Runge-Kutta method in Excel VBA

6 pages project ( determine the salt concentrations using a 2nd order or higher Runge-Kutta method in Excel VBA)

2

>Salt Concentration Data

(g/L)

(g/L)

0 0

5

8

.

3

0

0

5

1

.

3

9

4

79

7

8.0

7

.

2

7

7

8.9

8

7

6

1

1

9

9

1.1

8

8

1.2

8

.

89

4

7

.6

3

872

9

128

12

014

7

1.6

467

.3987

0

.54

761

7

31

1.9

328

1

195

2

2

467

8

3

.2428

2.3

391

2

.05

713

2.5

042

783

979

2.6

9

427

2

2.7

86

013

302

1.7833

125

7

2.9

5

87

3

414

087

032

3.1

929

425

438

3.2

.3047

.

04

074

7

334

3.4

438

047

3.5

.5732

277

823

3.6

863

8

3.7

21

.3716

2

826

3

3.9

552

3

4

489

953

43

891

818

68

4.2

1

643

879

4.3

2

883

381

225

834

016

762

4.6

796

155

881

4.7

064

511

977

4.8

8

4.9

517

904

817

5

004

585

811

5.1

348

67

566

5.2

652

373

567

5.3

363

53

56

365

566

308

253

325

5.6

326

867

5.7

96

37

5.8

8

157

5.9

355

6

495

851

6.1

513

793

195

6.2

642

616

6.3

873

04

6.4

887

996

052

6.5

689

472

948

6.6

8

19

6.7

267

514

979

6.8

346

683

6.9

324

423

7

65

7.1

193

24

7.2

475

674

7.3

478

046

7.4

621

143

626

7.5

172

7.6

515

113

7.7

488

847

7.8

47

973

286

7.9

5

703

8

786

559

8.1

4

319

8.2

308

99

21

8.3

81

386

8.4

954

8.5

997

479

472

8.6

027

8.7

085

231

8.8

787

8.9

632

756

955

9

439

976

9.1

149

9.2

083

451

9.3

752

55

921

9.4

025

9.5

09

536

9.6

093

27

9.7

921

575

54

9.8

095

838

591

9.9

59

198

10

287

336

948

331

815

387

10.3

156

231

843

215

953

502

864

665

593

825

862

624

252

005

965

219

11

034

291

11.1

442

768

725

165

788

265

781

028

484

362

103

594

778

293

927

993

863

281

611

646

585

124

536

12

945

155

733

198

543

939

076

67

223

358

885

12.4

824

294

514

12.5

337

238

173

399

127

12.7

972

569

348

12.9

16

646

13

13.1

671

774

1

298

745

112

143

089

063

099

725

554

638

14

669

164

71

902

67

273

563

773

009

43

681

073

376

384

14.9

869

15

544

966

832

153

662

875

765

746

777

444

992

382

98

973

.5959

573

16

248

721

16.1

512

934

868

711

458

301

683

222

764

285

16.9

188

534

17

142

76

827

17.2

561

52

326

17.4

706

243

912

75

17.7

921

17.8

01

18

611

072

52

865

18.3

959

078

66

18.5

289

499

415

968

421

401

906

154

646

19

99

487

31

788

667

726

775

19.5

102

043

809

974

205

524

20

20.1

20.4

61

20.6

389

38

302

632

20.8

839

858

20.9

21

21.2

102

791

627

7

694

21.8

97

082

315

22

465

852

22.3

545

249

216

476

22.6

343

206

241

229

22.9

286

653

23

941

073

876

175

23.3

757

23.4

432

512

864

721

527

878

876

24

829

41

985

24.2

941

735

24.6

251

24.8

25

062

312

362

25.3

25.4

757

032

136

399

516

257

663

909

251

26

447

675

722

999

541

044

684

449

672

26.7

574

26.8

532

26.9

831

27

053

348

27.1

749

78

27.3

147

58

26

27.4

019

469

678

065

9

405

847

28

921

382

9

984

472

163

687

28.5

082

746

459

653

29

992

944

671

244

29.4

347

29.5

98

362

29.7

808

902

535

29.9

30

681

30.1

702

553

22

30.3

246

246

641

932

322

30.5

346

997

846

779

804

579

31

372

747

996

96

032

94

303

533

31.7

704

58

31.9

32

181

32.1

078

299

32.4

811

4

32.5

32.6

471

32.7

836

32.8

32.9

6

33

412

223

33.3

33.4

33.5

773

33.7

095

43

34

113

507

34.2

34.3

942

34.4

65

681

34.5

34.6

16

34.7

869

113

856

35

35.2

751

35.4

299

35.5

963

355

35.6

949

519

35.8

35.9

142

36

109

36.1

45

36.2

75

36.3

36.4

36.5

36.6

139

36.7

237

36.8

36.9

37

37.1

37.2

37.4

018

37.5

378

37.7

37.8

37.9

357

38

562

38.1

91

38.2

788

38.4

38.6

38.7

38.9

39

39.1

105

39.3

39.4

69.3578

39.5

39.6

39.7

687

39.8

8

39.9

629

40

40.3

40.4

40.5

40.6

40.9

41

41.1

41.2

41.3

3

41.4

41.5

591

41.6

41.8

41.9

42

012

42.3

42.6

42.7

275

42.8

932

42.9

158

43

43.1

619

02

43.3

416

43.4

897

43.5

43.8

43.9

44

094

44.1

44.2

612

44.3

972

44.4

9

44.7

44.8

44.9

45

471

45.1

425

45.3

45.5

45.6

401

45.8

205

46

426

809

46.3

46.4

4

46.5

208

309

46.7

46.8

46.9

47

47.1

2

47.3

47.4

941

47.7

47.9

48

48.1

284

48.3

48.4

48.5

48.6

387

48.7

49

963

49.1

49.2

49.3

415

49.5

49.6

8.0566

49.8

49.9

50

50.2

50.3

50.4

50.5

50.6

50.7

50.9

51

51.1

51.3

51.8471

51.5

51.6

51.7

51.8

51.9

52

52.2

52.3

52.4

52.5 6.3617

52.7

52.8

52.9 5.3674

53

9.884

53.1

53.2

53.3

53.4

53.8

54

54.1

54.3

54.4

54.5

54.6

54.7

54.8

54.9

55

55.1

55.2

55.4

55.5

55.7

55.8

55.9

56

56.2

56.3

56.4

56.5

56.6

56.7

56.9

57

57.1

57.2

57.3

57.4

57.5

57.6

57.7

57.8

58

58.1

58.3

58.5

58.6

58.7

58.8

59

59.1

59.2

59.3

59.6

59.9

60

60.2

60.3

60.4

60.5

60.6

60.8

60.9

61

61.1

4.5293

61.2

61.3

2.8

61.5 3.1

61.6

61.8

62

4.8199

62.1

62.2

62.3

62.4

62.5

62.6

62.7

62.9

63

63.1

63.4

63.5

3

63.6

63.8

63.9

64

64.2

64.3

64.5

64.6

64.8

64.9

65

65.1

65.2

65.3

65.4

65.5

65.6

65.7

3.787

65.8

65.9

66

66.2

66.3

66.4

66.5

66.6

66.7

66.9

67

1

67.2

67.4

43

67.5

67.6

67.7

67.8 1.9347

67.9

