GAS DIFFUSION THROUGH UNSATURATED CEMENT-BASED MATERIALS

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1 GAS IFFUSION THROUGH UNSATURATE CEMENT-BASE MATERIALS Tha Hoa Vu (1,2), Faben Frzon (2) and Sylve Lorente (1) (1) Unversté de Toulouse ; UPS, INSA ; LMC (Laboratore Matéraux et urablté des Constructons) ; 135, avenue de Ranguel ; F Toulouse Cedex 04, France. (2) Commssarat à l Energe Atomque, EN, Waste Treatment and Condtonng Research epartment, F Bagnols sur Cèze, France. Abstract Cement-based materals possess a pore sze range varyng from nanometer to mllmeter. At dfferent relatve humdty, and when n equlbrum wth the exteror envronment, some of the pores are flled by the ntersttal soluton, and as a result authorze ether an aqueous dffusve transport or a gaseous one. The combnaton of ordnary dffuson (Fck regme) and Knudsen dffuson of hydrogen s examned through a model porous medum. The model system conssts of a bundle of cylndrcal tubes, each of whch s constructed from mercury ntruson porosmetry data. fferent confguratons are developed allowng to show the exstence of a saturaton threshold whch corresponds to the appearance of water curtans fllng frst the pores of smallest sze. The model quantfes the total thckness of the water curtans, and therefore account for the combnaton of gas dffuson and water dffuson for hydrogen molecules. The fast decrease wth the saturaton degree of the macroscopc hydrogen dffuson coeffcent of cement-based materals s descrbed by the model wth a good accuracy. 1. INTROUCTION Transfer trough porous meda s a fundamental topc that fnds applcatons n many doman such as envronmental, energetc, chemcal, and cvl engneerng. Several attempts were made to fnd relatonshps between dffuson coeffcents at the scale of the pore and at the scale of the materal, and to account for ntrnsc parameters lnked to the pores volume (porosty), the pore network geometry (tortuosty), and the lqud volume fracton fllng the pores (saturaton degree); e.g. Refs. [1], [2], [3]. More sophstcated s the model developed by Ncholson and Petropoulos [4] n whch molecular dffuson and Knudsen dffuson are ncluded, the model porous medum s assumed to be homogeneous on the macro scale and conssts of a bundle of ndependent cylndrcal tubes. Each of these s made up of a tortuous seres of a large number of short sectons whose radus s generated randomly by usng a probablty densty functon. The Advectve-ffusve Model [5] was developed n order to descrbe the transport of multphase fluds n porous meda. In ths model the dfferent transport mechansms (namely dffuson and convecton) can happen smultaneously. The resultng global transport s smply a lnear addton of dfferent phenomenologcal laws. An 261

2 even more complete descrpton, n the case of multcomponent gas transport through porous meda, s provded by the usty-gas-model (GM) [6], [7] whch consders the advecton, Knudsen dffuson, ordnary dffuson and ther combnatons. The term dusty-gas s used because the model treats the porous medum as a component of the gas mxture, consstng of gant, heavy molecules (lke dust n a gas) that are motonless and unformly dstrbuted n the space. The geness of ths work comes from the feld of nuclear engneerng and cvl engneerng. Today, the preferred soluton for a certan category of nuclear wastes conssts of usng cementtous materals as an envelope and physcal barrer of the waste. Any cement-based materal s a porous system wth pore szes rangng from nanometers to mcrometers. As such t can be nvaded by gaseous speces or lquds. Therefore to understand and predct dffusve transfer through unsaturated porous systems s of extreme mportance n ths specfc applcaton. The present work addresses the general ssue of gas dffuson through partally saturated porous materals by developng a model takng nto account for the ordnary dffuson, the Knudsen dffuson as well as ther combnaton va a gas knetc approach. Frstly, the dffuson coeffcent expressons were establshed at mcroscopc scale for a dry and then a wet pore. In order to determne the dffuson coeffcent at macroscopc scale, these expressons were next ntroduced nto a model mcrostructure whch s consttuted of parallel canals of varable cross-secton. Fnally, the applcaton of the model to hydrogen dffuson through unsaturated cement-based materals has gven a very good agreement wth expermental results. 