Article pubs.acs.org/Langmuir
On the Thermodynamics of Refrigerant + Heterogeneous Solid Surfaces Adsorption Azhar Bin Ismail,† Ang Li,† Kyaw Thu,† K. C. Ng,*,† and Wongee Chun‡ †
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576 Department of Nuclear and Energy, Jeju National University, Jeju, Korea
‡
ABSTRACT: This Article presents a theoretical framework for the understanding of pressurized adsorption systems using the statistical rate methodology. Utilizing results from the statistical rate theory, basic thermodynamic variables including enthalpy (ha), entropy (sa), and the specific heat capacity (cp,a) of the adsorbed phase are derived using the thermodynamic requirements of chemical equilibrium, Gibbs law, as well as Maxwell relations. A built-in constant (K) describes the adsorbed molecular partition function (q s ), and it captures the heterogeneous properties of the adsorbent + adsorbate pair at equilibrium states. Improved adsorbed-phase volume considerations were incorporated in the formulations of these variables where they could be utilized with relative ease for analyzing the energetic performances of any practical adsorption system. In this Article, we have demonstrated how derived thermodynamic quantities can bridge the information gap with respect to the states of adsorbed phase, as well as resolved some theoretical inconsistencies that were found in previously derived quantities. Experimentally, the adsorption isotherms of propane (refrigerant) on activated carbon powder (Maxsorb III) for temperatures from 5 to 75 °C and pressures up to 8 bar are presented, and they are used to illustrate the behaviors of the adsorbed-phase during uptakes, temperatures, and pressure excursions or changes.
1. INTRODUCTION When an adsorbate molecule is adsorbed on an adsorbent surface, the energy of the adsorbate molecules is reduced as it becomes more stabilized. This results in a phase transformation from gaseous to what is termed as the adsorbed phase.1 From a thermodynamic perspective, this phase is distinguishable, but there is uncertainty however in the precise location of the phase boundary.2 Thus, the adsorbed phase is considered to be a thermodynamic state, which is dependent on the adsorption uptake (q) in addition to the temperature (Te) and pressure (Pe).1 Its internal energy, enthalpy, entropy, heat of adsorption, and specific heat capacity are thermodynamic quantities that are necessary to analyze an adsorption system. This Article presents the theoretical framework in which thermodynamic quantities of the adsorbed phase, including the heats of adsorption, specific volume, entropy, specific heat capacity, as well as enthalpy, are developed. The formulations of the adsorbent + adsorbate pair are first determined from the uptake data. This framework is grounded upon the statistical rate theory with the rigor of thermodynamics.
molecule is attached to the adsorbent surface via van der Waals forces and loses a translational degree of freedom.5 In other words, an adsorbate molecule, which initially has three degrees of translational freedom in the gas phase,6,7 now has two degrees of freedom in the adsorbed phase. Adsorption is thus an exothermic process unless the adsorbed state is characterized by a very large additional entropy such as those attributed to vibrations. This is mathematically described by the Gibbs free energy, which governs thermodynamic systems at constant temperatures and pressures8 [because q = q(P,T)] as follows:
2. THERMODYNAMICS FRAMEWORK 2.1. Gibbs Free Energy. Physical adsorption is characterized by the release of heat, moving to a lower, more thermodynamically stable energy state, and is thus a spontaneous process.3,4 During adsorption, the adsorbate
(2)
© 2013 American Chemical Society
ΔGads = ΔHads − T ΔSads
(1)
where ΔHads = Ha − Hg. The chemical potentials of the associated gas and adsorbed phases are defined as:9 ⎛ ∂G ⎞ μa = ⎜ a ⎟ ⎝ ∂na ⎠T , P Received: August 29, 2013 Revised: November 4, 2013 Published: November 5, 2013 14494
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⎛ ∂Gg ⎞ ⎟⎟ μg = ⎜⎜ ⎝ ∂ng ⎠T , P
over the range of the adsorption energies. By describing χ(ε) with a suitable Gaussian function, several authors have successfully rearrived at the well-known Langmuir−Freundlich, Dubinin−Astakhov (DA), Dubinin−Radushkevich (DR), and Toth isotherm equations.16,17 Because the uptake of an adsorbent + adsorbate system is a function of its equilibrium temperature T and pressure P, εc could thus be regarded as a constant along an isostere where its equilibrium pressure P and its partial derivative ∂P/∂T are given by eqs 11 and 12, respectively.
(3)
Following the isothermal assumption, when the pressure of the adsorbent−adsorbate system is increased, the number of gas molecules in the system has to be increased by dng, altering the equilibrium state of the adsorbed phase and the gas phase of adsorbate molecules. When this happens, the thermodynamic systems consequently transfer molecules toward a lower free energy state until the chemical potentials of the gas phase and the adsorbed phase are equal. This follows: dG = (μa − μg ) dna = 0
(4)
μa = μg
(5)
P=
sa = sg +
⎛ ∂Sg ⎞ ⎛ ∂V ⎞ ⎛ ∂μ ⎞ ⎟⎟ dT + ⎜⎜ g ⎟⎟ dP + ⎜⎜ g ⎟⎟ dng dμg = −⎜⎜ ⎝ ∂ng ⎠T , P ⎝ ∂ng ⎠T , P ⎝ ∂ng ⎠T , P (7)
where dna = −dng, (∂Si/∂ni)P,T = si, and (∂Vi/∂ni)P,T = vi. Substituting these into eqs 4 and 5 reproduces the classical Clausius−Clayperon equation,6,14 which is
(9)
Hence, the expression for the adsorbed phase entropy along an isostere simplifies to: sa = sg −
⎛ ∂P ⎞ ⎜ ⎟ (v − v ) a ⎝ ∂T ⎠n g a
P(vg − va) T
ln KP
(13)
2.3. Adsorbed Phase Enthalpy (ha). One of the most important thermodynamic properties that needs to be determined in modeling the adsorption chiller is the adsorbed phase enthalpy, ha. In the derivation of ha, it is found that several authors obtained a latent heat of vaporization term of the adsorbate, hfg,10,18 even though the adsorbed phase is not related to the liquid phase.19 Further, in the derivation of the heat of adsorption, several considerations were made including the use of the ideal gas equation,20 and the assumption of the specific volume of the adsorbed phase to be either liquid, gaseous, or much smaller than the gaseous phase.21−23 The use of the excess adsorption uptake from the experimental data24,25 contributes to further inaccuracies of the derived isotherm properties. The determination of the thermodynamic quantities should therefore utilize absolute uptakes where volumetric corrections are considered. The ideal gas law postulates that the gas molecules are noninteracting and occupies inherent volume. In refrigerant-adsorption applications where low temperatures and high pressures are desired, both the volume and the interaction effects become strikingly apparent. Thus, nonideality of the adsorbed phase could not be ignored. In this work, we arrive at the change of enthalpy incorporating the correction to the adsorbed phase volume without the use of the ideal gas relations. Further, the adsorbed phase volume is calculated from the absolute instead of the excess adsorption uptake values. Invoking the Gibb’s free energy equation, the enthalpy of adsorption is given as:26
(6)
sg − sa dP = dT vg − va
(12)
The expression for the adsorbed phase entropy therefore becomes:
⎛ ∂S ⎞ ⎛ ∂V ⎞ ⎛ ∂μ ⎞ dμa = −⎜ a ⎟ dT + ⎜ a ⎟ dP + ⎜ a ⎟ dna ⎝ ∂na ⎠T , P ⎝ ∂na ⎠T , P ⎝ ∂na ⎠T , P
(8)
(11)
εc ⎛ ∂P ⎞ ⎛ ε ⎞ P ⎜ ⎟ exp⎜ − c ⎟ = − ln KP = 2 ⎝ RT ⎠ ⎝ ∂T ⎠n T KRT a
2.2. Adsorbed Phase Entropy (sa). Previous works that dealt with adsorbed phase properties have shown that its entropy is a function of the system’s temperature (T), pressure (P), as well as the adsorbate uptake (q).10−13 The total differential of this term (dsa ) during adsorption from mathematics is thus the addition of the partial change in entropy with respect to these three thermodynamic properties. This is expressed utilizing the derivatives of μa and μg, which are functions of T, P, and n that are related to the uptake q, as a total derivative given by:
−sa dT + va dP = −sg dT + vg dP
⎛ ε ⎞ 1 exp⎜ − c ⎟ ⎝ RT ⎠ K
(10)
Rudzinski and co-workers previously defined εc, which is a function of the adsorbent + adsorbate system’s equilibrium temperature T and pressure P given by −RT ln(K × P).15 R is the universal gas constant or the Boltzmann constant, in J mol−1 K−1. K is described by both qs, the molecular particle function of the adsorbate gas molecules, and μgo, the chemical potential of the gas phase at a reference pressure, given by zi exp(μgo/RT). εc has been incorporated in the homogeneous energy-surface model of the Langmuir isotherm equation, which could then elucidated as exp(ε − εc/RT)/[1 + exp (ε − εc/RT)]. The condensation approximation (CA) is then invoked so that the averaged surface coverage θt of an energetically heterogeneous surface could be mathematically described as an integral of the adsorption site distribution, χ(ε),
ΔHads(na) = T ΔSads
(14)
ha − hg = T (sa − sg)
(15)
Substituting eq 13 into eq 15: ha = hg + P(vg − va) ln KP
(16)
In other words, the change in enthalpy of adsorption is given by:15 ΔHads(na) = −P(vg − va) ln(K × P) 14495
(17)
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∂va /∂T at constant specific volumes and temperature, respectively.
