Problem 54750. Find the length of stream affected by a spill

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>When a contaminant is spilled into a stream, one might want to know how much of the stream is affected—e.g., the length over which the concentration exceeds a specified threshold. The concentration </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"C\"/></w:customXmlPr><w:r><w:t>C</w:t></w:r></w:customXml><w:r><w:t> is often computed as a function of time </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"t\"/></w:customXmlPr><w:r><w:t>t</w:t></w:r></w:customXml><w:r><w:t> and distance </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"x\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t> from the spill using the advection-dispersion equation:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"dC/dt + U dC/dx = K d^2C/dx^2\"/></w:customXmlPr><w:r><w:t>\\frac{\\partial C}{\\partial t} + U \\frac{\\partial C}{\\partial x} = K \\frac{\\partial^2 C}{\\partial x^2}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"U\"/></w:customXmlPr><w:r><w:t>U</w:t></w:r></w:customXml><w:r><w:t> is the mean velocity of the river and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"K\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t> is a dispersion coefficient, which describes spreading by several mechanisms. For an instantaneous spill of mass </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"M\"/></w:customXmlPr><w:r><w:t>M</w:t></w:r></w:customXml><w:r><w:t> mixed over the cross section (with area </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"A\"/></w:customXmlPr><w:r><w:t>A</w:t></w:r></w:customXml><w:r><w:t>) at </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"x = 0\"/></w:customXmlPr><w:r><w:t>x = 0</w:t></w:r></w:customXml><w:r><w:t>, the concentration can be shown—using some of the math needed for </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/51625\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 51625</w:t></w:r></w:hyperlink><w:r><w:t>—to be</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"C = (M/(A sqrt(4 pi K t))) exp(-(x-U t)^2/(4 K t))\"/></w:customXmlPr><w:r><w:t>C = \\frac{M}{A\\sqrt{4\\pi K t}} \\exp\\left(-\\frac{(x-U t)^2}{4 K t}\\right)</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to compute the length of stream affected by the spill. In other words, find the position </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"x = L_a\"/></w:customXmlPr><w:r><w:t>x = L_a</w:t></w:r></w:customXml><w:r><w:t> (say) beyond which the concentration never exceeds a threshold </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"C = C_t\"/></w:customXmlPr><w:r><w:t>C = C_t</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 282.7px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 141.35px; transform-origin: 407px 141.35px; vertical-align: baseline; "><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 378.317px 8px; transform-origin: 378.317px 8px; unicode-bidi: normal; ">When a contaminant is spilled into a stream, one might want to know how much of the stream is affected—e.g., the length over which the concentration exceeds a specified threshold. The concentration C<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 123.675px 8px; transform-origin: 123.675px 8px; unicode-bidi: normal; "> is often computed as a function of time t<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; "> and distance x<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.833px 8px; transform-origin: 168.833px 8px; unicode-bidi: normal; "> from the spill using the advection-dispersion equation:</div><div style="block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img 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alt="dC/dt + U dC/dx = K d^2C/dx^2" style="width: 126.5px; height: 36.5px;" width="126.5" height="36.5"></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; ">where U<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 113.958px 