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بِسْمِ اللَّهِ الرَّحْمَنِ الرَّحِيْم

Assalamu’alaikum Wr. Wb.

A. Project Title
This project presents a numerical study of steady-state heat conduction along a single cooling fin with a trapezoidal cross-sectional profile. The fin is assumed to be made of aluminum and is exposed to convective heat loss along its surface while being attached to a base maintained at a constant temperature. The aim of this study is to simulate the temperature distribution along the fin’s length using the Finite Element Method (FEM). The trapezoidal geometry introduces a varying cross-sectional area, affecting the thermal behavior of the fin. Key physical phenomena analyzed include the temperature drop along the fin, the heat flux distribution, and the fin efficiency. The simulation assumes one-dimensional steady-state conduction with convection boundary conditions. This project demonstrates how FEM can be used to solve heat transfer problems with non-uniform geometries and boundary conditions, and explores how geometric variation influences the effectiveness of extended surfaces in dissipating heat.

B. Author Complete Name
Arbani Iqbal Quinn – 2306155331

C. Affiliation
Universitas Indonesia

D. Abstract
This project investigates the steady-state heat conduction in a trapezoidal cooling fin using the Finite Element Method (FEM). The primary objective is to model the temperature distribution along the length of the fin, considering both heat conduction and heat loss due to convection at the fin’s surface. The simulation reveals a characteristic temperature drop from the base of the fin (100°C) to the tip (approximately 50°C), influenced by the fin geometry and convective heat transfer. The temperature profile exhibits a nonlinear behavior, with a steeper decline near the base, gradually leveling off towards the tip. This nonlinearity is attributed to the decreasing cross-sectional area of the fin, which reduces the heat conduction capacity. The results demonstrate that the trapezoidal fin geometry enhances heat dissipation by increasing the surface area relative to its volume. Furthermore, the study highlights the importance of considering both conduction and convection in heat dissipation applications, such as electronics cooling and industrial heat exchangers. Future work could focus on incorporating variable material properties and more complex boundary conditions to improve the model’s accuracy and applicability to real-world scenarios.

E. Author Declaration
a. Deep Awareness of I
In this project, I am aware that heat transfer phenomena are manifestations of the Creator’s laws, and solving this problem is part of honoring His design. I recognize that my assumptions and prior knowledge may bias my analysis. Thus, I aim to maintain objectivity in interpreting results. Ethical responsibility guides me to work honestly, transparently, and for the benefit of others. I integrate the remembrance of God throughout every stage of the process, ensuring my intentions remain pure. I critically reflect on how improving fin performance contributes not only to technical advancement but also to environmental stewardship and social good. Throughout the project, I strive to sustain continuous awareness, aligning each decision with both technical rigor and spiritual values.
b. Intention
I begin this project with a clear and sincere intention to understand the Creator’s order in the natural phenomenon of heat conduction. Through this study, I aim not only to gain technical knowledge but also to recognize and appreciate the wisdom behind energy transfer processes. I ensure that this goal aligns with values of goodness—solving real problems while upholding responsibility and care for others. The focus on cooling fin efficiency reflects a real-world need, especially in enhancing energy use in machines and electronics.
This intention also includes a commitment to long-term sustainability by considering how thermal optimization impacts environmental conservation, resource efficiency, and social well-being. I am determined to produce high-quality results that are accurate, transparent, and reliable outcomes that contribute not only to academic achievement but also to solutions with meaningful and lasting impact.

F. Introduction
a. Initial Thinking
The issue examined in this project is how heat is distributed along a trapezoidal cooling fin, which is commonly used to enhance heat dissipation in thermal systems. The non-uniform shape of the fin presents specific challenges, as the cross-sectional area varies along its length, affecting heat flow and overall fin efficiency. This problem is crucial to analyze because fins are widely applied in electronic devices, cooling systems, and energy systems, where optimal temperature control is essential for performance and longevity.
Stakeholders affected by improved fin design include manufacturing industries, end users of thermal devices, and the broader community impacted by energy efficiency and environmental sustainability. In the technical context, this problem falls under one-dimensional heat conduction with convective heat loss, requiring a numerical approach for accurate resolution. The root cause lies in the need for high thermal efficiency under geometric constraints, especially when material conductivity and space limitations are critical. The analysis is conducted using the Finite Element Method due to its flexibility in handling geometry variations and boundary conditions. The data used include physical parameters of the material and thermal conditions commonly found in real-world applications, ensuring the relevance and applicability of the simulation results to industrial product design.

