1 00:00:04,580 --> 00:00:09,250 At a pristine lake in Ontario, Canada, scientists had been adding low levels of estrogen to 2 00:00:09,250 --> 00:00:14,710 the lake and were watching for changes. The levels were approximately equal to the concentration 3 00:00:14,710 --> 00:00:19,000 of estrogen found in treated sewage discharge. 4 00:00:19,000 --> 00:00:23,680 The result: fathead minnow populations began to collapse! 5 00:00:23,680 --> 00:00:28,109 Many of the chemicals that we use and release, whether on purpose or by accident, can have 6 00:00:28,109 --> 00:00:34,070 severe effects on organisms and ecosystems. To predict the accumulation and levels of 7 00:00:34,070 --> 00:00:38,409 contaminants, we can use differential equations. 8 00:00:38,409 --> 00:00:43,220 This video is part of the Differential Equations video series. 9 00:00:43,220 --> 00:00:47,909 Laws that govern a system's properties can be modeled using differential equations. 10 00:00:47,909 --> 00:00:53,470 Hi, my name is David Griffith, and I'm a graduate student in the MIT/Wood's Hole Joint Program 11 00:00:53,479 --> 00:00:58,940 in Oceanography and the Department of Civil and Environmental Engineering at MIT. 12 00:00:58,940 --> 00:01:03,040 Before watching this video, you should be familiar with defining control volumes and 13 00:01:03,040 --> 00:01:06,930 applying conservation of mass. 14 00:01:06,930 --> 00:01:11,640 After watching this video, you will be able to: Construct a differential equation to estimate 15 00:01:11,640 --> 00:01:17,100 the concentration of a chemical in the environment; and Appreciate how informed estimates can 16 00:01:17,100 --> 00:01:21,150 help simplify and solve these differential equations. 17 00:01:25,800 --> 00:01:27,680 While everyday chemicals provide a range of 18 00:01:27,700 --> 00:01:32,789 benefits to society, they are often released into the environment and can cause unintended 19 00:01:32,789 --> 00:01:36,729 harm. Examples include acid rain from smokestack 20 00:01:36,729 --> 00:01:43,200 sulfur emissions, toxic groundwater from leaky gas station tanks, and endocrine disruption 21 00:01:43,200 --> 00:01:49,470 in fish populations due to treated and raw sewage discharges. 22 00:01:49,470 --> 00:01:53,070 It would be helpful if we could monitor the levels of potentially hazardous chemicals 23 00:01:53,070 --> 00:01:58,479 in the environment. But, given the tens of thousands of chemicals in daily use, it shouldn't 24 00:01:58,479 --> 00:02:01,890 be surprising that measuring all of them is not practical. 25 00:02:01,890 --> 00:02:08,110 So, what we need is a framework for simplifying a very complex world. We accomplish this through 26 00:02:08,110 --> 00:02:12,920 the use of mathematical models, which are simplified versions of reality. 27 00:02:12,920 --> 00:02:17,410 Many models take the form of differential equations, which describe how one variable, 28 00:02:17,410 --> 00:02:22,920 like the concentration of a chemical, changes with respect to another variable, like time. 29 00:02:22,920 --> 00:02:27,030 By applying conservation of mass, we can model the change in chemical concentration with 30 00:02:27,030 --> 00:02:34,030 time as being equal to the inputs minus the outputs. An input is the rate at which a chemical 31 00:02:34,680 --> 00:02:39,400 enters our control volume. An output is the rate at which the chemical leaves the control 32 00:02:39,400 --> 00:02:40,490 volume. 33 00:02:45,920 --> 00:02:48,160 To find the inputs and outputs of chemicals 34 00:02:48,160 --> 00:02:52,610 into a control volume, the first step is to identify the ways that chemicals enter and 35 00:02:52,610 --> 00:02:56,990 leave the control volume. If we think about the estrogen experiment 36 00:02:56,990 --> 00:03:01,959 in Ontario, the estrogen entered the lake from the scientist intentionally dumping estrogen 37 00:03:01,959 --> 00:03:07,020 from a motorboat and from the fish themselves. 