Shrinking core model for the reaction-diffusion problem in thermo-chemical heat storage Lan, S.; Zondag, H.A.; Rindt, C.C.M.

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1 Shrinking ore model for the reation-diffusion problem in thermo-hemial heat storage Lan, S.; Zondag, H.A.; Rindt, C.C.M. Published in: Proeedings of The 13th International Conferene on Energy Storage, May 2015, Beijing, China Published: 01/01/2015 Doument Version Aepted manusript inluding hanges made at the peer-review stage Please hek the doument version of this publiation: A submitted manusript is the author's version of the artile upon submission and before peer-review. There an be important differenes between the submitted version and the offiial published version of reord. People interested in the researh are advised to ontat the author for the final version of the publiation, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publiation after peer review. The final published version features the final layout of the paper inluding the volume, issue and page numbers. Link to publiation General rights Copyright and moral rights for the publiations made aessible in the publi portal are retained by the authors and/or other opyright owners and it is a ondition of aessing publiations that users reognise and abide by the legal requirements assoiated with these rights. Users may download and print one opy of any publiation from the publi portal for the purpose of private study or researh. You may not further distribute the material or use it for any profit-making ativity or ommerial gain You may freely distribute the URL identifying the publiation in the publi portal? Take down poliy If you believe that this doument breahes opyright please ontat us providing details, and we will remove aess to the work immediately and investigate your laim. Download date: 26. Aug. 2018

2 The 13th International Conferene on Energy Storage Greenstok 2015 C/T/E Energy Storage - 09 Shrinking Core Model for the Reation-Diffusion Problem in Thermo- Chemial Heat Storage Shuiquan Lan 1, Herbert A. Zondag 1, 2, Camilo C.M. Rindt 1,* 1 Eindhoven University of Tehnology, Department of Mehanial Engineering, P.O.Box 513, 5600 MB Eindhoven, The Netherlands. 2 ECN, Energy Researh Centre of the Netherlands, P.O. Box 1, 1755 ZG Petten, The Netherlands. * Corresponding C.C.M.Rindt@tue.nl Keywords: Thermohemial heat storage, Shrinking ore model, Reation diffusion interation, Lithium sulphate monohydrate, Thermal dehydration SUMMARY In this work, we develop a kineti model to study the dehydration reation of Li2SO4.H2O single partiles involving interation between the intrinsi hemial reation and the bulk diffusion. The mathematial framework of the model is based on the shrinking ore model. A variable-grid, finite-differene method with fully impliit formulation is used for solving the model. It is found that the Damkӧhler number Da (krr 0) / (De 0) plays an important role in determining the nature of the diffusion/reation dynamis. A very small Da value means that the overall reation is ontrolled by the intrinsi hemial reation at the interfae, while a very large Da value means that the overall reation is ontrolled by the diffusion of water through the produt phase. Moreover, the numerial results of frational onversion alulated in the model are in good agreement with the theoretial analysis under extreme ases in whih either diffusion (large Da) or reation (small Da) dominates the dehydration proess. With onsideration of numerial solutions at various Da values, it is onluded that both intrinsi reation and mass diffusion are important in determining the reation kinetis within a range of Da values between 0.1 and 10. INTRODUCTION Dehydration reations of salt hydrates are of prime importane in various industrial and environmental appliations suh as thermohemial heat storage. An adequate understanding of the mehanism and kinetis of the reations is essential to analyse and then to optimize the reation rate. In this work, the thermal dehydration of lithium sulphate monohydrate (see Eq. 1) is investigated. Li SO. H O(s) Li SO (s) H O( g) (1) This reation used to be reommended by the Kinetis Committee of the International Confederation of Thermal Analysis (ICTA) as a material for the study of the reation kinetis of solid-state reation. Therefore, its reation kinetis has been extensively studied using various kineti models, but signifiant differenes in the Arrhenius parameters have been reported (Brown et al. 1992; Brown et al. 1993; Ferhaud et al. 2012; Kanungo and Mishra 1

