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<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 3.2.1 Series Solution for Diffusion with a Second Order Reaction</Font></Text-field>
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<Text-field style="Text" layout="Normal">Consider diffusion with a second order reaction in a rectangular pellet.<Font superscript="true">[18]</Font>  The dimensionless concentration is governed by:</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="Text" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal">with the boundary condiitons:</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal">and</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZqci1JI21vR0YkNi1RIn5GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGQ0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRistSSNtaUdGJDYlUSJjRicvJSdpdGFsaWNHRjRGLy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMUYnRi9GL0YvLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGWUZRRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrLUZNNiQtRiM2JS1GUjYkUSQzLjJGJ0YvLUZSNiRRIy4zRidGL0YvRi9GLw==">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</Equation></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">where \316\246 is the Thiele modulus.  This boundary value problem is solved in Maple below for different values of \316\246.  The unknown initial condition c(0) is taken as an unknown constant cl.</Font></Text-field>
<Text-field style="Text" layout="Normal"> </Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
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<Text-field style="Text" layout="Normal">Enter the governing equation:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq:=diff(c(x),x$2)=Phi^2*c(x)^2;</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjZXFHNiIvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JImNHRiQ2I0kieEdGJC1JIiRHRig2JEYtIiIjKiYpSSRQaGlHRiRGMSIiIilGKkYxRjU=">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JImNHNiI2I0kieEdGK0YtRi0qJkkkUGhpR0YrIiIjRilGMA==</Equation></Text-field>
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<Text-field style="Text" layout="Normal">Enter the boundary condition at x = 1&quot;</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">  bc:=c(x)-1;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjYmNHNiIsJi1JImNHRiQ2I0kieEdGJCIiIkYqISIi">LCYtSSJjRzYiNiNJInhHRiUiIiIhIiJGKA==</Equation></Text-field>
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<Text-field style="Text" layout="Normal">The series solutions are obtained assuming that c(0) = cl:</Text-field>
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<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eq,c(0)=c1,D(c)(0)=0},{c(x)},type=series);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkkc29sRzYiLy1JImNHRiQ2I0kieEdGJCsrRilJI2MxR0YkIiIhLCQqKCMiIiIiIiNGMClJJFBoaUdGJEYxRjApRitGMUYwRjBGMSwkKigjRjAiIzdGMClGMyIiJUYwKUYrIiIkRjBGMEY6LUkiT0clKnByb3RlY3RlZEc2I0YwIiIn">Ly1JImNHNiI2I0kieEdGJSsrRidJI2MxR0YlIiIhLCQqJkkkUGhpR0YlIiIjRilGLiMiIiJGLkYuLCQqJkYtIiIlRikiIiQjRjAiIzdGMy1JIk9HJSpwcm90ZWN0ZWRHNiNGMCIiJw==</Equation></Text-field>
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<Text-field style="Text" layout="Normal">The order of the series is increased to get more accurate solutions:</Text-field>
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<Group labelreference="L8" drawlabel="true">
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Order:=8;</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmT3JkZXJHNiIiIik=">IiIp</Equation></Text-field>
