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Estimating the Heat Capacity of the Adsorbate−Adsorbent System from Adsorption Equilibria Regarding Thermodynamic Consistency Valentin Schwamberger*,† and Ferdinand P. Schmidt†,‡ †

Energy and Building Technology Group, Institute of Fluid Machinery, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Baden-Württemberg, Germany ‡ Fraunhofer Institute for Solar Energy Systems (ISE), 79110 Freiburg, Baden-Württemberg, Germany ABSTRACT: The heat capacity of the adsorbate−adsorbent system is a crucial physical property required for modeling adsorption cycles. It is specific to the considered adsorptive gas and to the corresponding adsorbent. In previous work, it has been shown that this heat capacity can be estimated using only the properties of the gaseous phase, the adsorption equilibria, and the isosteric heats of adsorption. As these quantities are required in any case for simulations, no additional effort arises. In this paper, the separated adsorbed phase heat capacities are computed at higher accuracies by employing reversible thermodynamic paths, which traverse equilibrium states only. As a result, the thermodynamic consistency of the model improves, since the residuals of the energy and entropy balances are significantly smaller. Results are given for the water−zeolite 13X adsorption pair.

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INTRODUCTION AND MOTIVATION A closed adsorption cycle, e.g., for adsorption heat pumps,1−3 is characterized by a fixed amount of a single adsorptive fluid, with its mass fractions changing over time between bulk liquid phase, gas phase and adsorbed phase (see cycle in Figure 1).

respective pure (bulk) phase, while the differences between the physical quantities of the pure and the loaded adsorbent are assigned to the adsorbed phase.4 In this manner, a heat capacity of the adsorbed phase may be defined. For the inert constituents, temperature-independent estimates of the heat capacities are sufficient for not violating the energy and entropy balances. (Temperature-dependent estimates are also possible.) For the adsorptive fluid and the adsorbed phase, this does not apply. Physically, their enthalpies are mutually dependent, thus they and the corresponding heat capacities cannot be specified independently. The link between adsorbed and gas phase enthalpies is given by the adsorption equilibria and the corresponding isosteric heat of adsorption. While the properties of the gaseous and the liquid phases are tabulated for most relevant adsorptive fluids at very high precision (e.g., for water7), precise experimental data of adsorption equilibria along cycles of interest are more difficult to find in literature. Typically, the number of measurements and the corresponding accuracies are significantly lower than those for the fluid phases. Moreover, calorimetric measurements of the adsorbed phase heat capacity are rarely available, since they would, for instance, require a sealing of samples into a vapor-tight casing for each value of adsorbate loading, which is a demanding experiment, or a correction with respect to the occurring desorption.8,9 Both approaches are very timeconsuming. Fixing the adsorbed phase heat capacity (e.g., by using measured error-prone data) besides using measured and/or modeled adsorption equilibria and reference data for the gas phase would violate the energy balance and the model consistency in general, because the system would be overdetermined. Any model assuming a temperature-independent

Figure 1. Idealized closed adsorption cycle (ABC: adsorber is cooled, CDA: adsorber is heated). During adsorption, gas is supplied by the evaporator. During desorption, gas is released to the condenser.

The reference system considered here contains a single adsorber, an evaporator, and a condenser. Modeling such cycles commonly requires the numerical solution of the equations regarding the energy balance and the adsorption equilibria. A closed energy and entropy balance is of great importance for thermodynamic analyses. To this end, the physical properties of the inert constituentse.g., the heat exchanger, the heat transfer fluid, and the (pure) adsorbentas well as those of the adsorptive fluid and of the adsorbed phase must be known. As it is standard in adsorption thermodynamics,4−6 the adsorbent is considered as being inert in the sense that the adsorbent simply provides a potential field affecting the adsorbed molecules. The interaction energy is assigned to the adsorbate. The same applies to truly noninert adsorbents undergoing a structural change upon adsorption: The adsorbent is modeled using the physical properties of its © 2013 American Chemical Society

Received: Revised: Accepted: Published: 16958

April 14, 2013 September 27, 2013 October 14, 2013 October 14, 2013 dx.doi.org/10.1021/ie4011832 | Ind. Eng. Chem. Res. 2013, 52, 16958−16965

