A single pass heat exchanger is to be designed to heat \(100,000 \mathrm{lbm}\) of water in an hour from \(60^{\circ} \mathrm{F}\) to \(100^{\circ} \mathrm{F}\) by condensation of water vapor at \(230^{\circ} \mathrm{F}\) on the shell side. Each tube has an inner diameter of \(1.2 \mathrm{in}\). and a wall thickness of \(0.12 \mathrm{in}\). The inner surface convection heat transfer coefficient is \(480 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) and the outer surface convection heat transfer coefficient is \(2000 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\). If the inlet velocity of water \(\left(c_{p c}=1 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right.\) and \(\rho=62.3 \mathrm{lbm} / \mathrm{ft}^{3}\) ) on the tube side is \(4 \mathrm{ft} / \mathrm{s}\), determine the required number of tubes and length of tubes. Assume the thermal resistance of the tubes is negligible.

Understanding heat transfer is critical when designing a heat exchanger. Heat transfer is essentially the movement of thermal energy from one substance to another. In the context of a heat exchanger, like the one in our exercise, the objective is to transfer heat from hot water vapor on the shell side to cold water within the tubes.

There are three modes of heat transfer: conduction, which is the transfer of heat through a solid medium; convection, which occurs in fluids and is driven by the movement of molecules within the fluid itself; and radiation, which involves emitting heat in the form of electromagnetic waves.

In the case of our heat exchanger, conduction is negligible due to the thinness of the tube walls, so the primary mechanism is convection. The hot water vapor condenses on the outer surface of the tubes, releasing latent heat, which is then transferred to the water inside the tubes through the tube walls. This process effectively raises the temperature of the water from the initial 60°F to the desired 100°F.

The convection heat transfer coefficient is a vital factor in determining how efficiently heat is exchanged between the fluid and the surface within the heat exchanger. In our exercise, we have two values for this coefficient: 480 BTU/h.ft².°F for the inner surface and 2000 BTU/h.ft².°F for the outer surface.

This coefficient measures the heat transfer per unit area per unit temperature difference between the surface and the fluid. A higher value of the convection heat transfer coefficient means that the surface is very effective at transferring heat.

The difference in the coefficients on the inner and outer surfaces indicates that the conditions for convective heat transfer are more favorable on the outside of the tubes. This is likely due to the phase change of water vapor to liquid, which typically results in a higher heat transfer coefficient. The exact values of these coefficients are determined by the nature of the fluid, the fluid dynamics, surface properties, and the temperature difference.

The log mean temperature difference (LMTD) is a concept used to quantify the average temperature driving force in a heat exchanger. In our exercise, we use the LMTD to find the required heat transfer area. LMTD is particularly useful when the temperature difference between the two fluids in a heat exchanger is not constant along the length of the exchanger.

To calculate the LMTD, we consider the temperature difference between the hot and cold fluids at both ends of the heat exchanger. In our problem:

• The temperature difference at one end, deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed (\(\Delta T_1\)) is 170°F (230°F minus 60°F)deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed �
• At the other end, (\(\Delta T_2\)) is 130°F (230°F minus 100°F)

We apply the logarithmic mean to account for the changing temperature gradient:
deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed deemed nnnnnnnnnnnnnnnn(\(\Delta T_{lm}\)) = \(\frac{\Delta T_1 - \Delta T_2}{\ln(\frac{\Delta T_1}{\Delta T_2})}\) = n149.1°FnThe LMTD helps ensure that the heat exchanger will perform effectively, as it properly evaluates the varying temperature differences along its length. In practice, using LMTD allows for more accurate sizing of the heat exchanger, which is crucial for optimizing performance and cost-efficiency in the system's design.