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width="387" height="10.5" align-x="5.25" align-y="12" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <reference> <originRef doc-id="feb7a6a0-07a5-4a5a-a222-c1ab6800481d" version-id="182364b1-ded3-4511-ab35-1cc54d53f972" branch-id="00000000-0000-0000-0000-000000000000" revision-num="4" is-modified="true" region-id="0" href="C:\Documents and Settings\Raymond Wagner\Desktop\current_model\utility\Aero Utility.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="00000000-0000-0000-0000-000000000000" version-id="00000000-0000-0000-0000-000000000000" branch-id="00000000-0000-0000-0000-000000000000" revision-num="0" is-modified="false" region-id="0" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> </reference> <link href=".\..\utility\Aero Utility.xmcd"/> <rendering item-idref="2"/> </region> <region region-id="8095" left="90" top="57.75" width="513.75" height="105.75" align-x="127.5" align-y="84" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#0f0" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="center" list-style-type="none" tabs="inherit"> <f family="Arial" charset="0" size="28"> <b> <inlineAttr line-through="false"> <c val="#000080">Von Mises'</c> </inlineAttr> </b> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="28"> <b> <inlineAttr line-through="false"> <c val="#000080">Representative Blade Propeller Model</c> </inlineAttr> </b> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"/> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="24"> <b> <inlineAttr line-through="false"> <c val="#000080">A Simple Parametric Propeller Model</c> </inlineAttr> </b> </f> </p> </text> </region> <region region-id="8387" left="258" top="170.25" width="188.25" height="12" align-x="268.5" align-y="180" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit">Last edited 3/9/2006 by Raymond Wagner</p> </text> </region> <region region-id="8331" left="96" top="324.75" width="505.5" height="113.25" align-x="96" align-y="336" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="12"> <b> <u> <inlineAttr line-through="false">Introduction</inlineAttr> </u> </b> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Our Problem is that we need to investigate a family of props in order to determine the best one for our airplane. This means we will need a map for each different prop.<sp count="2"/>We'll be lucky if we can get exactly what we need, so the following can be used to generate a representative family of maps using one propeller data set as a starting point and changing beta (<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">b</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sub>.75</sub> </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">)</inlineAttr> </f> and solidity<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.<sp count="2"/> </inlineAttr> </f> <inlineAttr font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10"> <b>Von Mises' </b> </f> </inlineAttr> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">Theory of flight</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, representative blade theory was used as a reference, pgs 302-309. Two prop data sets were used to correlate and check the behavior of this model. They are listed below. You should correlate this model to a real propeller that is close to what you want. For instance if you need a small model aircraft propeller, you should find some data on that type and recorrelate this model to it.<sp count="2"/>You get the idea.</inlineAttr> </f> </p> </text> </region> <region region-id="7933" left="36" top="521.25" width="648.75" height="15.75" align-x="53.25" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="14"> <b> <u> <inlineAttr line-through="false">Propeller Data for Correlation<sp count="122"/> </inlineAttr> </u> </b> </f> </p> </text> </region> <region region-id="7934" left="24" top="584.25" width="660.75" height="24" align-x="24" align-y="594" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Lets take a look at 2 data sets for 2 different props.<sp count="2"/>I will initially use the 25 deg propeller data set to see if I can get close to the 15 deg data set with the following method.<sp count="2"/>If I can do that then it would easily get from 15 deg <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">b</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> to our optimum prop.</inlineAttr> </f> <sp/> </p> </text> </region> <region region-id="7935" left="84" top="638.25" width="255" height="72" align-x="84" align-y="648" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"> <f family="Arial" charset="0"> <inlineAttr font-weight="normal" line-through="false">25 deg pitch , AF=90, 2 blade prop from NACA WR 286 (see Torrenbeek pg .....<sp count="2"/>I'll be using this data set in order to test the method. I'll see how close to the 15 deg pitch prop it gets.<sp count="2"/>You should not use this data set for our design. Use a data set which is as close to your final prop as you can find.<sp count="2"/> </inlineAttr> </f> </p> </text> </region> <region region-id="8040" left="414" top="638.25" width="165" height="36" align-x="414" align-y="648" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">15 deg pitch, Von Mises, pg 390. I think this is probably a good prop data set to start with.