Industrial & Engineering Chemistry Research

Article

In the following, the amount of adsorbent is fixed, thus dns = 0. Then changing to the independent variables T, P, and na leads to

adsorption enthalpy leads to the adsorbed phase heat capacity being equal to that of the gas, see, e.g., Cacciola and Restuccia.1 There are also equilibrium models fixing the adsorbed phase heat capacity to that of the liquid, see, e.g., Pons and Grenier.10 For further details, see Walton and LeVan.11 Furthermore, efforts were made to approximate the characteristics of measured heat capacities of adsorbate− adsorbent systems.12,13 The energy balance was not focused on in these works. The resulting estimate is based on partial molar quantities at the given temperature and loading only. As shown in Table 1, the consistency is not improved. Hence, a thermodynamically consistent computation of the adsorbed phase heat capacity is an obvious choice. It leads to the correspondence of the temperature-dependencies of the gas and of the heat of adsorption to that of the adsorbed phase heat capacity with respect to the energy balance. Besides direct effects on the regarded physical quantities, closed balances allow for an easier validation of a model and for an improved estimate of its numerical errors. The accuracies of the computed heat capacities depend on those of the corresponding adsorption equilibria (data/model) and on the validity of the method for computing the isosteric heat of adsorption from them. Such a consistent approach employing thermodynamic paths was first described by Walton and LeVan11,14 for computing the integral heat capacities of the adsorbed phase15 for the general case of multicomponent, real gas adsorption. However, the chosen integration paths disregard adsorption equilibria. This again leads to inconsistencies, except for the case of ideal gases. In this paper, a similar approach is presented. It employs thermodynamic paths traversing adsorption equilibrium states only and is universally valid for all kinds of adsorption equilibrium data and models. Finally, the different approaches are applied to heat transfer applications, e.g., adsorption heat pumps which operate at relatively low pressures, and the corresponding results are compared.

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⎛ ∂H ⎞ ⎛ ∂H ⎞ ⎛ ∂H ⎞ dHa = ⎜ a ⎟ dT + ⎜ a ⎟ dP + ⎜ a ⎟ dna ⎝ ∂P ⎠T , n ⎝ ∂T ⎠ P , n ⎝ ∂na ⎠ P , T a a ⎧ ⎪ ⎪ ⎛ ∂Va ⎞ ⎫ ⎜ ⎟ ⎬dP + (Tsa̅ + μa )dna V T = CP , na ,adT + ⎨ − ⎪ a ⎪ ⎝ ∂T ⎠ P , n ⎭ ⎩ a (2)

where CP,na,a ≡ T(∂Sa/∂T)P,na, sa̅ ≡ (∂Sa/∂na)P,T, and the usual Maxwell relation (∂Sa/∂P)T,na = −(∂Va/∂T)P,na is employed. If considering the equilibrium between adsorbate and the adsorptive gas only, additionally na = na(p,T). Hence, in the absence of a nonadsorbing gas P = p, and the system may be described using two independent variables only. Regarding energy balances, T and na are a convenient choice: ⎛ ∂H ⎞ ⎛ ∂H ⎞ dHa = ⎜ a ⎟ dT + ⎜ a ⎟ dna ⎝ ∂T ⎠n ⎝ ∂na ⎠T a ⎧ ⎛ ∂Ha ⎞ ⎛ ∂p ⎞ ⎫ ⎪ ⎪ = ⎨Cp , na ,a + ⎜ ⎟ ⎜ ⎟ ⎬dT ⎝ ⎠ ⎪ ⎝ ∂p ⎠T , n ∂T na ⎪ ⎩ ⎭ a ⎧⎛ ⎛ ∂H ⎞ ⎛ ∂p ⎞ ⎫ ⎪ ∂H ⎞ ⎪ + ⎨⎜ a ⎟ + ⎜ a ⎟ ⎜ ⎟ ⎬dna ⎪⎝ ∂na ⎠ ⎝ ∂p ⎠T , n ⎝ ∂na ⎠T ⎪ ⎩ ⎭ p,T a

Along the phase boundary, dμa = dμg applies, and therefore (∂Ha/∂na)p,T ≡ h̅a = hg − qst.4,5 Defining Cna,a ≡ (∂Ha/∂T)na yields ⎧ ⎛ ∂Ha ⎞ ⎛ ∂p ⎞ ⎫ ⎪ ⎪ ⎨ dHa = Cna ,adT + hg − qst + ⎜ ⎟ ⎬dna ⎟ ⎜ ⎪ ⎝ ∂p ⎠T , n ⎝ ∂na ⎠T ⎪ ⎩ ⎭ a

THEORETICAL DERIVATION

(4)