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<Group labelreference="L9" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eq,c(0)=c1,D(c)(0)=0},{c(x)},type=series);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkkc29sRzYiLy1JImNHRiQ2I0kieEdGJCstRilJI2MxR0YkIiIhLCQqKCMiIiIiIiNGMClJJFBoaUdGJEYxRjApRitGMUYwRjBGMSwkKigjRjAiIzdGMClGMyIiJUYwKUYrIiIkRjBGMEY6LCQqKCNGMCIjc0YwKUYzIiInRjApRitGOkYwRjBGQi1JIk9HJSpwcm90ZWN0ZWRHNiNGMCIiKQ==">Ly1JImNHNiI2I0kieEdGJSstRidJI2MxR0YlIiIhLCQqJkkkUGhpR0YlIiIjRilGLiMiIiJGLkYuLCQqJkYtIiIlRikiIiQjRjAiIzdGMywkKiZGLSIiJ0YpRjMjRjAiI3NGOS1JIk9HJSpwcm90ZWN0ZWRHNiNGMCIiKQ==</Equation></Text-field>
</Output>
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<Group labelreference="L46" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The solution obtained is converted to polynomial form and stored in ca:</Text-field>
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<Group labelreference="L10" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ca:=convert(rhs(sol),polynom);</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjY2FHNiIsKkkjYzFHRiQiIiIqKiNGJyIiI0YnKUkkUGhpR0YkRipGJylGJkYqRicpSSJ4R0YkRipGJ0YnKiojRiciIzdGJylGLCIiJUYnKUYmIiIkRicpRi9GNEYnRicqKiNGJyIjc0YnKUYsIiInRicpRiZGNEYnKUYvRjxGJ0Yn">LCpJI2MxRzYiIiIiKihJJFBoaUdGJCIiI0YjRihJInhHRiRGKCNGJUYoKihGJyIiJUYjIiIkRilGLCNGJSIjNyooRiciIidGI0YsRilGMSNGJSIjcw==</Equation></Text-field>
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<Group labelreference="L47" drawlabel="true">
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<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">We observe that the series solution obtained is a function of \316\246 and cl.  Now the boundary condition is evaluated using the solution obtained and substituting x = 1:</Font></Text-field>
</Input>
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<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqc:=subs(c(x)=ca,bc);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFjRzYiLCxJI2MxR0YkIiIiKiojRiciIiNGJylJJFBoaUdGJEYqRicpRiZGKkYnKUkieEdGJEYqRidGJyoqI0YnIiM3RicpRiwiIiVGJylGJiIiJEYnKUYvRjRGJ0YnKiojRiciI3NGJylGLCIiJ0YnKUYmRjRGJylGL0Y8RidGJ0YnISIi">LCxJI2MxRzYiIiIiKihJJFBoaUdGJCIiI0YjRihJInhHRiRGKCNGJUYoKihGJyIiJUYjIiIkRilGLCNGJSIjNyooRiciIidGI0YsRilGMSNGJSIjcyEiIkYl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqc:=subs(x=1,eqc);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFjRzYiLCxJI2MxR0YkIiIiKigjRiciIiNGJylJJFBoaUdGJEYqRicpRiZGKkYnRicqKCNGJyIjN0YnKUYsIiIlRicpRiYiIiRGJ0YnKigjRiciI3NGJylGLCIiJ0YnKUYmRjJGJ0YnRichIiI=">LCxJI2MxRzYiIiIiKiZJJFBoaUdGJCIiI0YjRigjRiVGKComRiciIiVGIyIiJCNGJSIjNyomRiciIidGI0YrI0YlIiNzISIiRiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L48" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">This is a nonlinear equation and cannot be solved explicitly:</Text-field>
</Input>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">solve(eqc,c1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsLEkjX1pHRiQiI3MqJkkkUGhpR0YnIiIjRipGLiIjTyomRi0iIiVGKiIiJCIiJyomRi1GM0YqRjEiIiIhI3NGNQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">However, the solution can be obtained for a particular value of \316\246:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">solve(subs(Phi=0.1,eqc));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">NiYkIis1Q1RdKiohIzVeJCQhK2EpZUBuIiEiKCQiKyxwWU1cRikkISswRmpsRUYpXiRGJyQhKyxwWU1cRik=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The 'solve' command takes too long to solve and gives complex roots.  Hence, the 'fsolve' command is used to solve for the real values.</Text-field>