2. MOEL ESCRIPTION For a dry materal, the gas dffuson coeffcent c, through one pore channel s the result of the combnaton of two dffuson mechansms: molecular dffuson, and Knudsen dffuson. The contrbuton of each mechansm depends on the pore sze, or the Knudsen number Kn whch s defned as the rato between the molecule mean free path λ and the pore dameter 2r. When the Knudsen number s greater than 10, Knudsen dffuson s the man mechansm, and c, = K,. For Knudsen numbers lower than 0.1, molecular dffuson prevals, and c, = M. In the Kn range , the two mechansms are of the same order of magntude. The effectve mean free path n a channel (λ e ) s therefore a functon of the pore radus (r) and of the mean free path n a unconfned space. In accordance wth [8] λ e can be wrtten = + (1) λ λ r e At ambent temperature and atmospherc pressure, the mean free path of hydrogen 2 molecules s about 131 nm ( λ= kt 2 πpσ from [9]). The equaton 1 shows that when the pore radus s of a few nanometers (Kn > 10) the pore sze lmts the molecule mean free path and λ r. For larger pore channels (radus of a few hundreds of nanometers), Kn < 0.1 and e λ e λ. Equaton 1 leads to = + c, M, K, (2) wth 262

3 3π M, = νλe 16 (3) 2r K, = ν 3 (4) The dffuson coeffcent M s calculated from [9], 1/ 2 3 kt RT M = (5) 2 8σ P Mπ where σ s the gas molecule dameter, k s the Boltzmann constant, T s the absolute temperature, P s the pressure, R s the gas constant, and M s the molar mass. The mean thermal molecular velocty s defned as 1/ 2 8RT ν = (6) πm Combnng Eqs. 5 and 6 wth the expresson of the mean free path [9] yelds the dffuson coeffcent M (Eq. 3). Consder now a materal partally saturated wth lqud. The saturaton degree S l s defned as the rato between the volume of lqud and the pore volume, and descrbes the level of lqud saturaton at the scale of the materal. At the scale of the pores, we vew the saturaton degree S l as the effect of m layers of lqud adsorbed to the pore walls, and consequently the space avalable for gas transport s rg, = r δ (7) The total thckness of the m layers s δ = m σh2o (8) where σ H 2 O s the water molecule dameter [8]. Because the dffuson coeffcent through lqud s four orders of magntude lower than the dffuson coeffcent of gas [10], dffuson through the lqud s neglgble when space s avalable for gas dffuson, namely when r g, > 0. Yet, because water flls frst the smallest pores, when the lqud saturaton degree ncreases the avalable space for gas transport vanshes, r g, = 0, and the gas molecules must dffuse through the lqud. In other words, the transfer of gas molecules occurs through lqud curtans that are the result of capllary condensaton [11]. Assume that ξ s the wdth of the lqud curtan n pore (Fg. 1). ξ depends on the saturaton degree, the pore radus r, the thckness of the layer of water molecules δ along the r axs, and the pore length L. In general case, the gas molecules dffuse n one pore through two meda: gas of thckness L ξ through the pore, and lqud of thckness ξ. Combnng the two, we obtan L 1 1 ξ = ( L ξ ) + (9) c, M K, l where l s the dffuson coeffcent through lqud. 263