Enthalpy of the Adsorbed Phase. The adsorbate− adsorbent system in its entirety can be described by standard thermodynamics. Following the literature,4,5,16,17 the adsorbate and the adsorbent may be separated to some extent. As described above, for an inert adsorbent, the interaction energy between the two components is assigned to the adsorbate. In the case of a noninert adsorbent, additionally the changes in its internal energy Us − U0s , in its entropy Ss − S0s , and in its volume Vs − V0s are implicitly included into the corresponding quantities of the adsorbate, as they usually cannot be separated. Berezin et al. conclude from their experimental data that KNaX zeolite, which is crystallographically comparable to 13X zeolite, is virtually inert under water adsorption.8 The differential enthalpy of the separated adsorbate Ha(Sa,P,ns,na) is4 dHa = TdSa + VadP − Φdns + μa dna

(3)

Due to the virtual incompressibility and the small volume of the adsorbate,4 particularly compared to the gas, (∂Ha/∂p)T,na is negligible: ⎛ ∂Ha ⎞ ⎛ ∂V ⎞ = Va − T ⎜ a ⎟ ≈ 0 ⎜ ⎟ ⎝ ∂T ⎠ p , n ⎝ ∂p ⎠T , n a a

(5)

With this approximation, it follows from eq 2 that Cp,na,a ≡ (∂Ha/∂T)p,na ≈ (∂Ha/∂T)na ≡ Cna,a (but Cp,na,a ≠ (∂Ha/∂T)p, where p is constant in equilibrium with the adsorptive gas only, since na changes with p). The differential of the approximate enthalpy of the adsorbate is dHa ≈ Cna ,adT + (hg − qst)dna

(6)

In order to abandon the approximation in eq 5, the volume of the adsorbate and its dependence on temperature and pressure could be estimated using the corresponding bulk liquid. In that case, the same estimate must be employed in the Clausius− Clapeyron equation. Thermodynamic Paths and Heat Capacity of the Adsorbed Phase. As the enthalpy is a function of state, enthalpy differences vanish along any closed path. Hence, appropriately chosen paths are employed for computing the

(1)

where −Φdns ≡ (μs − μ0s )dns is the energy corresponding to the change of the chemical potential of the adsorbent due to the presence of the adsorbate. In eq 1, the hydrostatic pressure is indicated by P, which comprises the partial vapor pressure of the adsorptive p and optionally the pressure of an additional inert (nonadsorbing) gas. 16959

dx.doi.org/10.1021/ie4011832 | Ind. Eng. Chem. Res. 2013, 52, 16958−16965


Article

Such equilibrium paths have been adopted here (see Figure 2). Along these paths, nonidealities in the gas phase are considered rigorously, whereas the finite volume and the thermal expansion of the adsorbate could in principle be considered (e.g., by approximating the adsorbate density with that of the bulk liquid) but have been neglected here (see eq 5). Since the temperature derivative of the adsorbed phase enthalpy is the quantity of interest (along CD), two isothermal pressurization paths at two different temperatures (BD, AC) and a connecting path traversing only pure gas states (vanishing equilibrium loading) during isobaric heating (AB) are considered. The latter is possible only at an extremely low pressure. For the convenience of the numerical calculations, an initial loading xi ≈ 0 can be chosen. This small amount of loading must be negligible with respect to the (minimum) loadings considered. Then, the initial pressure pi = p(xi,T) is finite. Additionally, the small change in loading Δx due to the isobaric, nonisosteric heating (AB) has been neglected, as Δx ≤ xi. The first path (see Figure 2 in gray) is split into two parts: First, isobaric heating of the gaseous phase at pi and vanishing loading xi (AB), then pressurization leading to simultaneous adsorption (BD). Starting with the molar enthalpy hg(pi, Ti) at point A, the molar enthalpy of the yet unadsorbed gas after the isobaric heating (at point B) is

enthalpy change of the adsorbate along an isostere, and its derivative with respect to the temperature at constant loading is the wanted integral isosteric heat capacity of the adsorbate. The approach given by Walton and LeVan11,14 is based on the following path: The total amount of gas is heated up isobarically and then pressurized isothermally at temperature T to the final pressure pf ≡ p(x,T). Then, the gas is adsorbed all at once, and the loading changes from zero to the final value x (see the wavy lines in Figure 2). Thermodynamically, this

hg (pi , T ) = hg (pi , Ti) +

Figure 2. Thermodynamic paths 1 (gray, ABD) and 2 (black, ACD) for computing the averaged heat capacity of the adsorbate between Ti = 40 °C and T = 60 °C at x = 0.2. The considered adsorption pair is water−zeolite 13X. Dashed lines represent the two-phase regions (AC, BD) where adsorbed and gaseous phases coexist. The associated vertical dotted lines in the planes parallel to the p-x plane indicate ongoing adsorption according to the equilibrium pressure p. The curve CD represents the isosteric state change of the adsorbate.

