ECH140Lecture2c - Undergraduate Lectures on ECH140...

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Undergraduate Lectures on ECH140: Mathema9cal Methods for Chemical Engineers September 2-­‐ 17, 2013 Chapter 1: Lecture 2c: Cooling of a Sphere Brian G. Higgins Department of Chemical Engineering and Materials Science University of California, Davis Class Web Site: h.ps://sites.google.com/site/ekayasolu9onsteachingvietnam/
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Problem Statement C p @ T @ t = k r 2 T q · n = h ( T - T 1 ) Sphere cooling a bath of fluid with temperature T 1 Temperature distribu9on in sphere is governed by Heat lost from sphere is given by Newton’s law of cooling n R 1 T 1 T 0 Ini9al temperature of sphere How does the temperature change with 9me? BGHiggins/UCDavis/Hanoi-­‐Vietnam/Sep2013
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Analysis q · n = h ( T - T 1 ) Select a control volume the encompasses the sphere, and then integrate over sphere n R 1 T 1 T 0 Ini9al temperature of sphere Z V C p @ T @ t dV = Z V k r 2 TdV C p d dt Z V TdV = k Z V r · ( r T ) dV Assume constant proper9es Apply Divergence Theorem k Z V r · ( r T ) dV = k Z A r T · n dA BGHiggins/UCDavis/Hanoi-­‐Vietnam/Sep2013
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Analysis Cont.
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  • Summer '13
  • LeThuy
  • Trigraph, SEPTA Regional Rail, web  site

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