Kinetic study of the dissolution of vanadyl sulfate and vanadium pentoxide in sulfuric acid aqueous solution Ranine El Hage, Fabien Chauvet, Béatrice Biscans, Laurent Cassayre, L. Maurice, Théodore Tzedakis
To cite this version:
Ranine El Hage, Fabien Chauvet, Béatrice Biscans, Laurent Cassayre, L. Maurice, et al.. Kinetic study of the dissolution of vanadyl sulfate and vanadium pentoxide in sulfuric acid aqueous solution. Chemical Engineering Science, Elsevier, 2019, 199, pp.123-136. 10.1016/j.ces.2019.01.024. hal- 02290658
HAL Id: hal-02290658 https://hal.archives-ouvertes.fr/hal-02290658 Submitted on 17 Sep 2019
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Official URL: https://doi.org/10.1016/j.ces.2019.01.024
To cite this version:
El Hage, Ranine and Chauvet, Fabien and Biscans, Béatrice and Cassayre, Laurent and Maurice, L. and Tzedakis, Théo Kinetic study of the dissolution of vanadyl sulfate and vanadium pentoxide in sulfuric acid aqueous solution. (2019) Chemical Engineering Science, 199. 123-136. ISSN 0009-2509
Any correspondence concerning this service should be sent to the repository administrator: [email protected] Kinetic study of the dissolution of vanadyl sulfate and vanadium pentoxide in sulfuric acid aqueous solution a a b c a R. El Hage , F. Chauvet , B. Biscans b, L Cassayre , L. Maurice , T. Tzedakis ,*
• Laboratoirede Génie Chimique, UMR CNRS 5503, CAMPUS Université Toulouse 111 - Paul Sabotier,FSI. 118, Route de Narbonne, 31062 Toulouse,France • Laboratoirede Génie Chimique, UMR CNRS 5503, CAMPUS/NP - ENSJAŒT, 4 allée Emile Manso, 31 432 Toulouse,France < UniversitéToulouse 111 - PaulSabotier, 118, Route de Narbonne, 31062 Toulouse, France
HIGHLIGHTS
• Vanadium(IV) sulfate VOS04 and vanadium (V) pentoxide V2 05 dissolution . • Temporal evolution of the concentrations of vo2• and VOi• at various temperatures. • Understanding the dissolution limitations: vrv -+mass transfer; vv -+ reaction with W. • Simple kinetic models to predict the evolution of vü2•and V02• concentrations.
ARTICLE INFO ABSTRACT
Keywords: The study deals with the 'vanadium (IV) sulfate' and 'vanadium (V) pentoxide' dissolution processesin 2 Vanadyl sulfate 3 M H2S0 4 aqueous media. Severa( measurements of the concentration of dissolved vo • and VOi were Vanadium pentoxide achieved in the range of O 40 •c, and allowed to understand thelimitations of the dissolution process
Dissolution kinetics ('endothermic' mass transport for V0S04 and 'exothermic' reactionwith the proton for V205 ). ln addi Vanadium redox battery lion, simple models were proposed (diffusion/accumulation for V0S0 and kinetic rate for V 0 ) and their Solubility 4 2 5 resolution leads to theoretical kinetic equations describing the temporal evolution of these concentra lions with satisfactory agreement with the experimental curves. Solubility's data and their temperature dependence were determined for both vanadium compounds involved
1. Introduction vanadium" redox tlow battery (VRFB) developed in the 1980's by Rychcik and Skyllas Kazacos (1988), Skyllas Kazacos et al. (1988) Renewable energy storage studies have expanded in the past and has been widely explored since (Cunha et al., 2015; Skyllas decades due to the rapid increase in energy consurnption, limited Kazacos et al., 2016, 2015, 2013). fossil fuels reserves and growing ecological concems of their The battery was introduced to overcome the problem of cross impact on health and environment. contamination of electrolytes, inducing capacity lasses reported Only a very limited number of systems enable energy storage primarily by NASA researches in the 1970s for the Fe Cr system such as thermal storage, pumped hydropower and compressed (Thaller, 1977, 1974). Given that the same element in different oxi air energy storage (Amirante et al., 201 7; Ding et al., 2009). Electro dation states is used in both compartments of the VRFB, no con chemical storage constitutes an interesting alternative and tamination of the active material will occur in case ions cross the recently the Redox Flow Battery (RFB) acquired a great importance ionic separator. The redox couples employed are v<111l/'.f11l and as they have the particularity of converting and storing energy by y{Vl/y{ 111l for the negative and the positive half cells respectively using electroactivespecies dissolved in electrolyte solutions (Wang (Sum et al., 1985; Sum and Skyllas Kazacos, 1985), and the only et al., 2013; Leung et al., 2012). The most spread RFB is the "ail redox reactions involved are thus the valence changes ofthe vana dium ions. The electrolyte solutions are usually prepared in 2 3 M * Corresponding author at: Chemical Engineering Laboratory, UMR CNRS 5503, sulfuric acid, even if other electrolytes have been investigated FSL Université Toulouse Ill - Paul Sabatier, 118, Route de Narbonne. 31062 (generation 2 and 3 of the VRB) (Skyllas Kazacos et al., 2007. The Toulouse,France. VRB is characterized by a long life span (>10,000 cycles) (Ashby E-mail address: [email protected](T. Tzedakis). and Polyblank, 2012), low maintenance cost and deep discharge
https://doi.org/10.1016/j.ces2019.01 .024 Nomenclature
A(765 nm) absorbance at the wave length of 765 nm PES polyether sulfone A Arrhenius pre exponential factor R° initial radius of the particle (m)
aVOSO4 solid activity of the solid compound R(t) radius of the particle at the dissolution time t (m) a constant (Leveque correlation) r initial reaction rate (mol L 1 s 1)
A1,A2,A3 and A4 constants (defined in Appendix A) rglobal total rate of the reaction of the dissolution of V2O5 (mol 2+ 1 1 Cbulk concentration of VO in the bulk (M) L s ) 2+ Csurface Csaturation superficial concentration of VO , assumed at RFB Redox Flow Battery dp u q saturation (M) Re Reynolds number m dp u l dp particle diameter at time t (lm) SEM scanning electron microscope D diffusion coefficient (m2/s) S surface of particles involved in the dissolution process 2 Ea activation energy (J/mol) (m ) eðÞt thickness of the spherical crown film, function of the Sc Schmidt number = m/D time (m) Sh Sherwood number (Sh kðtÞ dp = = D e absorptivity (cm 1 M 1) 2 þ a Re1 2 Sc1 3 ! Leveque correlationÞ
DRH reaction enthalpy (J/mol) t dissolution time (expressed in ‘s’ for VOSO4 and in ‘h’ for j number of iterations taken as 100 V2O5) k kinetic constant following the Arrhenius law (unit func u average rate of the liquid around the solid particle cal tion of the reaction order) culated from the stirring rate (m/s), assumed constant D 2+ kð Þ mass transfer coefficient of the VO around the solid during the experiment time t eð Þ t 1 2 particle of VOSO4 (m s) m kinematic viscosity (m /s) l=q 2+ m moles number of dissolved VOSO4 (to VO ) into the DU internal energy (J/mol) bulk VRFB Vanadium Redox Flow Battery 3 M molar mass of the solid VOSO4 5H2O (kg/mol) V solid volume present in the solution at time t (m ) N number of VOSO4 particles having a certain initial ra V l total volume of the solution (including both the liquid dius R° and the powder)