which is a positive term given that the product of (K × P) is negative. It has previously been expounded how an adsorbate molecule undergoes a phase change from a higher to a lower energy level during adsorption.27 This results in a heat release known as the isosteric heat of adsorption. Do (1998) defined isosteric heat as “the ratio of the infinitesimal change in the adsorbate enthalpy to the infinitesimal change in the amount adsorbed.”28 In general, this value is 30−100% larger than the adsorbate’s heat of vaporization29 but is much lower as compared to a chemical adsorption process. 2.4. Adsorbed Phase Specific Heat Capacity (cp,a). The adsorbed phase is distinct from the liquid and gas phases, and thus it is necessary to obtain an expression for its specific heat capacity at constant pressure cp,a. Until recently, the adsorbed phase has been assumed to be gaseous6 or liquid.30 Several specific heat capacity models thus assume values of the reference gas phase,31 while others gave that of the liquid phase.32 Further, there are several existing issues with available models in terms of thermodynamic consistency. Walton and Levan (2005) highlighted how these values will approach negative at higher loadings.33 Furthermore, the value of the specific heat capacity obtained would vary widely depending on which isotherm model is used. Several other expressions lead to a specific heat capacity term to be always lower than that in the gas phase. In deriving the specific heat capacity of the adsorbed phase, its definition, similar to that for the liquid and gaseous phases along the isoteres as per the previous works, is used:34 ⎛ ∂hg ⎞ ⎛ ∂h ⎞ ⎛ ∂H ⎞ ⎟ − ⎜ ads ⎟ cp ,a = ⎜ a ⎟ = ⎜ ⎝ ∂T ⎠n ⎝ ∂T ⎠n T ∂ ⎠n ⎝ a a
⎛ ∂ 2P ⎞ ⎛ ∂Hads ⎞ ⎛ ∂P ⎞ ⎜ ⎟ ≈ ⎜ ⎟ (vg − va) + T ⎜ 2 ⎟ (vg − va) ⎝ ∂T ⎠n ⎝ ∂T ⎠n ⎝ ∂T ⎠n a a a
⎡ ⎤ ⎛ ∂v ⎞ ⎛ ∂P ⎞ ⎛ ∂vg ⎞ + T ⎜ ⎟ ⎢⎜ ⎟ − ⎜ a ⎟ ⎥ ⎝ ∂T ⎠n ⎢⎝ ∂T ⎠ ⎝ ∂T ⎠n ⎥ a⎣ n a⎦ a
⎛ ∂Hads ⎞ (P ln KP)2 ⎛ ∂vg ⎞ ⎜ ⎟ = ⎜ ⎟ − (P ln KP) ⎝ ∂T ⎠n T ⎝ ∂P ⎠
⎛ ∂hg ⎞ ⎛ ∂q ⎞ ⎛ ∂hg ⎞ ⎟ ⎜ ⎟ ⎟ = −⎜ ⎜ ⎝ ∂q ⎠T ⎝ ∂T ⎠h ⎝ ∂T ⎠n
Similarly: ⎛ ∂hg ⎞ ⎟ ≈− ⎜ ⎝ ∂T ⎠n a
⎛∂ P⎞ ⎛ ∂Hads ⎞ ⎛ ∂P ⎞ ⎜ ⎟ = ⎜ ⎟ (vg − va) + T ⎜ 2 ⎟ (vg − va) ⎝ ∂T ⎠n ⎝ ∂T ⎠n ⎝ ∂T ⎠n a a
⎛ ∂vg ⎞ ⎛ vg ⎞ ⎜ ⎟ − P(ln KP)2 ⎜ ⎟ ⎝T ⎠ ⎝ ∂P ⎠T
From the expression of (∂P/∂T)na = εc/KRT exp(−εc/RT): 2
(29)
With these derived thermodynamic properties, an improved thermodynamic framework has been developed. The key difference from the expressions utilized by Kazi et al. (2011)12 is the involvement of the results from the statistical rate theory. While employing the same thermodynamic definitions, this model offers simplified expressions for isosteric heat and specific heat capacity at constant pressure. The enthalpy and entropy, as described by eqs 13 and 15, are functions of equilibrium pressures (P) and temperatures (T) at a specified uptake (q). It is calculated with respect to the gaseous phase enthalpy and entropy.
(21)
Because vg = vg(q, T), (∂vg/∂T)na = (∂vg/∂T)q may be derived using the cyclic relation:
(22)
Therefore:
( ∂v )T
(28)
⎡⎛ ∂v ⎞ vg ⎤ (P ln KP)2 g cp ,a = cp ,g + 2(P ln KP)⎢⎜ ⎟ − ⎥ − ⎢⎣⎝ ∂T ⎠ T ⎥⎦ T P (20)
a
P
(27)
Finally, the expression for the specific heat capacity of the adsorbed phase is derived as follows:
a
⎤ ⎡ ⎛ ∂v ⎞ ⎛ ∂P ⎞ ⎛ ∂vg ⎞ + T ⎜ ⎟ ⎢⎜ ⎟ − ⎜ a ⎟ ⎥ ⎝ ∂T ⎠n ⎢⎝ ∂T ⎠ ⎝ ∂T ⎠n ⎥ a⎣ n a⎦
ln KP + ⎛ ∂vg ⎞ T ⎜ ⎟ =− ∂P ⎝ ∂T ⎠q
= cp ,g
⎡⎛ ∂v ⎞ ⎛ ∂hg ⎞ vg ⎤ g ⎟ = cp ,g + P ln KP ⎢⎜ ⎟ − ⎥ ⎜ ⎢⎣⎝ ∂T ⎠ T ⎥⎦ ⎝ ∂T ⎠n P a
2
T
( ∂∂TP )h + TP ln KP ( ∂∂Ph )T
⎡⎛ ∂v ⎞ vg ⎤ g + P ln KP ⎢⎜ ⎟ − ⎥ ⎢⎣⎝ ∂T ⎠ T ⎥⎦ P
(19)
a
⎛ ∂vg ⎞ ⎛ ∂T ⎞ ⎛ ∂q ⎞ ⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎟⎟ = −1 ⎝ ∂T ⎠q⎝ ∂q ⎠v ⎝ ∂vg ⎠
(26)
a
From eq 27:
⎛ ∂P ⎞ ΔHads(na) = T ⎜ ⎟ (vg − va) ⎝ ∂T ⎠n
a
(25)
Using the cyclic relation:
From the change in enthalpy derived earlier:
⎛ ∂ 2P ⎞ P(ln KP) [2 + ln KP] ⎜ 2⎟ = ⎝ ∂T ⎠n T2
T
a
⎡⎛ ∂v ⎞ 2 v⎤ ⎢⎜ g ⎟ − g ⎥ + P(ln KP) vg ⎢⎣⎝ ∂T ⎠ T ⎥⎦ T P
(18)
a
(24)
( ∂∂TP )v
3. EXPERIMENTAL SECTION In the experiment, high-purity grade sample of propane supplied by Linde gas with an initial mole fraction purity of 0.995 was used. Its properties are evaluated using the Reference Fluid Thermodynamic and Transport Properties Database of NIST (REFPROP) with specific reference to the work of Lemmon et al.35The activated carbon,
(23)
The second terms of both the numerator and the denominator are much smaller than the first terms due to the presence of the 14496
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Figure 1. Schematic arrangement of the constant volume variable pressure experimental setup. Maxsorb III (by Kansai Coke Co., Japan), has been chosen to be the adsorbent in determining the adsorption characteristics of propane on Maxsorb III. The surface characteristics are determined using the AUTOSORB-1 MP microspore analyzer manufactured by Quantachrome Istruments, Boynton Beach, FL. It has a Brunauer−Emmet− Teller (BET) surface area of 3150 m2 g−1, total pore volume of 1.7 cm3 g−1, and an average pore diameter of 2.0 nm. Figure 1 illustrates the experimental setup of the constant volume variable pressure (CVVP) apparatus. It includes a stainless steel (SS 304) charging cell with internal volume of 1000 mL as well as an adsorption cell with an internal volume of 50 mL. 5.82 g of Maxsorb III with a packing density ranging between 0.1 and 0.12 g cm−3 was packed in the adsorption chamber. A water bath that is thermally stabilized using continuous circulating water from the constant temperature circulator (HAAKE F8-C35) is used to control the chamber temperature. The constant-temperature unit operates from 278.15 to 368.15 K with an accuracy of ±0.01 K. The uptakes are determined using both the temperature readings within the adsorption and charging cells using 1/3 DIN Pt 100 Ω RTDs as well as the pressure readings, which are read by two 0−5 MPa range KYOWA pressure transducers (PGS-50KA). The temperature sensors have an estimated uncertainty of ±0.15 K, while the pressure readings have an uncertainty of 0.1% of the rated full scale. The entire setup is evacuated prior to the experiments to a pressure of 0.005 kPa using a BOC Edwards direct drive vane vacuum pump. Regeneration for more than 24 h in the adsorption cell is carried out at temperatures over 393.15 K so that any residual gas in the cell is removed from the chamber. The propane gas is then introduced in the charging chamber. Both the pressure and the temperature are recorded when the pressure of the gas remains constant within the instrumental error. The pressure and temperature are used to evaluate the mass of gas adsorbed as per the work of Loh et al.25 The adsorption experiment is repeated with the bath maintained at temperatures ranging from 278.15 to 348.15 K and the charging cell charged to various pressures below propane’s saturation pressure (∼800 kPa) at the respective temperatures.
