Article pubs.acs.org/IECR
Enthalpy of Solid−Liquid Phase Change Confined in Porous Materials Yuping Wu and Tao Wang* State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing, 100084, China ABSTRACT: The confinement effect of the solid−liquid phase in porous materials was explained only qualitatively in previous research because the pore structure is complicated and difficult to describe quantitatively. In this work, fractal theory was adopted to characterize quantitatively the pore shape by surface fractal dimension. A quantitative model that disclosed the relationship among the solid−liquid phase-change enthalpy, pore size, pore shape, and interfacial groups was proposed for phasechange materials confined in a porous matrix. Furthermore, the model was applied to hydrated salts/silica composite and hydrated salts/expanded graphite composite; the results show that the fitted values agree well with the experimental values.
1. INTRODUCTION Because of the confinement effect, the phase-change enthalpy and phase-change temperature of the phase-change material (PCM) confined in a pore are different from those in bulk.1−3 Nevertheless, these contributions focus on only the influence of pore diameter. In fact, other parameters of the pore structure have effects on the phase-change performance, for example, the pore shape.4−6 Liu et al.7 investigated the phase-change behavior of tridecane−tetradecane mixtures confined in SBA15 with the pore shape of ordered cylinder and the capillary porous glass (CPG) with the pore shape of disordered connected pore and found that the mixtures confined in SBA-15 displayed only a peak in the differential scanning calorimeter (DSC) curve, but those confined in CPG showed more than two peaks. This indicated that different pore shape also has influence on the phase-change properties of PCM confined in a porous matrix. However, because of the complexities of the porous structure, these reported works explained the phenomenon only qualitatively. The confinement of solid−liquid phase change in porous materials depends not only on the pore structure but also on the interfacial groups. Guo et al.8 found that the confinement effect on the phasechange behavior of NaNO3 in porous silica could be eliminated effectively by removing the −OH on the silica surface through calcination. In a previous work,9 we reported that reducing the amount of interfacial Si−OH group is effective for enlarging the melting enthalpy of the hydrated salts/silica composites. Fractal theory is an applicable and potential tool for describing irregular phenomena, which was proposed by Mandelbrot.10 It is possible to quantitatively characterize irregular objects with statistical self-similarity through fractal dimension. Though the pore structure of the porous material is complicated and cannot be characterized by traditional Euclidean geometry, it is found that the pore void and pore surface have self-similarity.11,12 That is to say, the pore structure can be quantitatively characterized by fractal dimensions. In this © 2016 American Chemical Society
regard, it is supposed that the fractal theory may help to characterize the pore shape quantitatively. In this paper, the pore shape was quantitatively characterized by surface fractal dimension. On the basis of this, a quantitative model was established to reveal the relationship among the phase-change enthalpy, pore diameter, pore shape, and interfacial groups for the solid−liquid phase change confined in porous materials. In addition, this model was applied to two hydrated salt/porous matrix composites to validate its reliability.
2. MODELING To explain the phenomenon that the phase-change enthalpy of PCM confined in pores is smaller than that in the bulk, researchers proposed the confined phase-change model.13,14 It is assumed that all the pores of the matrix are filled with PCM and the PCM can be divided into two parts: one part is that close to the pore wall, which is adsorbed to form a constraint layer with a constant thickness and unable to experience the phase change; the other part is that in the pore core, also called free PCM, which can experience the phase change and has the same phase-change enthalpy in the bulk state. The phasechange enthalpy of the PCM/porous matrix composite depends on the fraction of the free PCM. However, the density of PCM in the pore core is probably different from that of the PCM in the bulk state because of the confinement effect. On the basis of these assumptions, the phase-change enthalpy of the composite should be ΔHE = ρp V ΔHB (1) where ΔHE (kJ/kg) is the experimental phase-change enthalpy of the composite, ρp (kg/m3) the density of PCM in the pore Received: October 7, 2016 Accepted: October 18, 2016 Published: October 18, 2016 11536
DOI: 10.1021/acs.iecr.6b03890 Ind. Eng. Chem. Res. 2016, 55, 11536−11541
Article
Industrial & Engineering Chemistry Research
Figure 1. Confined phase-change model considering the interfacial groups (illustrated using porous silica as the example).