68

68.2

68.3

68.4

68.5

68.7

68.9

69

6

69.1

69.2

69.3

69.5

69.6

69.8

2.678

69.9

70

70.1

70.2

70.5

70.6

70.7

70.8

70.9

71

71.1

71.2

71.4

71.5

71.7

71.8

71.9

72

72.1

72.2

72.3

72.4

2.2081

72.6

72.7

72.8

72.9

73

73.2

1.729

73.3

73.4

1.778

73.5

73.6

73.7 1.064

73.8

74

74.1

74.2

74.3

74.5

1.91

74.6

74.7

74.9

75

75.1

75.2

1.6444

75.3

75.4 1.071

75.5

75.6

75.7

75.8

76

76.1

1.3844

76.3

76.4

76.5

76.6

76.7

76.8

76.9

77

77.1

77.2

77.3

77.5

77.6

77.7

1.435

77.9

78

78.3

78.4

78.5 0.7013

78.6

78.8

79

79.1

79.3

79.4

79.5

79.6

79.7

79.8

80

80.1

80.2

80.3

80.4

80.5

80.7

1.17

81

81.1

81.2

81.3

81.4

81.5

81.6

0.9195

81.7

81.8

81.9

82

82.1

82.2

82.4

82.6

82.7

3

82.8

82.9

83

83.2

83.3

83.4

8

0.921

83.5

0.7845

83.6

83.7

83.8

83.9

84

0.8707

84.1

84.3

84.4

0.8694

84.6

84.7

84.9

0.719

85

85.1

64

85.3

85.4

85.5

8.9928

85.6

8.9095

85.8

85.9

86

86.1

86.2

86.3

0.735

86.4

11.2473

86.6

0.63

86.7

86.8

86.9

87

87.1

87.2

10.93

6

87.8

87.9

88 0.346

88.1

2

88.4

88.5

88.8

0.5267

89

89.1

89.2 0.3152 0.498

89.3

89.4 0.357

89.7

89.8

89.9

90

90.2

0.558

90.3

90.5

0.4516

90.8

0.5187

91

91.1

91.2

91.4

91.5

91.7

0.41

91.8

6.54

91.9

92

92.1

92.2

92.3

92.4

92.6

92.8

92.9

93

93.1

93.2

93.3

93.4

93.6

93.7

93.8

93.9

7

94

94.2

94.3

94.5

94.6

94.7

94.8

95

0.3348

0.3741

95.2

95.3

0.327

95.5 0.218

95.8

0.3548

96

96.4

96.5

0.2979

0.3329

96.7

0.291

0.2796

97

0.3287

97.2

97.3

0.2652

97.4

0.266

97.5

97.6

6.487

97.8 0.1892

98

98.1

98.2

98.3

98.4

98.5

98.6

98.7

98.9

0.2445

99

99.2

0.2364

99.3

99.4 0.1686

99.5

5.8714

99.6

99.8

0.2207

99.9

100

t (s) Cs

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1 Cs2 (g/L) Cs

3
0 32 9
0.1 31 2.

5 8 7 6
0.2 37 5.6 97 1

3.1 15 0.0

30
0.3 31

4 57 14 1

8.9 92 0.0

89
0.4 30

5.3 47 2

4.9 72 0.

17 67
0.5 30

6.3 79 30.

12 11 0.

28
0.6 30

3.9 38 4

1.9 13 0.5728
0.7 34 20 41 91 36 0.7

93
0.8 32

8.8 96 5

4.3 51 1.00

95
0.9 35 63 5

2.9 62 1.1 71
30

7.6 16 5

6.1 98 1.2 87
34

3.7 27 7

1.6 78 1.9

54
29 6.4 61 76 40 2.3 19
1.3 34

7.9 64 81 39 2.6
1.4 32

9.5 73 8

6.7 3.

10
1.5 28

9.6 46 7

7.8 2.7 44
33 6.8 86 3.96

58
1.7 33

7.3 25 90 24 3.4
1.8 286.00

65 92.

56 4.

99
27

4.7 9

3.2 66 5.7
32

6.5 52 101.13

88 4.6
2.1 29

7.2 42 102.5191 6.6 49
2.2 2

77 104.7865 5.7892
3

13.1 106.8712 7.8

23
2.4 2

70 83 132.0019 8.6
30

8.3 1

34.3 7.1
303.0

55 1

17.2 9.6

43
28

3.6 1

20.1 8.4
2.8 26 12

4.2 11.1 75
25

9.7 45 12

8.2 9.5446
25

7.7 1

29.5 12.7
29

6.9 15

3.5 10.3
2

48 1

60 74 11.1673
3.3 2

92.4 13

5.2 82 14.9
24

9.8 13

7.5 12.0866
2

50 1

61.1 16.1
24

5.9 165.0602 12.9 21
244.0799 173.

68 18
3.8 2

86.4 59 1

74.3 18.3 53
2

81.6 1

73.3 22 20.0116
2

38.4 1

75.3 20.8
4.1 2

41.6 15

7.4 16.9
2

29.7 85 1

52.4 19.5
2

70.5 154.0093 17.8
4.4 2

65.9 1

62.3 24.0223
4.5 2

34.2 1

56.9 2

4.8
2

22.6 1

87.8 1

9.1
2

42.8 1

92.8 20.9
2

57.7 94 166.0688 20.959
21

5.1 1

92.1 2

8.5
2

20.6 1

61.2 2

8.1
2

47.1 1

96.5 22.9
2

21.8 1

81.4 27.1
2

45.6 1

95.2 24.8
5.4 22

9.9 1

72.4 2

5.8
5.5 2

44.1 202.8333 2

6.2
203.7476 1

94.5 34.7
247.0949 1

95.3 34.4
2

45.1 198.074 35.6
241.0682 206.3411 37.1
21

9.4 203.3148 36.6
1

98.1 1

71.8 38.2
2

26.8 206.5068 30.3
203.6674 1

99.4 32.1
2

25.4 1

99.9 31.7
1

89.9 1

75.6 42.7
230.3

84 206.6299 42.9
1

97.4 1

76.3 43.3
1

93.8 1

78.5 34.7713
223.0237 1

78.8 35.4
21

9.2 202.1661 45.1389
1

83.3 208.1688 35.5
217.8785 1

81.3 37.8
208.2563 1

79.5 36.8
1

86.9 1

77.7 3

8.7
206.9936 202.9025 50.6
203.4518 1

79.7 51.9
1

82.6 208.7541 41.1
171.

69 2

12.5 40.4
1

74.1 181.6271 53.4
1

74.6 2

12.4 41.6831
201.2

80 1

73.7 56.079
1

73.8 17

9.3 44.4
194.5028 18

1.17 43.5
1

92.9 173.3019 45.6898
1

91.4 1

92.3 57.2
195.3934 1

72.9 58.0189
160.0836 1

76.5 60.9
162.3728 206.9035 48.1
1

59.3 1

91.2 62.4
1

91.1 171.8573 49.3
181.0415 205.8576 48.3
1

59.1 1

90.2 48.3244
1

57.8 1

99.3 51.1
1

83.6 200.8274 50.6174
1

57.1 207.9434 52.9
1

79.1 175.3993 66.6
1

51.3 1

76.8 69.1
1

79.6 1

70.7 67.4
173.357 19

6.54 70.6
1

75.2 205.3515 55.8
10.1 170.6804 1

74.9 71.4
10.2 1

48.7 202.3386 70.2
1

69.6 201.7423 56.2
10.4 1

46.4 175.2978 54.8
10.5 1

46.8 1

83.9 58.5
10.6 138.0855 1

99.8 72.2
10.7 1

41.5 1

97.3 57.6
10.8 143.5064 1

71.2 57.5
10.9 1

39.3 167.01 57.3
1

39.8 189.9447 77.5
1

37.2 1

94.6 68.5
11.2 1

55.2 1

90.3 61.5
11.3 1

36.5 1

68.2 77.3
11.4 1

32.6 1

94.3 79.3
11.5 1

30.1 1

66.9 63.6
11.6 1

53.2 176.8938 81.2
11.7 1

54.9 1

63.8 62.4936
11.8 1

32.7 1

64.9 80.3
11.9 131.7357 159.0435 64.8
1

37.5 1

89.7 64.5
12.1 1

44.7 1

85.4 66.4
12.2 1

46.3 1

84.6 84.0486
12.3 1

22.3 1

80.4 64.3
1

45.3 1

60.8 85.1
1

20.4 1

61.8 83.4
12.6 1

23.4 179.7558 67.9
136.8164 1

85.6 65.2
12.8 1

39.6 181.489 88.0456
139.8355 1

52.2 68.3
140.0247 154.8363 89.0739
1

17.4 1

82.2 7

0.719
13.2 1

25.3 1

53.3 71.1
13.3 134.3071 1

47.7 69.9
13.4 129.5644 178.0315 90.2479
13.5 110.1945 1

54.4 71.4516
13.6 109.3405 1

50.4 91.1842
13.7 132.6027 1

68.9 90.0201
13.8 108.8485 17

4.5293 92.0931
13.9 1

30.5 1

46.5 73.7023
1

17.7 1

45.8 92.9747
14.1 1

27.3 1

43.4 73.5
14.2 108.9916 141.1785 92.9988
14.3 125.4044 145.1234 71.7
14.4 121.0078 1