n°(V) initial moles number of V2O5 (mol) x stirring rate (rpm) + n°(H) initial moles number of H (mol) X moles number of V2O5 dissolved at time t q specific gravity of the powder (kg/m3) PTFE polytetrafluoroethylene
2þ 2 capability (Vijayakumar et al., 2013). The energy density of the bat VOSO4ðsÞ ¡ VO +SO4 ð3Þ tery is largely depending on the volume and concentration of the vanadium electrolytic solutions and numerous authors studied ¡ þ 2 ð Þ the effect of the composition of electrolyte solutions (Choi et al., (VO2)2SO4ðsÞ 2VO2 +SO4 4 2017; Skyllas Kazacos et al., 2016) on the performance of the As mentioned by Rahman and Skyllas Kazacos (2009), vana battery. dium (V) can precipitate after 1000 h at 50 °C and the redissolution Despite that, stable solutions with vanadium concentrations of the oxide V2O5 (Rahman, 1998) appears to be difficult. Numer higher than 2 M could not be achieved in the working conditions ous authors (Ivakin and Voronova, 1973; Vijayakumar et al., of the VRB, limiting the quantity of stored energy to around 2011; Kausar, 2002) conducted studies to understand the poor 3 40 kWh/m . thermal stability of V(V) at high temperatures. They showed that The electrolyte in the negative electrode compartment does not the precipitation process of V2O5 is endothermic and thus (III) seem to exhibit major limitations related to the solubility of V enhanced when temperature increases. (II) (II) and V , but one point to mention is that V is easily oxidized Moreover, the aqueous chemistry of vanadium (V), studied by by air (choi et al., 2013). One important limitation encountered several authors (Crans et al., 2004; Elvingson et al., 1998), inferred by the vanadium battery appears to be related to the precipitation the presence of many species as a function of the solution’s pH. It of compounds in the positive electrode: precipitation of the V(IV) at + was suggested that pervanadyl or dioxovanadium cation (VO2)is (V) low temperature and precipitation of the V compounds at high the main species formed in the lower pH range (0.5 1.3) (Baes temperature. and Mesmer, 1976). However, the presence of VO3 species was also This fact limits the working temperature range of the VRB reported in the concentration range of 1 M of V(V) in 5 M total sul ° between 10 and 40 C(Li et al., 2011). Precipitation of various fate (Kausar, 2002). Based on Raman spectroscopy analysis, the vanadium salts and oxides was studied by several research groups authors propose that VO3 species are surrounded by a densely and it was found that higher sulfuric acid concentrations stabilizes packed ionic environment of other V(V) and sulfate ions and these (V) the V solutions (Rahman, 1998) but decreases the solubility of ions could be involved in weak or unstable complex formation V(II),V(III) and V(IV). This is attributed to the common sulfate ion + with other ions. In the case of H ions depletion, VO3 tends to pre effect (Skyllas Kazacos et al., 2016): increasing the total sulfate cipitate as a disordered V2O5 aggregate (Kausar, 2002). Increasing concentration shifts the equilibria (1) (4) towards the left, leading the acid concentration enables to prevent this precipitation and to a lesser dissociation of the vanadium sulfate salts (Rahman and (V) + favor the stabilization of V in the form of VO2. However, this Skyllas Kazacos, 1998; Cheng, 1991) and to the precipitation of the induces a negative effect on the overall performance of the battery salts in solution. because of the increasing viscosity and an enhancement of the 2þ 2 ohmic drops resulting in the lowering of the energy efficiency VSO4ðsÞ ¡ V +SO4 ð1Þ (Choi et al., 2017). 3þ 2 Numerous works exist on the field of the vanadium battery, V2(SO4)3ðsÞ ¡ 2V + 3SO4 ð2Þ and one important objective is to avoid the vanadium pentoxide precipitation. Indeed according to Rahman (1998) and Rahman 2.2. Characterization methods and Skyllas Kazacos (2009) vanadium (V) oxide dissolution is dif ficult, in addition to being an endothermic dissolution (Ivakin and 2.2.1. UV visible spectrophotometry for vanadium (IV) analysis Voronova, 1973; Vijayakumar et al., 2011; Kausar, 2002). Neither The UV VIS spectrophotometer device is a Hewlett Packard the understanding of the dissolution phenomena of the vanadium Model 8453. compounds, nor elucidation of the corresponding mechanisms According to their oxidation state, the vanadium compounds was significantly explored; the published studies do not exhibit exhibit various absorption bands at the UV visible wavelengths the determination of the physical limitations in order to act range. However, no absorption band could be determined for the + upstream. Hence, the present work deals with the study of the VO2 species; this is why potentiometric titration was used (see dissolution mechanisms of two common vanadium compounds next section). in 3 M sulfuric acid, at various temperatures (0 40 °C), stirring For the V(IV) quantification, the absorbance was measured at a rates and initial particles size. Apart from being the most com wavelength of 765 nm (Choi et al., 2017) exclusively due to VO2+. mon concentration used for an operating battery with 2 M VOSO4, A preliminary calibration enables to obtain the A(765nm) =f the sulfuric acid concentration (3 M) is chosen here, as a compro ([VO2+]) correlation and leads to an absorptivity e value of 1 1 De mise between its positive effect on the dissolution of the vana 21.4 cm M , with deviations e of 0.6 and 0.25 from the values dium pentoxide and its negative effect on the dissolution of proposed by Choi et al., 2013 and Brooker et al.(2015) respectively. vanadium sulfate (Rahman, 1998). Note also that, by increasing the sulfuric acid concentration, the solubility of the oxide 2.2.2. Potentiometric titration of vanadium (V) increases, but this increases the viscosity of the solution leading A potentiometric titration method is used to determine the con to a pressure drop. Moreover, during the cycling of the battery + centration of the VO2 released from the dissolved V2O5 powder. (IV) (V) and more specifically during recharge (oxidation of V to V ), The titration agent is Mohr salt, supplied by Sigma Aldrich and pre + H ions are generated in the positive compartment leading to pared in sulfuric acid media (Le Flem, 1964). The redox systems an increase in acidity, therefore, it is not necessary to study + 2+ (III) (II) involved in this titration are thus VO2/VO and Fe /Fe and higher H2SO4 concentrations. the titration reaction is: To sum up, the general purpose of this study is to understand þ þ ðIIÞ ¡ 2þ ðIIIÞ ð Þ the phenomena governing the dissolution of vanadyl sulfate VOSO4 VO2 +2H +Fe VO +H2O+Fe 6 and vanadium pentoxide V O and to establish the corresponding 2 5 The titrations are carried out using a combined Pt Ag/AgCl/Cl kinetic laws, in order to have a better control on the accidental pre electrode. cipitation during VRB operations.
