Problem 57969. Compute flow in a partially full pipe

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:rPr><w:b/></w:rPr><w:t>Problem statement</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>When does the maximum flow occur in a pipe? Intuition might suggest that it occurs when the pipe is flowing full—i.e., when the depth of flow </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=\"h\"/></w:customXmlPr><w:r><w:t>h</w:t></w:r></w:customXml><w:r><w:t> equals the diameter </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=\"D\"/></w:customXmlPr><w:r><w:t>D</w:t></w:r></w:customXml><w:r><w:t> of the pipe. </w:t></w:r></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 that takes the ratio </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=\"h/D\"/></w:customXmlPr><w:r><w:t>h/D</w:t></w:r></w:customXml><w:r><w:t> and produces the flow rate as a fraction of the value for a full pipe. For example, when the input is 1/2, the output should be 1/2, and when the input is 1, the output should be 1. Assume that Manning’s equation applies and Manning’s roughness coefficient is constant. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>See Test 13 for the answer to the initial question.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Background</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Steady uniform flow in a channel is often computed with Manning’s equation, which results from a balance between the component of the fluid’s weight in the flow direction and friction on the walls of the channel. The shear stress on the channel walls is computed with an empirical relation. Manning’s equation for the flow </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=\"Q\"/></w:customXmlPr><w:r><w:t>Q</w:t></w:r></w:customXml><w:r><w:t> is</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=\"Q = (C/n)R^(2/3)S0^(1/2)A\"/></w:customXmlPr><w:r><w:t>Q = \\frac{C}{n}R^{2/3}S_0^{1/2}A</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:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> is Manning’s roughness coefficient, </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=\"R = A/P\"/></w:customXmlPr><w:r><w:t>R=A/P</w:t></w:r></w:customXml><w:r><w:t> is the hydraulic radius, </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=\"S0\"/></w:customXmlPr><w:r><w:t>S_0</w:t></w:r></w:customXml><w:r><w:t> is the slope of the channel, </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> is the cross-sectional area of the flow, 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=\"P\"/></w:customXmlPr><w:r><w:t>P</w:t></w:r></w:customXml><w:r><w:t> is the wetted perimeter (i.e., the perimeter of the solid wall of the channel that is touching the water.) The coefficient </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 1 for SI units and 1.5 (or 1.49) for U.S. customary units. </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: 392.8px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 196.4px; transform-origin: 407px 196.4px; vertical-align: baseline; "><div style="block-size: 21px; 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 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Problem statement</span></span></div><div style="block-size: 42px; 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 21px; text-align: left; transform-origin: 384px 21px; 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: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; "><span style="">When