where $ L_{1} $ and $ L_{\infty} $ are the thicknesses of the compressed cake at $ \theta_{c}=0 $ and $ \infty $ , respectively, $ B_{k} $ and $ \eta_{k} $ are the creep constants of the k-th stage creep, i is the number of drainage surfaces, and $ C_{e} $ is the modified consolidation coefficient (Shirato et al., 1967). The solid line shown in Figure 2(a) is the result of calculation derived by substituting the time variation of L calculated from Eq. (3) into Eq. (2). Fairly good agreement with the experimental data can be obtained by accounting for the three-stage creep phenomenon that the value of K in Eq. (3) is 3. The contributions A (defined as $ (1-B_{1}-B_{2}-B_{3}) $ ), $ B_{1} $ , $ B_{2} $ , and $ B_{3} $ of each consolidation stage to the overall consolidation are 0.186, 0.259, 0.337, and 0.218, respectively. In contrast to the case of treated carrots, the experimental data of $ U_{c} $ for raw carrots are not shown in the figure since it is impossible to determine the value of $ L_{\infty} $ in Eq. (3) because of the gradually ever-changing cake thickness.

### 2.2 Effect of temperature

The effect of temperature of a sample controlled during ultrasonic irradiation on the deliquoring performance is investigated by changing the temperature of the liquid in the thermostat bath. The time evolutions of the moisture content R of the cake are plotted in Figure 4 for the different values of sample temperature T in ultrasonic pretreatment under the conditions of the load power P of 40W and the sonication time t of 15min. The data for carrots pretreated at 40°C show a similar trend to those for untreated carrots, and the ultrasonic pretreatment shows little improvement in deliquoring due to expression. Even when ultrasonic is irradiated with being heated to 47°C, the deliquoring behavior is closely similar to that of carrots treated at 40°C until about 6 h after the onset of expression. However, shortly thereafter, the moisture content R begins to decrease abruptly and eventually decreases to 70% range in 24h. In the case of 50°C, the moisture content R decreases rapidly from the beginning of expression and finally reaches below 70% in 24h.

Fig. 4 Effect of sample temperature in ultrasonic pretreatment on time variation of cake moisture content in expression

Therefore, it is concluded that the sample temperature plays an extremely important role in accelerating the effect of ultrasonic pretreatment on the expression performance. It is necessary to heat the carrot above a minimum temperature threshold (around $ 50^{\circ} $ C) in ultrasonic irradiation in order to attain a desired effect in expression. The loosening of the network of a cell wall biopolymer such as pectin, especially in the middle lamella between adjacent cells, at elevated temperature during ultrasonic irradiation may weaken the cell wall strength. Several studies have reported that carrots lose cell turgor pressure rapidly through thermal texture degradation due to $ \beta $ -elimination reaction when the tissue internal temperature reaches $ 50^{\circ} $ C (Greve et al., 1994; Sila et al., 2006; Day et al., 2012). It is obvious that such changes in pectin structure due to heating during ultrasonic irradiation lead to a significant improvement in expression performance obtained in this study. However, the expression performance of carrots pretreated by heating alone without ultrasonic irradiation is similar to that of raw carrots, as shown in Figure 4, and therefore the ultrasonic treatment in combination with heating is highly effective in improving the expression performance.

### 2.3 Effect of sonication time

In Figure 5, the variations with time of the cake moisture content R in expression are shown for the different values of the sonication time t in ultrasonic pretreatment under the conditions of the load power P of 40 W and the sample temperature T of 50°C. As the sonication time t increases from 5 to 15 min, the cake moisture content R significantly reduces, especially at comparatively-early times. However, a further increase in the sonication time t does not lead to a more significant improvement in the expression performance.

### 2.4 Effect of ultrasonic energy

Figure 6 shows the variations with time of the cake moisture content R in expression for the different values of the

Fig. 5 Effect of sonication time in ultrasonic pretreatment on time variation of cake moisture content in expression