2204
Langmuir 2009, 25, 2204-2211
Theoretical Insight of Physical Adsorption for a Single-Component Adsorbent + Adsorbate System: I. Thermodynamic Property Surfaces Anutosh Chakraborty,† Bidyut Baran Saha,*,†,‡ Kim Choon Ng,† Shigeru Koyama,‡ and Kandadai Srinivasan§ Department of Mechanical Engineering, National UniVersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Interdisciplinary Graduate School of Engineering Sciences, Kyushu UniVersity, 6-1 Kasuga Koen, Kasuga Shi, Fukuoka 816-8580, Japan, and Frigrite Limited, 27 Grange Road, Cheltenham, Victoria 3192, Australia ReceiVed October 7, 2008. ReVised Manuscript ReceiVed NoVember 12, 2008 Thermodynamic property surfaces for a single-component adsorbent + adsorbate system are derived and developed from the viewpoint of classical thermodynamics, thermodynamic requirements of chemical equilibrium, Gibbs law, and Maxwell relations. They enable us to compute the entropy and enthalpy of the adsorbed phase, the isosteric heat of adsorption, specific heat capacity, and the adsorbed phase volume thoroughly. These equations are very simple and easy to handle for calculating the energetic performances of any adsorption system. We have shown here that the derived thermodynamic formulations fill up the information gap with respect to the state of adsorbed phase to dispel the confusion as to what is the actual state of the adsorbed phase. We have also discussed and established the temperature-entropy diagrams of (i) CaCl2-in-silica gel + water system for cooling applications, and (ii) activated carbon (Maxsorb III) + methane system for gas storage.
1. Introduction The potency of an adsorbent + adsorbate system is inter alia determined by its adsorption isotherms, thermodynamic property surfaces of energy and entropy, heat of adsorption, specific heat capacity, and adsorption kinetics. When gas molecules come into contact with a solid adsorbent surface, the gas molecules impinge against the surfaces and they behave nonideally as some molecules are captured by the field of force of the surface atoms named the van der Waals force while the uncaptured molecules depart from the surface. If the surface forces are relatively intense, molecules leaving the adsorbent surface are negligible, and the porous surface of the solid adsorbent would be covered with a layer of molecules. The adsorption is considered to be monolayer, and such a process occurs over wide ranges of pressures and temperatures. Adsorption isotherms and pore-size distribution of any singlecomponent adsorbent + adsorbate system are traditional methods for characterizing adsorbents and the nature of adsorbates, but characterization is incomplete without information on the thermodynamic properties of energy, enthalpy, and entropy of the adsorbent + adsorbate system, the specific heat capacity, the adsorbed phase volume, and the enthalpy of adsorption. The knowledge of thermodynamic property surfaces of an adsorbent + adsorbate system is important, because it enables the adsorption process to be analyzed. The early theoretical models for physical adsorption systems are attributed largely to the works of Hill1-4 and Everett.5-8 A * Corresponding author. E-mail:
[email protected]. † National University of Singapore. ‡ Kyushu University. § Frigrite Limited.
(1) Hill, T. L. J. Chem. Phys. 1949, 17, 520. (2) Hill, T. L. J. Chem. Phys. 1950, 18, 246. (3) Hill, T. L. Trans. Faraday Soc. 1951, 47, 376. (4) Hill, T. L.; Emmett, P. H.; Joyner, L. G. J. Am. Chem. Soc. 1951, 73, 5102. (5) Everett, D. H. Trans. Faraday Soc. 1950, 46, 453. (6) Everett, D. H. Trans. Faraday Soc. 1950, 46, 943. (7) Everett, D. H. Discuss. Faraday Soc. 1965, 40, 177. (8) Everett, D. H. Trans. Faraday Soc. 1950, 46, 957.
more general approach to adsorption with an exchange of heat and work has been developed by Guggenheim.9 These important contributions to the thermodynamics of physical adsorption by Everett, Hill, and Drain10 are summarized by Young and Crowell.11 The thermodynamics of physical adsorption on solid adsorbents is based on the concept of an inert adsorbent and the introduction of two new variables: surface area and spreading pressure, as suggested by Young and Crowell.11 This surface and statistical thermodynamics approach is standard in adsorption12-15 but the problem with this approach is that it requires a series of assumptions such as inert adsorbent, pure perfect gas, negligible volume of adsorbed phase, and is not easy to handle. In 2002, Myers16,17 developed the thermodynamic functions such as Gibbs free energy, enthalpy, and entropy on the basis of isothermal condition and concluded that the heat capacity of adsorbate is equal to its heat capacity in the perfect gas state. Chua et al.18 developed the thermodynamic property fields of an adsorbent-adsorbate system, which is based on the works of Feuerecker et al. originally, analyzed for an absorption (liquid-vapor) system.19 This method yields the energy and entropy balances for an adsorbent-adsorbate system as a function of pressure (P), temperature (T), and the amount of adsorbate (ma). (9) Guggenheim, E. A. Trans. Faraday Soc. 1940, 35, 397. (10) Drain, L. E.; Morrison, J. A. Trans. Faraday Soc. 1952, 48, 316. (11) Young, D. M.; Crowell, A. D. Physical Adsorption of Gases; Butterworths: London, 1952. (12) Bakaev, V. A. Surf. Sci. 2004, 564, 108. (13) Ramirez-Pastor, A. J.; Pereyra, V. D.; Riccardo, J. L. 1999, 15, 5707. (14) Grosman, A.; Ortega, C. Phys. ReV. B 2008, 78, 085433. (15) Ramirez-Pastor, A. J.; Eggarter, T. P.; Pereyra, V. D.; Riccardo, J. L. Phys. ReV. B 1999, 59, 11–027. (16) Myers, A. L. AIChE J. 2002, 48, 145. (17) Myers, A. L.; Monson, P. A. Langmuir 2002, 18, 10261. (18) Chua, H. T.; Ng, K. C.; Chakraborty, A.; Oo, N. M. Langmuir 2003, 19, 2254. (19) Feuerecker, G.; Scharfe, J.; Greiter, I.; Frank, C.; Alefeld, G. Measurement of thermophysical properties of aqueous LiBr-solutions at high temperatures and concentrations. Proceedings of the International Absorption Heat Pump Conference, 1994; pp 493-499.
