Analysis of the response surface design

CCD was used for the design of experiments and process optimization. In this way, four coded operating parameters were defined in five levels, namely − 2, − 1, 0, 1 and 2, to obtain the optimal regeneration yield. Pressure (100–300) bar, temperature (313–333 K), flow rate (2–6 g/min), and dynamic time (30–150 min) were considered as the operating parameters of the process to be studied. Table 2 reports the experiments responses at the different values of operating conditions maintaining constant the particle size at 1 mm. Among models presented in CCD, the quadratic model represents the best model for the prediction and optimization of the regeneration yield. Equation (5) represents the obtained quadratic polynomial model through which the yield was correlated, as a response to the coded independent parameters. Therefore, the RSM predictive model was obtained as follows:

This model was found to be significant at 95% level of confidence while the corresponding F-value and p-value were calculated to be 92.58 and 0.0001, respectively. Also, accuracy of the model can be tested by investigating the determination coefficient (R₂). The values of the coefficient of determination (R₂) and adjusted coefficient of determination (R_2adj.) were calculated to be 98.86 and 97.79%, respectively. The value of R₂ indicates a good agreement between the experimental and predicted response values. The value of adjusted R₂ revealed that only 2.21% of total variations failed to be explained by the model. The lack-of-fit measures the failure of the model to represent data in the experimental domain at points which are not included in the regression. The p-value of the lack-of-fit was higher than 0.05 indicating an excellent fit. Furthermore, Signal-to-noise ratio (SNR) is known as an adequate precision measure. In fact, SNR compares the range of predicted values at design points to the average prediction error. As a SNR greater than 4 represents desirable results, the achieved ratio (35.30) served as an adequate signal as it was much smaller than the actual effect size. Low C.V. (3.32%) indicates the reliability of the carried-out experiments. The ANOVA of quadratic model was carried out using the data tabulated in Table 3 with the purpose of examining significance of the variables as linear, quadratic and interaction coefficients of the RSM. Those variables and their interactions with higher regression coefficient and smaller p-value (p < 0.05) have a significant influence on the regeneration yield^41,42. Analysis results reported in Table 3, as well as, the parameters which statistically indicated highly significant impact on the yield were linear term of pressure, CO₂ flow rate and time (p < 0.0001), followed using the quadratic term of flow rate(p < 0.0001). The quadratic term of time (p = 0.0003), the linear term of temperature (p = 0.0037) as well as the quadratic term of pressure and temperature (p = 0.0005), represent significant influence on the yield.

Effect of process parameters on regeneration yield

In this section the effect of the different operational parameters on the supercritical regeneration of activated carbon were examined with three dimensional graphs, while the other variables were maintained at its respective fixed middle level(Pressure, 200 bar, Temperature, 318 K, flow rate,4 g/min and time 90 min), corresponding to zero code.

Figure 3 shows the 3D plot for the influence of pressure and temperature. It can be observed that the regeneration yield increased with increasing pressure and temperature from 100 to 300 bar and from 313 to 333 K, respectively. Based on CCD, pressure had the most important influence on the regeneration yield. Particularly, the increase of pressure leads to increase solvent power of CO₂ and therefore solubility of the solutes increased as well^9,43. On another hand, the increase in temperature had a slight positive effect on the regeneration yield (Fig. 4). At a pressure of 100 to 300 bar, with the increase in temperature, the solubility of solute increases, and it is due to the increment of the solute vapor pressure effect. therefore, the solubility of solute increases with the increase in temperature. On the other hand, the extract vapor pressure is raised with increasing temperature, which leads to an enhancement in the supercritical fluid diffusivity. Figure 5 denotes the 3D plot of the yield as a function of pressure and flow rate at 323 K and 90 min. Increasing the CO₂ flow rate from 1 (code =  − 2) to 4 g/min (code = 1) reduces the film thickness around the particles leading to lower resistance for mass transfer around the particles and accordingly enhancing the regeneration of the activated carbon. Whereas at more than 4 g/min with reduction of residence time (SC-CO₂– particles contact time) a negative effect of flow rate was observed on the regeneration yield. This contrast effect was also observed in Figs. 6, 8 and 9.

3D graphic surface for the effects of temperature and time on the yield.

3D graphic surface for the effects of flow and pressure on the yield.

3D graphic surface for the effects of temperature and flow on the yield.