∫T

T

dT ′c p ,g(T ′)

i

i

(7)

where cp,g(T) ≡ (∂hg/∂T)p. Introducing the loading x ≡ ma/ms = Mana/ms, the isothermal pressurization part of path 1 (BD) can be thought of as following each molecule through reversible processes of pressurization as gas (up to equilibrium pressure p(x′,T) for each loading x′), adsorption, and further pressurization as adsorbate. The molar enthalpy of the adsorbate ha ≡ Han−1 a ≠ h̅a at the end of path 1 at point D (no gas left) is then obtained as (see eq 4)

means that at all intermediate loadings xm < x, the gas pressure pf is higher than the corresponding equilibrium pressure pm ≡ p(xm,T) of the adsorbate. Hence, at all intermediate loadings, gas and adsorbate are not in equilibrium with each other. When brought into contact, the integral heat released is not obtained by integrating the isosteric heat of adsorption qst over the loading but is the so-called enthalpy of immersion.17 For an ideal gas, both quantities have the same value. Since the isosteric heat of adsorption is derived for gas−adsorbate equilibrium, it is not applicable for nonequilibrium adsorption.4 In effect, using the isosteric heat of adsorption in the paths of Walton and LeVan causes an energy deviation for real gases. This energy deviation can only be avoided by choosing reversible (quasi-stationary) adsorption paths for calculation of the adsorbed phase enthalpy, i.e., paths traversing only adsorption equilibria. Then, the differential enthalpy change is given by eq 4, and its integration along such a path leads to the corresponding integral enthalpy. If adsorbing an infinitesimal amount of gas under isobaric and isothermal conditions, the isosteric heat of adsorption is released. However, if the amount of adsorbing gas is noninfinitesimal (macroscopic), the pressure has to be changed simultaneously: The process can be divided into a repeated sequence of adsorbing an infinitesimally small amount of gas under isobaric and isothermal conditions, and then isothermally and isosterically increasing the pressure to the new equilibrium pressure according to the new loading.

haf ≡ ha(p(x , T ), T ) = hg (pi , T ) + ( ×

∫0

x

⎧ ⎪ dx′⎨ ⎪ ⎩

∫0

∫p

x

dx′)−1

p(x ′ , T )

i

⎛ ∂hg ⎞ ⎟ dp′⎜ ⎝ ∂p′ ⎠T

p(x , T )

− qst(x′, T ) +

∫p(x′,T)

⎫ ⎛ ∂h ⎞ ⎪ dp′⎜ a ⎟ ⎬ ⎝ ∂p′ ⎠x ′ , T ⎪ ⎭

(8)

From eq 5 or by direct usage of eq 6, it follows that haf ≈

1 x

∫0

x

dx′{hg (p(x′, T ), T ) − qst(x′, T )}

(9)

Hence, the enthalpy of the adsorbate is considered constant under pressurization, only the enthalpy of the gaseous phase (slightly) changes. Another aspect becomes apparent by regarding the definition of qst ≡ hg−h̅a: The right-hand side of eq 9 is just the mean of the partial molar enthalpy of the adsorbate h̅a ≡ (∂Ha/∂na)p,T with respect to the amount of the adsorbate na. The second path (see Figure 2 in black) comprises again pressurization and simultaneous adsorption (AC) and then isosteric heating of the adsorbate (CD). At the end of the first 16960



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isosteric heat capacity of the adsorbate cx,a(T) using first-order finite differences for the partial derivative and the trapezoidal rule for integration is

part of path 2, when the gas is completely adsorbed at point C, the molar enthalpy hia ≡ ha(p(x,Ti),Ti) becomes (see eq 4) 1 = hg (pi , Ti) + x

hai

∫0

x

⎧ ⎪ dx′⎨ ⎪ ⎩

∫p

p(x , T )