</p> </text> </region> <region region-id="8032" left="102" top="729" width="32.25" height="12.75" align-x="110.25" align-y="738" 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/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>5</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="8033" left="432" top="729" width="33" height="12.75" align-x="441" align-y="738" 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/math20"> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>6</ml:real> </ml:range> </ml:define> </math> 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<ml:real>.808</ml:real> <ml:real>.817</ml:real> <ml:real>.803</ml:real> <ml:real>-.2</ml:real> </ml:sequence> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="7944" left="222" top="794.25" width="159.75" height="24" align-x="222" align-y="804" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">efficiency is hard to curve fit so lets convert it to thrust coefficient:</p> </text> </region> <region region-id="7945" left="234" top="829.5" width="93.75" height="39.75" align-x="266.25" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define 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<ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="T15">vC</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="P15">vC</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">vη15</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">vJ15</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="8099" left="30" top="938.25" width="640.5" height="12" align-x="37.5" align-y="948" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This section plots the data and shows a typical spline curve fit of it. We will not be using the curve fit of this section, it is just to show how it is done.</p> </text> </region> <region region-id="8177" left="54" top="1010.25" width="105.75" height="24" align-x="54" align-y="1020" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Cubic spline fit thrust and power coefficient:</p> </text> </region> <region region-id="8178" left="258" top="1011" width="104.25" height="15" align-x="284.25" align-y="1020" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="P">d2C</ml:id> <ml:apply> <ml:id xml:space="preserve">lspline</ml:id> <ml:sequence> <ml:id xml:space="preserve">vJ</ml:id> <ml:id xml:space="preserve" subscript="P25">vC</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="8179" left="426" top="1010.25" width="126" height="12" align-x="426" align-y="1020" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">calculates second derivatives</p> </text> </region> <region region-id="8180" left="246" top="1029" width="149.25" height="15" align-x="285" align-y="1038" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" subscript="P.dat">C</ml:id> <ml:boundVars> <ml:id xml:space="preserve">J</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">interpl</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="P">d2C</ml:id> <ml:id xml:space="preserve">vJ</ml:id> <ml:id xml:space="preserve" subscript="P25">vC</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="8181" left="426" top="1028.25" width="93" height="12" align-x="426" align-y="1038" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">interpolates for any J:</p> </text> </region> <region region-id="8182" left="258" top="1053" width="105.75" height="15" align-x="285" align-y="1062" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="T">d2C</ml:id> <ml:apply> <ml:id xml:space="preserve">lspline</ml:id> <ml:sequence> <ml:id xml:space="preserve">vJ</ml:id> <ml:id xml:space="preserve" subscript="T25">vC</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="8183" left="246" top="1071" width="151.5" height="15" align-x="285.75" align-y="1080" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" subscript="T.dat">C</ml:id> <ml:boundVars> <ml:id xml:space="preserve">J</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">interpl</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="T">d2C</ml:id> <ml:id xml:space="preserve">vJ</ml:id> <ml:id xml:space="preserve" subscript="T25">vC</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="8184" left="114" top="1130.25" width="124.5" height="12" align-x="114" align-y="1140" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">now lets calculate efficiency:</p> </text> </region> <region region-id="8185" left="138" top="1152" width="220.5" height="84.75" align-x="171" align-y="1170" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" subscript="dat">η</ml:id> <ml:boundVars> <ml:id xml:space="preserve">J</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:ifThen> <ml:apply> <ml:and/> <ml:apply> <ml:greaterThan/> <ml:apply> <ml:id xml:space="preserve" subscript="P.dat">C</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:greaterThan/> <ml:apply> <ml:id xml:space="preserve" subscript="T.dat">C</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve" subscript="T.dat">C</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:apply> <ml:id xml:space="preserve">J</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="P.dat">C</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:apply> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:and/> <ml:apply> <ml:lessThan/> <ml:apply> <ml:id xml:space="preserve" subscript="P.dat">C</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:lessThan/> <ml:apply> <ml:id xml:space="preserve" subscript="T.dat">C</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve" subscript="P.dat">C</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve" subscript="T.dat">C</ml:id> <ml:id xml:space="preserve">J</ml:id> </ml:apply> <ml:id