# ANNEXE A3

## LA COMPRESSION UNIDIRECTIONNELLE A DEFORMATION CONSTANTE (CDC)

### Réf. : CONDITIONS OPERATOIRES

### Pressage à vis (TSPUC®)

#### Pressage VIs

#### Conditions Expérimentales

<table><thead><tr><th>Number</th><th>Reference</th><th>Material</th><th>temperature [°C]</th><th>Screw</th><th>rotation [rpm]</th><th>P feed [kPa]</th><th>P out [kPa]</th><th>Out-liquid mass Flow Ql [g/s]</th><th>Out-solid mass Flow Qs [g/s]</th><th>Dry content before process [%]</th><th>Dry content after process [%]</th></tr></thead><tbody><tr><td>1</td><td>RDAP6-T26</td><td>RDAP</td><td>26</td><td>HCR</td><td>6</td><td>75.8</td><td>63.2</td><td>21.8</td><td>6.7</td><td>8.1</td><td>17.8</td></tr><tr><td>2</td><td>RDAP</td><td>RDAP</td><td>26</td><td>HCR</td><td>6</td><td>0</td><td>90.5</td><td>9.1</td><td>2.6</td><td>8.1</td><td>19.1</td></tr><tr><td>3</td><td>RDAP21-T26</td><td>RDAP</td><td>26</td><td>HCR</td><td>21</td><td>0</td><td>73.1</td><td>11.4</td><td>6.3</td><td>8.1</td><td>15.6</td></tr><tr><td>4</td><td>RDAP36-T26</td><td>RDAP</td><td>26</td><td>HCR</td><td>36</td><td>0</td><td>73.1</td><td>11.3</td><td>7.7</td><td>8.1</td><td>14.7</td></tr><tr><td>5</td><td>RDAP08-T26</td><td>RDAP</td><td>26</td><td>HCR</td><td>0.8</td><td>0</td><td>96.3</td><td>2.04</td><td>0.34</td><td>8.1</td><td>22.1</td></tr><tr><td>6</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>7</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>8</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>9</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>10</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>11</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>12</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>13</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>14</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>15</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>16</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>17</td><td>F6-T18</td><td>F</td><td>18</td><td>HCR</td><td>6</td><td>75.8</td><td>136</td><td>3</td><td>8</td><td>10.3</td><td>13.5</td></tr><tr><td>18</td><td>CC6-T38</td><td>Cc</td><td>38</td><td>HCR</td><td>6</td><td>48.3</td><td>132.6</td><td>1</td><td>1.2</td><td>33.5</td><td>51.1</td></tr><tr><td>19</td><td>CC21-T37</td><td>Cc</td><td>37</td><td>HCR</td><td>21</td><td>48.3</td><td>92.9</td><td>6.3</td><td>14.2</td><td>28.7</td><td>43.9</td></tr><tr><td>20</td><td>CC6-T11</td><td>Cc</td><td>11</td><td>HCR</td><td>6</td><td>48.3</td><td>119.4</td><td>1.8</td><td>4.3</td><td>33.5</td><td>41.7</td></tr><tr><td>21</td><td>CC6-T32</td><td>Cc</td><td>32</td><td>HCR</td><td>6</td><td>75.8</td><td>169.1</td><td>0.9</td><td>1.1</td><td>29.3</td><td>43.9</td></tr><tr><td>22</td><td>Cr6-T27</td><td>Cr</td><td>27</td><td>HCR</td><td>6</td><td>0</td><td>109.5</td><td>3.8</td><td>2.3</td><td>9.8</td><td>16.4</td></tr><tr><td>23</td><td>Cr21-T27</td><td>Cr</td><td>27</td><td>HCR</td><td>21</td><td>0</td><td>109.5</td><td>6.5</td><td>7</td><td>9.8</td><td>15.6</td></tr><tr><td>24</td><td>Cr</td><td>Cr</td><td>27</td><td>HCR</td><td>6</td><td>0</td><td>139.3</td><td>3.6</td><td>2.7</td><td>9.8</td><td>17.6</td></tr><tr><td>25</td><td>GDAP6-T22</td><td>GDAP</td><td>22</td><td>HCR</td><td>6</td><td>0</td><td>116.1</td><td>9.2</td><td>4.3</td><td>8.8</td><td>16.3</td></tr><tr><td>26</td><td>GDAP</td><td>GDAP</td><td>22</td><td>HCR</td><td>6</td><td>0</td><td>141</td><td>7.7</td><td>4.4</td><td>8.8</td><td>16</td></tr><tr><td>27</td><td>GDAP6-T24</td><td>GDAP</td><td>24</td><td>LCR</td><td>6</td><td>0</td><td>109.5</td><td>9.9</td><td>4.4</td><td>8.8</td><td>17.2</td></tr><tr><td>28</td><td>GDAP6b-T24</td><td>GDAP</td><td>24</td><td>LCR</td><td>6</td><td>0</td><td>139.3</td><td>9.5</td><td>4</td><td>8.8</td><td>17.4</td></tr><tr><td>29</td><td>MIAP</td><td>MIAP</td><td>24</td><td>HCR</td><td>6</td><td>0</td><td>112.8</td><td>10.8</td><td>3.1</td><td>7.7</td><td>17.8</td></tr><tr><td>30</td><td>MIAP</td><td>MIAP</td><td>24</td><td>HCR</td><td>6</td><td>0</td><td>140.9</td><td>12.2</td><td>2.6</td><td>7.1</td><td>18.4</td></tr></tbody></table>

#### Pressage VIs

#### Résultats Expérimentaux

<table><thead><tr><th>Number</th><th>Reference</th><th>Material</th><th>Feed= Ql Mass flow [g/s]</th><th>Collecteur 1 Mass flow [g/s]</th><th>Collecteur 2 Mass flow [g/s]</th><th>Collecteur 3 Mass flow [g/s]</th><th>Collecteur 4 Mass flow [g/s]</th><th>Collecteur 5 Mass flow [g/s]</th><th>Collecteur 6 Mass flow [g/s]</th><th>Collecteur 7 Mass flow [g/s]</th><th>Qs Mass flow [g/s]</th></tr></thead><tbody><tr><td>1</td><td>RDAP6-T26</td><td>RDAP</td><td>28.66</td><td>5.46</td><td>4.00</td><td>3.29</td><td>3.30</td><td>2.89</td><td>1.91</td><td>1.11</td><td>6.70</td></tr><tr><td>2</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>3</td><td>RDAP21-T26</td><td>RDAP</td><td>17.76</td><td>3.10</td><td>1.24</td><td>1.24</td><td>1.93</td><td>1.75</td><td>1.51</td><td>0.69</td><td>6.30</td></tr><tr><td>4</td><td>RDAP36-T26</td><td>RDAP</td><td>18.96</td><td>3.74</td><td>1.44</td><td>1.27</td><td>1.78</td><td>1.44</td><td>1.14</td><td>0.45</td><td>7.70</td></tr><tr><td>5</td><td>RDAP08-T26</td><td>RDAP</td><td>2.50</td><td>0.73</td><td>0.23</td><td>0.16</td><td>0.22</td><td>0.30</td><td>0.35</td><td>0.17</td><td>0.34</td></tr><tr><td>6</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>7</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>8</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>9</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>10</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>11</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>12</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>13</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>14</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>15</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>16</td><td>RDAP</td><td>RDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>17</td><td>F6-T18</td><td>F</td><td>11.47</td><td>1.50</td><td>0.68</td><td>0.38</td><td>0.31</td><td>0.26</td><td>0.23</td><td>0.11</td><td>8.00</td></tr><tr><td>18</td><td>CC6-T38</td><td>Cc</td><td>2.34</td><td>0.08</td><td>0.01</td><td>0.00</td><td>0.05</td><td>0.22</td><td>0.54</td><td>0.26</td><td>1.20</td></tr><tr><td>19</td><td>CC21-T37</td><td>Cc</td><td>19.75</td><td>0.54</td><td>0.15</td><td>0.10</td><td>0.48</td><td>0.89</td><td>2.84</td><td>0.55</td><td>14.20</td></tr><tr><td>20</td><td>CC6-T11</td><td>Cc</td><td>6.05</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.61</td><td>0.82</td><td>0.32</td><td>4.30</td></tr><tr><td>21</td><td>CC6-T32</td><td>Cc</td><td>1.81</td><td>0.10</td><td>0.02</td><td>0.01</td><td>0.08</td><td>0.17</td><td>0.22</td><td>0.11</td><td>1.10</td></tr><tr><td>22</td><td>Cr6-T27</td><td>Cr</td><td>6.03</td><td>0.63</td><td>0.57</td><td>0.53</td><td>0.62</td><td>0.56</td><td>0.52</td><td>0.30</td><td>2.30</td></tr><tr><td>23</td><td>Cr21-T27</td><td>Cr</td><td>13.96</td><td>1.39</td><td>1.18</td><td>1.08</td><td>1.19</td><td>1.08</td><td>0.73</td><td>0.31</td><td>7.00</td></tr><tr><td>24</td><td>Cr6b-T27</td><td>Cr</td><td>6.24</td><td>0.70</td><td>0.56</td><td>0.53</td><td>0.57</td><td>0.48</td><td>0.42</td><td>0.28</td><td>2.70</td></tr><tr><td>25</td><td>GDAP6-T22</td><td>GDAP</td><td>13.43</td><td>3.13</td><td>1.88</td><td>1.51</td><td>1.59</td><td>1.02</td><td>0.00</td><td>0.00</td><td>4.30</td></tr><tr><td>26</td><td>GDAP</td><td>GDAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>27</td><td>GDAP6-T24</td><td>GDAP</td><td>14.72</td><td>2.18</td><td>2.26</td><td>1.84</td><td>1.66</td><td>1.45</td><td>0.88</td><td>0.05</td><td>4.40</td></tr><tr><td>28</td><td>GDAP6b-T24</td><td>GDAP</td><td>13.44</td><td>2.28</td><td>2.22</td><td>1.72</td><td>1.43</td><td>0.98</td><td>0.79</td><td>0.02</td><td>4.00</td></tr><tr><td>29</td><td>MIAP</td><td>MIAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>30</td><td>MIAP</td><td>MIAP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
Compression en cellule piston (MMCC©)

Produit C (Protéines lactiques coagulées)

Données et Résultats Bruts

<table><thead><tr><th>rate [cm/min]</th><th>sample number</th><th>vi Vinitial [cm3/g]</th><th>vf Vfinal [cm3/g]</th><th>T[C]</th></tr></thead><tbody><tr><td>0.254</td><td>C1</td><td>3.42</td><td>1.37</td><td>40-45</td></tr><tr><td>0.254</td><td>C2</td><td>3.46</td><td>1.3</td><td>40-45</td></tr><tr><td>0.762</td><td>C3</td><td>3.45</td><td>1.15</td><td>45</td></tr><tr><td>0.254</td><td>C4</td><td>3.55</td><td>1.29</td><td>26</td></tr><tr><td>0.254</td><td>C5</td><td>3.56</td><td>1.19</td><td>26</td></tr><tr><td>0.254</td><td>C6</td><td>3.58</td><td>1.49</td><td>11</td></tr><tr><td>0.254</td><td>C7</td><td>3.54</td><td>1.51</td><td>11</td></tr><tr><td>2.54</td><td>C8</td><td>3.45</td><td>1.36</td><td>45</td></tr></tbody></table>

C1

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>3.67</td><td>4.48</td><td>3.42</td><td>1.17</td><td></td></tr><tr><td>1.93</td><td>21.96</td><td>3.99</td><td>3.05</td><td>6.98</td><td></td></tr><tr><td>3.94</td><td>40.70</td><td>3.48</td><td>2.66</td><td>12.96</td><td></td></tr><tr><td>5.91</td><td>66.60</td><td>2.98</td><td>2.24</td><td>21.17</td><td></td></tr><tr><td>7.91</td><td>96.30</td><td>2.47</td><td>1.89</td><td>30.59</td><td></td></tr><tr><td>8.90</td><td>126.70</td><td>2.22</td><td>1.69</td><td>40.28</td><td></td></tr><tr><td>9.45</td><td>159.80</td><td>2.08</td><td>1.59</td><td>50.80</td><td>0.00</td></tr><tr><td>9.88</td><td>223.00</td><td>1.97</td><td>1.50</td><td>70.90</td><td>9.02</td></tr><tr><td>10.31</td><td>388.90</td><td>1.86</td><td>1.41</td><td>123.60</td><td>31.49</td></tr><tr><td>10.55</td><td>595.20</td><td>1.80</td><td>1.37</td><td>189.10</td><td>70.34</td></tr><tr><td>10.67</td><td>850.20</td><td>1.77</td><td>1.35</td><td>270.20</td><td>130.55</td></tr><tr><td>10.94</td><td>1285.40</td><td>1.70</td><td>1.30</td><td>408.50</td><td>264.75</td></tr><tr><td>11.18</td><td>1981.60</td><td>1.64</td><td>1.25</td><td>629.70</td><td>490.89</td></tr><tr><td>11.42</td><td>2806.00</td><td>1.58</td><td>1.20</td><td>891.70</td><td>760.13</td></tr></tbody></table>

C2

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>2.74</td><td>4.53</td><td>3.46</td><td>0.87</td><td></td></tr><tr><td>1.89</td><td>20.00</td><td>4.05</td><td>3.09</td><td>6.36</td><td></td></tr><tr><td>3.90</td><td>35.20</td><td>3.54</td><td>2.70</td><td>11.18</td><td></td></tr><tr><td>5.91</td><td>57.60</td><td>3.03</td><td>2.32</td><td>18.30</td><td></td></tr><tr><td>7.87</td><td>96.40</td><td>2.53</td><td>1.93</td><td>30.60</td><td></td></tr><tr><td>8.43</td><td>121.40</td><td>2.39</td><td>1.83</td><td>38.60</td><td></td></tr><tr><td>8.94</td><td>161.30</td><td>2.26</td><td>1.73</td><td>51.30</td><td></td></tr><tr><td>9.41</td><td>268.00</td><td>2.14</td><td>1.64</td><td>85.20</td><td>0.00</td></tr><tr><td>9.65</td><td>415.00</td><td>2.08</td><td>1.59</td><td>132.00</td><td>3.12</td></tr><tr><td>9.80</td><td>586.00</td><td>2.04</td><td>1.56</td><td>186.00</td><td>6.88</td></tr><tr><td>9.88</td><td>718.00</td><td>2.02</td><td>1.54</td><td>228.00</td><td>12.44</td></tr><tr><td>10.08</td><td>1002.00</td><td>1.97</td><td>1.50</td><td>318.00</td><td>20.51</td></tr><tr><td>10.35</td><td>1801.00</td><td>1.90</td><td>1.45</td><td>572.00</td><td>36.83</td></tr><tr><td>10.67</td><td>2670.00</td><td>1.82</td><td>1.39</td><td>848.00</td><td>70.90</td></tr><tr><td>10.87</td><td>3583.00</td><td>1.77</td><td>1.35</td><td>1138.00</td><td>108.90</td></tr><tr><td>11.14</td><td>4447.00</td><td>1.70</td><td>1.30</td><td>1413.00</td><td>228.00</td></tr></tbody></table>

C3

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.63</td><td>4.53</td><td>3.45</td><td>0.20</td><td></td></tr><tr><td>0.77</td><td>12.38</td><td>3.94</td><td>2.99</td><td>3.94</td><td></td></tr><tr><td>1.59</td><td>25.70</td><td>3.32</td><td>2.52</td><td>8.18</td><td></td></tr><tr><td>2.38</td><td>44.20</td><td>2.72</td><td>2.07</td><td>14.05</td><td></td></tr><tr><td>2.77</td><td>66.80</td><td>2.42</td><td>1.84</td><td>21.25</td><td></td></tr><tr><td>2.97</td><td>92.30</td><td>2.27</td><td>1.72</td><td>29.30</td><td></td></tr><tr><td>3.18</td><td>155.90</td><td>2.11</td><td>1.61</td><td>49.50</td><td>0.00</td></tr><tr><td>3.27</td><td>262.40</td><td>2.04</td><td>1.55</td><td>83.40</td><td>0.08</td></tr><tr><td>3.39</td><td>520.00</td><td>1.95</td><td>1.49</td><td>165.20</td><td>23.79</td></tr><tr><td>3.43</td><td>740.00</td><td>1.92</td><td>1.46</td><td>235.20</td><td>46.20</td></tr><tr><td>3.48</td><td>876.00</td><td>1.88</td><td>1.43</td><td>278.40</td><td>80.20</td></tr><tr><td>3.60</td><td>1203.90</td><td>1.79</td><td>1.37</td><td>382.60</td><td>259.80</td></tr><tr><td>3.66</td><td>2004.90</td><td>1.74</td><td>1.33</td><td>637.10</td><td>536.00</td></tr><tr><td>3.74</td><td>2669.20</td><td>1.68</td><td>1.28</td><td>848.30</td><td>732.70</td></tr><tr><td>3.82</td><td>3504.00</td><td>1.62</td><td>1.23</td><td>1113.70</td><td>980.70</td></tr><tr><td>3.98</td><td>4505.00</td><td>1.50</td><td>1.15</td><td>1431.90</td><td>1275.60</td></tr></tbody></table>

C4

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>3.65</td><td>4.65</td><td>3.55</td><td>1.16</td><td></td></tr><tr><td>1.89</td><td>22.35</td><td>4.17</td><td>3.18</td><td>7.10</td><td></td></tr><tr><td>4.41</td><td>44.80</td><td>3.53</td><td>2.69</td><td>14.24</td><td></td></tr><tr><td>6.97</td><td>90.15</td><td>2.88</td><td>2.20</td><td>28.60</td><td></td></tr><tr><td>8.23</td><td>136.20</td><td>2.56</td><td>1.95</td><td>43.30</td><td></td></tr><tr><td>8.86</td><td>175.80</td><td>2.40</td><td>1.83</td><td>55.80</td><td></td></tr><tr><td>9.37</td><td>231.30</td><td>2.27</td><td>1.73</td><td>73.50</td><td>0.00</td></tr><tr><td>9.65</td><td>278.90</td><td>2.20</td><td>1.68</td><td>88.60</td><td>0.75</td></tr><tr><td>9.88</td><td>339.20</td><td>2.14</td><td>1.63</td><td>107.80</td><td>11.10</td></tr><tr><td>10.12</td><td>441.70</td><td>2.08</td><td>1.59</td><td>140.30</td><td>31.27</td></tr><tr><td>10.35</td><td>627.10</td><td>2.02</td><td>1.54</td><td>199.30</td><td>67.70</td></tr><tr><td>10.51</td><td>812.00</td><td>1.98</td><td>1.51</td><td>258.10</td><td>95.30</td></tr><tr><td>10.63</td><td>1001.80</td><td>1.95</td><td>1.49</td><td>318.40</td><td>143.00</td></tr><tr><td>10.87</td><td>1612.80</td><td>1.89</td><td>1.44</td><td>512.50</td><td>308.50</td></tr><tr><td>11.10</td><td>2489.00</td><td>1.83</td><td>1.39</td><td>791.00</td><td>565.30</td></tr><tr><td>11.38</td><td>3269.00</td><td>1.76</td><td>1.34</td><td>1039.00</td><td>772.60</td></tr><tr><td>11.69</td><td>4293.00</td><td>1.68</td><td>1.28</td><td>1364.00</td><td>1152.30</td></tr></tbody></table>
<table><caption>C5</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.42</td><td>4.67</td><td>3.56</td><td>0.13</td><td></td></tr><tr><td>1.81</td><td>17.80</td><td>4.21</td><td>3.21</td><td>5.66</td><td></td></tr><tr><td>3.86</td><td>33.53</td><td>3.69</td><td>2.82</td><td>10.60</td><td></td></tr><tr><td>5.87</td><td>56.40</td><td>3.18</td><td>2.43</td><td>17.90</td><td></td></tr><tr><td>6.85</td><td>72.10</td><td>2.93</td><td>2.23</td><td>22.90</td><td></td></tr><tr><td>7.83</td><td>98.10</td><td>2.68</td><td>2.04</td><td>31.20</td><td></td></tr><tr><td>8.90</td><td>137.50</td><td>2.41</td><td>1.84</td><td>43.70</td><td>0.00</td></tr><tr><td>9.80</td><td>209.00</td><td>2.18</td><td>1.66</td><td>66.00</td><td>0.00</td></tr><tr><td>10.43</td><td>324.00</td><td>2.02</td><td>1.54</td><td>103.00</td><td>23.49</td></tr><tr><td>10.79</td><td>447.00</td><td>1.93</td><td>1.47</td><td>142.00</td><td>65.90</td></tr><tr><td>10.91</td><td>585.00</td><td>1.90</td><td>1.45</td><td>186.00</td><td>117.20</td></tr><tr><td>11.10</td><td>757.00</td><td>1.85</td><td>1.42</td><td>240.00</td><td>195.00</td></tr><tr><td>11.14</td><td>949.00</td><td>1.84</td><td>1.40</td><td>301.00</td><td>265.00</td></tr><tr><td>11.38</td><td>1299.00</td><td>1.78</td><td>1.35</td><td>412.00</td><td>437.00</td></tr><tr><td>11.61</td><td>1989.00</td><td>1.72</td><td>1.32</td><td>632.00</td><td>619.00</td></tr><tr><td>11.81</td><td>2753.00</td><td>1.67</td><td>1.28</td><td>875.00</td><td>861.00</td></tr><tr><td>12.09</td><td>3645.00</td><td>1.60</td><td>1.22</td><td>1158.00</td><td>1081.00</td></tr><tr><td>12.28</td><td>4290.00</td><td>1.55</td><td>1.18</td><td>1363.00</td><td>1178.00</td></tr></tbody></table>

<table><caption>C6</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>2.35</td><td>4.69</td><td>3.58</td><td>0.75</td><td></td></tr><tr><td>1.42</td><td>24.85</td><td>4.33</td><td>3.31</td><td>7.89</td><td></td></tr><tr><td>2.91</td><td>44.90</td><td>3.95</td><td>3.01</td><td>14.30</td><td></td></tr><tr><td>4.41</td><td>77.90</td><td>3.57</td><td>2.72</td><td>24.70</td><td></td></tr><tr><td>5.98</td><td>125.00</td><td>3.17</td><td>2.42</td><td>39.70</td><td></td></tr><tr><td>6.97</td><td>186.00</td><td>2.92</td><td>2.23</td><td>59.10</td><td></td></tr><tr><td>7.95</td><td>286.00</td><td>2.67</td><td>2.04</td><td>91.02</td><td>0.00</td></tr><tr><td>8.43</td><td>394.00</td><td>2.55</td><td>1.95</td><td>122.30</td><td>5.73</td></tr><tr><td>8.78</td><td>485.00</td><td>2.46</td><td>1.88</td><td>154.30</td><td>21.08</td></tr><tr><td>8.94</td><td>557.00</td><td>2.42</td><td>1.85</td><td>177.30</td><td>39.27</td></tr><tr><td>9.21</td><td>752.00</td><td>2.35</td><td>1.79</td><td>239.00</td><td>78.50</td></tr><tr><td>9.29</td><td>933.00</td><td>2.33</td><td>1.77</td><td>296.60</td><td>130.90</td></tr><tr><td>9.72</td><td>1385.00</td><td>2.22</td><td>1.69</td><td>440.30</td><td>249.10</td></tr><tr><td>9.96</td><td>2042.00</td><td>2.16</td><td>1.65</td><td>649.00</td><td>441.10</td></tr><tr><td>10.24</td><td>2922.00</td><td>2.09</td><td>1.60</td><td>928.90</td><td>675.30</td></tr><tr><td>10.47</td><td>3634.00</td><td>2.03</td><td>1.55</td><td>1155.10</td><td>909.00</td></tr><tr><td>10.79</td><td>4492.00</td><td>1.95</td><td>1.49</td><td>1427.90</td><td>1185.00</td></tr></tbody></table>