2.2.3. ICP analysis ICP analysis (Inductively Coupled Plasma; ULTIMA 2 ICP OES) 2. Material and methods was used for total vanadium aqueous concentration determination, in order to characterize the amount of water in the commercial 2.1. Dissolution protocol vanadyl sulfate powder VOSO4 xH2O. The chemicals used are vanadium (IV) sulfate oxide hydrate 2.2.4. Morphology and particle size distribution of the powders VOSO4 xH2O (99.9%), vanadium (V) pentoxide V2O5 (99.6%) from The scanning electron microscope (SEM) used to observe the Alfa Aesar, NormaPur H2SO4 from Sigma Aldrich, and deionized grains morphology is a PhenomWorld XL SEM. The solid powders water. (V O and VOSO xH O) were coated with a thin gold layer (using Dissolution experiments were achieved in a thermo regulated 2 5 4 2 an Emiteck K550X device) prior SEM observation. reactor (cylinder of an internal diameter = 2 cm) at constant tem A laser diffraction particle sizing technique, using a Malvern perature. An adequate volume of aqueous solution 3 M sulfuric Mastersizer 3000, was used to get the size distribution of the acid is poured in the reactor and stirred with a 1.5 cm magnetic grains. The Malvern Mastersizer is usually used for materials rang bar until reaching the selected temperature. Then, an excess of ing from hundreds of nanometers to several millimeters in size (Fu vanadium compound is added (m(vanadyl sulfate) = 4.559 g and m(- and Sun, 2001). The particle size is reported as a volume equivalent vanadium pentoxide) = 1.461 g), marking the initial time of the experi sphere diameter. ment. Stirring rate is kept constant during the experiment. At regular time intervals, aliquots are withdrawn from the solution with plastic syringes and are directly filtered twice with filters 3. Results and discussion resisting to the acid media. The first filtration is performed using a polyether sulfone (PES) filter with a pore diameter of 0.23 mm, 3.1. Dissolution of vanadium (IV) sulfate and the second one a polytetrafluoroethylene (PTFE) membrane filter (Millipore ready to use filters in plastic assembly) with a 3.1.1. Characterization of the initial powder pore diameter of 0.1 mm. The commercial vanadyl sulfate powder VOSO4 xH2O used for The recovered filtrate is diluted and analyzed with the appro the dissolution experiments is partially hydrated. Its water content priate method. x was determined by ICP at different concentrations, according to The vanadium compounds dissolution occurs according to reac the method of determination of the vanadium concentration in tions (3) and (5) for VOSO4 and V2O5 respectively. an aqueous solution. The results of three analyses are found to be in the range from 5.4 to 5.6; thus taking into account the uncer þ þ tainties of the analyses, the retained value of x is 5. Note that this V2O5ðsÞ +2H ¡2VO2 +H2O ð5Þ result is in agreement with an octahedral geometry, and also with These reactions were taken into account because the species the fact claimed by Selbin (1965), the hydrated vanadyl sulfate 2+ involved at the chosen concentration of H2SO4 (3 M) are in agree exists as [VO(H2O)5] , ‘the vanadyl ion’ in acidic solutions. In the ment with the E pH diagram indicated by Post and Robins (1976), following, VOSO4 xH2O is thus indicated as VOSO4 5H2O Fig. 1. and with the V(V) species distribution calculated from constants Fig. 1 presents SEM micrographs of the commercial powder. given by Baes and Mesmer (1976), Guzman et al. (2002). These images display aggregated and multi shaped particles of Fig . 1. SEM images of the vanadyl sulfate V0S04-5H:i()commercial powder; 15 kV-Point; 8SO full. ail sizes. Their dispersion during the dissolution in a stirred solu 3.1.2. Temperature dependence of the dissolution kinetics tion is expected to be easy, because of the dislocation occurring Fig. 3 gathers experimental measurements of vü2• concentra after the dissolution of the smallest particles. tion in the electrolyte during the dissolution, for temperatures in The VOSO4 5H 2O particle size distribution is presented in Fig.2 . the range from 5 to 30 °C. For ail the examined temperatures, a 2 The first graph (Fig. 2 1) provides the particle size distribution of similar temporal evolution of the concentration of vo • was the commercialpowder and clearly shows two distinct ranges: from obseived: the concentration increases rapidly at short reaction 5 to 350 µm and from350 µm to 4 mm. The commeràal powderwas times (t < 3 min) and then the dissolution slows down until equi sieved at 315 µm and separated into two fractions, forwhich particle librium is reached, after around 10 min When the temperature sizes are presented in Fig. 2 2 and 2 3. They confirm two distinct increases from 5 to 30 °C the initial dissolution rate increases as populations of particle diameter: dp < 315 µm and dp > 315 µm. well.
