Note
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Diffusion flame extinction strain rate#
This example computes the extinction point of a counterflow diffusion flame. A hydrogen-oxygen diffusion flame at 1 bar is studied.
The tutorial makes use of the scaling rules derived by Fiala and Sattelmayer (doi:10.1155/2014/484372). Please refer to this publication for a detailed explanation. Also, please don’t forget to cite it if you make use of it.
Requires: cantera >= 3.0, matplotlib >= 2.0
Creating the initial solution
Flame burning at alpha = 2.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 3.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 4.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 5.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 6.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 7.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 8.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 9.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 10.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 11.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 12.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 13.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 14.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 15.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 16.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 17.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 18.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 19.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 20.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 21.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 22.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 23.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 24.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 25.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 26.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 27.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 28.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 29.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 30.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 31.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 32.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 33.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 34.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 35.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 36.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 37.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 38.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 39.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 40.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 41.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 42.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 43.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 44.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 45.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 46.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 47.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 48.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 49.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 50.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 51.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 52.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 53.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 54.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 55.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 56.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 57.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 58.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 59.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 60.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 61.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 62.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 63.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 64.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 65.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 66.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 67.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 68.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 69.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 70.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 71.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 72.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 73.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 74.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 75.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 76.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 77.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 78.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 79.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 80.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 81.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 82.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 83.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 84.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 85.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 86.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 87.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 88.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 89.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 90.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 91.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 92.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 93.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 94.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 95.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 96.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 97.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 98.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 99.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 100.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 101.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 102.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 103.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 104.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 105.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 106.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 107.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 108.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 109.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 110.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 111.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 112.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 113.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 114.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 115.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 116.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 117.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 118.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 119.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 120.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 121.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 122.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 123.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 124.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 125.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 126.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 127.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 128.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 129.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 130.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 131.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 132.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 133.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 134.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 135.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 136.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 137.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 138.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 139.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 140.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 141.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 142.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 143.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 144.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 145.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 146.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 147.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 148.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 149.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 150.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 151.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 152.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 153.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 154.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 155.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 156.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 157.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 158.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 159.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 160.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 161.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 162.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 163.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 164.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 165.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 166.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 167.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 168.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 169.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 170.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 171.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 172.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 173.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 174.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 175.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 176.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 177.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 178.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 179.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 180.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 181.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 182.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 183.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 184.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 185.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 186.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 187.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 188.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 189.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 190.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 191.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 192.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 193.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 194.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 195.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 196.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 197.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 198.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 199.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 200.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 201.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 202.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 203.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 204.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 205.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 206.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 207.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 208.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 209.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 210.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 211.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 212.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 213.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 214.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 215.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 216.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 217.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 218.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 219.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 220.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 221.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 222.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 223.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 224.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 225.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 226.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 227.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 228.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 229.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 230.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 231.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 232.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 233.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 234.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 235.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 236.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 237.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 238.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 239.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 240.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 241.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 242.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 243.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 244.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 245.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 246.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 247.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 248.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 249.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 250.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 251.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 252.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 253.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 254.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 255.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 256.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 257.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 258.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 259.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 260.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 261.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 262.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 263.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 264.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 265.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 266.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 267.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 268.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 269.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 270.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 271.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 272.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 273.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 274.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 275.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 276.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 277.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 278.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 279.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 280.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame burning at alpha = 281.0000. Proceeding to the next iteration, with delta_alpha = 1.0
Flame extinguished at alpha = 282.0000. Restoring alpha = 281.0000 and trying delta_alpha = 0.02
Flame burning at alpha = 281.0200. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.0400. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.0600. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.0800. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.1000. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.1200. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.1400. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.1600. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.1800. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.2000. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.2200. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.2400. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.2600. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.2800. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.3000. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.3200. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.3400. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.3600. Proceeding to the next iteration, with delta_alpha = 0.02
Error: Did not converge at n = 300
*******************************************************************************
CanteraError thrown by OneDim::timeStep:
Took maximum number of timesteps allowed (500) without reaching steady-state solution.
*******************************************************************************
Flame burning at alpha = 281.3800. Proceeding to the next iteration, with delta_alpha = 0.02
Error: Did not converge at n = 301
*******************************************************************************
CanteraError thrown by OneDim::timeStep:
Took maximum number of timesteps allowed (500) without reaching steady-state solution.