10.1021/la803289p CCC: $40.75 2009 American Chemical Society Published on Web 01/13/2009
Thermodynamics of Adsorbent + Adsorbate Systems
Langmuir, Vol. 25, No. 4, 2009 2205
The main objectives of this paper are (i) to describe the mathematical formulations of various extensive thermodynamic properties such as enthalpy, entropy, internal energy of a singlecomponent adsorbent + adsorbate system from the rigor of classical thermodynamics; (ii) to evaluate the isosteric heat of adsorption (Qst) as a function of pressure, temperature, and the amount of adsorbate; (iii) to formulate the specific heat capacity of adsorbed phase (cpa); and (iv) to evaluate the adsorbed phase volume (Va). In this paper, we also develop the temperature-entropy maps for CaCl2-in-silica gel + water and activated carbon (Maxsorb III) + methane systems using the proposed simple thermodynamic formulations and experimentally measured adsorption isotherms and isosteric heat of adsorption data20-23 for practical interests and applications of cooling and gas storage.
dµa ) -
( ) ∂sa ∂x
At equilibrium, µa = µg.
24
( ) ∂sa ∂x
dT -
T,P
( ) ∂Va ∂x
dss )
( )
( )
( )
cp,s ∂ss ∂ss ∂Vs dT + dP ) dP dT ∂T P ∂P T T ∂T P
and the total differential of entropy in the adsorbed phase is
dsa )
( ) ∂sa ∂T
dT +
P,x
( ) ∂sa ∂P
dP +
T,x
( ) ∂sa ∂x
( )
dx
(2)
T,P
(3)
So we can write dµa ) dµg, or
dP -
T,P
( ) ∂µa ∂x
dx ) sgdT - VgdP
T,P
At constant amount of adsorbate,
( ) ∂sa ∂x
( ) ( ) ( ) ∂µa ∂x
dT ) sgdT - VgdP +
T,P
( ) ∂sa ∂x
T,P
)0
∂Va ∂x
dP
T,P
∂sa ∂x
2. Theoretical Insight of Physical Adsorption In a physisorption of an adsorbate onto micro- and macropore surfaces of an adsorbent, the adsorbed phase is held near the pores by the existence of van der Waals forces. For such a system, the extensive properties such as the entropy (s), enthalpy (h), and internal energy (u) are described in terms of pressure (P), temperature (T), and the amount of adsorbate uptake (x) where the effects of the isosteric heat of adsorption and the specific heat capacity are taken into account. Here x ) ma/Ms; Ms defines the mass of solid adsorbent. I. Derivation of Total Differential for Physical Adsorption Extensive Properties. Entropy (s). The total differential of entropy in the adsorbent-adsorbate system is the summation of its solid and adsorbed phases, i.e., ds ) dss + dsa. Subscripts “s” and “a” indicate the solid adsorbent and adsorbate, respectively. The extensive entropy of solid phase as a function of P and T is given by
( )
∂Va ∂µa dP + T,P ∂x T,P ∂x dµg ) -sgdT + VgdP dT +
T,P
T,P
) sg - Vg
) sg - (Vg - Va)
dP dP + Va dT dT
dP dT
(4)
The total differential entropy of the adsorbent-adsorbate system can be written by
ds )
(
) ( )
( )
cp,s cp,a ∂Vs ∂Va dT dP + T T ∂T P ∂T
dP +
p,x
dx (5) {s - (V - V ) dP dT } g
g
a
Enthalpy (h). Total differential of extensive specific enthalpy (dh) is written as
dh ) dhs + dha
(6)
where dhs is the total differential of the solid phase (adsorbent) enthalpy and dha is that of adsorbate phase enthalpy. Using the relation between derivatives, the total differential of the solid phase enthalpy is written as
dhs )
( )
( )
{( ) }
∂hs ∂hs ∂ss dT + dP ) cp,sdT + T ∂T P ∂P T ∂P
T
+ Vs dP
Using Maxwell‘s relationship, we have
dx
P,T
( ) ( ) { ( )}
(1)
where the first term of the right-hand side refers to the partial change of entropy with respect to temperature at constant pressure and the amount of adsorbate uptake and is written as (∂sa/∂T)P,x ≈ cp,a/T; the second term represents the change of entropy as a function of pressure at constant temperature and the amount of adsorbate in the adsorbed phase, and, using Maxwell relationship, this can be expressed as (∂sa/∂P)T,x ) -(∂Va/∂T)p,x; and the third term is defined as the change of entropy with respect to the amount of adsorbate. Expression of Partial Entropy of Adsorption (∂sa/∂x)P,T. It is well-known that the chemical potential of an adsorbed phase µa[ ) (∂u/∂x)s,V,µg]is a result for the partial change in internal energy (u) with the amount of adsorbate uptake (x). The total differentials of chemical potential in the adsorbed and gaseous phases are24 (20) Aristov, Y. I.; Glaznev, I. S.; Freni, A.; Restuccia, G. Chem. Eng. Sci. 2006, 61, 1453. (21) Tokarev, M. M.; Okunev, B. N.; Safonov, M. S.; Kheifets, L. I.; Aristov, Y. I. Russ. J. Phys. Chem. 2005, 79, 1490. (22) Saha, B. B.; Koyama, S.; El-Sharkawy, I. I.; Habib, K.; Srinivasan, K.; Dutta, P. J. Chem. Eng. Data 2007, 52, 2419. (23) Himeno, S.; Komatsu, T.; Fujita, S. J. Chem. Eng. Data 2005, 50, 369. (24) Chi, T. Adsorption Calculation and Modeling; Series in Chemical Engineering; Butterworh-Heinemann: Boston, 1994.