6 100
(1) l 4 l ::, ::, -.::o., 50 ë ni > 3 2 E u::,
o...J-1.0 -.,.....,..,...,..!lll:lil10.0R"lltl: ljU.L,Ll,1,1,L!,1,1,1,4.LLLl.1,LLL,Ll,l,Ll,,l,l,l,1,,1.L.LLl..l,Ll,.L,11.100.0 1000.0 ��10000.0,.,.J....,o Size ranges (µm)
8 .---r100. 10 100
(2) 8 (3) 6 l l E 41 � :::, QI 6 � :::, ->o E QI 4 -u 50 0 ::, E -u 5 � ::, 41 .!l! 4 ...n:J g :::, E g 'S :::, E 2 :::, u 2 u
0-i- -..-...-.-;i�\o\o+-....._...,,.�.1.+,1+.....,.,._l-'-"-....,...... ,...... ,_, ..,.,.,-1--0 1.0 100.0 1000.0 o10.0-i---,-- ,.ii;100.0,;;ci5::ii ctl,l-U-IJLW..U.U�WJ1000.0 l...-.�10000.0...J-o Size ranges (µm) Size ranges (µm)
Fig. 2. Laser diffractionanalysis (P • 3 bar; dry mode) showing the V0S04-5H20 size distribution: (1) commercial powder,(2) sieved powderd p < 315 µm; (3) sieved powder dp > 315 µm. 1vo2+1 (M) 3.13. Effectof stirringrate and particle size on dissolution kinetics 2 Severa! dissolution experiments were achieved by applying three stirring rates (500, 100 and 15 rpm), corresponding respec tively to (i) a well stirred homogeneous suspension, (ii) a mini 1.5 mum stirring rate enabling to uplift the particle and (iii) partially decanted solid. Besides, these experiments were carried out for two different particles size dp ( <315 µm and >315 µm). The corre sponding time evolution of vo 2• concentration is presented in Fig. 4. For a definedrange of particlessize (dp < 315 µm cuives 1 2 3 0.5 or dp > 315 µm curves 4 5 6, Fig. 4), it appearsthat a higherstir ring rate increases the dissolution kinetics, since higher concentra 2 tions are achieved in a shorter time. For instance, [Vo •1cdp < 315 ,.m: 0 -f'-...... -.---.- -.--r- ...... - ...... --r-...... -.-...... ro = 500 rpm: r = l') = 1.5 M and [Vü2•Jcd < 315 µm: ro = 100 rpm: r = 4 8 10 12 p 0 6 t (min) l') = 1.0 M. Under constant stirring, the dissolution rate increases by Fig. 3. Temporal evolution of the concentration of VO� released by the dissolution decreasing the partides size. Thus, for a given dissolution time, of a VOSO4-5H2O commercial powder, for various temperatures; [H2SO4) • 3 M; stirring 500rpm; aliquotsfiltered by 0.23 and 0.1 µm filters. and before reaching equilibrium, higher sait concentrations were measured with smaller particles. Indeed comparison of curves (1) The first concentration reported on the graph corresponds to and (4) obtained in two "well stirred" suspensions show that the 20 s after the introduction of the solid into the stirred solution. concentration of vo2• after 1 min reaction time increases from For this first measurement, at 5 °C the vanadium concentration ~1.2 to ~1.5 M when the size decreases from dp > 315 µm to reaches 0.7 M and this value doubles (13 M) at 30 °C. d < 315 µm. 2 p The concentration of the vo • measured at 20 s, will beused to A different shape is obtained for curves (3) and (6), which cor determine the initial dissolution rate r0 (defined as the ratio responds to the lowest stirring rate; they exhibit the slowest disso 1"°'�'°'). Assuming that the kinetic constant of the initial dissolution lution rate for both particle sizes. The slow agitation does not rate follows the Arrhenius law, Eq. (1) givesthe initial dissolution enable the motion of the particles which sediment at the bottom rate r°. of the vesse!, forming a 'coagulated' powder paste. Around each grain of this paste, the vü2• concentration reaches rapidly the sat ,,. k xf A Xe Ea/RT xf (1) uration concentration These facts are in agreement with a dissolu tion rate limited by mass transfer. For long reaction time, ail the where fis a function of the various operating parameters (activity ♦ curves tend to converge to the same saturation concentration of the solid vanadyl sulfate, diffusion coefficient of vü2 • size of (~1.7 to 1.8 M). solid particles ...). Sorneof these parameters depend on the temper ature and this dependence is expected to be studied in a future work. Here, in order to get a rough order of the magnitude of the activation energy, the experimental data were analyzed (assuming 3.1.4. Equilibrium data the function f independent of T) and leads to the following linear Aftera certain time of stirring (between 5 and 10 min), the solu regression: tion is saturated and the concentration of vo2• is constant. The 2266.7 equilibrium values are reported in Fig. 5, along with data from + 4.8579 (2) Skyllas Kazacos et al. (2016) and Rahman and Skyllas Kazacos T(K) (1998). This relation leads to an approximation of the activation energy A satisfactory agreement is observed between our results and of 18.8 kJ/mol for the dissolution of VOSO4 5H2O in 3 M sulfuric those of previous works, which confirms that the solubility of acid, a value relatively low compared to the activation energy of vü2•increases with temperature. For the highest comparable tem a dassical chemical reaction perature(30 °C), a significant difference of around 20% is observed.
2+ d <315µm [VQ2+] d >315 µm 1vo 1 P P (M) (M) 2 2 (1) i i � i � f ! (4) ! ! (2) u t (5), i#! 1 , t 1 f t �tt t(3/ t (6) ! ft t t (min) t (min) 0 0 0 5 JO 15 20 0 5 JO 15 20
Fig. 4. Temporal evolution of the concentration of vü2• released by the dissolution of a VOSO4-5H2Û commercial powder, for suspensions containing two different particle
sizes and subjected to various stirring rates co. T • 25 °C, [H�O4) • 3 M, VOSO4-5H2O powder, Left: dp < 315 µm: co(in rpm)for(1 ), (2)and( 3)are 500/triangles, 100/squares and 15/diamonds respectively; Right: dp > 315 µm: co(in rpm) for (4), (5) and(6) are 500/triangles, 100/squaresand 15/diamondsrespectively. ô' 4 3.1.5. Elucidation of the mechanism of the vanadyl sulfate dissolution • Skyllas Kazacos The results of this study show that, at constant temperature,the ♦Rahman dissolution of vanadyl sulfate in 3 M sulfuric acid is limited by ::1e'. 3 Athis work mass transport and that two main parameters (stirring rate and "' particle size) affect the dissolution rate. ·= • Let us considerthe following mechanism for the dissolution of g • the solid sait VOSO4 5H2 O: À À • ' • VQSQ4(solid) � VQSQ4(solvated/dissolvedonthesurface) (7)
2 Vû$Û4(solvattd/dissol-.dorc�rfaœ) t; (VQ + + SOi ) bothspttiesdissoclat
Even if the first dissoàation of H2SO4 to HSO4 is total (high S x D x (grad è),u,tac, of partiel• (5) KH,so,/Hso,), the second dissociation to S� remains partial with (Fick's 1st law) an equilibrium constant KHSo,iso! 0.013 at 25 •C(the concen To simplify, the VOSO 5H O solid particles are considered trations are in mol/kg) (Wallace, 1966). However, the released 4 2 spherical and a simple filmmodel around the particle is creating quantity of sulfate ions from HSO dissociation is not significant 4 the limitation to mass transfer (i.e. one directiontransfer), the pre compared to the quantity released fromthe dissolution of VOSO ; 4 vious equation can be simplifiedas following: this imply that [vo2•1 can be assumed equal to [SO¾ ]. Conceming the activity of the solid powder of VOSO4, we will assume it equal to 1. Thereby, the above equation can be written SxDx (aë)âR => as follows: surfac, of partiel•