Po, the model grounded on statistical rate theory considers the reference pressure to depend on both the adsorbent + adsorbate characteristics rather than the conventional belief that it is only dependent on the adsorbate saturation properties.15 Furthermore, the current study has focused on incorporating the improved adsorbed phase specific volume term, which is not only a simplified model, but has also been found to yield improved corrections of the determined uptakes.36 Dubinin (1975) approximates the adsorbed phase specific volume (va) using the following equation: va = v boil exp[α(T − Tboil)]
(30)
where vboil is the specific volume of the liquid at its normal boiling point Tboil and α is the thermal expansion of the superheated liquid. Further, the temperature dependence of α is given by:
( )
ln α=
b v boil
Tcri − Tboil
(31)
where b is the van der Waals volume as described by Akkimaradi et al.37 and Saha et al.38 for the thermal expansion of the adsorbed phase of subcritical adsorption of HFC 134a and R507A on Maxsorb specimens of activated carbon. Srinivasan et al., on the other hand, assumes that the adsorbed phase is an equilibrium phase36 similar to but not the same as the liquid phase. This model also considers the effects of bulk phase gas density on the adsorbed phase because they are interacting with one another. These effects are related to the Gibbs criterion where pressure effects are the same in both the bulk and the adsorbed phases. The analogy of LRD,36 which has been shown to exhibit applicability for phase changes between vapor and liquid states along the coexistence curve, is also introduced. This gives rise to an averaged total density of the bulk gas and adsorbed phase, which is related to temperature linearly given by:
4. RESULTS 4.1. Adsorption Isotherms. For higher accuracy, it is necessary to estimate the adsorbed phase volume and the reference pressure (Po) in the D−A equation. In determining 14497
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Langmuir 1 1 + = A − BT va vg
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Loh et al.,25 n-butane of Saha et al.,39 and R-32 of Ahmed et al. (unpublished data of Ahmed A. Askalany, Kutub Uddin, Bidyut B. Saha, Takahiko Miyzaki, Shigeru Koyama, and Kandadai Srinivasan) are also fitted using this model, and the parameters are furnished in Table 1. 4.2. Adsorbed Phase Specific Volume (va). In Figure 4, the adsorbed phase specific volume obtained from the experimental data is found to increase exponentially with temperature T. This trend agrees with the previous activated carbon + methane work of Kazi (2011),29 which also describes the adsorption uptakes utilizing the Dubinin−Astakhov equation.36 In this model, the adsorbed phase is regarded as a separate equilibrium phase, rather than that of the liquid phase. Hence, pressure effects that are significant in pressurized systems such as that of activated carbon + propane systems are automatically considered as per Gibbs criterion.40 The plot illustrates the increased density of the adsorbed phase as compared to the gaseous phase, which defines the adsorption process itself. While this phenomenon has been ignored by the gas-phase assumption,41 the liquid-phase assumption of the earlier works gave an underestimate of the adsorbed phase volume. The previous model utilized by Kazi et al. (2011), which incorporates a thermal expansion constant α at 0.0025 K as proposed by Amankwah and Schwarz,42 has been superimposed as well. Here, this model is observed to give lower va values as compared to the underestimated liquid-phase model. 4.3. Heats of Adsorption. The K values obtained through regressions are found to be 6.802 × 10−9 K Pa−1 and εo = 43 644 J/mol. The calorimetrically measured experimental data are also plotted together. This phenomenon is a result of the surface structure of the activated carbon, which consists of mesopores of varying widths. Upon adsorption, the propane molecules adsorb quickly unto high energy sites, before it is adsorbed on sites with decreasing energy. The heat released from the change in energy of the propane molecules is thus much higher at lower uptake values. After the smaller pores are occupied, propane molecules are now taking up positions in the larger pores, and thus the adsorption affinity becomes weaker. The heats of adsorption (Hads) for the Maxsorb III + propane pair are calculated using eq 17 and plotted in Figure 5 against the adsorbate loading. The uptake experimental points evaluated for heat of adsorption are also calculated from the equation developed by Kazi (2011).29 It is plotted alongside for comparison. In both models, a similar trend is observed. The heats of adsorption logarithmically decrease with increasing adsorbate loading, which reaches a limiting value of approximately 18 000 J mol−1 as the saturation uptake limit is reached. The formulation obtain in the current work, however, is easier to compute numerically. The isosteric change in enthalpy upon adsorption for the Maxsorb III/propane pair may now be plotted for various loadings as shown in Figure 6 utilizing eq 17. It is compared to the latent heat of vaporization of propane. Figure 7 shows the heats of adsorption for various refrigerants, which is a useful comparison in choosing the working fluid in the adsorption chillers. This is given the findings of Banker et al. that the heat of adsorption is the most significant component of heat inventories that severely affects the coefficient of performance.43 Here, we observe the relatively higher heats of adsorption values of hydrocarbon refrigerants nbutane and propane as compared to the rest of the refrigerants at the same temperature of 298 K.
(32)
The expansion coefficient may thus be expressed as: ⎡ αg ⎤ α = va⎢B − ⎥ vg ⎥⎦ ⎢⎣
(33)
The regressed values of wo, E, n, and α for the DA model are found to be 1.629 cm3/g, 8577 J/mol, 1.26, and 7.75 × 10−4 K−1 with an error of 3.38%. On the other hand, Srinivasan et al.’s model yielded a 2.72% error with wo, E, n, A, and B to be 1.629 cm3/g, 8700 J/mol, 1.28, 0.3586 g/cm3, and 2 × 10−4 g cm−3 K−1, respectively. The propane adsorption data25 have been superimposed on the regressed values using both models in Figure 2. The numerical coefficient thus for the adsorption
Figure 2. Final regression of the experimental adsorption isotherm data with 5% error bars of propane on Maxsorb III regressed with the improved D−A equation with adsorbed phase volume correction (dotted lines are from Dubinin’s adsorbed phase volume model; full lines are from Srinivasan’s adsorbed phase volume model).
parameters using both models regressed from the experimental data has an overall regression deviation within approximately 5% of experimental data as shown in Figure 3. The adsorption uptake data of assorted refrigerants HFC-134a and R507a of
Figure 3. Comparison of adsorption uptake deviations between experimental uptake and predicted values using the various models. □,Toth model. ■, DA without volume correction. ○, DA with Dubinin’s adsorbed phase volume model. ●, DA with Srinivasan et al.’s adsorbed phase volume model. 14498
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Table 1. Numerical Values of the Parameters Wo, E, A, B, K, and εo for the DA Model with Volume Correction That Have Been Regressed from the Experimental Data with Maxsorb III as Adsorbent refrigerant samples W0/cm3 g−1 E/J mol−1 n A/g cm−3 103·B/g cm−3 K−1 AAD/kg kg−1 K/kPa−1 εo/− a
R-32a
R134a
R507a
n-butane
propane
4.05 3939 1.15 0 −1.546 0.03 1.90638 × 10−07 19 929
1.506 7332 1.28 1.626 0.4354 0.03 2.13754 × 10−07 21 964
0.708 7547 1.36 3.57 2.12 0.03 2.86706 × 10−07 19 600
0.336504235 17436 1.0 3.7459110 4.55 0.03 4.07715 × 10−07 22 859
3.33 8700 1.28 0.358 0.278 0.03 4.94773 × 10−07 19 000
Unpublished data of Ahmed A. Askalany, Kutub Uddin, Bidyut B. Saha, Takahiko Miyzaki, Shigeru Koyama, and Kandadai Srinivasan.
Figure 4. Temperature dependence of the adsorbed phase specific volume (va) in the present model obtained from the experimental data points (△) with the full red line representing the best isotherm fit obtained from the modified DA equation. The dotted red line represents the correlation utilized by Kazi (2011). The full black line represents the saturated gas specific volume at that temperature, and the dotted black line represents the liquid phase at saturated temperatures, which were both approximations used by previous authors.
Figure 6. Heat of adsorption (Hads) plots for Maxsorb III/propane pair drawn against the temperature (T). The full black line represents the heat of vaporization (hfg) for propane over temperatures from 220 to 370 K.
Figure 5. Heat of adsorption (Hads) for Maxsorb III/propane pair drawn against the adsorbate loading qva/Wo at the measured isotherm temperatures. The data points are calculated from the model of Kazi (2011), while the dotted line is from eq 17 used in this work.
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significant, resulting in a higher value of (cp,g). This same trend is depicted in the adsorbed phase volume of va as shown earlier. 4.5. Adsorbed Phase Entropy and Enthalpy. Here, the entropy and enthalpy (Figure 9) are calculated using eqs 13 and
Figure 7. Heat of adsorption (Hads) plots for different refrigerants on activated carbon powder (Maxsorb III) at a temperature (T) of 298 K.
4.4. Adsorbed Phase Specific Heat Capacity. As determined in the earlier formulations, the adsorbed phase specific heat capacity (cp,a) is a function of uptake (q), temperature (T), as well as pressure (P). The specific heat capacity of the bulk gaseous refrigerant phase (cp,g) on the other hand only depends on the pressure (P) and temperature (T). The expression for this has thus been developed in eq 29. The adsorbed phase specific heat capacity (cp,a) is plotted against temperature in Figure 8. These were plotted on the basis of the
Figure 9. Enthalpy of the adsorbed phase (ha) of Maxsorb III/propane pair for temperatures between 230 and 370 K and pressures up to 8 bar. The black, red dotted, and full red lines represent the saturated liquid, adsorbed, and gaseous phase, respectively.
17. The standard of the International Institute of Refrigeration (IIR) is used where reference state values of 1 kJ/(kg·K) and 200 kJ/kg are used for entropy and enthalpy for the saturated liquid at 0 °C.44 This is also the standard used by ASHRAE for most refrigerants.45 The location of the adsorbed phase on both the entropy and the enthalpy diagram may be explained by the diatomic hypothesis46 where matter is organized in pairs. The fundamental difference between the known solid, liquid, and gas phases is their degrees of freedom in terms of translational particle motion. In a solid, the particles do not have any degrees of freedom for translational motion. This is because each direction of motion is occupied by intermolecular force. In the liquid phase, two-dimensional motion is possible for the particles, and they are still under the influence of intermolecular force. In the gas phase, however, all directions of motions in space are possible. Hence, upon adsorption, the gas molecule transits from the free gas (with three degrees of translational freedom) to the adsorbed film (with two degrees of translational freedom) due to the van der Waals forces from the solid surface and therefore loses translational entropy. These therefore explain the evolution of heat during the adsorption process.