∂ss ∂P
)-
T
∂Vs ∂T
P
∂Vs dhs ) cp,sdT + Vs - T ∂T
P
dP
(7)
On the other hand, the total differential of the adsorbed phase enthalpy is given by
dha )
( ) ∂ha ∂T
dT +
P,x
( ) ∂ha ∂P
dP +
T,x
( ) ∂ha ∂x
dx
P,T
(8)
Using the Gibbs equation, the Maxwell relation, and the relation between derivatives, the second term of the right-hand side of eq 8 is
( ) ∂ha ∂P
and
{( ) ∂sa ∂P
dP ) T
T,x
( ) ∂ha ∂x
T,x
} { ( )}
+ Va dP ) Va - T
∂Va ∂T
P,x
dP
dx ≈ {hg(P, T) - Qst(P, T, x)}dx
P,T
(ref 18), where Qst indicates the isosteric heat of adsorption as a function of P, T, and x. Equation 6 is now written as
2206 Langmuir, Vol. 25, No. 4, 2009
Chakraborty et al.
{ ( )} { ( )} ∂Vs ∂T ∂Va Va - T ∂T
dh ) (cp,s + cp,a)dT + Vs - T
dP )
dP +
P
P,x
du ) (dhs + dha) - (PdVs + VsdP + PdVa + VadP) The differential form of energy of the adsorbent solid + adsorbate phase becomes
{
( )} { ( ) }
∂Va dP + Va - T dP + {hg - Qst}dx ∂T P,x - (PdVs + VsdP + PdVa + VadP) (10)
II. Expression of Isosteric Heat of Adsorption. The Clausius-Clapeyron (C-C) equation relates the adsorption heat effects to the temperature dependence of the adsorption isotherm. Two approximations were introduced in deriving the C-C equation: (1) the bulk gas phase is considered ideal, and (2) the adsorbed phase volume is neglected. These two assumptions are reasonable at low pressures but may not be true at higher pressures. The isosteric heat of adsorption Qst is defined as the differential change in energy δq that occurs when an infinitesimal amount of adsorbate uptake δx is transferred at constant pressure P, temperature T, and the amount of adsorbent Ms or the constant adsorbent surface ranging from the bulk gas phase to the adsorbed phase.25,26
Qst )
( ) ( )
( ∂q∂x )
(11)
P,T,Ms
-dq ) Tds
(12)
where the total entropy s is the sum of the entropies of the different phases. These are gaseous, adsorbed, and solid phases, i.e., s ) sg + sa + ss. From a mass balance and assuming an inert adsorbent, we have
( ) ( )
( )
ma mg mg dma ) -dmg w d ) -d w dx ) -d (13) Ms Ms Ms where ma is the mass of adsorbate and mg is the mass of molecules in pure gaseous phase. Equation 12 is now written as
( ∂q∂x )
P,T,Ms
[( )
) -T
∂sg ∂x
+ P,T,Ms
( ) ∂sa ∂x
+ P,T,Ms
( ) ] ∂ss ∂x
P,T,Ms
(14)
Hence (∂ss/∂x)P,T,Ms ≈ 0. Using eqs 13, 14, and 4,
[( )
Qst ≈ -T
∂sa ∂x
P,T,Ms
Qst ) -T[Vg - Va]
]
- sg dP dT
( ) ( )
g
where (∂P/∂T)g indicates the change of pressure with respect to temperature at the pure gaseous phase of the system. III. Expression of Specific Volume of Adsorbed Phase. The adsorbed phase volume, which is calculated by eq 15 and experimentally measured isosteric heat of adsorption data, is given as
{
∂P ⁄T ( ∂P ∂T ) } ( ∂T ) g
x
where Vg is the specific volume of the gaseous phase. The term (∂P/∂T)x is calculated from experimentally measured adsorption isotherm data or equations. IV. Expression of Specific Heat Capacity of Adsorbed Phase. It is well-known that the adsorbed phase specific heat capacity (cp,a) is defined as the temperature derivative of the differential adsorbed phase enthalpy at constant surface coverage (x), i.e., cp,a ) (∂ha/∂T)x.27,28 Now we use a useful mathematical tool named functional determinants or Jacobians29 for the calculation of (∂ha/∂T)x, where ha and x for T are the determinants.