*******************************************************************************
Flame burning at alpha = 281.4000. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.4200. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.4400. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.4600. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.4800. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.5000. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.5200. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.5400. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.5600. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.5800. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.6000. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.6200. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.6400. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.6600. Proceeding to the next iteration, with delta_alpha = 0.02
Flame burning at alpha = 281.6800. Proceeding to the next iteration, with delta_alpha = 0.02
Flame extinguished at alpha = 281.7000. Restoring alpha = 281.6800 and trying delta_alpha = 0.0004
Flame burning at alpha = 281.6804. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6808. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6812. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6816. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6820. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6824. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6828. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6832. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6836. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6840. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6844. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6848. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6852. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6856. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6860. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6864. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6868. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6872. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6876. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6880. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6884. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6888. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6892. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6896. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6900. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6904. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6908. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6912. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6916. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6920. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6924. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6928. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6932. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6936. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6940. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6944. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6948. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6952. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6956. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6960. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6964. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6968. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6972. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6976. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6980. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6984. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6988. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame burning at alpha = 281.6992. Proceeding to the next iteration, with delta_alpha = 0.0004
Flame extinguished at alpha = 281.6996. Restoring alpha = 281.6992 and trying delta_alpha = 8e-06
Flame burning at alpha = 281.6992. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6992. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6992. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6992. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6992. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6992. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6993. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6994. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6995. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6995. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6995. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6995. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6995. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6995. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6995. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame burning at alpha = 281.6995. Proceeding to the next iteration, with delta_alpha = 8e-06
Flame extinguished at alpha = 281.6995. Restoring alpha = 281.6995 and trying delta_alpha = 1.6e-07
Flame extinguished at alpha = 281.6995. Abortion criterion satisfied.
----------------------------------------------------------------------
Parameters at the extinction point:
Pressure p=1.0000000000000002 bar
Peak temperature T=1552 K
Mean axial strain rate a_mean=1.58e+05 1/s
Maximum axial strain rate a_max=6.54e+05 1/s
Fuel inlet potential flow axial strain rate a_fuel=3.12e+05 1/s
Oxidizer inlet potential flow axial strain rate a_ox=1.01e+05 1/s
Axial strain rate at stoichiometric surface a_stoich=2.71e+05 1/s
from pathlib import Path
import numpy as np
import matplotlib.pyplot as plt
import cantera as ct
# PART 1: INITIALIZATION
# Set up an initial hydrogen-oxygen counterflow flame at 1 bar and low strain
# rate (maximum axial velocity gradient = 2414 1/s)
reaction_mechanism = 'h2o2.yaml'
gas = ct.Solution(reaction_mechanism)
width = 18.e-3 # 18mm wide
f = ct.CounterflowDiffusionFlame(gas, width=width)
# Define the operating pressure and boundary conditions
f.P = 1.e5 # 1 bar
f.fuel_inlet.mdot = 0.5 # kg/m^2/s
f.fuel_inlet.X = 'H2:1'
f.fuel_inlet.T = 300 # K
f.oxidizer_inlet.mdot = 3.0 # kg/m^2/s
f.oxidizer_inlet.X = 'O2:1'
f.oxidizer_inlet.T = 500 # K
# Set refinement parameters
f.set_refine_criteria(ratio=3.0, slope=0.1, curve=0.2, prune=0.03)
# Define a limit for the maximum temperature below which the flame is
# considered as extinguished and the computation is aborted
temperature_limit_extinction = max(f.oxidizer_inlet.T, f.fuel_inlet.T)
# Initialize and solve
print('Creating the initial solution')
f.solve(loglevel=0, auto=True)
# Define output locations
output_path = Path() / "diffusion_flame_extinction_data"
output_path.mkdir(parents=True, exist_ok=True)
hdf_output = "native" in ct.hdf_support()
if hdf_output:
file_name = output_path / "flame_data.h5"
file_name.unlink(missing_ok=True)
def names(test):
if hdf_output:
# use internal container structure for HDF
file_name = output_path / "flame_data.h5"
return file_name, test
# use separate files for YAML
file_name = output_path / f"{test}.yaml".replace("-", "_").replace("/", "_")
return file_name, "solution"
file_name, entry = names("initial-solution")
f.save(file_name, name=entry, description="Initial solution")
# PART 2: COMPUTE EXTINCTION STRAIN
# Exponents for the initial solution variation with changes in strain rate
# Taken from Fiala and Sattelmayer (2014)
exp_d_a = - 1. / 2.
exp_u_a = 1. / 2.
exp_V_a = 1.
exp_lam_a = 2.
exp_mdot_a = 1. / 2.
# Set normalized initial strain rate
alpha = [1.]
# Initial relative strain rate increase
delta_alpha = 1.
# Factor of refinement of the strain rate increase
delta_alpha_factor = 50.
