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Nonlinear water diffusion in nonsaturated concrete

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Abstract

The equations governing drying and wetting of concrete are formulated, assuming the diffusivity and other material parameters to be dependent on pore humidity, temperature and degree of hydration. By fitting of computer solutions for slabs, cylinders and spheres against numerous test data available in the literature it is found that the diffusion coefficient decreases about 10 to 20 times when passing from 0.9 to 0.6 pore humidity. The problem is thus strongly nonlinear. Dependence on temperature is found to agree satisfactorily with the rate process theory. Effect of temperature on equilibrium pore humidity is also studied. The aging effect is defined by means of an equivalent hydration period. To enable easy prediction of drying of simple bodies, charts for the solution in terms of nondimensional variables are presented. Finally, correlation to the diffusion in a saturated concrete is discussed.

Résumé

On présente les équations du séchage et de l'humidification du béton, étant supposé que le coefficient de diffusion et les autres paramètres du matériau sont déterminés par l'humidité interstitielle, la température et le degré d'hydratation. On examine le traitement numérique de l'équation de diffusion, et par comparaison des solutions calculées pour le séchage de dalles, cylindres et sphères, avec les nombreux résultats d'essais fournis par la littérature on démontre que le coefficient de diffusion diminue d'environ 20 fois lorsque l'humidité interstitielle passe de 0,9 à 0,6. On voit donc que le problème de la diffusion est nettement non linéaire. Ce résultat confirme aussi que la migration superficielle le long des couches constitue le mécanisme dominant de la diffusion. L'influence de la température se révèle en accord satisfaisant avec le concept d'énergie d'activation. L'effet de la température sur l'équilibre de l'humidité interstitielle est traduit par un coefficient hygrothermique dont on donne les valeurs approximatives. On tient compte des variations des paramètres des matériaux avec l'âge du béton par l'intermédiaire d'une période équivalente d'hydratation dont l'accroissement dépend de la température et de l'humidité interstitielle. On montre que l'influence de l'humidité interstitielle sur la perméabilité peut être très simplement déterminée en mesurant le transfert permanent de l'eau à travers des parois minces dont l'une des faces est exposée à différentes conditions d'humidité ambiante. Afin de faciliter la prédiction du séchage de corps simples, on présente des diagrammes pour la résolution du problème du séchage non linéaire en fonction de variables non dimensionnelles. Enfin on examine la diffusion de l'eau dans des éléments présentant des régions de béton saturé sous une surpression hydraulique, à côté de régions de béton non saturé. On montre que de part et d'autre de la limite entre ces deux régions, le coefficient de diffusion varie d'environ / 000 fois.

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Abbreviations

c, ĉ:

permeability given by Eqs. (1) and (3)

C, C1 :

diffusivity (Eqs. 8–11) and its value at H→1

H, H en , H s , H eq :

pore humidity (or relative vapor pressure), environmental humidity (Eq. 12), self-dessiccation (Eq. 4), and equivalent humidity (Eq. 22)

H c :

parameter in Eq. (20)

J:

mass flux of water (Eq. 1)

k:

inverse slope of desorption isotherm (Eq. 4)

L:

half thickness of a slab

n:

exponent in Eq. (20)

p1, p v :

pressure in pore water or pore vapor

Q, Q h :

activation energy for diffusion (Eq. 21) and for hydration (Eq. 7a)

r:

radius coordinate

R:

radius of a cylinder or a sphere

t, t e , t0 :

time, equivalent hydration period (Eq. 7) and instant of drying exposure

T:

absolute temperature

W, W e , W n :

mass of water contained in a unit volume of material, and its evaporable and non-evaporable parts (Eqs. 4, 9–11)

x:

coordinate across the thickness of slab

α0, α1 :

parameters in Eqs. (20) and (20a)

β = β T β H :

relative hydration rate (Eq. 7)

K:

hygrothermic coefficient (Eq. 4)

ϱ, ϱ g :

specific mass and specific weight of water

References

  1. Abrams, M.S., andGustaferro, A.H.—Fire endurance of concrete slabs as influenced by thickness, aggregate type, and moisture, Journal of the Portland Cement Association Research and Development Laboratoiries, Vol. 10, No. 2, May 1968, 9–24 (PCA Bulletin 223).

    Google Scholar 

  2. Abrams, M.S., andMonfore, G.E.—Application of a small probe-type relative humidity gage to research on fire resistance of concrete, Journal of the Portland Cement Association Research and Development Laboratories, Vol. 7, No. 3, Sept. 1965, 2–12 (PCA Bulletin 186).

    Google Scholar 

  3. Abrams, M.S., andOrals, D.L.—Concrete drying methods and their effect on fire resistance, in: Moisture of materials in relation to fire tests, STP No. 385, 1965, 52–73, publ. by American Society for Testing Materials (PCA Bulletin 181).

