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matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="true" strings-use-origin="false" zero-over-zero="error"> <compatibility multiple-assignment="MC12" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="mks" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="476" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="2097260"/> </miscellaneous> </settings> <regions> <region region-id="40" left="6" top="2.25" width="338.25" height="24" align-x="6" align-y="12" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Solve substrate concentration profile in a spherical gel (solution via "odesolve") Instructor: Nam Sun Wang </text> </region> <region region-id="122" left="6" top="30" width="436.5" height="9.75" align-x="6" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="2" display-width="435" display-height="8.25"/> </picture> <rendering item-idref="3"/> </region> <region region-id="109" left="6" top="44.25" width="282" height="12" align-x="6" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Substrate diffusion in a spherical bead in the dimensionless form: </text> </region> <region region-id="49" left="18" top="56.25" width="99" height="39.75" align-x="78.75" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">d</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">s</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">dr</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">ds</ml:id> <ml:id xml:space="preserve">dr</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="4"/> </region> <region region-id="51" left="144" top="68.25" width="25.5" height="12" align-x="144" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="192" left="180" top="69" width="33" height="12.75" align-x="201.75" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="5"/> </region> <region region-id="54" left="240" top="62.25" width="41.25" height="27.75" align-x="270" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">ds</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:id xml:space="preserve">dr</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="6"/> </region> <region region-id="2" left="6" top="98.25" width="236.25" height="12" align-x="6" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Dimensionless model parameters and rate expression: </text> </region> <region region-id="3" left="24" top="117" width="24" height="12.75" align-x="36" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">β</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="4" left="90" top="117" width="28.5" height="12.75" align-x="102" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Γ</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="187" left="168" top="117" width="24.75" height="12.75" align-x="180.75" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">φ</ml:id> <ml:real>7</ml:real> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="281" left="354" top="135" width="54.75" height="12.75" align-x="364.5" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">s</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:real>1</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="5" left="24" top="134.25" width="89.25" height="42.75" align-x="47.25" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">v</ml:id> <ml:boundVars> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">Γ</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">s</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="288" left="282" top="156" width="164.25" height="129.75" align-x="282" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="12"/> <rendering item-idref="13"/> </region> <region region-id="42" left="6" top="182.25" width="237" height="12" align-x="6" align-y="192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Transform the above equation into two 1st-order ODEs: </text> </region> <region region-id="198" left="6" top="201" width="24" height="12.75" align-x="17.25" align-y="210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="14"/> </region> <region region-id="7" left="24" top="211.5" width="51" height="29.25" align-x="54" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="15"/> </region> <region region-id="291" left="216" top="218.25" width="22.5" height="12" align-x="216" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="289" left="246" top="219" width="33" height="12.75" align-x="267.75" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math 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background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Plot of substrate profile </text> </region> <region region-id="312" left="312" top="315" width="38.25" height="12.75" align-x="333" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="313" left="378" top="315" width="54" height="12.75" align-x="399" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.786603234677985</ml:real> </result> </ml:eval> </math> <rendering item-idref="25"/> </region> <region region-id="55" left="6" top="440.25" width="437.25" height="12" align-x="6" align-y="450" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Compute the effectiveness factor, which is (observed rate / max rate without mass transfer limitation): </text> </region> <region region-id="56" left="24" top="458.25" width="64.5" height="42.75" align-x="36" align-y="474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">η</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="58" left="120" top="465" width="44.25" height="12.75" align-x="131.25" align-y="474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">η</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.3126064581307646</ml:real> </result> </ml:eval> </math> <rendering item-idref="27"/> </region> <region region-id="476" left="0" top="504" width="6000" height="6" align-x="0" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="324" left="6" top="510" width="436.5" height="9.75" align-x="6" align-y="510" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="28" display-width="435" display-height="8.25"/> </picture> <rendering item-idref="29"/> </region> <region region-id="325" left="6" top="524.25" width="432" height="48" align-x="6" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Another approach without decomposing into two 1st-order ODEs.<sp count="2"/>The boundary condition must be in "prime" form, e.g, s'(0), which is entered with CTRL-F7.<sp count="2"/>Furthermore, we need to extend the interval in "odesolve" to evaluate the derivative z(1) later.