<table><caption>C7</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>3.45</td><td>4.65</td><td>3.54</td><td>1.09</td><td></td></tr><tr><td>1.77</td><td>31.10</td><td>4.20</td><td>3.20</td><td>9.87</td><td></td></tr><tr><td>3.78</td><td>68.00</td><td>3.69</td><td>2.82</td><td>21.70</td><td></td></tr><tr><td>4.76</td><td>93.00</td><td>3.44</td><td>2.63</td><td>29.80</td><td></td></tr><tr><td>5.75</td><td>129.00</td><td>3.19</td><td>2.44</td><td>41.00</td><td></td></tr><tr><td>6.77</td><td>189.00</td><td>2.93</td><td>2.24</td><td>60.00</td><td></td></tr><tr><td>7.72</td><td>288.00</td><td>2.69</td><td>2.05</td><td>91.00</td><td>0.00</td></tr><tr><td>8.46</td><td>450.00</td><td>2.50</td><td>1.91</td><td>143.00</td><td>7.00</td></tr><tr><td>8.78</td><td>579.00</td><td>2.42</td><td>1.85</td><td>184.00</td><td>24.14</td></tr><tr><td>8.98</td><td>754.00</td><td>2.37</td><td>1.81</td><td>230.00</td><td>42.30</td></tr><tr><td>9.17</td><td>936.00</td><td>2.32</td><td>1.77</td><td>297.00</td><td>68.00</td></tr><tr><td>9.41</td><td>1250.00</td><td>2.26</td><td>1.73</td><td>397.00</td><td>106.00</td></tr><tr><td>9.53</td><td>1526.00</td><td>2.23</td><td>1.70</td><td>485.00</td><td>127.00</td></tr><tr><td>9.80</td><td>2468.00</td><td>2.16</td><td>1.65</td><td>784.00</td><td>178.00</td></tr><tr><td>10.12</td><td>3515.00</td><td>2.08</td><td>1.59</td><td>1117.00</td><td>266.00</td></tr><tr><td>10.51</td><td>4550.00</td><td>1.98</td><td>1.51</td><td>1445.00</td><td>458.00</td></tr></tbody></table>

<table><caption>C8</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>2.73</td><td>4.53</td><td>3.45</td><td>0.87</td><td></td></tr><tr><td>0.19</td><td>7.46</td><td>4.04</td><td>3.08</td><td>2.37</td><td></td></tr><tr><td>0.39</td><td>10.30</td><td>3.53</td><td>2.69</td><td>3.27</td><td></td></tr><tr><td>0.59</td><td>25.30</td><td>3.02</td><td>2.30</td><td>8.05</td><td></td></tr><tr><td>0.80</td><td>54.00</td><td>2.51</td><td>1.92</td><td>17.17</td><td>0.00</td></tr><tr><td>0.89</td><td>122.00</td><td>2.27</td><td>1.73</td><td>39.95</td><td>0.39</td></tr><tr><td>0.93</td><td>242.00</td><td>2.17</td><td>1.65</td><td>76.94</td><td>26.09</td></tr><tr><td>0.94</td><td>333.00</td><td>2.13</td><td>1.63</td><td>105.90</td><td>44.66</td></tr><tr><td>0.95</td><td>464.00</td><td>2.11</td><td>1.61</td><td>147.70</td><td>74.65</td></tr><tr><td>0.97</td><td>758.00</td><td>2.07</td><td>1.58</td><td>240.90</td><td>143.90</td></tr><tr><td>0.99</td><td>1100.00</td><td>2.01</td><td>1.54</td><td>370.00</td><td>288.50</td></tr><tr><td>1.00</td><td>1585.00</td><td>1.98</td><td>1.51</td><td>504.00</td><td>411.20</td></tr><tr><td>1.02</td><td>2370.00</td><td>1.93</td><td>1.47</td><td>753.30</td><td>582.00</td></tr><tr><td>1.05</td><td>3273.00</td><td>1.86</td><td>1.42</td><td>1040.20</td><td>903.00</td></tr><tr><td>1.08</td><td>4397.00</td><td>1.78</td><td>1.35</td><td>1397.60</td><td>1244.00</td></tr></tbody></table>
Produit F (pulpes de betteraves)

Données et Résultats Bruts

<table><thead><tr><th>rate [cm/min]</th><th>sample number</th><th>vi Vinitial [cm3/g]</th><th>vf Vfinal [cm3/g]</th></tr></thead><tbody><tr><td>0.254</td><td>F1</td><td>11.69</td><td>1.57</td></tr><tr><td>0.254</td><td>F2</td><td>11.72</td><td>1.78</td></tr><tr><td>0.762</td><td>F3</td><td>11.4</td><td>3.15</td></tr><tr><td>0.762</td><td>F4</td><td>11.48</td><td>2.38</td></tr><tr><td>2.54</td><td>F5</td><td>11.6</td><td>4.28</td></tr><tr><td>2.54</td><td>F6</td><td>11.66</td><td>4.08</td></tr><tr><td>7.62</td><td>F7</td><td>11.62</td><td>4.58</td></tr><tr><td>7.62</td><td>F8</td><td>11.27</td><td>4.38</td></tr></tbody></table>

F1

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PF1 [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>4.24</td><td>4.74</td><td>11.69</td><td>1.35</td><td></td></tr><tr><td>5.41</td><td>34.75</td><td>3.37</td><td>8.31</td><td>11.04</td><td></td></tr><tr><td>7.69</td><td>52.98</td><td>2.79</td><td>6.88</td><td>16.84</td><td></td></tr><tr><td>10.00</td><td>86.35</td><td>2.20</td><td>5.43</td><td>27.44</td><td></td></tr><tr><td>11.80</td><td>140.28</td><td>1.75</td><td>4.30</td><td>44.58</td><td></td></tr><tr><td>12.29</td><td>168.26</td><td>1.62</td><td>4.00</td><td>53.47</td><td>0.00</td></tr><tr><td>12.80</td><td>213.39</td><td>1.49</td><td>3.68</td><td>67.82</td><td>4.98</td></tr><tr><td>13.31</td><td>288.51</td><td>1.36</td><td>3.35</td><td>91.69</td><td>18.40</td></tr><tr><td>13.83</td><td>445.96</td><td>1.23</td><td>3.03</td><td>141.73</td><td>46.89</td></tr><tr><td>14.07</td><td>565.37</td><td>1.17</td><td>2.88</td><td>179.68</td><td>75.66</td></tr><tr><td>14.31</td><td>756.73</td><td>1.11</td><td>2.73</td><td>240.45</td><td>120.12</td></tr><tr><td>14.56</td><td>923.09</td><td>1.04</td><td>2.57</td><td>293.36</td><td>185.81</td></tr><tr><td>14.81</td><td>1255.05</td><td>0.98</td><td>2.42</td><td>398.85</td><td>277.36</td></tr><tr><td>15.06</td><td>1706.50</td><td>0.92</td><td>2.27</td><td>542.33</td><td>400.58</td></tr><tr><td>15.33</td><td>2228.60</td><td>0.85</td><td>2.09</td><td>708.25</td><td>548.75</td></tr><tr><td>15.57</td><td>2764.49</td><td>0.79</td><td>1.95</td><td>878.56</td><td>702.45</td></tr><tr><td>15.80</td><td>3330.68</td><td>0.73</td><td>1.80</td><td>1058.49</td><td>888.90</td></tr><tr><td>16.16</td><td>4250.58</td><td>0.64</td><td>1.57</td><td>1350.84</td><td>1152.69</td></tr></tbody></table>

F2

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PF1 [kPa]</th></tr></thead><tbody><tr><td>12.64</td><td>376.00</td><td>1.43</td><td>3.52</td><td>119.60</td><td>4.19</td></tr><tr><td>12.83</td><td>465.00</td><td>1.36</td><td>3.36</td><td>147.80</td><td>7.57</td></tr><tr><td>13.07</td><td>574.00</td><td>1.30</td><td>3.22</td><td>183.00</td><td>16.50</td></tr><tr><td>13.35</td><td>731.00</td><td>1.23</td><td>3.04</td><td>232.00</td><td>28.60</td></tr><tr><td>13.58</td><td>920.00</td><td>1.17</td><td>2.90</td><td>292.00</td><td>47.60</td></tr><tr><td>13.82</td><td>1200.00</td><td>1.10</td><td>2.73</td><td>381.00</td><td>75.30</td></tr><tr><td>14.09</td><td>1564.00</td><td>1.04</td><td>2.57</td><td>497.00</td><td>131.00</td></tr><tr><td>14.33</td><td>2051.00</td><td>0.98</td><td>2.42</td><td>652.00</td><td>209.00</td></tr><tr><td>14.61</td><td>2488.00</td><td>0.91</td><td>2.25</td><td>790.00</td><td>307.00</td></tr><tr><td>14.84</td><td>3045.00</td><td>0.85</td><td>2.10</td><td>967.00</td><td>435.00</td></tr><tr><td>15.12</td><td>3606.00</td><td>0.79</td><td>1.96</td><td>1146.00</td><td>543.00</td></tr><tr><td>15.35</td><td>4351.00</td><td>0.72</td><td>1.78</td><td>1382.00</td><td>697.00</td></tr></tbody></table>

F3

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PF1 [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>2.64</td><td>4.63</td><td>11.41</td><td>0.83</td><td></td></tr><tr><td>0.92</td><td>23.11</td><td>3.93</td><td>9.68</td><td>7.34</td><td></td></tr><tr><td>1.93</td><td>50.65</td><td>3.16</td><td>7.80</td><td>16.09</td><td></td></tr><tr><td>2.42</td><td>75.77</td><td>2.78</td><td>6.86</td><td>24.08</td><td></td></tr><tr><td>2.91</td><td>114.71</td><td>2.41</td><td>5.93</td><td>36.45</td><td></td></tr><tr><td>3.19</td><td>160.08</td><td>2.20</td><td>5.43</td><td>50.87</td><td></td></tr><tr><td>3.41</td><td>219.81</td><td>2.03</td><td>5.00</td><td>69.86</td><td>0.00</td></tr><tr><td>3.55</td><td>295.01</td><td>1.92</td><td>4.73</td><td>94.07</td><td>4.71</td></tr><tr><td>3.64</td><td>401.96</td><td>1.85</td><td>4.55</td><td>127.74</td><td>18.15</td></tr><tr><td>3.78</td><td>630.96</td><td>1.75</td><td>4.31</td><td>200.52</td><td>55.94</td></tr><tr><td>3.95</td><td>970.88</td><td>1.62</td><td>3.99</td><td>308.55</td><td>142.98</td></tr><tr><td>4.04</td><td>1659.40</td><td>1.55</td><td>3.81</td><td>527.04</td><td>320.83</td></tr><tr><td>4.16</td><td>2504.38</td><td>1.46</td><td>3.60</td><td>795.89</td><td>540.95</td></tr><tr><td>4.30</td><td>3735.42</td><td>1.35</td><td>3.33</td><td>1187.12</td><td>832.53</td></tr><tr><td>4.39</td><td>4225.25</td><td>1.28</td><td>3.15</td><td>1342.78</td><td>1157.25</td></tr></tbody></table>

F4

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PF1 [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>1.27</td><td>4.66</td><td>11.50</td><td>0.41</td><td></td></tr><tr><td>0.77</td><td>16.40</td><td>4.07</td><td>10.05</td><td>5.40</td><td></td></tr><tr><td>1.57</td><td>34.00</td><td>3.46</td><td>8.54</td><td>11.40</td><td></td></tr><tr><td>2.38</td><td>60.00</td><td>2.85</td><td>7.05</td><td>19.90</td><td></td></tr><tr><td>3.19</td><td>108.00</td><td>2.23</td><td>5.52</td><td>35.60</td><td>0.00</td></tr><tr><td>3.58</td><td>158.00</td><td>1.93</td><td>4.70</td><td>51.90</td><td>1.63</td></tr><tr><td>3.98</td><td>207.00</td><td>1.63</td><td>4.02</td><td>100.00</td><td>16.30</td></tr><tr><td>4.17</td><td>508.00</td><td>1.48</td><td>3.66</td><td>166.00</td><td>51.00</td></tr><tr><td>4.28</td><td>726.00</td><td>1.40</td><td>3.46</td><td>239.00</td><td>76.00</td></tr><tr><td>4.38</td><td>1049.00</td><td>1.32</td><td>3.27</td><td>344.00</td><td>155.00</td></tr><tr><td>4.49</td><td>1592.00</td><td>1.24</td><td>3.06</td><td>522.00</td><td>277.00</td></tr><tr><td>4.58</td><td>2233.00</td><td>1.17</td><td>2.89</td><td>732.00</td><td>465.00</td></tr><tr><td>4.69</td><td>3063.00</td><td>1.09</td><td>2.69</td><td>1004.00</td><td>659.00</td></tr><tr><td>4.75</td><td>3767.00</td><td>1.04</td><td>2.56</td><td>1234.00</td><td>888.00</td></tr><tr><td>4.86</td><td>4366.00</td><td>0.96</td><td>2.38</td><td>1431.00</td><td>1194.00</td></tr></tbody></table>
<table><caption>F5</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>V [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.74</td><td>4.71</td><td>11.60</td><td>0.24</td><td></td></tr><tr><td>0.18</td><td>6.70</td><td>4.25</td><td>10.48</td><td>2.13</td><td></td></tr><tr><td>0.39</td><td>22.65</td><td>3.73</td><td>9.19</td><td>7.19</td><td></td></tr><tr><td>0.57</td><td>46.68</td><td>3.25</td><td>8.01</td><td>14.83</td><td></td></tr><tr><td>0.68</td><td>65.39</td><td>2.98</td><td>7.34</td><td>20.78</td><td></td></tr><tr><td>0.78</td><td>96.01</td><td>2.73</td><td>6.72</td><td>30.51</td><td></td></tr><tr><td>0.88</td><td>155.98</td><td>2.47</td><td>6.10</td><td>49.57</td><td>0.00</td></tr><tr><td>0.94</td><td>239.45</td><td>2.32</td><td>5.72</td><td>76.09</td><td>13.22</td></tr><tr><td>0.98</td><td>346.62</td><td>2.22</td><td>5.49</td><td>110.15</td><td>38.22</td></tr><tr><td>1.01</td><td>536.73</td><td>2.14</td><td>5.27</td><td>170.57</td><td>72.75</td></tr><tr><td>1.03</td><td>717.70</td><td>2.09</td><td>5.16</td><td>228.08</td><td>112.68</td></tr><tr><td>1.08</td><td>1434.22</td><td>1.97</td><td>4.87</td><td>455.79</td><td>346.67</td></tr><tr><td>1.11</td><td>2218.98</td><td>1.90</td><td>4.69</td><td>705.19</td><td>551.52</td></tr><tr><td>1.13</td><td>3067.70</td><td>1.84</td><td>4.53</td><td>974.93</td><td>774.00</td></tr><tr><td>1.17</td><td>4350.50</td><td>1.74</td><td>4.28</td><td>1382.60</td><td>1220.59</td></tr></tbody></table>

<table><caption>F7</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>V [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>5.73</td><td>4.71</td><td>11.62</td><td>1.82</td><td></td></tr><tr><td>0.10</td><td>30.19</td><td>3.99</td><td>9.83</td><td>9.59</td><td></td></tr><tr><td>0.20</td><td>59.29</td><td>3.22</td><td>7.95</td><td>18.53</td><td></td></tr><tr><td>0.26</td><td>127.00</td><td>2.72</td><td>6.70</td><td>40.36</td><td>0</td></tr><tr><td>0.31</td><td>300.85</td><td>2.36</td><td>5.80</td><td>95.61</td><td>11.78</td></tr><tr><td>0.33</td><td>572.55</td><td>2.21</td><td>5.44</td><td>181.96</td><td>63.4</td></tr><tr><td>0.34</td><td>1008.22</td><td>2.15</td><td>5.29</td><td>320.41</td><td>135.52</td></tr><tr><td>0.35</td><td>1678.64</td><td>2.08</td><td>5.12</td><td>533.47</td><td>209.26</td></tr><tr><td>0.35</td><td>2455.01</td><td>2.01</td><td>4.95</td><td>786.56</td><td>324.91</td></tr><tr><td>0.36</td><td>3368.20</td><td>1.95</td><td>4.80</td><td>1070.41</td><td>495.05</td></tr><tr><td>0.37</td><td>4506.56</td><td>1.86</td><td>4.58</td><td>1432.18</td><td>1137.85</td></tr></tbody></table>

<table><caption>F6</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>V [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.95</td><td>4.72</td><td>11.66</td><td>0.30</td><td></td></tr><tr><td>0.18</td><td>7.00</td><td>4.26</td><td>10.50</td><td>2.25</td><td></td></tr><tr><td>0.38</td><td>20.00</td><td>3.75</td><td>9.30</td><td>6.60</td><td></td></tr><tr><td>0.59</td><td>40.00</td><td>3.22</td><td>7.95</td><td>12.90</td><td></td></tr><tr><td>0.68</td><td>61.00</td><td>2.99</td><td>7.30</td><td>19.60</td><td></td></tr><tr><td>0.78</td><td>94.00</td><td>2.74</td><td>6.70</td><td>27.00</td><td></td></tr><tr><td>0.89</td><td>132.00</td><td>2.47</td><td>6.09</td><td>42.00</td><td>0.00</td></tr><tr><td>0.94</td><td>176.00</td><td>2.32</td><td>5.72</td><td>56.00</td><td>1.84</td></tr><tr><td>0.99</td><td>233.00</td><td>2.21</td><td>5.40</td><td>74.00</td><td>8.64</td></tr><tr><td>1.01</td><td>304.00</td><td>2.15</td><td>5.30</td><td>96.00</td><td>23.80</td></tr><tr><td>1.04</td><td>382.00</td><td>2.09</td><td>5.20</td><td>122.00</td><td>35.90</td></tr><tr><td>1.07</td><td>601.00</td><td>2.01</td><td>4.97</td><td>191.00</td><td>89.50</td></tr><tr><td>1.07</td><td>791.00</td><td>1.99</td><td>4.90</td><td>251.00</td><td>132.00</td></tr><tr><td>1.13</td><td>1642.00</td><td>1.85</td><td>4.60</td><td>522.00</td><td>393.00</td></tr><tr><td>1.17</td><td>2794.00</td><td>1.75</td><td>4.30</td><td>888.00</td><td>691.00</td></tr><tr><td>1.21</td><td>4335.00</td><td>1.65</td><td>4.08</td><td>1378.00</td><td>1203.00</td></tr></tbody></table>

<table><caption>F8</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>V [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.42</td><td>4.58</td><td>11.30</td><td>0.13</td><td></td></tr><tr><td>0.06</td><td>14.60</td><td>4.09</td><td>10.10</td><td>4.65</td><td></td></tr><tr><td>0.13</td><td>32.00</td><td>3.58</td><td>8.80</td><td>10.14</td><td></td></tr><tr><td>0.20</td><td>63.00</td><td>3.08</td><td>7.60</td><td>20.11</td><td></td></tr><tr><td>0.23</td><td>92.00</td><td>2.83</td><td>6.98</td><td>29.50</td><td>0</td></tr><tr><td>0.26</td><td>140.00</td><td>2.58</td><td>6.40</td><td>44.00</td><td>3.35</td></tr><tr><td>0.29</td><td>241.00</td><td>2.35</td><td>5.80</td><td>76.00</td><td>59.00</td></tr><tr><td>0.31</td><td>412.00</td><td>2.22</td><td>5.46</td><td>130.00</td><td>120.00</td></tr><tr><td>0.31</td><td>533.00</td><td>2.18</td><td>5.38</td><td>169.00</td><td>160.00</td></tr><tr><td>0.32</td><td>699.00</td><td>2.13</td><td>5.26</td><td>222.00</td><td>203.00</td></tr><tr><td>0.33</td><td>865.00</td><td>2.10</td><td>5.18</td><td>275.00</td><td>244.00</td></tr><tr><td>0.35</td><td>1974.00</td><td>1.93</td><td>4.75</td><td>627.00</td><td>472.00</td></tr><tr><td>0.35</td><td>2735.00</td><td>1.90</td><td>4.69</td><td>869.00</td><td>510.00</td></tr><tr><td>0.36</td><td>3659.00</td><td>1.84</td><td>4.55</td><td>1163.00</td><td>659.00</td></tr><tr><td>0.37</td><td>4499.00</td><td>1.78</td><td>4.38</td><td>1429.00</td><td>1148.00</td></tr></tbody></table>
Produit Cr (Airelles)

Données et Résultats Bruts

<table><thead><tr><th>rate [cm/min]</th><th>sample number</th><th>vi Vinitial [cm3/g]</th><th>vf Vfinal [cm3/g]</th><th>Openings Filter (mic.)</th></tr></thead><tbody><tr><td>0.254</td><td>Cr1</td><td>12.45</td><td>3.39</td><td>500</td></tr><tr><td>0.254</td><td>Cr2</td><td>12.56</td><td>2.64</td><td>500</td></tr><tr><td>0.254</td><td>Cr3</td><td>13.1</td><td>2.9</td><td>100</td></tr><tr><td>0.254</td><td>Cr4</td><td>13.07</td><td>2.48</td><td>100</td></tr><tr><td>0.254</td><td>Cr5</td><td>13.06</td><td>3.01</td><td>53</td></tr><tr><td>0.254</td><td>Cr6</td><td>13.08</td><td>2.33</td><td>53</td></tr><tr><td>0.762</td><td>Cr7</td><td>13.08</td><td>3</td><td>500</td></tr><tr><td>2.54</td><td>Cr8</td><td>13.28</td><td>4.94</td><td>500</td></tr><tr><td>2.54</td><td>Cr9</td><td>13.06</td><td>5.1</td><td>53</td></tr></tbody></table>

Cr9

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.42</td><td>4.12</td><td>13.08</td><td>0.13</td><td></td></tr><tr><td>0.19</td><td>7.92</td><td>3.63</td><td>11.50</td><td>2.51</td><td></td></tr><tr><td>0.39</td><td>19.22</td><td>3.13</td><td>9.91</td><td>6.14</td><td></td></tr><tr><td>0.59</td><td>69.70</td><td>2.62</td><td>8.31</td><td>22.15</td><td>0.00</td></tr><tr><td>0.69</td><td>141.20</td><td>2.36</td><td>7.48</td><td>44.87</td><td>18.15</td></tr><tr><td>0.74</td><td>219.70</td><td>2.24</td><td>7.10</td><td>69.84</td><td>35.58</td></tr><tr><td>0.76</td><td>273.70</td><td>2.19</td><td>6.95</td><td>86.99</td><td>50.70</td></tr><tr><td>0.79</td><td>391.60</td><td>2.11</td><td>6.88</td><td>121.29</td><td>82.73</td></tr><tr><td>0.81</td><td>545.20</td><td>2.05</td><td>6.48</td><td>173.28</td><td>127.00</td></tr><tr><td>0.83</td><td>701.90</td><td>2.00</td><td>6.35</td><td>223.08</td><td>177.60</td></tr><tr><td>0.85</td><td>895.00</td><td>1.96</td><td>6.22</td><td>284.43</td><td>224.30</td></tr><tr><td>0.88</td><td>1344.00</td><td>1.88</td><td>5.97</td><td>427.12</td><td>391.00</td></tr><tr><td>0.92</td><td>2072.00</td><td>1.79</td><td>5.68</td><td>658.54</td><td>596.00</td></tr><tr><td>0.93</td><td>2771.00</td><td>1.75</td><td>5.55</td><td>1100.00</td><td>1057.50</td></tr><tr><td>0.96</td><td>3525.00</td><td>1.69</td><td>5.36</td><td>1120.50</td><td>1291.00</td></tr><tr><td>0.99</td><td>4873.00</td><td>1.61</td><td>5.10</td><td>1549.00</td><td>1375.20</td></tr></tbody></table>
<table><caption>Cr1</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>ν [cm3/g]</th><th>PI [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.21</td><td>3.94</td><td>12.45</td><td>0.07</td><td></td></tr><tr><td>0.79</td><td>7.88</td><td>3.74</td><td>11.82</td><td>2.51</td><td></td></tr><tr><td>1.85</td><td>16.94</td><td>3.47</td><td>10.97</td><td>7.39</td><td></td></tr><tr><td>3.86</td><td>41.38</td><td>2.96</td><td>9.36</td><td>13.14</td><td></td></tr><tr><td>5.83</td><td>71.59</td><td>2.46</td><td>7.77</td><td>22.75</td><td></td></tr><tr><td>6.85</td><td>111.76</td><td>2.20</td><td>6.98</td><td>35.52</td><td></td></tr><tr><td>7.87</td><td>200.93</td><td>1.94</td><td>6.15</td><td>63.86</td><td>0.00</td></tr><tr><td>8.35</td><td>304.81</td><td>1.82</td><td>5.76</td><td>96.87</td><td>2.43</td></tr><tr><td>8.58</td><td>380.57</td><td>1.76</td><td>5.58</td><td>120.94</td><td>13.58</td></tr><tr><td>8.82</td><td>495.42</td><td>1.70</td><td>5.37</td><td>157.44</td><td>27.96</td></tr><tr><td>9.09</td><td>652.88</td><td>1.63</td><td>5.17</td><td>207.48</td><td>51.52</td></tr><tr><td>9.25</td><td>790.64</td><td>1.59</td><td>5.02</td><td>251.26</td><td>84.60</td></tr><tr><td>9.57</td><td>939.16</td><td>1.51</td><td>4.78</td><td>298.46</td><td>152.03</td></tr><tr><td>9.80</td><td>1173.86</td><td>1.45</td><td>4.58</td><td>373.05</td><td>273.77</td></tr><tr><td>10.24</td><td>1846.43</td><td>1.34</td><td>4.24</td><td>596.79</td><td>432.73</td></tr><tr><td>10.55</td><td>2505.46</td><td>1.26</td><td>4.00</td><td>796.23</td><td>640.60</td></tr><tr><td>10.98</td><td>3497.65</td><td>1.15</td><td>3.66</td><td>1111.55</td><td>921.30</td></tr><tr><td>11.30</td><td>4491.93</td><td>1.07</td><td>3.39</td><td>1427.53</td><td>1174.66</td></tr></tbody></table>