( ) | ( ) ( ) | |( ) ( ) | ( ) ( ) ∂ha ) ∂T x
∂ha ∂T 0
x
∂ha ∂x 1
T
∂ha ) ∂T x ∂x ∂T x
∂ha ∂x T ) ∂(ha, x) ∂x ∂(T, x) ∂x T ∂(ha, x) ∂(T, x) ) ⁄ (16) ∂(T, P) ∂(T, P)
Hence, the Jacobians of ha and x for two variables P and T are the determinant, i.e.,
A change in the integral heat of adsorption -dq is defined as
Qst )
T
dP ∂P ∂P dx ∂P ∂P + + ) ≈ dT ∂T x ∂x T dT ∂T x ∂T
Va ) Vg - Qstexpt - TVg
du ) (cp,s + cp,a)dT + Vs P
x
dP + {hg - Qst}dx (9)
Internal Energy (u). The total differential of internal energy in an adsorption process is the summation of its solid and adsorbed phases, i.e.,
∂Vs T ∂T
∂P dT + ( ) dx ( ∂P ∂T ) ∂x
∂(ha, x) ) ∂(T, P)
|(( )) (( )) | ( ) ( ) ( ) ( ) ∂ha ∂T P ∂x ∂T P
∂ha ∂P T ) ∂ha ∂x ∂T ∂P T
P
∂ha ∂x ∂P T ∂P
∂x ∂T
T
P
and similarly we can show that
∂(T, x) ∂x ) ∂(T, P) ∂P
( )
T
Equation 16 is now written as
( ) ( ) ( )( ) ( ) ∂ha ∂ha ) ∂T x ∂T
-
P
∂ha ∂P
T
∂x ∂x ⁄ ∂T P ∂P
(17)
T
The thermodynamic quantity uptake x is a function of two variables, P and T, or x ) x(P,T); therfore, we are justified in considering P as a function of T and x, or P ) P(T,x), and T as a function of P and x, or T ) T(P,x). So the following relations uniquely relate all possible derivatives of these three functions:
∂T ∂P ∂x ∂x ) -1 w ( ) ) - ( ) ⁄ ( ) ( ∂P∂x ) ( ∂P ∂T ) ( ∂x ) ∂T ∂T ∂P T
(15)
For any adsorbent + adsorbate system, pressure, P, is a function of T and x: (25) Pan, H.; Ritter, J. A.; Balbuena, P. B. Langmuir 1998, 14, 6323. (26) Chakraborty, A.; Saha, B. B.; Koyama, S.; Ng, K. C. Appl. Phys. Lett. 2006, 89, 171901.
x
P
x
P
T
Being a property of a thermodynamic system, the exact behavior of cp,a could be obtained by (27) Al-Muhtaseb, S. A.; Ritter, J. A. J. Phys. Chem. B 1999, 103, 2467. (28) Chakraborty, A.; Saha, B. B.; Koyama, S.; Ng, K. C. Appl. Phys. Lett. 2007, 90, 171902. (29) Sychev, V. V. The Differential Equation of Thermodynamics, 2nd ed.; MIR Publishers: Moscow, 1992; p. 24.
Thermodynamics of Adsorbent + Adsorbate Systems
cp,a )
Langmuir, Vol. 25, No. 4, 2009 2207
( ) ( )( ) ∂ha ∂T
∂ha ∂P
+
P
T
∂P ∂T
(18)
x
The Gibbs laws as well as the Maxwell relations are invoked to transform the enthalpy changes with respect to pressure at constant temperature and uptake, and can be written as
( ) ∂ha ∂P
( )
∂Va ) -T T ∂T
( )
∂Va + Va + P P ∂P
( )
( )
∂Vg ≈ -T T ∂T
( )
∂Vg + Vg + P P ∂P
∂Vg ∂P +P ∂T x ∂P
P
T
∂P ∂T
x
(19)
From the thermodynamic viewpoint, the partial gradient (∂P/ ∂T)x could be extracted from the adsorption isotherms of the adsorbent + adsorbate system. Invoking the C-C equation, the mentioned partial gradient can be replaced by the measurable heat of adsorption,
)
where Qst is the isosteric heat of adsorption. The third term of the right-hand side of eq 19 approximately approaches
( )( )
P
∂Vg ∂P
T
∂Qst ∂P )∂T x ∂T
|
P
In an adsorbent + adsorbate system, Qst is found to be a function of vapor uptake, and it is also sensitive to temperature. Equation 19 is now written as
{ ( )}
cp,a ) cp,g + Qst
1 1 ∂Vg T Vg ∂T
P
-
∂Qst ∂T
|
(20) P
V. Thermodynamic Property Surfaces. The physical adsorption system is an ensemble of adsorbed gas plus adsorbent enclosed by a surface. The extensive thermodynamic quantities such as entropy (s), internal energy (u), and enthalpy (h) of a single-component adsorbent + adsorbate system at a given mass of adsorbent, Ms, are path independent, and their changes can be tracked by integrating in succession between the limits of initial pressure (Po) and P, initial temperature (To) and T, and xo (x ) 0) and x with T, x; P, x; and P, T being held constant, respectively, for these integration processes. Neglecting the pressure and uptake dependence of solid phase thermodynamic properties with respect to pressure and temperature, the entropy of the adsorbent + adsorbate system becomes