41.2 72.8
14.5 1

24.6 168.3018 96.7
14.6 118.0155 167.4051 93.7
14.7 111.5246 1

66.5 72.9463
14.8 105.1273 1

44.3 73.2
1

18.5 137.509 75.2355
9

8.0566 1

35.2 73.8077
15.1 97.2 1

41.9 74.0849
15.2 97.6 1

33.7 75.2217
15.3 98.9 134.4997 74.6244
15.4 99.2 1

46.9 99.2686
15.5 104.7948 1

56.3 75.1
15.6 109.1818 1

58.3 96.4
15.7 109.3674 1

57.4 76.5246
15.8 108.5198 1

56.5 100
15.9 107.6788 1

28.5 97.371
91.7 153.3776 75.4
109.0167 152.4872 80.1
16.2 92.2 1

47.3 101.822
16.3 98.4 1

39.5 98.5
16.4 90.8 1

49.8 103.2961
16.5 105.6762 144.7992 103.5772
16.6 103.8952 1

23.3 100.24
16.7 102.1355 1

26.7 102.3105
16.8 100.3968 124.6149 100.7534
99.6 1

46.7 77.3434
83.9264 1

40.5 104.8921
17.1 83.2 1

44.9 80.4527
100.0749 1

41.4 81.5
17.3 100.2103 119.5687 81.7
84.9 138.4236 79.1477
17.5 80.7 116.806 78.3
17.6 83.6586 135.4317 79.4
83.0103 1

34.5 83.336
92.8819 127.3854 103.9063
17.9 93.9 113.9312 106.8886
94.0359 1

35.8 104.2886
18.1 77.0426 1

32.5 81.1
18.2 89.1 116.7656 84.0932
76.6 1

33.3 103.8857
18.4 90.3183 115.2911 84.3
87.1 110.9014 107.9672
18.6 75.7 113.8264 82.7
18.7 76.7 1

29.9 108.2838
18.8 86.7 126.7171 94.2
18.9 76.4 1

29.4 106.691
86.1 1

27.4 105.8833
19.1 85.5 1

24.2 80.4576
19.2 70.7234 1

26.9 106.1412
19.3 77.9 105.3099 94.8
19.4 69.6314 106.9222 80.7345
82.9 103.9385 105.5407
19.6 83.0293 107.7956 82.8
19.7 68.7 101.4509 106.7011
19.8 82.4 119.819 85.8
19.9 81.1138 1

21.2 82.0777
79.7468 103.874 84.056
66.6733 109.771 85.0779
20.2 77.7885 109.0409 82.2711
20.3 75.7429 118.0623 83.2859
67.9982 117.2737 105.3732
20.5 66.7 112.2138 82.4334
65.5 95.5 110.2941
20.7 65.7 97.0045 86.3
69.3 95.3037 84.4
72.9769 96.7409 96.0501
72.4113 98.153 107.6208
21.1 63.0383 108.7759 108.6226
71.2933 109.0566 107.6977
21.3 73.4 101.2288 109.6537
21.4 72.1 95.512 84.6671
21.5 70.9 106.8423 82.7615
21.6 7

1.064 108.0949 107.8027
21.7 58.8 108.3392 109.7443
59.6 89.0109 110.7198
21.9 60.5 105.8845 84.7
66.9912 104.1906 81.8
22.1 57.6927 100.5977 108.8088
22.2 57.2456 88.4 84.7339
66.6805 103.9356 86.6
22.4 5

6.3617 103.2152 107.8267
22.5 55.9 88.5 83.7
54.8883 86.0555 82.7717
22.7 65.8 98.3 83.7181
22.8 65.3 82.1 105.8265
64.2 98.7 109.6465
64.3155 82.7786 105.7686
23.1 52.7 94.7 109.5796
23.2 54.6 95.8 110.5018
59.9 80.1681 86.4465
53.8 91.8 85.4506
23.5 50.614 80.781 85.4126
23.6 60.8241 78.4 85.3
23.7 60.3 81.3579 86.2
23.8 57.6874 92.6 83.367
23.9 58.8758 78.5223 106.3049
48.6 90.5 108.1563
24.1 48.3056 89.8 109.0457
50.5 76.8627 109.9309
24.3 49.6 86.9839 106.9906
24.4 55.0564 86.3645 106.9152
24.5 47.3457 76.0414 107.7908
51.6 72.286 109.6152
24.7 54.8107 86.1234 108.5766
53.8776 83.9239 106.5849
24.9 45.895 86.4673 84.6258
53.0458 83.5 81.7024
25.1 53.1 72.0593 109.1557
25.2 44.8 84.6193 85.3477
53.2887 71.0273 81.4775
52.8 71.2744 85.1845
25.5 51.5 71.5 104.9564
25.6 43.9 68.0085 84.0684
25.7 42.6 81.6147 106.6271
25.8 51.7 67.7 83.8
25.9 49.9 79.7077 84.7408
43.0565 78.4048 107.2185
26.1 41.3 68.4 106.1568
26.2 42.8476 77.2 106.9715
26.3 49.2 66.057 83.4137
26.4 42.6348 76.1 80.5
26.5 40.9 76.3004 84.1
26.6 39.7 76.4401 105.5158
46.0355 75.1935 93.2
39.1 65.0579 102.4496
47.4 73.4211 102.314
39.4 74.2 92.8876
38.6 73.0224 102.0367
27.2 44.7008 63.1 104.6739
38.9 71.9 78.6
39.4435 71.4311 8

2.2081
27.5 37.9 61.8002 104.225
27.6 38.4257 61.3 81.9
27.7 43.8 68.5889 102.9963
27.8 38.2349 69.3578 7

9.884
27.9 36.7 67.5 103.5992
43.1 57.6828 91.5
28.1 42.071 66.5939 101.4473
28.2 41.3548 58.6 80.2
28.3 40.6 65.6 80.1687
28.4 41.8 57.2191 104.5947
36.2 54.9842 90.8015
28.6 34.7931 63.5 79.7717
28.7 33.7729 64.8908 99.5
28.8 40.5857 54.3 100.279
28.9 35.0988 53.3765 78.4611
36.6597 52.3 90.0276
29.1 38.1 52.0111 79.0843
29.2 33.5 62.5 78.0461
29.3 39.0372 52.9509 102.9784
38.7 51.9929 77.7635
33.4 58.8992 75.8
29.6 37.087 52.3371 101.5209
32.9 50.2 75.5
29.8 37.2033 51.0133 100.2518
36.2314 57.1672 78.8087
36.9 58.3628 99.8678
31.9 50.9 77.6
30.2 31.7224 50.0432 98.5981
30.1512 58.1 97.5
30.4 36.1 55.5 74.5
35.8838 52.055 76.1326
30.6 34.6 55.7 98.6
30.7 29.8697 54.87 74.0802
30.8 29.9569 55.4 98.2
30.9 29.7247 54.5 75.5103
34.5147 45.1248 99.6038
31.1 33.0019 53.2469 74.3295
31.2 32.4 46.9213 75.8968
31.3 29.1205 52.4524 94.6682
31.4 28.2865 44.2 97.8
31.5 32.8 50.7 73.6
31.6 31.4432 45.5 76.9
27.3368 42.8103 75.0779
31.8 27.1249 49.5 72.3
26.9147 43.5755 73.8956
26.1255 42.3 72.035
27.3632 50.3 97.2392
32.2 26.2937 50.3876 97.0184
32.3 26.09 41.8254 93.4
26.1692 49.1 74.7
25.6871 47.4675 71.2158
25.211 41.3324 91.9
24.7407 47.1963 70.8
24.549 41.5875 71.5486
28.6892 4

6.487 72.2091
25.2442 47.4306 91.0813
33.1 29.3121 44.9331 93.3
33.2 29.085 39.9 70.8672
24.1371 37.9349 92.0686
23.6897 45.5965 91.0242
27.8973 39.0247 89.9833
33.6 23.5802 37.0818 70.1
22.8888 43.344 93.6
33.8 22.7115 42.6104 70.6414
33.9 23.2866 43.0936 93.1
23.6031 43.9667 72.7
34.1 26.1319 42.8406 89.4
26.9079 35.4303 92.4389
25.9712 41.8037 88.1
2

1.91 41.487 87.9
21.9856 42.3271 68.5979
22.5266 40.4788 69.2
24.7048 41.6878 87.2
34.8 21.2449 34.2257 87.0589
34.9 21.3119 40.3 67.8
20.9169 33.7083 88.1762
35.1 2

0.5267 34.1877 70.6683
20.8202 39.7653 89.2
35.3 23.5781 38.0151 68.7301
20.276 35.9 68.5457
24.3195 37.7 89.3
23.0341 32.9089 85.9
35.7 23.291 32.6588 88.8
20.0866 31.365 67.8055
23.1457 36.3 69.1567
21.2651 37.4 85.825
20.0453 36.4 84.0597
22.1929 31.0974 86.8
19.3203 35.8923 83.5935
18.7583 36.6177 83.3601
21.4764 35.6782 84.6377
18.2657 31.1445 66.3
18.5269 35.1366 64.6
21.3806 30.6717 65.9391
21.6114 29.7971 84.4309
18.4929 34.3393 85.6798
20.6921 29.029 63.1471
20.9191 33.5045 83.707
37.3 17.2975 28.2771 84.9426
17.5449 28.6781 62.6
19.8688 30.6013 82.2474
37.6 17.6495 33.4045 62.2
20.3073 28.6291 81.0375
2

