Question

In a counter flow heat exchanger, hot fluid enters at \(120^\circ\text{C}\) and leaves at \(80^\circ\text{C}\). Cold fluid enters at \(20^\circ\text{C}\) and leaves at \(60^\circ\text{C}\). Find the log mean temperature difference (LMTD).

• A. \(30^\circ\text{C}\)
• B. \(40^\circ\text{C}\)
• C. \(60^\circ\text{C}\)
• D. \(80^\circ\text{C}\)

Hint:

For any heat exchanger, if the temperature differences at both ends are equal ($\Delta T_{1} = \Delta T_{2}$), you do not need to use the logarithmic formula.
The LMTD is simply equal to this common temperature difference value ($\Delta T$).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to calculate the Log Mean Temperature Difference (LMTD) for a counter-flow heat exchanger with specified terminal temperatures of the hot and cold fluids.

Step 2: Key Formula or Approach:
The LMTD (\(\Delta T_{\text{lm}}\)) for a counter-flow heat exchanger is defined as:
\[ \Delta T_{\text{lm}} = \frac{\Delta T_{1} - \Delta T_{2}}{\ln(\Delta T_{1} / \Delta T_{2})} \] where:
\(\Delta T_{1} = T_{\text{h,in}} - T_{\text{c,out}}\) is the temperature difference at one end.
\(\Delta T_{2} = T_{\text{h,out}} - T_{\text{c,in}}\) is the temperature difference at the other end.

Step 3: Detailed Explanation:

• Identify the given temperature values:
Hot fluid inlet, \(T_{\text{h,in}} = 120^\circ\text{C}\).
Hot fluid outlet, \(T_{\text{h,out}} = 80^\circ\text{C}\).
Cold fluid inlet, \(T_{\text{c,in}} = 20^\circ\text{C}\).
Cold fluid outlet, \(T_{\text{c,out}} = 60^\circ\text{C}\).

• Calculate the temperature differences at both ends for a counter-flow arrangement:
At the left end:
\[ \Delta T_{1} = 120^\circ\text{C} - 60^\circ\text{C} = 60^\circ\text{C} \] At the right end:
\[ \Delta T_{2} = 80^\circ\text{C} - 20^\circ\text{C} = 60^\circ\text{C} \]
• Since the temperature differences at both ends are equal (\(\Delta T_{1} = \Delta T_{2} = 60^\circ\text{C}\)), the standard logarithmic formula yields an indeterminate form (\(0/0\)).

• Taking the mathematical limit as \(\Delta T_{1}\) approaches \(\Delta T_{2}\) reveals that:
\[ \lim_{\Delta T_{1} \to \Delta T_{2}} \Delta T_{\text{lm}} = \Delta T_{1} = \Delta T_{2} \]
• Therefore, when the temperature differences at both ends of a counter-flow heat exchanger are identical, the LMTD is simply equal to this constant temperature difference of \(60^\circ\text{C}\).

Step 4: Final Answer:
The log mean temperature difference (LMTD) is \(60^\circ\text{C}\).