core, V (m3/kg) the specific volume of the free PCM in the pore core of the composite, and ΔHB (kJ/kg) the phase-change enthalpy of PCM in the bulk. Because we assumed that all the pores of the matrix are filled with PCM, the mass faction of PCM in the composite can be written as ω = ρB V0
account the effects of the interfacial groups and the pore shape of the matrix. In fact, the interfacial groups on the pore wall are uneven, and the constraint layer might be influenced by the interfacial groups. That is, the constraint layer is uneven in pore wall, as displayed in Figure 1. Because of this, we equalized the uneven constraint layer into an even constraint layer with the thickness of d. The dependence of the constraint layer thickness on the intensity of interfacial groups was assumed to be a liner relationship as
(2)
where ω is the mass fraction of PCM in the composite, ρB (kg/ m3) the density of PCM in the bulk state, and V0 (m3/kg) the specific volume of the pore in the composite. According to eqs 1 and 2, it was found that ρp V ΔHE V = =k ωΔHB ρB V0 V0 (3)
d = k 0ε
where d (nm) is the thickness of the constraint layer; k0 (nm3) a constant; and ε (nm−2) the intensity of the interfacial group, which is presented as the average amount of the group per interfacial area. The pore shape factor is difficult to describe by traditional Euclidean geometry, because the pore shape in porous material is complicated and irregular. Therefore, we adopt fractal theory to solve this. It is known that the pore surface has the characteristic of self-similarity; therefore, it is reasonable to employ surface fractal dimension to characterize the pore shape quantitatively. On the basis of the above hypothesis, a modified model is derived as follows:
where V (m3/kg) is the specific volume of the pore core in the composite and k = ρp/ρB is the density ratio of PCM in the pore core to that of PCM in bulk state. The volume ratio of the free PCM in pore core to the total PCM in the composite depends on the diameters of the pore core and pore as well as the shape of the pore. In a previous work (Guo et al.15), this volume ratio was presented as
⎛ D0 − 2d ⎞n V =⎜ ⎟ V0 ⎝ D0 ⎠
⎛ D − 2k 0ε ⎞ Ds ΔHE = k·⎜ 0 ⎟ ΔHT D0 ⎝ ⎠
(4)
where D0 (nm) is the average pore diameter of the matrix; d (nm) the thickness of the constraint layer; and n a constant, which is dependent on the geometrical shape of the pore (for regular cylinder, n = 2; for regular sphere, n = 3; for irregular pore, n is not an integer). On the basis of eqs 3 and 4, Guo et al.15 proposed the relationship between the phase-change enthalpy of the composite and the average pore diameter of the matrix as ⎛ D0 − 2d ⎞n ΔHE = k·⎜ ⎟ ΔHT ⎝ D0 ⎠
(6)
(7)
where DS is the surface fractal dimension of the matrix.
3. SURFACE FRACTAL DIMENSION OF THE POROUS MATERIALS The surface fractal dimension of porous materials can be acquired by different methods, including nitrogen adsorption,16 small angle diffraction,17 scanning electron microscopy,18 and mercury penetration.19 Generally, researchers may choose the method to calculate the surface fractal dimension based on the pore size of porous materials. The nitrogen adsorption method may be chosen when the pore size of porous materials is in the mesoscale, while the method of mercury penetration is preferred when the pore size of porous materials is in the macroscale. Because the silica used in this work is a mesoporous matrix, its surface fractal dimension was calculated from nitrogen adsorption data using the method proposed by Wang et al.20 According to this method, the surface fractal dimension can be calculated through eq 8 as the slope of the line ln(A(X)) versus ln(B(X)):
(5)
where ΔHT = ωΔHB is the theoretical phase-change enthalpy. Though eq 5 can correlate the phase-change enthalpy with the average pore diameter of the porous matrix quantitatively, it has some problems that need to be solved. First, the factors that have effect on the thickness of the constraint layer were not taken into consideration. Second, the pore shape factor n was a fitted parameter, which was not known from the pore structure in advance. Therefore, we propose a new model for the confined phase change in porous materials, which takes into 11537
DOI: 10.1021/acs.iecr.6b03890 Ind. Eng. Chem. Res. 2016, 55, 11536−11541
Article
Industrial & Engineering Chemistry Research ln A(X) = const + Ds ln B(X)
graphite sample had the correlation coefficient as high as 0.9984 and provided the surface fractal dimension as 2.652.