Figure 8. Specific heat capacity of the adsorbed phase (cp,a) of Maxsorb III/propane pair for temperatures between 270 and 370 K and pressures up to 8 bar. The red lines represent the gaseous phase specific heat capacities at the same temperatures and pressures.
properties of propane including the isobaric specific heat of the gaseous phase (cp,g) from the measurements of Lemmon et al.35 together with the uptake measurements obtained in this work, which gave the required constants A, B, and K. The present formulations as depicted in Figure 8 resolved the limitations of previous models33,34 for adsorption of refrigerants on activated carbon in the moderate pressure ranges. These include negative values of specific heat capacities as well as specific heat capacities of the adsorbed phase being always lower than the isobaric specific heat capacities of the gaseous phase. It also represents a more realistic model, which is dependent on the gaseous phase properties rather than that of the saturated liquid. In Figure 8, the adsorbed phase specific heat capacities are close to (in terms of trend) but always higher than the specific heat of the gaseous phase. As the pressure increases, the deviation from the gaseous phase also increases. This is attributed to the second term, which is dependent on P, T, ln KP, and the ratio P/T. As pressure increases, the isothermal compression effects are more
5. CONCLUSION The theoretical framework related to the operation of the pressurized adsorption applications, which are the adsorbed phase thermodynamic properties, have been developed and described in this Article. These include the specific volume of the adsorbed phase, its entropy, enthalpy, as well as specific heat capacity. The previous models developed by Chakraborty et al. (2009) and Kazi (2011) have been simplified by using results obtained from statistical rate theory. Further, the elimination of the ideal gas assumption in the formulation of the heats of adsorption improves the accuracy of the adsorbed phase properties. An improved model to predict the specific volume of the adsorbed phase further provides a better description of the mechanism involved in the physiosorption process. 14500
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Symposium, Qatar, January 10−14, 2010; Elsevier: New York, 2010; p 187. (14) Mooney, R.; Keenan, A.; Wood, L. Adsorption of water vapor by montmorillonite. I. Heat of desorption and application of BET theory. J. Am. Chem. Soc. 1952, 74, 1367−1371. (15) Rudzinski, W.; Panczyk, T. Kinetics of isothermal adsorption on energetically heterogeneous solid surfaces: a new theoretical description based on the statistical rate theory of interfacial transport. J. Phys. Chem. B 2000, 104, 9149−9162. (16) Kumar, K. V.; de Castro, M. M.; Martinez-Escandell, M.; Molina-Sabio, M.; Rodriguez-Reinoso, F. A site energy distribution function from Toth isotherm for adsorption of gases on heterogeneous surfaces. Phys. Chem. Chem. Phys. 2011, 13, 5753−5759. (17) Rudzinski, W.; Panczyk, T. Kinetics of gas adsorption in activated carbons, studied by applying the statistical rate theory of interfacial transport. J. Phys. Chem. B 2001, 105, 6858−6866. (18) Chakraborty, A.; Saha, B. B.; Koyama, S.; Ng, K. C.; Srinivasan, K. Adsorption thermodynamics of silica gel−water systems. J. Chem. Eng. Data 2008, 54, 448−452. (19) Evans, R.; Marconi, U. M. B.; Tarazona, P. Fluids in narrow pores: Adsorption, capillary condensation, and critical points. J. Chem. Phys. 1986, 84, 2376. (20) Chakraborty, A.; Bidyut, B.; Koyama, M.; Takata, Y.; Ng, K. C. Isosteric heat of adsorption for an adsorbent + adsorbate system. (21) Niazmand, H.; Talebian, H.; Mahdavikhah, M. Bed geometrical specification effects on the performance of silica/water adsorption chillers. Int. J. Refrig. 2012. (22) Talebian, H.; Niazmand, H. Numerical modeling of combined heat and mass transfer in an adsorbent bed with rectangular fins. J. Math. 2012. (23) Bottani, E. J.; Tascón, J. M. Adsorption by Carbons: Novel Carbon Adsorbents; Access Online via Elsevier, 2011. (24) Klein, N.; Henschel, A.; Kaskel, S. n-Butane adsorption on Cu3(btc)2 and MIL-101. Microporous Mesoporous Mater. 2010, 129, 238−242. (25) Loh, W. S.; Ismail, A. B.; Xi, B.; Ng, K. C.; Chun, W. G. Adsorption isotherms and isosteric enthalpy of adsorption for assorted refrigerants on activated carbons. J. Chem. Eng. Data 2012, 57, 2766− 2773. (26) Michalkova, A.; Gorb, L.; Hill, F.; Leszczynski, J. Can the Gibbs free energy of adsorption be predicted efficiently and accurately: An M05-2X DFT study. J. Phys. Chem. A 2011, 115, 2423−2430. (27) Düren, T.; Bae, Y.-S.; Snurr, R. Q. Using molecular simulation to characterise metal−organic frameworks for adsorption applications. Chem. Soc. Rev. 2009, 38, 1237−1247. (28) Do Duong, D. Absorption Analysis: Equilibria and Kinetics; Imperial College Press: London, 1998; Vol. 2. (29) Rahman, K. A. Experimental and theoretical studies on adsorbed natural gas storage system using activated carbons. ?????? 2011. (30) Everett, D. Thermodynamics of adsorption from solution. Part 1. Perfect systems. Trans. Faraday Soc. 1964, 60, 1803−1813. (31) Chakraborty, A.; Saha, B.; Koyama, S.; Ng, K. Specific heat capacity of a single component adsorbent-adsorbate system. Appl. Phys. Lett. 2007, 90, 171902−171902−3. (32) Ye, X.; Qi, N.; LeVan, M. D. Prediction of adsorption equilibrium using a modified D−R equation: organic−water vapor mixtures on BPL carbon. Carbon 2003, 41, 2519−2525. (33) Walton, K. S.; LeVan, M. D. Adsorbed-phase heat capacities: Thermodynamically consistent values determined from temperaturedependent equilibrium models. Ind. Eng. Chem. Res. 2005, 44, 178− 182. (34) Kluitenberg, G. 5.2 Heat capacity and specific heat. Methods of Soil Analysis: Part 4 Physical Methods; Soil Science Society of America: Madison, WI, 2002; pp 1201−1208. (35) Lemmon, E. W.; McLinden, M. O.; Wagner, W. Thermodynamic properties of propane. III. A reference equation of state for temperatures from the melting line to 650 K and pressures up to 1000 MPa. J. Chem. Eng. Data 2009, 54, 3141−3180.
These thermodynamic quantities are dependent on the properties of the temperature, pressure, and the adsorption parameters of a given adsorbent + adsorbate system. This approach is not limited to pressurized conditions but may be extended to vacuum conditions when the Dubinin−Astakhov model is used. With these values, an adsorption system may then be designed with greater ease given the simplicity of the equations. In this work, the quantities have been calculated from the adsorption uptake data for the relevant temperatures and pressures in cooling and refrigeration applications. Thus, the analysis that follows is expected to predict better the experimental results of the adsorption chiller system.
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AUTHOR INFORMATION
Corresponding Author
*Phone: +65 65162214. E-mail: [email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS A.B.I. is supported by the National Research Foundation Singapore under its National Research Foundation (NRF) Environmental and Water Technologies (EWT) Ph.D. Scholarship Programme and administered by the Environment and Water Industry Programme Office (EWI).
