Question

In the heat transfer relation \(\dot{Q}=U A_{s} \Delta T_{\mathrm{lm}}\) for a heat exchanger, what is \(\Delta T_{\mathrm{lm}}\) called? How is it calculated for a parallel-flow and a counterflow heat exchanger?

Short answer

Answer: The term \(\Delta T_{\mathrm{lm}}\) is called the Logarithmic Mean Temperature Difference (LMTD) in the heat transfer equation for a heat exchanger. For both parallel-flow and counterflow heat exchangers, LMTD is calculated using the formula: \(\Delta T_{\mathrm{lm}} = \frac{(\Delta T_1 - \Delta T_2)}{\ln(\frac{\Delta T_1}{\Delta T_2})}\). In parallel-flow heat exchangers, \(\Delta T_1 = T_{h1} - T_{c1}\) and \(\Delta T_2 = T_{h2} - T_{c2}\), while in counterflow heat exchangers, \(\Delta T_1 = T_{h1} - T_{c2}\) and \(\Delta T_2 = T_{h2} - T_{c1}\).

Step-by-step solution

Definition of \(\Delta T_{\mathrm{lm}}\)

The \(\Delta T_{\mathrm{lm}}\) in the heat transfer equation of a heat exchanger is called the Logarithmic Mean Temperature Difference (LMTD).

Calculating LMTD for Parallel-flow Heat Exchanger

In a parallel-flow heat exchanger, the hot and cold fluids flow in the same direction. Let's denote the temperatures for the hot fluid as \(T_{h1}\) and \(T_{h2}\) for the inlet and outlet, respectively. For the cold fluid, the temperatures are denoted as \(T_{c1}\) and \(T_{c2}\) for the inlet and outlet. The LMTD for a parallel-flow heat exchanger is calculated using the following formula: $$ \Delta T_{\mathrm{lm}} = \frac{(\Delta T_1 - \Delta T_2)}{\ln(\frac{\Delta T_1}{\Delta T_2})} $$ where \(\Delta T_1 = T_{h1} - T_{c1}\) and \(\Delta T_2 = T_{h2} - T_{c2}\).

Calculating LMTD for Counterflow Heat Exchanger

In a counterflow heat exchanger, the hot and cold fluids flow in opposite directions. We use the same temperature notations as in the parallel-flow case. The LMTD for a counterflow heat exchanger is calculated using the same formula: $$ \Delta T_{\mathrm{lm}} = \frac{(\Delta T_1 - \Delta T_2)}{\ln(\frac{\Delta T_1}{\Delta T_2})} $$ However, in the counterflow case, the temperature differences are \(\Delta T_1 = T_{h1} - T_{c2}\) and \(\Delta T_2 = T_{h2} - T_{c1}\). In summary, the \(\Delta T_{\mathrm{lm}}\) is called the Logarithmic Mean Temperature Difference (LMTD), and it can be calculated for parallel-flow and counterflow heat exchangers using the same formula but with different temperature differences.