Moisture transport in concrete during wetting/drying cycles

Transcription

1 Cheitry and Material Reearch Vol.5 03 Special Iue for International Congre on Material & Structural Stability Rabat Morocco 7-30 Noveber 03 Moiture tranport in concrete during wetting/drying cycle A. Taher X. Cao I.S. Pop A.J.J van der Zanden and H.J.H Brouwer Eindhoven Univerity of Technology Departent of the Built Environent Eindhoven the Netherland Eindhoven Univerity of Technology Departent of Matheatic and Coputer Science Eindhoven The Netherland Abtact. An accurate decription of oiture tranport in concrete i ignificant to deterine the ervice life of a concrete tructure. Many author anaged to decribe the global oiture tranport in concrete during wetting/drying cycle by uing Fick law of diffuion. The difference between the odel of the variou author lie in the ue of the diffuion coefficient. There are author who are convinced that by uing two different diffuion coefficient one for wetting and the other for drying a better decription of the oiture tranport in concrete can be obtained. In thi paper thi approach i exained by iulating and coparing two odel one with a ingle diffuion coefficient and the other with two diffuion coefficient. A coparion i ade between the reult of a odel with two diffuion coefficient and a odel with a ingle diffuion coefficient where the diffuion coefficient i the average of the wetting and drying diffuion coefficient. The reult i coputed for one cycle of wetting and drying. The iulation how that there are difference in the reult of the odel. In order to validate the odel and to invetigate which of the odel decribe the oiture tranport ot accurately experiental work i needed. Introduction The durability of concrete i related to the decription of oiture tranport. Moiture tranport in arine environent where drying and wetting cycle occur lead chloride to penetrate into reinforced concrete tructure. When chloride reache the rebar corroion can appear and that decreae the ervice life tie of the tructure. By decribing the oiture tranport in uch a tructure the ervice life can be deterined. Many decription of oiture tranport in concrete can be found in the literature. For exaple the author [-5] decribe oiture tranport in concrete tructure by uing a ingle diffuion coefficient. In [6-8] the ae by uing two different diffuion coefficient for drying and wetting. The purpoe of thi paper i to copare thee two ethod of decribing oiture tranport in concrete tructure during drying/wetting cycle. In addition to the odeling an experiental et-up i propoed to validate the odel. Matheatical odel In thi paper the oiture tranport (one phae flow) in concrete i decribed baed on the Darcy law and a balance [9]: k v = kr pc (Darcy) () µ where v i the voluetric velocity (/) k the intrinic pereability of the concrete ( ) k r the relative pereability (-) µ the vicoity ( kg ) and p c the capillary preure ( kg ). ( φ) + v = 0 ( a balance ) () where φ i the poroity of the concrete (-) the water aturation (-) and t the tie (). Coonly k r and aturation ee [9]. By ubtituting eq.() into eq.() k ( ) ( kr pc ) = 0. t φ µ p c are increaing function of the Furtherore the following i aued in thi paper: The aple i hoogeneou. There i no flow at the boundarie except the boundary at the right. The initial aturation i contant. () For the boundary at the right a periodic repetition of (3) 86

2 Cheitry and Material Reearch Vol.5 03 Special Iue for International Congre on Material & Structural Stability Rabat Morocco 7-30 Noveber 03 wetting and drying cycle i aued. In view of thee auption the lateral flow i 0. Thi allow reducing the odel to one dienion. The aple occupie then the interval (0 L ). The following condition are ipoed = β ( ) while the initial and boundary condition are (3) at t = 0 : = i in (0 L ) (4) at x = 0 : k pc kr = 0 µ x t > 0 (5) at x = L : = b ( t) t > 0 (6) where i i the initial aturation b ( t) the aturation at the right boundary. In the context of a one-dienional odel eq.(3) becoe k pc ( φ) kr = 0. x µ x To ake the equation dienionle the following notation are introduced x t pc x := t := pc := L T P where L T P are characteritic value for the length tie and capillary preure. Subtitute eq.(8) into eq.(7) and et (7) (8) ( x0) = i (0 t) = 0 ( t) = b ( t) x (4) where i i a contant nuber between 0 (in cae of dried concrete) and (in cae of water aturated concrete). b ( t) i a periodic function which iulate wetting/drying cycle t (0 Tw ] + k Tp b ( t) = 0 t ( Tw Tp ] + k Tp (5) where T d i the dienionle drying tie T w i the dienionle wetting tie T p = T d +T w i the period of one cycle and k i any natural nuber. 3 Approache Two odeling approache for the oiture tranport in concrete are copared in thi paper. The firt odel ue the ae diffuion coefficient for both wetting and drying phae. The econd odel ue two diffuion coefficient + one for the drying D ( ) and one for the wetting D ( ). In the literature [8] thee following coefficient are deterined experientally T = φµ L kp then eq.(7) tranfor into = D( ). x x (9) (0) 0 6 D ( ) = 0 e (6) D ( ) = 0 ( ) (7) 6 + ( ) 0.08 which are plotted in Figure. Here dpc ( ) D( ) = kr ( ). () d By uing the Kirchhoff tranforation β ( ) = D( v) dv () 0 eq.(0) becoe 3. Standard odel: one diffuion coefficient A tandard odel conider the ae diffuion coefficient for both wetting and drying In view of eq.(6) and eq.(7) the following average are conidered D ( ) + D ( ) Da ( ) = (8) D ( ) = D ( ) D ( ). (9) g 87