"lvii+ X "Isa! X [ X cbulk C,u,tac, v<>2+] [s� ] =>K (J) s X D X avaso solid �7)buB< 4 (R(r) + e(rJ) R(r) S X k(t) X (Csar Cbu& (6) (4) ) The effect of the convectionwas examined on the mass transfer Îi,o2+ XJ'so2 coefficient k rJ, using the expression of Ranz and Marshall relative to where Â. . ( °"""'•"'"' 2 a forced convection around a solid sphere (Cussler, 1984) (also Because the concentrations of vo • at saturation are relatively known as the Leveque correlation providing the Sherwood number). high (1.2 1.7 M), the activity coefficientsdepend on the concentra 1 2 tion, white the activity of the vanadyl sulfatesolid is equal to 1; in a Sh Sherwood number 2 + a x Re 1 x Sc113 (7) future work, the Pitzer model will be used in order to express these activity coefficients.Here, in order to get a rough estimation of the This correlation is appropriated for spherical partides having sensitivity of vanadyl sulfate solubility against temperature, the the same size. ln order to improve its applicability on the present logarithmof the vü2• saturation concentration was plotted versus case, some assumptions will be considered: 1 /T,assuming À. independent fromthe concentration of the various speàes. The calculations will be achieved assuming the presence of four The linear regression analysis leads to the following relation: separated size ranges of the solid partides having an initial 982.2 radius R• = 20, 40, 500 and 1000µm representing respectively +3.8 20%, 35%, 35% and 10% of the total solid volume (see curve of T(inKJ Fig. 2 1, cumulated volume); with a relatively low correlation coefficient (R2 = 0.91) with the Ali the particles of the powder will be assumed spherical, experimental results,explaining the uncertaintiesdue to the depen despite the various shapes observed (Fig. 1 ). Considering Eq. dence of the activity coefficients on the concentration. (7), the constant "a" is generally assumed equal to 0.6. However, From the slope of the above equation ( 982.2 k/(2R)), we in the present study and because of the large size distribution, deduce the value of k = 16.3 kJ/mol. "a" will be taken as an adjusting factor, which effect will be The obtained value is (i) positive showing the endotherrnic examined. character of the vanadyl sulfate dissolution, and (ii) a relatively low value, compatible with the dissoàation of an ionic sait weakly The molar quantity of the dissolved vü2•"m" for N particles of bonded. initial radius R• can be expressed as: q V q 2 m N V M one particle M q 4 p ð 3 3Þ ð Þ N R RðtÞ 8 ] (M)
3 M 2+ (Refer to notation and symbols) Experimental 2+ m [VO 1 Optimal curve The concentration of VO in solution can be written asCbulk V ; l Simulation 1 V being the total volume of the suspension (including both the liq l Simulation 2 uid and the solid). Simulation 3 Combining Eqs. (6) (8) leads to Eq. (9) describing the variation of the radius R of the solid particles as a function of time, and rep Variation of "a" resenting the dissolution rate r of the vanadyl sulfate: 0 p p 0 5 01 3 3 3 3 A1 A2 R RðtÞ þ A3 RðtÞ A4ðR RðtÞ Þ RðtÞ t (min) r ) dR dt RðtÞ ð9Þ 2 where A1 to A4 are constants (refer to Appendix A for their devel oped expressions). dR ] (M) r ) ðÞ 2+ Eq. (9), written as dt fR, is solved numerically using the þ ð ; Þ Euler method (Rjþ1 Rj step f R t ) and enables to calculate a [VO 1 theoretical radius at time t, given that at t° =0,R=R°. Experimental reaction time 10 : Note that the step used is: number of iterations j 100 0 1min Optimal curve The value of the radius R of a certain particle of VOSO4 5H2O, Simulation 1 obtained at a certain time t, from the resolution of Eq. (9) enables Simulation 2 Simulation 3 Variation of "N" to obtain the theoretical values of the dissolved quantity of VO2+ 0 i.e. the moles number, using Eq. (8). 04812 Note that Eq. (9) is resolved 4 times, each for one of the chosen t (min) initial R° (20, 40, 500 and 1000 mm) of the solid particles and the 2+ total concentration of the dissolved vanadium sulfate (VO2+)in Fig. 6. Temporal evolution of the concentration of VO released by the dissolution of the VOSO 5H O commercial powder: dots = experimental curves; Continuous solution is deduced (sum of the four obtained values of molar 4 2 lines = simulated curves obtained with the non optimized parameter value, i.e. the ° quantity divided by V l). Also, for each R , a different value of the constant ‘‘a” in the Leveque equation (figure on the top), and the percentage of each adjusting factor ‘‘a” (Eq. (7)) was determined. particle size range which directly affects the number of total particles ‘‘N” of each R° Fig. 6 shows two examples of the iterative determination, at (figure on the bottom); Broken lines = simulated curves obtained with the 30 °C, of the optimal value of two parameters (the constant ‘‘a” optimized parameters of ‘‘a” (figure on the top), and ‘‘N” (figure on the bottom). in the Leveque equation and the percentage of each particle size range which directly affects the number of total particles N of each R°). The optimized value of ‘‘a” or ‘‘N”, is the one leading to a curve Both the stirring of the suspension and the size of the particles of the concentration of the dissolved vanadium sulfate which fits strongly affect the dissolution rate (mainly for dissolution times best with the experimental data curve. lower than 5 min), proving that the dissolution kinetics of the Following this iterative mode enables to evaluate the calculated vanadyl sulfate (to VO2+), is limited by the mass transfer of curves at each temperature and the results are presented in Fig. 7 VO2+ from the solid powder surface to the bulk, while the chem comparatively to the experimental data. Note that the viscosity ical reaction with the proton is not limiting. was calculated for each temperature (Fassulo, 1965), while the value of the diffusivity at 25 °C(Jiang et al., 2016) was used for all the examined temperatures. 3.2. Dissolution of vanadium pentoxide DC : = : A relatively good agreement ( C 0 1M 1 5M) is observed between the results of the model and the experimental 3.2.1. Characterization of the initial powder ones. Discrepancies are observed for short durations of the dissolu The commercial powder of V2O5 was characterized by SEM and tion (t < 1 min) because of the difficulty to standardize rapidly the laser diffraction particle sizing technique. The SEM micrograph in suspension and to achieve precise measurement at the first few Fig. 8 1 shows that the V2O5 particles form agglomerated sticks seconds after the stirring begins of various sizes. The size distribution (Fig. 8 2) shows a large dis Another difficulty comes from the dissolution of the smallest persion of particle size from 120 nm to 4 mm. particles, which exhibit fast kinetics. In fact, compared to the bigger The graph exhibits three peaks located respectively at 2, 80 and m particles, this introduces additional uncertainties because the num 900 m; therefore, in order to study the effect of particles size on ber of small particles of powder was not exactly taken into account. the dissolution kinetic, the initial powder was sieved using four m m m To sum up, the obtained results lead to the following different cut off threshold sieves: 315 m; 200 m; 120 m and m conclusions: 80 m. Thus, five different diameter ranges of powder were obtained: dp > 315 mm; 315 > dp(mm) > 200; 200 > dp(mm) > 120; Regarding the effect of temperature: 120 > dp(mm) > 80 and dp(mm) < 80, three of which were used to The initial dissolution rate increases with temperature. carry out dissolution experiments. The solubility of the vanadyl sulfate at saturation (when the equilibrium of dissolution is reached) increases slightly with 3.2.2. Effect of temperature on dissolution kinetics the temperature in the examined range of 5 30 °C, and this The effect of temperature on the dissolution of vanadium pen is compatible with the dissociation of an ionic salt weakly toxide was studied in the range 0 40 °C and the temporal evolu + bonded. tion of the dissolved VO2 concentration is presented in Fig. 9. + rvo2 1 (M) T=5°C 2 2
-Theoretical -Theoretical ■ Experimental ■ Experimental
t (min) t (min) 0 0 0 4 8 12 0 4 8 12
2+ + rvo 1 (M) T = 15°C 1v02 1 (M) 2 2
-Theoretical -Theoretical • Experimental ■ Experimental
t (min) t (min) 0 0 0 4 8 12 0 4 8 12
Fig. 7. Comparisonbetween the experimental data( reportecl from Fig. 3) and the mode) (calculated using Eq. (9)) of the temporal evolution of the concentration of v02• releasecl by the dissolution of a VOS04-SH20 commercial powder, at various temperatures.
