WEBVTT 00:00:00.030 --> 00:00:02.400 The following content is provided under a Creative 00:00:02.400 --> 00:00:03.820 Commons License. 00:00:03.820 --> 00:00:06.850 Your support will help MIT OpenCourseWare continue to 00:00:06.850 --> 00:00:10.520 offer high-quality educational resources for free. 00:00:10.520 --> 00:00:13.390 To make a donation or view additional materials from 00:00:13.390 --> 00:00:17.180 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.180 --> 00:00:18.430 ocw.mit.edu. 00:00:22.280 --> 00:00:23.310 SAL BARRIGA: Hi, I'm Sal. 00:00:23.310 --> 00:00:25.840 Today, we're going to solve Problem #5 of 00:00:25.840 --> 00:00:28.250 Exam 3 of Fall 2009. 00:00:28.250 --> 00:00:30.250 Now before you start this problem, there are a couple of 00:00:30.250 --> 00:00:32.360 things that you need to know in order for you to fully 00:00:32.360 --> 00:00:35.680 understand the science behind it. 00:00:35.680 --> 00:00:40.210 And so what you need to know is Fick's first law. 00:00:40.210 --> 00:00:43.030 Not just knowledge of it, but how it works. 00:00:43.030 --> 00:00:44.850 Also, Fick's second law. 00:00:44.850 --> 00:00:47.440 And not necessarily the equation form, but the 00:00:47.440 --> 00:00:50.110 solution to Fick's second law, which is a function of 00:00:50.110 --> 00:00:54.060 concentrations, and which we will refer to 00:00:54.060 --> 00:00:55.150 to solve the problem. 00:00:55.150 --> 00:01:00.240 And also know popular dislocations, how they help 00:01:00.240 --> 00:01:01.930 your material and whatnot. 00:01:01.930 --> 00:01:04.010 So the problem reads as follows. 00:01:04.010 --> 00:01:07.300 Specimens of steel are being surface hardened by the 00:01:07.300 --> 00:01:08.940 introduction of carbon. 00:01:08.940 --> 00:01:11.630 The concentration of carbon at the free surface of the steel 00:01:11.630 --> 00:01:14.480 is kept constant at c sub s. 00:01:14.480 --> 00:01:17.460 Chemical analysis of a number of specimens indicate that 00:01:17.460 --> 00:01:18.430 after a time-- 00:01:18.430 --> 00:01:19.640 t sub 1-- 00:01:19.640 --> 00:01:23.920 the carbon concentration has the value of c star at a depth 00:01:23.920 --> 00:01:25.710 from the surface-- x1-- 00:01:25.710 --> 00:01:28.900 shown below, which is shown on your graph. 00:01:28.900 --> 00:01:30.450 So this is given to you. 00:01:30.450 --> 00:01:33.090 So this pretty much is your concentration profile as a 00:01:33.090 --> 00:01:35.110 function of position. 00:01:35.110 --> 00:01:38.080 And it's telling you what's happening. 00:01:38.080 --> 00:01:42.470 So a good thing to do is draw a picture of the science 00:01:42.470 --> 00:01:44.730 that's happening here. 00:01:44.730 --> 00:01:45.830 And they tell you, first of all, I'm going to draw it 00:01:45.830 --> 00:01:48.710 underneath here that you have a rod of steel. 00:01:51.490 --> 00:01:55.980 And so this is steel. 00:01:55.980 --> 00:01:59.410 And I have a certain fixed concentration here at the 00:01:59.410 --> 00:02:00.320 surface of carbon. 00:02:00.320 --> 00:02:08.140 So I have some type of a carbon flux in my surface. 00:02:08.140 --> 00:02:10.740 And this is a visual image of how it looks. 00:02:10.740 --> 00:02:16.102 And what the problem says is that you have a certain 00:02:16.102 --> 00:02:19.170 concentration on your surface, which is given by c sub s. 00:02:19.170 --> 00:02:22.610 And then it starts telling you that chemical analysis was 00:02:22.610 --> 00:02:25.590 able to go ahead and measure this concentration at two 00:02:25.590 --> 00:02:29.400 different values inside your position at different times. 00:02:29.400 --> 00:02:33.470 So if I continue reading the problem, it says that chemical 00:02:33.470 --> 00:02:35.930 analysis of a number of specimens indicate that after 00:02:35.930 --> 00:02:37.190 a time-- t1-- 00:02:37.190 --> 00:02:39.510 the carbon concentration has a value of c star. 00:02:39.510 --> 00:02:42.100 So this one right here-- 00:02:42.100 --> 00:02:43.190 this is x1-- 00:02:43.190 --> 00:02:49.490 so this has a value of x1 at t1. 