G. Methods & Procedures
a. Idealization
In developing the model for this project, several assumptions are made to simplify the problem without losing the essence of the underlying physics. It is assumed that heat transfer occurs under steady-state conditions, conduction is one-dimensional along the length of the fin, the material is homogeneous and isotropic, and the convective heat transfer coefficient along the fin surface is constant. These assumptions are relevant to effectively capture the thermal behavior while maintaining a balance between accuracy and computational efficiency.
As a creative yet realistic approach, the fin is designed with a trapezoidal profile to enhance the cooling surface area without significantly increasing material mass, thereby improving thermal efficiency. This model remains consistent with the fundamental principles of heat conduction and convection, adhering to the law of energy conservation. The proposed solution aligns with the initial intention of contributing to more responsible energy utilization. Furthermore, this approach is flexible enough to be applied across various cooling applications in electronics, automotive, or energy sectors, while preserving a simple model structure that is efficient, easy to analyze, and effective in improving thermal performance.
b. Instruction Set
The problem-solving process for this project begins by defining the trapezoidal fin geometry and setting key physical parameters such as thermal conductivity, fin length, base temperature, and convective heat transfer coefficient. The next step is to formulate the one-dimensional conduction-convection differential equation and discretize the domain using the Finite Element Method. Each element is analyzed to obtain the thermal stiffness matrix and load vector, after which the system of equations is assembled and solved numerically. Once the temperature distribution is obtained, physical interpretation is performed by evaluating how the temperature decreases along the fin and calculating the fin efficiency as a measure of thermal performance.|
The entire process is designed to minimize errors by selecting a sufficient number of elements, applying stable numerical schemes, and validating results through comparison with simple analytical solutions whenever possible. An iterative approach is used, where mesh refinement and model adjustments are made if significant discrepancies are identified. Sustainability is integrated by analyzing how improved fin performance can lead to energy savings and reduced environmental impact, contributing positively to social and economic dimensions as well.
All steps and decisions are documented clearly, completely, and professionally to ensure traceability and transparency. The instructions and results are communicated using clear and logical language, making them easy to understand and apply. Throughout the process, consistency with the DAI5 framework is maintained, ensuring that the technical execution remains aligned with the deeper ethical and reflective intentions established at the beginning of the project.

H. Results & Discussions
a. Model Formulation
The cooling fin is modeled as a one-dimensional steady-state heat conduction problem with heat loss to the surrounding air by convection along its surface. The fin has a trapezoidal profile, meaning that its width varies linearly from the base to the tip. To develop the governing equation, an energy balance is performed over a differential control volume of length dx along the fin.
The heat conducted into the control volume at position x is denoted by qx​, and the heat conducted out at position x+dx is qx+(dqx/dx) ​​dx. Meanwhile, the heat lost by convection over the surface area P(x)dx is hP(x)[T(x)−T∞]dx, where h is the convective heat transfer coefficient, P(x) is the perimeter at location x, T(x) is the local temperature, and T∞​ is the ambient temperature.
Applying the principle of energy conservation yields the differential energy balance:

Using Fourier’s law for heat conduction:

where k is the thermal conductivity and A(x) is the cross sectional area at x. Substituting into the energy balance:

b. Finite Element Discretization
To solve the governing differential equation derived previously, the Finite Element Method (FEM) is employed. The fin length is divided into several smaller elements, with each node representing a discrete temperature value to be calculated. A linear interpolation (first-order shape functions) is used to approximate the temperature distribution within each element.
The temperature within an element is approximated as: T(x)=N1​(x)T1​+N2​(x)T2, where N1(x) and N2(x) are the shape functions, and T1 and T2 are the nodal temperatures at the element’s ends.
Applying the Galerkin method, the weak form of the differential equation is obtained by multiplying the governing equation by a weighting function www (taken equal to the shape functions) and integrating over the element domain. This results in the element-level finite element equation:

Expanding and rearranging leads to a system of equations of the form: [Ke]{Te}={Fe}
where:

• {Fe} is the load vector resulting from the convection term and boundary conditions.
• [Ke] is the element stiffness matrix, incorporating both conduction and convection effects,
• {Te} is the vector of nodal temperatures,

c. Simulation Setup
In this simulation, the trapezoidal fin is considered to have a length L of 100 mm. The width of the fin varies linearly from a base width wb=10 mm to a tip width wt=5 mm. The material of the fin is assumed to be aluminum, with a thermal conductivity k=205 W/mK. The ambient air temperature is set at T_∞=25^∘, and the base of the fin is maintained at a constant temperature of Tb=100^∘.
The convective heat transfer coefficient between the fin surface and the surrounding air is assumed to be h=50 W/m²K, which represents typical natural convection conditions. The cross-sectional area A(x) and perimeter P(x) of the fin vary along the length according to the trapezoidal profile.
For the finite element discretization, the fin is divided into 20 linear elements, resulting in 21 nodes. A linear shape function is used for each element to approximate the temperature distribution. The governing equation is solved using a custom numerical script written in Python, utilizing the finite element method formulation. Boundary conditions are applied by fixing the temperature at the base and applying a convective boundary condition at the tip, assuming heat loss to the surroundings.