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REFERENCES
(1) Do, D.; Do, H. Adsorption of supercritical fluids in non-porous and porous carbons: analysis of adsorbed phase volume and density. Carbon 2003, 41, 1777−1791. (2) Foster, D.; Orlandini, E.; Tesi, M. Surface critical exponents for models of polymer collapse and adsorption: the universality of the Theta and Theta’ points. J. Phys. A: Math. Gen. 1992, 25, L1211. (3) Radha, A.; Bomati-Miguel, O.; Ushakov, S. V.; Navrotsky, A.; Tartaj, P. Surface enthalpy, enthalpy of water adsorption, and phase stability in nanocrystalline monoclinic zirconia. J. Am. Ceram. Soc. 2009, 92, 133−140. (4) Zhang, H.; Penn, R. L.; Hamers, R. J.; Banfield, J. F. Enhanced adsorption of molecules on surfaces of nanocrystalline particles. J. Phys. Chem. B 1999, 103, 4656−4662. (5) Israelachvili, J. N. Intermolecular and Surface Forces, revised 3rd ed.; Academic Press: New York, 2011. (6) Ross, S.; Olivier, J. P. On Physical Adsorption; ACS Publications: Washington, DC, 1964. (7) Khelifa, A.; Derriche, Z.; Bengueddach, A. Adsorption of propene on NaX zeolite exchanged with Zn2+ and Cu2+. Appl. Catal., A 1999, 178, 61−68. (8) Mueller-Plathe, F. Calculation of the free energy for gas absorption in amorphous polypropylene. Macromolecules 1991, 24, 6475−6479. (9) Myers, A.; Prausnitz, J. M. Thermodynamics of mixed-gas adsorption. AIChE J. 1965, 11, 121−127. (10) Chakraborty, A.; Saha, B. B.; Ng, K. C.; Koyama, S.; Srinivasan, K. Theoretical insight of physical adsorption for a single-component adsorbent+ adsorbate system: I. Thermodynamic property surfaces. Langmuir 2009, 25, 2204−2211. (11) Ginoux, J.; Lang, J.; Bonnetain, L. Analytical method applying Polanyi’s theory to adsorption on synthetic zeolites. Adv. Chem. Ser. 1973, 121, 382−391. (12) Rahman, K. A.; Chakraborty, A.; Saha, B. B.; Ng, K. C. On thermodynamics of methane+ carbonaceous materials adsorption. Int. J. Heat Mass Transfer 2012, 55, 565−573. (13) Rahman, K. A.; Loh, W. S.; Chakraborty, A.; Saha, B. B.; Ng, K. C. Adsorption Thermodynamics of Natural Gas Storage onto Pitch-Based Activated Carbons; Proceedings of the 2nd Annual Gas Processing 14501
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(36) Srinivasan, K.; Saha, B. B.; Ng, K. C.; Dutta, P.; Prasad, M. A method for the calculation of the adsorbed phase volume and pseudosaturation pressure from adsorption isotherm data on activated carbon. Phys. Chem. Chem. Phys. 2011, 13, 12559−12570. (37) Akkimaradi, B.; Prasad, M.; Dutta, P.; Srinivasan, K. Effect of packing density and adsorption parameters on the throughput of a thermal compressor. Carbon 2002, 40, 2855−2859. (38) Saha, B. B.; El-Sharkawy, I. I.; Habib, K.; Koyama, S.; Srinivasan, K. Adsorption of equal mass fraction near an azeotropic mixture of pentafluoroethane and 1, 1, 1-trifluoroethane on activated carbon. J. Chem. Eng. Data 2008, 53, 1872−1876. (39) Saha, B. B.; Chakraborty, A.; Koyama, S.; Yoon, S.-H.; Mochida, I.; Kumja, M.; Yap, C.; Ng, K. C. Isotherms and thermodynamics for the adsorption of n-butane on pitch based activated carbon. Int. J. Heat Mass Transfer 2008, 51, 1582−1589. (40) McGrother, S. C.; Gubbins, K. E. Constant pressure Gibbs ensemble Monte Carlo simulations of adsorption into narrow pores. Mol. Phys. 1999, 97, 955−965. (41) Loh, W.; El-Sharkawy, I.; Ng, K.; Saha, B. Adsorption cooling cycles for alternative adsorbent/adsorbate pairs working at partial vacuum and pressurized conditions. Appl. Therm. Eng. 2009, 29, 793− 798. (42) Amankwah, K.; Schwarz, J. A modified approach for estimating pseudo-vapor pressures in the application of the Dubinin-Astakhov equation. Carbon 1995, 33, 1313−1319. (43) Banker, N.; Srinivasan, K.; Prasad, M. Performance analysis of activated carbon+ HFC-134a adsorption coolers. Carbon 2004, 42, 117−127. (44) Younglove, B. A.; McLinden, M. O. An International Standard Equation of State for the Thermodynamic; 1995. (45) Handbook, A. HVAC systems and equipment. American Society of Heating, Refrigerating, and Air Conditioning Engineers, Atlanta, GA, 1996. (46) Prebeg, Z. Diatomic hypothesis and calculation of condensation force. J. Theor. 2000, 2, 2000−2001.
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On the Thermodynamics of Refrigerant + Heterogeneous Solid Surfaces Adsorption Azhar Bin Ismail,† Ang Li,† Kyaw Thu,† K. C. Ng,*,† and Wongee Chun‡ †
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576 Department of Nuclear and Energy, Jeju National University, Jeju, Korea
‡
ABSTRACT: This Article presents a theoretical framework for the understanding of pressurized adsorption systems using the statistical rate methodology. Utilizing results from the statistical rate theory, basic thermodynamic variables including enthalpy (ha), entropy (sa), and the specific heat capacity (cp,a) of the adsorbed phase are derived using the thermodynamic requirements of chemical equilibrium, Gibbs law, as well as Maxwell relations. A built-in constant (K) describes the adsorbed molecular partition function (q s ), and it captures the heterogeneous properties of the adsorbent + adsorbate pair at equilibrium states. Improved adsorbed-phase volume considerations were incorporated in the formulations of these variables where they could be utilized with relative ease for analyzing the energetic performances of any practical adsorption system. In this Article, we have demonstrated how derived thermodynamic quantities can bridge the information gap with respect to the states of adsorbed phase, as well as resolved some theoretical inconsistencies that were found in previously derived quantities. Experimentally, the adsorption isotherms of propane (refrigerant) on activated carbon powder (Maxsorb III) for temperatures from 5 to 75 °C and pressures up to 8 bar are presented, and they are used to illustrate the behaviors of the adsorbed-phase during uptakes, temperatures, and pressure excursions or changes.
1. INTRODUCTION When an adsorbate molecule is adsorbed on an adsorbent surface, the energy of the adsorbate molecules is reduced as it becomes more stabilized. This results in a phase transformation from gaseous to what is termed as the adsorbed phase.1 From a thermodynamic perspective, this phase is distinguishable, but there is uncertainty however in the precise location of the phase boundary.2 Thus, the adsorbed phase is considered to be a thermodynamic state, which is dependent on the adsorption uptake (q) in addition to the temperature (Te) and pressure (Pe).1 Its internal energy, enthalpy, entropy, heat of adsorption, and specific heat capacity are thermodynamic quantities that are necessary to analyze an adsorption system. This Article presents the theoretical framework in which thermodynamic quantities of the adsorbed phase, including the heats of adsorption, specific volume, entropy, specific heat capacity, as well as enthalpy, are developed. The formulations of the adsorbent + adsorbate pair are first determined from the uptake data. This framework is grounded upon the statistical rate theory with the rigor of thermodynamics.
molecule is attached to the adsorbent surface via van der Waals forces and loses a translational degree of freedom.5 In other words, an adsorbate molecule, which initially has three degrees of translational freedom in the gas phase,6,7 now has two degrees of freedom in the adsorbed phase. Adsorption is thus an exothermic process unless the adsorbed state is characterized by a very large additional entropy such as those attributed to vibrations. This is mathematically described by the Gibbs free energy, which governs thermodynamic systems at constant temperatures and pressures8 [because q = q(P,T)] as follows:
2. THERMODYNAMICS FRAMEWORK 2.1. Gibbs Free Energy. Physical adsorption is characterized by the release of heat, moving to a lower, more thermodynamically stable energy state, and is thus a spontaneous process.3,4 During adsorption, the adsorbate
(2)
© 2013 American Chemical Society
ΔGads = ΔHads − T ΔSads
(1)
where ΔHads = Ha − Hg. The chemical potentials of the associated gas and adsorbed phases are defined as:9 ⎛ ∂G ⎞ μa = ⎜ a ⎟ ⎝ ∂na ⎠T , P Received: August 29, 2013 Revised: November 4, 2013 Published: November 5, 2013 14494
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⎛ ∂Gg ⎞ ⎟⎟ μg = ⎜⎜ ⎝ ∂ng ⎠T , P
over the range of the adsorption energies. By describing χ(ε) with a suitable Gaussian function, several authors have successfully rearrived at the well-known Langmuir−Freundlich, Dubinin−Astakhov (DA), Dubinin−Radushkevich (DR), and Toth isotherm equations.16,17 Because the uptake of an adsorbent + adsorbate system is a function of its equilibrium temperature T and pressure P, εc could thus be regarded as a constant along an isostere where its equilibrium pressure P and its partial derivative ∂P/∂T are given by eqs 11 and 12, respectively.
(3)
Following the isothermal assumption, when the pressure of the adsorbent−adsorbate system is increased, the number of gas molecules in the system has to be increased by dng, altering the equilibrium state of the adsorbed phase and the gas phase of adsorbate molecules. When this happens, the thermodynamic systems consequently transfer molecules toward a lower free energy state until the chemical potentials of the gas phase and the adsorbed phase are equal. This follows: dG = (μa − μg ) dna = 0
(4)
μa = μg
(5)
P=
sa = sg +
⎛ ∂Sg ⎞ ⎛ ∂V ⎞ ⎛ ∂μ ⎞ ⎟⎟ dT + ⎜⎜ g ⎟⎟ dP + ⎜⎜ g ⎟⎟ dng dμg = −⎜⎜ ⎝ ∂ng ⎠T , P ⎝ ∂ng ⎠T , P ⎝ ∂ng ⎠T , P (7)
where dna = −dng, (∂Si/∂ni)P,T = si, and (∂Vi/∂ni)P,T = vi. Substituting these into eqs 4 and 5 reproduces the classical Clausius−Clayperon equation,6,14 which is
(9)
Hence, the expression for the adsorbed phase entropy along an isostere simplifies to: sa = sg −
⎛ ∂P ⎞ ⎜ ⎟ (v − v ) a ⎝ ∂T ⎠n g a
P(vg − va) T
ln KP
(13)
2.3. Adsorbed Phase Enthalpy (ha). One of the most important thermodynamic properties that needs to be determined in modeling the adsorption chiller is the adsorbed phase enthalpy, ha. In the derivation of ha, it is found that several authors obtained a latent heat of vaporization term of the adsorbate, hfg,10,18 even though the adsorbed phase is not related to the liquid phase.19 Further, in the derivation of the heat of adsorption, several considerations were made including the use of the ideal gas equation,20 and the assumption of the specific volume of the adsorbed phase to be either liquid, gaseous, or much smaller than the gaseous phase.21−23 The use of the excess adsorption uptake from the experimental data24,25 contributes to further inaccuracies of the derived isotherm properties. The determination of the thermodynamic quantities should therefore utilize absolute uptakes where volumetric corrections are considered. The ideal gas law postulates that the gas molecules are noninteracting and occupies inherent volume. In refrigerant-adsorption applications where low temperatures and high pressures are desired, both the volume and the interaction effects become strikingly apparent. Thus, nonideality of the adsorbed phase could not be ignored. In this work, we arrive at the change of enthalpy incorporating the correction to the adsorbed phase volume without the use of the ideal gas relations. Further, the adsorbed phase volume is calculated from the absolute instead of the excess adsorption uptake values. Invoking the Gibb’s free energy equation, the enthalpy of adsorption is given as:26
(6)
sg − sa dP = dT vg − va
(12)
The expression for the adsorbed phase entropy therefore becomes:
⎛ ∂S ⎞ ⎛ ∂V ⎞ ⎛ ∂μ ⎞ dμa = −⎜ a ⎟ dT + ⎜ a ⎟ dP + ⎜ a ⎟ dna ⎝ ∂na ⎠T , P ⎝ ∂na ⎠T , P ⎝ ∂na ⎠T , P
(8)
(11)
εc ⎛ ∂P ⎞ ⎛ ε ⎞ P ⎜ ⎟ exp⎜ − c ⎟ = − ln KP = 2 ⎝ RT ⎠ ⎝ ∂T ⎠n T KRT a
2.2. Adsorbed Phase Entropy (sa). Previous works that dealt with adsorbed phase properties have shown that its entropy is a function of the system’s temperature (T), pressure (P), as well as the adsorbate uptake (q).10−13 The total differential of this term (dsa ) during adsorption from mathematics is thus the addition of the partial change in entropy with respect to these three thermodynamic properties. This is expressed utilizing the derivatives of μa and μg, which are functions of T, P, and n that are related to the uptake q, as a total derivative given by:
−sa dT + va dP = −sg dT + vg dP
⎛ ε ⎞ 1 exp⎜ − c ⎟ ⎝ RT ⎠ K
(10)
Rudzinski and co-workers previously defined εc, which is a function of the adsorbent + adsorbate system’s equilibrium temperature T and pressure P given by −RT ln(K × P).15 R is the universal gas constant or the Boltzmann constant, in J mol−1 K−1. K is described by both qs, the molecular particle function of the adsorbate gas molecules, and μgo, the chemical potential of the gas phase at a reference pressure, given by zi exp(μgo/RT). εc has been incorporated in the homogeneous energy-surface model of the Langmuir isotherm equation, which could then elucidated as exp(ε − εc/RT)/[1 + exp (ε − εc/RT)]. The condensation approximation (CA) is then invoked so that the averaged surface coverage θt of an energetically heterogeneous surface could be mathematically described as an integral of the adsorption site distribution, χ(ε),
ΔHads(na) = T ΔSads
(14)
ha − hg = T (sa − sg)
(15)
Substituting eq 13 into eq 15: ha = hg + P(vg − va) ln KP
(16)
In other words, the change in enthalpy of adsorption is given by:15 ΔHads(na) = −P(vg − va) ln(K × P) 14495
(17)
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∂va /∂T at constant specific volumes and temperature, respectively.