xml:space="preserve">J</ml:id> </ml:apply> </ml:apply> </ml:ifThen> <ml:otherwise> <ml:real>0</ml:real> </ml:otherwise> </ml:program> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="8186" left="402" top="1160.25" width="162.75" height="12" align-x="402" align-y="1170" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Thrust Power / Shaft Power Absorbed</p> </text> </region> <region region-id="8187" left="402" top="1202.25" width="199.5" height="12" align-x="402" align-y="1212" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e2e2e2" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Shaft Power Produced / Drag Power (windmill)</p> </text> </region> <region region-id="8196" left="36" top="1274.25" width="640.5" height="12" align-x="43.5" align-y="1284" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This section plots the data and shows a typical spline curve fit of it. We will not be using the curve fit of this section, it is just to show how it is done.</p> </text> </region> <region region-id="8197" left="660" top="1311" width="52.5" height="15" align-x="672.75" align-y="1320" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ffff" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="a">J</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>.02</ml:real> </ml:sequence> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="8198" left="138" top="1326" width="434.25" height="307.5" align-x="138" align-y="1326" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#e8e8e8" tag=""> <plot disable-calc="false" item-idref="19"/> <rendering item-idref="20"/> </region> <region region-id="8303" left="342" top="1508.25" width="90" height="24" align-x="342" align-y="1518" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="10"> <b> <inlineAttr line-through="false">Windmilling when </inlineAttr> </b> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="10"> <b> <inlineAttr line-through="false">values below zero</inlineAttr> </b> </f> </p> </text> </region> <region region-id="8199" left="12" top="1739.25" width="684" height="27.75" align-x="19.5" align-y="1752" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="14"> <b> <u> <inlineAttr line-through="false">Propeller Representative Blade Theory<sp count="111"/> </inlineAttr> </u> </b> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"> from: Theory of Flight, Von Mises, pg 302-309</p> </text> </region> <region region-id="8200" left="12" top="1794.75" width="168" height="14.25" align-x="12" align-y="1806" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="12"> <b> <inlineAttr line-through="false">Propeller Design Parameters:</inlineAttr> </b> </f> </p> </text> </region> <region region-id="8201" left="18" top="1820.25" width="173.25" height="24" align-x="29.25" align-y="1830" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">blade pitch at 3/4 radius of prop</p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">(usually measured from bottom of airfoil)</p> </text> </region> <region region-id="8202" left="300" top="1826.25" width="43.5" height="13.5" align-x="312" align-y="1836" show-border="false" show-highlight="true" 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/math20"> <ml:id xml:space="preserve">β</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="8203" left="18" top="1850.25" width="231.75" height="12" align-x="38.25" align-y="1860" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">difference between measured alpha and zero lift alpha</p> </text> </region> <region region-id="8204" left="300" top="1850.25" width="50.25" height="13.5" align-x="323.25" align-y="1860" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Δαp</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="8205" left="18" top="1874.25" width="218.25" height="12" align-x="47.25" align-y="1884" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">representative blade pitch, measured from zero lift:</p> </text> </region> <region region-id="8206" left="300" top="1874.25" width="58.5" height="15.75" align-x="315.75" align-y="1884" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="p">β</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">Δαp</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="7990" left="12" top="1914.75" width="243" height="14.25" align-x="21" align-y="1926" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="12"> <b> <inlineAttr line-through="false">Representative Propeller Airfoil properties:</inlineAttr> </b> </f> </p> </text> </region> <region region-id="8145" left="486" top="1938.75" width="176.25" height="42.75" align-x="486" align-y="1950" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#b4fee0" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="12"> <b> <inlineAttr line-through="false">When recorrelating to a new propeller data set follow these comments: </inlineAttr> </b> </f> </p> </text> </region> <region region-id="8007" left="300" top="1940.25" width="44.25" height="27.75" align-x="318" align-y="1956" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="α">cl</ml:id> <ml:apply> <ml:div/> <ml:real>.068</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="8012" left="18" top="1952.25" width="114" height="12" align-x="34.5" align-y="1962" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Average 2Dairfoil lift slope:</p> </text> </region> <region region-id="8009" left="18" top="1976.25" width="146.25" height="12" align-x="28.5" align-y="1986" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Approximate cl<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> roll off with alpha</inlineAttr> </f> </p> </text> </region> <region