3 The 13th International Conferene on Energy Storage Greenstok ). One of the main reasons is the use of different methods of data analysis. It is well known that the ompliated proess of dehydration is omposed of several stages, e.g., the intrinsi reation of bond breaking, formation of nulei and their growth, diffusion of gaseous produts, and heat transfer. The fat is that, in general, the reation kinetis is not to be determined by a single mehanism but by a ombination of several mehanisms. Unfortunately, kineti models for the dehydration rate are often simplified to a ertain mehanism due to the ease of appliation. Kineti studies based on them do not neessarily provide reliable kineti data. Figure 1: Illustration of the shrinking ore model that has been used to desribe the dehydration of a salt hydrate partile. A shell of one phase surrounds a ore of the other phase, with the interfae between the two phases moving inwards. We want to onsider the kinetis of the dehydration reation in a more fundamental mehanisti level, where the overall reation is dependent on intrinsi reation and mass diffusion. The idea of the shrinking ore (sharp interfae) model is used to desribe the thermal dehydration of Li2SO4.H2O salt hydrates. As shown shematially in Fig. 1, the dehydration proess of a salt hydrate rystal inludes a phase hange only at the sharp interfae of the ore. The partile size does not hange but the radius of the ore dereases as the reation proeeds. In the fully hydrated state, the salt hydrate rystal is in the α phase. The dehydration proess begins when water vapor at the surfae of the partile starts to esape from the rystal struture. The transformation of the salt hydrate (α phase) at the surfae promotes the formation of the dehydrated phase (β phase). Next, the adsorbed water that is lost at the surfae of the partile is replenished by diffusion of water moleules from the salt hydrate α phase to the surfae through the β phase. The dehydration proess is omplete when all the water from the salt rystal has been lost and the partile onsists only of the β phase. The shrinking ore model was first developed by Yagi and Kunii (Yagi and Kunii 1955; Yagi and Kunii 1961) to desribe the reations between gaseous speies and solid partiles, and has been developed to inorporate various physiohemial phenomena under a variety of onditions (Knorr et al. 2012). A summary of previous work on this model an be found in the artile of Gupta and Saha (Gupta and Saha 2003). There are a few appliations of this model to a solid-state reation where the interation between the interfae reation and mass diffusion is involved simultaneously. One example is about the thermal deomposition of SrCO3 ompats, but the alulation is limited to the pseudo-steady state (Arvanitidis et al. 1999). In other fields of appliation suh as the eletrohemistry of the disharging of a metal hydride eletrode (Deshpande et al. 2011; Subramanian et al. 2000), metallurgy of the phase transformation (Shuh 2000) and hydrogen storage (Saetre 2006), only the diffusion of gaseous produts is onsidered without reation at the moving interfae. Reently, Cui et al. (Cui et al. 2013) presented a partiularly omplete model to desribe the interfae-reation ontrolled diffusion in binary solids, suh as LixSi, whih an aount for the finite deformation kinetis, diffusion-stress interation and the interplay between bulk diffusion and intrinsi hemial reation. 2

