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<div id="gooSearch"> <script async src="https://cse.google.com/cse.js?cx=6a091aba9f3e1fba3"></script> <div class="gcse-searchbox-only"></div> </div> <div id="content"> <div> <h1> <a href="arithmetic-logarithmic-mean-temperature-d_436.html"> Arithmetic and Logarithmic Mean Temperature Difference </a> </h1> <h2> Arithmetic Mean Temperature Difference in Heat Exchangers - <i>AMTD</i> - and Logarithmic Mean Temperature Difference - <i>LMTD</i> - formulas with examples - Online Mean Temperature Calculator. </h2> <div id="adSlotDoc1" class="noPrint"> <div>  <div id="adngin-doc1-0"></div> </div> </div> <div> <p>According to <strong>Newton's Law of Cooling</strong> heat transfer rate is related to the instantaneous temperature difference between hot and cold media</p> <ul> <li>in a heat transfer process the temperature difference vary with position and time</li> </ul> <h3>Mean Temperature Difference</h3> <p>The mean <a href="temperature-d_291.html" title="Temperature">temperature</a> difference in a heat transfer process depends on the direction of fluid flows involved in the process. The primary and secondary fluid in an heat exchanger process may</p> <ul> <li>flow in the same direction - <strong>parallel flow or co-current flow</strong></li> <li>in the opposite direction - <strong>counter-current flow</strong></li> <li>or perpendicular to each other - <strong>cross flow</strong></li> </ul> <p style="text-align: center;"><img style="border: 0px initial initial;" title="Logarithmic mean temperature difference" alt="Logarithmic arithmetic mean temperature - difference lmtd amtd" src="https://www.engineeringtoolbox.com/docs/documents/436/logarithmic_mean_temperature_difference.png" width="528" height="333" ></p> <p>With saturation steam as the primary fluid the primary temperature can be taken as a constant since the heat is transferred as a result of a change of phase only. The temperature profile in the primary fluid is not dependent on the direction of flow.</p> <h3>Logarithmic Mean Temperature Difference -<i> LMTD</i></h3> <p>The rise in secondary temperature is non-linear and can best be represented by a logarithmic calculation. A logarithmic mean temperature difference is termed</p> <ul> <li><i>LMTD</i> <em>(or DT<sub>LM</sub>)</em> - Logarithmic Mean Temperature Difference</li> </ul> <p>LMTD can be expressed as</p> <blockquote> <p><em>LMTD = (dt<sub>i</sub> - dt<sub>o</sub>) / ln(dt<sub>i</sub> / dt<sub>o</sub>)</em><i>                                       (1)</i></p> <p><i>where</i></p> <p><em>LMTD =</em><i> Logarithmic Mean Temperature Difference (<sup>o</sup>F, <sup>o</sup>C</i>)</p> <p>For parallel flow:<i><br ></i></p> <p><em>dt<sub>i</sub> = t<sub>pi</sub> - t<sub>si</sub></em><i> = <strong>inlet </strong>primary and secondary fluid temperature difference (<sup>o</sup>F, <sup>o</sup>C)</i></p> <p><em>dt<sub>o</sub> = t<sub>po</sub> - t<sub>so</sub></em><i> = <strong> outlet</strong> primary and secondary fluid temperature difference (<sup>o</sup>F, <sup>o</sup>C)</i></p> <p>For counter flow:<i><br ></i></p> <p><em>dt<sub>i</sub> = t<sub>pi</sub> - t<sub>so</sub></em><i> = <strong>inlet </strong>primary and outlet secondary fluid temperature difference (<sup>o</sup>F, <sup>o</sup>C)</i></p> <p><em>dt<sub>o</sub> = t<sub>po</sub> - t<sub>si</sub></em><i> = <strong> outlet</strong> primary and inlet secondary fluid temperature difference (<sup>o</sup>F, <sup>o</sup>C)</i></p> </blockquote> <p>The Logarithmic Mean Temperature Difference is always less than the Arithmetic Mean Temperature Difference.