The fin array of Problem 3.142 is commonly found in compact...

The fin array of Problem $3.142$ is commonly found in compact heat exchangers, whose function is to provide a large surface area per unit volume in transferring heat from one fluid to another. Consider conditions for which the second fluid maintains equivalent temperatures at the parallel plates, $T_{o}=T_{L}$, thereby establishing symmetry about the midplane of the fin array. The heat exchanger is $1 \mathrm{~m}$ long in the direction of the flow of air (first fluid) and $1 \mathrm{~m}$ wide in a direction normal to both the airflow and the fin surfaces. The length of the fin passages between adjoining parallel plates is $L=8 \mathrm{~mm}$, whereas the fin thermal conductivity and convection coefficient are $k=200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ (aluminum) and $h=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively. (a) If the fin thickness and pitch are $t=1 \mathrm{~mm}$ and $S=4 \mathrm{~mm}$, respectively, what is the value of the thermal resistance $R_{t, o}$ for a one-half section of the fin array? (b) Subject to the constraints that the fin thickness and pitch may not be less than $0.5$ and $3 \mathrm{~mm}$, respectively, assess the effect of changes in $t$ and $S$.

The fin array of Problem $3.142$ is commonly found in compact heat exchangers, whose function is to provide a large surface area per unit volume in transferring heat from one fluid to another. Consider conditions for which the second fluid maintains equivalent temperatures at the parallel plates, $T_{o}=T_{L}$, thereby establishing symmetry about the midplane of the fin array. The heat exchanger is $1 \mathrm{~m}$ long in the direction of the flow of air (first fluid) and $1 \mathrm{~m}$ wide in a direction normal to both the airflow and the fin surfaces. The length of the fin passages between adjoining parallel plates is $L=8 \mathrm{~mm}$, whereas the fin thermal conductivity and convection coefficient are $k=200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ (aluminum) and $h=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively.
(a) If the fin thickness and pitch are $t=1 \mathrm{~mm}$ and $S=4 \mathrm{~mm}$, respectively, what is the value of the thermal resistance $R_{t, o}$ for a one-half section of the fin array?
(b) Subject to the constraints that the fin thickness and pitch may not be less than $0.5$ and $3 \mathrm{~mm}$, respectively, assess the effect of changes in $t$ and $S$.

Key Concepts

Thermal Resistance

Thermal resistance quantifies the opposition to heat flow through materials and interfaces, analogous to electrical resistance in circuits. In the context of heat exchangers with fin arrays, it represents the combined resistance offered by conduction through the fin material and convection to the surrounding fluid. Analyzing thermal resistance is crucial for predicting the efficiency of heat transfer and optimizing the design of compact exchangers.

Fin Array Geometry

Fin array geometry deals with the design parameters such as fin length, thickness, and spacing (pitch) that determine the available surface area for heat exchange. By adjusting these dimensions, one can significantly influence the performance of the heat exchanger. The geometric design needs to be balanced with manufacturing constraints and thermal performance requirements to maximize heat transfer while minimizing material usage and resistance.

Conjugate Heat Transfer

Conjugate heat transfer refers to the simultaneous occurrence of heat conduction within solids, such as the fins, and convection in the adjacent fluids. This integrated analysis is critical when assessing systems like compact heat exchangers, as it allows for the evaluation of temperature distributions and overall thermal resistance by considering both the internal conduction paths and the external convective processes.

Boundary Conditions and Symmetry

Establishing boundary conditions and utilizing symmetry in heat transfer problems simplifies the analysis by reducing the domain of interest. When a system exhibits symmetry, as with identical temperatures on opposite boundaries, it allows the model to consider only a portion of the geometry. This concept is particularly useful in designing and analyzing heat exchangers, as it helps determine the temperature profiles and associated thermal resistances more efficiently.

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