  4. Aleksandrovskii, S.V.—On thermal and hygrometric properties of concrete related to heat and moisture exchange (in Russian), Akad. Stroit. i Arkhitektury USSR (Moscow), Nauchno-Issled. Inst. Betona i Zhelezobetona (NIIZhB), Issled. Svoistv Betona, Zhelezob. Konstr., Trudy Inst., No. 4, 1959, 184–214.

    Google Scholar 

  5. Aleksandrovskii, S.V.—Analysis of plain and reinforced concrete structures for temperature and moisture effects (with account of creep) (in Russian), Stroyizdat, Moscow, 1966.

    Google Scholar 

  6. Bažant, Z.P.—Constitutive equation for concrete creep and shrinkage based on thermodynamics of multiphase systems, Materials and Structures (RILEM), Vol. 3, No. 13, 1970, pp. 3–36.

    Google Scholar 

  7. Bažant, Z.P.—Delayed thermal dilatations of cement paste and concrete due to mass transport, Nuclear Engineering and Design, Vol. 14, 1970, 308–318.

    Article  Google Scholar 

  8. Bažant, Z.P.—Thermodynamic theory of deformations of concrete with explanation of drying creep, American Concrete Institute Symp. on Designing for Effects of Creep, Shrinkage and Temperature, SP 27, p. 411, Detroit 1971; see alsoThermodynamic theory of concrete deformation at variable temperature and humidity, Report No. 69-11, Div. of Struct. Engineering and Struct. Mech., University of California, Berkeley, August 1969.

  9. Bažant, Z.P., andNajjar, L.J.—Drying of concrete as a nonlinear diffusion problem, Cement and Concrete Research, An International Journal, Vol. 1 (1971), 461–473.

    Article  Google Scholar 

  10. Brophy, J.M., Rose, R.M., andWulff, J.—The thermodynamics of structure, Vol. II of The Structure and Properties of Materials, ed. by Wulff, J. Wiley, New York, 1964.

    Google Scholar 

  11. Brunauer, S., Emmett, P.H., andTeller, E. —Adsorption of gases in multimolecular layers, J. Amer. Chemical Soc., Vol. 60, 1938, 309–319.

    Article  Google Scholar 

  12. Carlson, R.W.—Drying shrinkage of large concrete members, American Concrete Institute Journal, January–February, 1937, Proc. Vol. 33, p. 327.

    Google Scholar 

  13. Carlson, R.W., andDavis, R.E.—Discussion on the significance of pore pressure in hydraulic structures, Proc. ASCE, Vol. 74, 1532–1536, 1948.

    Google Scholar 

  14. Carslaw, H.S., andJaeger, J.C.—Conduction of Heat in Solids, Oxford, 1959, 2nd ed.

  15. Copeland, L.E., andBragg, R.H.—Selfdesiccation in portland cement pastes, Proc., Amer. Soc. for Testing Materials, No. 204, February 1955 (PCA Bulletin 52).

  16. Copeland, L.E., Kantro, D.L., andVerbeck, G.—Chemistry of hydration of portland cement, in: Chemistry of Cement, Proc. 4th Intern. Symp., Washington, D.C. 1960, National Bureau of Standards, Monograph 43, Vol. I, Paper IV-3, 429–465 (PCA Bulletin 153).

  17. Cottrell, A.H.—The mechanical properties of matter. John Wiley and Sons, New York, 1964.

    Google Scholar 

  18. Crank, J.—Mathematics of diffusion, Oxford University Press, London 1957.

    MATH  Google Scholar 

  19. Guggenheim, E.A.—Thermodynamics, classical and statistical, in: Encyclopedia of Physics, ed. by S. Flügge, Vol. III/2, Principles of Thermodynamics and Statistics, Springer, Berlin, 1959.

    Google Scholar 

  20. Hancox, N.L.—A note on the form of the rate of drying curve for cement paste and its use in analyzing the drying behavior of this material, RILEM Bulletin, No. 36, Sept. 1967, 197–201.

    Google Scholar 

  21. Hanson, J.A.—Effects of curing and drying environments on splitting tensile strength, American Concrete Institute Journal, Vol. 65, July 1968, 535–543 (PCA Bulletin D141).

    Google Scholar 

  22. Harmathy, T.Z.—Simultaneous moisture and heat transfer in porous systems with particular reference to drying, Industrial and Engng. Chemistry Fundamentals, Vol. 8, 92–103, February 1969, Amer. Chem. Soc.

    Article  Google Scholar 

  23. Helmuth, R.A., andTurk, D.H.—The reversible and irreversible drying shrinkage of hardened portland cement and tricalcium silicate paste, Journal of the Portland Cement Association Research and Development Laboratories, Vol. 9, No. 2, May 1967, 8–21 (PCA Bulletin 215).

    Google Scholar 

  24. Hilsdorf, H.K.—A method to estimate the water content of concrete shields, Nuclear Engineering and Design, Vol. 6, 1967, 251–263.

