A Numeical Hydaion Model of Poland Cemen Ippei Mauyama, Tesuo Masushia and Takafumi Noguchi ABSTRACT : A compue-based numeical model is pesened, wih which hydaion and micosucual developmen in Poland cemen-based maeials can be simulaed. Poposed model enables he pedicion of hydaion cuves as a funcion of he paicle size disibuion, chemical composiion of he cemen, wae o cemen aio and he acual cuing empeaue. Muual elaionship beween developing he micosucue and is effec on hydaion pocess is modeled explicily. In his conibuion modeling and validaion of his model ae discussed. KEYWORDS: Hydaion model, paicle size disibuion, micosucue, cemen composiion. INTRODUCTION Ove he pas few decades a numbe of sudies have been made on modeling of cemen hydaion in ode o gasp he ime dependen popeies of cemen based maeials (Kondo 968). In hose ecen sudies, simulaion of inicae and compound pocess of cemen hydaion, especially focused on mico-mechanics, wih he poenial of moden compue is aemped (van Beugel 995, Benz 99). Two poins seem o be helpful in aemping o skech ou wha makes i complex o simulae pocess of cemen hydaion and why i needs compue powe. One is physical aspec. Cemen pase sucue, which is composed of cemen paicles and wae, is deemined by paicle size disibuion (Bezjak, 980) and wae o cemen aio. In cemen hydaion pocess, cemen paicles ae ineconneced and make sucue of cemen maix. This physical aspec affecs he ae of cemen hydaion hough diffusion of ions (Knudsen 984). Second is chemical aspec. Cemen is mainly composed of icalcium silicae, bu is poly-mineal maeial a he same ime. Reacions of he componens ineac wih each ohe. And empeaue is much influenial on he ae of hydaion fom he chemical poins of view (Tomosawa 974). These wo aspecs have muual dependen elaionship hough diffusion of ions and maeial fomaion. This elaionship can no be solved wih simple equaion wih egad o space-ime poblem in cemen-based maeial. Bu using compue powe wih concep of discee even sysem makes i possible o find an answe o his poblem.. HYDRATION MODEL OF CEMENT PARTICLE. Basic Assumpions Poposed hydaion model is based on he fundamenal kineic model fo Poland cemen ha is developed by Tomosawa (Tomosawa 997). Tomoswa s model is expessed as a single equaion composed of fou ae deemining coefficiens which deemine he ae of fomaion and desucion of iniial impemeable laye, he acivaed chemical eacion pocess and he following elen diffusion conolled pocess. This peliminay appoach shows high poenial of simulaing of hydaion pocess. Fou coefficiens, howeve, ae jus fi paamee in Tomosawa s model and hey ae no pedicable Reseach Associae, Gaduae School of Engineeing, Hioshima Univesiy, Japan Gaduae suden, Gaduae School of Engineeing, The Univesiy of Tokyo, Japan Associae Pof., Gaduae School of Engineeing, The Univesiy of Tokyo, Japan
fom any infomaion of cemen popeies. This can be deduced by he fac ha paicle size disibuion and inepaicle conac, which play impoan ole in hydaion pocess, ae no consideed. In ode o ake ino accoun hese and ge consisen elaionship beween cemen popeies and coefficiens, econsucion and modificaion ae conduced fom he assumpions of Tomosawa s model wih addiional change. The assumpions ae lised as follows:. The cemen paicle iniiaes hydaion fom he momen ha i is bough ino conac wih wae.. The hydae fomed by hydaion adhees o he cemen paicle. And hydae will be coveing i up spheically unil inepaicle conac comes up. And unhydaed cemen keeps spheical shape as well. The new gel is fomed a he suface wih no esicion of inepaicle conacs. If he suface conacs wih he suface of anohe paicle, new gel is no longe poduced on i.. The hydae has a v gel imes as much as he oiginal cemen in volume. 4. The liquid phase, which is assumed o be wae, diffuses hough he hydae laye and eaches he suface of he cemen paicle (eacing fon) and chemically eacs wih cemen. This pocess coninues hough hydaion pocess. And pa of he hydae poduced a he eacing suface moves ou hough he laye of hydae. Hence, equi-mola coune diffusion of wae and hydae (pesumably ions) is assumed o be aking place in he hydae laye. 5. The diffusion coefficien of hydae laye fo wae is no diffeen beween oue poducs and inne poducs. This diffusion coefficien is affeced by ouosiy of gel as well as adius of gel poe in hydae. This phenomenon is expessed as a funcion of degee of hydaion. 6. The paicle size disibuion of cemen can be appoximaed by Rosin-Rammle funcion. And each paicle wih he same diamee has he same ae of hydaion. 