牛顿冷却定律

牛顿冷却定律 6.1 牛頓冷卻定律 Vensim 牛頓冷卻定律結構圖 F(0) F'(t) M-F(0)k M 【牛頓冷卻定律】:物體溫度的變率跟「該物體與周圍溫度之差」成正比。 ,FtkFtM,,,FF0,即 ; ，，，，，，，，0 用不定積分可得 kt,FtFMeM,,， ，，，，0 k其中表物體的啟始溫度,表周圍溫度,表與該物體有關之常數。 MF0 6.1.1、「流浪漢的屍體」 某冬晨,警局接到報案,在街頭發現一位流浪者的屍體,早上六點測量其體溫13?, 到早上七點時,其體溫已降到11?,若假設室外溫度約維持在10?,且人體正常體溫為37?。 問題1:試用 Vensim 建立「屍體溫度」、「室外溫度」、「溫差」、「變量」 與「常數」的模組。 答: 砰放 跑秖 放畉盽计 放 :依據前提條件,試問常數k為何, 問題2 1,, 答: 。 ,ln,,3,, 問題3:試用 Vensim 繪出「屍體溫度對時間的關係圖」。 答: 砰放 癸 丁 闽玒瓜 40 30 20 10 0 012345 丁( ) 砰放 (尼ん) 問題4:試用 Vensim 建立「屍體溫度對時間表」。 答: 時間 (時) 屍體溫度 1.21875 16.9128 2.53125 11.5938 01.252.5625 37 16.6755 11.5391 0.03125 36.073 1.28125 16.4463 2.59375 11.4862 0.0625 35.17791.31252.625 16.225 11.4352 0.09375 34.3135 1.34375 16.0113 2.65625 11.3859 0.125 33.4788 1.375 15.8049 2.6875 11.3384 0.15625 32.6727 1.40625 15.6056 2.71875 11.2924 0.1875 31.8943 1.4375 15.4132 2.75 11.248 0.21875 31.1427 1.46875 15.2273 2.78125 11.2052 0.25 30.4168 1.5 15.0479 2.8125 11.1638 14.8746 11.1239 0.28125 29.71591.531252.84375 0.3125 29.039 1.5625 14.7072 2.875 11.0853 0.34375 28.3853 1.59375 14.5456 2.90625 11.048 0.375 27.7541 1.625 14.3895 2.9375 11.012 0.40625 27.1446 1.65625 14.2388 2.96875 10.9773 0.4375 26.556 1.6875 14.0933 3 10.9437 0.46875 25.9876 1.71875 13.9528 3.03125 10.9113 0.5 25.4387 1.75 13.8171 3.0625 10.88 0.53125 24.9087 1.78125 13.686 3.09375 10.8498 0.5625 24.3969 1.8125 13.5595 3.125 10.8207 0.59375 23.90261.843753.15625 13.4373 10.7925 0.625 23.4253 1.875 13.3193 3.1875 10.7653 0.65625 22.9644 1.90625 13.2053 3.21875 10.739 0.6875 22.5193 1.9375 13.0953 3.25 10.7136 0.71875 22.0895 1.96875 12.989 3.28125 10.6891 0.75 21.6744 2 12.8864 3.3125 10.6655 0.78125 21.2736 2.03125 12.7873 3.34375 10.6426 0.8125 20.8866 2.0625 12.6916 3.375 10.6206 0.84375 20.5128 2.09375 12.5992 3.40625 10.5993 0.875 20.1519 2.125 12.51 3.4375 10.5787 0.90625 19.8034 2.15625 12.4238 3.46875 10.5588 0.9375 19.4668 2.1875 12.3406 3.5 10.5396 0.96875 19.1418 2.21875 12.2602 3.53125 10.5211 2.25 12.1826 3.5625 10.5032 1 2.28125 12.1077 3.59375 10.4859 18.8279 2.3125 12.0353 3.625 10.4693 1.03125 18.5249 2.34375 11.9655 3.65625 10.4531 1.0625 18.2322 2.375 11.898 3.6875 10.4376 1.09375 17.9496 2.40625 11.8328 3.71875 10.4226 1.125 17.6767 2.4375 11.7699 3.75 10.4081 1.15625 17.4131 2.46875 11.7091 3.78125 10.394 1.1875 17.1586 2.5 11.6505 3.8125 10.3805 3.84375 10.3675 4.25 10.2333 4.65625 10.1482 3.875 10.3548 4.28125 10.2253 4.6875 10.1431 3.90625 10.3427 4.3125 10.2176 4.71875 10.1382 3.9375 10.3309 4.34375 10.2101 4.75 10.1334 3.96875 10.3195 4.375 10.2029 4.78125 10.1288 4 10.3086 4.40625 10.1959 4.8125 10.1244 4.03125 10.298 4.4375 10.1892 4.84375 10.1201 4.0625 10.2877 4.46875 10.1827 4.875 10.116 4.09375 10.2779 4.5 10.1764 4.90625 10.112 4.125 10.2683 4.53125 10.1704 4.9375 10.1082 10.2591 4.56254.968754.15625 10.1645 10.1045 4.1875 10.2502 4.59375 10.1589 5 10.1009 10.2416 4.6254.21875 10.1534 問題5:發現流浪漢死亡時體溫為13?,依據上表可判定流浪漢已死亡多久, 答: 約2小時 。 問題6:流浪漢死亡的時間為何, 答: 約凌晨4:00AM 。 6.1.2、「咖啡杯的溫度」 今有一杯用95?熱水沖的即溶咖啡,放置在31?的房間中,如果5分鐘後,咖啡的溫度是63?。 問題1:試用 Vensim 建立「咖啡溫度」、「室內溫度」、「溫差」、「變量」 與「常數」的模組。 答: 〡柏放 跑秖 放畉盽计 ず放 問題2:依據前提條件,試問常數k為何, ln2 答: 。 5 :試用 Vensim 繪出「咖啡溫度對時間的關係圖」。 問題3 答: 〡柏放 癸 丁 闽玒瓜 100 75 50 25 0 0481216202428323640 丁(だ) 〡柏放 (尼ん) 問題4:試用 Vensim 建立「咖啡溫度對時間表」。 