− qst(x′, Ti) +

∫p(x′,T) i

p(x ′ , Ti)

i

⎛ ∂hg ⎞ ⎟ dp′⎜ ⎝ ∂p′ ⎠T

i

⎛ ∂h ⎞ ⎫ ⎪ dp′⎜ a ⎟ ⎬ ⎝ ∂p′ ⎠x ′ , T ⎪ ⎭ i

≈

1 x

∫0

cx ,a(T ) = ≈ (10)

∫T

dT ′cx ,a(T ′)

i

1 x

∫0

dx′[f (x′, T + ΔT ) − f (x′, T )]

N−1

⎡ ⎛ ⎣ N

⎞

⎛

⎞

∑ ⎢f ⎜⎝ kx , T + ΔT ⎟⎠ − f ⎜⎝ kx , T ⎟⎠ k=0

N

RESULTS AND DISCUSSION Comparison of the Approaches. As a test of the methodology described above, a simple adsorption cycle is considered (see Figure 3). The energy and entropy balances are

Equating eq 9 and eq 12 and using eq 11 yields dT ′cx ,a(T ′) =

x

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⎛ ∂h (p(x , T ), T ) ⎞ 1 ⎛ ∂Ha(p(x , T ), T ) ⎞ cx ,a(T ) ≡ ⎜ a ⎟ = ⎜ ⎟ ⎝ ⎠x ⎠x ∂T na ⎝ ∂T T

∫0

The computational efficiency is improved by precomputing cx,a(T) on a grid (xj,Tk), 1 ≤ j ≤ nj, 1 ≤ k ≤ nk. During simulation runs, bilinear interpolation between these values is performed.

(12)

where

∫T

⎛ ∂f (x′, T ) ⎞ dx′⎜ ⎟ ⎝ ∂T ⎠x ′

(15)

T

i

1 xΔT

x

⎛ (k + 1)x ⎞ ⎛ (k + 1)x ⎞⎤ , T + ΔT ⎟ − f ⎜ , T ⎟⎥ + f⎜ ⎝ ⎠ ⎝ ⎠⎦ N N

(11)

Additionally considering the isosteric heating in the second part of path 2 (CD), the final enthalpy hfa of the adsorbate is haf ≡ hai +

∫0

1 ≈ 2N ΔT

x

dx′{hg (p(x′, Ti), Ti) − qst(x′, Ti)}

1 x

x

dx′{hg (p(x′, T ), T )

− hg (p(x′, Ti), Ti) − [qst(x′, T ) − qst(x′, Ti)]}

Subsequent differentiation with respect to T while keeping x constant leads to the isosteric heat capacity of the adsorbed phase (adsorbate) cx,a(T): cx ,a =

1 x

∫0

x

⎡ ∂{hg (p(x′, T ), T ) − q (x′, T )} ⎤ st ⎥ dx′⎢ ⎢⎣ ⎥⎦ ∂T x′ (13)

This expression may be compared with that derived by Walton and LeVan,11 adapted to the notation used within the present paper: ca,WL

1 = cp ,g(T ) − x

∫0

x

⎛ ∂q x′, T ⎞ )⎟ st ( dx′⎜ ⎜ ⎟ ∂T ⎝ ⎠x ′

Figure 3. Idealized cycles corresponding to adsorption heat pumps, considering four condensation/adsorption temperatures.

inspected for different temperatures of the heat sources and the heat rejection unit. The thermal mass of the adsorber, of the (clean) adsorbent, and of the pipes is neglected. Only the heat corresponding to the sensible heating of the adsorbable component and the adsorbate as well as the (latent) isosteric heat of adsorption are accounted for. The adsorber is reversibly max heated from Tmin ads to Treg = 230 °C and then cooled down to min Tads . The condenser and the evaporator are idealized in the sense that they operate at fixed temperatures Tcd and Tev, respectively, where Tev = 10 °C. The pressures in both components are at the corresponding saturation pressures. Isosteric phases are considered until the adsorption equilibrium pressure decreases to the evaporator pressure (before adsorption) or increases to the condenser pressure (before desorption). Then adsorption or desorption phases follow, respectively. Throttlingthe condensate returning from the condenser to the evaporatoris modeled isenthalpically and adiabatically. Superheating the gaseous phase leaving the evaporator to the temperature of the adsorber and cooling down the gas leaving the adsorber to the temperature of the

(14)