s(P, T, x) ) s(Po, To, xo) +
∫TT o
( ) ∫( ) ∂ss ∂T P
Po
dT+
P,x
∂sa ∂P
∫TT o
( ) ∫( )
dP +
T,x
∂sa ∂T x
0
dT+
P,x
∂sa ∂x
dx
P,T
∫TT o
∫{
}
∫
[
∫
]
∫
( ) ∫{
[{
}
cp,s T 1 1 1 Qst dT+ T o T T T Vg ∂Qst Qst ∂Vg x dT - 0 sg dx ∂T P ∂T T
∫TT
∫
o
( )}
]
∫
{
}
The enthalpy for adsorbent + adsorbate system becomes
h(P, T, x) ) h(Po, To, xo) +
∫TT (cp,s + cp,a)dT + ∫PP
( )}
o
T h(P, T, x) ≈ h(P, T, xo) +
∂Va ∂T
P,x
o
dP +
o
( )}
-
o
P
o
]
∂Qst dT + ∂T
{
Va -
∫xx {hg - Qst}dx
∫TT (cp,s + cp,g)dT + ∫TT
1 ∂Vg Vg ∂T
Qst ∂P ∂ ln P )P ) ∂T x ∂T x TVg
( ) (
o
For simplicity,
T
{ ( ) }( ) ( ) ( ) ∂Vg ∂T
( )
s(P, T, x) ≈ s(P, T, xo) +
Substituting these transformations into eq 18, the cp of the adsorbed phase is now expressed in terms of the measurable variables, i.e.,
cp,a ) cp,g + Vg - T
∫
cp,s dT+ T
∫TT
{ ( )}
T
where the first term indicates thermal expansion of the adsorbed phase, and the third term defines the isothermal compressibility of the adsorbed phase on the adsorbents. Here Va is the specific volume of the adsorbed phase. For simplicity,
∂ha ∂P
cp,a dTT x P ∂Va dP dP + 0 sg - (Vg - Va) dx Po ∂T p,x dT T cp,s T 1 s(P, T, x) ) s(Po, To, xo) + T cp,g + dT+ T o T o T ∂Qst P ∂Va 1 1 ∂Vg dT - P dP + Qst o T Vg ∂T P ∂T ∂T p,x Qst x dx sg 0 T
s(P, T, x) ) s(Po, To, xo) +
[{ Qst
1 T
∫xx {hg - Qst}dx o
and the internal energy for an adsorbent + adsorbate system can be represented by
u(P, T, x) ) h(P, T, xo) +
( )}
1 ∂Vg Vg ∂T
P
-
∫TT (cp,s + cp,g)dT + ∫TT o
]
∂Qst dT + ∂T
o
[{ Qst
1 T
∫xx {hg - Qst}dx o
∫0V PdVa - ∫PP VadP a
o
3. Results and Discussion In this work, a combination of an adsorbent and a singlecomponent adsorbed adsorbate is taken as an adsorbent + adsorbate (or adsorbate + adsorbent) system, and thermodynamic equilibrium prevails between this system and single-component unadsorbed gas phase. We shall make use of experimental isotherm data20,21 of CaCl2-in-silica gel + water and Maxsorb III + methane to highlight the present findings. I. Thermodynamic Property Fields of CaCl2-in-Silica Gel + Water System. Recently, a new family of composite sorbents called selective water sorbents (SWSs) has been presented for adsorption cooling and heat pumping applications.30,31 It is based on a porous host matrix (silica, alumina, etc.) and an inorganic salt (CaCl2, LiBr, MgCl2, MgSO4, Ca(NO3)2, etc.) impregnated inside pores.32 Among the different SWSs, the SWS-1 L (“CaCl2 confined to KSK silica gel”) shows very high water sorption capacity (up to 0.8 g of water per 1 g of dry adsorbent). This composite adsorbent is synthesized by a dry impregnation of a (30) Aristov, Y. I.; Restuccia, G.; Cacciola, G.; Parmon, V. N. Appl. Therm. Eng. 2002) , 22, 191. (31) Saha, B. B.; Chakraborty, A.; Koyama, S.; Aristov, Y. I. Int. J. Heat Mass Transfer 2009, 52, 516. (32) Aristov, Y. I. J. Eng. Thermophys. 2007, 16, 63.
2208 Langmuir, Vol. 25, No. 4, 2009
Chakraborty et al.
Figure 2. Isosteric heat of adsorption as a function of surface coverage of a CaCl2-in-silica gel + water system for various temperatures. Here the dashed line indicates the average Qst value calculated by the To´th adsorption isotherm equation.