00:02:52.330 --> 00:02:57.800 And it also tells you that after a certain time-- t2, 00:02:57.800 --> 00:03:00.410 which is equal to 2 times t1-- 00:03:00.410 --> 00:03:03.750 the carbon concentration has a value of c star, so the same 00:03:03.750 --> 00:03:06.090 value at a depth of x2. 00:03:06.090 --> 00:03:12.560 So now, this thing says that at a depth of x2, at a time of 00:03:12.560 --> 00:03:17.910 t2, we measure the same concentration, c sub star in 00:03:17.910 --> 00:03:18.420 the specimen. 00:03:18.420 --> 00:03:26.080 And it tells you that x2 equals 2x1, and it also tells 00:03:26.080 --> 00:03:32.460 us that t2 equals 2t1. 00:03:32.460 --> 00:03:35.100 Now they tell us that for a reason, because it's probably 00:03:35.100 --> 00:03:40.190 expected for us to use this to solve the problem, so don't 00:03:40.190 --> 00:03:42.270 forget that. 00:03:42.270 --> 00:03:45.300 So the question is, under these conditions, would you 00:03:45.300 --> 00:03:49.310 describe the rate of carburization as one, steady 00:03:49.310 --> 00:03:50.770 state diffusion. 00:03:50.770 --> 00:03:52.850 Two, transient state diffusion. 00:03:52.850 --> 00:03:57.140 Or three, not governed by any diffusion, i.e., rate limited 00:03:57.140 --> 00:03:58.270 by some other process. 00:03:58.270 --> 00:04:00.290 Now you have to justify your choice. 00:04:00.290 --> 00:04:02.680 So this is pretty much a problem of process of 00:04:02.680 --> 00:04:03.920 elimination. 00:04:03.920 --> 00:04:06.065 And for part a, which is-- 00:04:06.065 --> 00:04:08.120 I'll label this part a-- 00:04:08.120 --> 00:04:09.610 it wants us to do three things. 00:04:09.610 --> 00:04:14.380 So the first thing, it asks us if it's steady state. 00:04:14.380 --> 00:04:20.220 So the first thing to ask yourself is, well, what's 00:04:20.220 --> 00:04:21.180 steady state? 00:04:21.180 --> 00:04:24.680 And the definition of steady state is that whatever 00:04:24.680 --> 00:04:28.080 phenomenon is happening doesn't vary with time. 00:04:28.080 --> 00:04:30.810 But the problem already tells you that it varies with time. 00:04:30.810 --> 00:04:33.200 It tells you that at a certain time you measure this and at 00:04:33.200 --> 00:04:36.140 another time you measure this, and you have this flux coming 00:04:36.140 --> 00:04:38.210 in, and this is what's happening. 00:04:38.210 --> 00:04:41.510 So from just reading the problem, I can already say 00:04:41.510 --> 00:04:45.840 that this process can't be steady state, because your 00:04:45.840 --> 00:04:48.460 concentration profile varies as a function of time. 00:04:48.460 --> 00:05:04.580 So for one, I would write, can't be steady state due to 00:05:04.580 --> 00:05:11.930 the concentration being a function of time. 00:05:11.930 --> 00:05:15.140 So one is out, so we can scratch that. 00:05:15.140 --> 00:05:17.730 It's not a steady state. 00:05:17.730 --> 00:05:19.180 Now two. 00:05:19.180 --> 00:05:22.060 This is where it gets a little bit more complicated. 00:05:22.060 --> 00:05:24.780 Two is a transient state diffusion. 00:05:24.780 --> 00:05:29.000 Well, how do we know whether or not this is 00:05:29.000 --> 00:05:29.780 governed with time? 00:05:29.780 --> 00:05:34.190 Well, you would argue, because this is not steady state, then 00:05:34.190 --> 00:05:35.920 it has to be time dependent. 00:05:35.920 --> 00:05:42.330 But, there are certain rules that you need to know to be 00:05:42.330 --> 00:05:44.040 able to make that justification. 00:05:44.040 --> 00:05:48.080 First of all, if it's time dependent, then whatever your 00:05:48.080 --> 00:05:51.140 concentration and whatever your position in your time is, 00:05:51.140 --> 00:05:55.460 has to fit into your solution to Fick's law. 00:05:55.460 --> 00:05:59.810 So if I look at my solution to Fick's law right here, I know 00:05:59.810 --> 00:06:03.300 that on the left side of my equation, I have the 00:06:03.300 --> 00:06:05.440 concentration at any point in x-- 00:06:05.440 --> 00:06:07.170 that's what c stands for-- 00:06:07.170 --> 00:06:09.060 minus the concentration at my surface, 00:06:09.060 --> 00:06:12.560 which is given as cs-- 00:06:12.560 --> 00:06:16.660 sorry, this is supposed to be sub s-- 00:06:16.660 --> 00:06:18.580 divided by the initial concentration. 