4 The 13th International Conferene on Energy Storage Greenstok 2015 In this paper a shrinking ore model involving the intrinsi hemial reation and bulk diffusion is developed. A variable-grid, finite-differene method with fully impliit formulation is used for solving the governing equations of the model. Numerial solutions under various situations regarding the ompetion between intrinsi reation and mass diffusion are presented for disussion. NUMERICAL MODEL The dehydration of a spherial salt hydrate partile is assumed to follow the sequene desribed in Fig. 1. The mass flux of the reversible reation onsidered is written as: J H O 2 kr (1 ) (2) eq where kr is the reation rate onstant at the interfae, is the water onentration at the interfae and eq is the equilibrium onentration. In terms of the propagation of the reation interfae position dr in a time interval dt, the mass flux of water is given by: J HO (3) 2 0 dr dt where r is the interfae position. Combining Eq. 2 and Eq. 3, we have: dr kr (1 ), r r (4) dt 0 eq As the partile is dehydrated, the ore of the fully hydrated material (α phase) shrinks. The onentration distribution (r) of water in the β phase is governed by Fik s seond law: r D (r ) t r r 2 2 e r r r (5), 0 where r is the spatial variable and De is the effetive diffusivity of water through the porous β phase. In order to preserve the mass balane at the interfae, the mass flux relative to the moving interfae must be onsidered. The amount of water generated from the interfaial reation must be equal to the amount transported away from the interfae. In partiular, mass flux of onvetion must be onverted from a fixed frame to the moving interfae with an absolute veloity dr / dt. So the boundary ondition at the moving interfae is given as: (r, t) dr k (1 ) D r dt r e eq r r r r (6), Replaing dr / dt and rearranging the equation, the boundary ondition at the interfae an be rewritten as: 3

5 The 13th International Conferene on Energy Storage Greenstok 2015 (r, t) kr (1 )(1 ) r D r r e eq 0, r r (7) The boundary ondition at the outer surfae of the partile is given as: (r,t), r r0 (8) g where g is the water onentration in the environment. For the model to be as general as possible, the following nondimensional spatial and temporal variables are defined: r ˆ ˆ D t r r ˆ rˆ r r r e g ;t ; ; eq g 0 The orresponding dimensionless forms of the above equations are: dr dt Da (1 ), r r (9) r (r ), r r 1 (10) t r r 2 2 with boundary onditions (r, t) r r r Da (1 )(M ) r r (11), (r, t) 0, r 1 (12) where the hats are dropped for onveniene, (eq g ) / eq, M (0 g ) / (eq g ) and Da (krr 0) / (De 0) is the Damkӧhler number expressing the ratio of the hemial reation rate to the mass transport rate. The present model is solved by numerial sheme proposed by Illingworth and Golosnoy (Illingworth and Golosnoy 2005). The solution is based on the finite differene method with fully implit formulation. The moving interfae is traked by a variable grid method using a Landau transformation so that the moving interfae is fixed at a grid point. A onsiderable advantage of this sheme is that finite differene equations are derived in suh a way that mass onservation during the alulation is ensured. RESULTS AND DISCUSSIONS In this setion, the shrinking ore model is solved for a spherial partile for various values of the Damkӧhler number. It is worth noting that in all alulations, only the reation rate onstant kr is tuned in order to ahieve a range of Da values. The water onentration profile 4

6 The 13th International Conferene on Energy Storage Greenstok 2015 as a funtion of dimensionless time for various Da values is shown in Fig. 2. Eah profile shows the water ontent in the β phase from the moving interfae to the partile surfae. The arrow in Fig. 2 indiates the diretion of propagation of the sharp interfae between α/β phases. The shape of eah profile at a ertain time level is attributed to the interation between the intrinsi reation and the mass diffusion. From the definition of the Damkӧhler number, a small Da value means that the interfaial hemial reation rate is relatively low ompared to the diffusion rate of water moleules in the β layer. In the ase of Da = 0.1, the interfae moves very slow ompared to the diffusion of water in the rystal. As a result, water released by reation at the interfae has suffiient time to drain away. In ontrast, for a large Da value, e.g. Da = 100, the intrinsi hemial reation is faster, and water annot esape from the partile quikly enough. This leads to an inrease of the water onentration near the interfae, whih in turn diminishes the propagation of the reation interfae. In other words, the Damkӧhler number plays a signifiant role in determining the nature of the diffusion/reation dynamis. A very small Da value means that the overall reation is ontrolled by the intrinsi hemial reation at the interfae, while a very large Da value means that the overall reation is ontrolled by the diffusion of water through the β phase. Figure 2: Profiles of water onentration distribution in a spherial partile for various Da values. Another remarkable feature in the present results is that the water onentration at the interfae dereases after it reahes a peak. The explanation is that as a strutural model, the rate of reation is essentially related to the reation area whih hanges during the ourse of transforming the spherial partile. As the reative area shrinks, the mass prodution at the interfae also delines. As a result, a large perentage of gaseous water an diffuse away from 5

