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Scale Analysis of Heat Transfer in Rectangular Fins

Introduction

Rectangular fins are widely used in engineering applications to enhance heat transfer from surfaces by increasing the surface area exposed to a fluid. Fins are typically made from high thermal conductivity materials, such as aluminum or copper, and are attached to surfaces that require efficient cooling. Heat is conducted through the fin material and convected away by the surrounding fluid. Understanding the heat transfer process in rectangular fins is essential for optimizing their design and performance in cooling systems, electronics, and heat exchangers.

Aluminium Heatsink

Governing Equations

The heat transfer in a fin can be modeled by the following differential equation, which represents the energy balance along the length of the fin:

${\frac {d}{dx}}\left(kA_{c}{\frac {dT}{dx}}\right)-hP(T-T_{\infty })=0$

where:

• $k$ is the thermal conductivity of the fin material,
• $A_{c}$ is the cross-sectional area of the fin,
• $T(x)$ is the temperature distribution along the fin,
• $h$ is the heat transfer coefficient between the fin and the fluid,
• $P$ is the perimeter of the fin in contact with the fluid,
• $T_{\infty }$ is the ambient temperature of the fluid,
• $x$ is the distance along the fin from the base.

Diagram of a fin used to depict dimensions in thermodynamics problems

The boundary conditions typically used for solving this equation are:

1. At the base of the fin ($x=0$): $T(0)=T_{b}$, where $T_{b}$ is the temperature of the base.
2. At the tip of the fin ($x=L$): the condition could be adiabatic (${\frac {dT}{dx}}=0$) or a convective boundary condition depending on the design.

Scale Analysis

In this step, we apply scale analysis by estimating the orders of magnitude of key parameters, simplifying the governing heat equation based on dominant terms.

a) Characteristic Length

The characteristic length $L$ is the dimension along the length of the fin. The thickness $t$ and width $w$ define the cross-sectional area of the fin as $A_{c}=wt$, and the perimeter is given by $P=2(w+t)$. These geometrical properties are crucial in describing the heat dissipation characteristics of the fin.

b) Biot Number (Bi)

The Biot number ($Bi$) is a dimensionless parameter that compares the conductive resistance within the fin to the convective resistance at its surface. It is defined as:

$Bi={\frac {hL}{k}}$

where $h$ is the convective heat transfer coefficient, $L$ is the characteristic length, and $k$ is the thermal conductivity of the fin material. When $Bi\ll 1$, the conductive heat transfer within the fin is much faster than the heat loss to the surrounding fluid, leading to a small temperature gradient inside the fin. This allows for the assumption of a quasi-one-dimensional temperature profile, meaning the temperature varies primarily along the length $x$ of the fin.

c) Dimensionless Temperature and Variables

To further simplify the governing equations, dimensionless variables are introduced:

• Dimensionless position: $\xi ={\frac {x}{L}}$
• Dimensionless temperature:

$\theta (\xi )={\frac {T(x)-T_{\infty }}{T_{0}-T_{\infty }}}$

where $T_{0}$ is the base temperature of the fin, and $T_{\infty }$ is the ambient temperature. Substituting these dimensionless variables into the heat conduction equation transforms it into its dimensionless form:

${\frac {d^{2}\theta (\xi )}{d\xi ^{2}}}-{\frac {hP}{kA_{c}}}\theta (\xi )=0$

d) Fin Parameter

To simplify the equation further, we introduce the fin parameter $m^{2}$:

$m^{2}={\frac {hP}{kA_{c}}}$

This reduces the heat conduction equation to:

${\frac {d^{2}\theta (\xi )}{d\xi ^{2}}}-m^{2}L^{2}\theta (\xi )=0$

The fin parameter $m^{2}$ represents the relationship between heat conduction and convection, playing a critical role in determining the temperature distribution along the fin.

Step 3: Solution and Boundary Conditions

The general solution to this second-order differential equation is:

$\theta (\xi )=C_{1}\cosh(mL\xi )+C_{2}\sinh(mL\xi )$

Boundary conditions are applied as follows:

• At the base of the fin ($\xi =0$): $T(0)=T_{0}$ or $\theta (0)=1$.
• At the tip of the fin ($\xi =1$): Depending on the thermal condition at the tip, either:

 * Insulated tip: ${\frac {d\theta }{d\xi }}=0$ at $\xi =1$.
 * Convectively cooled tip: $-kA_{c}{\frac {dT}{dx}}=hA_{\text{tip}}(T-T_{\infty })$ at $\xi =1$.

Step 4: Key Non-dimensional Groups

The effectiveness and efficiency of the fin are typically expressed using non-dimensional groups, such as the following:

• Fin effectiveness ($\epsilon$):

$\epsilon ={\frac {Q_{\text{fin}}}{hA_{b}(T_{0}-T_{\infty })}}$

where $Q_{\text{fin}}$ is the total heat transfer and $A_{b}$ is the base area of the fin.

• Dimensionless fin length ($mL$): This parameter determines the rate of exponential temperature decay along the fin's length. For long fins, the effectiveness is proportional to the perimeter-to-area ratio ${\frac {P}{A_{c}}}$, indicating that a higher perimeter-to-area ratio enhances heat dissipation.

Authors

• Devyan Mishra

Conclusion

Through scale analysis, we can simplify the complex heat transfer problem by focusing on key parameters like the Biot number $Bi$ and the fin parameter $mL$. In particular, for rectangular fins, the effectiveness depends significantly on the fin's geometry (i.e., the perimeter-to-area ratio) and material properties (such as thermal conductivity). Solving the governing equation allows for the prediction of the temperature profile along the fin, and the total heat transfer, under various boundary conditions.