which is a positive term given that the product of (K × P) is negative. It has previously been expounded how an adsorbate molecule undergoes a phase change from a higher to a lower energy level during adsorption.27 This results in a heat release known as the isosteric heat of adsorption. Do (1998) defined isosteric heat as “the ratio of the infinitesimal change in the adsorbate enthalpy to the infinitesimal change in the amount adsorbed.”28 In general, this value is 30−100% larger than the adsorbate’s heat of vaporization29 but is much lower as compared to a chemical adsorption process. 2.4. Adsorbed Phase Specific Heat Capacity (cp,a). The adsorbed phase is distinct from the liquid and gas phases, and thus it is necessary to obtain an expression for its specific heat capacity at constant pressure cp,a. Until recently, the adsorbed phase has been assumed to be gaseous6 or liquid.30 Several specific heat capacity models thus assume values of the reference gas phase,31 while others gave that of the liquid phase.32 Further, there are several existing issues with available models in terms of thermodynamic consistency. Walton and Levan (2005) highlighted how these values will approach negative at higher loadings.33 Furthermore, the value of the specific heat capacity obtained would vary widely depending on which isotherm model is used. Several other expressions lead to a specific heat capacity term to be always lower than that in the gas phase. In deriving the specific heat capacity of the adsorbed phase, its definition, similar to that for the liquid and gaseous phases along the isoteres as per the previous works, is used:34 ⎛ ∂hg ⎞ ⎛ ∂h ⎞ ⎛ ∂H ⎞ ⎟ − ⎜ ads ⎟ cp ,a = ⎜ a ⎟ = ⎜ ⎝ ∂T ⎠n ⎝ ∂T ⎠n T ∂ ⎠n ⎝ a a
⎛ ∂ 2P ⎞ ⎛ ∂Hads ⎞ ⎛ ∂P ⎞ ⎜ ⎟ ≈ ⎜ ⎟ (vg − va) + T ⎜ 2 ⎟ (vg − va) ⎝ ∂T ⎠n ⎝ ∂T ⎠n ⎝ ∂T ⎠n a a a
⎡ ⎤ ⎛ ∂v ⎞ ⎛ ∂P ⎞ ⎛ ∂vg ⎞ + T ⎜ ⎟ ⎢⎜ ⎟ − ⎜ a ⎟ ⎥ ⎝ ∂T ⎠n ⎢⎝ ∂T ⎠ ⎝ ∂T ⎠n ⎥ a⎣ n a⎦ a
⎛ ∂Hads ⎞ (P ln KP)2 ⎛ ∂vg ⎞ ⎜ ⎟ = ⎜ ⎟ − (P ln KP) ⎝ ∂T ⎠n T ⎝ ∂P ⎠
⎛ ∂hg ⎞ ⎛ ∂q ⎞ ⎛ ∂hg ⎞ ⎟ ⎜ ⎟ ⎟ = −⎜ ⎜ ⎝ ∂q ⎠T ⎝ ∂T ⎠h ⎝ ∂T ⎠n
Similarly: ⎛ ∂hg ⎞ ⎟ ≈− ⎜ ⎝ ∂T ⎠n a
⎛∂ P⎞ ⎛ ∂Hads ⎞ ⎛ ∂P ⎞ ⎜ ⎟ = ⎜ ⎟ (vg − va) + T ⎜ 2 ⎟ (vg − va) ⎝ ∂T ⎠n ⎝ ∂T ⎠n ⎝ ∂T ⎠n a a
⎛ ∂vg ⎞ ⎛ vg ⎞ ⎜ ⎟ − P(ln KP)2 ⎜ ⎟ ⎝T ⎠ ⎝ ∂P ⎠T
From the expression of (∂P/∂T)na = εc/KRT exp(−εc/RT): 2
(29)
With these derived thermodynamic properties, an improved thermodynamic framework has been developed. The key difference from the expressions utilized by Kazi et al. (2011)12 is the involvement of the results from the statistical rate theory. While employing the same thermodynamic definitions, this model offers simplified expressions for isosteric heat and specific heat capacity at constant pressure. The enthalpy and entropy, as described by eqs 13 and 15, are functions of equilibrium pressures (P) and temperatures (T) at a specified uptake (q). It is calculated with respect to the gaseous phase enthalpy and entropy.
(21)
Because vg = vg(q, T), (∂vg/∂T)na = (∂vg/∂T)q may be derived using the cyclic relation:
(22)
Therefore:
( ∂v )T
(28)
⎡⎛ ∂v ⎞ vg ⎤ (P ln KP)2 g cp ,a = cp ,g + 2(P ln KP)⎢⎜ ⎟ − ⎥ − ⎢⎣⎝ ∂T ⎠ T ⎥⎦ T P (20)
a
P
(27)
Finally, the expression for the specific heat capacity of the adsorbed phase is derived as follows:
a
⎤ ⎡ ⎛ ∂v ⎞ ⎛ ∂P ⎞ ⎛ ∂vg ⎞ + T ⎜ ⎟ ⎢⎜ ⎟ − ⎜ a ⎟ ⎥ ⎝ ∂T ⎠n ⎢⎝ ∂T ⎠ ⎝ ∂T ⎠n ⎥ a⎣ n a⎦
ln KP + ⎛ ∂vg ⎞ T ⎜ ⎟ =− ∂P ⎝ ∂T ⎠q
= cp ,g
⎡⎛ ∂v ⎞ ⎛ ∂hg ⎞ vg ⎤ g ⎟ = cp ,g + P ln KP ⎢⎜ ⎟ − ⎥ ⎜ ⎢⎣⎝ ∂T ⎠ T ⎥⎦ ⎝ ∂T ⎠n P a
2
T
( ∂∂TP )h + TP ln KP ( ∂∂Ph )T
⎡⎛ ∂v ⎞ vg ⎤ g + P ln KP ⎢⎜ ⎟ − ⎥ ⎢⎣⎝ ∂T ⎠ T ⎥⎦ P
(19)
a
⎛ ∂vg ⎞ ⎛ ∂T ⎞ ⎛ ∂q ⎞ ⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎟⎟ = −1 ⎝ ∂T ⎠q⎝ ∂q ⎠v ⎝ ∂vg ⎠
(26)
a
From eq 27:
⎛ ∂P ⎞ ΔHads(na) = T ⎜ ⎟ (vg − va) ⎝ ∂T ⎠n
a
(25)
Using the cyclic relation:
From the change in enthalpy derived earlier:
⎛ ∂ 2P ⎞ P(ln KP) [2 + ln KP] ⎜ 2⎟ = ⎝ ∂T ⎠n T2
T
a
⎡⎛ ∂v ⎞ 2 v⎤ ⎢⎜ g ⎟ − g ⎥ + P(ln KP) vg ⎢⎣⎝ ∂T ⎠ T ⎥⎦ T P
(18)
a
(24)
( ∂∂TP )v
3. EXPERIMENTAL SECTION In the experiment, high-purity grade sample of propane supplied by Linde gas with an initial mole fraction purity of 0.995 was used. Its properties are evaluated using the Reference Fluid Thermodynamic and Transport Properties Database of NIST (REFPROP) with specific reference to the work of Lemmon et al.35The activated carbon,
(23)
The second terms of both the numerator and the denominator are much smaller than the first terms due to the presence of the 14496
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Figure 1. Schematic arrangement of the constant volume variable pressure experimental setup. Maxsorb III (by Kansai Coke Co., Japan), has been chosen to be the adsorbent in determining the adsorption characteristics of propane on Maxsorb III. The surface characteristics are determined using the AUTOSORB-1 MP microspore analyzer manufactured by Quantachrome Istruments, Boynton Beach, FL. It has a Brunauer−Emmet− Teller (BET) surface area of 3150 m2 g−1, total pore volume of 1.7 cm3 g−1, and an average pore diameter of 2.0 nm. Figure 1 illustrates the experimental setup of the constant volume variable pressure (CVVP) apparatus. It includes a stainless steel (SS 304) charging cell with internal volume of 1000 mL as well as an adsorption cell with an internal volume of 50 mL. 5.82 g of Maxsorb III with a packing density ranging between 0.1 and 0.12 g cm−3 was packed in the adsorption chamber. A water bath that is thermally stabilized using continuous circulating water from the constant temperature circulator (HAAKE F8-C35) is used to control the chamber temperature. The constant-temperature unit operates from 278.15 to 368.15 K with an accuracy of ±0.01 K. The uptakes are determined using both the temperature readings within the adsorption and charging cells using 1/3 DIN Pt 100 Ω RTDs as well as the pressure readings, which are read by two 0−5 MPa range KYOWA pressure transducers (PGS-50KA). The temperature sensors have an estimated uncertainty of ±0.15 K, while the pressure readings have an uncertainty of 0.1% of the rated full scale. The entire setup is evacuated prior to the experiments to a pressure of 0.005 kPa using a BOC Edwards direct drive vane vacuum pump. Regeneration for more than 24 h in the adsorption cell is carried out at temperatures over 393.15 K so that any residual gas in the cell is removed from the chamber. The propane gas is then introduced in the charging chamber. Both the pressure and the temperature are recorded when the pressure of the gas remains constant within the instrumental error. The pressure and temperature are used to evaluate the mass of gas adsorbed as per the work of Loh et al.25 The adsorption experiment is repeated with the bath maintained at temperatures ranging from 278.15 to 348.15 K and the charging cell charged to various pressures below propane’s saturation pressure (∼800 kPa) at the respective temperatures.
