Exact Solution for the Time-Dependent Temperature Field in Dry Grinding: Application to Segmental Wheels

<table>Field temperature <svg height="14.3875" id="M353" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 59.737499 14.3875" width="59.737499" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.3875)">
<g transform="translate(72,-60.49)">
<text transform="matrix(1,0,0,-1,-71.95,62.83)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑇</tspan>
<tspan style="font-size: 12.50px; " x="6.8766499" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="11.040149" y="0">𝑡</tspan>
</text>
<text transform="matrix(1,0,0,-1,-57,67.83)">
<tspan style="font-size: 8.75px; " x="0" y="0">∗</tspan>
</text>
<text transform="matrix(1,0,0,-1,-51.92,62.83)">
<tspan style="font-size: 12.50px; " x="0" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="5.2012482" y="0">𝑥</tspan>
<tspan style="font-size: 12.50px; " x="12.077898" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="17.291649" y="0">𝑦</tspan>
<tspan style="font-size: 12.50px; " x="23.493137" y="0">)</tspan>
</text>
</g>
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</svg> for <svg height="13.55" id="M354" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 165.64999 13.55" width="165.64999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.55)">
<g transform="translate(72,-61.16)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑥</tspan>
<tspan style="font-size: 12.50px; " x="11.040149" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="16.241398" y="0">𝑦</tspan>
<tspan style="font-size: 12.50px; " x="22.442884" y="0">)</tspan>
<tspan style="font-size: 12.50px; " x="30.082218" y="0">∈</tspan>
<tspan style="font-size: 12.50px; " x="42.122608" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="46.286106" y="0">−</tspan>
<tspan style="font-size: 12.50px; " x="54.850662" y="0">𝛿</tspan>
<tspan style="font-size: 12.50px; " x="61.139671" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="66.340919" y="0">𝛿</tspan>
<tspan style="font-size: 12.50px; " x="72.629929" y="0">)</tspan>
<tspan style="font-size: 12.50px; " x="79.581596" y="0">×</tspan>
<tspan style="font-size: 12.50px; " x="90.359184" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="94.522682" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="100.77418" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="105.97543" y="0">𝛿</tspan>
<tspan style="font-size: 12.50px; " x="112.26443" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="115.75278" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="122.00427" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="128.25577" y="0">)</tspan>
</text>
</g>
</g>
</svg> in the case of upgrinding. Contours give the temperature in K.</table>

Mathematical Problems in Engineering

fig15

Figure 15

Figure 15: Exact Solution for the Time-Dependent Temperature Field in Dry Grinding: Application to Segmental Wheels 