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z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Solution for <f family="Arial" charset="0"> <inlineAttr line-through="false"> Green's Function </inlineAttr> </f> to <f family="Arial" charset="0"> <inlineAttr line-through="false">Transient Temperature Equation for Rectangle</inlineAttr> </f> <inlineAttr font-weight="normal" underline="false" line-through="false"> <f family="Arial" charset="0">S</f> </inlineAttr> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">urface Source Model Predicts higher temp and faster transients.</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">For x,<sp count="2"/>sources have edge Lox and width W. 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xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">erf</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="oy">L</ml:id> <ml:id xml:space="preserve">H</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="si">D</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="129"/> </region> <region region-id="4315" left="12" top="1382.25" width="32.25" height="15" align-x="22.5" align-y="1392" show-border="false" show-highlight="false" 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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" subscript="S6">G</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:real>12</ml:real> </ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:real>12</ml:real> </ml:apply> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="132"/> </region> <region region-id="4700" left="324" top="1382.25" width="221.25" height="15.75" align-x="413.25" align-y="1392" 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">PSources</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="pk">P</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tri">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:real>12</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="S6">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="133"/> </region> <region region-id="3880" left="186" top="1430.25" width="250.5" height="12" align-x="204.75" align-y="1440" 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"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Greens Transient Temperature Rise for Sources, </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-style="normal" line-through="false">D</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">T</inlineAttr> </f> <inlineAttr line-through="false"> <f family="Arial" charset="0">o</f> </inlineAttr> </text> </region> <region region-id="5080" left="54" top="1472.25" width="197.25" height="21.75" align-x="96.75" align-y="1488" 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">ISqkρCp</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tong">k</ml:id> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ρCp</ml:id> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:real>-1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="134"/> </region> <region region-id="5079" left="48" top="1502.25" width="214.5" height="21.75" align-x="90.75" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="true"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tong">k</ml:id> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ρCp</ml:id> <ml:apply> <ml:plus/> <ml:real>273</ml:real> <ml:real>180</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:real>-1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="135"/> </region> <region region-id="3878" left="6" top="1543.5" width="540" height="58.5" align-x="78" align-y="1578" 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" subscript="os">ΔT</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:id xml:space="preserve">PSources</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="136"/> </region> <region region-id="4997" left="6" top="1606.5" width="621.75" height="87.75" align-x="147" align-y="1656" 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" subscript="o">ΔT</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">H</ml:id> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="load">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">H</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="137"/> </region> <region region-id="3563" left="66" top="1730.25" width="414" height="48" align-x="93.75" align-y="1740" 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"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Remove Heat Equation Nonlinearity from Temperature Dependence of Conductivity</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">Kirchhoff Transformation:<sp count="2"/> </inlineAttr> </f> K, D must have value at Tamb <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">Tk Kirch is the Inverse of </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Q</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10">, </f> </inlineAttr> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">the Integral Transform of Conductivity, </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">l</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">si(T)</inlineAttr> </f> </text> </region> <region region-id="3884" left="228" top="1790.25" width="110.25" height="36.75" align-x="241.5" align-y="1812" 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:id xml:space="preserve">Θ</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="s">λ</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">T</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">λ</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">Ts</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="138"/> </region> <region region-id="2568" left="360" top="1788.75" width="136.5" height="41.25" align-x="436.5" align-y="1812" show-border="false" show-highlight="false" 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z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Solution for <f family="Arial" charset="0"> <inlineAttr line-through="false"> Green's Function </inlineAttr> </f> to <f family="Arial" charset="0"> <inlineAttr line-through="false">Transient Temperature Equation for Rectangle</inlineAttr> </f> <inlineAttr font-weight="normal" underline="false" line-through="false"> <f family="Arial" charset="0">S</f> </inlineAttr> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">urface Source Model Predicts higher temp and faster transients.</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">For x,<sp count="2"/>sources have edge Lox and width W. 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xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">erf</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="oy">L</ml:id> <ml:id xml:space="preserve">H</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="si">D</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="129"/> </region> <region region-id="4315" left="12" top="1382.25" width="32.25" height="15" align-x="22.5" align-y="1392" show-border="false" show-highlight="false" 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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" subscript="S6">G</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:real>12</ml:real> </ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:real>12</ml:real> </ml:apply> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="132"/> </region> <region region-id="4700" left="324" top="1382.25" width="221.25" height="15.75" align-x="413.25" align-y="1392" 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">PSources</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="pk">P</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tri">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:real>12</ml:real> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="S6">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="133"/> </region> <region region-id="3880" left="186" top="1430.25" width="250.5" height="12" align-x="204.75" align-y="1440" 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"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Greens Transient Temperature Rise for Sources, </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-style="normal" line-through="false">D</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">T</inlineAttr> </f> <inlineAttr line-through="false"> <f family="Arial" charset="0">o</f> </inlineAttr> </text> </region> <region region-id="5080" left="54" top="1472.25" width="197.25" height="21.75" align-x="96.75" align-y="1488" 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">ISqkρCp</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tong">k</ml:id> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ρCp</ml:id> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:real>-1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="134"/> </region> <region region-id="5079" left="48" top="1502.25" width="214.5" height="21.75" align-x="90.75" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="true"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="tong">k</ml:id> <ml:id xml:space="preserve" subscript="ambK">T</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ρCp</ml:id> <ml:apply> <ml:plus/> <ml:real>273</ml:real> <ml:real>180</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:real>-1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="135"/> </region> <region region-id="3878" left="6" top="1543.5" width="540" height="58.5" align-x="78" align-y="1578" 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" subscript="os">ΔT</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:id xml:space="preserve">PSources</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="136"/> </region> <region region-id="4997" left="6" top="1606.5" width="621.75" height="87.75" align-x="147" align-y="1656" 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" subscript="o">ΔT</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">H</ml:id> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ISqkρCp</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">τ</ml:id> </ml:boundVars> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">mil</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="j">z</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>4</ml:real> <ml:id xml:space="preserve" subscript="si">D</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="load">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="x">L</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve" subscript="y">L</ml:id> <ml:id xml:space="preserve">H</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">τ</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="137"/> </region> <region region-id="3563" left="66" top="1730.25" width="414" height="48" align-x="93.75" align-y="1740" 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"> <f family="Arial" charset="0"> <inlineAttr line-through="false">Remove Heat Equation Nonlinearity from Temperature Dependence of Conductivity</inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr line-through="false">Kirchhoff Transformation:<sp count="2"/> </inlineAttr> </f> K, D must have value at Tamb <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">Tk Kirch is the Inverse of </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Q</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10">, </f> </inlineAttr> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" underline="false" line-through="false">the Integral Transform of Conductivity, </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">l</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">si(T)</inlineAttr> </f> </text> </region> <region region-id="3884" left="228" top="1790.25" width="110.25" height="36.75" align-x="241.5" align-y="1812" 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:id xml:space="preserve">Θ</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="s">T</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="s">λ</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">T</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">λ</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">Ts</ml:id> <ml:id xml:space="preserve">T</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="138"/> </region> <region region-id="2568" left="360" top="1788.75" width="136.5" height="41.25" align-x="436.5" align-y="1812" show-border="false" show-highlight="false" 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