00:06:18.580 --> 00:06:21.300 So c sub naught is if there was already some carbon in my 00:06:21.300 --> 00:06:25.480 steel, minus, again, the surface concentration. 00:06:25.480 --> 00:06:27.590 This equals to an error function-- 00:06:27.590 --> 00:06:30.140 which is the solution to Fick's second law-- 00:06:30.140 --> 00:06:33.880 and this error function has the values of position, 00:06:33.880 --> 00:06:35.840 diffusion coefficient, and time. 00:06:35.840 --> 00:06:43.180 So in order to verify if it's a transient state, we're going 00:06:43.180 --> 00:06:45.800 to go ahead and take that equation and break it down. 00:06:45.800 --> 00:06:49.500 So if I assume that there's no carbon, I can go ahead and 00:06:49.500 --> 00:06:52.690 write this down. 00:06:52.690 --> 00:06:57.360 Assume c0 is 0, just to make the equation easy. 00:06:57.360 --> 00:07:00.410 I can, because there was no information given about that. 00:07:00.410 --> 00:07:02.390 I can go ahead and-- 00:07:02.390 --> 00:07:04.100 because we're trying to figure out if 00:07:04.100 --> 00:07:06.830 it's a transient state-- 00:07:06.830 --> 00:07:07.770 write this down. 00:07:07.770 --> 00:07:13.340 So c minus c sub s over negative c sub s. 00:07:13.340 --> 00:07:18.380 This equals to my error function, which is x divided 00:07:18.380 --> 00:07:19.630 by 2 squared of Dt. 00:07:22.050 --> 00:07:28.220 And I know that at x1 and time-- t1-- 00:07:28.220 --> 00:07:32.240 I have this value of my concentration, c sub star. 00:07:32.240 --> 00:07:36.800 So I'm going to go ahead and plug that in 00:07:36.800 --> 00:07:38.030 here into this equation. 00:07:38.030 --> 00:07:40.530 So I know that my concentration is c sub star. 00:07:40.530 --> 00:07:48.600 So that's what we measured, c sub star minus c sub s over 00:07:48.600 --> 00:07:50.680 negative c sub s. 00:07:50.680 --> 00:07:59.815 This equals to my error function, x1 over 2 square 00:07:59.815 --> 00:08:01.460 root of Dt1. 00:08:04.680 --> 00:08:09.510 So this is for the first time point, x1, t1. 00:08:09.510 --> 00:08:13.210 Now I want to go ahead and do the same thing 00:08:13.210 --> 00:08:15.980 for the second point. 00:08:15.980 --> 00:08:19.630 And if I do it for the second point, I know that just by 00:08:19.630 --> 00:08:25.330 looking at the equation, I have the same concentration at 00:08:25.330 --> 00:08:29.000 x2, and I have the same surface concentration. 00:08:29.000 --> 00:08:34.670 So what that tells me is that this value is going to be the 00:08:34.670 --> 00:08:41.430 same for x1, t1, and x2, t2. 00:08:41.430 --> 00:08:46.950 So if I go ahead and plug this into my equation. 00:08:46.950 --> 00:08:53.190 I can go ahead and look at this and say, this is for t 00:08:53.190 --> 00:08:57.020 equals 1 for that, and this is going to be-- 00:08:57.020 --> 00:09:00.752 or, t equals t sub 1, sorry-- and this is t sub 2. 00:09:00.752 --> 00:09:05.650 I have c star minus cs all over negative cs. 00:09:05.650 --> 00:09:14.280 This equals to my error function of x2 over 2 square 00:09:14.280 --> 00:09:15.530 root of Dt2. 00:09:19.110 --> 00:09:24.190 So I know that this value is the same as this value. 00:09:24.190 --> 00:09:28.850 So therefore, this should be the same as this. 00:09:28.850 --> 00:09:32.020 The argument in both your error functions. 00:09:32.020 --> 00:09:33.640 So we have to go ahead and prove that. 00:09:33.640 --> 00:09:34.870 Well, how are we going to do that? 00:09:34.870 --> 00:09:37.880 Do I know x2 as a function of x1? 00:09:37.880 --> 00:09:41.680 I do, because that's given in the data. 00:09:41.680 --> 00:09:52.220 And I know again, that x2 equals 2x1 and t2 equals 2t1. 00:09:52.220 --> 00:10:03.740 So what I want to know, does x1 over 2 root Dt1, does this 00:10:03.740 --> 00:10:12.590 equal to x2 over 2 root Dt2? 00:10:12.590 --> 00:10:15.560 Well, I'm going to go ahead and plug in my values that I 00:10:15.560 --> 00:10:25.490 know, and this becomes 2x1 divided by 2 times the square 00:10:25.490 --> 00:10:27.530 root of D-- 00:10:27.530 --> 00:10:30.100 and t2 happens to be 2t1. 