<sp count="2"/>Since Mathcad considers the following to be only one equation, the "odesolve" function does not contain the argument "s". </text> </region> <region region-id="251" left="6" top="573" width="24" height="12.75" align-x="17.25" align-y="582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="30"/> </region> <region region-id="270" left="24" top="583.5" width="238.5" height="37.5" align-x="213.75" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">r</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="31"/> </region> <region region-id="271" left="294" top="596.25" width="22.5" height="12" align-x="294" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="272" left="324" top="597" width="34.5" height="12.75" align-x="347.25" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s'</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="32"/> </region> <region region-id="273" left="372" top="597" width="33" height="12.75" align-x="393.75" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="33"/> </region> <region region-id="257" left="6" top="627" width="82.5" height="12.75" align-x="16.5" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:Odesolve method="adams/bdf (auto)"/> <ml:sequence> <ml:id xml:space="preserve">r</ml:id> <ml:real>1.5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="335" left="132" top="636" width="159.75" height="129.75" align-x="132" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="35"/> <rendering item-idref="36"/> </region> <region region-id="340" left="294" top="632.25" width="81.75" height="12" align-x="294" align-y="642" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Double check B.C. </text> </region> <region region-id="336" left="306" top="651" width="54.75" height="12.75" align-x="327.75" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.074053305529437258</ml:real> </result> </ml:eval> </math> <rendering item-idref="37"/> </region> <region region-id="337" left="372" top="651" width="39" height="12.75" align-x="393.75" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.0000019012410579</ml:real> </result> </ml:eval> </math> <rendering item-idref="38"/> </region> <region region-id="338" left="306" top="667.5" width="52.5" height="29.25" align-x="326.25" align-y="684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">z</ml:id> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="339" left="372" top="675" width="54" height="12.75" align-x="393" align-y="684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.7865775479586707</ml:real> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="327" left="6" top="770.25" width="212.25" height="12" align-x="6" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> The following does not work because of 1/r at r=0 </text> </region> <region region-id="328" left="6" top="783" width="24" height="12.75" align-x="17.25" align-y="792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="41"/> </region> <region region-id="329" left="24" top="793.5" width="144.75" height="37.5" align-x="120" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="42"/> </region> <region region-id="341" left="198" top="806.25" width="22.5" height="12" align-x="198" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="342" left="234" top="807" width="34.5" height="12.75" align-x="257.25" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s'</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="43"/> </region> <region region-id="343" left="282" top="807" width="33" height="12.75" align-x="303.75" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="333" left="6" top="831" width="82.5" height="12.75" align-x="16.5" align-y="840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="This function cannot be used here." optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:Odesolve method="adams/bdf (auto)"/> <ml:sequence> <ml:id xml:space="preserve">r</ml:id> <ml:real>1.5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="346" left="6" top="848.25" width="418.5" height="24" align-x="6" align-y="858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> However, one can cheat without affecting numerical accuracy by starting at a value slightly above r=0. </text> </region> <region region-id="365" left="6" top="873" width="24" height="12.75" align-x="17.25" align-y="882" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="46"/> </region> <region region-id="395" left="354" top="873" width="77.25" height="12.75" align-x="376.5" align-y="882" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">zero</ml:id> <ml:real>0.000000001</ml:real> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="366" left="24" top="883.5" width="144.75" height="37.5" align-x="120" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="48"/> </region> <region region-id="367" left="198" top="896.25" width="22.5" height="12" align-x="198" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="368" left="234" top="897" width="46.5" height="12.75" align-x="269.25" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s'</ml:id> <ml:id xml:space="preserve">zero</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="49"/> </region> <region region-id="372" left="330" top="897" width="33" height="12.75" align-x="351.75" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="50"/> </region> <region region-id="370" left="6" top="921" width="82.5" height="12.75" align-x="16.5" align-y="930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:Odesolve method="adams/bdf (auto)"/> <ml:sequence> <ml:id xml:space="preserve">r</ml:id> <ml:real>1.5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="51"/> </region> <region region-id="468" left="132" top="930" width="159.75" height="129.75" align-x="132" align-y="930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="52"/> <rendering item-idref="53"/> </region> <region region-id="469" left="300" top="926.25" width="81.75" height="12" align-x="300" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Double check B.C. </text> </region> <region region-id="470" left="312" top="945" width="66.75" height="12.75" align-x="345.75" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">zero</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.074053305529419577</ml:real> </result> </ml:eval> </math> <rendering item-idref="54"/> </region> <region region-id="471" left="378" top="945" width="39" height="12.75" align-x="399.75" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" 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xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="4"/> </region> <region