#### 2.1. Fitting the Model

The experimental design was adopted on the basis of coded levels from four independent variables (

Table 1) to minimize the number of experimental runs and the time needed for optimizing oil extraction conditions from oak silkworm pupae, resulting in a 31-set simplified experimental set (

Table 2). To obtain a regression equation that could predict the response within the given range, independent and dependent variables were analyzed. To evaluate the significance of each coefficient and indicate the interaction strength of each parameter, the ANOVA (

F-test) and

p-values are used (

Table 3). The model with a

p-value less than 0.001 was statistically significant, which implied the model was suitable for this experiment. Meanwhile, the “lack of fit” of this model with the

p-value of 0.21 was insignificant, indicating that the accuracy and general availability of the polynomial model are adequate. The coefficient of determination (

R^{2}) and adjusted coefficient of determination (

Adj. R^{2}), with values of 0.9178 and 0.8459, are also shown in

Table 3. Regression coefficients of intercept, linear, quadratic, and interaction terms of the model which are presented in

Table 4 were calculated by using a least squares technique.

To calculate the coefficients of the second-order polynomial equation and the obtained regression coefficients, multivariable linear regression was used and its significance determined by the Student

t test and

p-value (

Table 4). More significance will be concluded with a larger absolute

F-value and a smaller

p-value [

20]. Neglecting the non-significant parameters, the final predictive equation obtained is showed as below (

Equation (1)):

Table 4 and

Equation (1) showed that the factors most significantly affecting oil yield was the two linear term of pressure and time (

p < 0.001), followed by the three quadratic term of temperature, pressure and time (

p < 0.001). The interactions between pressure and temperature, temperature and SC-CO

_{2} flow rate also had highly significant effects on oil yield. Based on the above model, the optimal condition for oak silkworm pupal oil yield was: 28.03 MPa, 35.31 °C, 1.83 h, and 20.26 L/h, and the oil yield was 26.18% under this condition.

#### 2.2. Analysis of Response Surface

According to

Equation (1), three-dimensional response surface curves and contour plots were plotted to determine their optimum values and to analyze the interactions among the various selected factors for obtaining the maximum oil recovery. The plots were generated by plotting the response using the

z-axis against two independent variables while keeping the other two independent variables at their zero level.

The best way to visualize the influence of the independent variables on the dependent one is to draw surface response plots of the model [

21,

22]. The generated response surfaces developed using the fitted quadratic polynomial equation obtained from regression analysis are shown in

Figure 1.

Because the solubility of lipids depends largely on the balance between fluid density and solute vapor pressure, which were controlled by pressure and temperature, extraction pressure and temperature are the main parameters that influence extraction efficiency.

Figure 1A shows the interaction between extraction pressure and extraction temperature on oil recovery from oak silkworm pupae, while the extraction time and CO

_{2} flow rate are respectively fixed at 1.5 h and 21 L/h. At a given extraction temperature, the yield of oil significantly increased with increasing pressure and then decreased with increasing pressure after the temperature reached the center point. This trend became more obvious at lower temperatures. Similar phenomena were also reported for the extraction of

Passiflora seed oil [

23,

24] and yellow horn seed oil [

25] by SF-CO

_{2}. This influence may be due to the fact that an elevated extraction pressure at a given temperature will result in an increase in fluid density, which means an enhanced solubility of the oil [

6,

21]. The solubility of vegetable oils extracted by SC-CO

_{2} also varies considerably with temperature and pressure. The oil increases as the pressure increases, basically in the range of 345–550 bar. Solubility increases with an increase in temperature, when the pressure is higher than 345 bar. Conversely, this effect does not occur with pressures lower than 345 bar [

26]. This behavior is related to the density of SC-CO

_{2}. However, high pressure is not always recommended since the increased repulsing solute-solvent interactions from the highly compressed CO

_{2} at high-pressure levels will potentially induce complex extraction and difficult analysis [

27,

28].

Figure 1B describes the effect of extraction pressure and time on oil yield. The results indicated that the oil yield increased gradually with the increase of extraction time at a lower fixed extraction pressure, while the oil yield decreased gradually with the increase of extraction time at the higher extraction pressure. When the extraction pressure lay in the center point up and down, the oil yield increased rapidly with increasing extraction time. At this point oil yield decreased linearly with increased extraction time.

The interaction between extraction pressure and CO

_{2} flow rate shown in

Figure 1C reveals that the oak silkworm pupal oil yield increases slowly with an increase in CO

_{2} flow rate at a lower fixed extraction pressure; then, a decrease of CO

_{2} flow rate after the center point of pressure. Before the center extraction pressure, the CO

_{2} flow rate only led to a gradual increase in oil yield, especially beyond 21 MPa, when no obvious effect was observed.

Figure 1D illustrates the interaction between extraction time and extraction temperature on oil yield. It was observed that at a given temperature, especially at low or high temperatures, oil yield changed dramatically with extraction time. At the temperature center point, the oil yield rapidly increased with the extraction time.

Figure 1E shows the response surface and contour plots of the effect of extraction temperature and CO

_{2} flow rate on the oil yield, with a fixed extraction pressure and extraction time at 25 MPa and 1.5 h, respectively. The CO

_{2} flow rate displayed a positive effect on the oil yield at low temperature. However, no obvious effect of CO

_{2} flow rate on the oak silkworm pupal oil was observed at a high extraction temperature.

The response surface for the oil yield as related to time and CO

_{2} flow rate with a fixed extraction pressure of 25 MPa and temperature of 35 °C is shown as a three-dimensional plot in

Figure 1F. It can be seen that no significant effect on oil yield was observed when the extraction time was fixed and the CO

_{2} flow rate increased. However, the obvious decreasing trends in oil yield with the extraction time were displayed when the CO

_{2} flow rate was fixed. Under a given pressure, temperature or CO

_{2} flow rate (

Figure 1B,D and F), oil yields decreased slightly after 2.1 h extraction. The above phenomena were difficult to explain, however similar phenomena were also reported for the extraction of other oils by SF-CO

_{2}, e.g.,

Passiflora seed oil [

23,

24], yellow horn seed oil [

25] and almond oil [

29].