Question

LMTD for counter flow and parallel flow heat exchanger will be the same, when

• A. A cold fluid is heated to a certain temperature by condensing steam (isothermal fluid)
• B. The outlet temperature of both the hot and cold fluids are same
• C. The outlet temperature of hot fluid is less than the outlet temperature of the cold fluid
• D. One of the fluids undergoes a phase change

Hint:

LMTD Optimization Rules: - For standard sensible heating/cooling: \(\text{LMTD}_{\text{counter}} > \text{LMTD}_{\text{cross}} > \text{LMTD}_{\text{parallel}}\). - When one fluid undergoes a phase change (isothermal condition): \(\text{LMTD}_{\text{counter}} = \text{LMTD}_{\text{parallel}}\). The direction of flow does not affect the heat transfer rate.

Answer 1

The Correct Option is D

Concept: The Log Mean Temperature Difference (\(\text{LMTD}\) or \(\Delta T_{lm}\)) is used to determine the driving force for heat transfer in heat exchangers. It is calculated from the temperature differences between the two fluids at the ends of the exchanger: \[ \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln\left(\frac{\Delta T_1}{\Delta T_2}\right)} \] Where:
• For Parallel Flow: \(\Delta T_1 = T_{h,\text{in}} - T_{c,\text{in}}\) and \(\Delta T_2 = T_{h,\text{out}} - T_{c,\text{out}}\)
• For Counter Flow: \(\Delta T_1 = T_{h,\text{in}} - T_{c,\text{out}}\) and \(\Delta T_2 = T_{h,\text{out}} - T_{c,\text{in}}\) Generally, a counter-flow configuration provides a higher \(\text{LMTD}\) than a parallel-flow configuration for the same operating temperatures. However, specific thermodynamic conditions can make their performance identical.

Step 1: Analyzing the case where one fluid undergoes a phase change.
Consider a system where one of the fluids undergoes a phase change, such as steam condensing in the shell side or water boiling in the tubes. During a phase change, the fluid absorbs or releases latent heat at a constant temperature. Therefore, the temperature of this fluid remains completely uniform (isothermal) throughout the entire length of the heat exchanger: \[ T_{h,\text{in}} = T_{h,\text{out}} = T_{\text{condensation}} \quad \text{(for condensing steam)} \]

Step 2: Evaluating the parallel-flow LMTD profile.
Let the constant temperature of the isothermal hot fluid be \(T_h\). For a parallel-flow arrangement, the temperature differences at the inlets and outlets are: \[ \Delta T_1 = T_h - T_{c,\text{in}} \quad \text{and} \quad \Delta T_2 = T_h - T_{c,\text{out}} \] Substituting these into the general \(\text{LMTD}\) equation gives: \[ \Delta T_{lm,\text{parallel}} = \frac{(T_h - T_{c,\text{in}}) - (T_h - T_{c,\text{out}})}{\ln\left(\frac{T_h - T_{c,\text{in}}}{T_h - T_{c,\text{out}}}\right)} = \frac{T_{c,\text{out}} - T_{c,\text{in}}}{\ln\left(\frac{T_h - T_{c,\text{in}}}{T_h - T_{c,\text{out}}}\right)} \]

Step 3: Evaluating the counter-flow LMTD profile.
Now, let's write the expressions for a counter-flow arrangement with the same isothermal fluid: \[ \Delta T_1 = T_h - T_{c,\text{out}} \quad \text{and} \quad \Delta T_2 = T_h - T_{c,\text{in}} \] Substituting these into the \(\text{LMTD}\) equation gives: \[ \Delta T_{lm,\text{counter}} = \frac{(T_h - T_{c,\text{out}}) - (T_h - T_{c,\text{in}})}{\ln\left(\frac{T_h - T_{c,\text{out}}}{T_h - T_{c,\text{in}}}\right)} = \frac{T_{c,\text{in}} - T_{c,\text{out}}}{\ln\left(\frac{T_h - T_{c,\text{out}}}{T_h - T_{c,\text{in}}}\right)} = \frac{T_{c,\text{out}} - T_{c,\text{in}}}{\ln\left(\frac{T_h - T_{c,\text{in}}}{T_h - T_{c,\text{out}}}\right)} \] Comparing the results from Step 2 and Step 3 shows that \(\Delta T_{lm,\text{parallel}} = \Delta T_{lm,\text{counter}}\). Therefore, when one of the fluids undergoes a phase change (remaining at a constant temperature), the parallel-flow and counter-flow configurations yield the exact same \(\text{LMTD}\). This matches Option (D).