Water vapor adsorption isotherm expressions based on capillary condensation

Separation and Purification Technology 116 (2013) 95–100

Contents lists available at SciVerse ScienceDirect

Separation and Purification Technology journal homepage: www.elsevier.com/locate/seppur

Water vapor adsorption isotherm expressions based on capillary condensation Xiang Jun Liu a,⇑, Yun Fei Shi a, Mohammad Ali Kalbassi b, Richard Underwood c, Ying Shu Liu a a

School of Mechanical Engineering, University of Science & Technology Beijing, Beijing 100083, China Air Products PLC., Walton-on-Thames, Surrey KT12 4RZ, UK c Air Products and Chemicals, Inc., Allentown, PA 18195-1501, USA b

a r t i c l e

i n f o

Article history: Received 30 October 2012 Received in revised form 9 May 2013 Accepted 10 May 2013 Available online 20 May 2013 Keywords: Water vapor adsorption Capillary condensation Isotherm model

a b s t r a c t Four comprehensive isotherm expressions are proposed for various water vapor adsorption mechanisms. Water vapor adsorption in micropores and on macropore walls is described using the Langmuir and Freundlich models, respectively. Water vapor condensation in the pore core volume of macropores is modeled by assuming that the effective pore size follows a Gaussian and uniform distribution. These expressions are used to fit water adsorption data on various kinds of alumina. The results show that the proposed water vapor adsorption isotherm expressions fit the water adsorption data well. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Removal of water vapor from air is receiving increasing attention because of its emerging applications in air reparation, purification, and environmental protection [1,2]. Accurate and reliable models for characterizing water adsorption isotherms are required to improve water removal technologies and processes. Numerous studies have investigated water vapor adsorption on various porous adsorbents [2–5]. Isotherms at low relative humidity can be well-correlated with the commonly used monolayer models, such as Langmuir, Freundlich, and load ratio correlation models [6]. With increased relative humidity, monolayer adsorption is predominated by capillary condensation, and isotherms of water vapor on porous adsorbents over the full partial pressure range are either Type IV or Type II according to the Brunauer classification [7]. A mathematical description for water adsorption isotherms over the full partial pressure range is much more challenging. Water adsorption isotherms can be characterized over the full partial pressure range by extending the description of monolayer adsorption to multi-layer adsorption. Aranovich and Donohue [8] proposed a general expression for multi-layer adsorption (AD models) by extending monolayer models to multilayer models using an accessorial function that describes the adsorption in the second and subsequent layers. Zhang and Wang [9] proposed an extended Langmuir (EL) model by adding a third parameter in the normal ⇑ Corresponding author. Tel.: +86 10 62333792. E-mail address: [email protected] (X.J. Liu). 1383-5866/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.seppur.2013.05.020

Langmuir equation to describe water adsorption on alumina and 13X. Kim et al. [10] used the n-layer BET equation to correlate the adsorption equilibrium data of water vapor on alumina, zeolite 13X, and on the composite of zeolite and activated carbon. Desai et al. [3] used a combined Langmuir and BET model to correlate the adsorption equilibrium data for water vapor on several types of activated alumina. The adsorption potential theory is an alternative method to characterize water adsorption isotherms over the full partial pressure range. Kotoh et al. [4] proposed a dual mechanism adsorption potential (DMAP) model by extending the adsorption potential theory to situations with more than one adsorption mechanism. The performances of the aforementioned models are adsorbentdependent. They may work well for certain types of adsorbent but may also introduce several deviations for other kinds of adsorbents. Industrial water adsorbents, such as alumina and silica gel, are pellets with abundant porous structures. Capillary condensation of water vapor may occur at low H2O partial pressures. Moore and Serbezov [11] reported that capillary condensation is significant when the relative humidity is higher than 25% for F-200 activated alumina. Carniglia [12] divided A-201 alumina/H2O equilibrium isotherms into three zones according to H2O loading denoted by k and suggested that the capillary condensation should be considered at H2O loading k > 9. Machin and Stuckless [13] reported that capillary condensation is significant at relative pressures near 0.3 for Davison grade 03 silica gel. Capillary condensation plays a vital role in water vapor adsorption. A comprehensive model to describe water vapor adsorption on porous adsorbents should include the