(8)
in which −∫ A(X ) =
rc(X ) =
Nmax
N (X )
ln(X ) dN (X ) 2
rc (X )
,
B(X ) =
[Nmax − N (X )]1/3 rc(X ) (9)
−2σVL RT ln(X )
(10)
where X is the relative pressure, P/P0; N(X) (mol) the physisorption quantity; rc(X) (m) the mean curvature radius; σ (N/m) the surface tension between liquid and gas of nitrogen; T (K) the absolute temperature of the adsorption; and VL (m3/ mol) the molar volume of liquid nitrogen. As an example, the fitting result of ln(A(X)) versus ln(B(X)) for the silica sample S1 was displayed in Figure 2, with the correlation coefficient of fitting as high as 0.9982. A reasonable surface fractal dimension value of 2.769 was obtained. Figure 3. Fitting results from eq 11 for the expanded graphite E5.
According to the assumption of surface fractal dimension, its value should be between 2 and 3. As mentioned above, we considered the surface fractal dimension (DS) represented the shape of pore. When DS is approximately 2, it is supposed that the pore surface is simply composed of regular lines. When DS is approximately 3, it is supposed that the pore surface is composed of amounts of curves with different curvature. For example, the pore shape of the expanded graphite is close to quadrangle with the surface fractal dimension approaching 2 (as shown in Table 2). However, the pore shape of the silica is rather complex with the surface mainly composed of curves. Therefore, the surface fractal dimension of the silica is approximately 3 (as shown in Table 1). Figure 2. Fitting results from eq 8 for the silica S1.
4. RESULT AND DISCUSSION To validate the reliability of the proposed confined phasechange model, it was applied to the hydrated salts/silica composite and hydrated salts/expanded graphite composite. For all systems, the values of melting enthalpy were obtained by DSC and could be measured precisely to four significant figures with uncertainties less than 0.05%. To find the parameters k and k0 or d in the confined phasechange enthalpy model, the data of ΔHE, D0, DS, and ε were fitted with eq 7 by the well-known Levenberg−Marquardt method, which is one kind of global optimization. Once the fitting achieved the convergence, the sum of squares of residues (SSE) was provided. The value of SSE could present the uncertainty in the model. The fitted k and k0 or d were used to calculate ΔHE. Furthermore, the relative error between the experimental and calculated ΔHE values are given for comparison. 4.1. Hydrated Salts/Silica Composite. The PCM is the hydrated salt mixture of Na2HPO4·12H2O and Na2SO4·10H2O in a mass ratio of 4:1. Its melting enthalpy was determined by DSC as 221.4 kJ/kg. The hydrated salts/silica composite with different mass fraction of hydrated salts was prepared in three different procedures (S1, S2, and S3) described in previous work.9 D0 was derived based on the data of nitrogen adsorption experiments. Ds was calculated from the nitrogen adsorption data by the method of Wang et al.20 The intensity of OH was
