Question

In a heat exchanger, for the same terminal temperatures, the logarithmic mean temperature difference for counter flow is

• A. Appreciable greater than that for co-current flow
• B. Appreciable lower than that for co-current flow
• C. Almost equal to that for co-current flow
• D. Appreciable lower than that for cross flow

Hint:

Counter-current flow is the most thermodynamically efficient arrangement for heat exchange as it maximizes the temperature driving force. This allows for smaller heat exchanger areas to achieve the same rate of heat transfer compared to co-current flow.

Answer 1

The Correct Option is A

Step 1: Understand the concepts of co-current and counter-current flow in heat exchangers.
Co-current flow (Parallel flow): In this arrangement, both the hot and cold fluids enter the heat exchanger at the same end and flow in the same direction, leaving at the other end. The temperature difference between the two fluids is large at the inlet and decreases along the length of the heat exchanger. Counter-current flow: In this arrangement, the hot and cold fluids enter the heat exchanger at opposite ends and flow in opposite directions. The colder fluid exits near the inlet of the hotter fluid, and the hotter fluid exits near the inlet of the colder fluid. This setup allows for a more uniform temperature difference along the length of the heat exchanger and the possibility of heating the colder fluid to a temperature closer to the inlet temperature of the hotter fluid. Step 2: Recall the formula for Logarithmic Mean Temperature Difference (LMTD).
The LMTD (\(\Delta T_{lm}\)) is used to determine the temperature driving force for heat transfer in heat exchangers. It is defined as: $$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln \left( \frac{\Delta T_1}{\Delta T_2} \right)}$$ Where:
\(\Delta T_1\) is the temperature difference between the hot and cold fluids at one end of the heat exchanger.
\(\Delta T_2\) is the temperature difference between the hot and cold fluids at the other end of the heat exchanger.
Step 3: Analyze the LMTD for co-current and counter-current flow with the same terminal temperatures.
Let \(T_{h,i}\) and \(T_{h,o}\) be the inlet and outlet temperatures of the hot fluid, and \(T_{c,i}\) and \(T_{c,o}\) be the inlet and outlet temperatures of the cold fluid. The problem states that the terminal temperatures are the same for both flow arrangements. This means that \(T_{h,i}\), \(T_{h,o}\), \(T_{c,i}\), and \(T_{c,o}\) have the same values in both cases. For co-current flow:
\(\Delta T_1 = T_{h,i} - T_{c,i}\) (at the inlet)
\(\Delta T_2 = T_{h,o} - T_{c,o}\) (at the outlet)
For counter-current flow:
\(\Delta T_1 = T_{h,i} - T_{c,o}\) (at one end)
\(\Delta T_2 = T_{h,o} - T_{c,i}\) (at the other end)
Consider a scenario where \(T_{h,i}>T_{h,o}\) and \(T_{c,o}>T_{c,i}\) (heat transfer from hot to cold fluid). For the same terminal temperatures, in counter-current flow, the colder fluid entering at one end meets the hotter fluid leaving at the same end, leading to a larger temperature difference at this end compared to the outlet end of a co-current flow exchanger. Similarly, the hotter fluid entering at the other end meets the colder fluid leaving at that end, potentially maintaining a larger temperature difference along the exchanger. Mathematically, it can be shown that for the same terminal temperatures, the LMTD for counter-current flow is always greater than or equal to the LMTD for co-current flow. The equality holds only when one of the fluids undergoes a negligible temperature change. In practical heat exchangers where significant heat transfer occurs, the LMTD for counter-current flow is appreciably greater. Step 4: Evaluate the given options.
(1) Appreciable greater than that for co-current flow: This aligns with the analysis that counter-current flow generally provides a larger average temperature difference.
(2) Appreciable lower than that for co-current flow: This is incorrect based on the temperature profiles in the two flow arrangements.
(3) Almost equal to that for co-current flow: This is only true in limiting cases where one fluid's temperature change is very small.
(4) Appreciable lower than that for cross flow: The relationship between LMTD for counter-current and cross flow depends on the specific configuration and correction factors for cross flow, but counter-current flow generally achieves the highest LMTD for the same terminal temperatures. Therefore, for the same terminal temperatures, the logarithmic mean temperature difference for counter flow is appreciably greater than that for co-current flow.