<table><caption>Cr3</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>ν [cm3/g]</th><th>PI [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.11</td><td>4.14</td><td>13.11</td><td>0.03</td><td></td></tr><tr><td>2.01</td><td>18.52</td><td>3.63</td><td>11.49</td><td>5.89</td><td></td></tr><tr><td>4.02</td><td>30.10</td><td>3.12</td><td>9.89</td><td>9.56</td><td></td></tr><tr><td>6.02</td><td>46.73</td><td>2.61</td><td>8.28</td><td>14.85</td><td></td></tr><tr><td>7.48</td><td>69.04</td><td>2.24</td><td>7.09</td><td>21.94</td><td></td></tr><tr><td>8.98</td><td>135.88</td><td>1.86</td><td>5.89</td><td>43.18</td><td></td></tr><tr><td>9.49</td><td>183.24</td><td>1.73</td><td>5.48</td><td>58.23</td><td></td></tr><tr><td>9.80</td><td>237.78</td><td>1.65</td><td>5.24</td><td>75.56</td><td>0.00</td></tr><tr><td>10.00</td><td>285.55</td><td>1.60</td><td>5.08</td><td>90.75</td><td>8.19</td></tr><tr><td>10.24</td><td>282.17</td><td>1.54</td><td>4.87</td><td>125.00</td><td>33.64</td></tr><tr><td>10.51</td><td>498.57</td><td>1.47</td><td>4.67</td><td>158.45</td><td>62.83</td></tr><tr><td>10.63</td><td>647.61</td><td>1.44</td><td>4.55</td><td>205.81</td><td>100.56</td></tr><tr><td>10.91</td><td>890.22</td><td>1.37</td><td>4.34</td><td>282.91</td><td>168.37</td></tr><tr><td>11.14</td><td>1050.31</td><td>1.31</td><td>4.15</td><td>333.79</td><td>222.30</td></tr><tr><td>11.46</td><td>1519.30</td><td>1.23</td><td>3.89</td><td>482.83</td><td>378.75</td></tr><tr><td>11.77</td><td>2006.41</td><td>1.15</td><td>3.63</td><td>637.64</td><td>512.59</td></tr><tr><td>12.13</td><td>2680.40</td><td>1.06</td><td>3.38</td><td>851.83</td><td>705.88</td></tr><tr><td>12.40</td><td>3662.39</td><td>0.99</td><td>3.14</td><td>1163.90</td><td>1000.43</td></tr><tr><td>12.72</td><td>4467.85</td><td>0.91</td><td>2.90</td><td>1419.90</td><td>1234.65</td></tr></tbody></table>

<table><caption>Cr2</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>ν [cm3/g]</th><th>PI [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>1.37</td><td>3.97</td><td>12.50</td><td>0.44</td><td></td></tr><tr><td>1.06</td><td>6.60</td><td>3.70</td><td>11.70</td><td>2.10</td><td></td></tr><tr><td>2.32</td><td>18.00</td><td>3.38</td><td>10.70</td><td>5.80</td><td></td></tr><tr><td>3.62</td><td>28.00</td><td>3.05</td><td>9.60</td><td>9.20</td><td></td></tr><tr><td>4.84</td><td>36.00</td><td>2.74</td><td>8.70</td><td>11.60</td><td></td></tr><tr><td>6.10</td><td>52.00</td><td>2.42</td><td>7.60</td><td>16.80</td><td></td></tr><tr><td>6.77</td><td>63.00</td><td>2.25</td><td>7.10</td><td>20.30</td><td></td></tr><tr><td>7.36</td><td>79.00</td><td>2.10</td><td>6.60</td><td>25.30</td><td></td></tr><tr><td>7.87</td><td>97.00</td><td>1.97</td><td>6.20</td><td>31.00</td><td></td></tr><tr><td>8.43</td><td>124.00</td><td>1.83</td><td>5.80</td><td>39.00</td><td></td></tr><tr><td>8.90</td><td>172.00</td><td>1.71</td><td>5.40</td><td>54.00</td><td>0.00</td></tr><tr><td>9.41</td><td>261.00</td><td>1.58</td><td>5.00</td><td>83.00</td><td>23.60</td></tr><tr><td>9.65</td><td>344.00</td><td>1.52</td><td>4.80</td><td>109.00</td><td>41.60</td></tr><tr><td>9.88</td><td>450.00</td><td>1.46</td><td>4.60</td><td>143.00</td><td>66.00</td></tr><tr><td>10.16</td><td>609.00</td><td>1.39</td><td>4.40</td><td>193.00</td><td>100.00</td></tr><tr><td>10.39</td><td>820.00</td><td>1.33</td><td>4.20</td><td>260.00</td><td>154.00</td></tr><tr><td>10.67</td><td>1046.00</td><td>1.26</td><td>4.00</td><td>332.00</td><td>225.00</td></tr><tr><td>10.94</td><td>1394.00</td><td>1.19</td><td>3.80</td><td>443.00</td><td>319.00</td></tr><tr><td>11.18</td><td>1771.00</td><td>1.13</td><td>3.60</td><td>562.00</td><td>443.00</td></tr><tr><td>11.46</td><td>2211.00</td><td>1.06</td><td>3.40</td><td>702.00</td><td>561.00</td></tr><tr><td>11.69</td><td>2730.00</td><td>1.00</td><td>3.19</td><td>867.00</td><td>712.00</td></tr><tr><td>11.89</td><td>3287.00</td><td>0.95</td><td>3.00</td><td>1044.00</td><td>873.00</td></tr><tr><td>12.17</td><td>3856.00</td><td>0.88</td><td>2.80</td><td>1225.00</td><td>1058.00</td></tr><tr><td>12.36</td><td>4384.00</td><td>0.83</td><td>2.60</td><td>1393.00</td><td>1229.00</td></tr></tbody></table>

<table><caption>Cr4</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>ν [cm3/g]</th><th>PI [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.42</td><td>4.13</td><td>13.07</td><td>0.13</td><td></td></tr><tr><td>1.93</td><td>13.00</td><td>3.64</td><td>11.50</td><td>4.13</td><td></td></tr><tr><td>3.94</td><td>25.90</td><td>3.13</td><td>9.90</td><td>8.23</td><td></td></tr><tr><td>5.94</td><td>41.00</td><td>2.62</td><td>8.30</td><td>13.20</td><td></td></tr><tr><td>7.91</td><td>74.00</td><td>2.12</td><td>6.70</td><td>23.60</td><td></td></tr><tr><td>8.94</td><td>112.00</td><td>1.86</td><td>5.90</td><td>35.80</td><td></td></tr><tr><td>9.61</td><td>172.00</td><td>1.69</td><td>5.40</td><td>54.80</td><td>0.00</td></tr><tr><td>9.88</td><td>213.00</td><td>1.62</td><td>5.10</td><td>67.80</td><td>12.15</td></tr><tr><td>10.39</td><td>318.00</td><td>1.49</td><td>4.70</td><td>101.00</td><td>43.47</td></tr><tr><td>10.91</td><td>457.00</td><td>1.36</td><td>4.30</td><td>145.00</td><td>99.00</td></tr><tr><td>11.22</td><td>647.00</td><td>1.28</td><td>4.06</td><td>205.00</td><td>155.00</td></tr><tr><td>11.50</td><td>931.00</td><td>1.21</td><td>3.84</td><td>296.00</td><td>239.00</td></tr><tr><td>12.52</td><td>2631.00</td><td>0.95</td><td>3.00</td><td>836.00</td><td>793.00</td></tr><tr><td>12.83</td><td>3500.00</td><td>0.87</td><td>2.78</td><td>1112.00</td><td>1046.00</td></tr><tr><td>13.19</td><td>4672.00</td><td>0.78</td><td>2.48</td><td>1484.00</td><td>1340.00</td></tr></tbody></table>
Cr5

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>0.00</td><td>4.12</td><td>13.07</td><td>1.00</td><td></td></tr><tr><td>1.93</td><td>17.90</td><td>3.63</td><td>11.52</td><td>5.69</td><td></td></tr><tr><td>3.94</td><td>39.90</td><td>3.12</td><td>9.89</td><td>12.19</td><td></td></tr><tr><td>5.87</td><td>53.20</td><td>2.63</td><td>8.32</td><td>18.48</td><td></td></tr><tr><td>6.85</td><td>83.10</td><td>2.38</td><td>7.53</td><td>26.41</td><td></td></tr><tr><td>7.87</td><td>117.00</td><td>2.12</td><td>6.71</td><td>37.33</td><td></td></tr><tr><td>8.86</td><td>186.00</td><td>1.87</td><td>5.94</td><td>59.10</td><td></td></tr><tr><td>9.61</td><td>305.00</td><td>1.68</td><td>5.32</td><td>97.00</td><td></td></tr><tr><td>10.08</td><td>448.00</td><td>1.56</td><td>4.96</td><td>142.00</td><td></td></tr><tr><td>10.35</td><td>606.00</td><td>1.49</td><td>4.72</td><td>192.00</td><td>0.00</td></tr><tr><td>10.59</td><td>791.00</td><td>1.43</td><td>4.54</td><td>251.00</td><td>1.21</td></tr><tr><td>10.75</td><td>937.00</td><td>1.39</td><td>4.42</td><td>297.00</td><td>4.40</td></tr><tr><td>11.06</td><td>1243.00</td><td>1.31</td><td>4.16</td><td>395.00</td><td>13.79</td></tr><tr><td>11.46</td><td>1854.00</td><td>1.21</td><td>3.83</td><td>589.00</td><td>52.95</td></tr><tr><td>11.85</td><td>2697.00</td><td>1.11</td><td>3.52</td><td>857.00</td><td>192.00</td></tr><tr><td>12.13</td><td>3392.00</td><td>1.04</td><td>3.29</td><td>1078.00</td><td>568.00</td></tr><tr><td>12.48</td><td>4572.00</td><td>0.95</td><td>3.01</td><td>1453.00</td><td>1237.00</td></tr></tbody></table>

Cr7

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>1.99</td><td>4.13</td><td>13.08</td><td>0.63</td><td></td></tr><tr><td>0.80</td><td>18.80</td><td>3.52</td><td>11.15</td><td>5.99</td><td></td></tr><tr><td>1.60</td><td>36.00</td><td>2.91</td><td>9.20</td><td>11.00</td><td></td></tr><tr><td>2.40</td><td>69.00</td><td>2.30</td><td>7.30</td><td>22.00</td><td></td></tr><tr><td>2.80</td><td>108.00</td><td>2.00</td><td>6.30</td><td>34.00</td><td></td></tr><tr><td>3.01</td><td>156.00</td><td>1.84</td><td>5.80</td><td>49.00</td><td>0.00</td></tr><tr><td>3.20</td><td>239.00</td><td>1.69</td><td>5.30</td><td>76.00</td><td>11.00</td></tr><tr><td>3.29</td><td>338.00</td><td>1.62</td><td>5.12</td><td>107.00</td><td>21.00</td></tr><tr><td>3.35</td><td>416.00</td><td>1.58</td><td>5.00</td><td>132.00</td><td>25.00</td></tr><tr><td>3.44</td><td>567.00</td><td>1.51</td><td>4.70</td><td>180.00</td><td>33.00</td></tr><tr><td>3.52</td><td>743.00</td><td>1.45</td><td>4.60</td><td>236.00</td><td>48.00</td></tr><tr><td>3.58</td><td>983.00</td><td>1.40</td><td>4.40</td><td>312.00</td><td>64.00</td></tr><tr><td>3.70</td><td>1298.00</td><td>1.31</td><td>4.10</td><td>412.00</td><td>94.00</td></tr><tr><td>3.82</td><td>1923.00</td><td>1.22</td><td>3.86</td><td>611.00</td><td>155.00</td></tr><tr><td>3.96</td><td>2633.00</td><td>1.11</td><td>3.50</td><td>836.00</td><td>224.00</td></tr><tr><td>4.06</td><td>3588.00</td><td>1.04</td><td>3.30</td><td>1140.00</td><td>329.00</td></tr><tr><td>4.17</td><td>4855.00</td><td>0.95</td><td>3.00</td><td>1542.00</td><td>1338.00</td></tr></tbody></table>

Cr6

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>1.05</td><td>4.13</td><td>13.08</td><td>0.33</td><td></td></tr><tr><td>1.97</td><td>21.29</td><td>3.63</td><td>11.49</td><td>6.76</td><td></td></tr><tr><td>4.02</td><td>42.07</td><td>3.11</td><td>9.86</td><td>13.37</td><td></td></tr><tr><td>5.98</td><td>63.36</td><td>2.61</td><td>8.28</td><td>20.13</td><td></td></tr><tr><td>6.97</td><td>77.94</td><td>2.36</td><td>7.48</td><td>24.77</td><td></td></tr><tr><td>7.99</td><td>104.49</td><td>2.10</td><td>6.67</td><td>33.20</td><td>0.00</td></tr><tr><td>8.98</td><td>141.62</td><td>1.85</td><td>5.87</td><td>45.00</td><td>1.39</td></tr><tr><td>10.00</td><td>225.13</td><td>1.59</td><td>5.04</td><td>71.54</td><td>5.28</td></tr><tr><td>10.43</td><td>299.62</td><td>1.48</td><td>4.70</td><td>95.22</td><td>27.61</td></tr><tr><td>10.67</td><td>369.91</td><td>1.42</td><td>4.48</td><td>117.55</td><td>44.44</td></tr><tr><td>11.02</td><td>484.26</td><td>1.33</td><td>4.22</td><td>153.89</td><td>68.18</td></tr><tr><td>11.34</td><td>665.96</td><td>1.25</td><td>3.98</td><td>211.64</td><td>104.15</td></tr><tr><td>11.69</td><td>978.80</td><td>1.16</td><td>3.69</td><td>311.06</td><td>173.07</td></tr><tr><td>11.97</td><td>1251.03</td><td>1.09</td><td>3.47</td><td>397.57</td><td>251.73</td></tr><tr><td>12.32</td><td>1850.37</td><td>1.00</td><td>3.19</td><td>588.04</td><td>410.83</td></tr><tr><td>12.64</td><td>2599.54</td><td>0.92</td><td>2.91</td><td>826.13</td><td>599.70</td></tr><tr><td>12.95</td><td>3380.89</td><td>0.84</td><td>2.67</td><td>1074.76</td><td>845.97</td></tr><tr><td>13.39</td><td>4704.99</td><td>0.73</td><td>2.33</td><td>1495.25</td><td>1308.93</td></tr></tbody></table>

Cr8

<table><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>1.05</td><td>4.20</td><td>13.28</td><td>0.33</td><td></td></tr><tr><td>0.20</td><td>7.03</td><td>3.70</td><td>11.73</td><td>2.23</td><td></td></tr><tr><td>0.39</td><td>19.82</td><td>3.20</td><td>10.13</td><td>6.30</td><td></td></tr><tr><td>0.59</td><td>54.22</td><td>2.70</td><td>8.51</td><td>17.23</td><td>0.00</td></tr><tr><td>0.69</td><td>99.32</td><td>2.44</td><td>7.59</td><td>31.56</td><td>0.02</td></tr><tr><td>0.75</td><td>155.44</td><td>2.30</td><td>7.29</td><td>49.40</td><td>13.21</td></tr><tr><td>0.79</td><td>267.04</td><td>2.19</td><td>6.94</td><td>84.86</td><td>47.09</td></tr><tr><td>0.82</td><td>384.51</td><td>2.12</td><td>6.72</td><td>122.20</td><td>73.65</td></tr><tr><td>0.83</td><td>521.08</td><td>2.08</td><td>6.58</td><td>165.60</td><td>121.25</td></tr><tr><td>0.85</td><td>671.80</td><td>2.03</td><td>6.44</td><td>213.50</td><td>157.45</td></tr><tr><td>0.88</td><td>940.52</td><td>1.97</td><td>6.24</td><td>298.90</td><td>230.54</td></tr><tr><td>0.92</td><td>1473.58</td><td>1.86</td><td>5.90</td><td>468.30</td><td>298.31</td></tr><tr><td>0.95</td><td>1993.55</td><td>1.79</td><td>5.67</td><td>633.50</td><td>571.25</td></tr><tr><td>0.97</td><td>2562.39</td><td>1.73</td><td>5.48</td><td>814.30</td><td>742.58</td></tr><tr><td>1.00</td><td>3377.18</td><td>1.66</td><td>5.25</td><td>1073.20</td><td>949.04</td></tr><tr><td>1.04</td><td>4715.06</td><td>1.56</td><td>4.94</td><td>1498.40</td><td>1342.52</td></tr></tbody></table>
Produit RDAP (pommes red delicious) Produit GDAP (pommes golden delicious)
Données et Résultats Bruts

<table><caption>RDAP01inch min-T25 piston</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>5.62</td><td>3.57</td><td>17.94</td><td>56.67</td><td></td></tr><tr><td>0.28</td><td>21.90</td><td>3.50</td><td>17.57</td><td>59.14</td><td></td></tr><tr><td>0.60</td><td>33.00</td><td>3.42</td><td>17.16</td><td>59.14</td><td></td></tr><tr><td>1.00</td><td>37.00</td><td>3.32</td><td>16.64</td><td>64.07</td><td></td></tr><tr><td>1.29</td><td>48.00</td><td>3.24</td><td>16.28</td><td>66.53</td><td></td></tr><tr><td>1.57</td><td>48.00</td><td>3.17</td><td>15.93</td><td>71.46</td><td></td></tr><tr><td>2.00</td><td>48.00</td><td>3.06</td><td>15.37</td><td>78.85</td><td></td></tr><tr><td>2.39</td><td>48.00</td><td>2.96</td><td>14.88</td><td>86.24</td><td></td></tr><tr><td>2.71</td><td>80.00</td><td>2.88</td><td>14.47</td><td>96.10</td><td></td></tr><tr><td>3.18</td><td>103.00</td><td>2.76</td><td>13.87</td><td>113.35</td><td></td></tr><tr><td>3.63</td><td>142.00</td><td>2.65</td><td>13.30</td><td>133.06</td><td></td></tr><tr><td>4.03</td><td>222.00</td><td>2.55</td><td>12.78</td><td>179.28</td><td>0.00</td></tr><tr><td>4.50</td><td>269.00</td><td>2.43</td><td>12.19</td><td>191.49</td><td>21.11</td></tr><tr><td>4.80</td><td>321.00</td><td>2.35</td><td>11.80</td><td>211.54</td><td>50.84</td></tr><tr><td>5.09</td><td>376.00</td><td>2.28</td><td>11.43</td><td>239.44</td><td>88.42</td></tr><tr><td>5.35</td><td>465.00</td><td>2.21</td><td>11.10</td><td>271.58</td><td>128.66</td></tr><tr><td>5.56</td><td>574.00</td><td>2.16</td><td>10.83</td><td>302.78</td><td>165.72</td></tr><tr><td>5.75</td><td>731.00</td><td>2.11</td><td>10.59</td><td>335.66</td><td>203.22</td></tr><tr><td>5.96</td><td>920.00</td><td>2.06</td><td>10.32</td><td>375.70</td><td>247.26</td></tr><tr><td>6.16</td><td>1200.00</td><td>2.01</td><td>10.07</td><td>416.75</td><td>290.93</td></tr><tr><td>6.38</td><td>1200.00</td><td>1.95</td><td>9.79</td><td>469.61</td><td>345.43</td></tr><tr><td>6.57</td><td>1200.00</td><td>1.90</td><td>9.54</td><td>519.19</td><td>395.08</td></tr><tr><td>6.82</td><td>1200.00</td><td>1.84</td><td>9.23</td><td>586.71</td><td>460.83</td></tr><tr><td>6.98</td><td>1200.00</td><td>1.80</td><td>9.03</td><td>635.15</td><td>506.89</td></tr><tr><td>7.15</td><td>1200.00</td><td>1.75</td><td>8.80</td><td>691.62</td><td>559.58</td></tr><tr><td>7.31</td><td>1200.00</td><td>1.71</td><td>8.60</td><td>745.83</td><td>609.25</td></tr><tr><td>7.40</td><td>3045.00</td><td>1.69</td><td>8.49</td><td>773.98</td><td>634.73</td></tr><tr><td>7.51</td><td>3606.00</td><td>1.66</td><td>8.35</td><td>814.53</td><td>671.07</td></tr></tbody></table>