    Article  Google Scholar 

  25. Hughes, B.P., Lowe, I.R.G., andWalker, J. —The diffusion of water in concrete at temeperatures between 50 and 95 °C, British Journal of Applied Physics, Vol. 17, 1966, 1545–1552.

    Article  Google Scholar 

  26. Jensen, B.M.—The effect of temperature on the thermal dilatation of concrete conditioned to a given humidity, Graduate Student Research Report No. 416, Div. of Struct. Engineering and Struct. Mech., University of California, Berkeley, July 1969.

    Google Scholar 

  27. Kasi, S.S.H., andPihlajavaara, S.E.—An approximate solution of a quasi-linear diffusion problem, Publ. No. 153, The State Institute for Technical Research, Helsinki, 1969.

    Google Scholar 

  28. Lowe, I.R.G., Hugues, B.P. andWalker, J.—The diffusion of water in concrete at 30 °C, Cement and Concrete Research, vol. 1, 547–557, 1971.

    Article  Google Scholar 

  29. Murata, J.—Studies on the permeability of concrete, RILEM Bulletin (Paris), No. 29, December 1965, 47–54.

    Google Scholar 

  30. Pihlajavaara, S.E.—Introductory bibliography for research on drying of concrete, The State Institute for Technical Research, Helsinki, 1964 (71 pp.).

    Google Scholar 

  31. Pihlajavaara, S.E., andVäisänen, M.—Numerical solution of diffusion equation with diffusivity concentration dependent, Publ. No. 87, State Institute for Technical Research, Helsinki, 1965.

    Google Scholar 

  32. Pihlajavaara, S.E.—On the main features and methods of investigation of drying and related phenomena in concrete, Ph. D. Thesis, Publ. No. 100, State Institute for Technical Research, Helsinki, 1965.

    Google Scholar 

  33. Pihlajavaara, S.E.—A review of the research on drying of concrete, RILEM Bulletin (Paris), No. 27, June 1965, 61–63.

    Google Scholar 

  34. Pickett, G.—The effect of change in moisture content on the creep of concrete under a sustained load, American Concrete Institute Journal, Proc. Vol. 36, February 1942, 333–355; see alsoShrinkage stresses in concrete, American Concrete Institute Journal, Proc. Vol. 42, January–February 1946.

    Google Scholar 

  35. Powers, T.C., andBrownyard, T.L.—Studies of the physical properties of hardened portland cement paste, American Concrete Institute Journal, 1946, pp. 101–132, 249–336, 469–504, 1947, pp. 549–602, 669–712, 845–880, 933–992 (PCA Bulletin No 22).

  36. Powers, T.C.—Hydraulic pressure in concrete, Proc. ASCE, Paper No. 742, Vol. 81, July 1955, 742–745 (reprinted as PCA Bulletin 63).

    Google Scholar 

  37. Powers, T.C., Copeland, L.E., Hayes, J.C., andMann, H.M.—Permeability of portland cement paste, American Concrete Institute Journal, Vol. 51, November 1954, 285–298 (PCA Bulletin 53).

    Google Scholar 

  38. Powers, T.C., Copeland, L.E., andMann, H.M. —Capillary continuity and discontinuity in cement paste, Journal of the Portland Cement Association Research and Development Laboratories, Vol. 1, No. 2, May 1959, 38–48 (PCA Bulletin 110).

    Google Scholar 

  39. Powers, T.C.—A discussion of cement hydration in relation to the curing of concrete, Proc. of the Highway Research Board, 27, 1947, 178–188 (PCA Bulletin No. 25).

    Google Scholar 

  40. Von Rosenberg, D.U.—Methods for the numerical solution of partial differential equations, ed. Bellman, R., American Elsevier, 1969.

  41. Wierig, H.J.—Die Wasserdampfdurchlässigkeit von Zementmörtel und Beton, Zement-Kalk-Gips, November 9, 1965, 471–482.

  42. Wu, Y.C., andCopeland, L.E.—Thermodynamics of adsorption. Barium-sulfate-water systems, Solid surfaces and the gas-solid interface, Advances in Chemistry Series, American Chem. Soc., 1961 (PCA Bulletin 141).

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Author information

Authors and Affiliations

  1. Northwestern University, 60201, Evanston, Illinois

    Z. P. Bažant (Assoc. Prof. of Civil Engineering)

  2. Northwestern University, 60201, Evanston, Illinois

    L. J. Najjar (Graduate Research Assistant)

Authors
  1. Z. P. Bažant
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  2. L. J. Najjar
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Additional information

Subscript sat … for saturation, H=1; Primes in r′, t′, H′ refer to non-dimensional variables Eq. (18).

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Bažant, Z.P., Najjar, L.J. Nonlinear water diffusion in nonsaturated concrete. Mat. Constr. 5, 3–20 (1972). https://doi.org/10.1007/BF02479073

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