3 Cheitry and Material Reearch Vol.5 03 Special Iue for International Congre on Material & Structural Stability Rabat Morocco 7-30 Noveber 03 Here Da ( ) i the arithetic average of the diffuion coefficient D g () the logarithical average of the + diffuion coefficient D ( ) the diffuion coefficient for wetting and D ( ) the diffuion coefficient for drying. When olving the yte ign( ) i replaced by the regularization ign ( ) (ee Figure ) δ t t δ δ if < δ ignδ ( ) = if δ < < δ δ δ δ + if > δ (6) where 0 < δ << i the regularization paraeter. Fig.. Variou diffuion coefficient a a function of aturation The coefficient introduced in eq.(8) and (9) are then ued for the Kirchhoff tranforation in eq.(0). For the reulting odel eq.() an iplicit nuerical chee [0] i cobined with a linear iterative procedure []. 3. Hyteretic odel Here the hyteretic odel in [] i adapted = xx p( ). (0) In the above p( ) ha to include the witch between two diffuion coefficient in thi cae p( ) read p( ) = p ( ) + γ ( ) ign( ) () e t where pe ( ) γ ( ) and ign( t ) are defined a β ( ) + β ( ) pe ( ) = β ( ) β ( ) γ ( ) = and if > 0 ign( ) = if t < 0. / β () (3) (4) ( ) are the Kirchhoff tranforation for the wetting/drying diffuion coefficient / β + / 0 ( ) = D( z) dz. (5) Fig.. The graph of (a.) ign and (b.) ign δ Solving the above equation directly uing finite difference ethod lead to incorrect reult. Alternatively the invere φ δ ( r) of the regularized ign δ i ued a conidered in [3] r + δ + if r < δ δ φδ ( r) := δ r if r [ δ δ ] (7) r δ + if r > δ δ For the tie dicretization let t be the tie tep and p p( t) ( t) The iplicit dicretization of eq.(0)and eq.() are: 88

