Problem 45391. Calculate the sphericity of a Raschig ring

Created by Karl Ezra Pilario

Solve Later

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It was defined by Wadell in 1935 as the ratio of the 'surface area of a sphere having the same volume as the given particle' to the 'surface area of the given particle'. By definition, the maximum value of sphericity is 1, which is that of a perfect sphere. The more elongated a particle is, the lesser its sphericity.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>A Raschig ring is a common particle found in packed beds that are shaped like small pieces of a tube (see figure). As a hollow cylinder, it has a height H, inner radius R1, and outer radius R2. The sphericity of a Raschig ring is important to calculate as it can influence the performance of a packed bed for adsorption.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"-1\"/><w:attr w:name=\"width\" w:val=\"-1\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Make a function that takes three values: H, R1, and R2. Output the sphericity of this hollow cylinder rounded to 4 decimal places. You are ensured that:</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:pPr><w:r><w:t>H, R1, and R2 are all positive integers</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:pPr><w:r><w:t>R1 &lt; R2</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:pPr><w:r><w:t>All inputs have consistent units; and</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:pPr><w:r><w:t>No input value exceeds 100.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray 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"}]}

Sphericity is a measure of the roundness of any particle. It was defined by Wadell in 1935 as the ratio of the 'surface area of a sphere having the same volume as the given particle' to the 'surface area of the given particle'. By definition, the maximum value of sphericity is 1, which is that of a perfect sphere. The more elongated a particle is, the lesser its sphericity.A Raschig ring is a common particle found in packed beds that are shaped like small pieces of a tube (see figure). As a hollow cylinder, it has a height H, inner radius R1, and outer radius R2. The sphericity of a Raschig ring is important to calculate as it can influence the performance of a packed bed for adsorption.<img src = "https://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/RaschigRings005.JPG/296px-RaschigRings005.JPG">Make a function that takes three values: H, R1, and R2. Output the sphericity of this hollow cylinder rounded to 4 decimal places. You are ensured that:<ul><li>H, R1, and R2 are all positive integers</li><li>R1 &lt; R2</li><li>All inputs have consistent units; and</li><li>No input value exceeds 100.</li></ul>

Sphericity is a measure of the roundness of any particle. It was defined by Wadell in 1935 as the ratio of the 'surface area of a sphere having the same volume as the given particle' to the 'surface area of the given particle'. By definition, the maximum value of sphericity is 1, which is that of a perfect sphere. The more elongated a particle is, the lesser its sphericity.

A Raschig ring is a common particle found in packed beds that are shaped like small pieces of a tube (see figure). As a hollow cylinder, it has a height H, inner radius R1, and outer radius R2. The sphericity of a Raschig ring is important to calculate as it can influence the performance of a packed bed for adsorption.

<<https://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/RaschigRings005.JPG/296px-RaschigRings005.JPG>> 
 
Make a function that takes three values: H, R1, and R2. Output the sphericity of this hollow cylinder rounded to 4 decimal places. You are ensured that:

* H, R1, and R2 are all positive integers
* R1 < R2
* All inputs have consistent units; and 
* No input value exceeds 100.

Solve

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Nikolaos Nikolaou on 3 Apr 2020

Please clarify : Outside surface of a tube or the inner on aswell
Volume of the entire tube or minus the hollow part

Karl Ezra Pilario on 3 Apr 2020

For surface area: Both inner + outer are included. For volume: subtract the hollow part.

mahesh kumar on 29 Aug 2021

Also include small rings on either side.

Christian Schröder on 21 Oct 2022

EXCELLENT test suite!

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Problem 45391. Calculate the sphericity of a Raschig ring

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Problem 45391. Calculate the sphericity of a Raschig ring

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