答: 時間 (分) 咖啡溫度 26.7813 32.5498 28.0938 32.2915 25.5 26.812528.125 32.8518 32.5431 32.2859 25.5313 32.8438 26.8438 32.5364 28.1563 32.2803 25.5625 32.835826.87528.1875 32.5298 32.2748 25.5938 32.8278 26.9063 32.5232 28.2188 32.2693 25.625 32.8199 26.9375 32.5166 28.25 32.2638 25.6563 32.812 26.9688 32.51 28.2813 32.2583 25.6875 32.8042 27 32.5034 28.3125 32.2528 25.7188 32.7964 27.0313 32.4969 28.3438 32.2474 25.75 32.7886 27.0625 32.4905 28.375 32.242 32.484 32.2366 25.7813 32.780827.093828.4063 25.8125 32.7731 27.125 32.4776 28.4375 32.2313 25.8438 32.7654 27.1563 32.4712 28.4688 32.2259 25.875 32.7578 27.1875 32.4648 28.5 32.2206 25.9063 32.7502 27.2188 32.4584 28.5313 32.2153 25.9375 32.7426 27.25 32.4521 28.5625 32.2101 25.9688 32.735 27.2813 32.4458 28.5938 32.2048 26 32.7275 27.3125 32.4396 28.625 32.1996 26.0313 32.72 27.3438 32.4333 28.6563 32.1944 26.0625 32.7126 27.375 32.4271 28.6875 32.1893 26.0938 32.705227.406328.7188 32.4209 32.1841 26.125 32.6978 27.4375 32.4148 28.75 32.179 26.1563 32.6904 27.4688 32.4087 28.7813 32.1739 26.1875 32.6831 27.5 32.4026 28.8125 32.1688 26.2188 32.6758 27.5313 32.3965 28.8438 32.1637 26.25 32.6686 27.5625 32.3904 28.875 32.1587 26.2813 32.6613 27.5938 32.3844 28.9063 32.1537 26.3125 32.6541 27.625 32.3784 28.9375 32.1487 26.3438 32.647 27.6563 32.3724 28.9688 32.1437 26.375 32.6398 27.6875 32.3665 29 32.1387 26.4063 32.6327 27.7188 32.3606 29.0313 32.1338 26.4375 32.6256 27.75 32.3547 29.0625 32.1289 26.4688 32.6186 27.7813 32.3488 29.0938 32.124 26.5 32.6116 27.8125 32.343 29.125 32.1191 26.5313 32.6046 27.8438 32.3371 29.1563 32.1143 26.5625 32.5977 27.875 32.3314 29.1875 32.1094 26.5938 32.5907 27.9063 32.3256 29.2188 32.1046 26.625 32.5839 27.9375 32.3198 29.25 32.0999 26.6563 32.577 27.9688 32.3141 29.2813 32.0951 26.6875 32.5702 28 32.3084 29.3125 32.0903 26.7188 32.5634 28.0313 32.3028 29.3438 32.0856 26.75 32.5566 28.0625 32.2971 29.375 32.0809 29.4063 32.0762 30.3125 31.9489 31.2188 31.8367 29.4375 32.0716 30.3438 31.9448 31.25 31.833 29.4688 32.0669 30.375 31.9407 31.2813 31.8294 29.5 32.0623 30.4063 31.9366 31.3125 31.8258 29.5313 32.0577 30.4375 31.9326 31.3438 31.8223 29.5625 32.0531 30.4688 31.9285 31.375 31.8187 29.5938 32.0486 30.5 31.9245 31.4063 31.8151 29.625 32.044 30.5313 31.9205 31.4375 31.8116 29.6563 32.0395 30.5625 31.9165 31.4688 31.8081 29.6875 32.035 30.5938 31.9126 31.5 31.8046 32.0305 30.62531.531329.7188 31.9086 31.8011 29.75 32.026 30.6563 31.9047 31.5625 31.7976 32.0216 30.687531.593829.7813 31.9007 31.7942 29.8125 32.0172 30.7188 31.8968 31.625 31.7907 32.0128 30.75 31.893 31.6563 31.7873 29.8438 29.875 32.0084 30.7813 31.8891 31.6875 31.7839 29.9063 32.004 30.8125 31.8852 31.7188 31.7805 29.9375 31.9997 30.8438 31.8814 31.75 31.7771 29.9688 31.9953 30.875 31.8776 31.7813 31.7738 31.991 31.8738 31.7704 3030.906331.8125 30.0313 31.9867 30.9375 31.87 31.8438 31.7671 30.0625 31.9825 30.9688 31.8662 31.875 31.7638 30.0938 31.9782 31 31.8625 31.9063 31.7604 30.125 31.974 31.0313 31.8587 31.9375 31.7572 30.1563 31.855 31.7539 31.9697 31.062531.9688 30.1875 31.9655 31.0938 31.8513 32 31.7506 30.2188 31.9614 31.125 31.8476 30.25 31.9572 31.1563 31.844 30.2813 31.953 31.1875 31.8403 問題5:那麼欲使咖啡降溫到32?,需多少分鐘, 答: 約30分鐘 。