A house built on a riverside is to be cooled in summer by utilizing the cool water of the river, which flows at an average temperature of \(15^{\circ} \mathrm{C}\). A 15 -m-long section of a circular duct of 20 -cm diameter passes through the water. Air enters the underwater section of the duct at \(25^{\circ} \mathrm{C}\) at a velocity of \(3 \mathrm{~m} / \mathrm{s}\). Assuming the surface of the duct to be at the temperature of the water, determine the outlet temperature of air as it leaves the underwater portion of the duct. Also, for an overall fan efficiency of 55 percent, determine the fan power input needed to overcome the flow resistance in this section of the duct.

Understanding the Logarithmic Mean Temperature Difference (LMTD) is crucial for analyzing heat transfer in ducts and designing heat exchangers.

Essentially, the LMTD represents an average temperature difference between two fluids exchanging heat. For a heat exchanger operating with a steady flow, the LMTD is used instead of a simple arithmetic mean because it more accurately reflects the varying temperature difference along the length of the exchanger.

The formula for LMTD takes into account the inlet and outlet temperatures of the two fluids involved:
\[LMTD = \frac{(T_{in}-T_{cold})-(T_{out}-T_{cold})}{\ln\frac{T_{in}-T_{cold}}{T_{out}-T_{cold}}}\] where \(T_{in}\) is the initial temperature of the hot fluid, \(T_{out}\) is the final temperature of the hot fluid, and \(T_{cold}\) is the temperature of the cold fluid, which is constant in the exercise scenario. If you're studying this in a textbook problem, you should check if the heat transfer is uniform. If it's not, you may need to calculate LMTD for different sections individually.

When it comes to heat exchanger design, there are many factors to consider to ensure effective heat transfer between fluids. One must select appropriate materials, determine the right size and shape, and conduct a thorough analysis of the temperature changes that will occur.

Let's focus on one aspect of this complex process: ensuring the heat exchanger is properly designed to handle the calculated heat transfer. This involves selecting the right size of the duct or pipe, the material that can withstand the temperature and pressure, and configuring the system to maintain the efficiency over time.

For instance, in the exercise, a duct within a body of water acts as a simple heat exchanger. The design would require the duct to be long enough and have an adequate surface area to allow the desired amount of heat transfer. The duct material must also conduct heat well while resisting corrosion from the water.

Remember that every heat exchanger design must be tailored to its specific situation. Different applications require different approaches, be it for residential heating, industrial processes, or refrigeration.

In systems involving airflow through ducts, such as the situation described in our exercise, calculating the fan power input is critical. This calculation ensures that the fan installed can overcome the resistance of the air moving through the duct and maintain the necessary flow rate for heat transfer.

To calculate the power input required by the fan, one must consider the fan efficiency and the power needed to move the air at the given flow rate:
\[P = \frac{Q}{\eta}\] where \(P\) is the fan power input required, \(Q\) is the heat transfer rate, and \(\eta\) is the fan efficiency. The heat transfer rate can be further broken down into the mass flow rate of the air and the temperature difference it experiences.

In the given exercise, with a fan efficiency of 55 percent, the calculation reveals the amount of power the fan needs to provide to sustain the airflow. It's also important to note that the actual power consumption of the fan system will be higher due to inefficiencies and other losses. Understanding these calculations can help with the selection of a fan that is adequate for the required tasks in both residential and industrial applications.