2 100
q) :, E g 50
______..... _._,_, �.-1-0 LO 10.0 100.0 1000.0 10000.0 Size ranges[µm)
Fig. 8. (1) SEM pictures (10 kV-Point; SEO) and (2) size distribution obtainecl (by laser diffraction sizing technique; P • 3 bar, dry mode) of the vanadium pentoxide V�5 commercial powder.
The behavior is rather similar for each temperature: a rapid dis stirring; for the dissolution at O °C a higher reaction time is solution occurs at "short" reaction times (less than 1 h) and then required to reach the saturation. the concentration tends to reach a constant value. Note that, for Severa! differences can be observedcompared to the dissolution ail the examined temperatures (except O °C), the dissolution equi features ofVOSO4: libria of the vanadium pentoxide to VOi is achieved after5 h of The saturation concentrations (Csa,.l of VOi are about three A simplified reaction scheme is proposed below for the vana times lower than those of the vü2•; in addition, the c.. , dium pentoxide dissolution (R5 ): decreases when the temperature increases (contrary to the case of the v solution is not saturated, even after 5 h. Another experiment � ,/ H• +- /'f was carried out at 0 °C where the suspension of vanadium pen o-v - ·o 1 � / � toxide was left under stirring for one day. The concentration of H 0 H (ln) 0 the dissolved VOi was deterrnined and the following values 0 �.+ 1 were obtained: [VOibh = 0.788 M and [VOihsh = 0.783 M. This o-v - vo,· • H,o / .,,,_ (S.) last result tends to show that the mass transport is not the lim H (ln) �o iting step. The concentrationof the VOi measured for reaction times lower than five minutes (named t;nitia1), will be used to estimate approx Hydrogen ion attacks the oxide and the product obtained from 0 reaction (Sb) decomposesacco rding to reaction (S ); then by reac imatively the initial rate of the V2O5 dissolution r (defined as the c iv�'.;:�aj tion with W, it formsan interrnediate(ln) which dehydrates to lead ratio ). toVOi. Assuming, as for the vanadyl sulfate, that (i) the kinetic constant However, another reaction pursue of the intermediate (ln) of the initial dissolution rate follows the Arrhenius law, and (ii) the could be its polymerization according to various reactions and in others terrns of the expression of the dissolution rate are indepen the last case, the extemal surface of each solid particle could be dent of the temperature, then, the logarithmic analysis of the evo entirely covered by a passive layer that prevents the dissolution lution of the initial rate with the temperature leads to the and could be the reason for which the observed saturation concen following correlation: tration ofVOi remains low. To clarify this point, the following dissolution experiment was 8727.3 carried out: a solid liquid suspension of the commercial V 22.7 (10) 2O5 T(in K) sieved powder (200< dp < 315 µm) was introduced into a H2SO4 3 M solution at 25 °C and stirred during 5 h. The results are illus The slope of this linear relation enables the deterrnination of a trated in Fig. 11: the [VOi] increases to reach the saturation con roughly estimated value of the activationenergy of the dissolution centration (~0.55 M after t ~ 1 h) then remains constant. Theo process: Ea 73 kJ/mol. the suspension was filtered to removethe residual solid and a fresh This energy is more than four times higher than the value sample of the powder was added into the filtrate. The dissolution obtained for the dissolution of VOSO4 (18.8 kJ/mol). This could be experiment was pursued for 6 h, and the temporal evolution of explained by the fact that the hydration of the vanadium pentoxide the concentration of the vanadium is indicated into Fig. 11 (for (V which exhibits covalent bonds) followedby its reaction with 2O5 t > 22 h until t = 28 h). The renewal of the solid does not change ions is energetically more difficult to achieve, comparatively to W the dissolved amount of vanadium, implying that the equilibrium the simple dissociation of the ionic sait VOSO 4. was reached and no further dissolution canbe achieved. This test confirms that there is no passivating or inhibiting layer forrned 3.2.3. Effectof available surface area on dissolutionkinetics on the surfaceof the particles. The effect of the particle size on the dissolution rateofV 2Os was examined at 40 °C and 500 rpm stirring rate and the results are 3.2.4. Equilibrium data shown in Fig. 10. The saturation concentration of VOi, obtained for t > 5 h Practically the same concentrations were measured for the ( extracted from Fig.9 ), are reportedin Fig.12 , simultaneously with three different size ranges of the solid particles (80 < d < 120 µm, the values from Rahman (1998), Skyllas Kazacos et al. (2016). Note p that the values extracted from the bibliography were multiplied by 120 < dp < 200 µm and 200 < dp < 315 µm). These results tend to show that the dissolution rate does not depend on the solid/liquid 2 in order to obtain the concentration of VOi, instead of the solu exchange surface. Theoretically, the particle size affects the ini tial bility ofV2O5 used in the cited refe rences. The saturation concen rate of the dissolution that should increase. However, in our case, tration of VOi decreases when the applied temperature increases, owing to the low values of these dissolution rates, the curves do even if a shorter time is required to reach the equilibrium. not have enough resolution to show any difference. Moreover, The equilibrium constant of reaction (5) can be written as a these results support our previous conclusion: the mass transfer functionof the activities of the speàes in solution: does not represent a limitation to the dissolution ofV2Os but it is 2 rather the chemistry of the system. (Ovo;) x an,o Consequently, the effect of stirring on the dissolution was no t av,o,soUa x (aw )2 studied. 2 2 Since the dissolution is not significantly affected by the particles an,o x (rvo;) ( (VOf] size, the possibility of kinetic contrai by surface passivation was 2 ° ) (11) av,o,soUa x (rw ) x (W] (vof] evaluated. 0.8 0.8 1vo,·1 0 OC) " • Skyllas-Kazacos (M) J 0 f ! VOi In the present case, the concentrations of2 the at saturation are lower than the values obtained for vo • at saturation (Fig. 9: 0.4 0.7 M for stirring duration of 5 h). The effect of the concentra 0 50 tion on the activity coefficient will also be examined in a future [V work via the Pitzer mode!. 02+ ] For the time being, in order to get a preliminary rough estima (M) o.45 tion of the sensitivity of the vanadium pentoxide solubility against 1 ( 2 1 � x "11io >'vo+) i 1 1 the temperature, the term ' ' will be considered con 0.40 ' OV20slOl'dX( l'H+ ) stant and named f Theo a logarithmic analysis of the data at the equilibrium (Fig. 9) versus the reverse temperature was performed 0.35 according to the following equation: 3 0. 