does the maximum flow occur in a pipe? Intuition might suggest that it occurs when the pipe is flowing full—i.e., when the depth of flow </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">h</span><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: 65.35px 8px; transform-origin: 65.35px 8px; unicode-bidi: normal; "><span style=""> equals the diameter </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">D</span><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: 38.5px 8px; transform-origin: 38.5px 8px; unicode-bidi: normal; "><span style=""> of the pipe. </span></span></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: 109.933px 8px; transform-origin: 109.933px 8px; unicode-bidi: normal; "><span style="">Write a function that takes the ratio </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADsAAAAlCAYAAAD8+ZFYAAADwElEQVRoQ+1ZuaoUQRR97wvEJTJ0AQ1EQU1EY7dYVDQwcwkFFdREXEARzFxAQUyeIiaCogYmiiAaaCAmKn6AC+gH6DnQdzhTU8utnp4WmTdwmZ7XVbfuuXXurdP9Zmem6DM7RVhn5sH2tNtXsc5l2Jee1vtnO7sWAN/Blk8D2PMN0D197SrX8dYsd2I17BfscQcBfoKP07C7GV/LcG+pc62XnnE5sJvhYDvspDg6hOsbHseZMTtw7xFsCex7Aewq3D8D2xCMe9r8Xiz37uH6Dux1yq9nZ0k5A7wO1+/HBMvGxM8Rpx8m/YWM3YlrZRcBH5UYmQj6Hml8HrCk2m7YZ9gKZ4C5YT9wc38QcG78Qdy83gz4ie9FicGs/7nm3tsm5iHAHrAMbiHsWsVupIJnQNzZVMCxeZZs3iNVc03tCe5vbZyMjC2BVQqF9GmzyQycDDlVMdmSzSl7YbmmZv3A3A+VXQnsccy62MwsNZRS/Kytb7Caurfz2Hx7zuU/EsgJXF+y3yWwRgsW/TZxwsB5FP2GeRsWa+8YrKbuNdnenvEGa1j3HqJyDqztBDFqhjQAD7UsRwziJqzm6NIavOCkv85ho9ro2Vnl/xZM+AijI56N52C3YaTVkMMElykQuDMeGqoLpaS3Z2iNu8Ha+cp2v7IBqjtjTksdksHT13qYlkIiL4M/h82mVHI2URPkprFxnxMIjKCti2qXHmoCCQQeeRhOVTET9oxUokIB4mpQRjs6JVjSb8B9XGvdkuI5bWoB1HZzbTSehBqDVN66jh5VI3QSBmpNIKdoLPu18pDztDnyt/e4IoO4MfyM9JJUHTDAw82kmPi3uvDUa6085LKabE9Cwzn8PdLQUmCt+cQW0rooKZo28pCBarI9MpVM4NOO7Wp0TgysqpZYrWi9ljokpR0T533CMeorHUsJDZOTFB+xYPUpI1YrVq9GYe7eGliod9vIQwZeKxGVBeza+2DR5+QY2NIjndUrM06h8QAWk4BMAsVHjTwk2PCRjmd8LHiW0xWYScOiwoqBNTAx3mu9ki6saQqFWDA8Ou7DBkLcOJr55pH3TGqPQ9kgP8icTbjeJSAZp+stZQiWFLKnHNZrKPJJzbMwBvUcdisBtI08JBMOOBLCB/KvsFcw17sn81lqMI61o0PYxJh9FSJtfXU2b1Jg28jDzkClHE0CrHXTWnn4X4K1d0y9vgD3ZGoSO9tGHnpiHXtM12B5ND2E1bw9HBuE10HXYHk0LYD19p85L1CO6xpszdq9j50H23vKe1rwL6q22CbizNfIAAAAAElFTkSuQmCC" alt="h/D" style="width: 29.5px; height: 18.5px;" width="29.5" height="18.5"></span><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: 251.258px 8px; transform-origin: 251.258px 8px; unicode-bidi: normal; "><span style=""> and produces the flow rate as a fraction of the value for a full pipe. For example, when the input is 1/2, the output should be 1/2, and when the input is 1, the output should be 1. Assume that Manning’s equation applies and Manning’s roughness coefficient is constant. </span></span></div><div style="block-size: 21px; 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 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 150.917px 8px; transform-origin: 150.917px 8px; unicode-bidi: normal; "><span style="">See Test 13 for the answer to the initial question.</span></span></div><div style="block-size: 21px; 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 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 40.8333px 8px; transform-origin: 40.8333px 8px; unicode-bidi: normal; "><span style="font-weight: 700; ">Background</span></span></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: 369.8px 8px; transform-origin: 369.8px 8px; unicode-bidi: normal; "><span style="">Steady uniform flow in a channel is often computed with Manning’s equation, which results from a balance between the component