4 Fgure 1: Two phases n the pore: gas and water. Fgure 2: One pore famly constructon. 3. PARALLEL CHANNELS WITH VARIABLE CROSS SECTION The model s based on parallel channels wth varable cross secton. The channels have round cross-sectons, but the scale of the cross secton vares wth the materal thckness. Ths feature s modeled by statng that the channel radus s a lnear functon of the channel thckness, and decreases when x ncreases (Fg. 2). Consder frst the dry state. Because the pore radus r vares along the channel, the dffuson coeffcent calculated from Eq. 2 also vares wth r. c, (r) = where 16 a= 3πλ and 3 16 b= + 2 3π 1 M, + 1 K, 1 νr = ar + b Along the axe x, the radus r vares wth the poston, each secton of length dx presents a radus r and the correspondng dffuson coeffcent c (r). The average dffuson coeffcent along a dry channel of thckness L s 1 and c, c, L 1 1 = dx L (13) (r) 0 c, 1 rmax, J(r) = L dr (14) r c, (r) mn, where J(r) s the jacoban matrx. Next, consder the unsaturated case. When the materal s partally saturated wth water and the pore radus s lower than the water molecule layer thckness δ, the zone s flled wth water and consttutes a water curtan. If the pore radus s greater than δ, the space avalable for gas s dctated by Eq. 7. ξ s the total length along a channel where the pore radus s smaller than δ. For a pore famly, ths length s (10) (11) (12) 264

5 L Concrete n Aggressve Aqueous Envronments, Performance, Testng and Modelng H( r) dx H( r ) J(r)dr (15) ξ = δ = δ mn, 0 rmn, δ where the H( x ) s the Heavsde functon : H(x) = 0 for x < 0 and H(x) = 1 for x 0. At any radus r we have ( ) ν( r δ ) ( δ ) + ( ) = H δ r + H δ r c, l a r b Accordng to Eq. 14, the average gas dffuson coeffcent through the pore belongng to the famly s r ( r δ ) max, 1 1 a + b 1 = J(r)dr + ξ c, νl (17) r δ L δ l Fnally, the gas dffuson coeffcent calculated at the global scale of the materal s = 1 S φ N f (18) e ( l ) c, g, where φ s the porosty, N s the number of pores, f g s a volumetrc gas fracton. 4. APPLICATIONS The model presented n ths work was compared to expermental data that are avalable n the lterature [12]. For the sake of clarty we gve here some detals on the samples that were prepared for the study. The materals were cement pastes based on CEM-I (Lafarge Cements, Le Tel factory). The water/cement ratos were 0.35 and The materals were cast nto cylndrcal moulds of 40 mm dameter and 80 mm heght. They were demoulded after 24 hours, then cured at 20 C for 9 months n a Ca(OH) 2 water saturated soluton. Next, they were kept for almost 10 years n sealed desscators at controlled relatve humdtes (RH): 3%, 54.5%, 69.9%, 81.8%, 93.2% and 100%. The cylnders were then cut nto 3 slces pror to dfferent expermental tests. The central slces of 20 mm n heght were kept for dffuson tests. The remanng slces were used for measurng other propertes such as porosty, water content, and densty (Table 1). Table 1: propertes of the two cement-pastes [12] Cement type CEM I CEM I W/C rato ensty (kg/m 3 ) Water porosty The cement paste was analysed wth mercury ntruson porometry (MIP). Representatve results are presented n Fg. 3 for the case wth water/cement rato of As expected for cement pastes based on type I cement, the results of mercury porometry reveal the exstence of two major pore famles: pores wth a radus of order 4 nm, whch correspond to the cement hydrates porosty, and pores wth a radus of about 20 nm correspondng to the capllary porosty. Although the MIP was subject to crtcsm n the lterature especally due to the bottleneck effect [8][13][14], t stll remans the most accessble technque to characterze porous materals of large pore-sze range. In fact the bottleneck effect only becomes sgnfcant when t occurs n a large proporton of the pores n the network [15]. In the present work, ths effect has not been taken nto account, the vrtual mcrostructure was created from (16) 265