<table><caption>GDAP01inch min-T25 piston</caption><thead><tr><th>time [min]</th><th>P [lb]</th><th>Z0 [cm]</th><th>v [cm3/g]</th><th>P1 [kPa]</th><th>PFI [kPa]</th></tr></thead><tbody><tr><td>0.00</td><td>5.62</td><td>3.46</td><td>17.82</td><td>142.92</td><td></td></tr><tr><td>0.20</td><td>21.90</td><td>3.41</td><td>17.53</td><td>150.31</td><td></td></tr><tr><td>0.43</td><td>33.00</td><td>3.35</td><td>17.22</td><td>160.16</td><td></td></tr><tr><td>0.72</td><td>37.00</td><td>3.28</td><td>16.85</td><td>177.41</td><td></td></tr><tr><td>1.08</td><td>48.00</td><td>3.19</td><td>16.38</td><td>197.13</td><td></td></tr><tr><td>1.41</td><td>62.00</td><td>3.10</td><td>15.95</td><td>214.37</td><td></td></tr><tr><td>1.71</td><td>80.00</td><td>3.03</td><td>15.56</td><td>239.01</td><td></td></tr><tr><td>2.01</td><td>103.00</td><td>2.95</td><td>15.17</td><td>261.19</td><td></td></tr><tr><td>2.32</td><td>142.00</td><td>2.87</td><td>14.76</td><td>293.22</td><td>0.00</td></tr><tr><td>2.60</td><td>222.00</td><td>2.80</td><td>14.39</td><td>325.11</td><td>3.26</td></tr><tr><td>2.92</td><td>269.00</td><td>2.72</td><td>13.98</td><td>358.91</td><td>18.62</td></tr><tr><td>3.22</td><td>321.00</td><td>2.64</td><td>13.59</td><td>401.49</td><td>52.07</td></tr><tr><td>3.50</td><td>376.00</td><td>2.57</td><td>13.22</td><td>452.36</td><td>101.41</td></tr><tr><td>3.80</td><td>465.00</td><td>2.50</td><td>12.83</td><td>518.35</td><td>172.62</td></tr><tr><td>4.08</td><td>574.00</td><td>2.42</td><td>12.46</td><td>593.57</td><td>258.55</td></tr><tr><td>4.29</td><td>731.00</td><td>2.37</td><td>12.19</td><td>656.17</td><td>331.95</td></tr><tr><td>4.46</td><td>920.00</td><td>2.33</td><td>11.96</td><td>714.86</td><td>401.55</td></tr><tr><td>4.65</td><td>1200.00</td><td>2.28</td><td>11.72</td><td>785.14</td><td>485.35</td></tr><tr><td>4.82</td><td>1564.00</td><td>2.24</td><td>11.49</td><td>855.52</td><td>569.42</td></tr><tr><td>5.01</td><td>2051.00</td><td>2.19</td><td>11.24</td><td>939.09</td><td>669.14</td></tr><tr><td>5.17</td><td>2488.00</td><td>2.15</td><td>11.04</td><td>1014.36</td><td>758.68</td></tr><tr><td>5.29</td><td>3045.00</td><td>2.12</td><td>10.87</td><td>1078.36</td><td>834.50</td></tr><tr><td>5.34</td><td>3606.00</td><td>2.10</td><td>10.81</td><td>1103.23</td><td>863.90</td></tr></tbody></table>
Ref. : 3.1 CDC DANS LE CAS DE LA VARIATION LINEAIRE DU RAPPORT DE VIDE – SPM NON-LINEAIRE (CDC1)

$$ \frac{K}{\gamma_f} \frac{\partial^2 P_f}{\partial z^2} = \frac{1}{1 + \left( -\left\{ \frac{\rho_s dZ/dt}{W} \right\} t \left[ 1 - \frac{bW}{\rho_s dZ/dt} \left( \frac{z-0.5Z_0}{Z_0} \right) \right] + e_0 \right)^x} \quad \text{Eq. A3.1} $$ 
$$ \frac{\partial}{\partial t} \left( -\left\{ \frac{\rho_s dZ/dt}{W} \right\} t \left[ 1 - \frac{bW}{\rho_s dZ/dt} \left( \frac{z-0.5Z_0}{Z_0} \right) \right] + e_0 \right) $$

Une solution complète de cette équation, (Eq. 3.6), prend la forme :

$$ P_f = \frac{\gamma_f c}{K} \left\{ z Z_0 \left[ \frac{1+e_0-bt}{ct(bt)} \right] + \frac{z^2}{2ct} - \left[ \frac{Z_0(1+e_0)}{ct(bt)} \right] \cdot \left[ \frac{\frac{Z_0(1+e_0)}{bt} \ln(1+e_0) - z \ln\left(1+e_0-ct\left(1-\frac{1}{2}\frac{b}{c}\right)\right)}{\frac{Z_0(1+e_0-ct\left(1+\frac{1}{2}\frac{b}{c}\right))}{bt} \ln\left(1+e_0-ct\left(1+\frac{1}{2}\frac{b}{c}\right)\right)} \right] \right\} \quad \text{Eq. A3.2} $$

$$ \text{dans laquelle } c = \frac{\rho_s dZ/dt}{W} \qquad \text{Eq. A3.3} $$

Ref. : 3.2 CDC DANS LE CAS DES LARGES DEFORMATIONS – SPM LINEAIRE OU NON (CDC2)

Here is an example of the type of finite difference analysis used and the convergence criteria calculation.

Let's take the basic differential equation:

$$ \frac{\partial P_f}{\partial t} = C_v \cdot \frac{\partial^2 P_f}{\partial z^2} \qquad \text{Eq. A3.4} $$

$P_{fR}$, $t_R$, $z_R$ are arbitrary references for the fluid pressure, the time and the position within the SPM, therewith:

$$ \text{The nondimensional fluid pressure: } \bar{P}_f = \frac{P_f}{P_{fR}} \qquad \text{Eq. A3.5} $$

$$ \text{The nondimensional time: } \bar{t} = \frac{t}{t_R} \qquad \text{Eq. A3.6} $$

$$ \text{And the nondimensional position: } \bar{z} = \frac{z}{z_R} \qquad \text{Eq. A3.7} $$

With the proper substitution of Eqs. A3.5 and A3.6 in the left-hand side of Eq. A3.4
$$ \frac{\partial P_f}{\partial t} = \frac{P_{fR}}{t_R} \frac{\partial \bar{P}_f}{\partial \bar{t}} \qquad \text{Eq. A3.8} $$

With the proper substitution of Eqs. A3.5 and A3.7, in the right-hand side of Eq. A3.4

$$ C_v \frac{\partial^2 P_f}{\partial z^2} = C_v \frac{P_{fR}}{z_R^2} \frac{\partial^2 \bar{P}_f}{\partial \bar{z}^2} \qquad \text{Eq. A3.9} $$

Equalling Eqs. A3.8 and A3.9,

$$ \begin{gathered} \frac{P_{fR}}{t_R} \frac{\partial \bar{P}_f}{\partial \bar{t}} = \frac{C_v \cdot P_{fR}}{z_R^2} \cdot \frac{\partial^2 \bar{P}_f}{\partial \bar{z}^2} \\ \text{or} \quad \frac{1}{t_R} \frac{\partial \bar{P}_f}{\partial \bar{t}} = \frac{C_v}{z_R^2} \cdot \frac{\partial^2 \bar{P}_f}{\partial z^2} \end{gathered} \qquad \text{Eq. A3.10} $$

Adopting $t_R = \frac{z_R^2}{C_v}$ then Eq. A3.10 will be of the form

$$ \frac{\partial \bar{P}_f}{\partial \bar{t}} = \frac{\partial^2 \bar{P}_f}{\partial z^2} \qquad \text{Eq. A3.11} $$

Using Taylor's series:

with $\frac{\Delta t}{\Delta \bar{t}} = t_R = \frac{z_R^2}{C_v}$ so, $\Delta \bar{t} = \frac{C_v \Delta t}{z_R^2}$ then $\frac{\partial \bar{P}_f}{\partial \bar{t}}$ becomes

$$ \begin{gathered} \bar{P}_{f0,\bar{i}+\Delta\bar{i}} = \bar{P}_{f0,\bar{i}} + \Delta\bar{t} \cdot \left( \frac{\partial \bar{P}_f}{\partial \bar{t}} \right)_0 \\ \frac{\partial \bar{P}_f}{\partial \bar{t}} = \frac{1}{\Delta \bar{t}} \left( \bar{P}_{f0,\bar{i}+\Delta \bar{i}} - \bar{P}_{f0,\bar{i}} \right) \end{gathered} \qquad \text{Eq. A3.12} $$

<figure><img src="image_209.png" /></figure>

$\bar{P}_{f0,\bar{i}}$ being the fluid pressure at any position.

Similarly for the right-hand side of Eq. A3.11:

$$ \bar{P}_{f1,T_v} = \bar{P}_{f0+\Delta\bar{z},T_v} $$

$$ \bar{P}_{f3,T_v} = \bar{P}_{f0-\Delta\bar{z},T_v} $$

<figure><img src="image_210.png" /></figure>
$$
\begin{aligned}
\bar{P}_{f,1,\bar{i}} = \bar{P}_{f,0,\bar{i}} &+ \Delta \bar{z} \left( \frac{\partial \bar{P}_f}{\partial \bar{z}} \right)_0 + \frac{(\Delta \bar{z})^2}{2!} \left( \frac{\partial^2 \bar{P}_f}{\partial \bar{z}^2} \right)_0 + \frac{(\Delta \bar{z})^3}{3!} \left( \frac{\partial^3 \bar{P}_f}{\partial \bar{z}^3} \right)_0 + \frac{(\Delta \bar{z})^4}{4!} \left( \frac{\partial^4 \bar{P}_f}{\partial \bar{z}^4} \right)_0 + \dots \\
& + \frac{(\Delta \bar{z})^{(n-1)}}{(n-1)!} \left( \frac{\partial^{(n-1)} \bar{P}_f}{\partial \bar{z}^{(n-1)}} \right)_0 + R_n
\end{aligned}
$$

Eq. A3.13

$R_n$ being the residual part

$$
\begin{aligned}
\bar{P}_{f,3,\bar{i}} = \bar{P}_{f,0,\bar{i}} &- \Delta \bar{z} \left( \frac{\partial \bar{P}_f}{\partial \bar{z}} \right)_0 + \frac{(\Delta \bar{z})^2}{2!} \left( \frac{\partial^2 \bar{P}_f}{\partial \bar{z}^2} \right)_0 - \frac{(\Delta \bar{z})^3}{3!} \left( \frac{\partial^3 \bar{P}_f}{\partial \bar{z}^3} \right)_0 + \frac{(\Delta \bar{z})^4}{4!} \left( \frac{\partial^4 \bar{P}_f}{\partial \bar{z}^4} \right)_0 + \dots \\
& + \frac{(-\Delta \bar{z})^{(n-1)}}{(n-1)!} \left( \frac{\partial^{(n-1)} \bar{P}_f}{\partial \bar{z}^{(n-1)}} \right)_0 + R_n
\end{aligned}
$$

Eq. A3.14

Adding these last two equations for $\bar{P}_{f,1,\bar{i}}$ and $\bar{P}_{f,3,\bar{i}}$ :

$$
\begin{aligned}
\bar{P}_{f,1,\bar{i}} + \bar{P}_{f,3,\bar{i}} = 2\bar{P}_{f,0,\bar{i}} &+ 0 + \frac{2(\Delta \bar{z})^2}{2!} \left( \frac{\partial^2 \bar{P}_f}{\partial \bar{z}^2} \right)_0 + 0 + \frac{2(\Delta \bar{z})^4}{4!} \left( \frac{\partial^4 \bar{P}_f}{\partial \bar{z}^4} \right)_0 + \dots \\
& + \frac{2(\Delta \bar{z})^{(n-1)}}{(n-1)!} \left( \frac{\partial^{(n-1)} \bar{P}_f}{\partial \bar{z}^{(n-1)}} \right)_0 + 2R_n
\end{aligned}
$$

Eq. A3.15

$\Delta \bar{z}$ being small enough, the third and subsequent terms can be neglected.
Then Eq. 3.12 becomes Eq. 3.13.

$$
\bar{P}_{f,1,\bar{i}} + \bar{P}_{f,3,\bar{i}} = 2\bar{P}_{f,0,\bar{i}} + (\Delta \bar{z})^2 \cdot \left( \frac{\partial^2 \bar{P}_f}{\partial \bar{z}^2} \right)_0 \qquad \text{Eq. A3.16}
$$

therefore

$$
\frac{\partial^2 \bar{P}_f}{\partial z^{-2}} = \frac{1}{(\Delta \bar{z})^2} \left( \bar{P}_{f,1,\bar{i}} + \bar{P}_{f,3,\bar{i}} - 2 \bar{P}_{f,0,\bar{i}} \right) \qquad \text{Eq. A3.17}
$$

Eqs. A3.12 and A3.17 can be equated to yield:
$$ \frac{1}{\Delta \bar{t}} \left( \bar{P}_{f \ 0, \bar{t}+\Delta \bar{t}} - \bar{P}_{f \ 0, \bar{t}} \right) = \frac{1}{(\Delta \bar{z})^2} \left( \bar{P}_{f \ 1, \bar{t}} + \bar{P}_{f \ 3, \bar{t}} - 2 \bar{P}_{f \ 0, \bar{t}} \right) $$

$$ \text{or } \bar{P}_{f \ 0, \bar{t}+\Delta \bar{t}} = \frac{\Delta \bar{t}}{(\Delta \bar{z})^2} \left( \bar{P}_{f \ 1, \bar{t}} + \bar{P}_{f \ 3, \bar{t}} - 2 \bar{P}_{f \ 0, \bar{t}} \right) + \bar{P}_{f \ 0, \bar{t}} \qquad \text{Eq. A3.18} $$

Now one needs to determine the interval for $\frac{\Delta \bar{t}}{(\Delta \bar{z})^2}$ which provides suitable convergence, similarly to CRANK(1973) diffusion problem.

Since: $\bar{P}_{f \ 1, \bar{t}} \in [0, \bar{P}_{f \ 0, \bar{t}}]$, $\bar{P}_{f \ 3, \bar{t}} \in [0, \bar{P}_{f \ 0, \bar{t}}]$ then $(\bar{P}_{f \ 1, \bar{t}} + \bar{P}_{f \ 3, \bar{t}} - 2\bar{P}_{f \ 0, \bar{t}}) \in [-2\bar{P}_{f \ 0, \bar{t}}, 0]$ At the lower boundary limit $\bar{P}_{f \ 1, \bar{t}} = \bar{P}_{f \ 3, \bar{t}} = 0$, Eq. A3.18 becomes

$$ \bar{P}_{f \ 0, \bar{t}+\Delta \bar{t}} = \frac{\Delta \bar{t}}{(\Delta \bar{z})^2} (-2\bar{P}_{f \ 0, \bar{t}}) + \bar{P}_{f \ 0, \bar{t}} \qquad \text{Eq. A3.19} $$

$$ \text{provided } \bar{P}_{f \ 0, \bar{t}+\Delta \bar{t}} \le \bar{P}_{f \ 0, \bar{t}} \ge 0 \qquad \text{Eq. A3.20} $$

(let's be reminded that the fluid pressure reduces while the solid pressure increases)

$$ \bar{P}_{f \ 0, \bar{t}+\Delta \bar{t}} = \frac{\Delta \bar{t}}{(\Delta \bar{z})^2} (-2\bar{P}_{f \ 0, \bar{t}}) + \bar{P}_{f \ 0, \bar{t}} \ge 0 $$

$$ \bar{P}_{f \ 0, \bar{t}+\Delta \bar{t}} = \bar{P}_{f \ 0, \bar{t}} \cdot \left\{ 1 - \frac{2\Delta \bar{t}}{(\Delta \bar{z})^2} \right\} \ge 0 \qquad \text{Eq. A3.21} $$

$$ \text{so } \left\{ 1 - \frac{2\Delta \bar{t}}{(\Delta \bar{z})^2} \right\} \ge 0 \text{ which is the convergence condition} \qquad \text{Eq. A3.22} $$

$$ 1 \ge \frac{2\Delta \bar{t}}{(\Delta \bar{z})^2} \Rightarrow \frac{\Delta \bar{t}}{(\Delta \bar{z})^2} \le \frac{1}{2} \text{ therefore for Eq. A3.18 to converge, } \Delta \bar{t} \text{ and } \Delta \bar{z} \text{ must be} $$

chosen in such a way that the ratio $\frac{\Delta \bar{t}}{(\Delta \bar{z})^2}$ is less than 0.5.
At the piston or the outflow surface point 3 will be chosen as the mirror image of point 1; thus $\bar{P}_{f,1,\bar{i}} = \bar{P}_{f,3,\bar{i}}$. So Eq. A3.18 becomes:

$$ \bar{P}_{f,0,\bar{i}+\Delta\bar{i}} = \frac{\Delta\bar{t}}{(\Delta\bar{z})^2} (2\bar{P}_{f,1,\bar{i}} - 2\bar{P}_{f,0,\bar{i}}) + \bar{P}_{f,0,\bar{i}} \qquad \text{Eq. A3.23} $$

The illustrations of the finite difference calculation previously described are given Fig. A3.1 and Fig. A3.2.

<figure><img src="image_211.png" /><figcaption>Figure A3.1 : Finite difference calculation grid</figcaption></figure>
### Example

In this example we assume that both the base and the piston are porous allowing for drainage at the two boundaries, which yields a symmetrical distribution of the pressure as per the centre of the SPM. It is also supposed that the pressure is initially uniformly distributed along the SPM height. This case does not correspond to a real situation but has only been selected to illustrate as simply as possible the calculation method chosen in the corresponding section.

Further assuming that the condition $\frac{\Delta \bar{t}}{(\Delta \bar{z})^2} < 0.5$ is satisfied by taking

$$ \frac{\Delta \bar{t}}{(\Delta \bar{z})^2} = 0.325 \text{ for an SPM total height of 8 cm.} $$

$\bar{P}_f \text{ At } t=0:$

$$ \bullet \text{ At } \bar{z} = 0 $$

$$ \bar{P}_{f0,\bar{i}+\Delta\bar{i}} = 0 $$

$$ \bullet \text{ At } \bar{z} = 0.25 $$

$$ \bar{P}_{f0,\bar{i}+\Delta\bar{i}} = 100 $$

$$ \bullet \text{ At } \bar{z} = 0.5 $$

$$ \bar{P}_{f0,\bar{i}+\Delta\bar{i}} = 100 $$

$$ \bullet \text{ At } \bar{z} = 0.75 $$

$$ \bar{P}_{f0,\bar{i}+\Delta\bar{i}} = 100 $$

$$ \bullet \text{ At } \bar{z} = 1 $$

$$ \bar{P}_{f0,\bar{i}+\Delta\bar{i}} = 0 $$

then

$\bar{P}_f \text{ At } t=5 \text{ min:}$

$$ \bullet \text{ At } \bar{z} = 0 $$

$$ \bar{P}_{f0,\bar{i}+\Delta\bar{i}} = 0 $$

$$ \bullet \text{ At } \bar{z} = 0.25 $$

$$ \bar{P}_{f0,\bar{i}+\Delta\bar{i}} = \frac{\Delta \bar{t}}{(\Delta \bar{z})^2} \left( \bar{P}_{f1,\bar{i}} + \bar{P}_{f3,\bar{i}} - 2\bar{P}_{f0,\bar{i}} \right) + \bar{P}_{f0,\bar{i}} \qquad \text{Eq. A3.18} $$
$$ \bar{P}_{f, 0, \bar{i}+\Delta \bar{i}}=0.325(0+100-2(100))+100=67.5 $$

* At $\bar{z} = 0.5$ still referring to Eq. A3.18.

$$ \bar{P}_{f, 0, \bar{i}+\Delta \bar{i}}=0.325(100+100-2(100))+100=100 $$

* At $\bar{z} = 0.75$ still referring to Eq. A3.18.

$$ \bar{P}_{f, 0, \bar{i}+\Delta \bar{i}}=0.325(100+0-2(100))+100=67.5 $$

* At $\bar{z} = 1$

$$ \bar{P}_{f, 0, \bar{i}+\Delta \bar{i}}=0 $$

and further on for $t = 10 \text{ min}$

The corresponding data and graphic representation of $\bar{P}_f = f(\bar{z}, t)$:

<figure><img src="image_212.png" /><figcaption>Figure A3.2 : Numerical calculation and graphical representation $\bar{P}_f = f(\bar{z}, t)$</figcaption></figure>
ANNEXE A4

LA COMPRESSION AXIALE A ECOULEMENT RADIAL (CAR) ET LE
PRESSAGE PAR VIS

Réf. : 4.1.3 Application au pressage à vis

For radial drainage the basic differential equation can be written as:

$$
\frac{\partial P_f}{\partial t} = C_r \cdot \left( \frac{\partial^2 P_f}{\partial r^2} + \frac{1}{r} \frac{\partial P_f}{\partial r} \right) \qquad \text{Eq. A4.1}
$$

To solve the equation P<sub>f</sub> is assumed to be the product of 2 independent functions,
CRANK (1973):

$$
P_f = F(r)G(t) \tag{Eq. A4.2}
$$

$$
\text{So, } \frac{\partial P_f}{\partial t} = F(r) \frac{\partial}{\partial t} G(t) = F(r) G'(t)
$$

$$
\frac{\partial P_f}{\partial r} = \frac{\partial}{\partial r} F(r)G(t) = F'(r)G(t)
$$

And

$$
\frac{\partial^2 P_f}{\partial r^2} = \frac{\partial^2}{\partial r^2} F(r)G(t) = F''(r)G(t)
$$

Therefore

$$
\begin{gather*}
F(r)G'(t) = C_r \left( F''(r)G(t) + \frac{1}{r} \cdot F'(r)G(t) \right) = C_r \cdot G(t) \left( F''(r) + \frac{1}{r} \cdot F'(r) \right) \\
\text{or } \frac{\left( F''(r) + \frac{1}{r} \cdot F'(r) \right)}{F(r)} = \frac{G'(t)}{C_v G(t)} \qquad \text{Eq. A4.3}
\end{gather*}
$$

Independent right-hand side left-hand side can only be equal to a constant, -b²,
generally called separate constant, therefore:

$$
\frac{\left( F''(r) + \frac{1}{r} \cdot F'(r) \right)}{F(r)} = -b^2, \text{ so,}
$$

$$
r^2 F''(r) + r \cdot F'(r) + b^2 r^2 \cdot F(r) = 0 \qquad \text{Eq. A4.4}
$$

Eq. A4.4 is analogue to the Bessel function, then can be written as follows:

Which solution, according to standard mathematical tables, is given by:

$$
F(r) = r^{-p} \left( c_1 J_{q/r} \left( \frac{\alpha}{s} r^s \right) + c_2 Y_{q/r} \left( \frac{\alpha}{s} r^s \right) \right) \quad \text{where} \quad q = \sqrt{p^2 - \beta^2} \qquad \text{Eq. A4.5}
$$

then

$$
F(r) = 1 \cdot \left( c_1 J_0 \left( \frac{b}{1} r^1 \right) + c_2 Y_0 \left( \frac{b}{1} r^1 \right) \right) = (c_1 J_0 (br) + c_2 Y_0 (br)) \quad \text{Eq. A4.6}
$$

with J<sub>n</sub>(r) Bessel function of first kind of n order
$$J_n(r) = \frac{r^n}{2^n \Gamma(n+1)} \left\{ 1 - \frac{r^2}{2(2n+2)} + \frac{r^4}{2 \cdot 4(2n+2)(2n+4)} - \dots \right\} = \sum_{k=0}^{\infty} \frac{(-1)^k (r/2)^{n+2k}}{k! \Gamma(n+k+1)}$$

$$J_0(br) = \frac{1}{1 \cdot \Gamma(0+1)} \left\{ 1 - \frac{(br)^2}{2(2 \cdot 0 + 2)} + \frac{(br)^4}{2 \cdot 4(2 \cdot 0 + 2)(2 \cdot 0 + 4)} - \dots \right\} = \sum_{k=0}^{\infty} \frac{(-1)^k (br/2)^{2k}}{k! \Gamma(k+1)} \\ = \frac{1}{1 \cdot \Gamma(1)} \left\{ 1 - \frac{(br)^2}{2(2 \cdot 0 + 2)} + \frac{(br)^4}{2 \cdot 4(2 \cdot 0 + 2)(2 \cdot 0 + 4)} - \dots \right\}$$

and gamma function $\Gamma(n) = \int_0^\infty r^{n-1} e^{-r} dr$

Then $J_0(br)$ becomes

$$J_0(br) = \sum_{k=0}^{\infty} \frac{(-1)^k (br/2)^{2k}}{(k!)^2} = \sum_{k=0}^{\infty} \frac{(-1)^k (br)^{2k}}{(k!)^2 (2)^{2k}}$$

and $Y_n(r)$ Bessel function of second kind of n order

$$Y_n(r) = \frac{2}{\pi} \left\{ \ln(r/2) + \gamma \right\} J_n(r) - \frac{1}{\pi} \sum_{k=0}^{n-1} \frac{(n-k-1)!}{k!} (r/2)^{2k-n} - \frac{1}{\pi} \sum_{k=0}^{\infty} (-1)^k \left\{ \phi(k) + \phi(n+k) \right\} \frac{(r/2)^{2k+n}}{k!(n+k)}$$

$\gamma = 0.5772156...$ Euler constant

$$Y_0(br) = \frac{2}{\pi} \left\{ \ln(br/2) + \gamma \right\} J_0(br) - \frac{1}{\pi} \sum_{k=0}^{-1} \frac{(k-1)!}{k!} (br/2)^{2k} - \frac{1}{\pi} \sum_{k=0}^{\infty} (-1)^k \{2 \cdot \phi(k)\} \frac{(br/2)^{2k}}{(k!)^2}$$

$$Y_0(br) = \frac{2}{\pi} \left\{ \ln(br/2) + \gamma \right\} J_0(br) + \frac{2}{\pi} \left( \frac{(br)^2}{2^2} - \frac{(br)^4}{2^2 4^2} \left(1 + \frac{1}{2}\right) + \frac{(br)^6}{2^2 4^2 6^2} \left(1 + \frac{1}{2} + \frac{1}{3}\right) - \dots \right)$$

Substitute $J_0(r)$ and $Y_0(r)$ in Eq. A4.6 and finally:

Eq. A4.7

$$F(r) = c_1 \sum_{k=0}^{\infty} \frac{(-1)^k (br)^{2k}}{(k!)^2 (2)^{2k}} + c_2 \left[ \frac{2}{\pi} \left\{ \ln(br/2) + \gamma \right\} J_0(br) + \frac{2}{\pi} \left( \frac{(br)^2}{2^2} - \frac{(br)^4}{2^2 4^2} \left(1 + \frac{1}{2}\right) + \frac{(br)^6}{2^2 4^2 6^2} \left(1 + \frac{1}{2} + \frac{1}{3}\right) - \dots \right) \right]$$

$$F(r) = c_1 \sum_{k=0}^{\infty} \frac{(-1)^k (br)^{2k}}{(k!)^2 (2)^{2k}} + c_2 \left[ \frac{2}{\pi} \left\{ \ln(br/2) + \gamma \right\} \sum_{k=0}^{\infty} \frac{(-1)^k (br)^{2k}}{(k!)^2 (2)^{2k}} + \frac{2}{\pi} \left( \frac{(br)^2}{2^2} - \frac{(br)^4}{2^2 4^2} \left(1 + \frac{1}{2}\right) + \frac{(br)^6}{2^2 4^2 6^2} \left(1 + \frac{1}{2} + \frac{1}{3}\right) - \dots \right) \right]$$

The right-hand side of Eq. A4.3 can be written as.

$$\frac{\left( F''(r) + \frac{1}{r} \cdot F'(r) \right)}{F(r)} = \frac{G'(t)}{C_r G(t)}$$

$$G'(t) = -b^2 C_r G(t) \qquad \text{Eq. A4.8}$$

$$G(t) = A_3 \cdot e^{-b^2 C_r t} \qquad \text{Eq. A4.9}$$

where $A_3$ is a constant.