of the fluid’s weight in the flow direction and friction on the walls of the channel. The shear stress on the channel walls is computed with an empirical relation. Manning’s equation for the flow </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">Q</span><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: 7px 8px; transform-origin: 7px 8px; unicode-bidi: normal; "><span style=""> is</span></span></div><div style="block-size: 34.8px; 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 17.4px; text-align: left; transform-origin: 384px 17.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANMAAABGCAYAAABWrKEpAAAMkklEQVR4Xu2dV6stSxHH9/0CiuFJQcQAiopiRvRBwYCCKFcxPB1QzAhec3oypzcxoXARI+qLmAV9UMSMonAfDFwE9cXsB9D6cabuqV27Q/XsmbVmza6B4uyzVk9Pd3X9u2LPuuksr+RAcmARDty0SC/ZSXIgOXCWYEohSA4sxIEE00KMzG6SAwmmlIHkwEIcSDAtxMjsJjmQYIrLwH2k6T1M8//K3782/3+8/H2b0N/jXWbLPXEgwdReTQDyQqHnC91F6A9Cv59uear8+0+hnwrdbfrsUXsSjpzLGAcSTGV+PVQ+fp8QgOF6o9CXhf5omqOpvij0yOmzj8q/rxhjf7beEwcSTBdXEy30EaOJbpa/rTln70Aj/UTovhPg3r8n4ci5jHEgwXSeXwDp89NHmHSPEer5QC+VNh8TeobQN8bYn633xIEE043VxD/6gVlctI0162rrjkn4K6G7B4C3J9nJuTgOJJiuMwRz7XdCBBm4Xib08QFpQTuNtB/oetWmGmBJX28BNieYrjMRH+nlEz+J0N11Ad6eQhfM+3NCPzyFwW59jAmm61rpb2ahiNydUiCB8ePbPXiag486tmSQMP/9Cg2ebvr7tvxdC8AcU76/JQ9Ho0ZM8YOMM8F0dqYBBGX4wzYqPCWBwF/7vhDBEkxU/DyuyIZAsOVebuMAmAgpfQE0TQ2Mmr1rCy9pCbTq09Z+0Ej/Caazsy8Iw543Me2UTDwNy1+TsauZZqORTzCfl2SitLMjoLcbgGlwJRrZHJE92xbf7U5C0WjoG6Ttn4RYO3sx3scK3bny/dzxhe5LMJ2d/WPaiWHYz4VOpYoB4DxE6K1upXVzeHfhO9sUMNmdHXDeUriHdmioNaKVaBjGyWbWG68de8k8ZSOAJ1SkqEZlPV8sdBAz9TJgghEPEFJbnclu1b5u7Sz/M19S0cCCHOpiR35c4GG/lTZ+167VArJrU73REk6do9/ZS0P5mXyIZurxxftu9KX+G9+9SOhTQpq3Y/wPEnqi0AiYShFI5vxwM0Y1V6lOQSYvYw7iBvxYqAvIOWCCqa8V0oF+b1qBJ8m/7Agw/tWFxQ/IzFGaWDCN7I5LDRbT5JMTP7VPdtQvCWGuEGXEh8EEpWTJayI/DgVTK4kcdd4R3FuFeslrnvmmaSDfkX9/Of39kmnc6suVoqQR8Ns50v5HQjYCCeABpA1GqInKvXO1quYeQ3IxAia76Cy2HzyDtjb71pzWmvBvwcxTgWKM3m+zuyzf92oAAcqjhWrh/ajzzo78HiHMpnc64bW8fJf85y3TuJ/p2tmx17T+KJi8ecpYiD6W/C3dKOeCCZCiNEIWSxRMtl6t17Ha2AgFKryrHmtSfqDPbQCCTQKToFdCtPTQVCDpt8RfhOXr5qG1iCNAwTJoaaWa827nBJCI9KlWpM+nCPkwtD6Pe2sg17HXIowjYBpNMgOmuX6wjfIy/1IK4ZwcRMA0Wq9m2/eAt7RQzunPjpf7o6FxDUvffwHw4VCrKVTT6NYcfYG0L/k7fIambVU01HJLJd5ZzVJ6puVdyxRi7DWAj4BpJMmsJtocC8lXxMCbrnbrgUkFRstseuFWHmpr3NBOSwjbHJCM3KPqXHfYt3cAwhy/KrSE5rW7O8+v1QRaMJUEE8F+llArUDC6szMeHV8JLNY8bW2cjL0mjCNgGtkIAB6R2TnRWb3X5u66st8Ckz1eoEIWqeHyBaPdQYxI/Upt/VwRHCJPJbPmdZPALgEkpuM1f60iwZp5HnBsel8R6gUKRnZ2y2rAUDLTvPlZW2tkolayFAVTKclcEwdVAnPWSAMXzOVtQhpmr1kDd4yhBSa767R2TD8hD6ZTOZoAoN4hZGv0NDJFVO0RE2PxDXqaawTzti6w5Hf4AITXEBoY8r6ehqqtYz6ys+sctNyqBhRromKJ4LCPlPhEwRSNQDLezwq1giat9eE5/5o2OYuBbkSvBiZvM474Ph6EUR8kmnPpCepIbVqpL8waomE44IT7uX4h9Buh7wotHZyw0US/8fjTvH4dVCNdk3HZnV83BlvE2tvZuec50xwtGFjP5wrVzCW/eY4mSiNgikYgWSv8Rkqs5lTxa6BNNwRrNXTzVTUw+Xq1Ee1io2NMruu4TQLrQdgDTe37KHjn9r/kfTYXQr/sfv8WQhMSPdMj8QgouSgrIN409ePyESg04AeFalrDgoJxfFOIhDLJ0FcKtTYRH8SppU5KvIuAKRKBpG/mSPTYAwlzlBPRrTkoP9mwNJdnedKN6NXApOFtBjgaRJibt1lKM9ks+5KCv0ZfPvz6CXmIJr95HrthrTLaVnaXxuYTm6X8jL/PrgG1b+SYoibbXEBFwBQxTwESuTUfgGFOryp87ufOOEg8+/ycDfw0FUMNTHNLbPxOG6leXkNIT6VPu2lpCNfvhr2gQmSuPRMv0kekjQdUxD1QHtTaYuIR9GkFv7wl5cfaCx5oxLIURrc+YTOYVgLTZQBhnek5zmhkwfbUxm5a1jy1C9gThAg/os57pK9eG5uApm3NRUCAnyyESatXSRuiMbyWtWPQur/WuHrn03BNMKvfW+iEkijNATbXogSmudE4H7ToRj96q7Lz730+zpoX0fxNhEUjznukv0gbq3F75U+9/iImXq+P1ve2/g5/1V/W7G7KdEQzRYMPVgBG/SwmsJTPdNlo3mUWZuTeVglRNJE78rwl27aSsDzHWjdzy3lUJngJaCS/OXd+gJUx1pLd4YhexGeKgInFZ0Ba3ewLHiMTPXQ0j7zRGtdfpNO/Bjq2VRclW73kTwW6Xb2JAr0XpVUT9jJgmptkjjJBfa1WBDgc0auByYa3I0GEnmBEJreUZopE83hn+J8jg5rRBjDds3Off