region-id="8010" left="300" top="1976.25" width="106.5" height="21.75" align-x="331.5" align-y="1992" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">cl</ml:id> <ml:boundVars> <ml:id xml:space="preserve">αp</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve" subscript="α">cl</ml:id> <ml:id xml:space="preserve">αp</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>.11</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">αp</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="8015" left="18" top="2000.25" width="47.25" height="12" align-x="25.5" align-y="2010" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">stall angle:</p> </text> </region> <region region-id="8016" left="300" top="2006.25" width="57" height="15.75" align-x="325.5" align-y="2016" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="stall">α</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>14</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="8142" left="486" top="2006.25" width="147" height="12" align-x="486" align-y="2016" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#b4fee0" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">controls thrust break at low speed</p> </text> </region> <region region-id="8082" left="300" top="2030.25" width="160.5" height="52.5" align-x="341.25" align-y="2040" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" subscript="L.p">C</ml:id> <ml:boundVars> <ml:id xml:space="preserve">αp</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:ifThen> <ml:apply> <ml:lessThan/> <ml:id xml:space="preserve">αp</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve" subscript="stall">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cl</ml:id> <ml:apply> <ml:neg/> <ml:id xml:space="preserve" subscript="stall">α</ml:id> </ml:apply> </ml:apply> </ml:ifThen> <ml:ifThen> <ml:apply> <ml:greaterThan/> <ml:id xml:space="preserve">αp</ml:id> <ml:id xml:space="preserve" subscript="stall">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cl</ml:id> <ml:id xml:space="preserve" subscript="stall">α</ml:id> </ml:apply> </ml:ifThen> <ml:otherwise> <ml:apply> <ml:id xml:space="preserve">cl</ml:id> <ml:id xml:space="preserve">αp</ml:id> </ml:apply> </ml:otherwise> </ml:program> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="8020" left="18" top="2036.25" width="62.25" height="12" align-x="24" align-y="2046" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Lift coefficient:</p> </text> </region> <region region-id="8080" left="300" top="2091" width="38.25" height="15" align-x="319.5" align-y="2100" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="0">cd</ml:id> <ml:real>.01</ml:real> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="8115" left="486" top="2090.25" width="115.5" height="15" align-x="494.25" align-y="2100" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#b4fee0" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">cd<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sub>0</sub> </inlineAttr> </f> bias's C<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sub>P</sub> </inlineAttr> </f> up and down</p> </text> </region> <region region-id="8116" left="486" top="2111.25" width="111" height="30" align-x="495" align-y="2124" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#b4fee0" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sup>2</sup> </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f>term effects C<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sub>P</sub> </inlineAttr> </f> roll off as J decreases </p> </text> </region> <region region-id="8137" left="300" top="2114.25" width="105" height="21.75" align-x="342" align-y="2130" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" subscript="D.p">C</ml:id> <ml:boundVars> <ml:id xml:space="preserve">αp</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="0">cd</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2.2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">αp</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="8394" left="300" top="2151" width="36" height="15" align-x="324" align-y="2160" 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/math20"> <ml:id xml:space="preserve" subscript="p">AR</ml:id> <ml:real>7</ml:real> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="8399" left="18" top="2156.25" width="95.25" height="12" align-x="35.25" align-y="2166" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Propeller aspect ratio:</p> </text> </region> <region region-id="8343" left="570" top="2234.25" width="122.25" height="15.75" align-x="590.25" align-y="2244" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ffff" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="a">αp</ml:id> <ml:range> <ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>-25</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>-24</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>25</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="8344" left="6" top="2256" width="318.75" height="250.5" align-x="6" align-y="2256" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="32"/> <rendering item-idref="33"/> </region> <region region-id="8345" left="360" top="2256" width="333" height="250.5" align-x="360" align-y="2256" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="34"/> <rendering item-idref="35"/> </region> <region region-id="8346" left="228" top="2526" width="231" height="148.5" align-x="228" align-y="2526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="36"/> <rendering item-idref="37"/> </region> <region region-id="8402" left="480" top="2526" width="229.5" height="148.5" align-x="480" align-y="2526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="38"/> <rendering