Since $P_f = F(r)G(t)$ then combining Eqs. A4.7 and A4.9 provides:
$$P_f = (c_1 \sum_{k=0}^{\infty} \frac{(-1)^k (br)^{2k}}{(k!)^2 (2)^{2k}} + c_2 \left[ \frac{2}{\pi} \left\{ \ln(br/2) + \gamma \right\} \sum_{k=0}^{\infty} \frac{(-1)^k (br)^{2k}}{(k!)^2 (2)^{2k}} + \frac{2}{\pi} \left( \frac{(br)^2}{2^2} - \frac{(br)^4}{2^2 4^2} \left(1+\frac{1}{2}\right) + \frac{(br)^6}{2^2 4^2 6^2} \left(1+\frac{1}{2}+\frac{1}{3}\right) - \dots \right) \right]) \cdot A_3 \cdot e^{-b^2 C_r}$$

$$A_4 = c_1 A_3 \text{ and } A_5 = c_2 A_3$$

Eq. A4.10

$$P_f = (A_4 \sum_{k=0}^{\infty} \frac{(-1)^k (br)^{2k}}{(k!)^2 (2)^{2k}} + A_5 \left[ \frac{2}{\pi} \left\{ \ln(br/2) + \gamma \right\} \sum_{k=0}^{\infty} \frac{(-1)^k (br)^{2k}}{(k!)^2 (2)^{2k}} + \frac{2}{\pi} \left( \frac{(br)^2}{2^2} - \frac{(br)^4}{2^2 4^2} \left(1+\frac{1}{2}\right) + \frac{(br)^6}{2^2 4^2 6^2} \left(1+\frac{1}{2}+\frac{1}{3}\right) - \dots \right) \right]) \cdot e^{-b^2 C_r}$$

The constants can be evaluated from the boundary conditions.

Selecting the separate constant $b = \frac{\alpha}{r_w}$, where $r_w$ is the intern radius, gives:

$$P_{f,r,t} = (A_4 \sum_{k=0}^{\infty} \frac{(-1)^k \left(\frac{\alpha}{r_w} r\right)^{2k}}{(k!)^2 (2)^{2k}} + A_5 \left[ \frac{2}{\pi} \left\{ \ln\left(\frac{\alpha}{r_w} r/2\right) + \gamma \right\} \sum_{k=0}^{\infty} \frac{(-1)^k \left(\frac{\alpha}{r_w} r\right)^{2k}}{(k!)^2 (2)^{2k}} + \frac{2}{\pi} \left( \frac{\left(\frac{\alpha}{r_w} r\right)^2}{2^2} - \frac{\left(\frac{\alpha}{r_w} r\right)^4}{2^2 4^2} \left(1+\frac{1}{2}\right) + \frac{\left(\frac{\alpha}{r_w} r\right)^6}{2^2 4^2 6^2} \left(1+\frac{1}{2}+\frac{1}{3}\right) - \dots \right) \right]) \cdot e^{-\left(\frac{\alpha}{r_w}\right)^2 C_r}$$

Eq. A4.11

Using the Eq. A4.11, we can consider and resolve 2 experimental situations.
* The fluid is expressed vertically and flows centrally:

<figure><img src="image_213.png" /><figcaption>Figure A4.1: The fluid is expressed vertically and flows centrally</figcaption></figure>

Simulation:

(Here below is a simulation which is based on the same dimensions than the TPSU©).

for $r_e = 7.62 [cm]$, $r_w = 2.54 [cm]$, $C_r = 0.01 [cm^2 s^{-1}]$ and $P_{f0} = 1 [N \cdot cm^{-2}]$

<figure><img src="image_214.png" /><figcaption>Figure A4.2</figcaption></figure>

<figure><img src="image_215.png" /><figcaption>Figure A4.3</figcaption></figure>

<caption>Figure A4.2 and A4.3: Graphical representations based on Eq. A4.11 and the configuration described on Figure A4.1</caption>
* Peripheral fluid flow and solid mandrel in the centre.
(A section / collector of the TSPU is equivalent to the situation depicted below.)

<figure><img src="image_216.png" /><figcaption>Figure A4.4: Peripheral fluid flow and solid mandrel in the centre</figcaption></figure>

Simulation:

(Here below is a simulation, which is based on the same dimensions than the TSPUC.)

for $r_e = 7.62 \ [cm]$, $r_w = 2.54 \ [cm]$, $C_r = 0.01 \ [cm^2 s^{-1}]$ and $P_{f0} = 1 \ [N \cdot cm^{-2}]$, with
peripheral drainage at $r = r_e$

<figure><img src="image_217.png" /><figcaption>Figure A4.5</figcaption></figure>

<figure><img src="image_218.png" /><figcaption>Figure A4.6</figcaption></figure>

Figure A4.5 and A4.6: Graphical representations based on Eq. A4.11 and the
configuration described on Figure A4.4

As expected, for the same coefficient of consolidation $C_r$ and the same geometrical parameters $r_e, r_w$, the radial compression occurs much faster when the drainage occurs peripherally.
ANNEXE A5

LA COMPRESSION UNIDIRECTIONNELLE A PRESSION CONSTANTE (CPC)

PUBLICATION P1

Etablie sur la base du DEA soutenu en 1981 à l'INPL, cette communication
scientifique a fait l'objet du congrès NATO « Mathematical Models and Design
Methods in Solid -Liquid Separation » à Lagos / Algarve au Portugal en 1982. Elle
illustre avant tout l'usage et l'exploitation de la théorie développée dans le chapitre 2.
La version complète est fournie dans cette annexe A5.

LECLERC D. & REBOUILLAT S. *Mathematical Models and Design Methods in Solid-Liquid Separation : Dewatering by compression.*
NATO ASI Series, 356-391, (1982-1985).
I am unable to find any text in this image. The image appears to be a blurry depiction of a person on a horse.

This image does not contain any recognizable text.
# DEWATERING BY COMPRESSION

D. LECLERC and S. REBOUILLAT

Laboratoire des Sciences du Génie Chimique - ENSIC-CNRS - 1 rue Grandville - 54042 NANCY CEDEX (FRANCE)

## CONTENTS

### INTRODUCTION

#### FUNDAMENTAL EQUATIONS FOR CONSTANT PRESSURE COMPRESSION

- Basic Equations and the Terzaghi Model

- The Voigt Model and Secondary Compression

- Semi-Theoretical Equations of Compression

#### EXPERIMENTAL ILLUSTRATION OF BASIC EQUATIONS

- Experimental Procedure and Apparatus

- Analysis of Experimental Results

- Terzaghi Voigt Model

- Simplified Model

### SYMBOLS

- B constant representing the secondary consolidation, (-)

- C<sub>c</sub> constant in the relation e versus p<sub>s</sub>, (-)

- C<sub>e</sub> compression coefficient, (L<sup>2</sup>.T<sup>-1</sup>)

- C'<sub>e</sub> modified consolidation coefficient, (L<sup>2</sup>.T<sup>-1</sup>)

- E stiffness coefficient in the Voigt's model, (ML<sup>-1</sup>.T<sup>-2</sup>)

- E<sub>o</sub> constant in the relation e versus p<sub>s</sub>, (-)

- e,e<sub>x</sub> ratio of void volume to solid volume, (-)

- e<sub>i</sub> void ratio of filter cake surface, (-)

- e<sub>ia</sub> average void ratio of filter cake surface, (-)

- G viscous rubbing coefficient in Voigt's model, (-)

- gc gravitational acceleration, (LT<sup>-2</sup>)

- i number of drainage surface, (-)

- K<sub>e</sub> consolidation constant, (T<sup>-1</sup>)
$K, K_s$ Ruth's filtration coefficient, $(L^2.T^{-1})$

$L$ thickness of compressed cake at time $\theta_c$ or $\theta_c'$, (L)

$L_1$ thickness of filter cake at time $\theta_c = 0$, (L)

$L_1'$ hypothetical thickness of filter cake at time $\theta_c = 0$, (L)

$L_2$ theoretical final thickness of cake, (L)

$L_\infty$ final thickness of cake when $\theta_c = \infty$, (L)

$m$ ratio of wet to dry filter cake mass, (-)

$m_v$ volume change coefficient, $(M^{-1}.L.T^2)$

$n$ compressibility factor, $(ML^{-1}.T^{-2})$

$P$ applied pressure, $(ML^{-1}.T^{-2})$

$\rho_s$ local pressure of the solid, $(ML^{-1}.T^{-2})$

$\rho_x$ local pressure of the liquid, $(ML^{-1}.T^{-2})$

$s, s_i$ mass fraction solids, (-)

$T_c$ compression time factor, (-)

$T_c'$ modified compression time factor, (-)

$U_c$ average compression rate, (-)

$U_c'$ modified average compression rate, (-)

$u_x$ liquid velocity at the distance $x$ of the medium, $(L.T^{-1})$

$\bar{\alpha}$ or $\alpha_{av}$ average specific resistance, $(L.M^{-1})$

$\alpha_o, \alpha_x$ local specific resistance, $(L.M^{-1})$

$\epsilon_i$ porosity at the surface of the cake, (-)

$\epsilon_x$ local porosity of the cake, (-)

$\eta$ secondary compression coefficient, $(T^{-1})$

$\theta$ time, (T)

$\theta_f$ filtration time, (T)

$\theta_c$ compression time, (T)

$\theta_c'$ hypothetical compression time, (T)

$\theta_f'$ fictitious filtration time for a volume $(L_1'-L)\Omega$, (T)

$\mu$ viscosity of liquid, $(M.L^{-1}.T^{-1})$

$\rho$ density of liquid, $(M.L^{-3})$

$\rho_s$ density of solid, $(M.L^{-3})$

$\nu$ consolidation behaviour index concerning creep characteristic of materials, (-)

$\omega_o$ total solid volume per unit area, (L)

$\omega_x$ dry solid mass per unit area, $(M.L^{-2})$

$\Omega$ filter area, $(L^2)$

## INTRODUCTION

On an overall basis about 20 % of the combustion energy used by French industry is used in operations for eliminating water. This has led to a considerable development in techniques for dehydrating at low energetic cost.

Mechanical compression is particularly well suited to the present situation as it uses about 20 times less energy than thermal processes. This has led to the need for models of the kinetics of filtration and compression.
Here we present the principal analytic equations which describe water elimination by pressing and detail the main difficulties encountered in using earlier models.

It is certain that the complete mathematical expressions for the simulation of compression kinetics are a very important scientific basis for the design of pressing units. However their complexity makes them difficult to apply in industrial practice and reduces the possibilities of improving on the empirical approach to pressing processes used until now. For this reason the second part of our paper is on the use of simplified models. We will illustrate the use of these semi-theoretical models initially developed Shirato.

The use of these mathematical tools is illustrated by our experimental results obtained with flocculated suspensions of talc and kaolin.

# 1 FUNDAMENTAL EQUATIONS FOR CONSTANT PRESSURE COMPRESSION

Sludge dewatering by compression can be theoretically divided into two stages : first filtration, second consolidation. As the filtration step has been described previously, we will not deal with its theoretical interpretation.

Filtration stage : pressure applied on the piston of a filtration-compression cell (Fig. 1) causes an immediate increase of the hydraulic pressure which is uniformly distributed through the whole suspension starting a constant pressure filtration.

Transition point between the filtration phase and the consolidation phase can be determined by the Ruth equation (1). In a further section we will develop a methodology for experimental determination of the cake thickness $L_1$ at the end of filtration.

Compression stage : During this phase, the mechanism of dewatering is characterized by a continuous decrease of hydraulic pressure through the cake. When this pressure reaches zero, a slight decrease

<figure><img src="image_219.png" /><figcaption>Fig. 1. Simplified scheme for the filtration-compression cell.</figcaption></figure>
in cake thickness can still be observed. The kinetic of this phenomenon mainly depends on the plastic characteristic of the solid particles. It is called secondary compression.

Thus compression can be divided into two phases : primary compression and secondary compression. These two phases occur simultaneously but can be distinguished by the fact that secondary compression which is a characteristic of the compressed particles carries on at zero pressure.

## 1.1 Basic Equations and the Terzaghi Model

### 1.1.1 Simplified presentation of the Terzaghi analogy.
For better understanding of cake behaviour during compression, sludge can be imagined to be a spring on which a pressure $p_s$ is applied (Fig. 2).

On 1 : the spring is immersed in a cylinder full of water, no pressure is applied on the liquid. The local hydraulic pressure $p_x = 0$, and the local pressure applied to the cake (here to the spring), $p_s = 0$.

On 2 : a pressure $p$ is applied on a frictionless piston. A tap on the piston is shut. So that the piston cannot move down and the pressure $p$ is entirely supported by liquid. $p_x = p$ and $p_s = 0$.

On 3 : the tap is opened and the descent of the piston is slowed by the stiffness of the spring. Liquid pressure $p_x$ diminishes progressively as the liquid flows out. $p_s = p-p_x$.

On 4 : an equilibrium is reached. Liquid does not flow any more and the total applied pressure is supported by the compressed spring. $p_x = 0$ and $p_s = p$.

<figure><img src="image_220.png" /><figcaption>Fig. 2. Terzaghi's piston spring analogy.</figcaption></figure>
The analogy supposes a uniform pressure through the whole depth of the cylinder. In a cake, compression begins at the level of the drainage surface and progresses through the cake. Uniform pressure is only obtained at the end of compression phase. This constitutes an important limit to the validity of Terzaghi analogy.

### 1.1.2 Fundamental equations. These are based on four main hypotheses :

- complete saturation of the media

- unidirectional fluid flow and unidirectional compression

- properties of solid do not change with pressure.

There are three basic equations which apply to a thin layer (Fig. 3).

1. Darcy's law gives the approach velocity through a thin layer of particles whose mass is $dM = \Omega.d\omega_x$

$$u_x = - \frac{1}{\alpha_x \mu} \frac{\partial p_x}{\partial \omega_x} \qquad (1)$$

2. The consolidation law, which in soil mechanics gives the proportionality coefficient between change of liquid volume in a thin layer and the variation of applied pressure.

$$m_v = - \frac{1}{1+e_x} \cdot \frac{de_x}{dp_s} = \frac{1}{1+e_x} \cdot \frac{de_x}{dp_x} \qquad (2)$$

where $e_x$ is the void index, that is to say the ratio between the void volume and the solid volume. If $\epsilon$ is the porosity of the layer

$$e_x = \frac{\epsilon_x}{1-\epsilon_x}$$

3. A continuity equation : the change of flowrate through the layer is equal to the variation of the void index

<figure><img src="image_221.png" /><figcaption>Fig. 3. Mass balance.</figcaption></figure>
$$ \frac{\partial u_x}{\partial \omega_x} = - \frac{1}{\rho_s} \cdot \frac{\partial e_x}{\partial \theta} \qquad (3) $$

Substituting Eq. (2) in Eq. (1), we obtain by differentiation:

$$ \frac{\partial u_x}{\partial \omega_x} = - \frac{c_e}{\rho_s} \cdot \frac{\partial^2 e_x}{\partial \omega_x^2} \qquad (4) $$

$$ \text{from Eq. (3) and Eq. (4) : } \frac{\partial e_x}{\partial \theta} = c_e \frac{\partial^2 e_x}{\partial \omega_x^2} \qquad (5) $$

in which $c_e$ is a supposed constant coefficient which in soil mechanics is called the "compressibility coefficient"

$$ c_e = \frac{\rho_s}{\mu \alpha_x (1+e_x) m_v} = \frac{\rho_s^2}{(1+e_x)^2} c_v \qquad (5') $$

$c_v$ is a compressibility coefficient defined by Terzaghi.

Eq. (5) is similar to the characteristic equation of diffusion. It can be solved by Laplace transform. Analytical solutions are given by

$$ U_c = 1 - \exp \left( - \frac{\pi^2 T_c}{4} \right) \qquad (6) $$

for a sinusoidal distribution of initial hydraulic pressures through a cake and by

$$ U_c = 1 - \sum_{N=1}^{\infty} \frac{8}{\pi^2 (2N-1)^2} \exp \left\{ -\frac{(2N-1)^2 \pi^2}{4} T_c \right\} \qquad (7) $$

for an initially uniform distribution of hydraulic pressure.

Generally, it can be considered that the initial profile of hydraulic pressure is sinusoidal through the whole thickness of cakes obtained by filtration.

For concentrated sludges which are too concentrated to be filtered, it is considered that there is a linear initial profile of hydraulic pressure. These sludges are called "semi-solids sludges".

$$ U_c \text{ is a mean consolidation ratio } U_c = \frac{L_1 - L}{L_1 - L_2} \qquad (8) $$
with L1 = cake thickness at the beginning of compression phase

L2 = theoretical final thickness of compressed cake.

$$T_c \text{ is a reduced compression time} \qquad T_c = \frac{i^2 c \theta}{\omega^2} \qquad (9)$$

The Terzaghi model adapted by Shirato (1) (1971) to filtration-compression operations is useful for determining the consolidation ratio versus $\theta_c$ when initial conditions are known.

## 1.2 Voigt Model and Secondary Compression

Eq. (6) and Eq. (7) are frequently insufficient to get a complete and precise description of compression phenomena especially when secondary consolidation is important.

To solve this problem, Voigt introduced the notion of secondary compression due to the creep effects on the solid. Adapted by Shirato (2) (1974), to the filtration-compression operations, the Terzaghi-Voigt model gives more a precise description of compression phenomena.

Previously, we considered that during the compression phase, the local void ratio $e$ was only dependent of local pressure $p_s$ without taking into account the creep effect.

To take this effect into account it is necessary to consider that variation of void ratio is due partly to the variation of local pressure $p_s$ and partly to the effect of secondary compression characteristic of the solid under consideration.

In this case, the continuity equation becomes :

$$\frac{\partial e}{\partial \theta_c} + \frac{\partial e}{\partial p_s} \cdot \frac{\partial p_s}{\partial \theta_c} = - \frac{\partial}{\partial \omega} \left( \frac{\rho_s}{\mu \alpha} \cdot \frac{\partial p_s}{\partial \omega} \right) \qquad (10)$$

in which $\omega$ is the mass of solid per unit area of filtration, $\theta_c$ the consolidation time, $\rho_s$ the density of solid, $\mu$ the liquid viscosity and $\alpha$ the local specific resistance.

First term of Eq. (10) gives the variation of void fraction due to secondary compression and the second term the variation due to variation of applied pressure $p_s$ on the particles.

If we represent the Voigt model schematically by Fig. 4, the variation of void fraction vs time can be calculated by
<figure><img src="image_222.png" /><figcaption>Fig. 4. Rheological model for the consolidation.</figcaption></figure>

$$ \frac{\partial e}{\partial \theta_c} = \frac{1+e}{E} \cdot \frac{\partial}{\partial \theta_c} \int_{0}^{\theta_c} \frac{\partial}{\partial e} \left[ 1 - \exp \left( -\frac{E}{G} (\theta_c - \tau) \right) \right] (p_s - p_s(o)) \cdot d\tau \quad (11) $$

where $(p_s)_o$ is the local pressure of solid at $\theta_c = 0$.

From Eq. (10) and Eq. (11) :

$$ c_e \frac{\partial p_s^2}{\partial \omega^2} = \frac{\partial p_s}{\partial \theta_c} + \beta \eta \frac{\partial}{\partial \theta_c} \int_{0}^{\theta_c} (p_s - (p_s)_o) \exp(-\eta(\theta_c - \tau)) d\tau \quad (12) $$

$$ \text{with } \beta = \frac{1}{m_v E} $$

$$ \eta = \frac{E}{G} $$

$$ c_e = \frac{\rho_s}{\mu \alpha m_v (1+e)} $$

- $m_v$ is the proportionality coefficient previously defined in Terzaghi model

- E and G are respectively the viscosity coefficient and the rigidity in Voigt model.

To a first approximation, it can be considered that $\beta$, $\eta$ and $c_e$ are constant during consolidation. This makes it possible to solve Eq. (12) but gives too complex solutions. Generally, secondary compression presents far slower kinetics than primary compression so we can consider that

$$ \eta \ll c_e (i\pi/2\omega_o)^2 $$
The initial sinusoidal distribution of pressure through filter cakes conduces to a simplified solution

$$ U_c = B \left[ 1 - \exp(-n\theta_c) + (1-B) \right] \left[ 1 - \exp \left\{ - \frac{i^2 \pi^2 c_e}{4\omega_o^2} \theta_c \right\} \right] \quad (13) $$

By rearranging Eq. (13) and identification with Eq. (6) from Terzaghi, gives :

$$ U'_c = 1 - (1-B) \exp \left\{ - \frac{i^2 \pi^2 c_e}{4 \omega_o^2} \theta_c \right\} - B \exp(-n\theta_c) \quad (14) $$

For an initially uniform distribution of pressure solution of Eq. (12) leads to

$$ U_c = (1-B) \left\{ 1 - \sum_{N=1}^{\infty} \frac{8}{\pi^2 (2N-1)^2} \exp\left\{ -\frac{(2N-1)^2}{4} \pi^2 T_c \right\} + B \left[ 1 - \exp(-n\theta_c) \right] \right\} \quad (15) $$

where N as a positive integer.

## 1.3 Semi-Theoretical Equations of Compression

Previous fundamental equations are rather complicated and difficult to use for designing filtration compression equipment. Shirato (3)(4) has adapted a simplified model based on the shape of consolidation curves and presented first by Sivaram and Swamee.

