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In a parallel-flow heat exchanger, both the hot and cold fluids enter at the same end and flow in the same direction. This setup creates a temperature gradient along the length of the exchanger, with the highest temperature difference at the entrance and the least at the exit. The effectiveness of parallel-flow heat exchangers is often lower than counter-flow designs due to the rapid rate at which the temperature difference decreases.

From a practical standpoint, the temperature of the cold fluid can never exceed the exit temperature of the hot fluid since they are in close contact throughout the heat exchanger. This can be crucial when designing systems where the cold fluid needs to reach a specific temperature. To illustrate, let’s use an example similar to the one from the exercise where the hot water enters at \(75^\circ C\) and the cold water at \(20^\circ C\). Considering the specific heat and flow rates provided, the effectiveness of a parallel-flow heat exchanger is calculated, which determines how well the exchanger performs relative to its theoretical maximum performance.

On the other hand, a counter-flow heat exchanger has the fluids flowing in opposite directions. This arrangement maintains a more constant temperature gradient along the length of the exchanger, which can lead to a higher effectiveness when compared to parallel-flow designs.

Revisiting our exercise, if we utilize a counter-flow heat exchanger under the same operating conditions, there is a rise in effectiveness. The hot and cold water, while flowing in opposite directions, maximize the temperature difference utilized for heat exchange over the entire length of the exchanger. Consequently, the temperature of the cold fluid can exceed the exit temperature of the hot fluid in the parallel-flow case, making the counter-flow design more desirable for transferring more heat. The rise in effectiveness when switching from parallel to counter-flow, as seen in the exercise, signifies the improved utilisation of the heat exchange surface.

The overall heat transfer coefficient is a measure of a heat exchanger's ability to transfer heat between two fluids separated by a solid barrier. It is denoted by \(U\) and typically expressed in units of \(W/(m^2 \cdot K)\). The \(U\) value incorporates all modes of heat transfer: conduction through the exchanger material, and convection on both sides of the exchanger.

In our exercise, the value provided is \(750 W/(m^2 \cdot K)\) for the surface area of the exchanger is \(6.4 m^2\). The role of this coefficient is pivotal as it represents the efficiency of heat transfer in a specific design. The higher the value, the better the exchanger’s ability to transfer heat. This coefficient is crucial when calculating the effectiveness which is a key factor in evaluating heat exchanger performance.