4 Cheitry and Material Reearch Vol.5 03 Special Iue for International Congre on Material & Structural Stability Rabat Morocco 7-30 Noveber = p xx + + e p p ( ) = φ δ ( ) γ ( ) (8) where = 0... initially Further at at 0 =. x = 0 p + = 0 and x + x = L = (( + ) t). b i In the equation above eq.(8) and the left ide of eq.(9) are explicit for +. The right ide of eq.(9) i iplicit for + due to the nonlinear function of pe ( ). Therefore pe( + ) i iplified for a linear function a invetigated in [4] by a Taylor expanion given by + ' + pe ( ) = pe ( ) pe ( ) ( ). + (9) Then the dicretized for becoe = xx p (30) + ' + p pe ( ) pe ( ) ( ) = φ δ ( ). γ ( ) For dicretizing the patial derivative finite difference on a grid with x = / N ( N N ) are ued. 4 Nuerical reult The odel i ipleented in Matlab. Here only one cycle i coputed which i one day for wetting and ix day for drying. The initial condition ( c i ) in thi cae i aued to be 50% aturation. Further the following i ued in the 5 nuerical chee: δ = and x = 0 dt = 0. Fig. 3. Coparion of the odel The reult how that there i a large difference between the three odel. To validate the odel experiental work i needed which will be explained in the next ection. 5 Experient Mortar pecien with a water ceent ratio of 0.5 and ceent type of CEM I 4.5 N are ued in thi paper to validate the odel. Mortar pecien are prepared by cating the in PVC tube with a diaeter of 00. After one day the ortar i deoulded and cured for 8 day. The ide of the pecien i ealed with epoxy to enure a one dienional flux. At one of the two open urface of the pecien the condition i changed to iulate cycle. Wetting i iulated by contacting the urface with water during one day and drying by blowing dry air with a certain flow at the urface during 6 day. After one drying/wetting cycle (one week) the oiture content i deterined by weighing the pecien. Thi a i copared with the calculated a fro the oiture profile of the odel for validation. Figure 4 how the experiental etup. The coparion between the reult of the tandard odel with one diffuion coefficient and the reult of the hyteretic odel with two diffuion coefficient are hown in Figure 3. Fig. 4. Sealed ortar pecien encounter wetting cycle for one day and drying cycle for ix day 6 Concluion Two variou ethod (tandard ethod and hyteretic ethod) to decribe oiture tranport in concrete during wetting/drying cycle are ipleented in Matlab and the reult are copared. Meaured diffuion coefficient in the 89

5 Cheitry and Material Reearch Vol.5 03 Special Iue for International Congre on Material & Structural Stability Rabat Morocco 7-30 Noveber 03 literature i ued in the odel. One cycle of wetting/drying i copared. The duration of the wetting phae i one day and drying i ix day. Fro the obtained nuerical reult it can be concluded that uing the tandard odel with one diffuion coefficient or uing the hyteretic odel with two diffuion coefficient can have a large influence on the reult. After one cycle the hyteretic odel give clearly a higher aount of oiture content in the tructure a it i hown in Figure 3. Furtherore an experient et-up i deigned to validate the odel. Reference. S.J.H. Meijer J.M.J.M. Bijen R. de Bort A.L.A. Fraaij Coputational reult of a odel for chloride ingre in concrete including convection drying-wetting cycle and carbonation. Material and tructure; 38 pp (005).. J. Arfvidon A new algorith to calculate the iotheral oiture penetration for periodically varying relative huidity at the boundary. Nordic Journal of Building Phyic; (999). 3. M.J. Cunningha Moiture diffuion due to periodic oiture and teperature boundary condition Çöan approxiate teady analytical olution with non-contant diffuion coefficient. Building and Environent; 7 pp (99). 4. L. Pel Moiture tranport in porou building aterial (995). 5. A. Selander Hydrophobic Ipregnation of Concrete Structure: Effect on Concrete Propertie (00). 6. J.M. Torrenti L. Granger M. Diruy P. Genin Modeling concrete hrinkage under variable abient condition. ACI Material Journal; 96 (999). 7. V. Baroghel-Bouny Caractériation de pâte de cient et de béton-méthode analye interprétation (994). 8. C. Li K. Li Z. Chen Nuerical analyi of oiture influential depth in conrete during drying-wetting cycle. Tinghua Science & Technology; 3 pp (008). 9. F. G. Pinder G. W. Gray Eential of Multiphae Flow and Tranport in Porou Media John Wiley & Son Inc. (008). 0. F.A. Radu I.S. Pop P. Knabner Error etiate for a ixed finite eleent dicretization of oe degenerate parabolic equation Nueriche Matheatik; 09 pp (008).. F.A. Radu I.S. Pop P. Knabner Newton type ethod for the ixed finite eleent dicretization of oe degenerate parabolic equation in Nuerical Matheatic and Advanced Application A. Berudez de Catro D. Goez P. Quintela P. Salgado (Ed.) Springer-Verlag Heidelberg; pp (006).. A. Yu. Beliaev S. M. Haanizaeh A theoretical odel of hyterei and dynaic effect in the capillary relation for two-phae flow in porou edia. Tranport in Porou Media; 43 pp (00). 3. B. Schweizer Intability of gravity wetting front for Richard equation with hyterei Interface and Free Boundarie; 4 pp (0). 4. A. Rätz B. Schweizer Hyterei odel and gravity fingering in porou edia ZAMM - Journal of Applied Matheatic and Mechanic. (0). 90