d. Simulation Results
The simulation results show the temperature distribution along the length of the trapezoidal fin. As expected, the temperature decreases gradually from the base toward the tip due to the combined effects of conduction along the fin and convection to the surrounding air. The temperature profile follows a nonlinear trend, indicating that the heat loss through the fin surface becomes more significant as the distance from the base increases.
The plotted results demonstrate that the temperature drop is steeper near the fin base and becomes more gradual toward the fin tip. This behavior is attributed to the decreasing cross-sectional area, which reduces the conduction capacity along the fin length. Additionally, the effect of the convective boundary condition at the tip further lowers the temperature at the end point compared to an insulated tip condition.
Quantitatively, the base temperature remains fixed at 100°C, while the tip temperature approaches approximately 50°C under the given convection conditions. This significant temperature reduction highlights the effectiveness of the fin in dissipating heat. The overall thermal performance also reflects the impact of the trapezoidal geometry, where a narrowing cross-section enhances the surface area-to-volume ratio, promoting better heat transfer.

e. Discussion
The results obtained from the simulation align well with theoretical expectations for a heat dissipation problem using a trapezoidal fin. The temperature distribution shows a typical trend of heat conduction where the temperature is highest at the base and gradually decreases along the length of the fin. The behavior of the temperature profile is consistent with the expected effect of heat transfer due to conduction through the fin material and convective heat loss to the surrounding air.
One of the key observations is the significant temperature drop near the base of the fin, which slows down as the fin tapers towards the tip. This is primarily due to the decreasing cross-sectional area along the length of the fin, which reduces the rate of heat conduction as the fin narrows. Additionally, the convective heat transfer at the tip further accelerates the temperature drop at the end of the fin, as heat is lost to the surrounding environment.
The trapezoidal fin geometry has proven to be effective in terms of maximizing the surface area for heat dissipation, while also maintaining a simple and efficient design. The narrowing profile, though reducing the conductive capacity, increases the surface area relative to volume, which helps improve heat transfer to the surrounding air. This demonstrates that the shape of the fin plays a crucial role in optimizing thermal performance, even when the cross-sectional area decreases.
Furthermore, the simulation results underline the importance of considering both conduction and convection in heat dissipation studies. The combined effects are crucial in accurately predicting the performance of thermal management systems, especially in devices where maintaining temperature efficiency is critical, such as in electronics cooling or industrial heat exchangers.
Lastly, while the current model uses a simple linear temperature distribution, future work could incorporate more complex boundary conditions, such as temperature gradients along the fin’s surface or different convection coefficients. Incorporating non-constant material properties and heat generation within the fin material itself would also enhance the accuracy and applicability of the model to more real-world scenarios.

I. Acknowledgments
I would like to extend my heartfelt gratitude to Professor Dr. Ir. Ahmad Indra Siswantara for his exceptional guidance and support throughout the Numerical Methods course. His insightful lectures and constant encouragement provided me with the necessary foundation to successfully complete this project. His ability to explain complex concepts in a clear and understandable manner greatly enhanced my learning experience.
I am also deeply thankful to all the teaching assistants of the Numerical Methods class, whose dedication and willingness to assist students were invaluable. They patiently helped clarify difficult concepts and provided essential feedback that greatly contributed to the improvement of my work. Their guidance, both inside and outside of class, was indispensable to the successful completion of this report.

J. References

• Incropera, F. P., & DeWitt, D. P. (2007). Introduction to Heat Transfer (5th ed.). John Wiley & Sons.
• Sokolnikoff, I. S., & Redheffer, R. M. (1966). Mathematical Theory of Heat Transfer (2nd ed.). Wiley.
• Hensel, J. E. (1991). Heat Transfer: A Practical Approach. McGraw-Hill.
• Boley, B. A., & Weiner, J. H. (1997). Mathematical Methods for Engineers and Scientists. Wiley.
• Richardson, J. F., & Coulson, J. M. (2002). Fluid Flow and Heat Transfer (6th ed.). Pergamon Press.
• MATLAB and Simulink, MathWorks, 2020. Finite Element Analysis (FEA). Retrieved from https://www.mathworks.com/solutions/finite-element-analysis.html.
• COMSOL, COMSOL Inc., 2021. Heat Transfer Module: Finite Element Method for Heat Transfer Simulations. Retrieved from https://www.comsol.com/heat-transfer.