3. LMTD for Heat Exchanger Analysis

Cocurrent Flow:

Using Eqn.(5) in Eqn.(4), we get \[\begin{align*} \ln \frac{\Delta T_2}{\Delta T_1} &= -UA\left(\frac{T_{h1}-T_{h2}}{Q} + \frac{T_{c2}-T_{c1}}{Q} \right) \\ &= -\frac{UA}{Q}\left[(T_{h1}-T_{c1}) - (T_{h2}-T_{c2}) \right] = \frac{UA}{Q}(\Delta T_2 - \Delta T_1) \\ \Longrightarrow \quad Q &= UA\frac{\Delta T_2 - \Delta T_1}{\ln \dfrac{\Delta T_2}{\Delta T_1} }\tag*{(6)} \end{align*}\] Comparing Eqns.(1) and (6), we get \[\boxed{\Delta T_m = \frac{\Delta T_2 - \Delta T_1}{\ln \dfrac{\Delta T_2}{\Delta T_1}} = \Delta T_{\text{lm}} = \text{LMTD}}\] LMTD = Log Mean Temperature Difference