Question

How does the log mean temperature difference for a heat exchanger differ from the arithmetic mean temperature difference? For specified inlet and outlet temperatures, which one of these two quantities is larger?

Short answer

Answer: In this specific example, the AMTD (50°C) is larger than the LMTD (approx. 49.3°C).

Step-by-step solution

Understand Log Mean Temperature Difference (LMTD) and Arithmetic Mean Temperature Difference (AMTD)

For a heat exchanger, the temperature differences between the hot and cold fluids are essential in determining the overall heat transfer. There are two ways to calculate the average temperature difference: LMTD and AMTD. The Log Mean Temperature Difference (LMTD) is calculated using the following formula: \[ LMTD = \frac{\Delta T_1 - \Delta T_2}{\log{(\Delta T_1 / \Delta T_2)}}\] where: - \(\Delta T_1\) is the temperature difference between the hot and cold fluids at the inlet, - \(\Delta T_2\) is the temperature difference between the hot and cold fluids at the outlet. The Arithmetic Mean Temperature Difference (AMTD) is calculated using the following formula: \[ AMTD = \frac{\Delta T_1 + \Delta T_2}{2} \]

Compare LMTD and AMTD using specific inlet and outlet temperatures

Let's use specified inlet and outlet temperatures to see which of LMTD and AMTD is larger. Suppose we have the following inlet and outlet temperatures for the heat exchanger: - Inlet: \(\Delta T_1 = 100°C - 40°C = 60°C\) - Outlet: \(\Delta T_2 = 60°C - 20°C = 40°C\) Now, let's calculate the LMTD and AMTD: \[ LMTD = \frac{60°C - 40°C}{\log{(\frac{60°C}{40°C})}} \approx 49.3°C \] \[ AMTD = \frac{60°C + 40°C}{2} = 50°C \] Thus, in this example, the LMTD is smaller than the AMTD. Note that this may not be the case for every heat exchanger; the specific inlet and outlet temperatures determine which of the two measures is larger.