COMPLEX ENGINEERING PROBLEM

HEAT & MASS TRANSFER

Submitted by: Muhammad Adnan Khan

Reg : 205452

BEME-F-20
Supervised by: Dr. Zabd-ur-Rehman

Department of Mechanical Engineering,

Air University Aerospace & Aviation Campus
Kamra

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 Abstract
This project investigates the heat transfer performance of a tapered rectangular fin made of aluminum,
designed to enhance thermal management efficiency. The fin features a gradual reduction in cross-
sectional area along its length, facilitating improved heat dissipation while optimizing material usage.
Under steady-state conditions, the temperature distribution along the fin is analyzed both analytically
and numerically, considering fixed boundary conditions such as the base temperature of 100°C and
ambient temperature of 25°C. The study evaluates key performance metrics including the rate of heat
transfer (Q), fin efficiency (η), and fin effectiveness (ε). Various boundary conditions are examined to
understand their influence on heat transfer characteristics. The impact of the tapered geometry on
thermal performance is discussed, highlighting the advantages of unconventional fin shapes in
enhancing heat dissipation efficiency within practical constraints. The findings aim to provide insights
into the design optimization of fins for effective thermal management applications.

Keywords: tapered rectangular Fins, Heat transfer, Heat dissipation

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 Introduction
Introduction

Efficient heat dissipation from hot surfaces often relies on extended surfaces, or fins, which enhance
convective cooling by increasing the area exposed to ambient fluid. In this report, we analyze a
rectangular fin with a taper angle, exploring its temperature distribution, heat‐transfer rate,
efficiency, and effectiveness under natural‐convection conditions. The fin material, geometry, and
boundary conditions are chosen to reflect a realistic engineering application, and both analytical
approximations and numerical simulations are used for validation.

Natural Convection Overview

Natural (free) convection arises when temperature differences in a fluid generate buoyancy‐driven
flow, moving warmer, less‐dense fluid upward and cooler, denser fluid downward. This mechanism
requires no external fan or pump. In many practical situations—such as electronic cooling or outdoor
heat exchangers—natural convection is the dominant mode of heat removal. When a fin is hotter
than the surrounding air, the local air layer warms, rises, and is replaced by cooler air, thereby
sustaining a convective heat‐transfer process.

Material Thermal Conductivity

The thermal performance of a fin depends heavily on the conductivity of its material. A high‐
conductivity material, like aluminum, rapidly transports heat from the base to the tip, ensuring a
more uniform temperature distribution and minimizing the conduction resistance along the fin.
Consequently, aluminum is widely used for fins that must transfer significant heat loads under
natural‐convection conditions.

Tapered Rectangular Fins

A rectangular fin with a taper angle has a base thickness larger than its tip thickness, reducing
material volume while maintaining ample surface area near the base where heat flux is highest. As
the fin narrows toward the tip, the local cross‐section shrinks, leading to lower conduction resistance
near the root but diminished surface area toward the tip. This geometry can improve efficiency
compared to a uniform‐thickness fin while using less material. In this study, each fin tapers linearly
from its maximum base thickness down to a smaller tip thickness at a fixed angle.

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 Role of Fins in Heat Sinks

In many electronic and mechanical systems, fins are integrated into a heat sink—a component
designed to absorb heat from a hot source and transfer it to the surrounding air. By providing
extended surfaces, fins promote thermal interaction between the solid base and the fluid, enhancing
overall cooling performance. A well‐designed heat sink balances fin spacing, height, thickness, and
taper to optimize both natural‐convection flow patterns and material usage.

Problem Statement

This report focuses on a rectangular fin with a taper angle mounted on a baseplate. The fin is
made of aluminum, maintained at a fixed base temperature of 100 °C, while the ambient air remains
at 25 °C under natural‐convection conditions. The fin’s total wetted surface area is constrained to
0.01 m². We aim to determine:

• The steady‐state temperature distribution along the tapered fin.

• The total heat‐transfer rate from one fin and from an array of ten identical fins.
• The fin efficiency, defined as the ratio of actual heat transfer to the idealized maximum if the
fin were uniformly at the base temperature.
• The fin effectiveness, representing the heat transfer relative to a hypothetical bare area equal
to the fin’s base cross section.

Fixed Parameters:

• Surface area of the fin (A): 0.01 m²

• Fin material: Aluminium (Thermal conductivity, k = 205 W/m·K)
• Base temperature (T_b): 100°C
• Ambient temperature (T_∞): 25°C
• Convective heat transfer coefficient (h): 50 W/m²

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Methodology
Geometry and Material

• Fin Shape: A straight rectangular plate whose width decreases linearly from a larger base
value to a smaller tip value, forming a triangular side profile.
• Key Dimensions:
o Fin length, L: 0.2382 m
o Base width, Wb: 0.03593 m
o Tip width,Wt: 0.00603 m
o Tapered angle 3.58 degree

• Material: Aluminum, chosen for its high thermal conductivity. Properties (conductivity,
density, specific heat) are treated as constant.

