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title: Infinitely Long Fin Analysis
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# Infinitely Long Fin Analysis
Unit 1, Lecture 13

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## Infinitely Long Fin Analysis

- Course: Heat and Mass Transfer (DI05019071)
- Unit 1: Conduction
- Lecture 13: The Infinitely Long Fin

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Welcome back. Having derived the general fin equation, we now apply boundary conditions. Today we tackle the case of the infinitely long fin.
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## Lecture Agenda

- 1. Recap of the General Solution
- 2. Defining Boundary Conditions for an Infinite Fin
- 3. Solving for Constants C1 and C2
- 4. Temperature Distribution Profile
- 5. Heat Transfer Rate (Q_fin)

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We will use the boundary conditions to find our constants, derive the exact temperature profile, and calculate the heat removed by the fin.
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## The General Solution Recap

- Governing Equation: d²θ/dx² - m²θ = 0
- General Solution: θ(x) = C1 * e^(mx) + C2 * e^(-mx)
- Where: θ(x) = T(x) - T∞
- And m = √(hP/kAc)

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Remember this exponential solution. The actual physical behavior of the fin is governed by C1 and C2, which depend on the fin's length and tip.
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## Boundary Condition 1: The Fin Base

- At the base of the fin (attached to the wall):
- x = 0
- Temperature is equal to the base temperature: T(0) = Tb
- In terms of excess temperature:
- θ(0) = Tb - T∞ = θb
- Therefore: θb = C1*e^0 + C2*e^0 => θb = C1 + C2

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The first boundary condition is always at the base. The temperature there is fixed by whatever surface the fin is attached to.
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## Boundary Condition 2: The Infinite Tip

- For a very long fin, as x approaches infinity (x → ∞):
- The fin eventually cools down to the ambient fluid temperature.
- T(∞) = T∞
- In terms of excess temperature:
- θ(∞) = T∞ - T∞ = 0

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If a fin is infinitely long, it has enough surface area to dissipate all the heat it conducted. The tip temperature just equals the fluid temperature.
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## Evaluating Constant C1

- Apply Boundary Condition 2 to the general solution:
- θ(∞) = C1 * e^(m*∞) + C2 * e^(-m*∞) = 0
- Since e^(-m*∞) → 0, the second term vanishes.
- We are left with: C1 * e^(∞) = 0
- Since e^(∞) is infinity, for the equation to hold true, C1 must be 0.
- Result: C1 = 0

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Because the temperature cannot blow up to infinity at the end of the fin, the coefficient for the positive exponential term must be zero.
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## Evaluating Constant C2

- Substitute C1 = 0 into Boundary Condition 1:
- θb = C1 + C2
- θb = 0 + C2
- Result: C2 = θb
- Now substitute C1 and C2 back into the general solution.

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With C1 eliminated, C2 is simply the excess temperature at the base of the fin.
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## Temperature Distribution Equation

- θ(x) = θb * e^(-mx)
- Or in terms of absolute temperatures:
- (T(x) - T∞) / (Tb - T∞) = e^(-mx)
- This indicates an exponential decay of temperature along the length of the fin.

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The temperature drops exponentially. The rate of this drop is governed by 'm'. A larger m means a faster drop in temperature.
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## Heat Transfer Rate (Q_fin)

- How much heat does the fin transfer?
- Method 1: Integrate convection over the entire surface.
- Method 2: Heat entering the base by conduction must equal total heat convected.
- Use Fourier's Law at x = 0:
- Q_fin = -k * Ac * (dT/dx) at x=0

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It's mathematically much easier to evaluate the conduction at the base than to integrate convection over the entire infinite length.
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## Calculating Q_fin

- We know: θ(x) = θb * e^(-mx)
- Differentiate: dθ/dx = -m * θb * e^(-mx)
- At x = 0: (dT/dx)_x=0 = -m * θb
- Substitute into Fourier's Law:
- Q_fin = -k * Ac * (-m * θb) = k * Ac * m * θb

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We take the derivative of our temperature profile, evaluate it at x=0, and plug it into Fourier's Law.
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## Summary

- For an infinite fin, tip temperature T(∞) = T∞.
- Temperature profile is an exponential decay: θ(x) = θb * e^(-mx).
- Heat transfer rate: Q_fin = √(hPkAc) * (Tb - T∞).
- This is the theoretical maximum heat a fin of this cross-section can transfer.

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The infinitely long fin represents an ideal, providing the maximum possible heat transfer for a given base and cross-section.
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## Next Lecture Preview

- Topic: Efficiency and Effectiveness of Fins
- How do real, finite fins perform compared to our infinite ideal?
- We will define the metrics to evaluate fin performance.

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Real fins aren't infinitely long. Next time, we'll see how to evaluate the performance of practical, finite fins using Efficiency and Effectiveness.
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