1 00:00:00,390 --> 00:00:02,650 - So I have an interesting system over here. 2 00:00:02,650 --> 00:00:05,520 I have two compartments, on the left compartment 3 00:00:05,520 --> 00:00:09,750 I have a gas that is at a temperature of T sub a 4 00:00:09,750 --> 00:00:12,550 and on the right side of this I have gas 5 00:00:12,550 --> 00:00:15,050 that is a temperature of T sub b 6 00:00:15,050 --> 00:00:19,850 and they are separated by a wall of depth, d, 7 00:00:19,850 --> 00:00:22,020 or I guess you say of thickness, d, 8 00:00:22,020 --> 00:00:25,420 and the contact area of the wall, 9 00:00:25,420 --> 00:00:27,820 or the contact area of the gas onto the wall 10 00:00:27,820 --> 00:00:29,890 that area is A, and I'm just drawing 11 00:00:29,890 --> 00:00:31,520 a section of it, we're assuming that 12 00:00:31,520 --> 00:00:34,750 these two compartments are completely separated. 13 00:00:34,750 --> 00:00:36,390 Now what I am curious about, and we're 14 00:00:36,390 --> 00:00:38,580 going to assume that the temperature on the left 15 00:00:38,580 --> 00:00:40,950 is higher than the temperature on the right, 16 00:00:40,950 --> 00:00:42,290 and so because of that you're going to have 17 00:00:42,290 --> 00:00:45,850 a transfer of thermal energy from the left to the right, 18 00:00:45,850 --> 00:00:47,890 and that thermal energy that gets transferred 19 00:00:47,890 --> 00:00:51,320 we call that heat, and we'll denote that with the letter Q, 20 00:00:51,320 --> 00:00:53,420 I'm curious about how does the rate 21 00:00:53,420 --> 00:00:55,020 at which heat is transferred, or so 22 00:00:55,020 --> 00:00:57,550 how much heat is transferred per unit time, 23 00:00:57,550 --> 00:00:59,090 that's the rate at which heat is transferred, 24 00:00:59,090 --> 00:01:01,150 how would that change depending on 25 00:01:01,150 --> 00:01:03,680 how we change these different variables. 26 00:01:03,680 --> 00:01:07,560 So for example, if our, if our area, 27 00:01:07,560 --> 00:01:10,460 if our contact area were to go up 28 00:01:10,460 --> 00:01:13,150 what would that do for Q over t? 29 00:01:13,150 --> 00:01:16,490 Well then Q over t would also increase per unit time 30 00:01:16,490 --> 00:01:19,520 'cause I have, I have more area for these 31 00:01:19,520 --> 00:01:23,290 hot molec--these hot air particles or hot air molecules 32 00:01:23,290 --> 00:01:25,080 to bump into and they'll heat that wall 33 00:01:25,080 --> 00:01:26,820 and then there will be more heated wall 34 00:01:26,820 --> 00:01:29,890 to heat up the colder air particles. 35 00:01:29,890 --> 00:01:34,120 So in that case our rate of heat transfer would also go up. 36 00:01:34,120 --> 00:01:37,420 Well what if we, what if we, and obviously if I made 37 00:01:37,420 --> 00:01:40,520 my area smaller, maybe I should just write that explicitly, 38 00:01:40,520 --> 00:01:43,820 if I made my contact area smaller then my rate 39 00:01:43,820 --> 00:01:46,350 of heat transfer, my rate of heat transfer 40 00:01:46,350 --> 00:01:49,520 would go down, and that feels like common sense. 41 00:01:49,520 --> 00:01:52,320 Now what about, what about the thickness? 42 00:01:52,320 --> 00:01:54,850 If I were to make it, if I were to make 43 00:01:54,850 --> 00:01:57,150 the thickness larger, if I were to make this a thicker wall, 44 00:01:57,150 --> 00:01:59,490 what would that do to my rate of heat transfer? 