+9EqYTI7oiXEcgZU2jeosLVcsbt/FqmEe3wm8jplU7OLj1225+6JSTmWyVYfq2qm0gNTBaNvYXUoAOmHeo4cuBsTSZF+8axXOMCTF/rdBwpIfI+6FbyZ7qbt5KYdn61cRPa/4zQm4Uwzak+ITFcOtqzxjqpmR3hqw0UVTeRVjmRtelLZog6ttQujWa912DOKfVpo56tzcq220pAx8pFyWqxxdG1im11C7xWwH9hU55TnDqy/urT9RSF9mlzp9Uq9F7VuB4QwxciD6AnKO8tf1PDRlb4A0JzSjdGJr+ntj4nAg9vFiqd+5obWV2TX/hyaE1MezQQgkaymYtqCTRLTy50k/ChZgVqxE+fO0d4SsW//qpJ6ZyW7btULlXUnj0w0anatbyDG9XOMXVCiKFz8XNnvNP7CLLUrprP4O85tG/hx+vfXmvfCETbVk5I+9Kd3ptYah5Gkr1zRASNSP2hv0pjrrXVey/cEwGTqm0YMJKNZ5cpHWOYw4S8Zz8csGF/78yrFujWwW2RHT0wqZOISsS+ZJKYeuyOtQpqrXTGDFgzP7BFfuaY+hywZlMNTPTSC733n3TgFtEARG1YgAsn8vapgWaLt+IsH5id+bgAB64UmPRwFQekPi30n4lBlONzrkWPBZT4hvZ6jVC+CD4gVVe0yZUCU2+NAdsDhXiBoBYp8pLE3nmRXr/5/dXggE2C1sw8/7qzk+BMz2c6iUnkIE+OA1oV7ys/1o7mrcqoBNOq7M3OKxzo5ZnmvJ7r6MxOMB19Ca7kAFoVEDCk+8LHLXItwVRfFfJrWrCqb/ixhZmll+lvcY23OiZ9eQm1eVTQoK2oRqhVg2x1HneMK8F0fonI7j9bSN+Vxrda8mJzbnrXmmUvmxeeBQaob0QikHXym1OCqSwReqREDzlyTID3sF0T4ke59IWQmU9bAFF76SLBVF5JjTZRI/ZhoVuFtJTK5klO4W21e5HVzc8jwXRxiWztGFGl108aSRPReiJ4ztH8zQtEDnA+BxJMF3nnD+75IyZ6Cnmtyub5q5l3HpUDCaaL7LdH9kunSfXU5UnmQo4qbTt/eILp4gLrWZvSj7VZf6n20y87F5mcXo0DCabznLFgKb3vTU+CnuR5m4TBuhxIMJ3nr32/QelFGxoy15A4wYrou7jXXcns/egcSDCdXwIFS0nz2CgfIfHbhPgdIF4qsvXf7T26oF2FASSYbqyyPRpQeuuOfREK/hLRvA8Jncqrza6CPB91jgmmG+y3IfFSMtb/wkNG844qutt7eILpxprYN8rWfjUB7UQd2am8z3x7ErfjESWYdry4ObXDciDBdFh+59N2zIEE044XN6d2WA4kmA7L73zajjmQYNrx4ubUDsuBBNNh+Z1P2zEHEkw7Xtyc2mE5kGA6LL/zaTvmQIJpx4ubUzssB/4P3KP+ZXElbY0AAAAASUVORK5CYII=" alt="Q = (C/n)R^(2/3)S0^(1/2)A" style="width: 105.5px; height: 35px;" width="105.5" height="35"></span></div><div style="block-size: 64px; 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 32px; text-align: left; transform-origin: 384px 32px; 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; "><span style="">where </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><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: 112.55px 8px; transform-origin: 112.55px 8px; unicode-bidi: normal; "><span style=""> is Manning’s roughness coefficient, </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="R = A/P" style="width: 59px; height: 18.5px;" width="59" height="18.5"></span><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: 73.5167px 8px; transform-origin: 73.5167px 8px; unicode-bidi: normal; "><span style=""> is the hydraulic radius, </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="S0" style="width: 15.5px; height: 20px;" width="15.5" height="20"></span><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: 87.125px 8px; transform-origin: 87.125px 8px; unicode-bidi: normal; "><span style=""> is the slope of the channel, </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">A</span><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: 39.6667px 8px; transform-origin: 39.6667px 8px; unicode-bidi: normal; "><span style=""> is the cross-sectional area of the flow, and </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">P</span><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: 278.1px 8px; transform-origin: 278.1px 8px; unicode-bidi: normal; "><span style=""> is the wetted perimeter (i.e., the perimeter of the solid wall of the channel that is touching the water.) The coefficient </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">C</span><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: 182.383px 8px; transform-origin: 182.383px 8px; unicode-bidi: normal; "><span style=""> is 1 for SI units and 1.5 (or 1.49) for U.S. customary units. </span></span></div></div></div>