SolidWorks Thermal Analysis Procedure

1. 3D Modeling
o Create separate 3D parts for the fin and base.
o Assemble them to ensure full, accurate contact between fin root and base.
2. Material Assignment
o Open the Simulation tab and start a new Thermal Study.
o Assign aluminum (or the chosen material) to both fin and base using the SolidWorks
material library.
3. Apply Thermal Loads
o Base Temperature: Fix the base face at the specified temperature (e.g., 100 °C).
o Convection: Assign natural‐convection boundary conditions to all exposed surfaces
(sides and tip of the fin) with the appropriate ambient temperature (e.g., 25 °C) and
convective coefficient.
4. Mesh Generation
o Generate a finite‐element mesh using a maximum element size of to capture
temperature gradients accurately throughout the fin geometry.
5. Run the Simulation
o Solve the steady‐state Thermal Study.
o Wait for convergence; the solver will compute temperature and heat‐flux fields.
6. Post-Processing Results
o Temperature Distribution: Plot isotherms and extract temperature values along the
fin length.
o Heat Flux: Review heat‐flux vectors on the fin surfaces.
o Thermal Gradients: Examine temperature gradients through cross-sectional cut-
plots of the fin.

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Python based calculation
On Python we implements a one‐dimensional finite‐difference solution for a linearly tapered
rectangular fin under both insulated and convective tip conditions. First, the fin’s length, base width,
tip width, thickness, and material properties are defined. The domain is discretized into 100 nodes,
and at each node the local cross‐section and perimeter are computed based on the linear taper. A
tridiagonal coefficient matrix is assembled to enforce energy conservation at each interior node, while
the base temperature is fixed and either an adiabatic or convective boundary condition is applied at
the tip. Solving this linear system yields the temperature profile along the fin for both cases. The script
then calculates the base heat‐transfer rate Q, fin efficiency , and fin effectiveness ε by comparing the
actual heat flow to the ideal convective capacity.

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Direct Comparison of Python vs. SolidWorks Results

1. Python (1D Finite‐Difference)

o Heat Transfer Rates
▪ Insulated tip: ≈20.10W
▪ Convective tip: ≈20.09
o Efficiencies
▪ Insulated tip: ≈0.536
▪ Convective tip: ≈0.536
o Effectivenesses
▪ Insulated tip: ≈42.62
▪ Convective tip: ≈42.62

The efficiencies of the insulated tip and convective tip cases are nearly identical because the tip heat
loss is negligible due to the very small cross-sectional area at the tip of the tapered fin.

2. SolidWorks (3D Thermal‐Flow)

o If you probe the numerical output in SolidWorks, you will see a base heat‐flow on the
order of 20 W for both tip conditions (since the tip perimeter is very small in this
geometry).
o The temperature contours along the fin (in Kelvin) also convert to nearly the same
Celsius values:
▪ Base is at 373 K=100 °C
▪ The tip temperature in SolidWorks is about 358–360 K≈85–87 °C which falls
in line with the Python profile of roughly 86 °C at the tip.

Why There Are Small Differences

• Dimensionality
o Python uses a one‐dimensional assumption (through‐thickness gradients are
neglected).
o SolidWorks solves the full 3D conduction and natural‐convection around edges
and the tip, so local convection coefficients ( h ) can vary slightly.
• Mesh Resolution
o In Python, we used a uniform 100‐node grid with Δx=0.00238
o In SolidWorks, the mesh is unstructured and may be finer near the tip or edges,
capturing subtle temperature gradients that a uniform 100‐node 1D grid cannot.
• Boundary‐Condition Implementation
o In Python, the “convective‐tip” condition is lumped into a single algebraic row.
o SolidWorks naturally couples the tip face to the ambient air, including radiation (if
enabled) and the local flow field. If radiation is off, SolidWorks still resolves the 3D
airflow around that small tip face, which causes a slightly different local heat transfer
than the idealized 1D model.

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Results

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Conclusion
In this project, a detailed thermal analysis of a tapered (triangular) fin was conducted using both numerical
modeling (1D finite difference method in Python) and simulation (SolidWorks). The study focused on
temperature distribution, heat transfer rate, efficiency, and effectiveness under two different tip boundary
conditions: insulated and convective. The results demonstrated that the temperature distribution along the
fin is nearly identical for both tip conditions, with base heat transfer rates of approximately 20.10 W for the
insulated tip and 20.09 W for the convective tip. Fin efficiency and effectiveness were also nearly the same
for both cases, around 53.6% and 42.6, respectively. This negligible difference is attributed to the minimal
heat loss from the fin tip due to its small cross-sectional area. Overall, the findings highlight that for slender,
tapered fins, the tip condition has little impact on overall performance, and heat transfer is predominantly
governed by the base region. These insights are valuable for optimizing fin design in practical thermal
management applications.

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