7. Doman peiod in he iniial pocess of hydaion is assumed ha hee is a pocess in which he eacion esisance inceases wih he incease of degee of hydaion in each paicle (film fomaion) followed by a peiod in which he eacion esisance deceases wih inceasing hickness of oue poducs.. Paicle size disibuion in space Cemen paicle disibuion will make a big diffeence in cemen hydaion pocess. Defining degee of cemen hydaion as aio of eaced cemen volume o iniial cemen volume, in his sense, each paicle shows diffeen degee of hydaion and degee of hydaion of oal cemen pase should be accouned fo his diffeen degee of hydaion of each paicle. In he poposed model, i is assumed ha he cemen paicle disibuion can be expessed wih Rosin-Ramle funcion: n ( p) exp( bdp ) V d = () On he ohe hand, cemen paicles develop an inepaicle conacs as hydaion poceeds. This phenomenon is deemined by he eacion ae, iniial posiion and size of each cemen paicle in pase maix. Afe fomaion of inepaicle conac, cemen hydaion will be inhibied by decease of aea o suck available wae o hydae. In egad o his aspec, hese exis seveal models of locaion of cemen paicles. To cie insances, andom disibuion model (Kundes 997) and flocculaion model (Benz 000) ae poposed so fa. In his poposed model, i is assumed ha each cemen paicle has he same aio of available wae volume o is cemen volume and uni cubic cell is deemined as ceneed spheical cemen paicle suounded wae in cube. The size of unie cubic cell is accoding o enie wae o cemen aio of cemen pase. (see Figue ) This is one of he mehods o aange Figue. Schemaic epesenaion of paicle size disibuion model in cemen pase
he cemen paicles in space homogeneously. In his case, he cemen paicles ae locaed as pseudo- 6-neighbohood. The 6-neighbohood aangemen is no efficien. If he closes packing is assumed, he aangemen should be -neighbohood among he same size of paicles. Bu when flocculaion is consideed (Taylo 997), he aangemen of spheical cemen paicles, which assumes homogenous aangemen wih egad o volumeic aio, mus be less efficien and he numbe of neighboing paicles will be educed.. Hydaion model of each cemen paicle Accoding o he Tomosawa s model hydaion pocess is expessed as a single equaion ha is dominaed by hee diffeen ae-deemining phenomena, i.e. poducing iniial poecive laye, inwad diffusion of exenal wae equivalen o he ouwad diffusion of eacion poducs and chemical eacion on he suface of unhydaed suface (Tomosawa 997): d Cw ρc γ + Wa, g = dr ( v ) gel d R d + +, = () d R d k D k d e whee is adius of unhydaed cemen paicle, R is oal adius including he gel laye, D e is effecive diffusion coefficien of wae in he cemen gel, k is coefficien of eacion ae pe uni aea of eacion fon, γ is he soichiomeic aio by mass of wae o cemen, W ag, is he aio of wae enapped in he gel poe o cemen, ρ c is densiy of unhydaed cemen and v gel is volumeic aio of hydaed cemen pase gel o unhydaed cemen. Wih Eq.(), degee of hydaion of each paicle, α d is calculaed: 4 4 αd =.0 π π =.0 0 0 () whee α d is degee of hydaion of he cemen paicle whose diamee is d = 0. The oal degee of hydaion α is defined as accumulaion of each degee of hydaion ove he cemen paicles whose disibuion is accoding o Eq.(). The fis em of he denominao on he igh side of fome Eq.() elaes o he iniial eacion, indicaing is eacion esisance. This affecs only a vey ealy sage when oal degee of hydaion almos equals o 0, and in his case, he eacion ae is deemined by k d. Hee k d is assumed o be expessed as he sum of he em of incease of mass ansfe esisance as incease of oal degee of hydaion and he em of degease of mass ansfe esisance as incease of hickness beween of each hydaing cemen paicle: oiginal bounday and eacion fon 0 B k 4.0 d = + C.5 0 (4) α As he hydaion pocess pogesses he gel densiy has been found o incease (Relis 977). The effecive diffusion coefficien D is assumed o be decease wih incease of he gel densiy:.5 e De = DE ln (5) αd whee D E is iniial diffusion coefficien ha is dependen on he composie of cemen paicle..4 Sucual limiaion by inepaicle conac Cemen paicles ae expanding wih hydaion pocess by a faco v gel. Available space is occupied on a fis come and each paicles ae ineconneced wih hadening of cemen pase maix. This fomed cemen pase maix sucue has effecs on physical aspec of maix as well as hydaion pocess of cemen wih feedback fom i. In his egad, he sucual limiaion is modeled explicily.