item-idref="39"/> </region> <region region-id="8342" left="90" top="2688.75" width="504.75" height="14.25" align-x="110.25" align-y="2700" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="12"> <b> <inlineAttr line-through="false">Remember</inlineAttr> </b> </f> <f family="Arial" charset="0"> <b> <inlineAttr line-through="false">, </inlineAttr> </b> </f> this is not a real airfoil, but an average of the behavior of all of the airfoils making up the real propeller</p> </text> </region> <region region-id="8237" left="24" top="2747.25" width="684" height="27.75" align-x="31.5" align-y="2760" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="14"> <b> <u> <inlineAttr line-through="false">Propeller Representative Blade Theory<sp count="111"/> </inlineAttr> </u> </b> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"> from: Theory of Flight, Von Mises, pg 302-309</p> </text> </region> <region region-id="8238" left="24" top="2802.75" width="168" height="14.25" align-x="24" align-y="2814" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="12"> <b> <inlineAttr line-through="false">Propeller Design Parameters:</inlineAttr> </b> </f> </p> </text> </region> <region region-id="8239" left="30" top="2828.25" width="173.25" height="24" align-x="41.25" align-y="2838" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">blade pitch at 3/4 radius of prop</p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">(usually measured from bottom of airfoil)</p> </text> </region> <region region-id="8240" left="312" top="2834.25" width="43.5" height="13.5" align-x="324" align-y="2844" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">β</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="8241" left="30" top="2858.25" width="231.75" height="12" align-x="50.25" align-y="2868" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">difference between measured alpha and zero lift alpha</p> </text> </region> <region region-id="8242" left="312" top="2858.25" width="50.25" height="13.5" align-x="335.25" align-y="2868" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Δαp</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="8243" left="30" top="2882.25" width="218.25" height="12" align-x="59.25" align-y="2892" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">representative blade pitch, measured from zero lift:</p> </text> </region> <region region-id="8244" left="312" top="2882.25" width="58.5" height="15.75" align-x="327.75" align-y="2892" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="p">β</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">Δαp</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="8316" left="492" top="2882.25" width="144" height="12" align-x="492" align-y="2892" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#b4fee0" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">controls J operating range of prop</p> </text> </region> <region region-id="8309" left="30" top="2900.25" width="147.75" height="36" align-x="45" align-y="2910" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solidity:</p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">proportional to blade to disk area, </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">similar to activity factor</p> </text> </region> <region region-id="8310" left="222" top="2904" width="47.25" height="36" align-x="233.25" align-y="2922" 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/math20"> <ml:equal/> <ml:id xml:space="preserve">μ</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="blades">A</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>2</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">D</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="43"/> </region> <region region-id="8318" left="312" top="2912.25" width="35.25" height="13.5" align-x="324" align-y="2922" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">μ</ml:id> <ml:real>.014</ml:real> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="8312" left="492" top="2912.25" width="195" height="15" align-x="492" align-y="2922" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#b4fee0" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">controls high speed slope of both C<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sub>P</sub> </inlineAttr> </f> and C<f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sub>T</sub> </inlineAttr> </f>.</p> </text> </region> <region region-id="8245" left="60" top="2972.25" width="243" height="12" align-x="71.25" align-y="2982" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">representative blade is a the 3/4 of the radius of the prop</p> </text> </region> <region region-id="8246" left="336" top="2966.25" width="34.5" height="27.75" align-x="355.5" align-y="2982" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">r_d</ml:id> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>8</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="8247" left="408" top="2972.25" width="29.25" height="12" align-x="412.5" align-y="2982" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">pg 306</p> </text> </region> <region region-id="7793" left="6" top="3026.25" width="257.25" height="74.25" align-x="77.25" align-y="3036" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" subscript="T.