Here, cp,g(T) is the real gas heat capacity depending on both temperature and pressure. The real gas behavior is not fully reflected in eq 14, as only the pressure corresponding to the regarded final loading is considered, p = p(x,T). Finally, the link between cx,a(T) and cp,a(T) ≡ (∂ha/∂T)p shall be regarded. It can be calculated by considering the total differentials dha and dp: ⎛ ∂h ⎞ ⎛ ∂p ⎞ ⎛ ∂h ⎞ ⎛ ∂h ⎞ cx ,a ≡ ⎜ a ⎟ = ⎜ a ⎟ + ⎜ a ⎟ ⎜ ⎟ ⎝ ∂T ⎠x ⎝ ∂T ⎠ p ⎝ ∂p ⎠ ⎝ ∂T ⎠x T

Since x is not kept constant in (∂ha/∂p)T, the second term of the right-hand side does not vanish. Hence, cp,a cannot be approximated by cx,a, but both quantities generally have distinct values. Numerical Evaluation. Defining f(x,T) ≡ hg(p(x,T),T)− qst(x,T), a simple expression for the numerical estimate of the 16961



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Table 1. Comparison of the Accuracy of the Energy and Entropy Balances for Methods W (Described by Walton and LeVan14), ★ (Described in This Paper), C (Described by Chakraborty et al.12), and R (Reference, cx,a = 2500 J kg−1 K−1)a method

Tcd (°C)

Δq (J g−1)

qh (J g−1)

Δq/qh (‰)

Δs (J kg−1 K−1)

sh (J kg−1 K−1)

Δs/sh (‰)

W ★ C R

30 30 30 30

0.421 0.003 13.394 7.429

985 986 942 1007

0.427 0.003 14.222 7.379

1.20 0.00 37.59 19.58

2476 2478 2349 2530

0.483 0.000 16.003 7.739

W ★ C R

40 40 40 40

0.589 0.002 22.686 9.902

910 910 865 933

0.648 0.002 26.229 10.616

1.62 0.00 61.49 25.53

2219 2221 2093 2276

0.731 0.001 29.383 11.220

W ★ C R

60 60 60 60

0.835 0.000 47.782 13.972

733 734 679 761

1.139 0.000 70.362 18.368

2.11 0.01 121.25 34.09

1695 1698 1556 1760

1.246 0.008 77.935 19.364

W ★ C R

80 80 80 80

0.816 0.007 44.322 13.750

496 497 450 527

1.647 0.014 98.544 26.101

1.91 0.02 104.13 31.73

1102 1104 994 1172

1.729 0.022 104.751 27.081

= 230 °C and Tcd = Tmin ads for all cases. For regenerating the adsorber, qh and sh are the supplied heat and entropy, respectively. Smaller residuals Δq and Δs indicate improved consistency.

a max Treg

for method W, since the deviations of the gas phase from an ideal gas increase. Heat Capacities of the Adsorbed Phase. For the adsorption pair water−zeolite 13X, heat capacities of the adsorbed phase have been computed over a wide range of temperatures and loading levels using the described approach. The results are shown in Figures 4, 5, and 6. From the latter

condenser is regarded. Entropy is increased for each of these irreversible processes. The adsorption pair is water−zeolite 13X. For the properties of the gaseous and the liquid phases of water, data is taken from the IAPWS.7 Hence, real gas behavior is considered for all compared approaches. All paths (those given by Walton and LeVan and ours) are computed starting at pi ≈ 0. For the adsorption equilibrium data, Dubinin’s characteristic curve was employed. The corresponding enthalpies of adsorption are computed using the Clausius− Clapeyron equation (see eq 19). Details about both approaches can be found in Appendix A. The differences in the residuals of the energy and entropy balances between the approach of Walton and LeVan14 (method W) and that described in this paper (method ★) are shown in Table 1. The approach presented by Chakraborty et al.12,13 (method C) is also considered, using the equation ca,Ch(T ) = cp ,g(T ) +

{ T1 − α (T)}q p ,g

st

⎛ ∂q ⎞ − ⎜ st ⎟ ⎝ ∂T ⎠x

where p = p(x,T) is the final pressure. As a reference, the residuals for a constant heat capacity of 2500 J kg−1 K−1 are computed (method R). The results are included in Table 1. With respect to thermodynamic consistency, the residuals for the energy Δq and enthalpy Δs should both vanish for the considered closed cycles for all approaches. It should be noted that the relative residuals would decrease if the thermal mass of the adsorber was included. Although the deviations due to the pressurizations of the gaseous phase are rather small, the approach presented in this paper leads to significantly reduced errors compared with method W, by at least two orders of magnitude in the regarded cases. This shows both the accuracy of the described method as well as its advantages for validating adsorption models and estimating numerical errors. As expected, the energy residuals Δq/qh (Table 1, fifth column) rise with the condenser pressure