Po, the model grounded on statistical rate theory considers the reference pressure to depend on both the adsorbent + adsorbate characteristics rather than the conventional belief that it is only dependent on the adsorbate saturation properties.15 Furthermore, the current study has focused on incorporating the improved adsorbed phase specific volume term, which is not only a simplified model, but has also been found to yield improved corrections of the determined uptakes.36 Dubinin (1975) approximates the adsorbed phase specific volume (va) using the following equation: va = v boil exp[α(T − Tboil)]
(30)
where vboil is the specific volume of the liquid at its normal boiling point Tboil and α is the thermal expansion of the superheated liquid. Further, the temperature dependence of α is given by:
( )
ln α=
b v boil
Tcri − Tboil
(31)
where b is the van der Waals volume as described by Akkimaradi et al.37 and Saha et al.38 for the thermal expansion of the adsorbed phase of subcritical adsorption of HFC 134a and R507A on Maxsorb specimens of activated carbon. Srinivasan et al., on the other hand, assumes that the adsorbed phase is an equilibrium phase36 similar to but not the same as the liquid phase. This model also considers the effects of bulk phase gas density on the adsorbed phase because they are interacting with one another. These effects are related to the Gibbs criterion where pressure effects are the same in both the bulk and the adsorbed phases. The analogy of LRD,36 which has been shown to exhibit applicability for phase changes between vapor and liquid states along the coexistence curve, is also introduced. This gives rise to an averaged total density of the bulk gas and adsorbed phase, which is related to temperature linearly given by:
4. RESULTS 4.1. Adsorption Isotherms. For higher accuracy, it is necessary to estimate the adsorbed phase volume and the reference pressure (Po) in the D−A equation. In determining 14497
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Loh et al.,25 n-butane of Saha et al.,39 and R-32 of Ahmed et al. (unpublished data of Ahmed A. Askalany, Kutub Uddin, Bidyut B. Saha, Takahiko Miyzaki, Shigeru Koyama, and Kandadai Srinivasan) are also fitted using this model, and the parameters are furnished in Table 1. 4.2. Adsorbed Phase Specific Volume (va). In Figure 4, the adsorbed phase specific volume obtained from the experimental data is found to increase exponentially with temperature T. This trend agrees with the previous activated carbon + methane work of Kazi (2011),29 which also describes the adsorption uptakes utilizing the Dubinin−Astakhov equation.36 In this model, the adsorbed phase is regarded as a separate equilibrium phase, rather than that of the liquid phase. Hence, pressure effects that are significant in pressurized systems such as that of activated carbon + propane systems are automatically considered as per Gibbs criterion.40 The plot illustrates the increased density of the adsorbed phase as compared to the gaseous phase, which defines the adsorption process itself. While this phenomenon has been ignored by the gas-phase assumption,41 the liquid-phase assumption of the earlier works gave an underestimate of the adsorbed phase volume. The previous model utilized by Kazi et al. (2011), which incorporates a thermal expansion constant α at 0.0025 K as proposed by Amankwah and Schwarz,42 has been superimposed as well. Here, this model is observed to give lower va values as compared to the underestimated liquid-phase model. 4.3. Heats of Adsorption. The K values obtained through regressions are found to be 6.802 × 10−9 K Pa−1 and εo = 43 644 J/mol. The calorimetrically measured experimental data are also plotted together. This phenomenon is a result of the surface structure of the activated carbon, which consists of mesopores of varying widths. Upon adsorption, the propane molecules adsorb quickly unto high energy sites, before it is adsorbed on sites with decreasing energy. The heat released from the change in energy of the propane molecules is thus much higher at lower uptake values. After the smaller pores are occupied, propane molecules are now taking up positions in the larger pores, and thus the adsorption affinity becomes weaker. The heats of adsorption (Hads) for the Maxsorb III + propane pair are calculated using eq 17 and plotted in Figure 5 against the adsorbate loading. The uptake experimental points evaluated for heat of adsorption are also calculated from the equation developed by Kazi (2011).29 It is plotted alongside for comparison. In both models, a similar trend is observed. The heats of adsorption logarithmically decrease with increasing adsorbate loading, which reaches a limiting value of approximately 18 000 J mol−1 as the saturation uptake limit is reached. The formulation obtain in the current work, however, is easier to compute numerically. The isosteric change in enthalpy upon adsorption for the Maxsorb III/propane pair may now be plotted for various loadings as shown in Figure 6 utilizing eq 17. It is compared to the latent heat of vaporization of propane. Figure 7 shows the heats of adsorption for various refrigerants, which is a useful comparison in choosing the working fluid in the adsorption chillers. This is given the findings of Banker et al. that the heat of adsorption is the most significant component of heat inventories that severely affects the coefficient of performance.43 Here, we observe the relatively higher heats of adsorption values of hydrocarbon refrigerants nbutane and propane as compared to the rest of the refrigerants at the same temperature of 298 K.
(32)
The expansion coefficient may thus be expressed as: ⎡ αg ⎤ α = va⎢B − ⎥ vg ⎥⎦ ⎢⎣
(33)
The regressed values of wo, E, n, and α for the DA model are found to be 1.629 cm3/g, 8577 J/mol, 1.26, and 7.75 × 10−4 K−1 with an error of 3.38%. On the other hand, Srinivasan et al.’s model yielded a 2.72% error with wo, E, n, A, and B to be 1.629 cm3/g, 8700 J/mol, 1.28, 0.3586 g/cm3, and 2 × 10−4 g cm−3 K−1, respectively. The propane adsorption data25 have been superimposed on the regressed values using both models in Figure 2. The numerical coefficient thus for the adsorption
Figure 2. Final regression of the experimental adsorption isotherm data with 5% error bars of propane on Maxsorb III regressed with the improved D−A equation with adsorbed phase volume correction (dotted lines are from Dubinin’s adsorbed phase volume model; full lines are from Srinivasan’s adsorbed phase volume model).
parameters using both models regressed from the experimental data has an overall regression deviation within approximately 5% of experimental data as shown in Figure 3. The adsorption uptake data of assorted refrigerants HFC-134a and R507a of
Figure 3. Comparison of adsorption uptake deviations between experimental uptake and predicted values using the various models. □,Toth model. ■, DA without volume correction. ○, DA with Dubinin’s adsorbed phase volume model. ●, DA with Srinivasan et al.’s adsorbed phase volume model. 14498
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Table 1. Numerical Values of the Parameters Wo, E, A, B, K, and εo for the DA Model with Volume Correction That Have Been Regressed from the Experimental Data with Maxsorb III as Adsorbent refrigerant samples W0/cm3 g−1 E/J mol−1 n A/g cm−3 103·B/g cm−3 K−1 AAD/kg kg−1 K/kPa−1 εo/− a
R-32a
R134a
R507a
n-butane
propane
4.05 3939 1.15 0 −1.546 0.03 1.90638 × 10−07 19 929
1.506 7332 1.28 1.626 0.4354 0.03 2.13754 × 10−07 21 964
0.708 7547 1.36 3.57 2.12 0.03 2.86706 × 10−07 19 600
0.336504235 17436 1.0 3.7459110 4.55 0.03 4.07715 × 10−07 22 859
3.33 8700 1.28 0.358 0.278 0.03 4.94773 × 10−07 19 000
Unpublished data of Ahmed A. Askalany, Kutub Uddin, Bidyut B. Saha, Takahiko Miyzaki, Shigeru Koyama, and Kandadai Srinivasan.
Figure 4. Temperature dependence of the adsorbed phase specific volume (va) in the present model obtained from the experimental data points (△) with the full red line representing the best isotherm fit obtained from the modified DA equation. The dotted red line represents the correlation utilized by Kazi (2011). The full black line represents the saturated gas specific volume at that temperature, and the dotted black line represents the liquid phase at saturated temperatures, which were both approximations used by previous authors.
Figure 6. Heat of adsorption (Hads) plots for Maxsorb III/propane pair drawn against the temperature (T). The full black line represents the heat of vaporization (hfg) for propane over temperatures from 220 to 370 K.