0.3348 32.8964 64.0573
16.692 27.3032 79.8
18.2007 31.2178 62.9
19.3239 27.7644 63.4
16.4861 31.9026 81.2857
38.3 18.8477 26.1905 61.6
18.7016 26.8471 63.6394
38.5 18.5567 26.9253 80.5579
18.5866 25.3131 8

0.3152
15.8571 30.7016 61.6488
38.8 18.4706 29.0819 70.6459
18.1578 24.7367 60.5708
18.5222 29.7296 80.7489
15.5383 29.7728 78.402
39.2 15.5837 29.0077 62.1
17.6014 24.2546 79.3126
17.9547 24.3334
14.7383 27.5591 76.7491
15.2669 28.3899 78.5788
17.0621 23.2616 60.4
17.2463 27.7001 60.2
14.4437 23.6701 77.1
16.98 22.731 57.8609
40.1 16.8484 23.058 57.6789
40.2 13.9571 23.13 75.0844
13.6967 25.9143 60.0124
16.6107 23.2665 74.6097
16.1823 26.2475 67.0023
13.3807 26.2874 57.4384
40.7 15.9325 22.7353 74.5645
40.8 15.6626 22.3234 57.0716
13.3625 22.3878 74.0871
15.1325 21.5144 57.3646
15.0152 21.5814 74.2676
12.7705 21.646 72.062
15.065 24.9075 7

3.787
1

5.3674 21.315 58.5777
12.8917 2

0.7013 72.6
14.855 20.5423 73.0685
41.7 14.8764 23.7081 73.4738
14.3548 20.8876 55.8881
12.0936 19.8544 7

1.071
12.3998 23.3825 72.1131
42.1 12.5683 23.6325 54.7
42.2 12.2084 19.8267 54.5201
14.0676 22.2129 56.2347
42.4 13.7001 19.3132 69.2724
42.5 13.5939 22.9143 69.0415
11.4526 22.5315 70.062
11.3638 21.7429 69.8
11.7769 18.319 69.5
11.437 18.3801 55.1
13.4453 21.2451 55.5473
12.9739 21.2806 67.6
43.2 10.93 21.5117 55.1724
12.8941 18.2128 52.5
12.555 20.7933 54.1
10.9154 21.019 69.1753
43.6 10.9485 17.987 68.9393
43.7 10.9804 17.8486 53.0344
10.6635 20.349 69.069
12.5361 17.014 52.0729
10.8412 19.8515 66.2
10.644 20.251 68.359
11.7974 19.7297 65.1
10.3682 17.0399 67.2
10.3984 16.9087 52.357
44.5 11.7448 18.742 49.8303
44.6 1

1.435 16.6493 52.578
9.7263 16.6968 64.6203
10.187 18.8355 49.3113
10.0017 18.8634 65.9039
11.1911 18.3747 51.8
11.3139 18.4035 65.4
45.2 11.2262 15.5563 57.2039
11.2423 17.9532 49.5921
45.4 9.2107 15.6504 63.0494
9.6471 18.1732 50.3745
9.0685 16.3938 47.9
45.7 10.9978 17.4061 47.7701
10.5158 15.3349 62.7
45.9 9.253 14.7362 55.8012
9.7674 16.8478 63.9
46.1 8.7225 16.5602 48.7553
46.2 8.7511 16.902 48.5
8.7787 14.4423 61.6087
8.7106 14.3309 6

0.291
8.831 14.0675 60.6
46.6 8.6694 13.959 48.4
10.1746 14.303 48.7985
10.004 15.9855 61.5882
8.1962 13.6385 60.2888
9.8495 13.3862 46.1255
8.3386 13.5749 6

0.921
47.2 9.4305 13.4701 59.1024
9.4457 15.8095 46.1549
9.4601 12.8352 47.0438
47.5 7.8223 13.3023 46.8721
47.6 9.0553 15.0251 59.2
8.1294 12.5404 58.5542
47.8 7.6419 14.9324 44.2755
7.8354 12.4848 45.6698
9.1123 14.5666 44.4677
8.7928 12.8332 58.7
48.2 7.7369 14.6107 51.3261
8.657 13.9659 58.8069
8.833 12.0103 56.042
8.0409 12.1795 43.1448
7.4999 11.9556 56.6
7.2834 14.0554 56.9321
48.8 7.4626 12.1555 42.6657
48.9 8.574 1

1.9347 56.5089
8.1982 13.2284 56.7
8.1346 11.6261 56.0878
7.9954 11.6603 44.0101
6.9512 11.3241 49.2643
49.4 6.9723 13.313 56.4
8.1829 11.2711 56.2295
6.7909 12.8676 42.866
49.7 12.8876 55.8071
6.5407 12.788 54.6298
6.8505 10.9268 41.4197
6.6543 12.4746 42.2227
50.1 7.4547 12.3782 54.0126
7.5379 12.3974 53.8079
6.2911 10.4792 41.2701
6.5892 12.4326 52.9275
7.2951 10.0936 41.4277
6.4874 10.4607 52.5253
7.318 10.2694 42.0469
50.8 6.2528 10.2997 40.9555
6.1376 10.0026 41.2626
6.9506 10.249 53.1128
7.0281 11.7755 51.0689
51.2 6.5174 11.6845 51.3313
6.7902 11.4888 50.221
51.4 6.7376 11.5046
6.6854 10.8968 50.7414
6.8231 11.0185 39.2632
6.6449 9.4006 44.956
5.5981 9.1252 38.9606
6.7892 10.0607 50.8544
6.1241 10.5819 50.2131
52.1 5.5905 8.9152 38.9524
6.4516 8.9445 39.2421
5.4444 9.0704 50.5091
6.4708 10.6455 49.8751
10.2751 48.8093
52.6 5.3772 9.5287 49.0532
5.5095 8.6986 38.0516
5.7546 10.3201 48.6708
9.961 4