96

X.J. Liu et al. / Separation and Purification Technology 116 (2013) 95–100

Nomenclature b n N p ps q q12 q3 qcm qm r r0 rk rm rp R Smac

adsorption equilibrium constant in Langmuir equation, 1/kPa adsorption equilibrium constant in Freundlich equation number of experimental points partial pressure, kPa saturation vapor pressure for a plane liquid surface, kPa adsorbed amount per unit weight of sorbent, mol/kg amount adsorbed in micropores and on macropore walls, mol/kg amount adsorbed due to capillary condensation, mol/kg limit of the amount adsorbed due to capillary condensation, mol/kg monolayer adsorbed amount, mol/kg pore radius, m expected pore radius in Gaussian distribution, m Kelvin pore radius, m maximal pore radius in uniform distribution, m real pore radius, m general gas constant, J/mol K specific surface area of the macropores, m2/kg

description of capillary condensation occurring in the adsorbent pores. In this study, we divide the total pore volume of a porous adsorbent into three parts as follows: the pore volume of very small pores (micropores) where capillary condensation may not take place [14], the region next to the macropore walls, and the remnant pore core volume of the macropores. The Langmuir and Freundlich models are used to describe both the micropore adsorption and the surface adsorption on the macropore walls. Water vapor condensation in the pore core volume of macropores is described based on the Kelvin theory without considering the interferences between vapor condensation and surface adsorption. Consequently, four expressions for water vapor adsorption are derived and their applications are further developed. 2. Basic theory and assumptions Surface tension weakens the saturation vapor pressure in a small pore. Therefore, smaller pores in a porous adsorbent may be filled with liquid sorbate with lower pressure than the saturation vapor for pure plane liquid surface at the same temperature. This phenomenon is the so-called capillary condensation, which plays a vital role in water vapor adsorption. Kelvin proposed the following equation, which is widely used to illustrate the relations between the pore radius and the corresponding equilibrium vapor pressure [14,15]:

lnðp=ps Þ ¼ 

cv m RTr

ð1Þ

where ps is the saturation vapor pressure for a plane liquid surface, r is the pore radius, R is the universal gas constant (J/mol K), and vm and c are the liquid molecular volume (m3/mol) and surface tension (N/m), respectively. However, the size of pores in real porous adsorbents varies widely. Capillary condensation may not occur in very small pores because the concept of a liquid meniscus is no longer meaningful [14]. These pores are called micropores, and the volume of micropores in a porous adsorbent is denoted as V1 and expressed in m3/kg. Water vapor adsorption in micropores is usually interpreted by the ‘‘pore filling’’ mechanism [16]. The relations between the pore radius and the corresponding equilibrium vapor pressure are described in Eq. (1). Interestingly,

t T

thickness of the adsorbed layer, m temperature, K molar volume of sorbate, m3/mol total pore volume per unit of adsorbent, m3/kg volume of micropores in an adsorbent, m3/kg volume of the adsorbed layer on macropore walls, m3/ kg remnant pore core volume of the macropores, m3/kg

vm V0 V1 V2 V3

Greek letters c surface tension of liquid sorbate, N/m g modification factor in L–U and F–U models l1 parameter corresponding to adsorption potential, J/mol l2 parameter in uniform distribution models, J/kg r variance of pore radius in Gaussian distribution Subscripts cal calculated exp experimental

surface adsorption occurs on the macropore walls sooner than capillary condensation such that the effective pore radius becomes smaller prior to the occurrence of capillary condensation [17]. The effective pore radius available for condensation is related to the true radius as follows:

rk ¼ rp  t

ð2Þ

where t is the thickness of the adsorbed layer. The volume of this adsorbed layer (V2) is calculated as follows:

V 2 ¼ Smac  t

ð3Þ 2

where Smac is the specific surface area of the macropores (m /kg). The remnant pore core volume of the macropores (V3) can be calculated by subtracting the volume of the adsorbed layer on the pore walls. The total pore volume of a porous adsorbent with three parts can be written as follows:

V0 ¼ V1 þ V2 þ V3

ð4Þ

Water vapor adsorption in micropores and on macropore walls can be described by existing classical models [6]. We assume that the combined contributions from these two parts can be expressed by one model and that the surface adsorption and the vapor condensation of the macropores have no interferences. Therefore, a comprehensive adsorption isotherm model considering the joint contributions from these three parts can be formulated as follows:

q ¼ q12 ðpÞ þ q3 ðpÞ

ð5Þ

Water vapor adsorption in micropores and on macropore walls is dominant at low pressures, whereas capillary condensation is more significant at higher relative pressure. An expression to describe the amount adsorbed due to capillary condensation is the key in Eq. (5). The effective radius of the remnant pore volume (V3) is calculated as follows: rk ¼ r p  t. The amount adsorbed due to capillary condensation at vapor pressure (p) can be calculated by the distribution of the effective pore size of an adsorbent [f(rk)] as follows:

q3 ¼

1

vm

Z

rk

f ðr k Þdr k

ð6Þ

0

where q3 is the adsorbed amount per unit of adsorbent (mol/kg). rk can be obtained from Eq. (1) as follows:

97

X.J. Liu et al. / Separation and Purification Technology 116 (2013) 95–100

cv m

rk ¼ 

ð7Þ

RT lnðp=ps Þ

The variation of dV3/drk with rk is described by f(rk) in Eq. (6). The total pore core volume per unit of adsorbent is calculated as follows:

V3 ¼

Z

1

f ðr k Þdr k

ð8Þ

We name the above equation as the L–G model. Adsorption in micropores and on macropore walls can also be characterized by the Freundlich equation. Consequently, the F–G model can be obtained as follows:

q ¼ qm ðp=ps Þ1=n þ qcm erf



l1



RT lnðps =pÞ

ð16Þ

0

3.2. Fitting results based on four sets of water adsorption data 3. Expressions based on Gaussian distribution and their applications 3.1. Langmuir–Gaussian (L–G) model and Freundlich–Gaussian (F–G) model The effective pore radius of the macropore core volume (V3) for a practical water vapor adsorbent can be modeled by certain distributions. For certain adsorbents, we assume that the distribution of the effective pore radius can be approximately characterized by the Gaussian function as follows [18]:



p1ffiffiffiffi e r 2p

f ðr k Þ ¼ V 3 R1 0



p1ffiffiffiffi e





ffi

rk r0 p 2r

2 ð9Þ

2

ffi

rk r0 p 2r

dr k

r 2p

The amount adsorbed due to capillary condensation can be calculated as follows:

q3 ¼

¼

1

vm

Z

rk

f ðr k Þdr k ¼

0

V 3 erf



vm

r k r0 pffiffi 2r



V3

R rk

vm R 

0

1 0

r p ffiffi0 2r

 erf   1  erf prffiffi20r



p1ffiffiffiffi e r 2p



p1ffiffiffiffi e



rk r0 p 2r

ffi





ffi

r k r0 p 2r

r 2p

2 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X SE ¼ ðqcal  qexp Þ2 N N   1 X qcal  qexp  Rave ¼   N N qexp 

dr k dr k ð10Þ

For some adsorbents, we further assume that r0 ? 0. Eq. (10) can be reduced as follows:

q3 ¼

  rk erf pffiffiffi vm 2r V3

ð11Þ

We obtain the following equation by substituting Eq. (7) into Eq. (11):

q3 ¼

  1 cv m erf  pffiffiffi vm 2r RT lnðp=ps Þ V3

ð12Þ

A comprehensive adsorption isotherm model over the full partial pressure range can be obtained by substituting Eq. (12) into Eq. (5) as follows:

q ¼ q12 ðpÞ þ

  1 cv m erf  pffiffiffi vm 2r RT lnðp=ps Þ V3

ð13Þ

Eq. (13) can be further simplified as follows:

q ¼ q12 ðpÞ þ qcm erf



l1



RT lnðps =pÞ

ð14Þ

pffiffiffi where l1 ¼ cv m = 2r and qcm ¼ V 3 =v m . Eq. (14) has two additional parameters compared with the expression for q12 ðpÞ. Eq. (14) can be expressed by applying the commonly used Langmuir equation to describe the combined contributions from the adsorption in micropores and on macropore walls as follows:

q¼

  qm bp l1 þ qcm erf 1 þ bp RT lnðps =pÞ

The L–G and F–G models are used to fit water adsorption data using four kinds of alumina [2–5]. Fig. 1a and b show the comparisons between the calculated adsorption isotherms and the experimental data. The parameters used in the models that best fit the experimental data are listed in Table 1. Fig. 1a and b show that the experimental data for AA300, Nakarai, and F-200 alumina are well-fitted by the L–G and F–G models. The existing DMAP, nBET, EL, and AD-Langmuir models are feasible equations to describe water adsorption isotherms over the full partial pressure range. The three-parameter models include nBET, EL, and AD-Langmuir models. DMAP is a four-parameter model and its expression is similar to the proposed L–G and F–G equations. These four models are also used to fit the same water adsorption data to compare with the proposed L–G and F–G models. The fitting results of the DMAP, nBET, EL, and AD-Langmuir models are shown in Fig. 1c–f. The comparisons of the six models in terms of the standard error (SE) and average relative error (Rave) calculated by Eqs. (17) and (18) are summarized in Table 2.

ð15Þ

ð17Þ

ð18Þ

Table 2 and Fig. 1a–f show that the performances of each model for the different kinds of alumina absorbents are varied. Among the three three-parameter models (nBET, EL, and AD-Langmuir models), the EL model shows the smallest deviation except for type A1 alumina. However, all the four-parameter models (DMAP, L–G, and F–G models) show better performance than the threeparameter models. Furthermore, all these models, except for the AD-Langmuir model, show the largest deviations for type A1 alumina. The adsorption isotherm curve of A1 alumina is different from the other three alumina absorbents. H2O adsorption capacity rapidly increased when the relative humidity exceeded 70%, which indicates the significant occurrence of capillary condensation. Although the L–G and F–G models show very good ability to fit the experimental data of AA300, Nakarai, and F-200 alumina, some deviations for A1 alumina still exist. The Rave values of the L–G and F–G models for A1 alumina are 7.14% and 4.87%, respectively. Most deviations occur when the relative humidity is within 0.2–0.6. These deviations might be caused either by neglecting the interferences between the vapor condensation and the surface adsorption, by the differences between the real distributions of the effective pore size and the Gaussian assumption, or both. 4. Expressions based on uniform distribution and their applications 4.1. Langmuir–Uniform (L–U) Model and Freundlich–Uniform (F–U) Model Eq. (19) is calculated by assuming that the effective pore size is uniformly distributed as follows:

98

X.J. Liu et al. / Separation and Purification Technology 116 (2013) 95–100

Fig. 1. Fitting results of six models on various types of alumina: (a) fitting results of L–G model, (b) fitting results of F–G model, (c) fitting results of DMAP model, (d) fitting results of nBET model, (e) fitting results of EL model and (f) Fitting results of AD-L model.

f ðr k Þ ¼ V 3 =r m ; 0 < r < r m

ð19Þ

The adsorbed amount due to capillary condensation is calculated by substituting Eq. (19) to Eq. (6) as follows:

q3 ¼

1

vm

Z 0

rk

f ðr k Þdrk ¼

V 3 rk

v m rm

¼

cV 3 =r m RT lnðp=ps Þ

ð20Þ

The comprehensive adsorption isotherm model over the full partial pressure range can be described by combining the expression for adsorption in the micropores and on the macropore walls as follows:

q ¼ q12 ðpÞ þ

cV 3 =r m RT lnðp=ps Þ

ð21Þ

99

X.J. Liu et al. / Separation and Purification Technology 116 (2013) 95–100 Table 1 Fitting parameters of L–G and F–G models on various types of alumina. Adsorbent

Model

A1

LG FG LG FG LG FG LG FG

AA300 Nakarai F-200

Table 3 Fitting parameters and errors of L–U and F–U models on various types of alumina.

