A = Fläche Wärmetauscher
\[\Delta {\vartheta _m}\,\,Uebertemperatur\]
\[\begin{array}{l}d{{\dot Q}_1} = - {{\dot m}_1} \cdot c \cdot d{\vartheta _1} = U \cdot ({\vartheta _1} - {\vartheta _2}) \cdot da\\d{{\dot Q}_2} = - {{\dot m}_2} \cdot c \cdot d{\vartheta _2} = U \cdot ({\vartheta _1} - {\vartheta _2}) \cdot da\\d{\vartheta _1} = - \frac{{U \cdot ({\vartheta _1} - {\vartheta _2})}}{{{{\dot m}_1} \cdot c}}da\\d{\vartheta _2} = - \frac{{U \cdot ({\vartheta _1} - {\vartheta _2})}}{{{{\dot m}_2} \cdot c}}da\\\int\limits_{\Delta {\vartheta _0}}^{\Delta \vartheta } {\frac{{d({\vartheta _1} - {\vartheta _2})}}{{{\vartheta _1} - {\vartheta _2}}}} = - \frac{U}{c} \cdot (\frac{1}{{{{\dot m}_1}}} + \frac{1}{{{{\dot m}_2}}})\int\limits_0^a {da} \\\ln (\frac{{\Delta \vartheta }}{{\Delta {\vartheta _0}}}) = - \frac{U}{c} \cdot (\frac{1}{{{{\dot m}_1}}} + \frac{1}{{{{\dot m}_2}}})a\\\Delta \vartheta = \Delta {\vartheta _0} \cdot {e^{ - \frac{U}{c} \cdot (\frac{1}{{{{\dot m}_1}}} + \frac{1}{{{{\dot m}_2}}})a}}\\\Delta {\vartheta _A} = \Delta {\vartheta _0} \cdot {e^{ - \frac{U}{c} \cdot (\frac{1}{{{{\dot m}_1}}} + \frac{1}{{{{\dot m}_2}}})A}}\\\Delta {\vartheta _m} = \int\limits_0^A {\frac{{\Delta \vartheta }}{A}} \cdot da = (\Delta {\vartheta _0} \cdot \frac{1}{{ - \frac{U}{c} \cdot (\frac{1}{{{{\dot m}_1}}} + \frac{1}{{{{\dot m}_2}}})a}} \cdot {e^{ - \frac{U}{c} \cdot (\frac{1}{{{{\dot m}_1}}} + \frac{1}{{{{\dot m}_2}}})a}})_0^A\\\Delta {\vartheta _m} = \frac{1}{{ - \frac{U}{c} \cdot (\frac{1}{{{{\dot m}_1}}} + \frac{1}{{{{\dot m}_2}}})a}} \cdot \Delta {\vartheta _0} \cdot ({e^{ - \frac{U}{c} \cdot (\frac{1}{{{{\dot m}_1}}} + \frac{1}{{{{\dot m}_2}}})A}} - 1)\\\Delta {\vartheta _m} = \frac{{\Delta {\vartheta _A} - \Delta {\vartheta _0}}}{{ - \frac{U}{c} \cdot (\frac{1}{{{{\dot m}_1}}} + \frac{1}{{{{\dot m}_2}}})A}}\\\Delta {\vartheta _m} = \frac{{\Delta {\vartheta _A} - \Delta {\vartheta _0}}}{{\ln (\frac{{\Delta {\vartheta _A}}}{{\Delta {\vartheta _0}}})}}\end{array}\]