8.9095
6.119 48.718
5.3407 8.7076 47.2501
5.9687 10.0046 36.8886
5.535 9.7467 36.7437
5.1077 8.3281 46.2749
53.5 5.722 8.2637 47.7688
53.6 5.7858 8.3761 47.998
53.7 5.741 9.1862 35.7526
5.803 9.2019 46.7918
53.9 5.758 7.8387 35.0583
5.5561 7.949 46.0117
5.6691 9.0749 36.8273
54.2 4.8511 8.9205 45.6479
4.7623 7.599 36.5358
5.3859 8.7831 44.4775
4.6887 7.6463 45.9116
4.6524 8.7293 35.2995
4.6163 8.9046 45.5468
5.2208 8.5144 34.2229
4.3984 8.7673 34.879
4.4613 8.4623 43.8194
4.5711 7.3766 34.994
5.1564 8.3318 34.4623
55.3 4.7374 7.0311 35.1043
4.7007 8.2801 33.4097
4.3844 8.216 43.3356
55.6 4.3967 7.0956 44.3177
4.133 8.0144 43.7557
4.7845 8.0267 32.4935
4.9735 6.8584 43.7848
4.8003 8.0494 42.4707
56.1 4.6741 6.6801 43.4323
4.8146 6.7725 42.1284
3.9883 6.7201 32.9671
4.0444 7.5903 42.5346
4.6172 7.7427 40.876
4.0248 6.2859 32.9383
3.9511 6.5145 32.8047
56.8 3.8783 6.3953 31.9374
4.5594 6.1411 40.5821
3.8185 7.1091 41.5094
4.4066 6.1136 32.2746
3.7595 6.1996 30.6984
3.852 7.1438 30.5733
3.621 6.2352 41.1951
4.2317 6.9034 31.0377
4.2385 6.9792 30.2002
4.3235 5.771 40.692
3.705 6.0444 31.0108
57.9 4.1407 5.8714 31.5856
3.5326 6.7029 39.4949
3.4291 5.9051 39.6809
58.2 4.1208 6.5996 30.1584
3.4511 6.5485 31.0701
58.4 3.4244 5.7691 38.8498
4.0626 6.6283 38.6898
3.2982 5.5007 39.2124
3.4545 5.4581 29.2031
3.2834 6.4755 29.4208
58.9 3.3295 6.4254 29.9728
3.8366 5.2744 37.8981
3.5249 5.4061 30.0595
3.1478 6.2773 29.9351
3.644 6.2287 36.767
59.4 3.7535 6.1243 28.3681
59.5 3.7585 5.9654 29.2359
3.6616 5.0341 37.6199
59.7 3.095 4.9951 32.577
59.8 3.1711 5.8823 35.6857
3.0472 5.8908 29.0759
3.4508 5.0945 36.9979
60.1 3.0328 5.5871 36.8438
3.4946 5.0159 27.7569
3.3712 4.8199 35.9016
3.3451 5.7183 28.4748
2.9714 4.7455 27.4106
3.3875 5.4253 31.3747
60.7 3.2679 5.4849 34.6802
2.8102 4.5353 26.7569
3.3094 4.6502 26.6447
3.2533 4.6638 27.4586
3.1678 33.7952
2.7541 4.4942 33.9592
3.208 5.3319 26.2002
61.4 5.1944 34.5845
4.3906 27.1887
3.134 5.1616 34.5948
61.7 3.1673 5.1216 29.9557
2.6285 5.1286 34.3038
61.9 3.005 4.3024 33.8621
2.5316 26.6204
2.7911 4.8737 25.3298
2.5202 4.158 32.5549
2.9953 4.7985 29.2047
2.5631 4.0939 33.4433
2.8679 4.1505 28.958
2.4967 4.6879 32.5838
2.4241 3.9125 31.8714
62.8 2.8282 4.0979 25.4461
2.3604 4.7082 32.1708
2.8366 4.5019 31.4672
2.3498 4.0034 25.1232
63.2 2.7671 3.8888 24.4542
63.3 2.3644 4.5225 24.63
2.6487 4.117 31.4924
2.678 3.84 24.6981
2.732 3.6886 24.5928
63.7 2.5876 4.4243 31.6416
2.2252 3.6317 30.6846
2.2323 3.7224 23.4607
2.6002 3.7329 23.6321
64.1 2.5561 3.587 27.0472
2.6074 4.1395 29.6246
2.4931 4.0307 30.8385
64.4 2.1237 3.9994 29.3713
2.5241 3.9685 22.8646
2.1829 3.3752 23.0313
64.7 2.1204 3.3863 30.0494
2.4206 3.3969 23.0964
2.0203 3.4073 23.5198
2.116 3.8171 28.8831
2.0554 3.7876 28.4998
2.193 3.2571 29.667
1.9801 3.8712 29.0255
2.375 3.1716 29.1562
2.2495 3.3219 22.4095
1.9557 3.8166 25.3555
1.9195 22.2165
1.9674 3.1086 28.6562
1.9313 3.7286 22.2753
2.1843 3.1279 22.4282
66.1 2.2491 3.6377 24.8125
2.2114 3.477 27.1756
2.0131 3.1215 27.3039
1.8577 3.0973 26.9407
1.8631 3.5586 27.3117
1.9666 2.9532 21.3661
2.049 2.8666 27.0752
66.8 2.0525 2.9393 27.6798
1.9213 3.2928 27.3199
1.792 2.8317 26.724
67.1 1.778 3.242 27.3205
1.7456 2.8186 20.8152
67.3 1.7134 2.736 21.1956
1.9404 2.8656 2

0.63
1.9987 2.8434 25.6805
1.7285 2.7323 19.7575
1.697 3.0942 25.6885
3.1287 19.5855
1.6353 3.1335 20.6471
1.605 2.7061 25.5819
68.1 1.8375 2.9994 22.7412
1.5628 2.551 25.8116
1.5679 2.6437 25.4733
1.8122 2.5674 24.6885
1.6964 3.046 19.8878
68.6 1.6832 2.6102 19.8007
1.7871 2.8626 19.9354
68.8 1.5412 2.4347 24.4792
1.6444 2.4158 24.3718
1.729 2.9298 24.2648
1.5219 2.4842 19.5878
1.7029 2.4125 19.2851
1.4506 2.8622 24.1622
69.4 1.471 2.5819 24.4855
1.4282 2.3825 23.7363
1.4327 2.3641 18.9481
69.7 1.6224 2.6989 18.4407
1.4106 17.9374
1.3844 2.2847 23.3209
1.6001 2.612 23.6364
1.6027 2.6895 18.5342
1.4119 2.6687 23.6357
70.3 1.342 2.624 23.3249
70.4 1.3316 2.5559 23.0164
1.5391 2.1805 22.9147
1.5559 2.2107 22.8134
1.3295 2.4969 22.7125
1.3333 2.5239 20.1893
1.4919 2.4584 18.0899
1.2988 2.4165 17.8097
1.4827 2.443 17.3323
1.4575 2.3568 17.6522
71.3 1.5008 2.1158 17.179
1.259 2.4088 22.0174
1.4776 2.0612 22.3111
71.6 1.2662 2.3716 22.4069
1.2432 2.0078 16.4881
1.4173 2.2707 16.4148
1.3802 2.1256 22.1095
1.1628 2.2568 16.8433
1.3589 2.1974 21.5322
1.3611 2.2842 21.8157
1.136 2.1841 16.2417
1.3651 16.3574
72.5 1.3173 2.2113 16.846
1.3071 2.1941 18.6344
1.2969 2.1372 21.1484
1.1533 2.1801 20.6847
1.2046 2.1632 16.1797
1.0997 2.0488 15.5584
73.1 1.2691 2.033 15.4889
1.2357 15.6011
1.0743 2.0207 20.7686
1.2282 20.6757
1.0692 1.9895 15.7507
1.0495 1.9554 15.502
1.9772 19.5124
1.213 1.7235 19.7782
73.9 1.0587 1.9285 15.2946
1.2054 1.8053 20.1264
1.196 1.6838 19.165
1.0017 1.6885 19.5994
1.0047 1.9047 18.9934
74.4 1.1255 1.8549 15.2984
0.9785 19.3366
0.9815 1.5506 19.5904
1.1205 1.8463 15.2627
74.8 0.9768 1.5606 14.8565
1.1238 1.801 14.7896
1.1049 1.6701 18.9058
1.1166 1.5578 14.4903
0.9065 14.7566
0.9195 1.4848 18.4865
1.4895 14.2954
0.9053 1.5261 14.3945
0.8983 1.4824 14.3296
0.9204 1.5025 17.9933
0.9133 1.5694 17.7507
75.9 1.0206 1.5572 14.1365
1.0317 1.4524 17.5908
1.0143 1.6712 18.3073
76.2 1.0157 13.9458
1.0079 1.4189 14.0405
1.0092 1.4229 17.7461
0.8648 1.6051 17.6659
0.9846 1.3272 17.4303
0.8515 1.6096 13.4783
0.9427 1.3212 16.9644
0.9619 1.5703 17.6552
0.7968 1.2866 17.5752
0.9123 1.5461 17.4955
0.7845 1.4074 16.9619
0.8041 1.5362 17.3372
77.4 0.7639 1.2471 16.9584
0.8843 1.5125 16.4333
0.7856 1.5007 14.8715
0.8707 12.7314
77.8 0.757 1.4104 16.3578
0.8572 1.4528 12.9094
0.8911 1.2167 16.3554
78.1 0.7395 1.181 12.9376
78.2 0.8694 1.224 12.8788
0.7914 1.4083 15.9892
0.8245 1.4102 15.9164
1.3865 16.1295
0.7268 1.1991 15.7718
78.7 0.8208 1.1898 12.5883
0.8145 1.1681 16.0508
78.9 0.8308 1.1344 15.5572
0.8169 1.2847 15.6257
0.6842 1.1169 12.0825
79.2 0.7748 1.301 11.7509
0.6809 1.1355 12.3853
0.6611 1.0793 15.6164
0.7929 1.2475 15.6815
0.7582 1.086 15.6098
0.7736 1.2514 12.0254
0.7535 1.0462 11.9705
79.9 0.7547 1.0609 13.3885
0.735 1.1885 11.9945
0.7224 1.1793 13.2662
0.7304 1.0253 11.885
0.7248 1.1943 14.591
0.7393 1.1741 13.0849
0.6269 1.1541 11.0712
80.6 0.5956 0.9723 14.9101
0.6894 1.0076 14.8417
80.8 0.7036 14.5167
80.9 0.6917 1.1082 11.1253
0.5837 0.9739 14.2566
0.7001 0.956 11.1502
0.5684 1.1032 11.2251
0.6831 0.9208 14.4378
0.6715 0.9339 11.2473
0.5676 0.9166 10.9468
0.6673 13.7445
0.5771 0.9124 10.9697
0.5545 0.8856 10.5512
0.5383 1.0741 10.5026
0.635 1.056 10.6975
0.6359 1.0575 12.1003
0.6252 1.0111 13.7308
82.3 0.6319 0.8992 10.5504
0.604 1.0331 13.6047
82.5 0.5194 0.8573 13.3044
0.623 0.8507 13.4796
0.5226 0.9909 1