7 The 13th International Conferene on Energy Storage Greenstok 2015 the partile during the same time period, whih in return will aelerate the hemial reation at the interfae. To further investigate the limiting mehanism of the reation kinetis between the intrinsi reation and the bulk diffusion proess, the dimensionless interfae position and normalized dimensionless interfae veloity as a funtion of dimensionless time for the different values of Da are plotted in Fig. 3. At low values of Da, the interfae position in Fig. 3(a) exhibits a linear dependene on time, whih is onsistent with the zero-order kinetis model used in the intrinsi reation. In ontrast, at high values of Da the proess is diffusion ontrolled, where the plot of the interfae position against time is urved. Figure 3: (a) Dimensionless interfae position for various Da values; (b) Dimensionless interfae veloity for various Da values (right). In Fig. 3(b), the normalized dimensionless interfae veloity is shown, whih is defined by: 1 dr v 1 Da dt. As disussed above, the veloity is onstant at a small Da value orresponding to a linear r profile. With the inrease of the Da value, the dimensionless veloity dereases rapidly, whih is attributed to the diffusion limitation. In the ase of Da = 1, the transition from a reation to a diffusion ontrolled proess an be observed. The overall kinetis of the dehydration reation is initially ontrolled by the interfaial hemial reation during whih the interfae veloity is onstant. As the interfae moves away from the partile surfae, water vapor annot diffuse out of the rystal effiiently, whih leads to the aumulation of water moleules. Consequently, the movement of interfae position is slown down gradually beause the propagation of the reation front is proportional to the differene of water onentration from the equilibrium onentration (see Eq. 4). The shape fator disussed above also explains the inrease of the interfae veloity in Fig. 3(b) after it reahes a minimum value. It is interesting to note that the interfae veloity for Da = 1 almost inreases to its initial value at the end of the reation. The reation kinetis will again be governed by the intrinsi reation after it is governed by the intrinsi reation in the initial stage of the dehydration proess and by the mass diffusion during the seond stage. By integrating the water onentration, the frational onversion of the salt hydrate an be plotted as a funtion of dimensionless time for different values of Da, see Fig. 4. Besides the numerial results at various Da values in solid lines, the line marked by + is the solution from the zero-order reation model and the marked line by o is the analytial solution of a pure diffusion proess. As expeted, our alulation results ompare well with the theoretial analysis under the extreme ases in whih either diffusion (large Da) or reation (small Da) 6

8 The 13th International Conferene on Energy Storage Greenstok 2015 dominates the dehydration proess. In the ase of Da = 1, the shape of frational onversion is nearly linear as in the ase of Da = 0.1. However, it is a result of the ombined influene of interfaial reation, mass diffusion and partile shape, instead of pure interfaial reation as in Da = 0.1. Based on the previous disussion, it is safe to say that the range of Da values for whih both intrinsi reation and mass diffusion are important in determining the reation kinetis is between 0.1 and 10, outside of whih only a single mehanism, either the intrinsi reation or the mass diffusion, is dominant in the reation kinetis. Figure 4: Frational onversion as a funtion of dimensionless time for various values of Da during dehydration of a spherial partile. The marked lines by o and + are analytial solutions for pure diffusion and zero-order reation, respetively. CONCLUSIONS In this work a shrinking model is developed for desribing the dehydration reation of Li2SO4.H2O single partiles. It inludes the interation between the intrinsi hemial reation and the mass diffusion simultaneously. The Damkӧhler number is found to distinguish the ontrolling step during the transformation. A very small Da value means that the overall reation is ontrolled by the intrinsi hemial reation at the interfae, while a very large Da value means that the overall reation is ontrolled by the diffusion of water in the produt phase. Frational onversion alulated from the model agrees with the theoretial analysis under the extreme ases in whih either diffusion (large Da) or reation (small Da) dominates the dehydration proess. Within the range of Da = 0.1 and Da = 10, both the intrinsi reation and the mass diffusion are important in determining the reation kinetis. In future work, experiments are designed to validate the model and to examine the dehydration reation in a muh more fundamental way. Information gained on the mehanisms of the dehydration reation of Li2SO4.H2O monorystals ould form the basis for the development of a more rigorous kineti model. REFERENCES C. Ferhaud, H. Zondag, R. d. Boer, C. Rindt, Charaterization of the sorption proess in thermohemial materials for seasonal solar heat storage appliation, in: Proeedings of the 12th International Conferene on Energy Storage, Innostok, Lleida, Spain, C. Shuh, Modeling gas diffusion into metals with a moving-boundary phase transformation, Metallurgial and Materials Transations A 31 (10) (2000)