The pore size of the expanded graphite was in the micrometer scale, so we calculated its surface fractal dimension from the mercury penetration data by the method of Zhang et al.21 This method evaluated the thermodynamic relation of porous media in the process of mercury penetration and established the following model: ⎛ V 1/3 ⎞ ⎛W ⎞ ln⎜ 2n ⎟ = Ds ln⎜⎜ n ⎟⎟ + C ⎝ rn ⎠ ⎝ rn ⎠
(11)
in which n
Wn =
∑ PiΔVi
(12)
i=1 3
where Wn (J) is the intruded work; Pi (Pa) and Vi (m ) are the applied pressure and intruded pore volume at step i, respectively; rn (m) is the pore radius; C is a constant. On the basis of this, the surface fractal dimension can be
( ) versus
obtained through eq 11 as the slope of the line ln
Wn rn 2
Vn1/3
( ). As displayed in Figure 3, the fitting for an expanded
ln
rn
11538
DOI: 10.1021/acs.iecr.6b03890 Ind. Eng. Chem. Res. 2016, 55, 11536−11541
Article
Industrial & Engineering Chemistry Research Table 1. Data for the Confined Phase Change in the Silica Matrixa
a
mass fraction, ω
preparation procedure
experimental enthalpy, ΔHE (kJ/kg)
average pore diameter, D0 (nm)
surface fractal dimension, DS
intensity of interfacial OH, ε (nm−2)
50% 50% 50% 60% 60% 60% 70% 70% 70%
S1 S2 S3 S1 S2 S3 S1 S2 S3
6.752 18.94 29.39 21.98 32.49 47.88 22.99 45.86 64.14
4.476 5.536 6.340 5.871 6.534 7.235 5.364 7.159 7.519
2.769 2.769 2.706 2.780 2.735 2.680 2.786 2.763 2.752
4.328 4.292 3.881 4.328 4.292 3.881 4.328 4.292 3.881
PCM: Na2HPO4·12H2O and Na2SO4·10H2O in mass ratio of 4:1 with 221.4 kJ/kg.
calculated by thermogravimetric (TG) analysis9 and was considered to be influenced only by the preparation procedures. On the basis of these known data (Table 1), the k and k0 in eq 7 were fitted to be 1.286 and 0.6888, respectively. According to the fitting result, the sum of squares of residues (SSE) for the PCM/silica is 0.0024, which indicates the uncertainty in the model. The result for the hydrated salts/silica composite was displayed as ⎛ D − 0.6888ε ⎞ Ds ΔHE = 1.286·⎜ 0 ⎟ ΔHT D0 ⎝ ⎠
measured by mercury porosimetry. The surface fractal dimension, D s , was calculated based on the mercury porosimetry data using the method of Zhang et al.21 The PCM is the hydrated salt mixture of Na2HPO4·12H2O and Na2SO4·10H2O in mass ratio of 1:1. Its melting enthalpy was measured by DSC as 226.9 kJ/kg. The hydrated salts/expanded graphite composites with the same ω of 85% were prepared according to the procedure described in previous work.22 The DSC measurements were conducted to obtain the melting enthalpy values of the hydrated salts/expanded graphite composites. It was found from Fourier transform infrared (FTIR) analysis that there was no obvious difference of interfacial groups among the expanded graphite samples prepared by different microwave conditions (Figure 5). It
(13)
According to eq 13, the fitted phase-change enthalpies were calculated and compared with the experimental values as shown in Figure 4. It was shown that the fitted phase-change
Figure 5. FTIR spectra of the different expanded graphite samples. Figure 4. Comparison of experimental and fitted values of the phasechange enthalpy of hydrated salts/silica composite.
