process, where $ U_{c} $ is the average consolidation ratio indicating a measure of the degree of consolidation and is defined by Shirato et al. (1967, 1974)

$$ U_{c}=\frac{L_{1}-L}{L_{1}-L_{\infty}} $$

where $ L_{1} $ , L, and $ L_{\infty} $ are the thicknesses of the compressed cake at $ \theta_{c}=0 $ , $ \theta_{c} $ , and $ \infty $ , respectively. Fig. 13 depicts the typical results obtained in the ultrahigh-pressure expression experiment conducted at the pressure of 15 MPa. The plots can be approximated by three straight lines connecting the different slopes except for the very early stage of consolidation. The slope of the straight line becomes smaller in the range of larger consolidation time. This suggests that there exists a three-staged creep phenomenon with the different values of the creep constant. Therefore, it is expected that the experimental data are not able to be described by neither the modified Terzaghi model (Shirato et al., 1967) nor the Terzaghi-Voigt model (Shirato et al., 1974).

The data indicated in Fig. 13 are illustrated in Fig. 14 in the form of a plot of the average consolidation ratio $ U_{c} $ vs. $ \sqrt{\theta_{c}} $ . The experimental data are evaluated by the modified Terzaghi and multi-stage Voigt combined model developed in order to describe the multi-stage creep phenomenon. According to the model, the

Fig. 13. Evaluation of creep effect.

Fig. 14. Typical results for variation of average consolidation ratio with consolidation time.

time evolution of $ U_{c} $ is described as Iritani et al. (2010)

$$ \begin{aligned}U_{c}=&\frac{L_{1}-L}{L_{1}-L_{\infty}}=\left(1-\sum_{k=1}^{K}B_{k}\right)\left\{1-\exp\left(-\frac{\pi^{2}}{4}\times\frac{i^{2}C_{e}\theta_{c}}{\omega_{0}^{2}}\right)\right\}\\&+\sum_{k=1}^{K}B_{k}\{1-\exp(-\eta_{k}\theta_{c})\}\end{aligned} $$

where $ B_{k} $ and $ \eta_{k} $ are the creep constants of the kth stage creep, and i is the number of drainage surfaces. The term $ C_{e} $ is referred to as the modified consolidation coefficient and is defined by Shirato et al. (1967)

$$ C_{e}=\frac{1}{\mu\alpha\rho_{s}(-d e/d p_{s})} $$

where $ \mu $ is the liquid viscosity, and $ \alpha $ is the local specific flow resistance of the compressed cake. Eq. (5) is derived based on the assumption that $ C_{e} $ is constant throughout the cake at any instant during consolidation although this assumption is not strictly valid (Shirato et al., 1967). Consequently, Eq. (5) may be used as an approximation with a proper mean value of $ C_{e} $ considered constant. The model is made up of series combination of the Terzaghi spring analogy and the multi-stage Voigt elements connected in series. The first term in Eq. (5) represents primary consolidation of filter cake based on the modified Terzaghi model, and the second term evaluates multi-stage creep effects. This equation is basically the same as the one for expression of agrofood presented by Lanoisellé et al. (1996), Grimi et al. (2010).

The solid line depicted in the figure represents the values calculated by using Eq. (5). Fairly good agreement with the experimental data can be obtained by accounting for the three-stage creep phenomenon. The dotted line represents the contribution of each consolidation stage to the overall consolidation. It is obvious that the fraction of the moisture removed during primary consolidation, A, defined as $ (1-B_{1}-B_{2}-B_{3}) $ , accounts for the greatest proportion of consolidation, more than 60%, and that the fraction of the moisture removed during the third creep period, $ B_{3} $ , is extremely small.

The time variation of moisture content R of the compressed cake on the mass basis is obtained from the time variation of the cake thickness L based on the relation

$$ R=\frac{100(L-\omega_{0})\rho}{(L-\omega_{0})\rho+\omega_{0}\rho_{s}} $$

In Fig. 15, R is plotted against the consolidation time $ \theta_{c} $ during ultrahigh-pressure expression for a variety of expression pressures $ p_{c} $ . The experimental data are obtained from Eq. (7), using both the time variation of cake thickness L and the solid volume $ \omega_{0} $ per unit cross-sectional area measured with an infrared-ray moisture

Fig. 15. Variation of cake moisture content with consolidation time.