Question

Under what conditions can a counterflow heat exchanger have an effectiveness of 1 ? What would your answer be for a parallel-flow heat exchanger?

Short answer

Answer: A counterflow heat exchanger can have an effectiveness of 1 when the temperature difference between the hot fluid inlet and the cold fluid outlet is infinitely small, meaning the hot fluid cools down completely, transferring all its heat energy to the cold fluid, while the cold fluid heats up completely, reaching the initial temperature of the hot fluid. In contrast, a parallel-flow heat exchanger can never have an effectiveness of 1 because the temperature difference between the hot and cold fluids remains at every point throughout the heat exchanger, preventing the maximum possible heat transfer from being achieved.

Step-by-step solution

Counterflow heat exchanger

To find the condition for a counterflow heat exchanger to have an effectiveness of 1, we need to analyze the temperature profiles of the hot and cold fluids in the heat exchanger. In a counterflow heat exchanger, the hot and cold fluids move in opposite directions. The effectiveness (ε) is given by: ε = (actual heat transfer) / (maximum possible heat transfer) For an effectiveness of 1, the actual heat transfer must be equal to the maximum possible heat transfer.

Analyzing the temperature profiles

For the counterflow heat exchanger, the effectiveness can reach 1 under the following condition: the temperature difference between the hot fluid inlet and the cold fluid outlet must be infinitely small. In mathematical terms: \[\lim_{\Delta T \to 0}(T_{H, inlet} - T_{C, outlet}) = 0\] This means that the hot fluid must cool down completely, transferring all of its heat energy to the cold fluid, while the cold fluid must heat up completely, reaching the initial temperature of the hot fluid.

Parallel-flow heat exchanger

Next, we will analyze the conditions for a parallel-flow heat exchanger to have an effectiveness of 1. In a parallel-flow heat exchanger, the hot and cold fluids flow in the same direction. Using the same equation for effectiveness as in step 1, we will analyze the temperature profiles to find the condition for an effectiveness of 1 in a parallel-flow heat exchanger.

Analyzing the temperature profiles for a parallel-flow heat exchanger

In a parallel-flow heat exchanger, the effectiveness can never reach 1. This is because the temperature difference between the hot and cold fluids reduces as the fluids flow through the heat exchanger. At the outlet, the hot fluid will always have a higher temperature than the cold fluid, so there will always be a temperature difference between the hot and cold fluids at every point throughout the heat exchanger. Consequently, the maximum possible heat transfer can never be achieved, and the effectiveness cannot be equal to 1 in a parallel-flow heat exchanger. In conclusion, a counterflow heat exchanger can have an effectiveness of 1 under the condition that the temperature difference between the hot fluid inlet and the cold fluid outlet is infinitely small, while a parallel-flow heat exchanger can never have an effectiveness of 1.