Parameter

Parameter

qm (mol/kg)

bps or n

qcm (mol/kg)

l1/RT

1.2918 1.8996 2.1435 5.8238 1.3954 3.5595 3.5308 7.4370

106.95 4.8617 45.147 2.5802 50.183 2.8472 32.941 2.9997

82.447 197.89 17.421 19.726 14.236 15.650 22.617 26.118

0.01203 0.004391 0.3037 0.2188 0.2555 0.1934 0.2398 0.1514

A1 AA300 Nakarai F-200

L–U F–U L–U F–U L–U F–U L–U F–U

qm (mol/kg)

n or bps

g

l2/RT

1.8698 1.9000 2.2007 7.0009 0.8302 1.9730 1.5266 4.1968

18.425 4.8608 31.585 2.4533 23.541 2.7470 30.913 3.0768

1.000 1.000 0.8662 0.9615 0.7811 0.8009 0.7276 0.7752

0.9753 0.9710 5.763 2.898 5.436 4.657 9.574 7.167

SE

Rave (%)

0.5627 0.5458 0.2251 0.2067 0.2997 0.2925 0.1314 0.2052

7.07 4.86 3.21 3.97 3.44 3.74 0.464 0.603

Table 2 Standard errors and average relative errors of fitting water vapor adsorption data with six models for various types of alumina.

A1 AA300 Nakarai F-200

SE Rave SE Rave SE Rave SE Rave

(%) (%) (%) (%)

L–G

F–G

DMAP

n-BET

EL

AD-L

0.3407 7.14 0.1602 2.35 0.3627 3.57 1.059 2.85

0.5527 4.87 0.1651 3.74 0.1521 3.11 0.2495 1.30

1.2919 7.30 0.1547 2.60 0.1772 2.70 0.1822 1.10

2.648 10.7 0.3224 9.82 0.3372 9.07 0.3778 2.31

2.934 10.2 0.2872 2.79 0.1913 3.89 0.4172 1.88

0.7393 7.51 0.6997 11.1 2.269 14.8 2.531 8.10

Eq. (21) can be re-formatted as follows:

q ¼ q12 ðpÞ þ

l2 RT lnðps =pÞ

ð22Þ

where l2 ¼ cV 3 =rm . This equation is based on the assumption that the effective pore size of the adsorbent is uniformly distributed within the range 0 to rm. The corresponding equilibrium vapor pressure (pm) for the pores at the maximum radius (rm) might be less than ps in some cases. The model can further be modified as follows:

q ¼ q12 ðpÞ þ

l2 RT lnðps =gpÞ

ð23Þ

The L–U and F–U models can be obtained by substituting the Langmuir or Freundlich equations for the q12 ðpÞ term as follows:

q¼

qm bp l2 þ 1 þ bp RT lnðps =gpÞ

q ¼ qm ðp=ps Þ1=n þ

l2 RT lnðps =gpÞ

ð24Þ

ð25Þ

4.2. Fitting results based on four sets of water adsorption data The L–U and F–U models are used to fit the same sets of water adsorption data used in Section 3.2. The best-fit parameters and results are shown in Table 3 and Fig. 2a and b. Fig. 2 and Table 3 show that all the four sets of water adsorption data are well-correlated with the L–U and F–U models. The main difference between these uniform-type models and the Gaussiantype models is the treatment on maximum loading. The Rave values of the L–U and F–U models for A1 alumina are 7.07% and 4.86%, respectively. Rave decreases slightly, but the deviations occur mainly when the relative humidity is within 0–0.2, which is the low range for relative humidity. Rave of the L–U and F–U models for A1 alumina when the relative humidity ranges from 0.2 to 1.0 is 2.86% and 4.03%, respectively. The fitting accuracy for type A1 alumina is improved when the relative humidity is within the

Fig. 2. Fitting results of F–U and L–U models on various types of alumina: (a) fitting results of L–U model (b) fitting results of F–U model.

middle and high ranges, which indicates that the L–U and F–U models are more suitable than the L–G and F–G models to describe water vapor adsorption on A1 alumina under middle and high relative humidity. 5. Conclusions Four comprehensive isotherm expressions are proposed to account for various water vapor adsorption mechanisms. These equations are used to fit the water adsorption data of four kinds of