</Input>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">fsolve(subs(Phi=0.1,eqc),c1=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiQkISswRmpsRSEiKCQiKzVDVF0qKiEjNQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L51" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Next, the constant cl is solved for different values of \316\246:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cc[1]:=fsolve(subs(Phi=0.1,eqc),c1=1)[2];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2NjRzYiNiMiIiIkIis1Q1RdKiohIzU=">JCIrNUNUXSoqISM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cc[2]:=fsolve(subs(Phi=1,eqc),c1=1)[2];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2NjRzYiNiMiIiMkIitETHRDciEjNQ==">JCIrREx0Q3IhIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cc[3]:=fsolve(subs(Phi=2,eqc),c1=1)[2];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2NjRzYiNiMiIiQkIismZllwWSUhIzU=">JCIrJmZZcFklISM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cc[4]:=fsolve(subs(Phi=5,eqc),c1=1)[2];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJI2NjRzYiNiMiIiUkIisnKiplQXMiISM1">JCIrJyoqZUFzIiEjNQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L52" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Next, plots are made by substituting the values for c1 and \316\246:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p1:=plot(subs(c1=cc[1],Phi=0.1,ca),x=0..1,thickness=3,color=red,axes=boxed):</Text-field>
</Input>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p2:=plot(subs(c1=cc[2],Phi=1,ca),x=0..1,thickness=3,color=green,axes=boxed):</Text-field>
</Input>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p3:=plot(subs(c1=cc[3],Phi=2,ca),x=0..1,thickness=3,color=gold,axes=boxed):</Text-field>
</Input>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p4:=plot(subs(c1=cc[4],Phi=5,ca),x=0..1,thickness=3,color=blue,axes=boxed):</Text-field>
</Input>
</Group>
<Group labelreference="L61" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pt:=textplot([[0.5,subs(x=0.5,c1=cc[1],Phi=0.1,ca)-0.05,[c1=cc[1],Phi=0.1]],[0.5,subs(x=0.5,c1=cc[2],Phi=1,ca)-0.08,[c1=cc[2],Phi=1]],[0.5,subs(x=0.5,c1=cc[3],Phi=2,ca)-0.1,[c1=cc[3],Phi=2]],[0.5,subs(x=0.5,c1=cc[4],Phi=5,ca)-0.1,[c1=cc[4],Phi=5]]]):</Text-field>
</Input>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display({p1,p2,p3,p4,pt},labels=[x,c],title=&quot;Figure Exp. 3.2.1.&quot;);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="300" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">[encoded data, 40968 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L53" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">We observe that as \316\246 increases, the concentration profiles become steeper.  We have obtained the series solution for the given nonlinear boundary value problem.  There is no guarantee ht the solution has converged.  Accuracy of the solution can be analyzed by increasing the number of terms in the series solution and checking the values for cl:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Order:=10;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkmT3JkZXJHNiIiIzU=">IiM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=dsolve({eq,c(0)=c1,D(c)(0)=0},{c(x)},type=series);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="Pkkkc29sRzYiLy1JImNHRiQ2I0kieEdGJCsvRilJI2MxR0YkIiIhLCQqKCMiIiIiIiNGMClJJFBoaUdGJEYxRjApRitGMUYwRjBGMSwkKigjRjAiIzdGMClGMyIiJUYwKUYrIiIkRjBGMEY6LCQqKCNGMCIjc0YwKUYzIiInRjApRitGOkYwRjBGQiwkKigjRjAiJC8mRjApRjMiIilGMClGKyIiJkYwRjBGSS1JIk9HJSpwcm90ZWN0ZWRHNiNGMCIjNQ==">Ly1JImNHNiI2I0kieEdGJSsvRidJI2MxR0YlIiIhLCQqJkkkUGhpR0YlIiIjRilGLiMiIiJGLkYuLCQqJkYtIiIlRikiIiQjRjAiIzdGMywkKiZGLSIiJ0YpRjMjRjAiI3NGOSwkKiZGLSIiKUYpIiImI0YwIiQvJkY+LUkiT0clKnByb3RlY3RlZEc2I0YwIiM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L27" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ca:=convert(rhs(sol),polynom);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkjY2FHNiIsLEkjYzFHRiQiIiIqKiNGJyIiI0YnKUkkUGhpR0YkRipGJylGJkYqRicpSSJ4R0YkRipGJ0YnKiojRiciIzdGJylGLCIiJUYnKUYmIiIkRicpRi9GNEYnRicqKiNGJyIjc0YnKUYsIiInRicpRiZGNEYnKUYvRjxGJ0YnKiojRiciJC8mRicpRiwiIilGJylGJiIiJkYnKUYvRkNGJ0Yn">LCxJI2MxRzYiIiIiKihJJFBoaUdGJCIiI0YjRihJInhHRiRGKCNGJUYoKihGJyIiJUYjIiIkRilGLCNGJSIjNyooRiciIidGI0YsRilGMSNGJSIjcyooRiciIilGIyIiJkYpRjUjRiUiJC8m</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqc:=subs(c(x)=ca,bc