Problem 50631. Cosine effect in solar power towers

Created by Daniel Pereira

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>In Solar thermal power plants with central receiver (</w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Solar_power_tower\"><w:r><w:t>https://en.wikipedia.org/wiki/Solar_power_tower</w:t></w:r></w:hyperlink><w:r><w:t>) the mirrors (called </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Heliostat\"><w:r><w:t>heliostats</w:t></w:r></w:hyperlink><w:r><w:t>) do not aim at the sun (vector </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=\"S\"/></w:customXmlPr><w:r><w:t>\\vec{s}</w:t></w:r></w:customXml><w:r><w:t>) or the tower (vector </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>\\vec{t}</w:t></w:r></w:customXml><w:r><w:t>), but the bisector of these two directions (vector </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=\"N\"/></w:customXmlPr><w:r><w:t>\\vec{n}</w:t></w:r></w:customXml><w:r><w:t>):</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=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"265\"/><w:attr w:name=\"width\" w:val=\"464\"/><w:attr w:name=\"verticalAlign\" w:val=\"baseline\"/><w:attr w:name=\"altText\" w:val=\"\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The solar irradiance that bounces off the mirror is reduced by an amount proportional to reflectivity (assume 100% for this problem) and to the cosine of the angle between the normal vector to the heliostat and the other vector (sun or tower).</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>The aim of this problem is to calculate the value of the cosine of the angle (</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=\"cos(theta)\"/></w:customXmlPr><w:r><w:t>\\cos\\theta</w:t></w:r></w:customXml><w:r><w:t>) given the </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>normalized</w:t></w:r><w:r><w:t> vectors </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=\"S\"/></w:customXmlPr><w:r><w:t>\\vec{s}</w:t></w:r></w:customXml><w:r><w:t> 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=\"T\"/></w:customXmlPr><w:r><w:t>\\vec{t}</w:t></w:r></w:customXml><w:r><w:t> that are known for every heliostat at every moment.</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></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:i/></w:rPr><w:t>Hints:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>Vectors are 3×1, with norm = 1.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>Make use of scalar product to calculate cosines, taking advantage of the normalized 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<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: 562.367px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 281.183px; transform-origin: 407px 281.183px; vertical-align: baseline; "><div style="block-size: 52px; 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 26px; text-align: left; transform-origin: 384px 26px; 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: 161px 8px; transform-origin: 161px 8px; unicode-bidi: normal; "><span style="">In Solar thermal power plants with central receiver (</span></span><a target='_blank' href = "https://en.wikipedia.org/wiki/Solar_power_tower"><span style=""><span style="">https://en.wikipedia.org/wiki/Solar_power_tower</span></span></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: 63.5px 8px; transform-origin: 63.5px 8px; unicode-bidi: normal; "><span style="">) the mirrors (called </span></span><a target='_blank' href = "https://en.wikipedia.org/wiki/Heliostat"><span style=""><span style="perspective-origin: 30px 8px; transform-origin: 30px 8px; ">heliostats</span></span></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: 97px 8px; transform-origin: 97px 8px; unicode-bidi: normal; "><span style="">) do not aim at the sun (vector </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAA4CAYAAACYCio/AAABD0lEQVRYhe2XYQ2DMBBGn4c6mAEMoAAFOMABDrCABiTgAQvVgAX243qhafgDg5Zk95Imo8nWx7XpvgPDMAzDMH7CA01pCYAa2HiJTMuLZDpEprv6A/VNow8iGzABn7Mi24NjOCMy3zT8gcTpqtxBEwQ8slVFcMCCVMb9tQTA+AYJkHujuIRhGMYVNH/oJVaR+UIbkH/XBQlBPnzecopoJvXRnAvP/vAbDzGzVyN++yHMZWNkT15x6KmQamVDexetSpEIqEyRzEjhJBYH5L6UCMiZUJGVjFWZDhaLD26WFtOFxapkPj64WdoGXTDt1rR/WXNIpAvqFnzYr/W0Uo/RIgm9RqqirWWxVtIwDON1fAHnd3ACjCJApQAAAABJRU5ErkJggg==" alt="S" style="width: 17px; height: 28px;" width="17" height="28"></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: 68px 8px; transform-origin: 68px 8px; unicode-bidi: normal; "><span style="">) or the tower (vector </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAA8CAYAAAADm2gpAAABBElEQVRoge2XUQ2DMBBAn4c6wAAGUIACHMwBDrAwDUjAQy2gYRa6j/bSbh/LYMB1yb3kQvgovLTHcQeGYRiGYfzEACyA0xaBKFKFjAM8FcmsRKFmzwMaoDsoViAAD2LubGJMi8+IFWi/FZGsPyJKCc+OXTkKnyQmFJN2SBK9lkA1Ei3xK1GVEBF1CcMwjL/HATMVtIV34v9FDZckQrqq0JN70kBsMaVXvYyR2AqWIjI+qLSD0pd6jZeXyG5MmhJ9IXJpXrwzFSKqyKiwaEo48m7cNEVkXAhsGCHPQKqpalmHXMjmdD+iUMga8rEMKeaPK06i47Wsryj9/qX3WIhHYhiGYQhPGKFtN+aS55YAAAAASUVORK5CYII=" alt="T" style="width: 17px; height: 30px;" width="17" height="30"></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: 154px 8px; transform-origin: 154px 8px; unicode-bidi: normal; "><span style="">), but the bisector of these two directions (vector </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAA4CAYAAACYCio/AAABK0lEQVRYhe2XbQ2DMBBAnwcczAAGpmAKcICDOcACGioBD1hAwyx0P9oLB2PL2LjAsntJ/9CEe+314wqO4ziO43zFAFz2lgA4A5GDyFQcSKYmydSf/uC8UbtmkQgE4LRWJBq2Zo1It1EbFiRWz8oWXLLAQErVLhRAT5qZ4q8lANojSEA6N3aXcBzHWcP8Iivyt9IyaEkqdALj9R6UQMP0yq+sRGS0YRZMLrjAeO3H/M0UPXJyQF2TSl9rLdIzpqWZBSyViGn1XqhAUgpqKqazZYYOFHmsP2X9dNYiLa+r8RtfvmPeRQLdFvr0+jA/R17Nhjym5utmc/SrbWnEsptkF9UYpajj+YhPTLdthdGBprftUlp02vosa1I8l4zPyGcLscn9VysJx3Gcn+AOM9qMINg1yfYAAAAASUVORK5CYII=" alt="N" style="width: 17px; height: 28px;" width="17" height="28"></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: 4.5px 8px; transform-origin: 4.5px 8px; unicode-bidi: normal; "><span style="">):</span></span></div><div style="block-size: 270.5px; 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 135.25px; text-align: left; transform-origin: 384px 135.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img class="imageNode" style="vertical-align: baseline" src="data:image/png;base64,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" data-image-state="image-loaded"></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="">The solar irradiance that bounces off the mirror is reduced by an amount proportional to reflectivity (assume 100% for this problem) and to the cosine of the angle between the normal vector to the heliostat and the other vector (sun or tower).</span></span></div><div style="block-size: 60px; 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 30px; text-align: left; transform-origin: 384px 30px; 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: 236.5px 8px; transform-origin: 236.5px 8px; unicode-bidi: normal; "><span style="">The aim of this problem is to calculate the value of the cosine of the angle (</span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="cos(theta)" style="width: 33.5px; height: 18px;" width="33.5" height="18"></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: 35.5px 8px; transform-origin: 35.5px 8px; unicode-bidi: normal; "><span style="">) given the </span></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="font-weight: 700; ">normalized</span></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: 27px 8px; transform-origin: 27px 8px; unicode-bidi: normal; "><span style=""> vectors </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAA4CAYAAACYCio/AAABD0lEQVRYhe2XYQ2DMBBGn4c6mAEMoAAFOMABDrCABiTgAQvVgAX243qhafgDg5Zk95Imo8nWx7XpvgPDMAzDMH7CA01pCYAa2HiJTMuLZDpEprv6A/VNow8iGzABn7Mi24NjOCMy3zT8gcTpqtxBEwQ8slVFcMCCVMb9tQTA+AYJkHujuIRhGMYVNH/oJVaR+UIbkH/XBQlBPnzecopoJvXRnAvP/vAbDzGzVyN++yHMZWNkT15x6KmQamVDexetSpEIqEyRzEjhJBYH5L6UCMiZUJGVjFWZDhaLD26WFtOFxapkPj64WdoGXTDt1rR/WXNIpAvqFnzYr/W0Uo/RIgm9RqqirWWxVtIwDON1fAHnd3ACjCJApQAAAABJRU5ErkJggg==" alt="S" style="width: 17px; height: 28px;" width="17" height="28"></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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAA8CAYAAAADm2gpAAABBElEQVRoge2XUQ2DMBBAn4c6wAAGUIACHMwBDrAwDUjAQy2gYRa6j/bSbh/LYMB1yb3kQvgovLTHcQeGYRiGYfzEACyA0xaBKFKFjAM8FcmsRKFmzwMaoDsoViAAD2LubGJMi8+IFWi/FZGsPyJKCc+OXTkKnyQmFJN2SBK9lkA1Ei3xK1GVEBF1CcMwjL/HATMVtIV34v9FDZckQrqq0JN70kBsMaVXvYyR2AqWIjI+qLSD0pd6jZeXyG5MmhJ9IXJpXrwzFSKqyKiwaEo48m7cNEVkXAhsGCHPQKqpalmHXMjmdD+iUMga8rEMKeaPK06i47Wsryj9/qX3WIhHYhiGYQhPGKFtN+aS55YAAAAASUVORK5CYII=" alt="T" style="width: 17px; height: 30px;" width="17" height="30"></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: 163.5px 8px; transform-origin: 163.5px 8px; unicode-bidi: normal; "><span style=""> that are known for every heliostat at every moment.</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: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; "><span style=""></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: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; "><span style="font-style: italic; ">Hints:</span></span></div><ul style="block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 20.4333px; transform-origin: 391px 20.4333px; margin-top: 10px; margin-bottom: 20px; "><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 98px 8px; transform-origin: 98px 8px; unicode-bidi: normal; "><span style="font-style: italic; ">Vectors are 3×1, with norm = 1.</span></span></li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; "><span style="block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 292px 8px; transform-origin: 292px 8px; unicode-bidi: normal; "><span style="font-style: italic; ">Make use of scalar product to calculate cosines, taking advantage of the normalized vectors.</span></span></li></ul></div></div>

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Last Solution submitted on May 15, 2023

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Rafael S.T. Vieira on 12 Nov 2022

It's so weird, but there is a difference in the result from using the law of cosines and the function arc cosine. And the error is significant, but I am glad the author has considered this.

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Problem 50631. Cosine effect in solar power towers

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Problem 50631. Cosine effect in solar power towers

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