Figures 4, 7, and 8 illustrate the effect of dynamic time and flow rate on the regeneration yield. As can be seen in Figs. 4, 7, and 8 the regeneration yields gradually increase with increasing the dynamic time, achieving its highest value about 150 min. This behavior has been reported by other researchers and explained in terms of the increase of the ratio between SC-CO₂ and the solute as the dynamic time increase³⁵. On another hand, the residence time decrease as flow rate increase, while the external mass transfer coefficient increased, so these opposite phenomena canceled their effects out leading to the almost constant yield. Moreover, the increase of the flow rate could reduce the contact time between the solvent and the activated carbon. This phenomenon could be related to the channeling effect, where SC-CO₂ at high flow rates would just flow around the samples with no ability to diffuse through the pores within the samples. Furthermore, the increased flow rate could cause the sample compaction limiting the amount of CO₂ able to being in direct contact with the sample matrix. Similar results were reported by some researchers^{33,44,45,46,47,48}. In addition, the perturbation plot (Fig. 9) revealed the significant effect of all process variables on the extraction. A perturbation plot does not show the effect of interactions and it is like one factor-at-a-time experimentation. The perturbation plot helps to compare the effect of all independent variables at a particular point in the design space. The response is plotted by changing only one factor over its range while holding the other factors constant.

3D graphic surface for the effects of pressure and time on the yield.

3D graphic surface for the effects of flow and time on the yield.

Perturbation plot of operational parameters obtained through CCD on the yield.

The optimal conditions to obtain the highest yield of regeneration from activated carbon were determined at 285 bar, 333 K, 4 g/min, and 147 min and the predicted yield was 93.75%. Average actual yield (94.25%) was in good agreement with the estimated value, revealing the ability of the developed model in terms of extraction process prediction and optimization.

Gas chromatography analysis

The solute composition determined for the extract, at the optimum condition, by gaseous chromatography (GC) were shown in Table 4. Thirty-two components were identified in the extract obtained using SC-CO₂ which comprised 95.2% of the extract. The main components extracted by SC-CO₂ were N-C6 (4.90), N-C7 (9.45%), N-C8 (9.04%), N-C9 (11.90%), N-C10 (12.40%) and N-C11 (4.73%).

Iodine results

Iodine number is a measure of iodine molecules adsorbed in the pores of a particle, which indicates the pore volume capacity and extent of micro pore distribution in the activated carbon. Iodine number can be correlated with the adsorbent ability to adsorb low molecular weight substances. Iodine adsorbed results in extracted and non-extracted activated carbon samples were shown in Table 5. Results of physical and adsorptive characteristics denote that SC-CO₂ appears to be a suitable way for the regeneration of activated carbon.

Evaluating the structure of activated carbon

As was previously mentioned the structure and surface morphology of non-extracted and extracted activated carbon samples was evaluated by scanning electron microscopy (SEM). Figure 10a,b presents two micrographs of activated carbon before (a) and after supercritical regeneration (b). The micrograph of the unprocessed sample shows an uneven and rough surface covered uniformly with a layer of solute. The structure of the treated sample is more porous and the surface is clearly deflated and depleted of solute due to the high recovery efficiency obtained by the extraction with supercritical carbon dioxide.

(a,b) SEM image of (a) un-process, (b) process of activated carbon.

Mathematical modeling

The experimental desorption was correlated with a kinetic model proposed by Tan and Liou^17,49. This model assumes linear desorption kinetics in the adsorbed phase, and has a high applicability for correlating desorption behavior using SC-CO₂ with only one fitting parameter. The following assumptions were considered in the formulation of the model:

1. 1.

   The system was isothermal and isobaric.

2. 2.

   The physical properties of the SC-CO₂ were constant during the extraction process.

3. 3.

   The extraction model was expressed as an irreversible desorption process.

4. 4.

   All particles were considered to be spherical and the solutes were uniformly distributed in their structures.

5. 5.

   The volume of the solid matrix (particles) was not changed during the extraction process.

6. 6.

   The solvent flow rate was constant along the bed and uniformly distributed without radial dispersion.

7. 7.

   Axial dispersion was neglected

A schematic diagram of the particles and bed was presented in Fig. 11. According to the above assumptions, the material balances for the solute in the solid and bulk phases are as follows:

A schematic diagram of the particles and bed.

Solid phase:

The concentration at the exit of the packed bed can then be obtained by:

K is the adjustable parameter of the kinetic model.