8px; transform-origin: 113.958px 8px; unicode-bidi: normal; "> is the mean velocity of the river and K<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 202.158px 8px; transform-origin: 202.158px 8px; unicode-bidi: normal; "> is a dispersion coefficient, which describes spreading by several mechanisms. For an instantaneous spill of mass M<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 125.242px 8px; transform-origin: 125.242px 8px; unicode-bidi: normal; "> mixed over the cross section (with area A<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.05px 8px; transform-origin: 12.05px 8px; unicode-bidi: normal; ">) at <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAkCAYAAADFGRdYAAAC7UlEQVRoQ+2YPU8VQRSG4ReI2lkRPwoqKDQ0WmgBJLYmQPwBaENFTNBSMdFAQ4GY2KsBWiOaSIGFBEK0svCjtFMw/AB9X3MmOTs7s3Puyl1IdjY52Xt3vs48886Zs9vbk68kgd5kjVyhJ0MyiCBDypAMBAxVspIypCiBsyg5I6XvU5zapqQJAHkA24HtwvphfDYLW4X9DAFrE6QlALgNewi7p2BM4fcybB12MwSqLZAciG+AMBwA8RrPRgMA/7FsA6TTmOcX2MkYBNlyz0VdQ7h/0tuuDZCcijjv67BXgbjDQE6V8fK3YyuUtI2JXxQA53D/HoDER79EbYR1vo6S3JH5ObCfL+PZgS/RiCNH8fiPDLqH+6kKB1xcYpUCzKrtNojKlOqINPKlyPI1KaMDXK3YKjnfGB8GDoHUD8NYHIY+fqwBqbAtqyBRPb9FOV8FhjsdTuA/c40nsHFZoapVclyous1DgHQLfTw19KPH60RJhf6tgfsOHHokTlGKL2ELsBcGR3UVrqzrp8OmheqL+BcKwH6fGhJ9ZuIYu/Qca0HSg1FBb2A6IfufCXezrfabyeJYxWBzKLsr5bUgsa01AHZz0p323eh2o3Mu+lNJlzr19gjrWxdXn26FhNIakzhHxh8GaQbAC7Dgy2ACRtOnG91xh441cJdOaisknbVy4FjmmhJM06cb/XEvtvxdeuVQDjvFlXaKBZLLNXiqRVP3FB0pb/p047B6YSZlR/ju6nyqlF6kIHF7bMGmYTxyQ3GJKzUPSyWSRo5dqeb8jp1w7mQLfiUIQeLW4vUB9gy2Anssz/x8aQbPuYePezrAxJjbiF8C/C2nvxJcQXnpS6UPyY8ZfgKmZUluqQStK7Ko2Sl935C2V3Hn5xDC4xz4ShWNsz4kUr0vjd8pBWm/qKZrsFh5zTk00ozzuwEjpD7YvoB7i3s0XKRiUiOeH/dBMiTDCmVIGZKBgKFKVlKGZCBgqJKVlCEZCBiqZCUZIP0F1sSXJS9IuzgAAAAASUVORK5CYII=" alt="x = 0" style="width: 36.5px; height: 18px;" width="36.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59.5083px 8px; transform-origin: 59.5083px 8px; unicode-bidi: normal; ">, the concentration can be shown—using some of the math needed for <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/51625">Cody Problem 51625</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.5583px 8px; transform-origin: 22.5583px 8px; unicode-bidi: normal; ">—to be</div><div style="block-size: 40.1px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 20.05px; text-align: left; transform-origin: 384px 20.05px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img src="data:image/png;base64,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" alt="C = (M/(A sqrt(4 pi K t))) exp(-(x-U t)^2/(4 K t))" style="width: 204.5px; height: 40px;" width="204.5" height="40"></div><div style="block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 313.617px 8px; transform-origin: 313.617px 8px; unicode-bidi: normal; ">Write a function to compute the length of stream affected by the spill. In other words, find the position <img src="data:image/png;base64,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" alt="x = L_a" style="width: 42.5px; height: 20px;" width="42.5" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.3417px 8px; transform-origin: 44.3417px 8px; unicode-bidi: normal; "> (say) beyond which the concentration never exceeds a threshold <img src="data:image/png;base64,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" alt="C = C_t" style="width: 46px; height: 20px;" width="46" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; ">. </div></div></div>

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ChrisR on 15 Jun 2022

I feel compelled to point out that the precision I requested in the tests is unrealistic in practice. These lengths do not need to be known to the centimeter, and dispersion coefficients are usually good to a factor of 2.

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Problem 54750. Find the length of stream affected by a spill

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Problem 54750. Find the length of stream affected by a spill

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