Figure 1. Temperature-entropy (T-s) diagram of a CaCl2-in-silica gel + water system for understanding the adsorption cooling cycle. Here Q indicates energy, and the subscript “e” is for evaporator, “ads” is for adsorption, “des” denotes desorption, and “c” indicates condenser.
mesoporous KSK silica gel (average pore size 15 nm) with a saturated aqueous solution of CaCl2 with subsequent drying at 150 °C. The salt content is 33.7 wt %. Furthermore, according to the SWS-1 L water sorption isobars,33 most of the adsorbed water can be removed at temperatures of 80-100 °C. We have calculated the entropy, isosteric heat of adsorption, and adsorbed phase volume of a CaCl2-in-silica gel + water system, and these are essential for the analyses of an adsorption cooling cycle. The temperature-entropy (T-s) maps for the adsorption of water on KSK silica gel are shown in Figure 1. The entropy of the adsorbed phase is presented here for various pressures and water vapor uptakes. From this graph, one can easily calculate the energetic performances of an adsorption cooling cycle in terms of entropy flow, and also plot a cooling cycle. The thermal compression process (A-B-C-D) is plotted on the thermodynamic entropy surfaces. As a result of the cooling load, the entropy of evaporation ∆sfg() sE - sH) involves in the evaporator, and the evaporated water vapor is adsorbed in the CaCl2-in-silica gel surfaces of bed 1 (E-A). During the regeneration phase (lines A-B-C), the pressure in the adsorber rises from the evaporator pressure (Pe) to the condenser pressure (Pc) by heating the desorber or bed 2, and desorption of water vapor from the silica gel occurs by connecting bed 2 with the condenser. The amount of adsorbed uptake falls from xads to xdes, and the entropy changes from sA to sC. At the condenser, the water vapor is condensed, and energy is released to the environment. The entropy changes from sC to sG. Finally the condensed liquid goes to the evaporator and completes the cycle H-E-A-B-C-F-G. On the other hand, during the adsorption phase (lines C-D-A), the adsorber (bed 1) is cooled, and the pressure falls from Pc to Pe. Then, the refrigerant vapor is adsorbed on the adsorbent, and the amount of adsorbate increases up to xads. During the next cycle, bed 1 is in desorption mode, and bed 2 works in the adsorption process. The adsorption cooling cycle is the amalgamation of (33) Aristov, Y. I.; Tokarev, M. M.; Cacciola, G.; Restuccia, G. React. Kinet. Catal. Lett. 1996, 59, 325.
“adsorption-triggered evaporation” and “desorption-resulted condensation” and is described elsewhere in detail.34-36 For calculating the adsorption isotherms of a CaCl2-in-silica gel + water system,32 the Dubinin-Astakhov (D-A) model is employed. Using D-A model and eq 15, the isosteric heat of adsorption is calculated, and this is given by
( )
Qst ) hfg + E ln
xm x
1⁄n
( dP dT )
+ TVg
g
(21)
where hfg indicates the latent heat of vaporization, E () 306 kJ/kg) is the characteristic energy, xm () 1.25) denotes the maximum uptake, and n indicates the heterogeneity constant. Here n ) 1.75 for the silica gel + water system. The derivative (dP/dT)g is calculated as a function of pressure and temperature from the National Institute of Standards and Technology (NIST) standard reference database.37 The isosteric heat of adsorption (Qst) of water vapor on CaCl2-in-silica gel for various uptakes and temperatures varying from 303 to 363 K is shown in Figure 2. It is found from Figure 2 that the isosteric heat of adsorption decreases with the increase of the amount of adsorbate. The calorimetrically measured experimental data are also furnished in Figure 2. The Qst33 values calculated from the C-C equation as a function of water vapor uptakes are also plotted in Figure 2 for comparison. At lower surface coverage, the Qst obtained by the proposed formalism is close to the data taken from C-C formula, but, at higher surface coverage, the proposed Qst matches well with the calorimetric measured Qst values. From Figure 2, one may observe that Qst is found to be very high at lower loading or Henry’s region compared to Qst at higher uptakes. Silica gel consists mainly of mesopores with different widths, and, at first, water vapor adsorbs rapidly onto sites of high energy, and molecules then adsorb onto sites of decreasing energy as adsorption progresses. The adsorbate molecules first penetrate into narrower pores and provide stronger interaction between the adsorbate and the adsorbent. This implies a higher value of (34) Saha, B. B.; Chakraborty, A.; Koyama, S.; Srinivasan, K.; Ng, K. C.; Dutta, P. Appl. Phys. Lett. 2007, 91, 111902. (35) Saha, B. B.; Chakraborty, A.; Koyama, S.; Ng, K. C.; Sai, M. A. Philos. Mag. 2006, 86, 3613. (36) Wang, L. W.; Wang, R. Z.; Lu, Z. S.; Chen, C. J.; Wu, J. Y. Chem. Eng. Sci. 2006, 61, 3761. (37) Lemmon, E. W.; Mclinden, M. O.; Huber, M. L. Reference fluid thermodynamic and transport properties. NIST Standard Reference Database, version 7; National Institute of Standards and Technology: Washington, DC, 2002.