Cemen paicle is assumed be spheical and each paicle has cubic space in accodance wih wae o cemen aio. This assumpion means ha he each paicle is even in ems of available space fo expanding. Thee ae hee ype of mode wih degee of expanding (Figue ). Mode is he sae ha he cemen paicle does no make conac wih sufaces of cell, mode is he sae ha he cemen paicle sas o make conac wih sufaces of cell and he conaced pas have cicula shape and mode is he sae ha he cemen paicle makes conac wih sufaces of cell widely and he conaced pas ae conneced each ohe. (i) Iniial sae of cemen paicle and cell Volume of cemen paicle in he cell whose size is in a side is deemined wih he cemen densiy ρ c, he wae densiy ρ w and wae o cemen aio W / C : Vc = c W / C ρ + ρ w Raio of adius of cemen paicle o he lengh of cubic side 0 holds: V c 0 = 4π / (ii) Suface and volume In mode, mode and mode, he aio of adius of hydaed cemen paicle o he lengh of cubic side saisfies 0 < 0.5, 0.5 < / and / < / especively. The suface ha is in coniguiy wih wae is epesened by a funcion of : 0.5 S () = 4π, S () 4 π π / / =, S() = 8 dxdy (8) x y / /4 x And he volume of hydaed cemen is epesened by a funcion of : V() 4 V() = π, = + 6 0.5 / 4 V() = π 6π + 4 0.5 x / 4 x π + Ac cos 4 0.5 x dx Each hee equaion is shown accoding o an ode of mode, mode and mode. The sucual limiaion of hydaion pocess is inoduced o Eq.() by using he educion faco d Cw ρc( γ + Wa, g) = Cs d R + + k D k d e Mode Mode Mode Figue. -dimensional expession of sucual limiaion of expanding hydae gel (6) (7) (9) C s : 4
whee ( v ) gel dr 4π d = (0) d S( R ) d C s is defined as: C s SR = () 4π R whee C s means he effec of educion of wae hough gel suface ha is conaced wih wae and SR ( ) is he same funcion of Eq.(8)..5 Tempeaue effec on hydaion pocess Cuing empeaue effec on he ae of chemical eacion, i.e. hydaion pocess. Tempeaue effec is inoduced o each mass ansfe coefficien and eacion faco in his poposed model (Tomosawa 997). I is assumed ha B in Eq.(4), k and D e in Eq.() follow Aheniu s law. Hence, wih he values of B 9, k 9 and D e9 ha ae given fo 9 K, he coefficiens a T K ae expessed as follows: B = B9 exp β / T / 9 De = De9 exp β / T / 9 () k = k exp E/ R / T / 9,9. MODEL PARAMETERS Fo he deeminaion of he model paamees k 9, D E 9 (in Eq.(5)), B 9, C 9 and E/ R an evaluaion was conduced. Degee of hydaion is deemined by he aio of amoun of hea libeaion o maximum hea libeaion pediced by Woods equaion (Woods 9). In his evaluaion moe han 0 hydaion ess (i.e. Tomosawa 997-) wee involved, compising 9 diffeen ypes of Poland cemen wih C S conens anging fom 0% o 70%. Cuing empeaues vaied fom 0 C o 60 C, paicle size disibuion wih blaine value anging fom 00 m /kg o 550 m /kg. k is expessed as a funcion of he C S and C A conens: The 9 0.975 0 9 8.05 0 CS CA k = w + w () whee w CS and w CA ae mass conens of C S and C A especively. This paamee indicaes he eacion ae pe uni aea of eacion fon and his value dominaes he iniial ae of hydaion pocess. I seems easonable o suppose ha he k 9 is associaed wih C S and C A conens showing high eacion speed befoe doman peiod. In his evaluaion he value of k is anging fom.8 0-8 mm/h o 5.6 0-8 mm/h. D is expessed as a funcion of he C S conens: 9 The E 9 4.04 = wc S (4) DE9. 0 The D E 9 value epesens effecive diffusion coefficien of cemen gel in iniial sage. This aedeemining value may be affeced by he C S conens. This D E 9 is found fom 0 0 - mm /h o 80 0 - mm /h. Fo he facos B 9, C 9 and E/ R wih consan value is applicable in Table. Model paamees in poposed model Coefficien B 9 [x0-0 mm/h] C 9 [x0-7 /mm h] β [K - ] β [K - ] E/ R [K - ] Value 0. 5.0 7579 564 5