Figure 4. Heat capacity of the adsorbate in dependence of temperature and loading computed using the method derived above. The dashed black line indicates constant pressure using x(psat(80 °C),T). The region right above that line is not required in order to compute the results in Table 1.

two, it can be seen that the method described by Walton and LeVan (dashed) deviates from that described in this paper (solid) for higher pressures (and correspondingly for higher loadings and temperatures). The values are valid only in the range where the temperature invariance of the adsorption potential is fulfilled and where it is 16962



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CONCLUSIONS In this paper, an improved methodology for computing the heat capacity of the adsorbed phase has been derived, using only adsorption equilibrium data and the corresponding heats of adsorption as well as the properties of the gaseous phase. By using thermodynamic paths traversing only states of adsorption equilibrium, the accuracy regarding model consistency (energy and entropy balance) improves significantly. Real gas behavior is included rigorously. The error regarding the true physical value of the heat capacity depends on the accuracy of the adsorption equilibrium data. The methodology is applicable for all kinds of adsorption equilibrium models.

■

APPENDIX A: ADSORPTION EQUILIBRIA AND HEAT OF ADSORPTION The methodology for evaluating adsorbed phase heat capacities described above is independent of the model used for describing the adsorption equilibria. However, for the numerical simulation presented above, adsorption isotherms are computed from a characteristic curve according to Polanyi’s theory.18 Besides being well suited for microporous adsorbents like zeolites,19 another advantage of Polanyi’s approach is that the two-dimensional functional relation x(p,T) between the loading x, pressure p, and temperature T is reduced to the onedimensional characteristic curve A(W). Here,

Figure 5. Heat capacity of the adsorbate in dependence of temperature computed using the methods derived above (solid) and by Walton− LeVan (dashed, mostly covered by the solid curves).

A(p , T ) ≡ RT ln(psat (T )/p)

(16)

defines the adsorption potential, and W ≡ x/ρa(T) is the specific adsorbed volume. Furthermore, p and psat(T) are the equilibrium and the saturation pressure at temperature T, respectively. The mass density of the adsorbate is indicated by ρa(T), which generally is not well-known. In this paper, ρa is estimated with the density of the bulk liquid at the saturation pressure corresponding to temperature T. The reduction to the characteristic curve is valid only if the partial derivative of the adsorption potential A with respect to the temperature keeping the filling θ = x/x* = W/W* constant vanishes. Here, x* indicates the maximum loading for the given temperature and W* the maximum adsorption space (where W* is assumed to be temperature-invariant):

Figure 6. Heat capacity of the adsorbate in dependence of loading computed using the methods derived above (solid) and by Walton− LeVan (dashed).

⎛ ∂A ⎞ ⎜ ⎟ = 0 ⎝ ∂T ⎠θ

based on experimental data. For high loadings and temperatures (upper right corner in Figure 4, and correspondingly for the curves of high loadings at high temperatures in Figure 5 and for those of high temperatures at high loadings in Figure 6), the number of measurements is small, and it is assumed that the temperature invariance is violated. Thus, the heat capacities within this rangesome of them are even negativeshould not be considered. These problems originate from using the characteristic curve as an adsorption equilibrium model. A similar observation was discussed by Walton and LeVan.11 However, it should be kept in mind that negative adsorbed phase heat capacities need not lead to negative heat capacities of the complete adsorbate−adsorbent system, depending on the mass fraction the adsorbent occupies. In the range where the characteristic curve is backed by plenty of experimental data and the temperature invariance assumption is fulfilled, the computed heat capacities lie between those of liquid water and ice. From a thermodynamic point of view, this is reasonable.

(17)

In this case, a smaller number of measurements is sufficient compared to an experimental determination of the complete field x(p,T). Although this assumption is not valid for any adsorptive−adsorbent pair, and the temperature range covered within some error bounds varies significantly, the corresponding approximative adsorption equilibria are sufficient for many applications. The explicit forms of the characteristic curves were analyzed thoroughly,19,20 and the Dubinin−Radushkevich and Dubinin− Astakhov equations are used frequently. However, in this paper, a generalized approach is employed, described by Núñez et al.:21,22 A larger variety of functions is allowed for describing the characteristic curve, leading to fits with smaller deviations from the measurements, in the sense of squared errors. Of course, it is crucial to avoid overfitting the data, regarding the errors of the experimental data and the fulfillment of the temperature invariance assumption (see eq 17). The experimental data for water−zeolite 13X and the corresponding 16963


Industrial & Engineering Chemistry Research characteristic curve used throughout this paper are shown in Figure 7.

