Question

The temperature difference between the hot and cold fluids in a heat exchanger is given to be \(\Delta T_{1}\) at one end and \(\Delta T_{2}\) at the other end. Can the logarithmic temperature difference \(\Delta T_{\operatorname{lm}}\) of this heat exchanger be greater than both \(\Delta T_{1}\) and \(\Delta T_{2}\) ? Explain.

Short answer

Answer: No, the logarithmic mean temperature difference (\(\Delta T_{\operatorname{lm}}\)) cannot be greater than both \(\Delta T_1\) and \(\Delta T_2\). It will always lie between \(\Delta T_1\) and \(\Delta T_2\) when \(\Delta T_1 > \Delta T_2\).

Step-by-step solution

Recall the formula for logarithmic temperature difference

The logarithmic mean temperature difference for a heat exchanger is given by the formula: $$\Delta T_{\operatorname{lm}} = \frac{\Delta T_1 - \Delta T_2}{\ln\frac{\Delta T_1}{\Delta T_2}}$$ Where \(\Delta T_1\) and \(\Delta T_2\) are the temperature differences of the hot and cold fluids at both ends of the heat exchanger, respectively.

Analyze the formula's inequality

We are tasked with identifying if the logarithmic mean temperature difference, \(\Delta T_{\operatorname{lm}}\), can be greater than both \(\Delta T_1\) and \(\Delta T_2\). To do this, let's first compare \(\Delta T_{\operatorname{lm}}\) to \(\Delta T_1\): $$\Delta T_{\operatorname{lm}} > \Delta T_1$$ This inequality simplifies to: $$\frac{\Delta T_1 - \Delta T_2}{\ln\frac{\Delta T_1}{\Delta T_2}} > \Delta T_1$$ Similarly, we can compare \(\Delta T_{\operatorname{lm}}\) to \(\Delta T_2\): $$\Delta T_{\operatorname{lm}} > \Delta T_2$$ This inequality simplifies to: $$\frac{\Delta T_1 - \Delta T_2}{\ln\frac{\Delta T_1}{\Delta T_2}} > \Delta T_2$$

Analyze the inequalities

After analyzing these two inequalities, we can see that they both depend on the values of \(\Delta T_1\) and \(\Delta T_2\). This means that the answer to the question will vary depending on the chosen temperature differences. However, it should be noted that: 1. If \(\Delta T_1 > \Delta T_2\), then \(\Delta T_1 - \Delta T_2 > 0\) and \(\ln\frac{\Delta T_1}{\Delta T_2} > 0\), meaning that the logarithmic mean temperature difference, \(\Delta T_{\operatorname{lm}}\), will always be positive and lie between \(\Delta T_1\) and \(\Delta T_2\). 2. If \(\Delta T_1 < \Delta T_2\), then \(\Delta T_1 - \Delta T_2 < 0\) and \(\ln\frac{\Delta T_1}{\Delta T_2} < 0\), meaning that the logarithmic mean temperature difference, \(\Delta T_{\operatorname{lm}}\), will always be negative. However, we cannot consider this situation for a heat exchanger, as it is not physically possible for the temperature to decrease in the direction of heat flow. So, considering the possible scenarios, we conclude that:

Conclusion

In a heat exchanger, the logarithmic mean temperature difference \(\Delta T_{\operatorname{lm}}\) cannot be greater than both \(\Delta T_1\) and \(\Delta T_2\). It will always lie between \(\Delta T_1\) and \(\Delta T_2\) when \(\Delta T_1 > \Delta T_2\).