0 +---�--�------[VO;J,., ) constant ln�) (12) 1 n ( ° 0.5 x ( 0 2 3 4 [W] [V01lsar 5 t (h) 6 The linear regression analysis of the experimental results leads Fig. 10. Temporal evolutions of the dissolvedVOi concentration (released by the dissolution of a V2 O5 co mmercial powder) for various initial sizes of the solid to the following equation ln 9.15 + n;72 with a particles of V� 5 obtained by sieving; [H:i$O4) • 3 M, stirring 500rpm, sieved V� 5 iff" ri.. ) powder, T •40 °C. (1) 80 [V02+l (M) [V02·1 (M) 0.6 0.8 r -' - +- - ½ ,+...... 0.6 �;-f 0.4 � i� I � 0.4 ' ° *' ° T = 10 C 0.2 I T=0 C ' f// I 0.2 �, I ►, ' 0 0 0 2 t (h) 4 6 0 2 t (h) 4 6 [V02•] (M) [VOz+] (M) 0.8 0.5 0.4 1- ,__ ,_- +- -+ 0.6 _ --+- -+- -!- _, 0.3 f ,t � 0.4 1 ° T=25 C 0.2 l T=40°C •I' l 0.2 I 0.1 1 I 0 0 0 2 t (h) 4 6 0 2 t (h) 4 6 Fig. 13. Comparison between the experimental data (diamo nds, imported from Fig. 9) andthe model(broken lines. calculated using Eq.(15)) for the temporal evolution of the concentration of VOi released by the dissolution of a V.,05 commercial powder, at various temperatures. Table 1 and more rapid, comparatively to that of the V2O5 oxide (which is Iteratively estimated values of the kinetic constants k(5) and k( 5) for the studied covalently bonded). temperatures, and their corresponding activation energies (estimations). Their initial dissolution rate increases with temperature, for ° 2 1 1 1 Temperature ( C) k(5) (mol s L k( 5) (mol s both compounds VOSO4 and V2O5, and the rough estimations of g) L) the activation energies at the beginning of the dissolution (18.8 kJ/ 4 4 0 0.9 10 2.3 10 mol for VOSO4 and 73 kJ/mol for V2O5) clearly show that the tem 4 4 10 2.6 10 3.4 10 perature influence two different phenomena: the mass transport 25 4.8 10 4 13 10 4 for the vanadyl sulfate, and the reaction with the proton for the 40 7 10 4 41 10 4 vanadium pentoxide. Activation energy (estimated by Ea Ea Ea/RT 2+ k=Ae ) (5) = 34.5 kJ/mol ( 5) = 52.9 kJ/mol The saturation concentration of the VO is reached after few minutes of stirring (<10 min), while several hours (>5h) were + required for the VO2 released by the oxide V2O5 to reach saturation (IV) Substituting it in Eq. (14) and considering the reaction orders (and more than 1 day at 0 °C). In addition the solubility of the V + + with respect to V2O5,H and VO2, to be equal to their stoichiomet salt is higher than the solubility of the oxide (at least two times, ric coefficients, the expression of global rate of the reaction is: even three for certain temperatures): VO2+ concentrations of 2 M can be achieved from the dissolution of VOSO4 at T 30 °C, while ! + 2=3 for the oxide dissolution the VO saturation concentration reaches dðXÞ ðÞp 0:5 2 Si Ni 34 M ° rðÞ kð Þ 0.45 M at 40 C. 5 Si 5 r q dt Na Moreover, conversely to the V(IV), in the case of vanadium pen nono 2 2 ð Þ ð Þ toxide decreasing the temperature causes the saturation concen = n H 2 X S 2 X S + 2 3 i i tration of VO2 to increase: from 0.42 to 0.78 M respectively at 40 ð Þ n V XRi k 5 Vsuspension Vsuspension and 0 °C, and the whole dissolution process appears to be exother 0 ð15Þ mic (k = 37.8 kJ/mol). Concerning the kinetic models, the dissolution of VOSO4 con Eq. (15), solved numerically using the Euler method, provides sists of the dissociation in the acid media, followed by its disper + the variation of the quantity of VO2 as a function of time. The only sion in the bulk. The whole process is limited by mass transport unknown parameters are k(5) and k( 5), the kinetic constants of the (more specifically diffusion and convection). The accumulation flux forward and backward reaction. They will be adjusted in order to of VO2+ was expressed using Fick’s Ist law and the Sherwood corre + determine the theoretical evolution of the VO2 concentration lation and the simulated results enable to reproduce experimental which correctly correlates with the experimental one. measurements and to validate the mass transport limitation. For Fig. 13 shows the correlation between the calculated values of V2O5, the dissolution rate appears to be limited by a chemical reac + dissolved VO2 (broken lines) and the experimental data (dia tion (acidic attack followed by the breaking bonds). A simple reac monds). The values of the kinetic constants, enabling to get a sat tion scheme was proposed for the dissolution and its rate was isfactory agreement between theory and experiment, are indicated expressed assuming the chemical reaction as an elementary reac in Table 1. tion. The model agrees with the experimental data and the result The results show that the backward kinetic constant k( 5) is ing kinetic constants show a backward reaction ( 5) more more sensitive to the temperature than the forward k(5). Besides, sensitive to temperature and having a rate in certain conditions for all the examined temperatures, k( 5) is higher than the k(5). This higher than the rate of the forward reaction (5). means that the dissolution of the oxide is allowed only for ‘rela Note that the obtained results were very sensitive to the sulfu þ tively’ low concentrations of VO2 , enabling to keep the term of ric acid concentration which changes during the battery operation the backward rate lower than this of the forward rate. (increases during the charge and decreases during the discharge). Note also that, because the order against the H+ concentration The 3 M of sulfuric acid was chosen as a compromise between its (in the forward term of the rate) is equal to 2, the acidic media is positive effect on the dissolution of the vanadium pentoxide and beneficial to the dissolution of the oxide and this is also in agree its negative effect on the dissolution of vanadium sulfate. ment with the results of Rahman and Skyllas Kazacos (2009). To sum up, in the absence of chemical additives enabling to keep the V(V) under dissolved form and avoid precipitation of the vanadium pentoxide, it is required to operate in strong acidic con 4. Conclusion ditions, with a ‘‘discharged battery” instead of a ‘‘battery com pletely charged”. Moreover, a charged vanadium battery will Numerous works exist in the field of the vanadium battery and exhibit a more