means that the intensities of the interfacial group of all expanded graphite samples are same. According to eq 6, the constraint layer thickness of PCM in the expanded graphite with different pore structures should be the same. On the basis of the known data (Table 2), the relationship for the hydrated salts/silica composite was obtained (eq 14). According to the fitting results, the SSE for the PCM/expanded graphite system is 0.0086, which presents the uncertainty in the model. As can be seen, the thickness of the constraint layer, d, was 13.35 nm. The average pore diameter of expanded graphite ranged from 1000 to 5000 nm. That is, the ratio that the constraint layer accounted for was less than 2.67%. This proved that the constraint layer in the hydrated salts/expanded graphite composite took up only a small fraction. Because of this, the phase-change enthalpy of the composite did not
enthalpies agreed well with the experimental results. The relative error between them was less than 10%, which demonstrated the reliability of the model. In addition, the thicknesses of the constraint layers in the composites prepared by three procedures were calculated to be 1.49 nm (S1), 1.48 nm (S2), and 1.34 nm (S3). This further confirmed that the intensity of OH in the hydrated salts/silica composite by procedure S3 was smaller than that prepared by the other two procedures.9 This was consistent with the results of NMR and TG analysis.9 4.2. Hydrated Salts/Expanded Graphite Composite. The average pore diameters of the expanded graphite samples, which were expanded by different microwave conditions, were 11539
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Table 2. Data for the Confined Phase Change in the Expanded Graphitea
AUTHOR INFORMATION
Corresponding Author
sample
experimental enthalpy, ΔHE (kJ/kg)
average pore diameter, D0 (nm)
surface fractal dimension, DS
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10
164.1 166.6 172.0 178.8 178.9 179.7 181.1 183.6 184.2 188.3
1198.4 1891.1 2165.4 980.1 1838.3 1522.8 4761.5 1869.0 5893.5 2734.6
2.692 2.659 2.664 2.726 2.652 2.693 2.591 2.656 2.612 2.612
*Phone: +86 10 62784877. Fax: +86 10 62784877. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Coasne, B.; Galarneau, A.; Pellenq, R. J. M.; Di Renzo, F. Adsorption, Intrusion and Freezing in Porous Silica: the View from the Nanoscale. Chem. Soc. Rev. 2013, 42 (9), 4141. (2) Maheshwari, P.; Dutta, D.; Sharma, S. K.; Sudarshan, K.; Pujari, P. K.; Majumder, M.; Pahari, B.; Bandyopadhyay, B.; Ghoshray, K.; Ghoshray, A. Effect of Interfacial Hydrogen Bonding on the Freezing/ Melting Behavior of Nanoconfined Liquids. J. Phys. Chem. C 2010, 114 (11), 4966−4972. (3) Liu, X. X.; Wang, Q.; Huang, X. F.; Yang, S. H.; Li, C. X.; Niu, X. J.; Shi, Q. F.; Sun, G.; Lu, K. Q. Liquid−Solid Transition of Confined Water in Silica-Based Mesopores. J. Phys. Chem. B 2010, 114 (12), 4145−4150. (4) Wang, C. Y.; Feng, L. L.; Li, W.; Zheng, J.; Tian, W. H.; Li, X. G. Shape-stabilized Phase Change Materials Based on Polyethylene Glycol/Porous CarbonComposite: The Influence of the Pore Structure of the Carbon Materials. Sol. Energy Mater. Sol. Cells 2012, 105, 21−26. (5) Hamada, Y.; Koga, K.; Tanaka, H. Phase Behavior and Fluid− solid Surface Tension of Argon in Slit Pores and Carbon Nanotubes. Phys. A 2009, 388 (12), 2289−2298. (6) Jähnert, S.; Vaca Chávez, F.; Schaumann, G. E.; Schreiber, A.; Schönhoff, M.; Findenegg, G. H. Melting and Freezing of Water in Cylindrical Silica Nanopores. Phys. Chem. Chem. Phys. 2008, 10 (39), 6039. (7) Yan, X.; Wang, T. B.; Gao, C. F.; Lan, X. Z. Mesoscopic Phase Behavior of Tridecane−Tetradecane Mixtures Confined in Porous Materials: Effects of Pore Size and Pore Geometry. J. Phys. Chem. C 2013, 117 (33), 17245−17255. (8) Guo, Q.; Wang, T. Study on preparation and thermal properties of sodium nitrate /silica composite as shape-stabilized phase change material. Thermochim. Acta 2015, 613, 66−70. (9) Wu, Y. P.; Wang, T. The dependence of phase change enthalpy on the pore structure and interfacial groups in hydrated salts/silica composites via sol−gel. J. Colloid Interface Sci. 2015, 448, 100−105. (10) Mandelbrot, M. B. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 1967, 156, 636−638. (11) Tang, H. P.; Wang, J. Z.; Zhu, J. L.; Ao, Q. B.; Wang, J. Y.; Yang, B. J.; Li, Y. N. Fractal Dimension of Pore-structure of Porous Metal Materials Made by Stainless Steel Powder. Powder Technol. 2012, 217, 383−387. (12) Watt-Smith, M. J.; Rigby, S. P.; Ralph, T. R.; Walsh, F. C. Characterisation of Porous Carbon Electrode Materials Used in Proton Exchange Membrane Fuel Cells Via Gas Adsorption. J. Power Sources 2008, 184 (1), 29−37. (13) Lopez, J.; Caceres, G.; Del Barrio, E. P.; Jomaa, W. Confined Melting in Deformable Porous Media: A First Attempt to Explain the Graphite/Salt Composites Behaviour. Int. J. Heat Mass Transfer 2010, 53 (5−6), 1195−1207. (14) Denoyel, R.; Pellenq, R. J. M. Simple Phenomenological Models for Phase Transitions in a Confined Geometry. 1: Melting and Solidification in a Cylindrical Pore. Langmuir 2002, 18 (7), 2710− 2716. (15) Guo, Q.; Wang, T. Influence of SiO2 pore structure on phase change enthalpy of shape-stabilized polyethylene glycol/silica composites. J. Mater. Sci. 2013, 48, 3716−3721. (16) El Shafei, G. M. S.; Philip, C. A.; Moussa, N. A. Fractal Analysis of Hydroxyapatite from Nitrogen Isotherms. J. Colloid Interface Sci. 2004, 277 (2), 410−416. (17) Hurd, A. J.; Schaefer, D. W.; Smith, D. M.; Ross, S. B.; Le Méhauté, A.; Spooner, S. Surface Areas of Fractally Rough Particles
PCM: Na2HPO4·12H2O and Na2SO4·10H2O in mass ratio of 1:1 with 226.9 kJ/kg; PCM mass fraction ω, 85%).
a
decrease too much compared with that in bulk. This was consistent with the experimental results. The fitted and the experimental values are compared in Figure 6. As can be seen, the relative error was less than 7%, which further confirmed the reliability of the model. ⎛ D − 26.67 ⎞ Ds ΔHE = 0.8137·⎜ 0 ⎟ ΔHT D0 ⎝ ⎠
Article
(14)
Figure 6. Comparison of experimental and fitted values of the phasechange enthalpy of hydrated salts/expanded graphite composite.
5. CONCLUSIONS A quantitative model for the solid−liquid phase change confined in porous materials was elaborated to disclose the relationship among the phase-change enthalpy, pore diameter, pore shape characterized using surface fractal dimension, and the intensity of interfacial groups. It was revealed that the phase-change enthalpy was influenced by not only the pore diameter but also the pore shape and the intensity of interfacial groups. The quantitative relationship was validated by the hydrated salts/silica composite and hydrated salts/expanded graphite composite. The fitted phase-change enthalpies agreed well with the experimental values, proving the reliability of the model. 11540
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Industrial & Engineering Chemistry Research Studied by Scattering. Phys. Rev. B: Condens. Matter Mater. Phys. 1989, 39, 9742. (18) Wang, Y.; Diamond, S. A Fractal Study of the Fracture Surfaces of Cement Pastes and Mortars Using a Stereoscopic SEM Method. Cem. Concr. Res. 2001, 31 (10), 1385−1392. (19) Zeng, Q.; Luo, M.; Pang, X.; Li, L.; Li, K. Surface Fractal Dimension: An Indicator to Characterize the Microstructure of Cement-based Porous Materials. Appl. Surf. Sci. 2013, 282, 302−307. (20) Wang, F. M.; Li, S. F. Determination of the Surface Fractal Dimension for Porous Media by Capillary Condensation. Ind. Eng. Chem. Res. 1997, 36 (5), 1598−1602. (21) Zhang, B. Q.; Li, S. Determination of the Surface Fractal Dimension for Porous Media by Mercury Porosimetry. Ind. Eng. Chem. Res. 1995, 34 (4), 1383−1386. (22) Wu, Y. P.; Wang, T. Hydrated salts/expanded graphite composite with high thermal conductivity as a shape-stabilized phase change material for thermal energy storage. Energy Convers. Manage. 2015, 101, 164−171.
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