Thermodynamics of Adsorbent + Adsorbate Systems
Figure 3. A plot of adsorbed phase volume against surface coverage for a CaCl2-in-silica gel + water system. Hence, (i) the thin solid line indicates Va for E ) 400 kJ/kg, n ) 1.45, and xm ) 1.25; (ii) the dashed line indicates Va for E ) 350 kJ/kg, n ) 1.4, and xm ) 1.25; and (iii) the thick solid line indicates Va for E ) 305 kJ/kg, n ) 1.75, and xm ) 1.25.
Qst at lower loadings. After completely filling the smaller pores, water molecules are gradually accommodated in larger pores, in which the adsorption affinity becomes weaker. A plot of adsorbed phase volume (Va) against surface coverage (x/xm) is shown in Figure 3. Va decreases with higher uptake values. The D-A isotherm model is employed to evaluate (∂P/ ∂T)x. It is found that the D-A isotherm parameters such as E and n are sensitive to (∂P/∂T)x. In Figure 3, three curves of adsorbed phase volume are plotted against water vapor uptake at 313 and 333 K for different values of E and n. For higher E () 400 kJ/kg), Va is found to be higher at lower surface coverage. These curves also indicate the surface heterogeneity of the CaCl2in-silica gel adsorbent. Using the proposed specific heat capacity expression, the specific heat capacity (cp,net) of a single-component silica gel (type CaCl2-in-silica gel) + water system is analyzed. The specific heat capacities of a CaCl2-in-silica gel + water system are plotted against the amount of adsorbate uptakes (x) for 313 and 333 K, respectively, and these are shown in Figure 4. The experimentally measured cp,net values are superimposed in Figure 4 for comparison. It can be observed that the proposed specific heat capacity agrees satisfactorily with the experimentally measured data, while the liquid-phase based cp,l and the gaseous phase based cp,g form the upper and lower bounds of the experimental data. The specific heat capacity model of Aristov et al.,38 for a CaCl2-in-silica gel + water system is also plotted in Figure 4, where the specific heat is approximated by curve fitting analysis. The specific heat capacity is expressed as cp,net ) a + b(T - 273), where a and b depend on the amount of adsorbate uptakes. II. Thermodynamic Property Fields of Maxsorb III + Methane. It is well-known that pitch-based activated carbon is produced by a direct chemical activation route in which oxidative stabilized pitch derived from ethylene tar oil is reacted with potassium hydroxide under various activation conditions. Abundant oxygen-containing functional groups (C-OH, C-O-C, CdO, COOR, etc.) are found to exist on its surface. The temperature-entropy (T-s) maps of activated carbon (Maxsorb III) + methane system for pressures ranging from 10 to 5000 kPa, and the amount of CH4 uptakes varying from 0.01 kg/kg to 0.27 kg/kg is shown in Figure 5. The entropy of the adsorbed (38) Aristov, Y. I.; Tokarev, M. M.; Cacciola, G.; Restuccia, G. Russ. J. Phys. Chem. 1997, 71, 327.
Langmuir, Vol. 25, No. 4, 2009 2209
Figure 4. The specific heat capacity of a CaCl2-in-silica gel + water system for various water vapor uptakes at 313 and 333 K. The error bars ((5%) are shown here. The cp,net of Aristov et. al38 is approximated as cp,net(T,x) ) a(x) + b(x) · (T - 273), with a(x) ) a0 + a1x, b(x) ) b0 + b1x + b2x2, where a0 ) 0.637 J/(g · K); a1 ) 2.91 J/(g · K); b0 ) 0.00191 J/(g · K2), b1 ) -0.0582 J/(g · K2); b2 ) 0.5346 J/(g · K2).
Figure 5. A representation of T-s maps of a functional activated carbon (Maxsorb III) + methane system as a function of P, T, and x to understand the gas storage process. Here, dotted lines are the entropy of pure gaseous phase for various pressures.
phase (sa) is higher than that of the gaseous phase (sg) and the adsorption of methane occurs above the critical ranges. The entropy of evaporation (i.e., 4.5745 kJ/kg K) of methane at boiling point (111.67 K) is lower than the entropy of adsorbed phase at 200 K and 5000 kPa. The amount of sa increases at lower pressures and higher temperatures. So the storage of methane is suitable at low temperature and high pressures. It is also found from Figure 5 that the amount of methane uptake is 0.15 kg/kg of Maxsorb III at room temperature and pressure at 500 kPa. The fabrication of new highly porous carbonaceous materials with higher specific surface area is necessary to store more methane at room temperature.39 A plot of Qst and Va as a function of surface coverage for various temperatures is shown in Figure 6. For the Maxsorb III + methane system, initially both Qst and Va decrease with
2210 Langmuir, Vol. 25, No. 4, 2009
Chakraborty et al.
ln P ) ln Ps -
1 E (-ln θ) n RT
(B)
where θ () x/xm) is the surface coverage.
( ∂ ∂Tln P ) ) ( x
)
1 ∂ ln Ps E + 2 (-ln θ) n ∂T x RT
(C)
Here, ln Ps ) -hfg/RT + sfg/R for T < Tcrit, Tcrit is the critical temperature, and hfg and sfg indicate the enthalpy and the entropy of vaporization, respectively. Neglecting the nonideality of gaseous and the adsorbed phase volume, and using C-C relations, the isosteric heat of adsorption is given by Figure 6. Isosteric heat of adsorption and adsorbed phase volume as a function of surface coverage for the adsorption of methane on Maxsorb III.
increasing surface coverage, and this is due to the fact that the contribution of the heat of polymolecular adsorption to the heat of adsorption is still perceptible, but after further increase in surface coverage (x/xm) by pressurization, it is almost compression of methane that is involved in the pores of Maxsorb III. The adsorbed phase entropy is close to the entropy of vaporization, which indicates that the thermodynamic properties of capillarycompressed methane adsorbate are close to those of the normal methane in compressed form.