</p> <div id="adInContent1" class="noPrint"> <div>  <div id="adngin-incontent1-0"></div> </div> </div> <h3>Arithmetic Mean Temperature Difference - <i>AMTD</i></h3> <p>An easier but less accurate way to calculate the mean temperature difference is the</p> <ul> <li><em>AMTD (or DT<sub>AM</sub>)</em> - Arithmetic Mean Temperature Difference</li> </ul> <p>AMTD can be expressed as:</p> <blockquote> <p><em>AMTD = (t<sub>pi</sub> + t<sub>po</sub>) / 2 - (t<sub>si</sub> + t<sub>so</sub>) / 2</em><i>                                  (2)</i></p> <p><i>where</i></p> <p><em>AMTD</em><i> = Arithmetic Mean Temperature Difference (<sup>o</sup>F, <sup>o</sup>C)</i></p> <p><em>t<sub>pi</sub></em><i> = primary inlet temperature (<sup>o</sup>F, <sup>o</sup>C)</i></p> <p><em>t<sub>po</sub></em><i> = primary outlet temperature (<sup>o</sup>F, <sup>o</sup>C)</i></p> <p><em>t<sub>si</sub></em><i> = secondary inlet temperature (<sup>o</sup>F, <sup>o</sup>C)</i></p> <p><em>t<sub>so</sub></em><i> = secondary outlet temperature (<sup>o</sup>F, <sup>o</sup>C)</i></p> </blockquote> <p>A linear increase in the secondary fluid temperature makes it more easy to do manual calculations. <i>AMTD</i> will in general give a satisfactory approximation for the mean temperature difference when the smallest of the inlet or outlet temperature differences is more than half the greatest of  the inlet or outlet temperature differences.</p> <p>When heat is transferred as a result of a change of phase like condensation or evaporation the temperature of the primary or secondary fluid remains constant. The equations can then be simplified by setting</p> <blockquote> <p><em>t<sub>p1</sub> = t<sub>p2</sub></em></p> <p><i> or </i></p> <p><em>t<sub>s1</sub> = t<sub>s2</sub></em></p> </blockquote> <h3>Arithmetic and Logarithmic Mean Temperature Difference Calculator</h3> <p>The calculator below can be used to calculate Arithmetic and Logarithmic Mean Temperature Difference of counter-flow an parallel-flow heat exchangers.</p> <form action="#" id="temperatureCalculate" name="temperatureCalculate"> <p style="padding-left: 30px;"><label><input value="80" id="tpi" name="tpi" type="number" ><i> t<sub>pi</sub> - primary flow - inlet temperature (<sup>o</sup>F, <sup>o</sup>C)</i></label></p> <p style="padding-left: 30px;"><label><input value="60" id="tpo" name="tpo" type="number" > <i>t<sub>po</sub> - primary flow - outlet temperature (<sup>o</sup>F, <sup>o</sup>C)</i></label></p> <p style="padding-left: 30px;"><label><input value="0" id="tsi" name="tsi" type="number" > <i>t<sub>si</sub> - secondary flow - inlet temperature (<sup>o</sup>F, <sup>o</sup>C)</i></label></p> <p style="padding-left: 30px;"><label><input value="20" id="tso" name="tso" type="number" > <i>t<sub>so</sub> - secondary flow - outlet temperature (<sup>o</sup>F, <sup>o</sup>C)</i></label></p> <p style="padding-left: 30px;"><label><input value="c" id="c" name="flows" type="radio" > Counter-flow </label><label><input checked="checked" value="p" id="p" name="flows" type="radio" > Parallel-flow</label></p> <p style="padding-left: 30px;"><input id="tempCalc" value="Calculate!" type="button" ></p> </form> <div id="result" style="padding-left: 30px;"></div> <div id="adInContent2" class="noPrint"> <div>  <div id="adngin-incontent2-0"></div> </div> </div> <h3>Logarithmic Mean Temperature Difference Chart</h3> <p