Water \(\left(c_{p}=1.0 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)\) is to be heated by solarheated hot air \(\left(c_{p}=0.24 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)\) in a double- pipe counterflow heat exchanger. Air enters the heat exchanger at \(190^{\circ} \mathrm{F}\) at a rate of \(0.7 \mathrm{lbm} / \mathrm{s}\) and leaves at \(135^{\circ} \mathrm{F}\). Water enters at \(70^{\circ} \mathrm{F}\) at a rate of \(0.35 \mathrm{lbm} / \mathrm{s}\). The overall heat transfer coefficient based on the inner side of the tube is given to be \(20 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\). Determine the length of the tube required for a tube internal diameter of \(0.5 \mathrm{in}\).

Understanding the Log Mean Temperature Difference (LMTD) is crucial for analyzing heat exchangers. It is a theoretical measure used to compare the temperature difference between the hot and cold fluids at each end of the heat exchanger.

The formula for LMTD is:\[\begin{equation}LMTD = \frac{\Delta T_{1} - \Delta T_{2}}{\ln(\frac{\Delta T_{1}}{\Delta T_{2}})}\end{equation}\]where \(\Delta T_{1}\) and \(\Delta T_{2}\) are the temperature differences at either end of the exchanger. If the temperatures of both fluids change linearly, which rarely happens, the LMTD would be a simple average of these differences.

However, since the temperature of fluids often varies, using LMTD provides an effective temperature difference for calculating the heat transfer rate over the length of the exchanger. In scenarios like a counterflow heat exchanger, where fluids flow in opposite directions, the LMTD method comprehensively accounts for the varying temperatures and enhances the accuracy of the design and analysis process.

The heat transfer rate, often denoted by \(Q\), is a measure of the thermal energy transferred per unit time. It is a fundamental concept in designing and evaluating the performance of heat exchangers.

Calculating \(Q\) involves the specific heat capacities of the fluids (\(c_p\)), their mass flow rates (\(m\)), and the change in temperature (\(\Delta T\)) as the fluids pass through the exchanger:\[\begin{equation}Q = m c_p \Delta T\end{equation}\]In our example, the heat transfer rate calculation for both air and water was crucial to determining the change in temperature of the water, which could then be used to further compute other parameters such as the required tube length. It's essential to consider that in an ideal heat exchange process without energy losses, the heat lost by the hot fluid is equal to the heat gained by the cold fluid.

Counterflow heat exchangers are designed in a way that the two fluids involved move in opposite directions. This setup increases the efficiency of heat transfer compared to parallel flow exchangers.

In a counterflow heat exchanger, the hot fluid exits the exchanger at a lower temperature, while the cold fluid exits at a higher temperature, thus transferring heat more effectively. This occurs because the temperature gradient between the fluids, which drives the heat transfer, remains relatively high along the entire length of the heat exchanger.

When considering the design of a counterflow exchanger, it becomes evident that not only does it often provide better performance, but it may also require less space, as illustrated by the compact design in our exercise example. The efficiency of a counterflow design is particularly evident in the improved LMTD, which can result in a more compact heat exchanger for the same heat transfer rate.

The overall heat transfer coefficient, often symbolized by \(U\), is a comprehensive measure of the heat transfer capability of a heat exchanger. It takes into account the heat transfer by conduction through the materials involved, as well as the convective heat transfer from the fluids. The unit of \(U\) is typically expressed in \(\mathrm{Btu/h\cdot ft^2 \cdot{ }^\circ F}\).

The value of \(U\) depends on the properties of the fluids, the materials used in the construction of the heat exchanger, the condition of the heating surfaces, and the type of flow. A higher \(U\) represents a more effective heat exchanger, able to transfer more heat between the fluids with a smaller surface area.

In the exercise, the given value of \(U\) based on the inner side of the tube is a determined factor in calculating the necessary length of the tube for the desired rate of heat transfer. By combining the overall heat transfer coefficient with the LMTD and the heat transfer area, we can accurately determine the tube length required to achieve the heating or cooling needed in a specific application.