$ R_{e} $ final moisture content of cake on mass basis (wt%)

$ R_{f} $ ratio of added aluminum to solid mass in sludge

s mass fraction of solids in sludge

$ U_{c} $ average consolidation ratio

$ \nu $ filtrate volume per unit medium area (m³/m²)

$ w_{\infty} $ final wet cake mass per unit cross-sectional area, corresponding expression pressure (kg)

$ w_{d} $ dry cake mass per unit cross-sectional area (kg)

$ \alpha $ specific flow resistance of compressed cake (m/kg)

$ \varepsilon $ porosity of compressed cake

$ \varepsilon_{1} $ empirical constant in Eq. (1) (kg $ ^{4} $ /m $ ^{4} $ /s $ ^{2\lambda} $ )

$ \eta $ creep constant (s $ ^{-1} $ )

$ \mu $ liquid viscosity (Pa s)

$ \theta $ time (s)

$ \theta_{c} $ consolidation time (s)

$ \lambda $ empirical constant in Eq. (1)

$ \rho $ density of liquid (kg/m³)

$ \rho_{s} $ true density of solids (kg/m³)

$ \omega_{0} $ total solid volume per unit cross-sectional area (m³/m²)

## Acknowledgements

This work has been partially supported by Grant-in-Aid for Scientific Research from The Ministry of Education, Culture, Sports, Science and Technology, Japan and from The Ministry of the Environment, Japan, and by The Steel Foundation for Environmental Protection Technology. The authors wish to acknowledge with sincere gratitude the financial support leading to the publication of this article. The authors also wish to express their sincere appreciation to the Ueda Sewage Treatment Works for their generous contribution of the activated sludge used in this research.

## References

Bergins, C., 2004, Mechanical/thermal dewatering of lignite. Part 2: A rheological model for consolidation and creep process. Fuel 83, 267-276.

Chang, J.L., Lee, D.J., 1998. Ternary expression stage in biological sludge dewatering. Water Res. 32 (3), 905–914.

Christensen, M.L., Keiding, K., 2007. Creep effects in activated sludge filter cakes. Powder Technol. 177, 23–33.

Chu, C.P., Lee, D.J., Huang, C., 1998. The role of ionic surfactants in compression dewatering of alum sludge. J. Colloid Interface Sci. 206 (1), 181–188.

Grimi, N., Vorobiev, E., Lebovka, N., Vaxelaire, J., 2010. Solid-liquid expression from denatured plant tissue: filtration-consolidation behavior. J. Food Eng. 96 (1), 29–36.

Grace, H.P., 1953. Resistance and compressibility of filter cakes, Part 1. Chem. Eng. Prog. 49 (6), 303–318.

Huang, S., Chen, J.L., Chiang, K.Y., Wu, C.C., 2010. Effects of acidification on dewaterability and aluminum concentration of aluminum sludge. Sep. Sci. Technol. 45 (8), 1165–1169.

Iritani, E., Mukai, Y., Katagiri, N., Yoshii, I., 2003. Formation of gel emulsion by filtration-consolidation of O/W emulsions. J. Chem. Eng. Jpn. 36 (5), 590–596.

Iritani, E., Matsumoto, S., Katagiri, N., 2008. Formation and consolidation of filter cake in microfiltration of emulsion-slurry. J. Membr. Sci. 318, 56–64.

Iritani, E., Sato, T., Katagiri, N., Hwang, K.J., 2010. Multi-stage creep effect in consolidation of tofu and okara as soft colloids. J. Chem. Eng. Jpn. 43 (2), 140-149.

Iritani, E., Katagiri, N., Sano, Y., 2011. Properties of microfiltration of humic acid solution flocculated with PACl. Kagaku Kogaku Ronbunshu 37 (3), 246–250.

lwata, M., Igami, H., Murase, T., Yoshida, H., 1991. Analysis of electroosmotic dewatering. J. Chem. Eng. Jpn. 24 (1), 45–50.

Kang, S.M., Kishimoto, M., Shioya, S., Yoshida, T., Suga, K., 1990a. Properties of extracellular polymer having an effect on expression of activated sludge. J. Ferment. Bioeng. 69 (2), 111–116.