style="text-align: center;"><a title="Logarithmic mean temperature difference" href="https://www.engineeringtoolbox.com/docs/documents/436/logarithmic_mean_temperature_difference.pdf"><img loading="lazy" title="LMTD - Logarithmic mean temperature difference - chart" alt="LMTD - Logarithmic mean temperature difference - chart" src="https://www.engineeringtoolbox.com/docs/documents/436/logarithmic_mean_temperature_difference_2.png" width="746" height="868" ></a></p> <ul> <li><a title="LMTD - Logarithmic mean temperature difference - Chart" href="https://www.engineeringtoolbox.com/docs/documents/436/logarithmic_mean_temperature_difference_2.png">Logarithmic Mean Temperature Difference Chart (pdf)</a></li> </ul> <h3>Example - Arithmetic and Logarithmic Mean Temperature, Hot Water Heating Air</h3> <p>Hot water at <i>80<sup> o</sup>C</i> heats air from from a temperature of <i>0<sup> o</sup>C</i> to <i>20<sup> o</sup>C</i> in a parallel flow heat exchanger. The water leaves the heat exchanger at <i>60<sup> o</sup>C</i>.</p> <p>Arithmetic Mean Temperature Difference can be calculated as</p> <blockquote> <p><em>AMTD</em><i> = ((80<sup> o</sup>C) + (60<sup> o</sup>C)) / 2 - ((0<sup> o</sup>C) + (20<sup> o</sup>C)) / 2</i></p> <p><i>    = <span style="text-decoration: underline;">60</span> <sup>o</sup>C</i></p> </blockquote> <p>Logarithmic Mean Temperature Difference can be calculated as</p> <blockquote> <p><em>LMTD</em><i> = ((60<sup> o</sup>C) - (20<sup> o</sup>C)) - ((80<sup> o</sup>C) - (0<sup> o</sup>C))) / ln(((60<sup> o</sup>C) - (20<sup> o</sup>C)) / ((80<sup> o</sup>C) - (0<sup> o</sup>C)))</i></p> <p><i>    = <span style="text-decoration: underline;">57.7</span> <sup>o</sup>C</i></p> </blockquote> <h3>Example - Arithmetic and Logarithmic Mean Temperature, Steam Heating Water</h3> <p>Steam at <i>2 bar</i> gauge heats water from <i>20<sup> o</sup>C to 50<sup> o</sup>C</i>. The saturation temperature of steam at <i>2 bar</i> gauge is <i>134<sup> o</sup>C</i>.</p> <p><strong>Note!</strong> that steam condenses at a constant temperature. The temperature on the heat exchangers surface on the steam side is constant and determined by the steam pressure.</p> <blockquote> <ul> <li><a href="saturated-steam-properties-d_101.html" title="Saturated steam properties table">Steam Table</a></li> </ul> </blockquote> <p>Arithmetic Mean Temperature Difference can be calculated like</p> <blockquote> <p><em>AMTD</em><i> = ((134<sup> o</sup>C) + (134<sup> o</sup>C)) / 2 - ((20<sup> o</sup>C) + (50<sup> o</sup>C)) / 2</i></p> <p><i>    = <span style="text-decoration: underline;">99</span> <sup>o</sup>C</i></p> </blockquote> <p>Log Mean Temperature Difference can be calculated like</p> <blockquote> <p><em>LMTD</em><i> = ((134<sup> o</sup>C) - (20<sup> o</sup>C) - ((134<sup> o</sup>C) - (50<sup> o</sup>C))) / ln(((134<sup> o</sup>C) - (20<sup> o</sup>C)) / ((134<sup> o</sup>C) - (50<sup> o</sup>C)))</i></p> <p><i>    = <span style="text-decoration: underline;">98.24</span> <sup>o</sup>C</i></p> </blockquote> </div> <div id="adSlotDoc2" class="noPrint"> <div>  <div id="adngin-doc2-0"></div> </div> </div> <h2> Related Topics </h2> <ul> <li> <h3> <a href="heat-loss-pipes-tanks-t_15.html" title="pipes tubes tanks vessels insulation heat loss foam rockwool fiberglass transfer coefficient water oil thermal temperature"> Heat Loss and Insulation </a> </h3> Heat loss from pipes, tubes and tanks - with and without insulation. 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