majoiy cases (See Table ). The B 9, C 9 and E/ R ae ahe sensiive o he behavio befoe doman peiod. 0%-vaiaion of hee value may cause 5% of degee of hydaion a 7 hous. The β and β ae insensiive. 0% -vaiaion of hese values may leads % of diffeence in degee of hydaion a 7 hous. In Figue Compaison of simulaion esul wih expeimenal daa of hea libeaion is shown. The expeimenal hea libeaion is value of he end of each expeimen. Simulaion esuls ae good ageemen wih expeimenal esuls. 4. CONCLUSION Enie hydaion pocess of each cemen paicle is modeled by a single kineic equaion wih assuming 0 0 00 00 00 400 500 ha paicle size disibuion, inepaicle conac wih cell concep and ha each cemen paicle has he same aio of available wae volume o is cemen volume. Enie sysem of cemen pase is pesened as accumulaion of hem. Simulaion esuls shows good ageemen wih expeimenal daa and is accuacy is bee han 0% befoe 7 hous fom mixing. 5. REFERENCES Benz, D. P. and Gaboczi, E. J. (99). Pecolaion of Phases in a Thee-Dimensional Cemen Pase Micosucual model, Cem. Conc. Res, Vol., pp.5-44 Benz, D. P. (000). CHEMHYDD: A Thee-Dimensional Cemen Hydaion and Micosucue Developmen Modelling Package. Vesion.0, NISTR 6485, Naional Insiue of Sandads and Technology Bezjak, A. (980). On he Deeminaion of Rae Consans fo Hydaion Pocesses in Cemen Pase, Cem. Conc. Res. Vol.0,, pp.55-56 Knudsen, T. (984). The Dispesion Model fo Hydaion of Poland Cemen I., Geneal Conceps Cem. Conc. Res., Vol.4, pp.6-60 Kondo, R. and Ueda, S. (968). Kineics and Mechanism of he Hydaion of Cemens, Fifh Inenaional Symposium on he Popeies of Cemen Pase and Concee Tokyo, II-4, pp.0-48 Kundes, E. A. B. (997). Simulaion of volume changes in hadening cemen-based maeials, Ph.D hesis, TU Delf, Relis, M. and Sooka, I. (977). Vaiaion in Densiy of Poland Cemen Hydaion Poducs Cem. Conc. Res, Vol.7, pp.67-680, 977 Taylo, H.F.W. (997). Cemen Chamisy nd Ediion, Thomas Telfod, pp0- Tomosawa, F. (974). A Hydaion model of cemen Poc. of Annual Meeing on Cemen Technology, Cemen Associaion of Japan, Vol.8, pp.5-57 Tomosawa, F. (997). Developmen of a kineic model fo hydaion of Cemen Poceedings of he 0h Inenaional congess on he chemisy of cemen, Gohenbug, Sweden, ii05, Tomosawa, F., Noguchi, T. and Hyeon, C. (997-). Simulaion Model fo Tempeaue Rise and Evoluion of Themal Sess in Concee Based on Kineic Hydaion Model of Cemen Poceedings of 0h Inenaional Congess of Chemisy of Cemen, Vol,4, pp.4iv07, van Beugel, K. (995). Numeical Simulaion of Hydaion and Micosucual Developmen in Hadening Cemen-Based Maeial (I) Theoy and (II) Applicaion Cem. Conc. Res., Vol.5, pp.9- and 5-50. Woods, H. (9). Effec of Cemen Composiion on Moa Sengh Engineeing News Recod, pp.404-407 Simulaion esul [J/g] 500 400 00 00 00 Expeimenal Hea libeaion [j/g] Figue. Pedicive accuacy of hea libeaion by poposed model (Compaison is made a he end of expeimen.) 6