# Limit of the refinement: Minimum normalized strain rate increase
delta_alpha_min = .001
# Limit of the Temperature decrease
delta_T_min = 1 # K
# Iteration indicator
n = 0
# Indicator of the latest flame still burning
n_last_burning = 0
# List of peak temperatures
T_max = [np.max(f.T)]
# List of maximum axial velocity gradients
a_max = [np.max(np.abs(np.gradient(f.velocity) / np.gradient(f.grid)))]
# Simulate counterflow flames at increasing strain rates until the flame is
# extinguished. To achieve a fast simulation, an initial coarse strain rate
# increase is set. This increase is reduced after an extinction event and
# the simulation is again started based on the last burning solution.
# The extinction point is considered to be reached if the abortion criteria
# on strain rate increase and peak temperature decrease are fulfilled.
while True:
n += 1
# Update relative strain rates
alpha.append(alpha[n_last_burning] + delta_alpha)
strain_factor = alpha[-1] / alpha[n_last_burning]
# Create an initial guess based on the previous solution
# Update grid
# Note that grid scaling changes the diffusion flame width
f.flame.grid *= strain_factor ** exp_d_a
normalized_grid = f.grid / (f.grid[-1] - f.grid[0])
# Update mass fluxes
f.fuel_inlet.mdot *= strain_factor ** exp_mdot_a
f.oxidizer_inlet.mdot *= strain_factor ** exp_mdot_a
# Update velocities
f.set_profile('velocity', normalized_grid,
f.velocity * strain_factor ** exp_u_a)
f.set_profile('spread_rate', normalized_grid,
f.spread_rate * strain_factor ** exp_V_a)
# Update pressure curvature
f.set_profile('lambda', normalized_grid, f.L * strain_factor ** exp_lam_a)
try:
f.solve(loglevel=0)
except ct.CanteraError as e:
print('Error: Did not converge at n =', n, e)
T_max.append(np.max(f.T))
a_max.append(np.max(np.abs(np.gradient(f.velocity) / np.gradient(f.grid))))
if not np.isclose(np.max(f.T), temperature_limit_extinction):
# Flame is still burning, so proceed to next strain rate
n_last_burning = n
file_name, entry = names(f"extinction/{n:04d}")
f.save(file_name, name=entry, description=f"Solution at alpha = {alpha[-1]}")
print('Flame burning at alpha = {:8.4F}. Proceeding to the next iteration, '
'with delta_alpha = {}'.format(alpha[-1], delta_alpha))
elif ((T_max[-2] - T_max[-1] < delta_T_min) and (delta_alpha < delta_alpha_min)):
# If the temperature difference is too small and the minimum relative
# strain rate increase is reached, save the last, non-burning, solution
# to the output file and break the loop
file_name, entry = names(f"extinction/{n:04d}")
f.save(file_name, name=entry, description=f"Flame extinguished at alpha={alpha[-1]}")
print('Flame extinguished at alpha = {0:8.4F}.'.format(alpha[-1]),
'Abortion criterion satisfied.')
break
else:
# Procedure if flame extinguished but abortion criterion is not satisfied
# Reduce relative strain rate increase
delta_alpha = delta_alpha / delta_alpha_factor
print('Flame extinguished at alpha = {0:8.4F}. Restoring alpha = {1:8.4F} and '
'trying delta_alpha = {2}'.format(
alpha[-1], alpha[n_last_burning], delta_alpha))
# Restore last burning solution
file_name, entry = names(f"extinction/{n_last_burning:04d}")
f.restore(file_name, entry)
# Print some parameters at the extinction point, after restoring the last burning
# solution
file_name, entry = names(f"extinction/{n_last_burning:04d}")
f.restore(file_name, entry)
print('----------------------------------------------------------------------')
print('Parameters at the extinction point:')
print('Pressure p={0} bar'.format(f.P / 1e5))
print('Peak temperature T={0:4.0f} K'.format(np.max(f.T)))
print('Mean axial strain rate a_mean={0:.2e} 1/s'.format(f.strain_rate('mean')))
print('Maximum axial strain rate a_max={0:.2e} 1/s'.format(f.strain_rate('max')))
print('Fuel inlet potential flow axial strain rate a_fuel={0:.2e} 1/s'.format(
f.strain_rate('potential_flow_fuel')))
print('Oxidizer inlet potential flow axial strain rate a_ox={0:.2e} 1/s'.format(
f.strain_rate('potential_flow_oxidizer')))
print('Axial strain rate at stoichiometric surface a_stoich={0:.2e} 1/s'.format(
f.strain_rate('stoichiometric', fuel='H2')))
# Plot the maximum temperature over the maximum axial velocity gradient
plt.figure()
plt.semilogx(a_max, T_max)
plt.xlabel(r'$a_{max}$ [1/s]')
plt.ylabel(r'$T_{max}$ [K]')
plt.savefig(output_path / "figure_T_max_a_max.png")
Total running time of the script: (0 minutes 54.809 seconds)