6 the MIP data onto that a numercal ntruson test was performed. The comparson as shown n Fg. 3 s to assure the representatvty of the vrtual mcrostructure to the actual one. Samples of materal were frst placed n chambers wth controlled temperature and relatve humdty untl equlbrum was reached. The temperature was mantaned constant at 20 C, whle the montored relatve humdty ranged from 3% to 100%. Then the samples were tested n hydrogen dffuson cells [12]. The saturaton degree correspondng to each equlbrum state was calculated for a gven relatve humdty, and the effectve dffuson coeffcent of hydrogen was obtaned from Fck s frst law of dffuson and measurements of the hydrogen flow rate through each sample. The results are presented n Fg. 4, whch shows how the hydrogen effectve dffuson coeffcent drops abruptly when the water saturaton degree exceeds 70%. In order to descrbe the materal pore network wth a greater accuracy, we decded to work wth 5 pore famles, n accordance wth Fg. 3. Each famly s characterzed by ts man pore radus r m,, and the boundares of the range of rad r max, and r mn,. Fgure 3 also gves the total volume of mercury used for a gven pore famly ( V t, ), from whch results the local porosty φ. We assumed that the varaton from r mn, to r m,, and from r m, to r max, s lnear. Therefore each pore famly corresponds to an assembly of two truncated cones, accordng to the sketch proposed n Fg. 2. From the total volume of a pore famly V t, and the volume of the elementary pore of the famly, we can calculate the correspondng total number of pores N. One pore channel s constructed as an assembly of two consecutve famles and +1. The boundary between these two pore famles s fxed for all the pore channels. For example, n the case of a pore network made of 5 dfferent knds of pore channels, we have N 1 confguratons correspondng to the pore famly 1, and N 2 confguratons correspondng to the pore famly 2. The pore channel 1 s made of the assembly of famles 1 and 2. The number of pore channels s therefore N 1. As a consequence, there are (N 1 N 2 ) pore channels 2 made of the pore famles 2 and 3, (N 2 N 3 ) pore channels 3 made of the pore famles 3 and 4, (N 3 N 4 ) pore channels 4, and (N 4 N 5 ) pore channels 5. Note that n the case of the pore channel 5, only one sort of pore geometry s used n the constructon, because of the hgh number of correspondng pores. The comparson between the artfcal pore network and the mercury ntruson porometry data s presented n Fg. 3. The agreement between the constructed network and the actual one s good. Fgure 3: Mercury ntruson porometry results for a type I cement paste and a water:cement rato of 0.45, and arfcal pore network 266

7 Fgure 4 shows the results n terms of hydrogen dffuson coeffcent as a functon of the water saturaton degree together wth the expermental data. We present n Fg. 4 the results obtaned wth the boundary postoned at L/2, L/3, L/4, and L/5. The fgure shows that the boundary poston has a very weak effect on the dffuson coeffcent. The theoretcal results reproduce wth a very good accuracy the sudden drop n the dffuson coeffcent when the water saturaton degree reaches 80%. The sudden drop s due to the fllng of the pores wth water. The pore channels made of pores wth small rad exhbt frst the change n the dffuson mode: gaseous dffuson becomes gas dffuson through water because of the lqud curtans. More realstc s the case where the length of the pore channels s dfferent than the materal thckness. The effectve dffuson coeffcent s calculated as 1 e = c,φ 2 (19) τ rather than wth Eq. 23. τ s the pore network tortuosty defned as the rato between the pore channels length and the materal thckness [16]. The results are summarzed n Fg. 9 where the tortuosty factor vares between 1 and 5. When the materal water saturaton degree s hgh (n the range %), better agreement wth the expermental data s found wth hgh values of the tortuosty. When the saturaton degree s lower than 70%, the results gven by the model are closer to the expermental data when the tortuosty factor s equal to one, whch means when the model does not account for tortuosty