αp">C</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="p">β</ml:id> <ml:id xml:space="preserve">αp</ml:id> <ml:id xml:space="preserve">μ</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:localDefine> <ml:id xml:space="preserve">γ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="p">β</ml:id> <ml:id xml:space="preserve">αp</ml:id> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:id xml:space="preserve">λ</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">r_d</ml:id> </ml:apply> </ml:localDefine> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve" subscript="L.p">C</ml:id> <ml:id xml:space="preserve">αp</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve" subscript="D.p">C</ml:id> <ml:id xml:space="preserve">αp</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">λ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">μ</ml:id> </ml:apply> </ml:program> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="7794" left="282" top="3026.25" width="257.25" height="74.25" align-x="352.5" align-y="3036" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" subscript="P.αp">C</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="p">β</ml:id> <ml:id xml:space="preserve">αp</ml:id> <ml:id xml:space="preserve">μ</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:localDefine> <ml:id xml:space="preserve">γ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="p">β</ml:id> <ml:id xml:space="preserve">αp</ml:id> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:id xml:space="preserve">λ</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">r_d</ml:id> </ml:apply> </ml:localDefine> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve" subscript="L.p">C</ml:id> <ml:id xml:space="preserve">αp</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve" subscript="D.p">C</ml:id> <ml:id xml:space="preserve">αp</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">γ</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">λ</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:id xml:space="preserve">μ</ml:id> </ml:apply> </ml:program> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="7795" left="570" top="3026.25" width="118.5" height="50.25" align-x="621" align-y="3036" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" 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z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="p">β</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>14</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="8257" left="576" top="3156" width="108.75" height="30.75" align-x="592.5" align-y="3174" 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/math20"> <ml:equal/> <ml:id xml:space="preserve">αp</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="p">β</ml:id> <ml:apply> <ml:id xml:space="preserve">atan</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve" subscript="αp">J</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">r_d</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="52"/> </region> <region region-id="8308" left="588" top="3218.25" width="96" height="12" align-x="594.75" align-y="3228" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Ref Von Mises pg 303</p> </text> </region> <region region-id="8340" left="48" top="3288" width="360" height="253.5" align-x="48" align-y="3288" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="53"/> <rendering item-idref="54"/> </region> <region region-id="8283" left="438" top="3306" width="243" height="169.5" align-x="438" align-y="3306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="55"/> <rendering item-idref="56"/> </region> <region region-id="7797" left="996" top="3302.25" width="162.75" height="12" align-x="996" align-y="3312" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Thrust Power / Shaft Power Absorbed</p> </text> </region> <region region-id="8289" left="516" top="3326.25" width="130.5" height="12" align-x="523.5" align-y="3336" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0"> <b> <inlineAttr line-through="false">airfoil stall occures at low J</inlineAttr> </b> </f> </p> </text> </region> <region region-id="7798" left="996" top="3344.25" width="199.5" height="12" align-x="996" align-y="3354" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Shaft Power Produced / Drag Power (windmill)</p> </text> </region> <region region-id="8274" left="198" top="3452.25" width="90" height="24" align-x="198" align-y="3462" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="10"> <b> <inlineAttr line-through="false">Windmilling when </inlineAttr> </b> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0" size="10"> <b> <inlineAttr line-through="false">values below zero</inlineAttr> </b> </f> </p> </text> </region> <region region-id="8282" left="468" top="3500.25" 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background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Δαp</ml:id> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>4</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="70"/> </region> <region region-id="7892" left="600" top="4160.25" width="48" height="15.75" align-x="615" align-y="4170" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="p">β</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.24434609527920614</ml:real> </result> </ml:eval> </math> <rendering item-idref="71"/> </region> <region region-id="8292" left="6" top="4280.25" width="139.5" height="12" align-x="6" align-y="4290" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Helical zero thrust relationships:</p> </text> </region> <region region-id="8293" left="162" top="4272" width="93.75" height="30.75" align-x="196.5" align-y="4290" 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/math20"> <ml:function> <ml:id