stable behavior if stored in low (T < 20 °C) temper one important objective is to avoid the vanadium pentoxide pre ature conditions, rather than in high temperature conditions cipitation, because its dissolution seems difficult. In addition, the (T > 40 °C). Also, taking into account the results of the bibliography, published studies do not exhibit the determination of the physical it is better to avoid maintaining the battery in recharged state for a limitations and phenomena to be overcomed, in order to act long time. upstream. The aim of present work was the study of the dissolution mechanisms of the vanadyl sulfate VOSO4 and vanadium pentox Declaration of interests ide V2O5 in 3 M sulfuric acid, under various operative conditions, and also to bring knowledge on the understanding of the phenom The authors declare that there is no conflict of interest. ena governing these dissolutions. Several experimental data giving the temporal evolution of the dissolved vanadium concentrations versus various operating Acknowledgments parameters were acquired and supplied. Moreover, kinetic laws describing both dissolution limitations phenomena were deter This study was supported by the Agence Nationale de la mined and explained. As expected these compounds exhibit differ Recherche, France. The authors would like to acknowledge Brigitte ent behaviors towards dissolution. The general results show that Dustou, Laure Latapie and Sandrine Desclaux for the technical sup the dissolution of VOSO4 5H2O, an ‘ionic salt’, appears to be easier port they provided for the accomplishment of this work. Appendix A the exponents a, b and c are the reaction orders, respectively + + for: a site constituted by V2O5,H and VO2; The combination of Eqs. (6) (8) enables to obtain the expres k(5) and k( 5) the forward and backward kinetic constants dR r sion of dt (Eq. (9)) also noted expressing the variation of the par respectively. ticles radius over time and representing the dissolution rate of the Nr = the number of sites accessible to H+, at time t RðN S Þ VOSO 5H O. To reach this expression, the following calculation surface of the particles atthetimet j j 4 2 r þ r þ r =N 1 N 2 N 3 surface occupied by one molecule r steps were followed: in addition:‘‘nV ”, and ‘‘nH ” represents respectively the initial + kðtÞ de 1 1 D 1 1 moles number of V O and H , 3 3 2 5 Sh 2 þ a Re2 Sc ! kðtÞ ð2 þ a Re2 Sc Þ D 2R ‘‘X” the moles number of V2O5 dissolved at time t, by con sumption of 2X moles of dm + + H , and releasing 2X moles of VO2; ð Þ þS kðtÞ ðCsat CbulkÞ dt R!1 þ d½VO dð2XÞ dðXÞ q 1 2 Si 1 Si Si 4 N 3 3 ðÞ þ d p R R r 5 Si 3 M 2 D 1 1 2 dt 2 V suspensiondt V suspensiondt 4pR 2 þ a Re2 Sc3 nono dt 2R b c a n H 2 ðXÞ 2 ðXÞ ðNriÞ Si Si ð Þ kð5Þ b kð 5Þ c 14 4 q N 3 3 Na V V suspension ðCsat p R R Þ suspension 3 M V l Developing the above equation leads to: Moreover, the following changes are required in order to trans form Eq. (14) into a resolvable form: 1 R dR MDCsat 4 ND 3 3 aMCsat 2 ðÞ1 1 p þ ðÞu 2 3m 6 2 q q R R q 2 D R p dt 3 V l 2 Si 4 Ni 2 Nr R ðAÞ i r r i 4 aN 1 2 ðÞ1 3 3 1 p ðÞ2 u 2D3m 6 ðR R ÞR2 Besides the mass of the V2O5 introduced into the suspension is: 6 V l mV O mR þmR þ m p p 2 5 1 2 R3 3 3 þ ð 3 3Þ The mass of V2O5 particles having Ri as diameter can be dR A1 A2 R R A3 R A4 R R R r ) ð9Þ expressed by the two following equations: dt R mRi Molecular weight of V 2O5 mol number of particles having Ri M nVRi A A A A 4pR3 1 2 3 4 and m qV q i Ri solideRi 3 sat 1 1 M D C 4 p D N aMC 2 ð 1Þ 2 ðÞ 1 p 3 sat ð uÞ2 3m 6 4 p aN ðÞu 2 3m 6 4 R q 3 V l q 2 D 2 D q i ) 2 6 Vl Combining both equations leads to: M nVRi 3 no1 3Mn 3 VRi ð Þ Ri 4pq B Substituting (B) into (A) gives the number of sites having, at time t, R as initial diameter: Appendix B i no= 2 2=3 p 3Mn 1 3 34ðÞp 0 5Mn 4 Ni VRi ) Ni VRi Nri r 4pq Nri r q þ a þ b 1 d½VO Nr moles of H þ 2 ð Þ v :) rðÞ5 kð5Þ kð 5Þ And substituting nVRi by n V XRi gi es 2 dt Na V suspension ! 2=3 þ c 0:5 moles of VO Ni 34ðÞp M 2 2 ð13Þ Nr n X 3 ðCÞ i r q V Ri V suspension The following nomenclature will be used to treat this equation Substituting (C) into the expression of the rate (Eq. (14)) leads to: + and to determine the theoretical concentration of VO2: For the considered weight of the V2O5 particles having as ini dðXÞ Si ðÞ tial diameter: 10, 60 and 1000 mm (from Fig. 8), it will be noted: r 5 Si V suspensiondt 0 ! 1a 2=3 ; ; ðÞp 0:5 S1 S2 and S3: their average initial surface. @ Ni 34 M 2=3A ð Þ k 5 n V XRi S1,S2 and S3: their average surface at the time t. Na r q ; : R1 R2 and R3 their average initial radii. nob noc R1,R2 and R3: their average radii at the time t. n H 2ðXÞ 2ðXÞ Si Si b kð 5Þ c V V also: suspension suspension 0 !1a N1; N2 and N3 : are the number of particles having as initial 2=3 dðXÞ ðÞp 0:5 diameter: 10, 60 and 1000 mm Si @ Ni 34 M A rðÞ kð Þ 5 Si 5 r q Nr1; Nr2 and Nr3 : are the number of sites (i.e. the number of dt Na + the V2O5 molecules) accessible to the proton H , at time t for nob noc m a n H 2 ðXÞ 2 ðXÞ the particles having as initial diameter: 10, 60 and 1000 m 2=3 Si Si n V XR kð 5Þ r°: is the surface occupied by one molecule of V2O5 i b 1 c 1 V suspension V suspension Na: Avogadro number The size of all these equations appears to be so high!! why??? Fu, Q., Sun, W., 2001. Mie theory for light scattering by a spherical particle in an To solve this equation, the reaction orders are required and they absorbing medium. Appl. 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Studies of V(IV) and V(V) species in vanadium cell electrolyte. nono April 2002, A thesis submitted as part of the requirements for the degree of 2 2 Doctor of Philosophy (Ph.D) M.Sc (Chemistry), University of New South Wales, n H 2 ðXÞ 2 ðXÞ Si Si ð Þ Sydney 2052, Australia. kð 5Þ 15 Le Flem, G., 1964. Le système V2O5 – ThO2. Faculté des Sciences de l’Université de Vsuspension Vsuspension Bordeaux. Note that three equations must solved for the three particle Leung, P., Li, X., Ponce-de-León, C., Berlouis, L., John-Low, C.T., Walsh, F.C., 2012. + Progress in redox flow batteries, remaining challenges and their applications in sizes chosen, each one providing the mole number of VO2 produced energy storage. RSC 2, 10125–10156. https://doi.org/10.1039/c2ra21342g. by the dissolution of the corresponding family of these particles as Li, L., Kim, S., Wang, W., Vijayakumar, M., Nie, Z., Cheng, B., Zhang, J., Xia, G., Hu, J., indicated in the following table. 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