0.357
0.5855 0.8193 11.7154
0.498 0.9756 10.3787
0.5106 0.8963 13.1166
83.1 0.5121 0.9516 12.825
0.5838 0.8207 10.3507
0.5042 0.8056 12.8214
0.558 12.7622
0.565 9.7543
0.5502 0.9239 12.5316
0.4887 0.9082 12.8109
0.5417 0.7749 12.6398
0.4863 0.7689 9.6865
0.4571 12.5232
0.5544 0.8969 12.4652
84.2 0.5501 0.8654 12.2977
0.5409 0.8668 9.8364
0.453 0.8521 12.0754
84.5 0.5276 9.6372
0.5187 0.7439 12.1794
0.5243 0.7853 11.9084
84.8 0.4343 0.7091 9.0768
0.4452 11.9045
0.5075 0.8285 8.9928
0.4663 0.7003 9.0564
85.2 0.41 0.7176 12.0541
0.4866 0.6895 11.9982
0.4555 0.6767 8.8271
0.4249 0.8043
0.4888 0.7981 9.2597
85.7 0.4183 0.7774
0.4636 0.7785 8.7662
0.4776 0.6724 11.4649
0.4043 0.6672 8.5839
0.3925 0.7042 8.7451
0.3981 0.6289 11.4058
0.4629 11.2531
0.3793 0.6467 11.2999
86.5 0.4516 0.6144
0.3817 11.2932
0.4446 0.6116 8.6993
0.4493 0.6336 11.1883
0.4418 0.6022 11.1362
0.3781 0.6566 8.2891
0.3632 0.7102 8.3464
0.3683 0.6982 8.3075
87.3 0.4126 0.6864
87.4 0.3548 0.5919 10.5953
87.5 0.3482 0.6695 10.7342
87.6 0.3455 0.6705 10.4029
87.7 0.346 0.6653 10.3544
0.3515 0.5615 8.1706
0.4087 0.6122 7.8552
0.5589 8.1865
0.3434 0.5425 10.3455
88.2 0.403 0.5443 7.9279
88.3 0.327 0.5935 7.8002
0.3859 0.6183 10.1109
0.3901 0.5434 10.0637
88.6 0.3303 0.6088 7.7808
88.7 0.377 0.6213 8.0114
0.3147 0.6051 10.0118
88.9 0.3122 7.9367
0.3718 0.5957 9.9183
0.3074 0.5298 9.8719
7.8258
0.3161 0.5985 7.616
0.5884 9.6477
89.5 0.3642 0.5784 9.8598
89.6 0.3285 0.4935 8.5335
0.3488 0.5801 9.6826
0.3558 0.5651 9.6373
0.2985 0.5607 7.5727
0.3471 0.5563 7.5372
90.1 0.3002 0.4746 7.0851
0.2979 9.1259
0.3329 0.5537 7.1013
90.4 0.2871 0.5494 9.3692
0.3339 0.4751 7.2802
90.6 0.2857 7.1647
90.7 0.3287 0.4579 9.2379
0.2693 0.4495 9.0332
90.9 0.2672 6.8234
0.2652 0.5099 9.0284
0.266 0.5155 8.9065
0.3191 0.521 8.8646
91.3 0.2706 0.4276 7.0899
0.3027 0.4943 7.0565
0.3004 0.4904 8.5841
91.6 0.2812 0.4224 8.9321
0.2539 6.6482
0.2547 0.4068
0.2555 0.4709 8.8066
0.248 0.405 6.631
0.2867 0.4195 6.5998
0.2925 0.4644 8.4563
0.2476 0.4348 8.4165
0.2774 0.4012 8.3768
92.5 0.2464 0.4024 6.4019
0.2445 0.3822 6.2976
92.7 0.2788 0.4467 6.5629
0.2792 0.3847 6.3118
0.2796 0.3776 8.1812
0.227 0.4405 6.2524
0.2302 0.433 6.1505
0.2681 0.3689 6.4816
0.2266 0.3741 6.236
0.2567 0.435 7.9901
93.5 0.2207 0.3643 8.0234
0.2551 0.4322 8.1268
0.2603 0.3509 7.8071
0.2156 0.3637 6.0902
0.218 0.3608 7.6639
0.2496 0.4152 7.6276
94.1 0.2153 0.4157 7.6606
0.2113 0.4088 7.8305
0.2279 0.4093 7.7251
94.4 0.2148 0.4061 7.7566
0.2379 0.3883 7.5845
0.2338 0.3962 7.4138
0.2364 0.3391 7.5129
0.2039 0.3293 5.8751
94.9 0.2045 0.3267 5.7144
0.1964 7.4068
95.1 0.2356 5.9237
0.1976 0.3816 7.4678
0.2003 7.4325
95.4 0.2281 0.3143 5.5155
0.366 5.4894
95.6 0.1895 0.3195 5.4634
95.7 0.2228 0.3037 7.2287
0.1866 0.3046 5.6027
95.9 0.1892 5.5128
0.1817 0.3521 7.1894
96.1 0.1882 0.3526 5.4606
96.2 0.1868 0.3531 6.9965
96.3 0.1756 0.3026 5.4711
0.2032 0.3476 6.9302
0.2017 5.3578
96.6 0.1791 5.2711
0.1985 0.3365 5.307
96.8 0.1707 5.1604
96.9 0.1694 6.8278
0.1755 5.2919
97.1 0.1686 0.3142 5.2668
0.1728 0.2673 6.5522
0.1805 5.1576
0.1612 5.1921
0.1777 0.2756 5.2849
0.1763 0.2878
97.7 0.1837 0.3085 5.1766
0.3117 5.152
97.9 0.1843 0.2559 5.7612
0.1863 0.293 6.4219
0.1781 0.2935 6.3913
0.1582 0.2472 5.1114
0.1569 0.2508 4.9176
0.1789 0.2461 6.413
0.1742 0.2469 4.9268
0.155 0.2397 6.4078
0.1521 0.2484 6.211
98.8 0.1494 0.2622 4.7464
0.1673 4.7788
0.1423 0.2401 6.1227
99.1 0.1412 0.2382 4.8965
0.1557 6.0103
0.1699 0.2371 4.7961
0.2677 6.0604
0.1642 0.2682
0.1418 0.2217 6.0559
99.7 0.1378 0.2689 4.5467
0.1605 5.9983
0.1607 0.2575 5.9697
0.1375 0.2173 4.6383