100

X.J. Liu et al. / Separation and Purification Technology 116 (2013) 95–100

alumina. The fitting results show that the proposed models fit well with the practical water adsorption data. The fitting accuracy can be further improved by using a more suitable expression for the effective pore size distribution. Acknowledgements This work was mainly supported by the collaboration project between Air Products and Chemicals, Inc. and University of Science & Technology Beijing. Appendix A. Expressions of the DMAP, nBET, EL, and AD-Langmuir models DMAP model [4]: q ¼ qs1 expð EA1 Þ þ ðqs2  qs1 Þ expð EA2 Þ nBET model [10]: EL model [9]:

q qm

q qm

BET ðp=ps Þ ¼ C1ðp=p Þ s

1ðnþ1Þðp=ps Þn þnðp=ps Þnþ1 1þðC BET 1Þðp=ps ÞC BET ðp=ps Þnþ1

bp ¼ 1þbp enp=ps

AD-Langmuir model [8]:

q qm

bp ¼ ð1þbpÞð1p=p Þd s

References [1] Y. Wang, M.D. LeVan, Adsorption equilibrium of carbon dioxide and water vapor on zeolites 5A and 13X and silica gel: pure components, J. Chem. Eng. Data 54 (2009) 2839–2844. [2] C. Nédez, J.P. Boitiaux, C.J. Cameron, B. Didillon, Optimization of the textural characteristics of an alumina to capture contaminants in natural gas, Langmuir 12 (1996) 3927–3931.

[3] R. Desai, M. Hussain, D.M. Ruthven, Adsorption of water vapour on activated alumina. I – Equilibrium behavior, Can. J. Chem. Eng. 70 (1992) 699–706. [4] K. Kotoh, M. Enoeda, T. Matsui, M. Nishikawa, A multilayer model for adsorption of water on activated alumina in relation to adsorption potential, J. Chem. Eng. Japan 26 (1993) 355–360. [5] A. Serbezov, Adsorption equilibrium of water vapor on F-200 activated alumina, J. Chem. Eng. Data 48 (2003) 421–425. [6] R.T. Yang, Gas Separation by Adsorption Processes, Butterworths, Boston, 1987. [7] S. Brunauer, L.S. Deming, W.E. Deming, E. Teller, On a theory of the van der waals adsorption of gases, J. Am. Chem. Soc. 62 (1940) 1723–1732. [8] G.L. Aranovich, M.D. Donohue, A new approach to analysis of multilayer adsorption, J. Colloid Interface Sci. 173 (1995) 515–520. [9] P.K. Zhang, L. Wang, Extended Langmuir equation for correlating multilayer adsorption equilibrium data, Separat. Purif. Technol. 70 (2010) 367–371. [10] J. Kim, C. Lee, W. Kim, J. Lee, et al., Adsorption equilibria of water vapor on alumina, zeolite 13x, and a zeolite x/activated carbon composite, J. Chem. Eng. Data 48 (2003) 137–141. [11] A. Serbezov, J.D. Moore, Y. Wu, Adsorption equilibrium of water vapor on Selexsorb-CDX commercial activated alumina adsorbent, J. Chem. Eng. Data 56 (2011) 1762–1769. [12] S.C. Carniglia, W.L. Ping, Alumina desiccant isosteres: thermodynamics of desiccant equilibria, Ind. Eng. Chem. Res. 28 (1989) 1025–1030. [13] W.D. Machin, J.T. Stuckless, Capillary-condensed water in silica gel, J. Chem. Soc. Faraday Trans. 81 (1985) 597–600. [14] S.J. Gregg, K.S.W. Sing, Adsorption, Surface Area and Porosity, second ed., Academic Press, New York, 1982. [15] W. Thomson, The equilibrium of vapour at a curved surface of liquid, London, Edin. Dublin Philos. Mag. 42 (1871) 448–452. [16] M.M. Dubinin, Generalization of the theory of volume filling of micropores to nonhomogeneous microporous structures, Carbon 23 (1985) 373–380. [17] D.D. Do, Adsorption Analysis: Equilibria and Kinetics, Imperial College Press, London, 1998. [18] E.T. Jaynes, Probability Theory: The Logic of Science, Cambridge University Press, New York, 2003.