynomial denominator is truncated after n1n and n2n2 terms, respectively. 2.2. Multi-temperature extension A general method based on a series expansion is used here to develop a family of isotherms at diﬀerent temperatures from a known isotherm at one temperature by examining an important property—the isosteric heat of adsorption, qst, as given by the Clausius–Clapeyron type equation o ln p qst ¼ Rg T 2 ð20Þ oT n
where T, p, and Rg are the temperature, pressure, and gas constant. If qst is constant, the result of integrating Eq. (20) is well known (e.g., ). However, as observed in experiments over a wide temperature and loading range, qst is not a constant, but a function of temperature and loading. Before investigating this relationship, we recall the two-variable Taylor series expansion about a reference point (x0, y0) f ðx; yÞ ¼ f ðx0 ; y 0 Þ þ fx0 ðx x0 Þ þ fy0 ðy y 0 Þ 1h 0 2 f ðx x0 Þ þ 2fxy0 ðx x0 Þðy y 0 Þ þ 2! xx i 2 þfyy0 ðy y 0 Þ þ
where fx0 and fy0 are ﬁrst-order partial derivatives, and fxx0 , fxy0 and fyy0 are second-order partial derivatives, all evaluated at the reference point (x0, y0). Rearranging Eq. (21) gives 1 f ðx; yÞ ¼ f ðx0 ; y 0 Þ fx0 x0 fy0 y 0 þ fxx0 x20 2 1 0 2 0 0 þ fxy x0 y 0 þ fyy y 0 þ ðfx fxx0 x0 fxy0 y 0 Þx 2 1 1 0 0 þ ðfy fyy y 0 fxy0 x0 Þy þ fxx0 x2 þ fyy0 y 2 2 2 þ fxy0 xy þ
or f ðx; yÞ ¼ d0 þ d1 x þ d2 y þ d3 x2 þ d4 y 2 þ d5 xy þ
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where d0 signiﬁes the six leading terms on the right side of Eq. (22) and the remaining values of d correspond one to one in order with the remaining terms of Eq. (22). Values of d depend on the reference point selected and are independent of x and y. Thus, the isosteric heat of adsorption as a function of temperature and loading can be expressed as 1 1 1 1 st q n; ¼ d 0 þ d1 n þ d2 þ d3 n2 þ d4 2 þ d5 n þ T T T T ð24Þ For a Taylor series truncated at ﬁrst-order, substituting Eq. (24) with only the ﬁrst three terms on the right side into Eq. (20), and integrating at constant n gives ln p ¼
d0 þ d1 n d2 þ þ g ð nÞ Rg T 2Rg T 2
where g(n) is a temperature independent function of n. With a reference adsorption isotherm at a particular temperature Tref, given by pref = pref(n, Tref), the adsorption isotherm at any other temperature T can be obtained from d0 þ d1 n 1 1 d2 1 1 þ p ¼ pref ðn;T ref Þexp Rg T T ref 2Rg T 2 T 2ref ð26Þ
where the d parameters are constants independent of temperature and loading. Note that Eq. (26) incorporates the well known form for a constant qst given only by the term involving d0. The terms involving d1 and d2 extend this to allow for some dependence of qst on loading and temperature. Additional terms can be easily added.
determined by performing a nonlinear least-squares regression on experimental data with the objective function, e, deﬁned as M 2 X e¼ prexp;i prcal;i ð27Þ i¼1
where M is the total number of data points, and prexp;i and prcal;i are relative pressures measured experimentally and calculated by the model, respectively. The average percentage deviation, , was calculated as an index for model ﬁtting quality using M r p prcal;i 1 X exp;i 100 ¼ ð28Þ M i¼1 prexp;i Water adsorption isotherms and model descriptions are shown in Fig. 1 for BPL, PVDC, and UO3-1 activated carbons and in Fig. 2 for NC100, polymeric type E, and ACF activated carbons. The equilibrium data points for polymeric type E system are representations of original experiment results, which contain many more points . The model depicts HenryÕs law behavior at low loadings accurately and describes the full range of the isotherms very well. The model gives successful portrayals for water adsorption on various activated carbons with quite diﬀerent pore and surface structures using a small number of parameters. The model parameters and the average percentage deviations are given in Tables 1 and 2. Higher values of n0, the HenryÕs law constants, for the BPL and PVDC systems compared to the UO3-1 system reveal that the surfaces of BPL and PVDC are more oxidized on a per weight basis, which gives a higher aﬃnity for the water molecules.