region-id="51" left="144" top="68.25" width="25.5" height="12" align-x="144" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="192" left="180" top="69" width="33" height="12.75" align-x="201.75" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="5"/> </region> <region region-id="54" left="240" top="62.25" width="41.25" height="27.75" align-x="270" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">ds</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:id xml:space="preserve">dr</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="6"/> </region> <region region-id="2" left="6" top="98.25" width="236.25" height="12" align-x="6" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Dimensionless model parameters and rate expression: </text> </region> <region region-id="3" left="24" top="117" width="24" height="12.75" align-x="36" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">β</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="4" left="90" top="117" width="28.5" height="12.75" align-x="102" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Γ</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="187" left="168" top="117" width="24.75" height="12.75" align-x="180.75" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">φ</ml:id> <ml:real>7</ml:real> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="281" left="354" top="135" width="54.75" height="12.75" align-x="364.5" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">s</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:real>1</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="5" left="24" top="134.25" width="89.25" height="42.75" align-x="47.25" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">v</ml:id> <ml:boundVars> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">Γ</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">s</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="288" left="282" top="156" width="164.25" height="129.75" align-x="282" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="12"/> <rendering item-idref="13"/> </region> <region region-id="42" left="6" top="182.25" width="237" height="12" align-x="6" align-y="192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Transform the above equation into two 1st-order ODEs: </text> </region> <region region-id="198" left="6" top="201" width="24" height="12.75" align-x="17.25" align-y="210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="14"/> </region> <region region-id="7" left="24" top="211.5" width="51" height="29.25" align-x="54" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="15"/> </region> <region region-id="291" left="216" top="218.25" width="22.5" height="12" align-x="216" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="289" left="246" top="219" width="33" height="12.75" align-x="267.75" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="16"/> </region> <region region-id="8" left="24" top="241.5" width="212.25" height="29.25" align-x="53.25" align-y="258" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">r</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> <region region-id="290" left="246" top="249" width="32.25" height="12.75" align-x="267" align-y="258" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="18"/> </region> <region region-id="242" left="6" top="276.75" width="113.25" height="29.25" align-x="30.75" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:matrix> <ml:apply> <ml:Odesolve method="adams/bdf (auto)"/> <ml:sequence> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:matrix> <ml:id xml:space="preserve">r</ml:id> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="26" left="126" top="306" width="159.75" height="129.75" align-x="126" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="20"/> <rendering item-idref="21"/> </region> <region region-id="205" left="312" top="297" width="54.75" height="12.75" align-x="333.75" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.074053305529437258</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="310" left="378" top="297" width="39" height="12.75" align-x="399.75" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.0000204014499832</ml:real> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="48" left="6" top="308.25" width="99.75" height="12" align-x="6" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Plot of substrate profile </text> </region> <region region-id="312" left="312" top="315" width="38.25" height="12.75" align-x="333" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="313" left="378" top="315" width="54" height="12.75" align-x="399" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.786603234677985</ml:real> </result> </ml:eval> </math> <rendering item-idref="25"/> </region> <region region-id="55" left="6" top="440.25" width="437.25" height="12" align-x="6" align-y="450" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Compute the effectiveness factor, which is (observed rate / max rate without mass transfer limitation): </text> </region> <region region-id="56" left="24" top="458.25" width="64.5" height="42.75" align-x="36" align-y="474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">η</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="58" left="120" top="465" width="44.25" height="12.75" align-x="131.25" align-y="474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">η</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.3126064581307646</ml:real> </result> </ml:eval> </math> <rendering item-idref="27"/> </region> <region region-id="476" left="0" top="504" width="6000" height="6" align-x="0" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="324" left="6" top="510" width="436.5" height="9.75" align-x="6" align-y="510" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="28" display-width="435" display-height="8.25"/> </picture> <rendering item-idref="29"/> </region> <region region-id="325" left="6" top="524.25" width="432" height="48" align-x="6" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Another approach without decomposing into two 1st-order ODEs.<sp count="2"/>The boundary condition must be in "prime" form, e.g, s'(0), which is entered with CTRL-F7.<sp count="2"/>Furthermore, we need to extend the interval in "odesolve" to evaluate the derivative z(1) later.