Figure 5. Heat of adsorption (Hads) for Maxsorb III/propane pair drawn against the adsorbate loading qva/Wo at the measured isotherm temperatures. The data points are calculated from the model of Kazi (2011), while the dotted line is from eq 17 used in this work.
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significant, resulting in a higher value of (cp,g). This same trend is depicted in the adsorbed phase volume of va as shown earlier. 4.5. Adsorbed Phase Entropy and Enthalpy. Here, the entropy and enthalpy (Figure 9) are calculated using eqs 13 and
Figure 7. Heat of adsorption (Hads) plots for different refrigerants on activated carbon powder (Maxsorb III) at a temperature (T) of 298 K.
4.4. Adsorbed Phase Specific Heat Capacity. As determined in the earlier formulations, the adsorbed phase specific heat capacity (cp,a) is a function of uptake (q), temperature (T), as well as pressure (P). The specific heat capacity of the bulk gaseous refrigerant phase (cp,g) on the other hand only depends on the pressure (P) and temperature (T). The expression for this has thus been developed in eq 29. The adsorbed phase specific heat capacity (cp,a) is plotted against temperature in Figure 8. These were plotted on the basis of the
Figure 9. Enthalpy of the adsorbed phase (ha) of Maxsorb III/propane pair for temperatures between 230 and 370 K and pressures up to 8 bar. The black, red dotted, and full red lines represent the saturated liquid, adsorbed, and gaseous phase, respectively.
17. The standard of the International Institute of Refrigeration (IIR) is used where reference state values of 1 kJ/(kg·K) and 200 kJ/kg are used for entropy and enthalpy for the saturated liquid at 0 °C.44 This is also the standard used by ASHRAE for most refrigerants.45 The location of the adsorbed phase on both the entropy and the enthalpy diagram may be explained by the diatomic hypothesis46 where matter is organized in pairs. The fundamental difference between the known solid, liquid, and gas phases is their degrees of freedom in terms of translational particle motion. In a solid, the particles do not have any degrees of freedom for translational motion. This is because each direction of motion is occupied by intermolecular force. In the liquid phase, two-dimensional motion is possible for the particles, and they are still under the influence of intermolecular force. In the gas phase, however, all directions of motions in space are possible. Hence, upon adsorption, the gas molecule transits from the free gas (with three degrees of translational freedom) to the adsorbed film (with two degrees of translational freedom) due to the van der Waals forces from the solid surface and therefore loses translational entropy. These therefore explain the evolution of heat during the adsorption process.
Figure 8. Specific heat capacity of the adsorbed phase (cp,a) of Maxsorb III/propane pair for temperatures between 270 and 370 K and pressures up to 8 bar. The red lines represent the gaseous phase specific heat capacities at the same temperatures and pressures.
properties of propane including the isobaric specific heat of the gaseous phase (cp,g) from the measurements of Lemmon et al.35 together with the uptake measurements obtained in this work, which gave the required constants A, B, and K. The present formulations as depicted in Figure 8 resolved the limitations of previous models33,34 for adsorption of refrigerants on activated carbon in the moderate pressure ranges. These include negative values of specific heat capacities as well as specific heat capacities of the adsorbed phase being always lower than the isobaric specific heat capacities of the gaseous phase. It also represents a more realistic model, which is dependent on the gaseous phase properties rather than that of the saturated liquid. In Figure 8, the adsorbed phase specific heat capacities are close to (in terms of trend) but always higher than the specific heat of the gaseous phase. As the pressure increases, the deviation from the gaseous phase also increases. This is attributed to the second term, which is dependent on P, T, ln KP, and the ratio P/T. As pressure increases, the isothermal compression effects are more
5. CONCLUSION The theoretical framework related to the operation of the pressurized adsorption applications, which are the adsorbed phase thermodynamic properties, have been developed and described in this Article. These include the specific volume of the adsorbed phase, its entropy, enthalpy, as well as specific heat capacity. The previous models developed by Chakraborty et al. (2009) and Kazi (2011) have been simplified by using results obtained from statistical rate theory. Further, the elimination of the ideal gas assumption in the formulation of the heats of adsorption improves the accuracy of the adsorbed phase properties. An improved model to predict the specific volume of the adsorbed phase further provides a better description of the mechanism involved in the physiosorption process. 14500
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Symposium, Qatar, January 10−14, 2010; Elsevier: New York, 2010; p 187. (14) Mooney, R.; Keenan, A.; Wood, L. Adsorption of water vapor by montmorillonite. I. Heat of desorption and application of BET theory. J. Am. Chem. Soc. 1952, 74, 1367−1371. (15) Rudzinski, W.; Panczyk, T. Kinetics of isothermal adsorption on energetically heterogeneous solid surfaces: a new theoretical description based on the statistical rate theory of interfacial transport. J. Phys. Chem. B 2000, 104, 9149−9162. (16) Kumar, K. V.; de Castro, M. M.; Martinez-Escandell, M.; Molina-Sabio, M.; Rodriguez-Reinoso, F. A site energy distribution function from Toth isotherm for adsorption of gases on heterogeneous surfaces. Phys. Chem. Chem. Phys. 2011, 13, 5753−5759. (17) Rudzinski, W.; Panczyk, T. Kinetics of gas adsorption in activated carbons, studied by applying the statistical rate theory of interfacial transport. J. Phys. Chem. B 2001, 105, 6858−6866. (18) Chakraborty, A.; Saha, B. B.; Koyama, S.; Ng, K. C.; Srinivasan, K. Adsorption thermodynamics of silica gel−water systems. J. Chem. Eng. Data 2008, 54, 448−452. (19) Evans, R.; Marconi, U. M. B.; Tarazona, P. Fluids in narrow pores: Adsorption, capillary condensation, and critical points. J. Chem. Phys. 1986, 84, 2376. (20) Chakraborty, A.; Bidyut, B.; Koyama, M.; Takata, Y.; Ng, K. C. Isosteric heat of adsorption for an adsorbent + adsorbate system. (21) Niazmand, H.; Talebian, H.; Mahdavikhah, M. Bed geometrical specification effects on the performance of silica/water adsorption chillers. Int. J. Refrig. 2012. (22) Talebian, H.; Niazmand, H. Numerical modeling of combined heat and mass transfer in an adsorbent bed with rectangular fins. J. Math. 2012. (23) Bottani, E. J.; Tascón, J. M. Adsorption by Carbons: Novel Carbon Adsorbents; Access Online via Elsevier, 2011. (24) Klein, N.; Henschel, A.; Kaskel, S. n-Butane adsorption on Cu3(btc)2 and MIL-101. Microporous Mesoporous Mater. 2010, 129, 238−242. (25) Loh, W. S.; Ismail, A. B.; Xi, B.; Ng, K. C.; Chun, W. G. Adsorption isotherms and isosteric enthalpy of adsorption for assorted refrigerants on activated carbons. J. Chem. Eng. Data 2012, 57, 2766− 2773. (26) Michalkova, A.; Gorb, L.; Hill, F.; Leszczynski, J. Can the Gibbs free energy of adsorption be predicted efficiently and accurately: An M05-2X DFT study. J. Phys. Chem. A 2011, 115, 2423−2430. (27) Düren, T.; Bae, Y.-S.; Snurr, R. Q. Using molecular simulation to characterise metal−organic frameworks for adsorption applications. Chem. Soc. Rev. 2009, 38, 1237−1247. (28) Do Duong, D. Absorption Analysis: Equilibria and Kinetics; Imperial College Press: London, 1998; Vol. 2. (29) Rahman, K. A. Experimental and theoretical studies on adsorbed natural gas storage system using activated carbons. ?????? 2011. (30) Everett, D. Thermodynamics of adsorption from solution. Part 1. Perfect systems. Trans. Faraday Soc. 1964, 60, 1803−1813. (31) Chakraborty, A.; Saha, B.; Koyama, S.; Ng, K. Specific heat capacity of a single component adsorbent-adsorbate system. Appl. Phys. Lett. 2007, 90, 171902−171902−3. (32) Ye, X.; Qi, N.; LeVan, M. D. Prediction of adsorption equilibrium using a modified D−R equation: organic−water vapor mixtures on BPL carbon. Carbon 2003, 41, 2519−2525. (33) Walton, K. S.; LeVan, M. D. Adsorbed-phase heat capacities: Thermodynamically consistent values determined from temperaturedependent equilibrium models. Ind. Eng. Chem. Res. 2005, 44, 178− 182. (34) Kluitenberg, G. 5.2 Heat capacity and specific heat. Methods of Soil Analysis: Part 4 Physical Methods; Soil Science Society of America: Madison, WI, 2002; pp 1201−1208. (35) Lemmon, E. W.; McLinden, M. O.; Wagner, W. Thermodynamic properties of propane. III. A reference equation of state for temperatures from the melting line to 650 K and pressures up to 1000 MPa. J. Chem. Eng. Data 2009, 54, 3141−3180.
These thermodynamic quantities are dependent on the properties of the temperature, pressure, and the adsorption parameters of a given adsorbent + adsorbate system. This approach is not limited to pressurized conditions but may be extended to vacuum conditions when the Dubinin−Astakhov model is used. With these values, an adsorption system may then be designed with greater ease given the simplicity of the equations. In this work, the quantities have been calculated from the adsorption uptake data for the relevant temperatures and pressures in cooling and refrigeration applications. Thus, the analysis that follows is expected to predict better the experimental results of the adsorption chiller system.
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AUTHOR INFORMATION
Corresponding Author
*Phone: +65 65162214. E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS A.B.I. is supported by the National Research Foundation Singapore under its National Research Foundation (NRF) Environmental and Water Technologies (EWT) Ph.D. Scholarship Programme and administered by the Environment and Water Industry Programme Office (EWI).
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