In a one-shell and eight-tube pass heat exchanger, the temperature of water flowing at rate of \(50,000 \mathrm{lbm} / \mathrm{h}\) is raised from \(70^{\circ} \mathrm{F}\) to \(150^{\circ} \mathrm{F}\). Hot air \(\left(c_{p}=0.25 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)\) that flows on the tube side enters the heat exchanger at \(600^{\circ} \mathrm{F}\) and exits at \(300^{\circ} \mathrm{F}\). If the convection heat transfer coefficient on the outer surface of the tubes is \(30 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\), determine the surface area of the heat exchanger using both LMTD and \(\varepsilon-\mathrm{NTU}\) methods. Account for the possible fouling resistance of \(0.0015\) and \(0.001 \mathrm{~h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F} /\) Btu on water and air side, respectively.

When engineering a heat exchanger, ensuring efficient energy transfer between fluids is essential. This is where the Logarithmic Mean Temperature Difference (LMTD) comes into play. It is a crucial calculation in the heat exchanger design process because it accurately characterizes the temperature driving force for heat exchange over the length of the heat exchanger.

To clarify, the LMTD is a weighted average temperature difference between the hot and cold fluids at the inlet and outlet of the exchanger. This value is necessary since the temperature difference is not constant throughout the unit. In our example, the LMTD is calculated using differences in inlet and outlet temperatures of hot air and water. The equation for LMTD is given by:
\[LMTD = \frac{\Delta{T_1} - \Delta{T_2}}{\ln(\frac{\Delta{T_1}}{\Delta{T_2}})}\]
This result provides a basis for determining the size and cost-efficiency of the heat exchanger by informing engineers of the required surface area to achieve the desired heat transfer.

Why is LMTD Preferred Over Arithmetic Mean?

While an arithmetic mean might seem a simpler way to average temperature differences, it does not account for the logarithmic relationship between the heat transfer rate and temperature difference across a heat exchanger. LMTD, therefore, provides a more accurate reflection of the true thermal driving force over the entire heat exchanger.

Another key factor in the design of heat exchangers is the overall heat transfer coefficient, which is a measure of how well heat is transferred through the materials of the heat exchanger. It is denoted by 'U' and typically has the units of Btu/(h*ft²*°F) in the Imperial system or W/(m²*K) in the SI system.

The overall heat transfer coefficient takes into account the heat transfer due to conduction through the heat exchanger material and the convection to and from the fluid. In simple terms, it quantifies the resistance to heat flow through the heat exchanger.
\[\frac{1}{U} = \frac{1}{h} + R_{f,water} + R_{f,air}\]
In the exercise provided, 'h' represents the convection heat transfer coefficient, while 'R_f,water' and 'R_f,air' represent the fouling resistances on the water and air side, respectively. Fouling resistances are important considerations in the calculation of 'U' because they account for the reduction in heat transfer due to the buildup of unwanted materials on the heat exchanger's surfaces.

The Importance of Accurate 'U' Values

The overall heat transfer coefficient crucially affects the design size and material choice for the heat exchanger. For instance, if 'U' is too low, the heat exchanger may need to be larger to compensate for the poor heat transfer efficiency, thus increasing costs. Conversely, higher 'U' values signify better heat transfer performance, allowing for a smaller, more cost-effective exchanger design.

For a more advanced and precise approach to analyzing heat exchangers, engineers often turn to the ε-NTU method. This approach is particularly useful when the fluids' inlet temperatures or the flow rates are not fixed. 'ε' stands for effectiveness, which measures the heat exchanger's ability to transfer the maximum possible heat, whereas 'NTU' stands for the Number of Transfer Units, a dimensionless quantity indicating the heat exchanger's size relative to the heat transfer rate.

The ε-NTU method involves complex calculations based on different configurations of heat exchangers (counterflow, parallel flow, crossflow, etc.) and accounts for both fluids' specific heat capacities. The effectiveness, 'ε', can be interpreted as a percentage of the potential maximum heat transfer achieved by the heat exchanger.
In our exercise, although the ε-NTU method wasn't shown in its entirety due to its complexity, it inherits the notion that precision in design is attainable by considering all variables that influence heat exchange. Furthermore, the ε-NTU method can accommodate changes in conditions or requirements, which lends a flexible advantage to design and operational adjustments.

When to Use the ε-NTU method

This method is most beneficial when one does not have precise temperature data for both fluids throughout the heat exchanger or when the heat exchanger operates under varying conditions. It's also helpful when considering different heat exchanger designs and predicting their performance under varied conditions, making it a versatile tool for thermal engineering.