effects. Such behavor fnds an explanaton when consderng the tortuosty as a factor representng the accessblty of gas to dffuson. Indeed when the saturaton degree s low the gaseous phase s nterconnected and gas dffuson occurs along the most drect route. At hgh saturaton levels, the obstacles created by water force the gaseous speces to dffuse through tortuous paths. In sum, dffuson occurs along the paths of least resstance [17]. Fgure 4: Hydrogen effectve dffuson coeffcent for varous locatons of the nterface between two pore famles. Fgure 5: Hydrogen effectve dffuson coeffcent as a functon of the tortuosty factor. 5. CONCLUSION In ths paper we presented a model for calculatng the effectve dffuson coeffcent of gaseous speces through unsaturated porous systems wth an applcaton to hydrogen dffuson through cement-based materals. We developed a method to create artfcal porous structures that resemble the actual one, based on results provded by mercury ntruson porometry. The artfcal pore network s 267

8 based on the dstrbuton of local porostes correspondng to local geometres of pore systems through an assembly of parallel pore channels wth varable cross sectons. The dffuson model s vald for partally saturated porous materals. It accounts for the hydrogen dffuson through water obstacles that fll partally the pores wth the smallest dmensons. The model allows us to predct the abrupt decrease n the effectve dffuson coeffcent, whch s measured when the water saturaton degree s around 70%. It was demonstrated that the results depend on tortuosty provded that tortuosty s not perceved as an ntrnsc parameter descrbng the porous geometry but as a factor accountng for the accessblty to dffuson to the gaseous speces. REFERENCES [1] Mllngton, R.J., Gas dffuson n porous meda, Scence 130 (1959) [2] Cure, J. A., Gaseous dffuson n porous meda: Part 1- A non-steady state method, J. Phys. : Appl. Phys, 11 (1960) [3] Cure, J. A., Gaseous dffuson n porous meda: Part 3 - Wet granular materals, J. Phys. : Appl. Phys. 12 (1961) [4] Ncholson,., Petropoulos, J. H., Capllary models for porous meda: I. Two-phase flow n a seral model, J. Phys. : Appl. Phys, 1 (1968) [5] Pruess, K., Oldenburg, C., Mords, G., TOUGH2 User s Gude, Verson 2.0, Lawrence Berkeley Natonal Laboratory Report LBNL-43134, Berkeley, CA, [6] Evans, III, R.B., Watson, G.M., Mason, E.A., Gas dffuson n porous meda at unform pressure, J. Chem. Physcs 35 (1961) [7] Mason, E.A, Malnauskas, A.P., Gas Transport n Porous Meda: the usty Gas Model, (Chem. Eng. Monograph 17, Elsever, New-York, 1983). [8] Schüth, F., Sng, K.S.W., Wetkamp, J., Handbook of Porous Solds, (Wley, Hoboken, 2002). [9] Aryanpour, G., Abbas, M.H., Computer smulaton of ordnary gas transfer n tubes, J. Porous Meda 8(4) (2005) [10] Ho, C.K., Webb, S.W, Gas transport n porous meda, (Sprnger, ordrecht, 2006). [11] Lu, T., Cho, K.F., L, Y., Capllary rse between cylnders, J. Phys. : Appl. Phys, 40 (2007) [12] Vdal, R., Expermental determnaton of the hydrogen dffuson coeffcent n a cementtous materal (n French), Report on ESS Elaboraton, caractérsaton et tratement des surfaces, Paul Sabater Unversty, Toulouse, France, [13] amond, S., Mercury porosmetry, an napproprate for the measurement of pore sze dstrbutons n cement-based materals, Cement and Concrete Research 30 (2000) [14] Moro, F. and Böhn, H., Ink-bottle effect n mercury ntruson porosmetry of cement-based materals, J. Collod and Interface Scence 246 (2002) [15] Ncholson,., Petropoulos, J. H., Capllary models for porous meda: III. Two-phase flow n a three-dmensonal network wth Gaussan radus dstrbuton, J. Phys. : Appl. Phys, 4 (1971), [16] Bejan, A., ncer, I., Lorente, S., Mguel A.F., Res, A.H., Porous and Complex Flow Structures n Modern Technologes, (Sprnger Verlag, New-York, 2004). [17] Bejan, A., Lorente, S., esgn wth Constructal Theory, (Wley, Hoboken, 2008). 268