The adjustable parameter was computed by minimizing the errors between experimental and calculated yield values. The average absolute relative deviation (AARD), described using the following equation was applied to evaluate the adjustable parameter:

The mathematical modeling was applied to study the impact of pressure, temperature and flow rate as on the regeneration process. Figure 12 and Table 6 show the modeling results by using the kinetics model. Fig 12a shows a positive effect of the pressure on the regeneration process at fixed temperature, flow rate, particle size and dynamic time. This effect was related to the increase of the SC-CO₂ density as pressure increase. As was expected, the increase of density leads to an increase in the solubility of the solutes in SC-CO₂. On the other side, increasing pressure decreased the diffusion coefficient of CO₂. The decreasing effect of mass transfer coefficient may lead decreasing the yield. Nevertheless, the effect of increasing density and solubility overcomes the decreasing effect of diffusivity on the final yield of extraction^22,50. The adjustable parameter of the model (K) and AARD values for the different extraction runs were reported in Table 6. The effect of temperature on the regeneration yield of activated carbon was shown in Fig. 12b. The yield increased with increasing the temperature at fixed pressure and flow rate. As previously mentioned, this may be due to the increase of the vapor pressure of the solute. In the present section, the effect of temperature on the regeneration yield at points of 318, 328 K and optimum temperature were modeled. The flow rate of carbon dioxide was another parameter that was investigated. The experimental data and results of model in three flow rates including 1, 2 and 4 (optimum) were shown in Fig. 12. Based on the results indicated in Fig. 12c, the yield was higher at optimal points. As can be seen in Fig. 12 the experimental data were well described with the kinetics model. Values of average absolute relative deviation are in the range 5.91-9.04%.

Effect of process parameters on the yield of extraction versus time.

A single‑sphere model (SSM)

Single sphere model (SSM) proposed by Crank⁵¹ with an assumption that particle size is one of the main factors on the diffusivity step in the extraction process. In addition, diffusion is assumed to occur in the sphere surface area of a particle solute. Dissemination of SC-CO2 in sphere equations is conducted to determine the diffusiveness of solvent to dissolve in matrices. Other assumptions made for an SSM model are: (1) The resistance of mass transfer is zero, (2) all of the particle sizes are homogenous, (3) the main factor in the extraction process is intraparticle of mass transfer, (4) the solute is in the inert porous sphere, and lastly, (5) all of the solutes in the bed will be extracted, and the extracted component will be dissolved in the particles by a process similar to diffusion. Equation 13 exhibits the diffusion equation for a constant coefficient.

where Mt is the total amount of the diffusing substances entered on the sheet at a specific time, M∞ is the particular quantity after countless time, D_eff is the diffusivity, R is the radius of particle solute, and t is the time.

The single sphere model is usually applied to determine the diffusivity coefficient and mass transfer between solvent and solute. Compared to another kinetic model, the single sphere model is easily applied, particularly the shrinking core model⁵² and the broken intact cell⁵³ due to one adjustable parameter. This is because one adjustable parameter is fit enough to determine the mass transfer process of experimental data. Furthermore, a single-sphere model can assess the effect of the parameter on the diffusivity process between solvent and solute⁵⁴.

The single sphere model was fitted to the experimental data with effective diffusivity as a fitting parameter, using average absolute deviation (AARD). Table 7 shows the AARD and effective diffusivity. The minimum %AARD was 2.08% at 285 bar and 333 K, while the maximum %AARD was 19.04% at 200 bar and 323 K. As shown in Fig. 13 the model successfully fitted the exponential trends of the extraction process. Putra et al.¹ applied single sphere model for fitting the experimental data of modified supercritical carbon dioxide with error below 5% with the highest diffusivity coefficient was 6.794 × 10^‐12 m²/s at operating condition was 10 MPa, 40 °C and particle size 425 µm. Moreover, Aris et al.² determined Momordica charantia extract yield with different mean particle size as well as diffusion coefficient, De, in the extraction process with and without co-extractant. Based on the results, mean particle size of 0.3 mm gave the highest extract yield, 3.32% and 1.34% with and without co-extractant respectively. Whereas, the value of De at 0.3 mm mean particle size, with and without co-extractant are 8.820 × 10⁻¹² and 7.920 × 10⁻¹² m²/s respectively.

Effect of process parameters on the yield of extraction versus time.