38 00:03:07,020 --> 00:03:11,599 Once released into the lake, there are many ways estrogens can leave the lake. A stream 39 00:03:11,599 --> 00:03:17,430 could carry the estrogens out of the lake. Also, physical, chemical, and biological processes 40 00:03:17,430 --> 00:03:23,760 within the lake could degrade the estrogens to inactive byproducts. 41 00:03:23,760 --> 00:03:28,280 In my own research, I am interested in the fate of estrogens in the treated sewage that 42 00:03:28,280 --> 00:03:32,040 is discharged into Massachusetts Bay. 43 00:03:32,040 --> 00:03:34,110 To start, I had two initial questions: 44 00:03:34,110 --> 00:03:39,239 1) would the estrogen levels in the bay be high enough to detect with our instruments? 45 00:03:39,239 --> 00:03:43,819 And 2) how much water would I need to collect? 46 00:03:43,819 --> 00:03:49,569 To answer these questions, I needed to estimate the concentration of estrogen in the bay. 47 00:03:49,569 --> 00:03:53,989 This is where the differential equation comes in. We can set up the equation by setting 48 00:03:53,989 --> 00:04:01,489 the time change in estrogen concentration (dC/dt) equal to inputs minus outputs. 49 00:04:02,800 --> 00:04:08,180 Note that our definition of outputs includes both physical flows and chemical reaction 50 00:04:08,180 --> 00:04:15,160 rates. What are the possible inputs? Because Massachusetts Bay receives 360 million gallons 51 00:04:15,160 --> 00:04:20,548 of treated sewage discharge from the city of Boston every day, we can assume here that 52 00:04:20,548 --> 00:04:25,499 sewage is the dominant input. Now, can you think of any removal processes 53 00:04:25,500 --> 00:04:33,340 (that is, outputs) besides water leaving the bay? Pause the video here and write down a couple. 54 00:04:40,159 --> 00:04:42,240 There are many possible outputs, including 55 00:04:42,249 --> 00:04:50,469 the rates of dilution, sedimentation, biodegradation, loss to the atmosphere, and photodegradation. 56 00:04:51,539 --> 00:04:55,779 Observe that some of these estrogen outputs remove estrogens from the control volume, 57 00:04:55,779 --> 00:05:02,779 while others chemically transform estrogens into other compounds. 58 00:05:03,169 --> 00:05:07,979 At this point we may decide to neglect terms that we suspect will be small given what we 59 00:05:07,979 --> 00:05:14,979 know about the characteristics of the particular chemical and environment that we're modeling. 60 00:05:15,139 --> 00:05:19,979 For example, we might neglect loss to the atmosphere because estrogen's structure means 61 00:05:19,979 --> 00:05:25,469 it has a low vapor pressure. In addition, the fact that sewage is discharged 62 00:05:25,469 --> 00:05:31,219 from a pipe at the seafloor and will be trapped below the sunlit surface allows us to neglect 63 00:05:31,219 --> 00:05:35,308 photodegradation. We include the inputs and outputs into the 64 00:05:35,308 --> 00:05:41,339 differential equation like so. We're going to make two more simplifying assumptions 65 00:05:41,339 --> 00:05:46,689 to make this equation easier to solve. Remember, the goal is not to have a perfect model of 66 00:05:46,689 --> 00:05:51,129 estrogens in the bay, but to make a quick computation to estimate how much water I should 67 00:05:51,129 --> 00:05:56,599 collect before I will have levels of estrogen that I can detect. 68 00:05:56,599 --> 00:06:01,789 The two simplifying assumptions we make are that: 1. The distribution of estrogens in 69 00:06:01,789 --> 00:06:08,789 the control volume is uniform. That is, the bay is "well mixed. 