00:10:33.220 --> 00:10:38.250 So those two cancel each other out, and I end up getting that 00:10:38.250 --> 00:10:45.225 this is actually equal to x1 over square root of 2Dt1. 00:10:48.270 --> 00:10:55.400 Now this and this are not the same. 00:10:55.400 --> 00:10:59.300 Because this has a square root of 2, where the other 00:10:59.300 --> 00:11:02.130 one is just a 2. 00:11:02.130 --> 00:11:05.730 So because it did not satisfy the solution to Fick's second 00:11:05.730 --> 00:11:09.940 law, we can say that it's not case two either. 00:11:09.940 --> 00:11:14.770 This is not transient state diffusion. 00:11:14.770 --> 00:11:17.380 So therefore, by the process of elimination, it has to be 00:11:17.380 --> 00:11:19.080 governed by some other phenomena. 00:11:19.080 --> 00:11:20.450 And that is the answer to this problem. 00:11:20.450 --> 00:11:25.190 The fact that it's governed by something that we don't have 00:11:25.190 --> 00:11:26.590 enough information to answer. 00:11:26.590 --> 00:11:29.390 We don't have enough data points to answer. 00:11:29.390 --> 00:11:32.580 So that's part a, which is good. 00:11:32.580 --> 00:11:41.350 Now part b asks, if the steel specimens in part a were 00:11:41.350 --> 00:11:44.480 replaced with steel specimens of identical composition but 00:11:44.480 --> 00:11:47.900 with a dislocation density 10 times greater than that of the 00:11:47.900 --> 00:11:50.260 steel in part a, how would the rate of 00:11:50.260 --> 00:11:51.760 uptake of carbon change? 00:11:51.760 --> 00:11:53.650 Explain. 00:11:53.650 --> 00:11:56.340 Again, this requires you to know properties of 00:11:56.340 --> 00:11:57.620 dislocations, and what 00:11:57.620 --> 00:11:59.640 dislocations do to your materials. 00:11:59.640 --> 00:12:00.890 Your metals, for example. 00:12:04.420 --> 00:12:05.610 What is a dislocation? 00:12:05.610 --> 00:12:09.430 Well, a dislocation is-- if you have a bunch 00:12:09.430 --> 00:12:12.050 of a plane of atoms-- 00:12:12.050 --> 00:12:15.230 what a line dislocation is, or an edge dislocation is, this 00:12:15.230 --> 00:12:18.110 line defect such that it just terminates. 00:12:18.110 --> 00:12:20.010 So you have a plane of atoms that just comes to a point and 00:12:20.010 --> 00:12:21.090 then it stops. 00:12:21.090 --> 00:12:23.200 And what do you create here? 00:12:23.200 --> 00:12:27.240 Well it creates void, and void facilitates diffusion. 00:12:27.240 --> 00:12:28.490 It increases-- 00:12:30.540 --> 00:12:33.970 in our case, we're talking about carbon 00:12:33.970 --> 00:12:36.000 diffusion into steel-- 00:12:36.000 --> 00:12:41.430 so I would argue that the more dislocations that you have, 00:12:41.430 --> 00:12:45.500 the greater the diffusion should be. 00:12:45.500 --> 00:12:47.660 Because you have more void-- more spaces-- 00:12:47.660 --> 00:12:51.540 for your carbon atom to jump into and from. 00:12:51.540 --> 00:12:56.260 So for part b, just by arguing the fact that dislocations 00:12:56.260 --> 00:13:00.090 represent defects in your crystal-- 00:13:00.090 --> 00:13:03.900 specifically a defect that can create void-- 00:13:03.900 --> 00:13:09.050 then with that argument, the fact that more void means 00:13:09.050 --> 00:13:14.030 higher diffusion, I would conclude that-- 00:13:14.030 --> 00:13:15.190 for this part-- 00:13:15.190 --> 00:13:39.340 that my rate of diffusion of carbon should increase when 00:13:39.340 --> 00:13:50.390 using steel of a greater dislocation density. 00:13:58.220 --> 00:14:02.025 So the rate of carbon uptake or diffusion of carbon into 00:14:02.025 --> 00:14:05.620 your steel increases as you increase 00:14:05.620 --> 00:14:07.920 your dislocation density. 00:14:07.920 --> 00:14:12.600 And with that, you're to be able to solve a lot of these 00:14:12.600 --> 00:14:15.110 problems. And these are common in 3.091. 00:14:15.110 --> 00:14:19.150 And again, knowledge of Fick's first and second law, how to 00:14:19.150 --> 00:14:23.910 use the solutions to Fick's first, second law, and knowing 00:14:23.910 --> 00:14:26.150 what a dislocation is, that will help you a lot in solving 00:14:26.150 --> 00:14:27.550 these problems.