3. Results and discussion
BPL (25°C) 25
3.1. Water adsorption on various activated carbons
20 n (mol/kg)
Six sets of adsorption equilibrium data for water on various activated carbons with large diﬀerences in surface area, pore size distribution, and surface chemical composition were chosen to test the proposed model. These water/carbon equilibrium data were measured on BPL at 25 C by Rudisill et al. , PVDC at 20 C by Bradley and Rand , UO3-1 at 20 C by Kraehenbuehl et al. , NC100 at 25 C by Cossarutto et al. , polymeric type E at 25 C by Terzyk et al. , and ACF at 100 C by Kaneko et al. . Shapes of the isotherms diﬀer greatly in adsorption characteristics such as the slope in the HenryÕs law region, the pressure and sharpness of the rise for ﬁlling of the micropores, and the saturation loading. The proposed model given by Eq. (16) was used to analyze all experimental data. The n parameters were
PVDC (20°C) UO3-1 (20°C) Model
pr Fig. 1. Water vapor isotherms for activated carbons: BPL, PVDC and UO3-1.
N. Qi, M.D. LeVan / Carbon 43 (2005) 2258–2263
NC100 (25°C) Polymeric Type E (25°C) ACF (100°C) Model
pr Fig. 2. Water vapor isotherms for activated carbons: NC100, polymeric type E and ACF.
Table 1 Model parameters for water vapor adsorption on activated carbons: BPL, PVDC and UO3-1
n0 n1 n2 n3 n4
BPL (25 C)
PVDC (20 C)
UO3-1 (20 C)
BPL (75 C)
1.780 0.1370 0.04559 1.866 · 103
4.308 0.2079 0.08219 2.640 · 103
0.1343 0.7161 0.01541 8.848 · 104 2.474 · 105 0.91
0.03227 0.03856 5.157 · 104 5.056 · 105
The water adsorption isotherm on BPL at 75 C was chosen as the reference isotherm. The proposed equilibrium model, Eq. (16), was ﬁtted to this reference isotherm as shown by the solid curve in Fig. 3. The model gives excellent description of the experimental data. Model parameters and the average percentage deviation for the reference isotherm description are listed in Table 1. With the reference isotherm at 75 C known in pressure explicit form pref(n, Tref), the multi-temperature model of Eq. (26) with the Taylor series truncated after the ﬁrst-order terms was applied to the water adsorption equilibrium data measured at 25 C, 50 C, 100 C, and 125 C. The result is shown by the dashed curves in Fig. 3. It can be seen that the proposed multi-temperature model describes the experimental data satisfactorily over this wide range of temperatures and loadings. The proposed multi-temperature model with the Taylor series truncated at higher orders, e.g., the secondorder (using ﬁve parameters), was used to analyze the same data in order to study model eﬃciency. No substantial improvement in model description accuracy was observed by the addition of more parameters. The model with the Taylor series truncated at the ﬁrst-order using only three parameters (d0 = 3.618 · 104, d1 = 9.876 · 101 and d2 = 3.335 · 106) gives good precision in data correlation ( = 12).
22 20 18
Table 2 Model parameters for water vapor adsorption on activated carbons: NC100, polymeric type E and ACF Polymeric type E (25 C)
ACF (100 C)
1.363 0.1786 0.02647 8.025 · 104
1.361 1.257 0.1814 0.01865 6.231 · 104 7.5
0.02364 0.01518 9.415 · 104 5.388 · 105
n0 n1 n2 n3 n4
NC100 (25 C)
25°C 50°C 75°C 100°C 125°C Model,Tref Model,T
12 10 8 6 4
3.2. Multi-temperature water adsorption on BPL carbon Water adsorption equilibrium data on BPL activated carbon at 25 C, 50 C, 75 C, 100 C, and 125 C, measured by Rudisill et al. , were used to demonstrate the performance of the proposed multi-temperature model. The d parameters were determined by nonlinear regression . The average percentage deviation, , was calculated using Eq. (28).
2 0 0.0
pr Fig. 3. Water vapor isotherms on BPL activated carbon. For clarity, pr, relative pressures (p/psat), for each isotherm are shifted by 0.1 consecutively from the 25 C isotherm. The solid curve is the model description of the Tref = 75 C reference isotherm. Dashed curves are model descriptions at other temperatures.
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4. Conclusions A new equation has been developed to describe water adsorption equilibria on activated carbon. The model is consistent with HenryÕs law at low loading, is mathematically simple, and is explicit in pressure. It depicts the full range of isotherms with high accuracy using a small number of parameters. The model was used to analyze water adsorption equilibrium data on BPL, PVDC, UO3-1, NC100, polymeric type E, and ACF activated carbons, which have large diﬀerences in surface area, surface chemical properties, and pore structure. A new versatile model has been proposed to generate adsorption isotherms at other temperatures from a reference isotherm known at a speciﬁc temperature. The model description of the multi-temperature water adsorption equilibria on BPL activated carbon is in good agreement with experimental data using very few parameters. Acknowledgement We are grateful to the National Aeronautics and Space Administration for the support of this research under award NCC2-1127.
    
  
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