xml:space="preserve" subscript="75">θ</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="0">J</ml:id> 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disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" subscript="0">J</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="75">θ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real font="0">.75</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:id xml:space="preserve" subscript="75">θ</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="74"/> </region> <region region-id="8296" left="504" top="4286.25" width="85.5" height="15.75" align-x="541.5" align-y="4296" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve" subscript="75">θ</ml:id> <ml:real>.82</ml:real> 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subscript="0">J</ml:id> </ml:apply> <ml:id xml:space="preserve">Δαp</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="76"/> </region> <region region-id="8298" left="504" top="4310.25" width="85.5" height="15.75" align-x="541.5" align-y="4320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve" subscript="75">β</ml:id> <ml:real font="0">.82</ml:real> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">deg</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>23.188857857078236</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="degree"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="77"/> </region> <region region-id="8299" left="642" top="4311" width="77.25" height="15" align-x="686.25" align-y="4320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve" subscript="0">J</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>19</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.81130282525418973</ml:real> </result> </ml:eval> </math> <rendering item-idref="78"/> </region> <region region-id="8300" left="456" top="4338" width="147" height="39" align-x="484.5" align-y="4356" 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/math20"> <ml:function> <ml:id xml:space="preserve" subscript="1">J</ml:id> 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<math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="p">N</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>10800</ml:real> <ml:id xml:space="preserve">rpm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="88"/> </region> <region region-id="8320" left="366" top="4713" width="40.5" height="15" align-x="382.5" align-y="4722" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ffff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="p">D</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>15</ml:real> <ml:id xml:space="preserve">in</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="89"/> </region> <region region-id="8321" left="366" top="4736.25" width="35.25" height="13.5" align-x="378" align-y="4746" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">μ</ml:id> <ml:real>.016</ml:real> </ml:define> </math> <rendering item-idref="90"/> </region> <region region-id="7546" left="12" top="4796.25" width="68.25" height="12" align-x="21" align-y="4806" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0"> <b> <inlineAttr line-through="false"> <c val="#000">Pitch at 3/4 r/d</c> </inlineAttr> </b> </f> </p> </text> </region> <region region-id="8356" left="156" top="4796.25" width="172.5" height="15.75" align-x="224.25" align-y="4806" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve" subscript=".75">pitch</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="p">D</ml:id> <ml:id xml:space="preserve" subscript="p">β</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>.75</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="p">D</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="p">β</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">deg</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="91"/> </region> <region region-id="7555" left="366" top="4796.25" width="103.5" height="15.75" align-x="433.5" align-y="4806" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#80ff80" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve" subscript=".75">pitch</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="p">D</ml:id> <ml:id xml:space="preserve" subscript="p">β</ml:id> </ml:sequence> </ml:apply> <ml:unitOverride> <ml:id xml:space="preserve">in</ml:id> </ml:unitOverride> <result xmlns="http://schemas.mathsoft.com/math20"> <unitedValue> <ml:real>7.4157145939579676</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="inch"/> </unitMonomial> </unitedValue> </result> </ml:eval> </math> <rendering item-idref="92"/> </region> <region region-id="7664" left="330" top="4818" width="189" height="15" align-x="342.75" align-y="4830" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ff80ff" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Times New Roman" charset="0" size="14"> <inlineAttr font-weight="normal" line-through="false"> <c val="#f00">Should be "Beta" not "Beta Prime"</c> </inlineAttr> </f> </p> </text> </region> <region region-id="7462" left="6" top="4832.25" width="242.25" height="12" align-x="9" align-y="4842" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In order to convert the independent variable from <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p to J </inlineAttr> </f> </p> </text> </region> <region region-id="7699" left="6" top="4862.25" width="69" height="12" align-x="24" align-y="4872" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial" charset="0"> <b> <inlineAttr line-through="false"> <c val="#000">Advance Ratio</c> </inlineAttr> </b> </f> </p> 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