Cs1 0 0.1 0.2 0.30000000000000004 0.4 0.5 0.6 0.7 0.79999999999999993 0.89999999999999991 0.99999999999999989 1.0999999999999999 1.2 1.3 1.4000000000000001 1.5000000000000002 1.6000000000000003 1.7000000000000004 1.8000000000000005 1.9000000000000006 2.0000000000000004 2.1000000000000005 2.2000000000000006 2.3000000000000007 2.4000000000000008 2.5000000000000009 2.600000000000001 2.7000000000000011 2.8000000000000012 2.9000000000000012 3.0000000000000013 3.1000000000000014 3.2000000000000015 3.3000000000000016 3.4000000000000017 3.5000000000000018 3.6000000000000019 3.700000000000002 3.800000000000002 3.9000000000000021 4.0000000000000018 4.1000000000000014 4.2000000000000011 4.3000000000000007 4.4000000000000004 4.5 4.5999999999999996 4.6999999999999993 4.7999999999999989 4.8999999999999986 4.9999999999999982 5.0999999999999979 5.1999999999999975 5.2999999999999972 5.3999999999999968 5.4999999999999964 5.5999999999999961 5.6999999999999957 5.7999999999999954 5.899999999999995 5.9999999999999947 6.0999999999999943 6.199999999999994 6.2999999999999936 6.3999999999999932 6.4999999999999929 6.5999999999999925 6.6999999999999922 6.7999999999999918 6.8999999999999915 6.9999999999999911 7.0999999999999908 7.1999999999999904 7.2999999999999901 7.3999999999999897 7.4999999999999893 7.599999999999989 7.6999999999999886 7.7999999999999883 7.8999999999999879 7.9999999999999876 8.0999999999999872 8.1999999999999869 8.2999999999999865 8.3999999999999861 8.4999999999999858 8.5999999999999854 8.6999999999999851 8.7999999999999847 8.8999999999999844 8.999999999999984 9.0999999999999837 9.1999999999999833 9.2999999999999829 9.3999999999999826 9.4999999999999822 9.5999999999999819 9.6999999999999815 9.7999999999999812 9.8999999999999808 9.9999999999999805 10.09999999999998 10.19999999999998 10.299999999999979 10.399999999999979 10.499999999999979 10.599999999999978 10.699999999999978 10.799999999999978 10.899999999999977 10.999999999999977 11.099999999999977 11.199999999999976 11.299999999999976 11.399999999999975 11.499999999999975 11.599999999999975 11.699999999999974 11.799999999999974 11.899999999999974 11.999999999999973 12.099999999999973 12.199999999999973 12.299999999999972 12.399999999999972 12.499999999999972 12.599999999999971 12.699999999999971 12.799999999999971 12.89999999999997 12.99999999999997 13.099999999999969 13.199999999999969 13.299999999999969 13.399999999999968 13.499999999999968 13.599999999999968 13.699999999999967 13.799999999999967 13.899999999999967 13.999999999999966 14.099999999999966 14.199999999999966 14.299999999999965 14.399999999999965 14.499999999999964 14.599999999999964 14.699999999999964 14.799999999999963 14.899999999999963 14.999999999999963 15.099999999999962 15.199999999999962 15.299999999999962 15.399999999999961 15.499999999999961 15.599999999999961 15.69999999999996 15.79999999999996 15.899999999999959 15.999999999999959 16.099999999999959 16.19999999999996 16.299999999999962 16.399999999999963 16.499999999999964 16.599999999999966 16.699999999999967 16.799999999999969 16.89999999999997 16.999999999999972 17.099999999999973 17.199999999999974 17.299999999999976 17.399999999999977 17.499999999999979 17.59999999999998 17.699999999999982 17.799999999999983 17.899999999999984 17.999999999999986 18.099999999999987 18.199999999999989 18.29999999999999 18.399999999999991 18.499999999999993 18.599999999999994 18.699999999999996 18.799999999999997 18.899999999999999 19 19.100000000000001 19.200000000000003 19.300000000000004 19.400000000000006 19.500000000000007 19.600000000000009 19.70000000000001 19.800000000000011 19.900000000000013 20.000000000000014 20.100000000000016 20.200000000000017 20.300000000000018 20.40000000000002 20.500000000000021 20.600000000000023 20.700000000000024 20.800000000000026 20.900000000000027 21.000000000000028 21.10000000000003 21.200000000000031 21.300000000000033 21.400000000000034 21.500000000000036 21.600000000000037 21.700000000000038 21.80000000000004 21.900000000000041 22.000000000000043 22.100000000000044 22.200000000000045 22.300000000000047 22.400000000000048 22.50000000000005 22.600000000000051 22.700000000000053 22.800000000000054 22.900000000000055 23.000000000000057 23.100000000000058 23.20000000000006 23.300000000000061 23.400000000000063 23.500000000000064 23.600000000000065 23.700000000000067 23.800000000000068 23.90000000000007 24.000000000000071 24.100000000000072 24.200000000000074 24.300000000000075 24.400000000000077 24.500000000000078 24.60000000000008 24.700000000000081 24.800000000000082 24.900000000000084 25.000000000000085 25.100000000000087 25.200000000000088 25.30000000000009 25.400000000000091 25.500000000000092 25.600000000000094 25.700000000000095 25.800000000000097 25.900000000000098 26.000000000000099 26.100000000000101 26.200000000000102 26.300000000000104 26.400000000000105 26.500000000000107 26.600000000000108 26.700000000000109 26.800000000000111 26.900000000000112 27.000000000000114 27.100000000000115 27.200000000000117 27.300000000000118 27.400000000000119 27.500000000000121 27.600000000000122 27.700000000000124 27.800000000000125 27.900000000000126 28.000000000000128 28.100000000000129 28.200000000000131 28.300000000000132 28.400000000000134 28.500000000000135 28.600000000000136 28.700000000000138 28.800000000000139 28.900000000000141 29.000000000000142 29.100000000000144 29.200000000000145 29.300000000000146 29.400000000000148 29.500000000000149 29.600000000000151 29.700000000000152 29.800000000000153 29.900000000000155 30.000000000000156 30.100000000000158 30.200000000000159 30.300000000000161 30.400000000000162 30.500000000000163 30.600000000000165 30.700000000000166 30.800000000000168 30.900000000000169 31.000000000000171 31.100000000000172 31.200000000000173 31.300000000000175 31.400000000000176 31.500000000000178 31.600000000000179 31.70000000000018 31.800000000000182 31.900000000000183 32.000000000000185 32.100000000000186 32.200000000000188 32.300000000000189 32.40000000000019 32.500000000000192 32.600000000000193 32.700000000000195 32.800000000000196 32.900000000000198 33.000000000000199 33.1000000000002 33.200000000000202 33.300000000000203 33.400000000000205 33.500000000000206 33.600000000000207 33.700000000000209 33.80000000000021 33.900000000000212 34.000000000000213 34.100000000000215 34.200000000000216 34.300000000000217 34.400000000000219 34.50000000000022 34.600000000000222 34.700000000000223 34.800000000000225 34.900000000000226 35.000000000000227 35.100000000000229 35.20000000000023 35.300000000000232 35.400000000000233 35.500000000000234 35.600000000000236 35.700000000000237 35.800000000000239 35.90000000000024 36.000000000000242 36.100000000000243 36.200000000000244 36.300000000000246 36.400000000000247 36.500000000000249 36.60000000000025 36.700000000000252 36.800000000000253 36.900000000000254 37.000000000000256 37.100000000000257 37.200000000000259 37.30000000000026 37.400000000000261 37.500000000000263 37.600000000000264 37.700000000000266 37.800000000000267 37.900000000000269 38.00000000000027 38.100000000000271 38.200000000000273 38.300000000000274 38.400000000000276 38.500000000000277 38.600000000000279 38.70000000000028 38.800000000000281 38.900000000000283 39.000000000000284 39.100000000000286 39.200000000000287 39.300000000000288 39.40000000000029 39.500000000000291 39.600000000000293 39.700000000000294 39.800000000000296 39.900000000000297 40.000000000000298 40.1000000000003 40.200000000000301 40.300000000000303 40.400000000000304 40.500000000000306 40.600000000000307 40.700000000000308 40.80000000000031 40.900000000000311 41.000000000000313 41.100000000000314 41.200000000000315 41.300000000000317 41.400000000000318 41.50000000000032 41.600000000000321 41.700000000000323 41.800000000000324 41.900000000000325 42.000000000000327 42.100000000000328 42.20000000000033 42.300000000000331 42.400000000000333 42.500000000000334 42.600000000000335 42.700000000000337 42.800000000000338 42.90000000000034 43.000000000000341 43.100000000000342 43.200000000000344 43.300000000000345 43.400000000000347 43.500000000000348 43.60000000000035 43.700000000000351 43.800000000000352 43.900000000000354 44.000000000000355 44.100000000000357 44.200000000000358 44.30000000000036 44.400000000000361 44.500000000000362 44.600000000000364 44.700000000000365 44.800000000000367 44.900000000000368 45.000000000000369 45.100000000000371 45.200000000000372 45.300000000000374 45.400000000000375 45.500000000000377 45.600000000000378 45.700000000000379 45.800000000000381 45.900000000000382 46.000000000000384 46.100000000000385 46.200000000000387 46.300000000000388 46.400000000000389 46.500000000000391 46.600000000000392 46.700000000000394 46.800000000000395 46.900000000000396 47.000000000000398 47.100000000000399 47.200000000000401 47.300000000000402 47.400000000000404 47.500000000000405 47.600000000000406 47.700000000000408 47.800000000000409 47.900000000000411 48.000000000000412 48.100000000000414 48.200000000000415 48.300000000000416 48.400000000000418 48.500000000000419 48.600000000000421 48.700000000000422 48.800000000000423 48.900000000000425 49.000000000000426 49.100000000000428 49.200000000000429 49.300000000000431 49.400000000000432 49.500000000000433 49.600000000000435 49.700000000000436 49.800000000000438 49.900000000000439 50.000000000000441 50.100000000000442 50.200000000000443 50.300000000000445 50.400000000000446 50.500000000000448 50.600000000000449 50.70000000000045 50.800000000000452 50.900000000000453 51.000000000000455 51.100000000000456 51.200000000000458 51.300000000000459 51.40000000000046 51.500000000000462 51.600000000000463 51.700000000000465 51.800000000000466 51.900000000000468 52.000000000000469 52.10000000000047 52.200000000000472 52.300000000000473 52.400000000000475 52.500000000000476 52.600000000000477 52.700000000000479 52.80000000000048 52.900000000000482 53.000000000000483 53.100000000000485 53.200000000000486 53.300000000000487 53.400000000000489 53.50000000000049 53.600000000000492 53.700000000000493 53.800000000000495 53.900000000000496 54.000000000000497 54.100000000000499 54.2000000000005 54.300000000000502 54.400000000000503 54.500000000000504 54.600000000000506 54.700000000000507 54.800000000000509 54.90000000000051 55.000000000000512 55.100000000000513 55.200000000000514 55.300000000000516 55.400000000000517 55.500000000000519 55.60000000000052 55.700000000000522 55.800000000000523 55.900000000000524 56.000000000000526 56.100000000000527 56.200000000000529 56.30000000000053 56.400000000000531 56.500000000000533 56.600000000000534 56.700000000000536 56.800000000000537 56.900000000000539 57.00000000000054 57.100000000000541 57.200000000000543 57.300000000000544 57.400000000000546 57.500000000000547 57.600000000000549 57.70000000000055 57.800000000000551 57.900000000000553 58.000000000000554 58.100000000000556 58.200000000000557 58.300000000000558 58.40000000000056 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11.1502 11.225099999999999 14.437799999999999 11.247299999999999 10.9468 13.7445 10.9697 10.5512 10.502599999999999 10.6975 12.100300000000001 13.7308 10.5504 13.604699999999999 13.304399999999999 13.4796 10.3573 11.715400000000001 10.3787 13.1166 12.824999999999999 10.3507 12.821400000000001 12.7622 9.7543000000000006 12.531599999999999 12.8109 12.639799999999999 9.6865000000000006 12.523199999999999 12.465199999999999 12.297700000000001 9.8363999999999994 12.0754 9.6372 12.179399999999999 11.9084 9.0768000000000004 11.904500000000001 8.9928000000000008 9.0564 12.0541 11.998200000000001 8.8270999999999997 8.9928000000000008 9.2597000000000005 8.9094999999999995 8.7661999999999995 11.4649 8.5838999999999999 8.7451000000000008 11.405799999999999 11.2531 11.299899999999999 11.247299999999999 11.293200000000001 8.6992999999999991 11.1883 11.136200000000001 8.2890999999999995 8.3463999999999992 8.3074999999999992 10.93 10.5953 10.7342 10.402900000000001 10.3544 8.1706000000000003 7.8552 8.1865000000000006 10.345499999999999 7.9279000000000002 7.8002000000000002 10.110900000000001 10.063700000000001 7.7808000000000002 8.0114000000000001 10.011799999999999 7.9367000000000001 9.9183000000000003 9.8719000000000001 7.8258000000000001 7.6159999999999997 9.6477000000000004 9.8597999999999999 8.5335000000000001 9.6826000000000008 9.6372999999999998 7.5727000000000002 7.5372000000000003 7.0850999999999997 9.1258999999999997 7.1013000000000002 9.3691999999999993 7.2801999999999998 7.1646999999999998 9.2378999999999998 9.0332000000000008 6.8234000000000004 9.0283999999999995 8.9064999999999994 8.8645999999999994 7.0899000000000001 7.0564999999999998 8.5840999999999994 8.9321000000000002 6.6482000000000001 6.54 8.8065999999999995 6.6310000000000002 6.5998000000000001 8.4563000000000006 8.4164999999999992 8.3767999999999994 6.4019000000000004 6.2976000000000001 6.5629 6.3117999999999999 8.1812000000000005 6.2523999999999997 6.1505000000000001 6.4816000000000003 6.2359999999999998 7.9901 8.0234000000000005 8.1267999999999994 7.8071000000000002 6.0902000000000003 7.6638999999999999 7.6276000000000002 7.6605999999999996 7.8304999999999998 7.7251000000000003 7.7565999999999997 7.5845000000000002 7.4138000000000002 7.5129000000000001 5.8750999999999998 5.7144000000000004 7.4067999999999996 5.9237000000000002 7.4678000000000004 7.4325000000000001 5.5155000000000003 5.4893999999999998 5.4634 7.2286999999999999 5.6026999999999996 5.5128000000000004 7.1894 5.4606000000000003 6.9965000000000002 5.4710999999999999 6.9302000000000001 5.3578000000000001 5.2710999999999997 5.3070000000000004 5.1604000000000001 6.8277999999999999 5.2919 5.2667999999999999 6.5522 5.1576000000000004 5.1920999999999999 5.2849000000000004 6.4870000000000001 5.1765999999999996 5.1520000000000001 5.7611999999999997 6.4218999999999999 6.3913000000000002 5.1113999999999997 4.9176000000000002 6.4130000000000003 4.9268000000000001 6.4077999999999999 6.2110000000000003 4.7464000000000004 4.7788000000000004 6.1227 4.8964999999999996 6.0103 4.7961 6.0603999999999996 5.8714000000000004 6.0559000000000003 4.5467000000000004 5.9983000000000004 5.9696999999999996 4.6383000000000001