9 The 13th International Conferene on Energy Storage Greenstok 2015 I. Arvanitidis, S. Seetharaman, X. Xiao, Effet of heat and mass transfer on the thermal deomposition of SrCO3 ompats, Metallurgial and Materials Transations B 30 (5) (1999) M.E. Brown, A.K. Galwey, A.L.W. Po, Reliability of kineti measurements for the thermal dehydration of lithium sulphate monohydrate.: Part 1. isothermal measurements of pressure of evolved water vapour, Thermohimia ata 203 (1992) M.E. Brown, A.K. Galwey, A.L.W. Po, Reliability of kineti measurements for the thermal dehydration of lithium sulphate monohydrate.: Part 2. Thermogravimetry and differential sanning alorimetry, Thermohimia ata 220 (1993) P. Gupta, R.K. Saha, Mathematial modeling of nonatalyti fluid-solid reations-generalized mathematial modeling of fluid-solid non-atalyti reations using finite volume method: Nonisothermal analysis., Journal of hemial engineering of Japan 36 (11) (2003) R. Deshpande, Y.T. Cheng, M.W. Verbrugge, A. Timmons, Diffusion indued stresses and strain energy in a phase-transforming spherial eletrode partile, Journal of the Eletrohemial Soiety 158 (6) (2011) A718 A724. S.B. Kanungo, S.K. Mishra, Thermal dehydration harateristis of hydrated salts: A ritial review, Journal of the Indian Chemial Soiety 83 (2006) S. Yagi, D. Kunii, Studies on ombustion of arbon partiles in flames and fluidized beds, Symposium (International) on Combustion 5 (1) (1955) S. Yagi, D. Kunii, Fluidized-solids reators with ontinuous solids feedi: Residene time of partiles in fluidized beds, Chemial Engineering Siene 16 (34) (1961) T.C. Illingworth, I.O. Golosnoy, Numerial solutions of diffusion-ontrolled moving boundary problems whih onserve solute, Journal of Computational Physis 209 (2005) T. Knorr, M. Kaiser, F. Glenk, B.J. Etzold, Shrinking ore like fluid solid reationsa dispersion model aounting for fluid phase volume hange and solid phase partile size distributions, Chemial Engineering Siene 69 (1) (2012) T. Saetre, A model of hydrogen diffusion in spherial partiles, International journal of hydrogen energy 31 (14) (2006) V.R. Subramanian, H.J. Ploehn, R. E. White, Shrinking ore model for the disharge of a metal hydride eletrode, Journal of the Eletrohemial Soiety 147 (8) (2000) Z. Cui, F. Gao, J. Qu, Interfae-reation ontrolled diffusion in binary solids with appliations to lithiation of silion in lithium-ion batteries, Journal of the Mehanis and Physis of Solids 61 (2) (2013)