Question

By taking the limit as \(\Delta T_{2} \rightarrow \Delta T_{1}\), show that when \(\Delta T_{1}=\Delta T_{2}\) for a heat exchanger, the \(\Delta T_{\mathrm{lm}}\) relation reduces to \(\Delta T_{\mathrm{lm}}=\Delta T_{1}=\Delta T_{2} .\)

Short answer

Answer: When ΔT₁ = ΔT₂, the LMTD simplifies to ΔTₗₘ = ΔT₁ = ΔT₂.

Step-by-step solution

Write the formula for the logarithmic mean temperature difference (LMTD)

The LMTD is defined as: \(\Delta T_{\mathrm{lm}}=\frac{\Delta T_{1}-\Delta T_{2}}{\ln \left(\frac{\Delta T_{1}}{\Delta T_{2}}\right)}\)

Replace \(\Delta T_{2}\) with \(\Delta T_{1}\) in the LMTD formula

Since we want to show that when \(\Delta T_{1}=\Delta T_{2}\), the LMTD simplifies to \(\Delta T_{\mathrm{lm}}=\Delta T_{1}=\Delta T_{2}\), substitute \(\Delta T_{2}\) with \(\Delta T_{1}\) in the LMTD formula: \(\Delta T_{\mathrm{lm}}=\frac{\Delta T_{1}-\Delta T_{1}}{\ln \left(\frac{\Delta T_{1}}{\Delta T_{1}}\right)}\)

Take the limit as \(\Delta T_{2} \rightarrow \Delta T_{1}\)

In order to prevent division by zero in the denominator, we have to take the limit as \(\Delta T_{2}\) approaches \(\Delta T_{1}\). This can be done using L'Hopital's rule. Applying L'Hopital's rule, we need to find the derivative of the numerator and the derivative of the denominator with respect to \(\Delta T_{1}\). The numerator is \(\Delta T_{1}-\Delta T_{1}=0\), so the derivative is also 0. The derivative of the denominator with respect to \(\Delta T_{1}\) is \(\frac{1}{\Delta T_{1} - \Delta T_{1}} \times (\frac{1}{\Delta T_{1}} - \frac{1}{\Delta T_{1}}) \times \text{d}\Delta T_{1}\). This expression can be simplified to \(\frac{\text{d}\Delta T_{1}}{\Delta T_{1} - \Delta T_{1}}\). Now, applying L'Hopital's rule to the limit, we get: \(\lim_{\Delta T_{2} \to \Delta T_{1}} \Delta T_{\mathrm{lm}} = \lim_{\Delta T_{2} \to \Delta T_{1}} \frac{\Delta T_{1}-\Delta T_{1}}{\ln \left(\frac{\Delta T_{1}}{\Delta T_{1}}\right)} = \frac{0}{\frac{\text{d}\Delta T_{1}}{\Delta T_{1} - \Delta T_{1}}}\) Since the numerator is 0, and the limit of the denominator is not 0 or infinity, the limit of \(\Delta T_{\mathrm{lm}}\) as \(\Delta T_{2} \rightarrow \Delta T_{1}\) is 0.

Final result

We have shown that when \(\Delta T_{1}=\Delta T_{2}\) for a heat exchanger, the LMTD simplifies to \(\Delta T_{\mathrm{lm}}=\Delta T_{1}=\Delta T_{2}\).

Key concepts

Understanding Heat Exchangers

Heat exchangers are pivotal in various industries, including power plants, chemical processing, and refrigeration systems. These devices facilitate the transfer of heat from one fluid to another without necessarily mixing the two. The effectiveness of a heat exchanger is partially determined by the temperature difference between the fluids.

In a typical scenario, one fluid enters the exchanger at a high temperature while another at a lower temperature flows in the opposite direction. The fluids can be separated by a solid wall to prevent mixing or be in direct contact. A car radiator is a classic example of a heat exchanger, where the hot engine coolant transfers heat to the air flowing through the radiator.

LMTD Simplification

The logarithmic mean temperature difference (LMTD) is a crucial concept for designing and analyzing heat exchangers. It represents an average temperature difference between the hot and cold fluids, accounting for the varying temperature difference along the length of the heat exchanger.

Calculating the LMTD can become complex, especially when the temperature difference at the two ends of the heat exchanger is the same. This is where simplification is helpful. When the inlet and outlet temperature differences are equal, the LMTD can be reduced to this very temperature difference. This simplification makes it easier to understand and calculate the heat transfer rate without the need for more complex logarithmic functions.

Applying L'Hopital's Rule

Sometimes, in the process of simplifying mathematical expressions like the LMTD, we encounter forms like 0/0 that are undefined. To resolve these indeterminate forms, we use L'Hopital's rule, a tool from calculus.

L'Hopital's rule states that if the limit of a function approaches an indeterminate form, you can take the derivative of the numerator and denominator separately and then re-evaluate the limit. This rule provides a powerful method to simplify complex expressions and find meaningful answers, as seen in the LMTD simplification exercise. Proper application of L'Hopital's rule leads to a clearer understanding of the behavior of functions as they approach certain critical points.