);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFjRzYiLC5JI2MxR0YkIiIiKiojRiciIiNGJylJJFBoaUdGJEYqRicpRiZGKkYnKUkieEdGJEYqRidGJyoqI0YnIiM3RicpRiwiIiVGJylGJiIiJEYnKUYvRjRGJ0YnKiojRiciI3NGJylGLCIiJ0YnKUYmRjRGJylGL0Y8RidGJyoqI0YnIiQvJkYnKUYsIiIpRicpRiYiIiZGJylGL0ZDRidGJ0YnISIi">LC5JI2MxRzYiIiIiKihJJFBoaUdGJCIiI0YjRihJInhHRiRGKCNGJUYoKihGJyIiJUYjIiIkRilGLCNGJSIjNyooRiciIidGI0YsRilGMSNGJSIjcyooRiciIilGIyIiJkYpRjUjRiUiJC8mISIiRiU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqc:=subs(x=1,eqc);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PkkkZXFjRzYiLC5JI2MxR0YkIiIiKigjRiciIiNGJylJJFBoaUdGJEYqRicpRiZGKkYnRicqKCNGJyIjN0YnKUYsIiIlRicpRiYiIiRGJ0YnKigjRiciI3NGJylGLCIiJ0YnKUYmRjJGJ0YnKigjRiciJC8mRicpRiwiIilGJylGJiIiJkYnRidGJyEiIg==">LC5JI2MxRzYiIiIiKiZJJFBoaUdGJCIiI0YjRigjRiVGKComRiciIiVGIyIiJCNGJSIjNyomRiciIidGI0YrI0YlIiNzKiZGJyIiKUYjIiImI0YlIiQvJiEiIkYl</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">fsolve(subs(Phi=0.1,eqc),c1=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiUkIStoaDQoZSUhIigkIStZdSpHRiRGJSQiKzVDVF0qKiEjNQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L54" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">We observe that there are three different solutions of which only the last one makes sense:</Text-field>
</Input>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cc2[1]:=fsolve(subs(Phi=0.1,eqc),c1=1)[3];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJJGNjMkc2IjYjIiIiJCIrNUNUXSoqISM1">JCIrNUNUXSoqISM1</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">fsolve(subs(Phi=1,eqc),c1=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">JCIrOnB4QXIhIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L55" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">For higher values of \316\246, 'fsolve' produces only the correct solution:</Font></Text-field>
</Input>
</Group>
<Group labelreference="L33" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cc2[2]:=fsolve(subs(Phi=1,eqc),c1=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJJGNjMkc2IjYjIiIjJCIrOnB4QXIhIzU=">JCIrOnB4QXIhIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cc2[3]:=fsolve(subs(Phi=2,eqc),c1=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJJGNjMkc2IjYjIiIkJCIrZjhKV1chIzU=">JCIrZjhKV1chIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cc2[4]:=fsolve(subs(Phi=5,eqc),c1=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="PiZJJGNjMkc2IjYjIiIlJCIrcTs5YDshIzU=">JCIrcTs5YDshIzU=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L56" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The constants c1 obtained using eight terms and ten terms in the series are compared:</Text-field>
</Input>
</Group>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i to 4 do print(cc[i],cc2[i]);od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiQkIis1Q1RdKiohIzVGIw==</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiQkIitETHRDciEjNSQiKzpweEFyRiU=</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiQkIismZllwWSUhIzUkIitmOEpXV0Yl</Equation></Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiQkIisnKiplQXMiISM1JCIrcTs5YDtGJQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L57" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">For \316\246=0.1 and 1, we observe that the solution has converged to the third digit.  For \316\246=2, the solution has converged to the first two digits.  For \316\246=5, we observe that the solution has not converged.  Hence, more terms are required in the series for higher values of \316\246.  The series solution obtained using Maple may or may not converge.  The convergence of the solution depends on the parameters (as illustrated in this example), and the nonlinearity of the problem.  In the next example the non-isothermal reaction in a rectangular pellet is analyzed.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>