This empirical model is easily introduce in the case of linear profile of hydraulic pressures (semi-solid sludges). It can still be used with sinusoidal profiles by using a ficticious cake with uniform porosity distribution.

### 1.3.1 Empirical model of Sivaram and Swamee.
In the case of a linear profile of hydraulic pressures (semi-solid sludges), the shape of the curves giving consolidation ratio versus $\sqrt{\theta_c}$ has two main characteristics :

- when secondary compression is negligible, it is a straight line up to $U_c = 0.53$,

- then asymptotic to $U_c = 1$ for large values of $\sqrt{\theta_c}$.

Sivaram and Swamee proposed the empirical equation :
$$ U_c = \frac{L_1 - L}{L_1 - L_\infty} = \sqrt{\frac{\frac{4}{\pi} T_c}{\left[1 + \left(\frac{4}{\pi} T_c\right)^{2.8}\right]^{1/2.8}}} \quad (16) $$

with a reduced compression time : $T_c = \frac{c_e i^2 \theta_c}{\omega_o^2}$

This equation fits the primary compression of semi-solid sludges with a precision of about 3 %. Eq. (7) is the full description of this

$$ U_c = 1 - \sum_{N=1}^{\infty} \frac{8}{(2N-1)^{2\frac{2}{\pi}}} \exp \left\{ - \frac{(2N-1)^{2\frac{2}{\pi}}}{4} \cdot \frac{c_e i^2}{\omega_o^2} \theta_c \right\} \quad (7) $$

It is clear that Eq. (16) is easier to use than Eq. (7) but it cannot be applied to secondary compression. Shirato therefore proposed a modification to the Sivaram and Swamee equation.

### 1.3.2 Adaptation of Sivaram and Swamee equation to compression of semi-solid sludges with secondary compression.
When secondary compression occurs the variation of consolidation ratio versus $\sqrt{\theta_c}$ retains the same shape but deviate earlier from the initial straight line. To fit this deviation, Shirato proposed to replace the Sivaram and Swamee coefficient 2.8 by a constant $v$ which would characterize the importance of secondary compression. Then

$$ U_c = \sqrt{\frac{\frac{4}{\pi} T_c}{\left[1 + \left(\frac{4}{\pi} T_c\right)^v\right]^{1/v}}} \quad (17) $$

The validity of this Eq. (17) will be discussed later in the experimental part of the paper.

### 1.3.3 Extension of empirical equation to the description of filter cake compression.
In contrast to semi-solid sludges, filter cakes have a sinusoidal profile of hydraulic pressure and the consolidation ratio versus $\sqrt{\theta_c}$ curves present a S shape.

The origin of those differences can be found in the fact that filtration does not lead to a uniform cake structure. Oldest layers near filter media are partially compressed and then porosity increases progressively from the media to the inlet surface of cakes, where the cakes are wetter.
To use empirical equations for consolidation of filter cake, it is necessary to avoid the S shape. This can be done by imagining that the compression of the true cake can be replaced by the compression of a fictitious semi-solid cake with an initial uniform porosity equal to porosity $ε_i$ of the inlet layer (Fig. 5). Then we will have to replace the initial thickness of cake $L_1$ by $L'_1$ and to define a mean modified consolidation coefficient

$$U'_{c} = \frac{L'_{1} - L}{L'_{1} - L_{\infty}} \qquad (18)$$

Then Shirato's equation can be written :

$$U'_{c} = \sqrt{\frac{\frac{4}{\pi} T'_{c}}{\left[1 + \left(\frac{4}{\pi} T'_{c}\right)^{\nu}\right]^{1/\nu}}} \qquad (19)$$

$$T'_{c} \text{ is a fictitious reduced time ; } T'_{c} = C'_{e} i^{2} \theta'_{c} / \omega_{o}^{2} \qquad (20)$$

$θ'_c$ is a fictitious time of consolidation which can be determined by the following considerations.

- Fictitious initial filter cake depth $L'_1$ can be calculated from

$$L'_{1} = (1+e_{i})\omega_{o} \qquad (21)$$

$e_i$ is the value of void ratio in a thin layer of cake where cake pressure is zero. It is extremely difficult to determine this value experimentally. So we will use the $E_o$ value from empirical equation previously used by Tiller (5) for the determination of the void ratio versus $p_s$

$$e = E_o - C_c \ln p_s$$

<figure><img src="image_223.png" /><figcaption>Fig. 5. Schematic illustration of the non-uniform void ratio distribution through a filter cake.</figcaption></figure>
Theoretically $E_0$ is the void ratio value for $p_s = 1$ and depends on the units of pressure used. To be near $p_s = 0$, we will choose to express $p_s$ in hectopa.

From Tiller, this constitutes a reasonable approximation of $E_0$.

- Fictitious time $\theta'_c$. Fictitious time $\theta'_f$ is equal to true time of consolidation more an hypothetical time $\theta'_f$ necessary to filter a sludge volume equivalent to $L'_1-L_1$ :

$$ \theta'_c = \theta'_f + \theta_c \quad (22) $$

$\theta'_f$ is the fictitious time necessary to obtain a volume of filtrate per unit surface $L'_1-(L_1/i)$ from sludge with concentration $s_i$ equal to the concentration at inlet surface of real cake.

$$ K_{si} \text{ is Ruth's coefficient :} \quad L'_1-L_1 = i\sqrt{K_{si}\theta'_f} \quad (23) $$

$K_{si}$ is Ruth's coefficient :

$$ K_{si} = \frac{2p(1-m s_i)}{\mu \alpha_{av} \rho s_i} $$

with $\alpha_{av}$ : mean specific resistance

$m$ : ratio between mass of wet cake and mass of solid in cake

$\mu, \rho$ : viscosity and density of filtrate.

Combining Eq. (22) and Eq. (23) with elimination of $\theta'_f$ :

$$ \theta'_c = \theta_c + \frac{L_1-L}{iK_{si}} = \theta_c + \frac{\mu\alpha_{av}\rho s_i^2}{2i^2\rho} (e_i - e_{iav}) \quad (24) $$

$e_{iav}$ is the mean void ratio in true cake

$$ e_{iav} = \frac{\rho_s (m-1)}{\rho} \quad (25) $$

## 2 EXPERIMENTAL ILLUSTRATION OF BASIC EQUATIONS

### 2.1 Experimental Procedure and Apparatus

The theoretical study of dewatering by expression leads to a relation for the average consolidation ratio $U_c$ versus $\theta_c$.

$U_c$ is the fractional approach, at time $\theta_c$ to the limiting value $U_{c\infty}$ at $\theta_{\infty}$. The degree of compression $U_c$ can be compared with the
transfer efficiency, and is given by :

$$U_c = \frac{L_1 - L}{L_1 - L_\infty}$$

where $L_1$, $L_\infty$, L have their conventional signification.

The variation of $U_c$ in terms of $\theta_c$ is measured by experiments in the filtration compression cell represented on Fig. 6.

This cell is fitted with a dial gauge to give the filtrate volume obtained between the beginning $\theta_o$ and $\theta_f$. The gauge is also used to follow the decrease in the thickness of the cake during the compression period.

The filtration compression cell, as indicated on Fig. 6, has a lips-ring placed at the end of the piston to prevent slurry leakage. Friction between the walls of the cylinder and the piston are considerably reduced by three teflon guides placed on the piston. An hydraulic jack moves the piston and a pressure up to 20 bar can be obtained. One or two porous plates are used on this equipment.

<figure><img src="image_224.png" /><figcaption>Fig. 6. Filtration-compression cell.</figcaption></figure>
The variation L versus θf or θc is directly accessible with experiment realized with this cell. The values L1 and L∞, previously mentioned are obtained graphically from the curve L = f(θf,c). This allows us to get experimental variations Uc vs θc for different suspensions.

These types of experiments will be used to illustrate the applicability of theoretical models. These mathematical tools are studied in order to simplify industrial data analysis and provisions.

## 2.2 Analysis of Experimental Results

To plot Uc vs √θc curves, we require two values L1 and L∞ which are obtained as follows.

### 2.2.1 Thickness of the cake at the beginning of the expression period : L1. The transition point (L1,θ1) characteristic of the end of the filtration period is determined graphically from the classical Ruth's equation

$$ (V_f+V_o)^2 = K(\theta_f+\theta_o) \quad (26) $$

where V_o and θ_o are fictitious values which correspond to the filter support (V_f ≈ Ω·ΔL).

During the filtration period, the ratio ΔL/Δ√θf is constant. The transition point is represented by the intersection of the two linear points of the curve. An illustration is given by Fig. (7) where (A), (B), (C) are different transition points for various experiments carried at constant pressure.

### 2.2.2 Theoretical final thickness L2. According to the analytical relationship for Uc vs θc, agreement between experiment and theory is possible when the two variables Cc and L∞ are well determined. The mathematical relations (6) and (7) have characteristics which are very useful to obtain the theoretical value L2 and to the operational variable Cc. L2 and Cc are determined by plotting (L1-L) vs √θc. This transformation of the exponential curve is illustrated on Fig. (8) and leads to the following relations :

- For a linear initial distribution of the hydraulic pressure

$$ \frac{A'C'}{A'B'} = 1.15 \quad \text{and} \quad C_e = \frac{0.848\omega_o^2}{i^2 \cdot \theta_{90}} \quad (27) $$
<figure><img src="image_225.png" /><figcaption>Fig. 7. Graphical determination of the transition point (A,B,C)</figcaption></figure>

<figure><img src="image_226.png" /><figcaption>Fig. 8. Graphical determination of the theoretical thickness for the infinitesimal compressed layer</figcaption></figure>
- For a sinusoidal initial distribution of the hydraulic pressure

$$ \frac{\overline{AC}}{AB} = 1.08 \text{ and } c_e = \frac{0.933\omega_o^2}{i^2\theta_{90}} \quad (28) $$

These relations are verified when $U_c = 0.90$. Thus, point C corresponds to a compression ratio $U_c = 0.90$ and is defined as indicate in Fig. (8).

Generally good agreement is obtained between $L_\infty$ and $L_2$. We obtained an average ratio of 0.982 for expression of kaolin suspensions under different pressions and for various values of $\omega_o$. However precautions must be taken when secondary consolidation is important.

2.2.3 The two particular shapes of the curves $U_c$ vs $\sqrt{\theta_c}$. Fig. (9) gives two examples of experimental curves of $U_c = f(\sqrt{\theta_c})$ for flocculated and non-flocculated kaolin. The fundamental difference between these two curves is in their respective form.

Non-flocculated kaolin gives a S shaped compression curve. Thus, as is predicted by the analytic expressions for the pressing of filter cakes, the filtration compression of non-flocculated kaolin leads to a sinusoidal initial hydraulic pressure distribution through the cake.

Recently we showed that the compression of flocculated kaolin sludges gives a linear variation of dehydration with time during the primary dewatering phase. This leads us to think that the flocculated kaolin behaves the same as semi-solid sludges. This hypothesis is supported by the fact that the flocculation of kaolin leads to a rapid sedimentation. Thus in this case the compression process is performed on a sludge obtained by eliminating the excess water by simple drainage. In this case we have substituted for s the term M which corresponds to the mass of dry solid involved.

We will now present the analysis of these two types of curve. We mention the principal difficulties in this modelisation and show the necessity having a simple mathematical model to simulate behaviour of filter cakes and sludges under expression.

## 2.3 Terzaghi-Voigt Model

Over the last ten years analytical expression for consolidation have been established and perfected to give greater precision on design. This experimental investigation on the dehydration of filter cakes and pressing of semi-solid sludges has shown the extra precision brought by these developments and allowed us to test and outline
<figure><img src="image_227.png" /><figcaption>Fig. 9. Comparison of dewatering behaviour of flocculated and non-flocculated kaolin.</figcaption></figure>

the applicability of the theoretical expressions.

The Terzaghi (6) model was initially developed to describe the primary compression phase which was the only compression really envisaged at that time. We will show that this model is effectively limited to this initial phase.

The theoretical development of the Voigt's model led to considering the compression to be composed of two simultaneous steps and allowed the introduction of secondary compression (7). We will show
that the Terzaghi-Voigt combined model gives a good prediction of the two phases of compression.

### 2.3.1 Filter-cakes compression.
Figs. (10, 11, 12) give consolidation curves $U_c$ vs $\sqrt{\theta_c}$. Fig. (10) illustrates the evolution of the dewatering kinetics with the initial thickness $L_1$. Fig. (11) shows the influence of the pressure and Fig. (12) completes this experimental illustration.

In order to get the theoretical values of $U_c$, according to Eq. (14), three operational values: $B$, $\eta$ which depend on secondary consolidation and $C_e$ for the primary compression.

In most cases, $U_c$ variation is much less during primary compression than during the secondary consolidation. This means that for large $\theta_c$ values, $\eta \ll C_e (i\pi/2\omega_0)^2$ and the Eq. (14) may be written as follows:

$$U_c \sim 1 - B \exp(-\eta\theta_c)$$

The variables $B$ and $\eta$ may be obtained by linearisation of this expression. Then by subtracting the fraction of secondary consolidation to the overall process, a simple mathematical treatment will lead to $C_e$. (Fig. 13).

$$1-(U_c)_{\text{corr}} = \frac{U_c - (1-B \exp(-\eta\theta_c))}{(1-B)} = \exp \left[ - \frac{i^2 \pi^2 C_e}{4\omega_0^2} \cdot \theta_c \right] \quad (29)$$

The left hand side of this relation can be obtained graphically from the experiments, so the linearisation of Eq. (29) will permit calculation of the operational value $C_e$.

Shirato presents a similar treatment and incorporates the average corrected consolidation ratio

$$(U_c)_{\text{corr}} = \frac{(L_1-L)-B(L_1-L_2) \cdot (1-\exp(-\eta\theta_c))}{(1-B)(L_1-L_2)}$$

This notation has the advantage of giving the expression of the primary consolidation process initially developed by Terzaghi

$$(U_c)_{\text{corr}} = 1 - \exp \frac{-i^2 \pi^2 C_e}{4\omega_0^2} \cdot \theta_c$$

This relation is equivalent to Eq. (6) initially developed on
<figure><img src="image_228.png" /><figcaption>Fig. 10. Kinetics of the compression process versus cake thickness.</figcaption></figure>

<figure><img src="image_229.png" /><figcaption>Fig. 11. Influence of the hydraulic pressure upon the water elimination velocity by pressing.</figcaption></figure>
<figure><img src="image_230.png" /><figcaption>Fig. 12. Influence of the initial mass fraction solids in slurry upon the dewatering kinetics.</figcaption></figure>

<figure><img src="image_231.png" /><figcaption>Fig. 13. Graphical determination of the secondary consolidation characteristics (B, η) and the primary compression coefficient C<sub>e</sub>.</figcaption></figure>
the basis of Terzaghi theory.

Value of the coefficient $C_e$ can be obtained by two methods. The first is the linearisation mentioned previously and the second is the calculation of this value with graphical and mathematical treatment of the expression of $(U_c)_{corr}$. This latter value is based on the calculation of $C_e$ from the Eq. (27). These two methods are illustrated with an example based on experiments with the filtration-compression cell. Figs. (14) and (15) show the good agreement of experimental values with theoretical calculations. A good accuracy is achieved with the overall process of primary and secondary consolidation. As illustrated on Fig. (14), the contribution of Voigt elements is important. The Terzaghi model gives a good accuracy during the primary period of consolidation.

Experimental results indicate that the dewatering mechanism by primary consolidation is well described by calculations based on the Terzaghi model. However a better understanding and a more accurate description are given by the Terzaghi-Voigt theory.

Incorporation of all these theoretical considerations by Shirato improved the efficiency in the calculations.

Influence of the initial thickness of the cake on the rate of dewatering by pressing. This influence is illustrated on Fig. (10). Fig. (16) completes this description and shows that the mechanical compression under 10 bars of a filter cake 1 cm thick is kinetically equivalent to the expression under 2 bars of a thickness cake of 0.5 cm.

### 2.3.2 Expression of semi-solid sludges illustrated by mechanical dewatering of flocculated sludges.
Eqs. (7) and (15) describe the kinetics of the consolidation process when the initial distribution of hydraulic pressures is linear. We recently showed that the dewatering of flocculated sludges by mechanical pressing leads to such a distribution. So we will illustrate the behaviour of semi-solid materials with this example. Figs. (17, 18, 19) represent different experiments performed to demonstrate typical behaviour of semi-solid materials during compression.

Variation of the dehydration kinetics with applied pressure is represented on Fig. (17). The initial thickness of sludge has an important influence on the pressing process. Fig. (18) illustrates this effect. The method used to analyse results is very similar to the analytical treatment previously described for filter cakes. In order to model the overall process we need the values of $B$ and $n$ which depend on secondary consolidation. As previously reported, these variables are obtained directly from the curve $\ln(1-U_c) = f(\theta_c)$. Secondly, by reducing the Eq. (15), we will get the simplified expression which will give $(U_c)_{corr}$.
<figure><img src="image_232.png" /><figcaption>Fig. 14. Determination of the primary compression coefficient. Validity of the Terzaghi-Voigt model.</figcaption></figure>

<figure><img src="image_233.png" /><figcaption>Fig. 15. Validity of the Terzaghi-Voigt model.</figcaption></figure>
<figure><img src="image_234.png" /><figcaption>Fig. 16. Influence of the cake thickness upon the dewatering kinetics.</figcaption></figure>
<figure><img src="image_235.png" /><figcaption>Fig. 17. Evolution of the dewatering kinetics for flocculated slurries vs pressure.</figcaption></figure>

<figure><img src="image_236.png" /><figcaption>Fig. 18. Influence of the initial slurry thickness upon the dewatering kinetics by compression.</figcaption></figure>
<figure><img src="image_237.png" /><figcaption>Fig. 19. Reproducibility of compression test for the filtration-compression cell.</figcaption></figure>
$$ (U_c)_{corr} = \frac{(L_1-L)-B(L_1-L_2)(1-\exp(-\eta\theta_c))}{(1-B)(L_1-L_2)} = 1 - \sum_{N=1}^{\infty} \left\{ \frac{8}{(2N-1)^2} \cdot \exp\left[-\frac{(2N-1)^2}{4} \cdot \frac{c_e i^2}{\omega_o^2} \cdot \theta_c\right] \right\} \quad (30) $$

This equation is similar to Eq. (7) which is based on the Terzaghi theory. The operational variable $c_e$ is determined graphically from Eq. (27), from the $(U_c)_{corr}$ vs $\sqrt{\theta_c}$ curve.

The three variables $B$, $\eta$, $c_e$ are known and we can now consider the modelling. In order to demonstrate that semi-solid materials and more particularly flocculated sludges behave as Eq. (15) predicts, we modelise with the two following possibilities (Fig. 20)

- sinusoidal initial distribution of hydraulic pressure, Eq. (14)

- linear distribution of hydraulic pressure, Eq. (15).

Obviously Eq. (14) does not simulate the behaviour of flocculated sludges under compression. However when the applied pressure is high and the initial thickness $L_1$ sufficiently small Eq. (14) may be used.

But according to Fig. (21), we can observe the difference in the shape of the theoretical and the experimental curves. Because of a greater simplicity, this modelisation based on Eq. (14) is sometimes useable (8).

These considerations demonstrate that our simulation of the behaviour of flocculated sludges is correct and that such materials may be used to illustrate the applicability and use of basic Eq. (15).

Fig. (22) strengthens these considerations and shows the good agreement between theoretical and experimental values. The accuracy brought by Voigt elements in Terzaghi theory is also clearly demonstrated. In this case, the difference between the simulation based on Eqs. (7) and (15) is less observable because flocculated sludges have a different behaviour during secondary consolidation.

## 2.4 Simplified Model

### 2.4.1 Analytical relations previously developed and illustrated by experimental results are a very important scientific basis for design. However their complexity makes them difficult to apply in in-
<figure><img src="image_238.png" /><figcaption>Fig. 20. Simulation of compression curves for flocculated slurries of kaolin</figcaption></figure>

(1) sinusoidal hydraulic pressure distribution

(2) linear hydraulic pressure distribution.

<figure><img src="image_239.png" /><figcaption>Fig. 21. Simulation of behaviour of flocculated slurries of kaolin using the equations of the filtration cake compression.</figcaption></figure>
<figure><img src="image_240.png" /><figcaption>Fig. 22. Modelisation of compression curves for flocculated slurries.</figcaption></figure>

dustrial practice and leads to maintain a certain empiricism during
the application of pressing technology.

In order to answer real industrial problems, it is important to
have simplified models. For this reason, we have chosen to present
analytical expressions directly adaptable to a simple numerical
treatment.

Thus we will detail the utilization of Sivaram and Swamee equation for modelling pressing process. First illustrations will be given for semi-solid materials because their behaviour under expression is generally simulated by Eq. (15) which is the most difficult to apply in industrial practise. As an example, we will give our own results recently obtained for flocculated sludges. Then we will detail the application of modified Sivaram and Swamee equation in filter cake dewatering.

Previously, we developed the basic equations which have been
used by Shirato for the adaptation of soils mechanics, theory to the
modelling of the behaviour of filter cakes and semi-solid materials under mechanical compression. Thus we will only outline the main difficulties encountered during the utilization of these equations. Instead of overloading our presentation with numerical or theoretical calculations, we will present and described some figures which really demonstrate the simplicity brought by these new mathematical tools.

### 2.4.2 Modelling of semi-solid materials expression, illustrated by experiments on flocculated kaolin sludges. Fig. (23) gives theoretical curves, obtained according to Eq. (17) in which $v$ is the compression index and $T_c$ is a dimensionless time related to the compression time by

$$ \frac{4}{\pi} \cdot T_c = K_e \cdot \theta_c = \frac{4i^2}{\pi\omega_0^2} \cdot C_e \cdot \theta_c \quad (31) $$

The index of compression $v$ is characteristic of the secondary consolidation phase. For $v = 2.8$, the theoretical curve is analogous to the curve obtained with Eq. (7). That means a negligible seconda-

<figure><img src="image_241.png" /><figcaption>Fig. 23. Theoretical diagram $U_c$ vs $\sqrt{T_c}$.</figcaption></figure>
ry compression phase. The secondary consolidation appear on this theoretical curves when $v < 2.8$.

Analysis of experimental results. The first step consists in the determination of $K_e$ and $C_e$. The $U_c$ vs $\sqrt{\theta_c}$ experimental curves present a linear curve during the primary consolidation. Thus, to obtain the operational variables, we have only to use the square of the slope of this linear part. Obviously, $C_e$ will be calculated from Eq. (31).

The second step consists in the determination of the index $v$. When $K_e$ or $C_e$ are known, the $U_c$ vs $\sqrt{\theta_c}$ curves can be plotted. A simple comparison with these one and the diagram will give to us $v$.

Figs. (24) and (25) show the agreement between theoretical calculations and experimental results. On Fig. (25), we can observe the analogy between calculations based on Terzaghi-Voigt model (Eqs. 7 and 15) and theoretical values obtained with modified Sivaram and Swamee (Eq. 17).

Our experimental results illustrate the great simplicity brought by simplified models. These considerations are very important for design and industrial application of dewatering and pressing processes.

### 2.4.3 Simulation of filter cakes behaviour, under compression with simplified models.
Fig. (5) illustrated the problem encountered by Shirato in order to extend the application of preceding Eq. (17). Theoretical considerations have been previously detailed. They justify the present method.

The principal stages will only be mentioned to outline precautions which should be taken.

1. Determination of the fictitious thickness $L'_1$. $L'_1$ is given by the Eq. (21) which is approximatively equivalent to : $L'_1 = (1+E_0)\omega_0$. Fig. (26) gives our experiments results used to evaluate $E_0$ in the chosen example. This procedure is general.

2. Determination of the fictitious compression time $\theta'_c$. This time is given by Eq. (24) in which the variables $e_{iav}$ and $\alpha_{av}$ are determined by classical filtration experiments. Fig. (27) shows the variation of the average specific resistance with $\Delta P$ for the example chosen as illustration.

