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This work analyzes heat transfer through a wall containing triangular fins partially embedded in its volume. The coupled heat diffusion equations governing each constituent are solved numerically using an iterative finite volume method. A well bracketed effectiveness of the combined system that suits wide range of applications is analytically derived. Good agreement between the numerical and the analytical results is attained. It is found that the fin-root can act simultaneously as a heat sink and heat source for the wall. The heat transfer rate through the combined system is clearly seen to be maximized at a specific fin-root length. The maximum reported heat transfer rate through the triangular rooted-finned wall is found to be at most 90% above that for the rootless fin case at wall Biot number of 1.54. This percentage is noticed to decrease as the wall Biot number decreases. At that Biot number, the maximum heat transfer rate through the combined system reaches 150% above that through the plain wall. As a result of this work, it is recommended to utilize the triangular rooted-fin as a heat transfer enhancer for high mechanical strength structures exposed to highly convective fluid streams.

This work analyzes heat transfer through a wall containing triangular fins partially embedded in its volume. The coupled heat diffusion equations governing each constituent are solved numerically using an iterative finite volume method. A well bracketed effectiveness of the combined system that suits wide range of applications is analytically derived. Good agreement between the numerical and the analytical results is attained. It is found that the fin-root can act simultaneously as a heat sink and heat source for the wall. The heat transfer rate through the combined system is clearly seen to be maximized at a specific fin-root length. The maximum reported heat transfer rate through the triangular rooted-finned wall is found to be at most 90% above that for the rootless fin case at wall Biot number of 1.54. This percentage is noticed to decrease as the wall Biot number decreases. At that Biot number, the maximum heat transfer rate through the combined system reaches 150% above that through the plain wall. As a result of this work, it is recommended to utilize the triangular rooted-fin as a heat transfer enhancer for high mechanical strength structures exposed to highly convective fluid streams.

Conduction; Convection; Fins; Augmentation; Roots; Thermal System

Heat transfer through finned surfaces, like finned tubes, was widely studied in the literature [

To the authors’ best knowledge, the thermal performance of the rooted-finned wall systems was first studied by Khaled [

In the next section, the two-dimensional heat diffusion equations governing the wall and the triangular rooted-fin materials are appropriately non-dimensionalized. Then, the corresponding boundary conditions and different suitable performance indicators are recognized. Next, an approximate simplified engineering model is casted for the first time in this work and a bracketed equation for the triangular rooted-finned wall effectiveness is identified. After that, the numerical and the approximate analytical computations of the performance indicators are compared. Eventually, an extensive parametric study is performed in order to fully characterize the thermal performance of the proposed system. Finally, the main, concluding remarks, are emphasized.

Consider a two dimensional slab wall having a uniform thickness W and height H. The x-axis is taken along the wall thickness starting from the left surface while the y-axis is aligned along its height starting from the wall center-line as shown in _{o} as seen from _{f}-axis of the fin is directed along the fin length starting from _{f}-axis of the fin is aligned along the fin thickness starting from the fin center line at

where

The quantities T, T_{f}, T_{L} and T_{¥} are the wall temperature field, fin temperature field, left surface temperature

The quantity h is the convection heat transfer coefficient of the fluid stream. When the flow is made parallel to the depth of

The triangular rooted-finned wall effectiveness [

When there is no fin, the wall right surface temperature,

As such, e_{W} can be computed from the following expression:

Define the rooted-finned wall second performance indicator

In spite of the complexity of the present system,

the combined wall and fin-root system. This combined system is bounded by

Recall that the thermal resistance [

Meanwhile, the convective thermal resistance of the right isothermal surface of the wall per unit depth is equal to:

Finally, recall that the total heat transfer rate through the system shown in

Using Equation (10) and the previous recalled formulations, the triangular rooted-finned wall effectiveness

where S is given by:

Using results available in any of heat transfer textbooks [

where Z based on the external-fin portion geometry is given by:

The two-dimensional heat diffusion equations given by Equations (1) and (2) are coupled via their boundary conditions represented by Equations (4), (5) and (6). These systems of equations have been solved using an iterative finite volume method [^{−12}. Furthermore, the solution convergence has been ensured by checking the value of heat transfer rate at the left surface ^{−5}. As such, the conservation of energy was satisfied,

Comparisons between the numerical results (2D solution) and analytical results (1D solution, Equation (14)) are shown in Figures 3 and 4. Excellent agreement are seen when

The condition given by Equation (18) requires that the variation in the temperature of the wall across the root length and the variation in the temperature of the fin-root are negligible. This indicates negligible transverse conduction across the wall-root interface as compared to the axial conduction through the fin root-base. As such, the analytical 1D model given by Equation (14) is valid.

As the root length increases, the wall conduction resistance decreases while the total convection resistance increases. This phenomenon is capable of minimizing the total thermal resistance of the combined system at a

specific root length. As such, the total heat transfer rate can be maximized at this length.

The problem of computing heat transfer rate through a system composed of a wall and triangular rooted-fin is theoretically investigated. Appropriate forms of the coupled heat diffusion equations for the wall and the triangular rooted-fin systems were solved using a finite volume method. Approximate closed form solution for the present problem based on a wide range of engineering applications was derived. Good agreement was obtained between the numerical and the closed form solutions when the wall Biot number based on the fin-root length is smaller than the one-half. The following remarks were concluded: 1) there is a critical fin-root length that maximizes the total heat transfer rate; 2) there is a critical fin-root length that makes the total system effectiveness independent on Biot number; 3) the maximum heat transfer rate through the combined system relative to that for the rootless fin case increases as the wall Biot number increases; 4) the maximum heat transfer rate through the triangular rooted-finned wall can be at most 90% above that for the rootless fin case at

The authors acknowledge the full support of this work by King Abdulaziz City for Science and Technology (KACST) under project no. 8-ENE192-3.

a_{H}, a_{t}: (wall height, fin-root base thickness) to wall width ratio; Equations (3g) and (3h).

Bi: wall Biot number; Equation (8).

H: wall height [m].

h: convection heat transfer coefficient [W/m^{2}∙K].

k, k_{f}: (wall, fin) thermal conductivity [W/m∙K].

q_{L}: triangular rooted-finned wall heat transfer rate [W].

L_{o}: fin-root length [m].

T, T_{f}: (wall, fin) temperature field [K].

T_{L}, T_{R}: wall (left, right) average surface temperature [K].

T_{¥}: free stream temperature [K].

t: fin-root base thickness [m].

W: wall width (and) fin length [m].

x, x_{f}: (wall, fin) axial coordinate [m].

X_{o}, x_{o}: (dimensionless, dimensional) axial location of the fin-root base [m].

X, X_{f}: dimensionless (x, x_{f})-coordinate; Equations (3b) and (3c).

y, y_{f}: (wall, fin) transverse coordinate [m].

Y, Y_{f}: dimensionless (y, y_{f})-coordinate; Equations (3d) and (3e).

g: triangular rooted-finned wall second performance indicator; Equation (13).

q, q_{f} : (wall, fin) dimensionless temperature; Equations (3a) and (3b).