45 00:01:59,490 --> 00:02:01,250 Well then I would have, I would have more 46 00:02:01,250 --> 00:02:02,950 things that I would have to heat up to get it 47 00:02:02,950 --> 00:02:06,530 to a certain temperature before I can, 48 00:02:06,530 --> 00:02:09,720 which then can heat up the particles on the right, 49 00:02:09,720 --> 00:02:11,250 and obviously this is a continuous process, 50 00:02:11,250 --> 00:02:13,390 it'll always be happening, but there will just be more 51 00:02:13,390 --> 00:02:15,720 stuff to heat up and it's going to take longer 52 00:02:15,720 --> 00:02:17,690 and more of that, and more of that 53 00:02:17,690 --> 00:02:19,520 more of that kinetic energy, that average 54 00:02:19,520 --> 00:02:21,550 kinetic energy is going to get dissipated 55 00:02:21,550 --> 00:02:25,210 in this wall so if this wall becomes thicker, 56 00:02:25,210 --> 00:02:27,200 if the wall becomes thicker then the 57 00:02:27,200 --> 00:02:30,150 rate of heat transfer is going to go down, 58 00:02:30,150 --> 00:02:32,920 or if you, if the wall became thinner, 59 00:02:32,920 --> 00:02:36,890 if this depth decreased then the rate of heat transfer, 60 00:02:36,890 --> 00:02:39,550 then the rate of heat transfer would go up. 61 00:02:39,550 --> 00:02:40,780 So you could say the rate of heat transfer 62 00:02:40,780 --> 00:02:43,240 is going to be inversely proportional 63 00:02:43,240 --> 00:02:46,120 to the thickness of this wall. 64 00:02:46,120 --> 00:02:47,850 Now what else could we think about? 65 00:02:47,850 --> 00:02:51,450 Well we could think about the temperature differential, 66 00:02:51,450 --> 00:02:55,880 the temperature differential, that's T sub a minus, 67 00:02:55,880 --> 00:03:00,400 minus T sub b, minus T sub b. 68 00:03:00,400 --> 00:03:02,890 Well if this temperature differential, if this 69 00:03:02,890 --> 00:03:05,090 temperature differential were to go up, 70 00:03:05,090 --> 00:03:06,720 well what's going to happen? 71 00:03:06,720 --> 00:03:09,320 Well it's common sense that well if this is super hot, 72 00:03:09,320 --> 00:03:11,890 if this is super hot over here, this is way hotter 73 00:03:11,890 --> 00:03:13,080 than what we have on the right, 74 00:03:13,080 --> 00:03:15,320 well we're going to have more heat transferred 75 00:03:15,320 --> 00:03:17,650 so the rate of heat transfer, you're going to have 76 00:03:17,650 --> 00:03:19,050 more heat transferred per unit time, 77 00:03:19,050 --> 00:03:22,890 your rate of heat transfer is going to go up. 78 00:03:22,890 --> 00:03:25,810 And likewise, if this differential were to go down 79 00:03:25,810 --> 00:03:26,920 and you could take the extreme case 80 00:03:26,920 --> 00:03:28,620 if there was no differential, 81 00:03:28,620 --> 00:03:31,550 if T sub a was the same as T sub b 82 00:03:31,550 --> 00:03:33,920 then you would have no heat transfer 83 00:03:33,920 --> 00:03:36,050 frankly in any unit of time. 84 00:03:36,050 --> 00:03:37,920 So it makes sense that the rate of heat transfer 85 00:03:37,920 --> 00:03:42,050 is going to be proportional to the temperature differential. 86 00:03:42,050 --> 00:03:44,680 So how can we encapsulate all of this intuition 87 00:03:44,680 --> 00:03:47,770 into maybe a formula for describing 88 00:03:47,770 --> 00:03:51,350 thermal conductivity, for thinking about 89 00:03:51,350 --> 00:03:53,720 how quickly some, how quickly this heat 90 00:03:53,720 --> 00:03:56,350 will be transferred, the rate of heat transfer? 91 00:03:56,350 --> 00:03:59,650 Well we could say the rate of heat transfer, 92 00:03:59,650 --> 00:04:02,050 and this is really, hopefully you know comes out of 93 00:04:02,050 --> 00:04:04,250 a little bit of common sense or intuition 94 00:04:04,250 --> 00:04:05,970 of what would happen here, the rate of 95 00:04:05,970 --> 00:04:09,190 heat transfer, I could say it's going to be proportional to 96 00:04:09,190 --> 00:04:11,150 well, what are the things it's going to be proportional to? 97 00:04:11,150 --> 00:04:13,090 It's going to be proportional to the area, 98 00:04:13,090 --> 00:04:14,990 the more surface area we have on this wall, 99 00:04:14,990 --> 00:04:17,820 more contact area, the more heat we're going 