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ACKNOWLEDGMENTS

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NOMENCLATURE

Article

The authors thank G. Füldner, M. Schicktanz, and Felix Ziegler for fruitful discussions. Funding by the State of BadenWürttemberg (ZO-III Initiative) through the Project Management Agency Karlsruhe (PTKA) is gratefully acknowledged.

Roman Letters

A = adsorption potential (J mol−1) cp = molar heat capacity (p const) (J K−1 mol−1) cx = molar heat capacity (loading const) (J K−1 mol−1) Cna = heat capacity (amount adsorbed const) (J K−1) CP,na = heat capacity (P and loading const) (J K−1) H = enthalpy (J) h = molar enthalpy (J mol−1) h̅ = partial molar enthalpy (J mol−1) M = molar mass (kg mol−1) m = mass (kg) n = amount of substance (mol) P = hydrostatic pressure (Pa) p = pressure (Pa) psat = saturation pressure (Pa) q = specific heat (J kg−1) qst = isosteric heat (J mol−1) qvap = enthalpy of vaporization (J mol−1) R = ideal gas constant (J mol−1 K−1) S = entropy (J K−1) s = specific entropy (J kg−1 K−1) s ̅ = partial molar entropy (J mol−1 K−1) T = temperature (K) Tmin ads = minimum adsorption temperature (K) Tmax reg = maximum regeneration temperature (K) U = internal energy (J) V = volume (m3) v = molar volume (m3 mol−1) W = adsorbed volume per mass of adsorbent (m3 kg−1) W* = maximum adsorption space (m3 kg−1) x = loading; mass adsorbed per mass of adsorbent (1) x* = maximum loading at given temperature (1)

Figure 7. Measurement data and corresponding fit of the characteristic curve for Baylith WE 894 (zeolite 13X with some zeolite A as a binding component). The gray shaded area shows the uncertainty computed for the fit (twice the standard deviation). Measurement data was taken from Núñez21 (temperature within 8 and 250 °C).

Assuming the gas to be ideal and the volume of the adsorbed phase to be negligible, the isosteric heat of adsorption may be computed using ⎛ ∂A ⎞ ⎟ qst = qvap + A − αA ,aT ⎜ ⎝ ∂ln W ⎠T

(18)

19,22

as it is proposed by Dubinin. However, the coefficient of thermal expansion of the adsorbate αA,a is known precisely only rarely. (Here, αA,a = va−1(∂va/∂T)A, hence keeping the adsorption potential A constant, not p.) Thus, for improved accuracies, the Clausius−Clapeyron equation is employed throughout this paper for computing the isosteric heat of adsorption. This heat is released during an isobaric and isothermal phase transition. 4,5 With the assumption that the volume of the adsorbed phase can be neglected, i.e., vg − va ≈ vg, the isosteric heat qst is given by ⎛ ∂p ⎞ qst = vgT ⎜ ⎟ ⎝ ∂T ⎠x

Greek Letters

αA = coefficient of thermal expansion (A const) (K−1) αp = coefficient of thermal expansion (p const) (K−1) Φ = change of the chemical potential of the adsorbent due to the presence of the adsorbate (J mol−1) μ = Chemical potential (J mol−1) ρ = mass density (kg m−3) θ = filling of the maximum adsorption space (1)

(19)

Here, the gas is not assumed to be ideal. The computation of the isosteric heat of adsorption qst according to eq 19 relies on the numerical evaluation of the isosteres and the corresponding derivatives, which is only slightly more complex than using eq 18 and can be done for smooth characteristic curves computationally efficiently and more accurately than by using an estimated value of αA,a (i.e., the isobaric coefficient of thermal expansion αp of liquid water).

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Subscripts/Superscripts

0 = pure adsorbent a = adsorbate cd = condenser Ch = Chakraborty ev = evaporator f = final g = gas h = supplied i = initial m = intermediate s = adsorbent WL = Walton−LeVan

AUTHOR INFORMATION

Corresponding Author

*Phone: +49 721 608-43495. Fax: +49 721 608-43529. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 16964



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Article

REFERENCES

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