Kang, S.M., Kishimoto, M., Shioya, S., Yoshida, T., Suga, K., 1990b. Dewatering of extracellular polymer and activated sludges by thermochemical treatment. J. Ferment. Bioeng. 69 (2), 117–121.

Kawasaki, K., Matsuda, A., Ide, T., Murase, T., 1990a. Measurement of cell density and floc property of excess activated sludge conditioned by freezing and thawing process, using centrifugal settling. Kagaku Kogaku Ronbunshu 16 (2), 401–403.

Kawasaki, K., Matsuda, A., Murase, T., 1990b. The effect of freezing and thawing process on the expression characteristics and final moisture content of excess activated sludge. Kagaku Kogaku Ronbunshu 16 (6), 1241–1246.

Kawasaki, K., Matsuda, A., Murase, T., 1996. The effect of bound water on filtration and consolidation characteristics of excess activated sludge treated by freezing and thawing treatment. Kagaku Kogaku Ronbunshu 22 (2), 294-300.

La Heij, E.J., Kerkhof, P.J.A.M., Herwijn, A.J.M., Coumans, W.J., 1996a. Fundamental aspects of sludge filtration and expression. Water Res. 30 (3), 697-703.

La Heij, E.J., Kerkhof, P.J.A.M., Kopinga, K., Pel, L., 1996b. Determining porosity profiles during filtration and expression of sewage sludge by NMR Imaging. AlChE J. 42 (4), 953–959.

Lanoiselle, J.L., Vorobyov, E.L., Bouvier, J.M., Piar, G., 1996. Modeling of solid/liquid expression for cellular materials. AlChE J. 42 (7), 2057-2068.

Lee, D.J., Wang, C.H., 2000. Theories of cake filtration and consolidation and implications to sludge dewatering. Water Res. 34 (1), 1–20.

Matsuda, A., Kawasaki, K., Mizukawa, Y., 1992. Measurement of bound water in excess activated sludges and effect of freezing and thawing process on it. J. Chem. Eng. Jpn. 25 (1), 100–103.

Mihoubi, D., Vaxelaire, J., Zagrouba, F., Bellagi, A., 2003. Mechanical dewatering of suspension. Desalination 158, 259-265.

Nakakura, H., Nishigaki, F., Matsuoka, T., Sambuichi, M., Osasa, K., 1996. Dewatering of gels by mechanical compression. Nippon Shokuhin Kagaku Kogaku Kaishi 43(6), 674-682.

Petryk, M., Vorobiev, E., 2013. Numerical and analytical modeling of solid-liquid expression from soft plant materials. AIChE J. 59 (12), 4762-4770.

Shirato, M., Murase, T., Kato, H., Fukaya, S., 1967. Studies on expression of slurries under constant pressure. Kagaku Kogaku 31 (11), 1125–1131.

Shirato, M., Murase, T., Tokunaga, A., Yamada, O., 1974. Calculations of consolidation period in expression operations. J. Chem. Eng. Jpn. 7 (3), 229-231.

Shirato, M., Iwata, M., Wakita, M., Murase, T., Hayashi, N., 1987. Constant-rate expression of semisolid materials. J. Chem. Eng. Jpn. 20 (1), 1–6.

Tiller, E.M., Cooper, H., 1962, The role of porosity in filtration: Part V. Porosity variation in filter cakes. AlChE J. 8 (4), 445–449.

Tiller, E.M., Lu, W.M., 1972. The Role of porosity in filtration VIII: Cake nonuniformity in compression-permeability cells. AlChE J. 18 (3), 569–572.

Tiller, F.M., Yeh, C.S., 1987. The role of porosity in filtration. Part XI: Filtration followed by expression. AIChE J. 33 (8), 1241-1256.

Tiller, F.M., Cleveland, T., Lu, R., 1999. Pumping slurries forming highly compactible cakes. Ind. Eng. Chem. Res. 38 (3), 590–595.

Venter, M.J., Kuipers, N.J.M., de Haan, A.B., 2007. Modelling and experimental evaluation of high-pressure expression of cocoa nibs. J. Food Eng. 80, 1157–1170.

Yoshida, H., 1993. Practical aspects of dewatering enhanced by electro-osmosis. Drying Technol. 11 (4), 787-814.