100 00:04:17,820 --> 00:04:19,649 to have transferred per unit time, 101 00:04:19,649 --> 00:04:23,420 so it's going to be proportional to that contact area. 102 00:04:23,420 --> 00:04:24,680 It's also going to be proportional to 103 00:04:24,680 --> 00:04:26,990 the temperature differential so let's multiply 104 00:04:26,990 --> 00:04:30,190 this times the temperature differential, 105 00:04:30,190 --> 00:04:34,690 so T sub a minus T sub b, 106 00:04:34,690 --> 00:04:36,370 minus T sub b, 107 00:04:36,370 --> 00:04:38,050 and it's inversely proportional to 108 00:04:38,050 --> 00:04:41,150 the thickness of the wall so all of that 109 00:04:41,150 --> 00:04:43,450 over the thickness of the wall. 110 00:04:43,450 --> 00:04:44,820 And now another thing that you might be saying, 111 00:04:44,820 --> 00:04:46,780 okay I have this proportionality constant 112 00:04:46,780 --> 00:04:48,920 but wouldn't this be different for different materials? 113 00:04:48,920 --> 00:04:50,920 For example, if this was a metal wall 114 00:04:50,920 --> 00:04:52,990 wouldn't this conduct the heat quicker 115 00:04:52,990 --> 00:04:54,750 than if this was a wood wall? 116 00:04:54,750 --> 00:04:56,960 And you would be correct, a metal wall would 117 00:04:56,960 --> 00:04:59,790 and so this K, this K right over here, 118 00:04:59,790 --> 00:05:02,650 this is dependent, this is dependent on 119 00:05:02,650 --> 00:05:05,150 the material, so what is the wall made of. 120 00:05:05,150 --> 00:05:06,740 So material... 121 00:05:06,740 --> 00:05:10,580 material of the wall, 122 00:05:10,580 --> 00:05:12,250 and you can actually, you can actually 123 00:05:12,250 --> 00:05:14,560 measure this thing and different materials 124 00:05:14,560 --> 00:05:16,850 will have different thermal conductivities, 125 00:05:16,850 --> 00:05:18,610 which this, which this variable 126 00:05:18,610 --> 00:05:21,120 right over here would actually represent. 127 00:05:21,120 --> 00:05:22,720 So going through a little bit of intuition 128 00:05:22,720 --> 00:05:24,350 we were able to come up with what looks like 129 00:05:24,350 --> 00:05:27,150 a fancy formula, and you will sometimes see 130 00:05:27,150 --> 00:05:29,920 this formula, formula for thermal conductivity 131 00:05:29,920 --> 00:05:33,320 through a solid barrier but it really comes out of 132 00:05:33,320 --> 00:05:35,120 hopefully common sense, the rate of, 133 00:05:35,120 --> 00:05:37,420 the amount of heat transferred per time 134 00:05:37,420 --> 00:05:39,020 is going to be proportional to, 135 00:05:39,020 --> 00:05:40,620 and the proportionality constant is going 136 00:05:40,620 --> 00:05:41,450 to be dependent on the material. 137 00:05:41,450 --> 00:05:43,550 Styrofoam for example would be very low here, 138 00:05:43,550 --> 00:05:45,830 that's why coolers are made out of Styrofoam, 139 00:05:45,830 --> 00:05:47,890 and it's going to be dependent on the area, 140 00:05:47,890 --> 00:05:49,420 it's going to be proportional to the area, 141 00:05:49,420 --> 00:05:50,920 the temperature differential, and then 142 00:05:50,920 --> 00:05:53,320 inversely proportional to the thickness. 143 00:05:53,320 --> 00:05:56,010 So if you wanted to really insulate something 144 00:05:56,010 --> 00:05:58,520 you would want to minimize the surface area 145 00:05:58,520 --> 00:06:00,750 and you would want to, and you would want to 146 00:06:00,750 --> 00:06:03,010 maximize the thickness and you would want to 147 00:06:03,010 --> 00:06:06,290 have something with a very low thermal conductivity, 148 00:06:06,290 --> 00:06:10,860 so a thick Styrofoam wall that's maybe shaped in a sphere 149 00:06:10,860 --> 00:06:13,740 might be a pretty good container for keeping 150 00:06:13,740 --> 00:00:00,000 something hot or for keeping something cool.