3. Expression of the modified average consolidation ratio $U'_c$.

4. Transformation of the curves $U_c = f(\sqrt{\theta_c})$ to $U'_c = f(\sqrt{\theta'_c})$. $U'_c$ is given by Eq. (19) in which $T_c$ is a fictitious dimensionless time. $T'_c$ is bound to $C_e$ by the relation :
<figure><img src="image_242.png" /><figcaption>Fig. 24. Experimental illustration of the semi-theoretical equation $U_c$ vs $\sqrt{\theta_c}$ for flocculated kaolin (Eq. 16).</figcaption></figure>

<figure><img src="image_243.png" /><figcaption>Fig. 25. Simulation of behaviour towards compression of flocculated slurries using simplified semi-theoretical models (Eq. 16).</figcaption></figure>
<figure><img src="image_244.png" /><figcaption>Fig. 26. Variation of void ratio for infinite compression vs pressure.</figcaption></figure>

<figure><img src="image_245.png" /><figcaption>Fig. 27. Variation of the average specific resistance for kaolin versus pressure drop.</figcaption></figure>
<figure><img src="image_246.png" /><figcaption>Fig. 30. Recapitulatory of theoretical models of the dewatering by mechanical compression.</figcaption></figure>
$$T'_{c} \equiv i^{2}C'_{e}\theta'_{c}/\omega_{o}^{2}$$

To get $C'_e$ and model the experimental curve, we must transform $U_c$ vs $\sqrt{\theta_c}$ curves to $U'_c$ vs $\sqrt{\theta'_c}$. The following considerations lead directly to this transformation :

$$U'_{c} = \frac{L'_{1}-L}{L'_{1}-L_{2,\infty}}$$

$L'_1$ is calculated according Eq. (21) and so $U'_c$ can be obtained for each of the values of $\theta'_c$ calculated previously. Thus the transformation is possible and this passage from $U_c$ vs $\sqrt{\theta_c}$ to $U'_c$ vs $\sqrt{\theta'_c}$ is illustrated by Figs. (10) and (28).

5. Modelling $U'_c$ vs $\sqrt{\theta'_c}$ curves. For small $\theta'_c$ values, Eq. (17) can be written :

$$U'_{c} = \sqrt{\frac{4}{\pi} \cdot \frac{i^2 C'_{e}}{\omega_{o}^2} \cdot \theta'_{c}} \quad (32)$$

Thus the slope of the early stage of the curve will lead to $C'_e$.

By reducing the $U'_c$ vs $\sqrt{\theta'_c}$ curve, we will be able to plot $U'_c$ vs $\sqrt{T'_c}$ curve and make the modelling possible. The index $\theta$ will be given by direct comparison of these curves with Fig. (22).

Figs. (28) and (29) illustrate the agreement between these theoretical considerations and the experimental results. However we must mention that the extension of the semi-solid simplified equation brings less simplifications than those reported for modelling the behaviour of semi-solid sludges.

REFERENCES

1. Shirato M., Murase T., Kato H. and Fukaya S., Fundamental analysis for expression under constant pressure. Filtration and Separation 7 (1970) 277-282.

2. Shirato M., Murase T., Tokunaga A. and Yamada O., Calculations of consolidation period in expression operations, Journal of Chemical Engineering of Japan 7 (1974) 229-231.

3. Shirato M., Murase T., Atsumi K., Aragaki T. and Noguchi T., Industrial expression equation for semi-solid materials of solid-liquid mixture under constant pressure. Journal of Chemical Engineering of Japan 12 (1979) 51-55.
4. Shirato M., Murase T. and Atsumi K., Simplified computational method for constant pressure expression of filter cakes. Journal of Chemical Engineering of Japan 13 (1980) 397-401.

5. Tiller F.M., Stewart Haynes, J.R. and Wei-Ming Lu. The role of porosity in filtration VII. Effect of side-wall friction in compression-permeability cells. AIChE Journal 18 (1972) 13-20.

6. Shirato M., Murase T., Negawa M. and Moridera H. Analysis of expression operations. Journal of Chemical Engineering of Japan 4 (1971) 263-

7. Shirato M., Murase T., Moridera H. and Uryu M. Expression of unsaturated mixtures under constant pressure. Journal of Chemical Engineering of Japan 7 (1974) 229-231.

8. Leclerc D. and Minery E. Comparaison du comportement des boues floculées et non floculées lors de leur déshydratation par compression mécanique. La Tribune Cebedeau 33 (1980) 441-449.
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# ANNEXE A6

## LA COMPRESSION UNIDIRECTIONNELLE A PRESSION CONSTANTE (CPC)

### PUBLICATION P5

Etablie sur la base du chapitre « Recherche d'Optima Techniques, Economiques et Energétiques d'Opérations de Pressage » de la thèse de Docteur-Ingénieur soutenue à l'INPL en 1983, cette publication est une synthèse de la méthode développée à cette époque.

REBOUILLAT S., LECLERC D. & BALUAIS G. *La déshydratation par filtration-pressage. Incidence des adjuvants de pressabilité sur le pressage des pulpes de betteraves*. Société Française de Filtration, Bulletin Technique n° 10, 72-84, (1985).

Version complète.
L'UF et la MFT apportent des avantages techniques importants :

- obtention d'un vin très bien clarifié, en une seule opération et à partir d'un produit très chargé

- c'est un procédé continu.

Par contre l'introduction de cette technique modifie le schéma traditionnel de traitement et de stabilisation des vins, en particulier au niveau de la prévention des précipitations tartriques.

Si les coûts de traitement ne sont pas prohibitifs, l'UF et la MFT devraient se développer rapidement dans le milieu viticole dans les prochaines années.

LA DESHYDRATATION PAR FILTRATION-PRESSAGE. INCIDENCE DES ADJUVANTS DE PRESSABILITE SUR LE PRESSAGE DES PULPES DE SOUTIEN.

S. REBOUILLAT, D. LECLERC, G. SALUAIS

Laboratoire des Sciences du Génie Chimique

E.N.S.I.C. / C.N.F.S.,1 rue Grandville, 54042 Nancy Cedex

# 1. INTRODUCTION

La réduction des coûts opératoires passe par la diminution de la consommation en énergie combustible et ceci concerne particulièrement le domaine des Industries Agricoles et Alimentaires (I.A.A.). C'est pourquoi on assiste actuellement à un développement important des techniques de déshydratation par pressage mécanique qui consomment environ 20 fois moins d'énergie que les procédés thermiques.

De ce point de vue, l'exemple du pressage des pulpes de betteraves est significatif et mesure bien l'enjeu économique. La production française de betteraves à sucre durant la campagne 1981-82 a été de 33 Mt, la teneur moyenne en matière sèche des pulpes étant de 8 %MS. En fin d'opération de déshydratation, la production de pellets (88% MS) a atteint 1,5 Mt ce qui correspond à l'élimination de 6 Mt d'eau/an. Les pulpes surpressées avaient une teneur moyenne en matières sèches de 23% MS. Un gain de 2% sur la siccité des pulpes avant séchage thermique (23 → 25 %MS) correspondait à 30 000 TEP/an d'économie soit environ 40 t/TEP/an.

Dans cette branche d'activité, en dix ans (1972-82), le taux de matières sèches des pulpes surpressées a augmenté de 5 points en passant de 18 à 23% ce qui a permis de diminuer la consommation spécifique de fuel de 37% (figure 1a.). Au cours de la même période le prix du fuel a été multiplié par 1,1 mais la réduction de la consommation a limité l'augmentation du coût de séchage thermique (figure 1b.).

L'effort réalisé est important et se poursuit notamment par l'optimisation de la chaîne de déshydratation (pressage+séchage) au niveau des séchoirs (récupération des excédents énergétiques par recompression des huées,...). Mais si l'on se limitent à la partie pressage de cette chaîne, deux voies sont possibles et complémentaires pour améliorer cette étape :

- modifier l'état physico-chimique des pulpes par adjonction de certains composés (sulfate d'alumine, sels de calcium...) qui en agissent sur le pH, augmentent la perméabilité des parois cellulosiques.

- rechercher des innovations technologiques au niveau du matériel de pressage, ces nouveaux "separateurs" permettant d'obtenir des pulpes ayant des siccités supérieures à 30%. En effet, les matériels les plus couramment utilisés dans les sucreries ne peuvent fournir régulièrement que des pulpes surpressées au environ de 22 %MS.
Evolution de la siccité des pulpes et de la consommation de fuel durant la dernière décennie

<figure><img src="image_247.png" /><figcaption>Figure 1a.</figcaption></figure>

Evolution des coûts du fuel et du séchage durant la dernière décennie

<figure><img src="image_248.png" /><figcaption>Figure 1b.</figcaption></figure>

## 2. REVUE DES MATERIELS DE PRESSAGE DANS LES IAA.

La liste de matériels qui va être présentée dans la suite de cet exposé ne se prétend pas exhaustive. Elle mentionne les matériels classiques que l'on peut d'ailleurs rencontrer dans des domaines variés, extérieurs aux IAA.

Les équipements de filtration-pressage sont souvent classés en distinguant les appareils alimentés par une suspension pompable de ceux qui nécessitent une alimentation par un produit possédant une stucture semi-solide.

### 2.1 Suspensions pompables

Les filtres presses sont des appareils bien connus dans le domaine des IAA. Toutefois pour obtenir des gâteaux de forte siccité, ces matériels ont dûs être modifiés afin de pouvoir, presser le matériau après la phase de filtration. Pour cela deux types de dispositifs sont apparus.

Dans le premier type d'appareils, un vérin rapproche les plateaux ce qui réduit le volume des chambres et comprime donc fortement le gâteau (figure 2). Les pressions ainsi exercées au pressage sont normalement de 20 bars mais peuvent aller jusqu'à 400 bars pour certains produits comme le beurre de cacao. Sur ce type de filtre, les cycles n'excédent pas 10 min à condition de former des gâteaux d'environ 10 mm. Le débatissage est automatisé avec bascule des plateaux.

L'autre technologie consiste à utiliser des plateaux chambrés (figure 3) c'est à dire composés de membranes en élastomères qui, gonflées avec de l'eau ou de l'air sous pression, vont comprimer le gâteau et réduire ainsi son humidité. Ce système permet d'atteindre des pressions d'environ 16 bars au sein du gâteau.

### 2.2 Produits pâteux ou pulpeux

Pour ce genre de produits obtenus naturellement ou après conditionnement, la technologie des filtres à bandes presseuses peut être appliquée. Dans la première partie de la bande on procède à l'égouttage du produit à traiter puis la compression

Figure 2. Filtre presse à plateaux

<figure><img src="image_249.png" /></figure>

Figure 3. Principe de fonctionnement des filtres à membrane presseuse

<figure><img src="image_250.png" /></figure>
<figure><img src="image_251.png" /><figcaption>Figure 5. Organe principal des presses à double vis</figcaption></figure>

<figure><img src="image_252.png" /><figcaption>Figure 6. Filtre à bande presseuse</figcaption></figure>

<figure><img src="image_253.png" /></figure>

s'opère par passage du gâteau déposé entre des rouleaux (figure 6). Les pressions exercées dans ce type de filtre restent modestes (5 à 10 bars maxi) car il y a souvent un phénomène de flageo du matériau et des problèmes de résistance mécanique de la toile. Sur certains de ces filtres, après la phase d'écoulement, on peut réaliser un lavage de la pulpe (non représenté à la figure 6) très efficace. Cette technologie s'applique parfaitement lorsqu'on ne recherche pas un produit très sec et qu'un procédé continu présente un intérêt.

À côté de ces matériels de filtration-pressage, les presses à vis, simple ou double, sont très souvent rencontrées ( figures 4 et 5). Ces appareils fonctionnent en continus, le mouvement des parois (vis) provocant la compression. Etant donné qu'il n'y a pas formation de gâteau, les fines particules sont souvent entraînées avec l'eau ce qui donne des effluents relativement riches en matère en suspension. En raison des effets de cisaillement et des frottements parois-solide, ces machines sont relativement consommatrices en énergie (1 à 5 kWh/t HS).

### 3. APPROCHE ET MODELISATION DU PRESSAGE

Pour dimensionner un filtre presseur du type présenté aux figures 2 et 3, nous avons développé une méthodologie basée sur des essais de laboratoire effectués sur cellule de filtration-compression. Cette cellule, schématisée à la figure 7, est une petite presse unidirectionnelle équipée de capteurs qui permettent un enregistrement continu de l'épaisseur du gâteau et de la pression appliquée. Le vérin qui actionne le piston est asservi afin de définir avec précision le profil d'application de la pression.

L'efficacité du pressage est traduite par le taux de compression :

$$U_c = (L_1 - L) / (L_1 - L_{\infty})$$

où $L_1$ est l'épaisseur initiale du gâteau, $L$ l'épaisseur à l'instant $t_c$ et $L_{\infty}$ l'épaisseur théorique finale.

En faisant les trois hypothèses simplificatrices suivantes: saturation complète du milieu, écoulement et compression unidirectionnels, loi de Darcy applicable, on peut déterminer les expressions analytiques du taux de compression $U_c$ en fonction du temps de compression $t_c$ à partir de modèles analogiques, en distinguant le cas où le gâteau est obtenu après une phase de filtration (gâteau de filtration) et le cas où le produit initial est plus concentré (produit semi-solide) et donc où il n'a pas de filtration avant pressage. Les différents modèles sont résulés au tableau I. Les expressions sont plus en noirs complèces suivant que l'on prend en compte la compression primaire seule (modèles élastiques) où les phases de compression primaires et secondaire (modèles viscoelastiques). Dans le cas des produits semi-solides un modèle empirique à deux paramètres est souvent utilisé. Ces deux paramètres sont : le coefficient de compression primaire et $\nu$ paramètre d'ajustement qui dépend du produit et
# Tableau I. Résumé des modèles de consolidation unidirectionnelle

## GATEAU DE FILTRATION

<table><tr><td>Modèle élastique</td><td>Uc = 1 - exp [- K₁.Ce.tC]</td></tr><tr><td>Modèle visco-élastique</td><td>Uc = 1 - (1-B).exp[-K₁.Ce.tC] - B.exp[-η.tC]</td></tr></table>

$$K_1 = (\pi iA/2M)^2$$

## PRODUIT SEMI-SOLIDE

<table><tr><td>Modèle élastique</td><td>Uc = 1 - Σ<sub>N=1</sub><sup>∞</sup> (8/π²(2N-1)²).exp[-K₂Ce.tC]</td></tr><tr><td>Modèle visco-élastique</td><td>Uc = B.(1-exp[-η.tC]) + (1-B).(1 - Σ<sub>N=1</sub><sup>∞</sup> (8/π²(2N-1)²).exp[-K₂Ce.tC])</td></tr><tr><td>Modèle empirique</td><td>Uc = [Ke.tC / ((1+(Ke.tC)²)¹/√V)]<sup>0.5</sup></td></tr></table>

$$K_2 = (2N-1)^2 \cdot K_1$$

$$Ke = (2iA/M)^2/\pi.Ce$$

avec

A : aire de la surface filtrante (m²)

B : fraction du taux de compression obtenu par compression secondaire (-)

Ce : coefficient de compression primaire (kg².m⁻⁴.s⁻¹)

i : nombre de surface de drainage

Ke : coefficient de compression (s⁻¹)

M : masse de solide déposé (kg)

N : entier positif (-)

t<sub>C</sub> : temps de compression (s⁻¹)

U<sub>C</sub> : taux moyen de compression (-)

η : coefficient de consolidation secondaire (s⁻¹)

φ : paramètre d'ajustement

<figure><img src="image_254.png" /><figcaption>Figure 7. Cellule de filtration-compression</figcaption></figure>

Figure 8.

<figure><img src="image_255.png" /><figcaption>Modélisation du pressage des pulpe de betteraves</figcaption></figure>

Figure 9.

<figure><img src="image_256.png" /><figcaption>Modélisation du pressage des pulpe de betteraves</figcaption></figure>
peu des conditions opératoires.

La validation de ce modèle est faite, par exemple, aux figures 8a et 8b avec des pulpes fraîches ayant des tailles de cossettes de 8 et 16 mm. Ces essais sont réalisés à 8 bars avec une charge de 3 kgMS/m².

### 4. INFLUENCE DES PARAMETRES OPERATOIRES SUR LE PRESSAGE DES PULPES DE BETTERAVES.

Pour optimiser une séparation solide-liquide, il faut fixer les facteurs qui interviennent et faire ressortir ceux qui sont prépondérants afin de cerner le ou les optimums dans un plan d'expérience.

Dans le cas des pulpes de betteraves, nous avons étudié l'influence du vieillissement des pulpes, de la charge (quantité de solide à presser / m²), de la pression, de la taille des cossettes, de la vitesse de pressage et de la teneur en adjuvant de pressabilité[1]. Nous ne présenterons ici que quelques résultats significatifs.

* Le vieillissement des pulpes est un paramètre important lorsque l'on entreprend une expérimentation dans un laboratoire hors du site. A titre indicatif, le même essai est répété après conservation de la pulpe 48 h à 10°C. Les performances sont 20% meilleures lorsque la pulpe est fraîche. Ceci impose donc que l'expérimentation soit réalisée à proximité d'une sucrerie.

* Plus les gâteaux sont minces, meilleures sont les performances. Lorsque l'on réduit la charge de 50 %, on peut gagner 3 points sur la siccité des pulpes.

* La pression dans le domaine 3 à 17 bars et la taille des cossettes dans la gamme 8 à 16 mm ne semblent pas être des facteurs prépondérants.

* La vitesse d'application de la pression, par contre, est très influente. Si l'on divise par 20 cette vitesse, la quantité d'eau éliminée double.

* Un apport de chlorure de calcium à raison de 0,5 kg/tB permet un accroissement de 38% de la cinétique de pressage (figure 9).

### 5. OPTIMISATION DE L'APPORT D'ADJUVANT DE PRESSABILITE.

Des essais réalisés sur une presse à vis industrielle (presse à simple vis STORD, FERRIANI 800t/j) ont permis d'établir les courbes de variation de la figure 10.

Un gain de 2,2% peut être obtenu sur ce type de presse avec une vitesse de rotation de 2,5 tr/min.

La séquence simplifiée de pressage -séchage des pulpes de betterave est représentée à la figure 11a. L'ensemble constitué de deux éléments I et II fonctionne en régime permanent. Les différents flux sont :

<figure><img src="image_257.png" /><figcaption>**Figure 10.** Siccité ($x_i$ %) et gain de siccité $x_i$ en fonction de la dose d'adjuvant et de la vitesse de rotation de la presse à vis (d'après La Sucrerie Belge, 96, 1977).</figcaption></figure>

<figure><img src="image_258.png" /></figure>
**INTRANTS** : $\dot{M}$, débit de pulpes fraîches; $\dot{A}$, débit d'adjuvant de pressabilité ; $\dot{W}$, flux d'énergie mécanique ; $\dot{Q}$, flux d'énergie termique

**EXTRANTS** : $\dot{P}$, débits de pellets ; $\dot{M}_i$, débit de pulpes surpressées ; $\dot{R}_p$, débit de jus de presse ; $\dot{R}_s$, débit d'eau évaporée

Les pulpes sont arrosées à la sortie du diffuseur avec une solution de sulfate de calcium obtenue par acidification des écumes.

On cherche à optimiser la siccité $x_i$ valeur intermédiaire qui représente la siccité dû à l'apport d'adjuvant en considérant la séquence fictive de la figure 11b.

La formulation du problème est résumée ci-dessous :

$$ \begin{gathered} \text{Rendement} \qquad \eta_e = \frac{\dot{P}}{\dot{M}_i x_e} \\ \eta_e = \eta_{max} \frac{\dot{A}}{\dot{A}_0 + \dot{A}_e} \end{gathered} $$

### Coût Opératoire Unitaire en Francs

$$ \begin{gathered} \text{COUF (F/tP)} = \frac{\dot{A} \cdot v_a}{\dot{P}} + \frac{\dot{Q} \cdot v_e}{\dot{P}} + \frac{C_f}{\dot{P}} \\ \text{Coûts} = \text{adjuvant} + \text{énergie} + \text{fixe} \end{gathered} $$

### Coût Opératoire Unitaire en Energie Primaire

$$ \begin{gathered} \text{COUEP (tep/tP)} = \frac{\dot{A} \cdot CE_a}{\dot{P}} + \frac{\dot{Q} \cdot CE_e}{\dot{P}} + \frac{CE_f}{\dot{P}} \\ \text{Consommations} = \text{adjuvant} + \text{énergie} + \text{annexe} \end{gathered} $$

avec

$\eta_e$ = rendement de séparation dû à l'adjuvant (figure 12)
$x_e$ = teneur en matière séches des pulpes
$A_0$ = consommation spécifique (kg CaCl₂/tB)
$v_a, v_e$ = valeur de l'adjuvant (F/kg) et de l'énergic (F/kWh)
$CE_a, CE_e$ = contenu énergétique de l'adjuvant (tep/kg) et de l'énergie (tep/kWh)

$C_F$ = coûts fixes

$CE_F$ = contenu énergétique des coûts fixes

<figure><img src="image_259.png" /><figcaption>Figure 11a. Séquence pressage séchage des pulpes</figcaption></figure>

<figure><img src="image_260.png" /><figcaption>Figure 11b. Séquence fictive de pressage-séchage</figcaption></figure>

<figure><img src="image_261.png" /><figcaption>Figure 11c. Rendement de pressage $\eta_e$, dû à l'adjuvant en fonction de la dose d'adjuvant</figcaption></figure>
P. LEGOFF [2,3] a proposé d'appeler "diagrammes CARactéristiques Energie Coût (CAREC) les graphes donnant la variation du coût opératoire unitaire (COUF) en fonction de la consommation opératoire en énergie primaire (COUEP). Cette courbe met en évidence les minimums monétaire et énergétique (figure 13). Le décideur technique, au sein d'une entreprise de déshydratation rechercha à minimiser le coût opératoire en se plaçant au voisinage du point COUFopt de la CAREC. Dans un programme d'économie d'énergie, les animateurs rechercheraient au contraire la consommation minimale d'énergie primaire et s'approcheraient donc du point COUEPopt. Dans notre cas, les deux tendances se rejoignent car les deux optimums sont proches, la parabole CAREC étant relativement pointue.

## 6. CONCLUSION

Durant la campagne 1981/82, la siccité moyenne des pulpes était de 23%. L'optimum monétaire en francs et l'optimum énergétique en TEP (Tonne Equivalent Pétrole) correspondent, pour un pressage avec adjuvant, à des pulpes ayant une siccité de 25 %MS. Ce calcul est très approché mais donne une indication sur les coûts liés à l'adjonction, au cours du pressage, de produits chimiques ayant un contenu énergétique identique aux produits courants (4,3 TEP/t). Dans un avenir proche, des conceptions nouvelles en matière de pressage vont intervenir. P. LEGOFF [2], en 1980 dans une étude qui portait sur l'optimisation de la chaîne complète de pressage- séchage des pulpes de betteraves, estimait l'optimum énergétique pour une siccité des pulpes de 32 %MS. Depuis 1980, le prix de l'énergie ayant évolué à la hausse, l'optimum doit se situer actuellement entre 35 et 40 %MS ce qui montre le bien fondé des recherches importantes menées par plusieurs centres actuellement pour atteindre ces performances.

## BIBLIOGRAPHIE

1. S. REBOUILLAT ; Déshydratation par filtration-pressage ; Thèse de Docteur Ingénieur ; INPL / ENSIC ; Nancy, 1983.

2. P. LE GOFF ; Energétique Industrielle ; Tome 2 ; Technique et Documentation ; Paris, 1980.

3. P. LE GOFF ; Energétique Industrielle ; Tome 3 ; Technique et Documentation ; Paris, 1982.

<figure><img src="image_262.png" /><figcaption>Figure 13. Optimisation de l'apport d'adjuvant de pressabilite. CARactéristique Energie Coût.</figcaption></figure>
ANNEXE A7

LA COMPRESSION UNIDIRECTIONNELLE A DEFORMATION CONSTANTE
(CDC)

PUBLICATION P7

REBOUILLAT S. & PELEG M. *Selected physical and mechanical properties of commercial apple cultivars*. Journal of Texture studies 19 (3), 217-230, (1988). Abstract provided in A7.