Problem statement
When does the maximum flow occur in a pipe? Intuition might suggest that it occurs when the pipe is flowing full—i.e., when the depth of flow  equals the diameter  of the pipe. 
Write a function that takes the ratio  and produces the flow rate as a fraction of the value for a full pipe. For example, when the input is 1/2, the output should be 1/2, and when the input is 1, the output should be 1. Assume that Manning’s equation applies and Manning’s roughness coefficient is constant. 
See Test 13 for the answer to the initial question.
Background
Steady uniform flow in a channel is often computed with Manning’s equation, which results from a balance between the component of the fluid’s weight in the flow direction and friction on the walls of the channel. The shear stress on the channel walls is computed with an empirical relation. Manning’s equation for the flow  is

where  is Manning’s roughness coefficient,  is the hydraulic radius,  is the slope of the channel,  is the cross-sectional area of the flow, and  is the wetted perimeter (i.e., the perimeter of the solid wall of the channel that is touching the water.) The coefficient  is 1 for SI units and 1.5 (or 1.49) for U.S. customary units.

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Christian Schröder on 8 Apr 2023

Is something broken on Cody right now? I submitted a solution (several iterations of it, in fact) well over an hour ago, and the problem shows up as solved for me, but no solutions are visible and my profile doesn't show that I solved this either.

Dyuman Joshi on 9 Apr 2023

Yes, there seems to be a problem. No solutions have been displayed for 2 solvers, nor is your comment visible on the comment section of Recent activity.

I'll inform the devs about this.

Ramon Villamangca on 10 Apr 2023

Hi Dyuman, I tried solving other problems, and my solutions, although correct, is also not counted. Are the devs working on this?

Christian Schröder on 10 Apr 2023

@Dyuman thank you.

@Ramon it appears work's being done on this; Solutions for this problem are showing up for me now, and recent comments are starting to show up again as well (albeit incomplete and with the wrong timestamps).

Arunika Oyshi on 10 Apr 2023

We are currently looking into this issue. Thank you for being patient!

Christian Schröder on 10 Apr 2023

Thanks, Arunika!

Arunika Oyshi on 10 Apr 2023

@Christian, Dyuman and Ramon, I've got some updates. So an unexpected issue occurred on 4/9/23 which caused a pretty big backlog of updates in our queue. That queue is slowly being processed, so the delayed updates to your profiles will self-resolve in the next few hours.

We are currently working on speeding this up, but aside from an inconvenient delay, we assure you your progress will not be lost. We apologize for this inconvenience and we hope you understand!

Dyuman Joshi on 10 Apr 2023

Looks like things are back to normal now, Thanks Arunika!

Christian Schröder on 11 Apr 2023

Totally understood, Arunika. Thanks again to you and everyone for fixing this so quickly!

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Problem 57969. Compute flow in a partially full pipe

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Problem 57969. Compute flow in a partially full pipe

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