2. The concentration is 70 00:06:08,809 --> 00:06:15,809 constant in time, that is the inputs and outputs are balanced. We could solve this equation 71 00:06:16,059 --> 00:06:20,289 without assuming that the concentration is constant in time, but this would make our 72 00:06:20,289 --> 00:06:26,300 solution a lot more complex, so for OUR back of the envelope calculation, this is OK. 73 00:06:30,980 --> 00:06:33,499 Let's look at the terms in our differential 74 00:06:33,499 --> 00:06:39,099 equation. The input rate is represented by the mass flux, which is the mass per time, 75 00:06:39,099 --> 00:06:46,099 Q, of estrogen leaving the wastewater treatment plant, divided by the volume, V, of the bay. 76 00:06:46,820 --> 00:06:51,860 Dilution rate is represented by the dilution rate constant K dilution times C. 77 00:06:53,760 --> 00:06:57,960 Sedimentation rate is determined in terms of the concentration of estrogens that are 78 00:06:57,969 --> 00:07:04,159 attached to solid particles, since only the estrogen on solid particles will sink. 79 00:07:04,159 --> 00:07:09,569 Keep in mind that the solid particle estrogens are represented by some fraction fs of the 80 00:07:09,569 --> 00:07:11,779 total concentration. 81 00:07:11,779 --> 00:07:17,819 So the sedimentation rate is represented by the sedimentation rate constant, k_sed, times 82 00:07:17,819 --> 00:07:24,819 the fraction of the concentration of estrogen attached to solid particles. 83 00:07:24,820 --> 00:07:32,120 Biodegradation rate is represented by the biodegradation rate constant k_deg times C. 84 00:07:32,129 --> 00:07:38,909 Assuming steady state we get this equation. Grouping like terms, and solving for C we 85 00:07:38,909 --> 00:07:44,149 get this. Now we can isolate C and solve the equation 86 00:07:44,149 --> 00:07:49,529 using what we know or can estimate about the chemical or the system. 87 00:07:49,529 --> 00:07:53,749 Once we solved for the concentration of estrogen in the bay, we determined that we needed to 88 00:07:53,749 --> 00:07:58,319 collect 20 liters of water in order to make the necessary measurements. 89 00:08:02,960 --> 00:08:07,140 Here you see me on Massachusetts bay collecting water samples. 90 00:08:07,149 --> 00:08:13,050 We would collect 8 samples of 20L of water per day, extract and concentrate each sample 91 00:08:13,050 --> 00:08:18,039 down to 100 micro liters before high enough levels of estrogens could be experimentally 92 00:08:18,039 --> 00:08:25,039 measured using a mass spectrometer. The model was key in helping me plan my sampling strategy. 93 00:08:26,020 --> 00:08:30,169 It turns out that this quick estimate was pretty good, and I've collected and was able 94 00:08:30,169 --> 00:08:33,990 to detect estrogens in Massachusetts bay. 95 00:08:33,990 --> 00:08:38,590 Now I can use these measurements to refine my model. For example, I might remove the 96 00:08:38,590 --> 00:08:43,049 assumption that the concentration is unchanging in time. 97 00:08:43,049 --> 00:08:48,430 By comparing measured and predicted concentrations, we can also hypothesize about which processes 98 00:08:48,430 --> 00:08:54,070 are dominant and discover potential missing processes that affect estrogen concentration 99 00:08:54,070 --> 00:08:58,980 in the bay. Once these hypotheses are well formed, we 100 00:08:58,980 --> 00:09:05,020 can then design new experiments to test them. And can continue to refine the model. 101 00:09:05,020 --> 00:09:10,080 Ultimately, it's the iterative process of taking measurements and refining our model 102 00:09:10,080 --> 00:09:15,210 that allows us to answer important questions about whether estrogen levels are likely to 103 00:09:15,210 --> 00:09:18,670 pose a hazard in the bay. 104 00:09:22,480 --> 00:09:24,820 We have demonstrated how differential equations 105 00:09:24,830 --> 00:09:29,750 and simplifying assumptions can be used to help us predict the concentration of a contaminant 106 00:09:29,750 --> 00:09:35,750 in a coastal bay. We have shown a simple example, but the approach can be applied to any chemical 107 00:09:35,750 --> 00:09:42,320 in any control volume and at any scale. So, the next time you are marveling at the complexity 108 00:09:42,320 --> 00:09:47,360 of the world around you, you can imagine how you might construct a model using differential 109 00:09:47,360 --> 00:09:50,180 equations to help you understand it better.