Les propriétés mécaniques des produits biologiques ne sont pas systématiquement considérées comme essentielles, sans doute du fait de leur variabilité et du manque de procédures établies. Les résultats que nous présentons, à partir de douze variétés de pommes, sont importants quant à la prise en compte de la déformation vraie, par rapport à la déformation apparente, qui prend encore plus d'importance lors de déformations importantes de matériaux non linéaires, comme le sont beaucoup de produits de ce type. Les précautions expérimentales au niveau de la préparation des échantillons et de la mesure du fluage proprement dit sont décrites. L'analyse statistique n'est pas aussi défavorable que l'on pourrait imaginer avec des produits naturels. Le taux de déformation n'influe pas significativement dans le domaine des faibles vitesses. Ce genre d'étude peut s'avérer très utile pour l'anticipation des tests de compression en cellule. Dans tous les cas la caractérisation des produits, même à forte variabilité intrinsèque, devrait rester une préoccupation pour faciliter la progression scientifique. Cette publication, reproduite en annexe séparée, est disponible sur demande. Le résumé des résultats se trouve dans cette annexe A7.
RESEARCH NOTE

SELECTED PHYSICAL AND MECHANICAL PROPERTIES
OF COMMERCIAL APPLE CULTIVARS<fnref n="1" />

S. REBOUILLAT<fnref n="2" />

Laboratory of Chemical Engineering Science (LSQC)
University of Nancy
Nancy, France

and

M. PELEG

Department of Food Engineering
University of Massachusetts
Amherst, MA 01003

Manuscript received November 9, 1987; in final form February 22, 1988.

ABSTRACT

The compressive modulus of deformability, strength and strain of failure of ap-
ple flesh of 12 commercial apple cultivars were determined using a Universal
Testing Machine. The deformability modulus, calculated from the linear part of
the corrected stress-Hencky's strain relationship was between 1.5 to 2.3 MPa
with a coefficient of variation of 9-26%. The apparent upper limit of the strain
in the linear region was on the order of 0.08-0.2. The compressive strength was
on the order of 160-280 kPa, with the Baldwin apples a notable exception with
strength twice as large. Failure usually occurred at a strain of 0.12-0.16 with
the exception of the Baldwin and Granny Smith apples for which failure occurred
at a strain of 0.25 and 0.2, respectively. The variability in the failure parameters

<footnote n="1">¹Contribution of the Massachusetts Agricultural Experiment Station at Amherst</footnote>

<footnote n="2">²Currently with DuPont de Nemours International S.A., Case Postale 50-52, Route des Acacias CH-1211, Geneva, Switzerland</footnote>
expressed as the coefficient of variation was usually on the order of 15-20%. Neither the total soluble solid content nor the fruit density could serve as a predictor of the flesh mechanical properties. The effects of the specimen dimensions and the deformation rate (in the range of 0.5-2 cm min⁻¹) had only minor influence on the flesh mechanical behavior.

## INTRODUCTION

The mechanical properties of apple flesh have received much attention in the food literature. This is primarily because of the economic significance of apple, but also because the flesh is a relatively uniform tissue that enables testing the applicability of mechanical theories to the data interpretation. The latter included failure criteria (Hamann 1983; Miles and Rehkugler 1973; McLaughlin 1987), viscoelastic effects (Sharma and Rafie 1983; Mittal et al. 1987; Calzada and Peleg 1978). The role of turgor pressure and water activity (Lin and Pitt 1986; Bourne 1986) and the role of the specimen dimensions and friction (Hamann 1983). In most cases, the reported data refer to fruits of a single apple cultivar or a limited number of cultivars. It is therefore of interest to compare the behavior of fruits of various cultivars under the same test conditions. This communication is a report of such comparison, made with fruits as they were offered to the consumer in a local supermarket. It was done to determine the common and different mechanical features of the different apples and to assess their range of variation within the experimental lot size.

## EXPERIMENTAL

Apples of twelve commercial cultivars, mainly locally grown, were purchased at a local supermarket in the middle of their harvest season. They are listed in alphabetical order in Table 1. Individual fruit was characterized by its total soluble solids (TSS), determined by a refractometer and expressed in °Brix, density determined by flotation, and the diameter at the equator, measured by a caliper. External color was measured with a Hunter Lab color difference meter at equally spaced locations, covering about 2/3 of the total area, and the results were averaged. Summary of these physical measurements is given in Table 1.

Individual fruits were peeled, halved, and their seeds removed. Cylindrical specimens were bored from the flesh using cork borers of different diameters. The cylinders were trimmed to the desired height by a pair of parallel blades. The nominal dimensions of the specimens were as follows:
ANNEXE A8

LA COMPRESSION UNIDIRECTIONNELLE A DEFORMATION CONSTANTE
(CDC)

PUBLICATION P8

REBOUILLAT S., CAMPANELLA O.H. & PELEG M. *Mechanical characteristics of raw and processed peat*. Powder Technology 51, 273-275, (1987).

La tourbe est une source d'énergie mise en valeur dans certains pays comme l'Irlande, la Finlande et l'URSS. Sa déshydratation initiale subit les aléas climatiques et sa valorisation s'en trouve affectée. La littérature scientifique est peu abondante quant aux caractéristiques physiques et mécaniques de ce produit. Nos travaux sur la compressibilité et la relaxation de la tourbe sont donc uniques. Des modèles simples de caractérisation du fluage et de la relaxation sont développés et laissent entrevoir la possibilité de mieux utiliser la relaxation pour atteindre des paramètres de compressibilité ; ceci dans la mesure ou les tests de compression garantissent une certaine réversibilité. Il serait donc intéressant de tenter d'analyser la relaxation suivant la compression en cellule-piston, de nombreuses données ont été recueillies dans le cadre de ce projet. Leur exploitation dépasse l'objectif de ce rapport. La publication P8 figure en totalité dans cette annexe A8.
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Short Communication

Mechanical Characteristics of Raw and
Processed Peat

S. REBOUILLAT, O. H. CAMPANELLA
and M. PELEG

*Department of Food Engineering, University of
Massachusetts, Amherst, MA 01003 (U.S.A.)*

(Received February 23, 1987; in revised form March
16, 1987)

Peat is a product of decaying organic
matter and is apparently one of the initial
stages in the formation of coal. It is usually
considered as a low-grade energy source,
primarily because of its high moisture
content. In certain areas though, notably
in the U.S.S.R., Finland and Ireland, it is
used to fire power stations and as a domestic
and industrial fuel [1]. Currently, in the
U.S.A., peat is mainly used in agriculture
and horticulture [2]. It is, however, con-
sidered a potential energy resource and
raw material for a variety of chemical and
fermentation processes [2]. There is little
in the literature on the mechanical properties
of peat and the effect of dewatering and
processing on its physical characteristics.

The objective of this note is to demon-
strate the applicability of methods previously
used to characterize food and other powders
[3] to raw and dewatered peat.

Experimental

The bulk density of raw (78% moisture)
and industrially dewatered (49% moisture)
Canadian peat was determined using a metal
cell with a known volume. The cell was then
mounted on an Instron Universal Testing
Machine, model TM, and its contents com-
pressed, at a crosshead speed of 1.0 cm
min⁻¹, recording the force-displacement
curves [3]. In other experiments, the peat
was compressed, at the same crosshead speed,
to a predetermined deformation where the
crosshead was locked in position and the
force relaxation recorded.

The compressive force-deformation curves
were transformed to bulk density (ρ<sub>B</sub>)-
normal stress (σ<sub>N</sub>) relationships. The latter,
as has been the case with a variety of powders
at this stress range [4], could be described by

$$
\log_{10} \rho_B = a + b \log_{10} \sigma_N \quad (1)
$$

where *a* and *b* are empirical constants. The
constant *b* representing the change in bulk
density by the applied stress has been referred
to as the powder compressibility.

The recorded force relaxation curves were
normalized and linearized by [5]

$$
\frac{F(0)t}{F(0) - F(t)} = k_1 + k_2 t \quad (2)
$$

where F(0) is the force at time zero, F(t)
is the decaying force after time t and k₁
and k₂ are constants. The reciprocal of
constant k₂ in eqn. (2) represents an asymp-
totic level of the stress decay and can be used
to calculate an asymptotic residual modulus
Eₐ from [3]

$$
E_A = \frac{F(0)}{A\epsilon} \left(1 - \frac{1}{k_2}\right) \quad (3)
$$

where A is the cross-sectional area of the
specimen and ε the strain.

Results and discussion

The fit of eqns. (1) and (2) by which the
peat compressibility and asymptotic modulus
were calculated is demonstrated in Figs. 1 and
2, respectively. Statistical analysis of all the
experimental data yielded the following
regression coefficients. For the compression
results, the range was 0.976 < r² < 0.994,
and for the relaxation data 0.996 < r² <
0.999, with a corresponding significance level
of 0.001 or better. Consequently, the
methodology used for food powders under
low normal stresses (i.e., up to 10⁵ Pa or
1 kgf cm⁻²) could also be used for peat.

The mechanical parameters of raw and
dewatered peat are listed in the Table. It
shows that both the compression and relaxa-
tion tests were highly reproducible, thus
<table><caption>TABLE

Mechanical characteristics of raw and dewatered peat</caption><thead><tr><th rowspan="2">Type</th><th colspan="2">Density</th><th colspan="2">Compressibility<sup>a</sup></th><th rowspan="2">Henky's strain<br>(—)</th><th colspan="4">Relaxation parameters</th></tr><tr><th>Mean<br>(g cm<sup>-3</sup>)</th><th>C.O.V.<br>(%)</th><th>Mean<br>(—)</th><th>C.O.V.<br>(%)</th><th>Density</th><th>Mean<br>(g cm<sup>-3</sup>)</th><th>C.O.V.<br>(%)</th><th>Asymptotic modulus<sup>b</sup><br>Mean<br>(kPa)</th><th>C.O.V.<br>(%)</th></tr></thead><tbody><tr><td rowspan="2">Raw</td><td rowspan="2">0.63</td><td rowspan="2">1</td><td rowspan="2">0.11</td><td rowspan="2">10</td><td>0.27</td><td>0.81</td><td>1</td><td>21</td><td>3</td></tr><tr><td>0.49</td><td>1.02</td><td>0.6</td><td>53</td><td>5</td></tr><tr><td rowspan="2">Dewatered</td><td rowspan="2">0.53</td><td rowspan="2">3.5</td><td rowspan="2">0.06</td><td rowspan="2">12</td><td>0.23</td><td>0.66</td><td>0.9</td><td>50</td><td>13</td></tr><tr><td>0.39</td><td>0.79</td><td>0.7</td><td>173</td><td>4</td></tr></tbody></table>

<sup>a</sup>The parameter b in eqn. (1).

<sup>b</sup>The parameter E<sub>A</sub> calculated using eqn. (3).

<figure><img src="image_263.png" /><figcaption>Fig. 1. Typical linearized compression curves of raw (untreated) and dewatered (treated) peat. ρ<sub>B</sub> is the density and σ<sub>N</sub> the applied compressive stress. (See eqn. (1).)</figcaption></figure>

<figure><img src="image_264.png" /><figcaption>Fig. 2. Typical linearized relaxation curves of raw (untreated) and dewatered (treated) peat. F(0) is the initial force and F(t) the force after time t in relaxation. (See eqn. (2).)</figcaption></figure>

strengthening the view that the reported parameters are useful to characterize the mechanical properties of the peat. The bulk density of the processed peat was lower than that of the raw peat. This was mainly a result of the differences in water content. Therefore, the initial density could not serve as a cohesiveness indicator, as is the case in proper powders [3, 6].

The difference in compressibility, however, was of the same character that is observed in food powders, i.e., the more moist the powder, the larger was its compressibility. In peat though, since the moisture content is by far higher than that found even in very wet powders, the moisture-compressibility relation is most probably a measure of the plasticity of the material itself rather than the openness of the bed structure as a result

of attractive interactions between the particles. As could be expected, the relaxation pattern also indicated that dewatering reduced the material plasticity. This was evident from the higher asymptotic modulus of the processed peat. The magnitude of these moduli is an index of solidity, where zero means that the sample cannot sustain unrelaxed stress and is therefore a liquid by definition. It is therefore clear that dewatered peat particles, both individually and in bulk, have significantly more rigid structure than in their original wet state.

Support for this conclusion also comes from the observation that the asymptotic residual modulus of the dewatered peat was higher than that of raw peat even when the differences in density are taken into account. This excludes the alternative explanation that
the compactness of the structure was the
primary cause of the higher observed
moduli.

Conclusions

It has been shown that the compressibility
and stress relaxation pattern of peat can be
described by simple relationships (eqns.
(1) and (2)) whose applicability, as well as
parameters, can be tested and determined
by linear regression. No attempt was made
to analyze peat of different sources and/or
after different kinds of treatments. The good
fit of the mathematical equations and the
high reproducibility of the tests, however,
suggest that the described methodology can
be used as a convenient means to characterize
different peats and the effects of dewatering
and other processes on their mechanical
properties.

Acknowledgement

The work was supported by the Massa-
chusetts Agricultural Experiment Station in
Amherst. The authors express their thanks
to Mr. A. B. Allen for contributing the peat
for the study as well as background material.

References

1 L. A. Kristoferson and V. Bokalders, *Renewable Energy Technologies*, Pergamon, Oxford, 1986.

2 A. B. Allen, Private communication, 1987.

3 A. J. Dobbs, M. Peleg, R. E. Mudgett and R. Rufner, *Powder Technol.*, **32** (1982) 69.

4 J. Malave, G. V. Barbosa-Canovas and M. Peleg, *J. Food Sci.*, **50** (1985) 1473.

5 M. Peleg and R. Moreyra, *Powder Technol.*, **23** (1979) 277.

6 A. H. Hollenbach, M. Peleg and R. Rufner, *Powder Technol.*, **35** (1983) 51.
# ANNEXE A9

## LA DEFORMATION DES MILIEUX POREUX SATURES

## ELEMENTS DE REPONSE AUX QUESTIONS LES PLUS FREQUEMMENT POSEES (QFP)

### L'Influence de la Température sur la Compression :
Écoulement Hydrodynamique et Déformation de la
Structure ?

<table><thead><tr><th rowspan="2">v<br/>[cm3/g]</th><th>T: 11 [°C]</th><th>T: 45 [°C]</th><th rowspan="2">Ce 45 / Ce 11</th><th>T: 11 [°C]</th><th>T: 45 [°C]</th><th rowspan="2">η 45 / η 11</th></tr><tr><th>Ce 11<br/>[g<sup>2</sup>.cm<sup>-4</sup>.min<sup>-1</sup>]</th><th>Ce 45<br/>[g<sup>2</sup>.cm<sup>-4</sup>.min<sup>-1</sup>]</th><th>η 11<br/>10<sup>-3</sup> [Pa.s]</th><th>η 45<br/>10<sup>-3</sup> [Pa.s]</th></tr></thead><tbody><tr><td>1.5</td><td>0.32</td><td>7.1</td><td>22</td><td>1.259</td><td>0.586</td><td>0.5</td></tr></tbody></table>

22

<figure><img src="image_265.png" /></figure>

2
Initiale

<table><thead><tr><th>Reference</th><th>Z [cm]</th><th>MS [%]</th><th>ν [cm3/g]</th><th>M<sub>T</sub> [g]</th><th>M<sub>s</sub> [g]</th><th>w [-]</th><th>V<sub>T</sub> [cm3]</th><th>M<sub>f</sub> [g]</th><th>γ<sub>s</sub> [g/cm3]</th></tr></thead><tbody><tr><td>F1</td><td>1.62</td><td>22.5</td><td>4.00</td><td>232</td><td>52</td><td>3.45</td><td>208</td><td>180</td><td>1.84</td></tr><tr><td>C5</td><td>2.18</td><td>47.8</td><td>1.66</td><td>360</td><td>172</td><td>1.09</td><td>281</td><td>188</td><td>1.86</td></tr><tr><td>Cr2</td><td>1.71</td><td>17.0</td><td>5.40</td><td>241</td><td>41</td><td>4.88</td><td>220</td><td>200</td><td>2.01</td></tr></tbody></table>

La Compressibilité du Solide ?

<table><thead><tr><th>Reference</th><th>Z [cm]</th><th>MS [%]</th><th>ν [cm3/g]</th><th>M<sub>T</sub> [g]</th><th>M<sub>s</sub> [g]</th><th>w [-]</th><th>V<sub>T</sub> [cm3]</th><th>M<sub>f</sub> [g]</th><th>γ<sub>s</sub> [g/cm3]</th></tr></thead><tbody><tr><td>F1</td><td>0.64</td><td>30.4</td><td>1.57</td><td>106</td><td>52</td><td>2.29</td><td>82</td><td>54</td><td>1.87</td></tr><tr><td>C5</td><td>1.55</td><td>49.0</td><td>1.19</td><td>280</td><td>172</td><td>1.04</td><td>205</td><td>107</td><td>1.77</td></tr><tr><td>Cr2</td><td>0.83</td><td>30.5</td><td>2.64</td><td>127</td><td>41</td><td>2.28</td><td>108</td><td>87</td><td>1.90</td></tr></tbody></table>

Finale
L/D (hauteur de l'échantillon/diamètre cellule)
rapport critique ?

<figure><img src="image_266.png" /></figure>
<figure><img src="image_267.png" /><figcaption>Pressoir Olives et Noix</figcaption></figure>

<figure><img src="image_268.png" /><figcaption>Pressoir à Corde</figcaption></figure>

<figure><img src="image_269.png" /><figcaption>Scène de pressage 1500</figcaption></figure>

<figure><img src="image_270.png" /><figcaption>Pressoir du
Village</figcaption></figure>

<figure><img src="image_271.png" /></figure>
En hommage à Henry Philibert Gaspard Darcy (1803-1858)
A l'approche du 200<sup>ème</sup> anniversaire de sa naissance à Dijon ...

<figure><img src="image_272.png" /><figcaption>Schéma de l'appareil utilisé par Darcy pour déterminer la loi de l'écoulement de l'eau
à travers le sable.</figcaption></figure>
INSTITUT NATIONAL
POLYTECHNIQUE
DE LORRAINE

AUTORISATION DE SOUTENANCE DE THESE
DU DOCTORAT D'ETAT

000

VU LES RAPPORTS ETABLIS PAR :

Monsieur HERMIA Jacques, Professeur, Université Catholique de Louvain, Belgique

Monsieur LECLERC Dominique, Professeur, LSGC-ENSIC

Monsieur ROUX Jean-Claude, Professeur, Ecole Française de Papeterie et des Industries Graphiques, Saint Martin d'Hères

Le Président de l'Institut National Polytechnique de Lorraine, autorise :

Monsieur REBOUILLAT Serge

à soutenir devant un jury de l'INSTITUT NATIONAL POLYTECHNIQUE DE LORRAINE, une thèse intitulée :

«Contribution à l'étude fondamentale de la déformation de milieux poreux saturés».

en vue de l'obtention du grade de :

DOCTEUR D'ETAT
SCIENCES

Spécialité : «CHIMIE-PHYSIQUE» GENIE DES PROCESSES

NANCY BRABOIS
2, AVENUE DE LA
FORET DE HAYE
BOITE POSTALE 3
F - 5 4 5 0 1
VANCEUVRE CEDEX

Fait à Vandoeuvre le, 28 octobre 2002

Le Président de l'IN.P.L.

L. SCHUETZENNECKER

Le Secrétaire Général

J.Y. RIVIERE

TEL. 33/03.83.59.59.51

FAX. 33/03.83.59.59.52

<figure><img src="image_273.png" /></figure>
## THESE DE DOCTORAT D'ETAT - SCIENCES - INPL - NANCY - Serge REBOUILLAT

### RESUME

Environnement, énergie et développement durable sont intrinsèquement liés. C'est dans ce contexte que se situent les travaux rapportés dans ce mémoire riche de 700 références.

Un potentiel important de réduction de la consommation d'énergie existe dans les industries minières, agroalimentaires, papetières, électroniques, pharmaceutiques et biotechnologiques, qui ont toutes à traiter de séparations solides-liquides ô combien pesantes. Le type de séparation solide-liquide considéré dans ce travail scientifique prend en compte des produits et dérivés provenant de ces six secteurs. La démarche scientifique présente un intérêt fondamental pour la caractérisation des écoulements hydrodynamiques et des déformations rapides. Elle pourrait s'étendre à la déformation des sols lors des catastrophes « dites naturelles ».

Plus axée sur la compréhension fondamentale et la composante pratique des déformations multidimensionnelles de milieux poreux saturés, cette étude privilégie quatre modes de compression.

Plusieurs cellules de compression multi-mode (MMCC©) et une unité pilote de pressage à vis (TSPU©) ont été conçues et favorablement testées pour la compression multidimensionnelle.

La compression à pression constante, CPC, et les modèles proposés conviennent à l'étude du comportement des milieux saturés d'origine minérale comme le talc et le kaolin. La compression à déformation constante, CDC, sans hypothèse excessivement restrictive, permet de mettre à disposition trois approches théoriques adaptables à une large gamme de situations. Un des apports importants de la CDC réside dans la rapidité des tests. Une base est ainsi établie pour l'étude à gradient de pression contrôlé. Les études de la relaxation, qui suit la compression CDC, ouvrent de nouvelles voies de recherche.

En matière de compression avec écoulement multi-axial, sont développées la compression axiale avec écoulement radial central ou périphérique et de façon plus générale la compression axiale avec écoulement tridimensionnel puis son extension au pressage à vis. On note un bon degré de prévisibilité des performances du pilote (TSPU©) à partir de la cellule de laboratoire, MMCC©.

Enfin une application de la thermodynamique statistique et le développement du calcul de l'énergie consommée par unité de volume comprimé semblent très prometteurs.

## TITLE: Fundamental Study of the Deformation of Saturated Porous Materials

### ABSTRACT

This thesis describes a method, which uses a new theory to access the dynamic compression coefficient of biological and food materials. A multi-mode large diameter expression cell has been built to study the dynamic compression characteristics of saturated porous materials.

An important theoretical background has been developed regarding multidimensional expression such as the one occurring during axial compression and radial flow. This set of equations is also useful to model a screw pilot press designed and instrumented for this study. Both the coefficients of compression obtained during piston-cell compression and during screw pressing agree, leading the way to more meaningful engineering and scientific database.

It has also been shown that the pressure relaxation pattern can be described by simple equations. Finally, relationships between the expression parameters and the Arrhenius law are shown.

Energy wise the dewatering of bio-products remains a huge potential area for energy saving as being reported in some of the 700 publications included in this thesis. It is also applicable to the new dynamic dimension, which take so-called natural catastrophes.

### DISCIPLINE: SCIENCES

### MOTS-CLES

Compression, consolidation, déformation, pressage, compactage, essorage, filtration, perméation, écoulement, dynamique, radial, axial, multi-axial, cellule de compression multi-mode, optima énergétiques et économiques, milieux poreux saturés, produits agroalimentaires, biotechnologie, poudres minérales, boues organiques, pulpes, tourbes, céramique, protéines, plastique, élastique, visco-élastique, rhéologie, loi de Darcy généralisée.

### KEYWORDS

Compression, consolidation, deformation, filtration, permeability, radial, multi-axial, multi-mode compression cell, energy, saturated porous materials, food and agricultural products, biotechnology, mineral powders, pulps, peat, proteins, visco-elastic, rheology, generalised Darcy law.

ADRESSE : Laboratoire des Sciences du Génie Chimique, ENSIC, rue Henri Déglin, 54001 Nancy Cedex

DIRECTEUR DE RECHERCHE : Pr. Dominique LECLERC, Université Henri Poincaré, Nancy