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        <Metadata-attribute name="Title" value="Optimizing the Efficiency of a Regenerative Rankine Cycle"/>
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<Task/><Text-field style="Title" layout="Title">Maximizing the Efficiency of a Regenerative Rankine Cycle</Text-field>
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<Input><Text-field style="Text" layout="Normal">Rankine cycles are a thermodynamic process that turn heat into mechanical work. They're often employed in generating electrical power, with high pressure steam used to rotate a turbine.</Text-field><Text-field style="Text" layout="Normal">The thermodynamic efficiency of a Rankine cycle is  <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>where <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EpYFdfX25ldGBGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EpYFdfX25ldGBGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==</Equation> is the net work done by the system, and Q is the heat added. Operating parameters are chosen to maximize efficiency - these may be the temperatures or pressures at various points in the system.</Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">A typical regenerative Rankine cycle is illustrated below.</Text-field>
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style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal">In this application, we will </Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal" leftmargin="36" bullet="dot">define a procedure that calculates the cycle efficiency as a function of the pump outlet pressures at points 2 and 4</Text-field><Text-field style="Text" layout="Normal" leftmargin="36" bullet="dot">and then find the pump outlet pressures that maximize the efficiency of the cycle. </Text-field><Text-field style="Text" layout="Normal"></Text-field>
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<Section collapsed="false" isCollapsible="false" drawButton="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field bookmark="bkmrk1" style="Heading 1" layout="Heading 1">Cycle Efficiency</Text-field></Title><Text-field style="Text" layout="Normal">This Code Edit region contains a procedure that calculates the cycle efficiency as a function of pump output pressures. The procedure calculates
</Text-field><Text-field style="Text" layout="Normal" leftmargin="36" bullet="dot">the enthalpies, entropies, specific volumes and temperatures at every point,</Text-field><Text-field style="Text" layout="Normal" leftmargin="36" bullet="dot">the fraction of water removed at the high and low pressure turbine extraction points,</Text-field><Text-field style="Text" layout="Normal" leftmargin="36" bullet="dot">and the work done by the pumps.</Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">The procedure can be modified to return any of these values; right now, it only returns the efficiency.</Text-field><Text-field style="Text" layout="Normal"></Text-field><CodeEditor-ExecGroup view="presentation" hide-output="false" labelreference="L4" display="code"><EC-CodeEditor id="CodeEditRegion1" visible="false" pixel-width="2102" pixel-height="1928" code-language="text/maple" autoexpand="true" wrapping="false" show-border="true" code-line-numbers="true">#Procedure to Calculate Efficiency as a Function of Pressures at Points 2 and 4
eta:=proc(P2, P4)
local P1, x1, h1,v1,s1,T1, Wp2, h2, v2, s2, T2, P3, x3, h3, v3, T3, s3, Wp1, h4, v4, s4, T4, P5, T5, h5, Q, s5, v5, s6, P6, h6, v6, Wt1, s7, P7, h7, T7,v7, Wt2, x, Wnet, eta:

uses ThermophysicalData:-CoolProp;

try 

   #Fixed condenser pressure and boiler temperature
   P1:=7000:
   T5:= 750:

   #Low Pressure Pump
   x1 := 0:
   h1 := PropsSI(&quot;enthalpy&quot;, &quot;P&quot;, P1, &quot;Q&quot;, x1, &quot;water&quot;):
   v1 := 1/PropsSI(&quot;D&quot;, &quot;P&quot;, P1, &quot;Q&quot;, x1, &quot;water&quot;):
   s1 := PropsSI(&quot;entropy&quot;, &quot;P&quot;, P1, &quot;Q&quot;, x1, &quot;water&quot;):
   T1 := PropsSI(&quot;T&quot;, &quot;P&quot;,P1,&quot;Q&quot;, x1, &quot;water&quot;):
   Wp2 := v1*(P1-P2):
   h2 := h1-Wp2:
   v2 := 1/PropsSI(&quot;D&quot;, &quot;P&quot;, P2, &quot;enthalpy&quot;, h2, &quot;water&quot;):
   s2 := PropsSI(&quot;entropy&quot;, &quot;P&quot;, P2, &quot;enthalpy&quot;, h2, &quot;water&quot;):
   T2 := PropsSI(&quot;T&quot;, &quot;P&quot;, P2, &quot;enthalpy&quot;, h2, &quot;water&quot;):

   #High Pressure Pump
   P3 := P2:
   x3 := 0:
   h3 := PropsSI(&quot;enthalpy&quot;, &quot;P&quot;, P3, &quot;Q&quot;, x3, &quot;water&quot;):
   v3 := 1/PropsSI(&quot;D&quot;, &quot;P&quot;, P3, &quot;Q&quot;, x3, &quot;water&quot;):
   T3 := PropsSI(&quot;T&quot;, &quot;P&quot;, P3, &quot;Q&quot;, x3, &quot;water&quot;):
   s3 := PropsSI(&quot;entropy&quot;, &quot;P&quot;, P3, &quot;Q&quot;, x3, &quot;water&quot;):
   Wp1 := -v3*(P4-P3):
   h4 := h3-Wp1:
   v4 := 1/PropsSI(&quot;D&quot;, &quot;P&quot;, P4, &quot;enthalpy&quot;, h4, &quot;water&quot;):
   s4 := PropsSI(&quot;entropy&quot;, &quot;P&quot;, P4, &quot;enthalpy&quot;, h4, &quot;water&quot;):
   T4 := PropsSI(&quot;temperature&quot;, &quot;P&quot;, P4, &quot;enthalpy&quot;, h4, &quot;water&quot;):

   #Boiler
   P5 := P4:
   h5 := PropsSI(&quot;enthalpy&quot;, &quot;T&quot;, T5, &quot;P&quot;, P5, &quot;water&quot;):
   Q := h5-h4:
   s5 := PropsSI(&quot;entropy&quot;, &quot;P&quot;, P5, &quot;enthalpy&quot;, h5, &quot;water&quot;):
   v5 := 1/PropsSI(&quot;D&quot;, &quot;P&quot;, P5, &quot;enthalpy&quot;, h5, &quot;water&quot;):

   #Turbine High Pressure
   s6 := s5:
   P6 := P2:
   h6 := PropsSI(&quot;enthalpy&quot;, &quot;P&quot;, P6, &quot;entropy&quot;, s6, &quot;water&quot;):
   v6 := 1/PropsSI(&quot;D&quot;, &quot;P&quot;, P6, &quot;entropy&quot;, s6, &quot;water&quot;):
   Wt1 := h5-h6:

   #Turbine Low Pressure
   s7 := s6:
   P7 := P1:
   h7 := PropsSI(&quot;enthalpy&quot;, &quot;P&quot;, P7, &quot;entropy&quot;, s6, &quot;water&quot;):
   T7 := PropsSI(&quot;T&quot;, &quot;P&quot;, P7, &quot;entropy&quot;, s7, &quot;water&quot;):
   v7 := 1/PropsSI(&quot;D&quot;, &quot;entropy&quot;, s7, &quot;P&quot;, P7, &quot;water&quot;):
   Wt2 := h6-h7:

   #Feedwater Heater
   x := (h2-h3)/(h2-h6):

   #Cycle Statistics
   Wnet := Wt1+(1-x)*(Wt2+Wp2)+Wp1:
   eta := Wnet/Q:
   return eta;

catch:

   return 0;
   
end try;

end proc:</EC-CodeEditor></CodeEditor-ExecGroup>
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<Section collapsed="false" isCollapsible="false" drawButton="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field bookmark="bkmrk2" style="Heading 1" layout="Heading 1">Optimization</Text-field></Title>
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