<sp count="2"/>Since Mathcad considers the following to be only one equation, the "odesolve" function does not contain the argument "s". </text> </region> <region region-id="251" left="6" top="573" width="24" height="12.75" align-x="17.25" align-y="582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="30"/> </region> <region region-id="270" left="24" top="583.5" width="238.5" height="37.5" align-x="213.75" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">r</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="31"/> </region> <region region-id="271" left="294" top="596.25" width="22.5" height="12" align-x="294" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="272" left="324" top="597" width="34.5" height="12.75" align-x="347.25" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s'</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="32"/> </region> <region region-id="273" left="372" top="597" width="33" height="12.75" align-x="393.75" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="33"/> </region> <region region-id="257" left="6" top="627" width="82.5" height="12.75" align-x="16.5" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:Odesolve method="adams/bdf (auto)"/> <ml:sequence> <ml:id xml:space="preserve">r</ml:id> <ml:real>1.5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="335" left="132" top="636" width="159.75" height="129.75" align-x="132" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="35"/> <rendering item-idref="36"/> </region> <region region-id="340" left="294" top="632.25" width="81.75" height="12" align-x="294" align-y="642" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Double check B.C. </text> </region> <region region-id="336" left="306" top="651" width="54.75" height="12.75" align-x="327.75" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.074053305529437258</ml:real> </result> </ml:eval> </math> <rendering item-idref="37"/> </region> <region region-id="337" left="372" top="651" width="39" height="12.75" align-x="393.75" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.0000019012410579</ml:real> </result> </ml:eval> </math> <rendering item-idref="38"/> </region> <region region-id="338" left="306" top="667.5" width="52.5" height="29.25" align-x="326.25" align-y="684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">z</ml:id> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="339" left="372" top="675" width="54" height="12.75" align-x="393" align-y="684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.7865775479586707</ml:real> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="327" left="6" top="770.25" width="212.25" height="12" align-x="6" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> The following does not work because of 1/r at r=0 </text> </region> <region region-id="328" left="6" top="783" width="24" height="12.75" align-x="17.25" align-y="792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="41"/> </region> <region region-id="329" left="24" top="793.5" width="144.75" height="37.5" align-x="120" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="42"/> </region> <region region-id="341" left="198" top="806.25" width="22.5" height="12" align-x="198" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="342" left="234" top="807" width="34.5" height="12.75" align-x="257.25" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s'</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="43"/> </region> <region region-id="343" left="282" top="807" width="33" height="12.75" align-x="303.75" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="333" left="6" top="831" width="82.5" height="12.75" align-x="16.5" align-y="840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="This function cannot be used here." optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:Odesolve method="adams/bdf (auto)"/> <ml:sequence> <ml:id xml:space="preserve">r</ml:id> <ml:real>1.5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="346" left="6" top="848.25" width="418.5" height="24" align-x="6" align-y="858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> However, one can cheat without affecting numerical accuracy by starting at a value slightly above r=0. </text> </region> <region region-id="365" left="6" top="873" width="24" height="12.75" align-x="17.25" align-y="882" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="46"/> </region> <region region-id="395" left="354" top="873" width="77.25" height="12.75" align-x="376.5" align-y="882" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">zero</ml:id> <ml:real>0.000000001</ml:real> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="366" left="24" top="883.5" width="144.75" height="37.5" align-x="120" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">φ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="48"/> </region> <region region-id="367" left="198" top="896.25" width="22.5" height="12" align-x="198" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> B.C.: </text> </region> <region region-id="368" left="234" top="897" width="46.5" height="12.75" align-x="269.25" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s'</ml:id> <ml:id xml:space="preserve">zero</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="49"/> </region> <region region-id="372" left="330" top="897" width="33" height="12.75" align-x="351.75" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="50"/> </region> <region region-id="370" left="6" top="921" width="82.5" height="12.75" align-x="16.5" align-y="930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:Odesolve method="adams/bdf (auto)"/> <ml:sequence> <ml:id xml:space="preserve">r</ml:id> <ml:real>1.5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="51"/> </region> <region region-id="468" left="132" top="930" width="159.75" height="129.75" align-x="132" align-y="930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="52"/> <rendering item-idref="53"/> </region> <region region-id="469" left="300" top="926.25" width="81.75" height="12" align-x="300" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Double check B.C. </text> </region> <region region-id="470" left="312" top="945" width="66.75" height="12.75" align-x="345.75" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">zero</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.074053305529419577</ml:real> </result> </ml:eval> </math> <rendering item-idref="54"/> </region> <region region-id="471" left="378" top="945" width="39" height="12.75" align-x="399.75" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.0000199518844797</ml:real> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="472" left="312" top="961.5" width="52.5" height="29.25" align-x="332.25" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">z</ml:id> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">r</ml:id> </ml:apply> </ml:lambda> 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