Time (s)

Salt Concentration (g/L)

Université d’Ottawa

Faculté de génie

Département de
génie chimique et biologique

University of Ottawa
Faculty of Engineering

Department of

Chemical and Biological Engineering

Fall 2019 1

CHG1371
PROJECT 2 DESCRIPTION

1. Background
Consider the following system of three continuous stirred tanks, connected in series and filled with water:

To study the flow and mixing pattern in this system, salt was dissolved in the first tank to obtain a desired initial

salt concentration. Fresh water was then allowed to flow through the system, resulting in the dissolved salt to

flow through all three tanks and then eventually leaving the system. Each tank outlet stream is monitored with

an on-line thermal conductivity detector to provide instantaneous salt concentration measurements, ???. By

plotting this data over time (Figure 1), we can determine how long it takes for all the salt to leave the system,

and the maximum salt concentrations held by each tank throughout the run.

Figure 1. Salt concentration measurements from the experimental system

0

50

100

150

200

250

300

350

400

450

0 20 40 60 80 100

Sa
lt

C
o

n
ce

n
tr

at
io

n
(

g/
L)

Time (s)

Cs1

Cs2

Cs3

500 L

??1 (g/L)

250 L

??2 (g/L)

1000 L

??3

(g/L)

50 (L/s)

?? = 0

50 (L/s)

??1

50 (L/s)

??2

50 (L/s)
??3

CHG1371: Numerical Methods and Engineering Computation in Chemical Engineering

Winter 2020 2

It is also possible to numerically generate these salt concentration plots using differential equations. Assuming

that pure water is fed to the first tank and that each tank is perfectly mixed (i.e. the salt concentration in a tank

is uniform and equal to the concentration in the outlet stream from that tank), the following expressions for the

change in salt concentration over time can be determined for each tank:

???1

??
= −

�̇�

?1
??1 (Eq 1)

???2

??
=

�̇�

?2
(??1 − ??2) (Eq 2)

???3

??
=
�̇�

?3
(??2 − ??3) (Eq 3)

Here, ?? and ??? are the volume and salt concentration of each tank, and �̇� is the flowrate of water. The initial

salt concentration in tanks 1, 2 and 3 are 450 g/L, 0 g/L and 0 g/L, respectively. The tank volumes and water

flowrate are provided in the Block Flow Diagram.

2. Task
For this project, you will determine the salt concentrations for each tank over time, by solving the system of

ordinary differential equations using a 2nd order or higher Runge-Kutta method. You will then compare and

discuss the differences between the numerical data with the experimental findings in Figure 1. The experimental

plot data is available in the provided Excel sheet. Hint – what is the solubility of salt in water?

The numerical solution must be solved in Microsoft Excel using VBA. The solution should be flexible and robust

enough that different parameters (e.g. different �̇�, ?? and initial salt concentrations) could be easily changed

and a new solution would be automatically calculated.

3. Evaluation

A report will accompany the solution, a soft copy in format and must be submitted electronically prior to

the due date of Friday April 3rd, 2020 at 16:00. It is expected that the report should be about 6 pages, 1.5 spaced

and including figures, broken down as follows:

• 1 Page for Introduction/Background

• 2 Pages for VBA code design and Excel sheet set-up

• 2 Pages for Results/Discussion/Validation

• 1 Page for Improvements & Extensions/Conclusions

The VBA code should be included as text in an Appendix (not included in the number of pages). This project

should be done in groups of 2, and each member must share equal responsibility for the code and the report.

The deadline for joining a group on BrightSpace is March 20th, 2020 at 16:00. If you do not join a group by this

time, you will be randomly assigned. More information on the expectations of the report and the marking

scheme can be found in the Project Guidelines & Expectations document.

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