The formulations of thermodynamic property surfaces, namely, the internal energy, enthalpy, and entropy of a single component adsorbent + adsorbate system, are the basic foundations of any adsorbate-adsorbent system. The theoretical framework for the isosteric heat of adsorption has been derived on the basis of classical thermodynamics, and it is distinctive from the conventionally accepted C-C form. Relatively good agreement has been obtained between the experimental data and the proposed Qst formulation. The thermodynamic equation of the adsorbed phase volume and the specific heat capacity for a singlecomponent adsorbent + adsorbate system are found to be sensitive to changes in temperature, pressure, and the amount of adsorbate uptake. They have immense importance, as they directly affect the computation of the energy and entropy balances of the adsorbed phase. Such key thermodynamic quantities are essential in the development of adsorption thermodynamics theory, and they would be useful in the design and analysis of the adsorption cooling cycle, natural gas storage, and, more recently, hydrogen storage.40 Acknowledgment. The authors would like to thank King Abdullah University of Science & Technology (KAUST) for the generous financial support through the project (WBS R265-000286-597).
Appendix: Derivation of the Specific Heat Capacity 1. For CaCl2-in-Silica Gel + Water System
( )
[ { ( )} ] n
(A)
where E indicates the activation energy, Ps defines the pressure at saturated conditions, n is the heterogeneity constant, and R represents the gas constant. Equation A is written as
1 n
(D)
Hence, eq 15 is represented by
( )
xm Qst ) hfg + E ln x
1 n
( ∂P ∂T )
+ TVg
(E)
g
Equation 20 is written as
{ ( )} ( ) { ( )} ( )
cp,a ) cp,g + Qst
) cp,g + Qst
1 1 ∂Vg T Vg ∂T
1 1 ∂Vg T Vg ∂T
P
∂hfg(T) ∂P - Vg ∂T p ∂T g
∂2P ∂T2
g
-
( )
( ∂P ∂T )
- cp,g + cp,l - Vg
P
g
2
TVg
∂P ≈ ∂T2 g
{ ( )}
cp,l + Qst
1 1 ∂Vg T Vg ∂T
P
(F)
where cp,l indicates the specific heat capacity of the liquid phase. The specific heat capacity of a CaCl2-in-silica gel + water system is given by
{ ( ) }]
[
cp,a ) cp,s + x cp,l + Qst
1 1 ∂Vg T Vg ∂T
(G)
P
2. For activated carbon (Maxsorb III) + methane system
( ) ( ) ( )
T > Tcrit f Ps )
xm Qst ) 2RT + E ln x
T 2 P Tcrit crit 1 n
∂P ∂T
+ TVg
g
The specific heat capacity of the adsorbed phase is
{ ( )} ( )
cp,a ) cp,g + Qst
The amount of water vapor uptake is calculated by the D-A equation and is given by
Ps RT ln E P
)
TVg
4. Conclusions
x ) xm exp -
(
1 xm ∂ ln Ps Qst|cc ) RT + E(-ln θ) n ≈ hfg + E ln ∂T x x 2
1 1 ∂Vg T Vg ∂T
P
( ∂P ∂T )
- 2R - Vg
g
2
∂P ≈ ∂T2 g 1 1 ∂Vg cp,g + Qst T Vg ∂T
TVg
{ ( )} P
- 2R (H)
The specific heat capacity for a single-component adsorbent + adsorbate system is represented as
Thermodynamics of Adsorbent + Adsorbate Systems
Langmuir, Vol. 25, No. 4, 2009 2211
cp,t ) cp,s + xcp,a
(I)
29
It should be noted here that
( ) ∫( )
cp,g s(Po, To, xo) + T dTo T
∫
T
∫P
P
Po
h(Po, To, xo) +
∂Va ∂T
P o
∂Va ∂T
∫TT cp,gdT + ∫PP o
o
h(Po, To, xo) +
∫TT cp,gdT + ∫PP o
o
{ ( )} Va - T
∂Va ∂T
P,x
dP ≈ h(P, T, xo)
dP ) s(Po, T, xo) -
For simplicity,
p,x
dP
p,x
≈ s(P, T, xo)
{ ( )} ∫{ ( )} Va - T
h(Po, T, xo) +
∂Va ∂T
P
Po
P,x
∂Vg ∂T
P,x
o
{ ( )} Va - T
∂Va ∂T
P,x
dP ≈
∫PP o
{ ( )} Vg - T
∂Vg ∂T
P,x
dP
LA803289P
dP ≈
Vg - T
∫PP
dP
(39) Chakraborty, A.; Saha, B. B.; Koyama, S.; Ng, K. C.; Yoon, S. H. Appl. Phys. Lett. 2008, 92, 201911. (40) Schlapbach, L.; Zut¨tel, A. Nature 2001, 414, 353.
