A pin fin of uniform, cross-sectional area is fabricated of an aluminum alloy (160 W/m-K). The fin diameter is D=7 mm, and the fin is exposed to convective conditions characterized by 220 W/m²K. It is reported that the fin efficiency is 7, 0.55. Determine (a) the fin length & and the fin effectiveness ry. Account for tip convection.
(a) L mn
(b)
The Correct Answer and Explanation is :
To solve this problem, we need to use the principles of heat transfer in a fin, considering both the fin efficiency and effectiveness. First, let’s break down the problem into two parts:
Given data:
- Thermal conductivity of the material (k): 160 W/m·K
- Diameter of the fin (D): 7 mm (0.007 m)
- Convective heat transfer coefficient (h): 220 W/m²·K
- Fin efficiency (η): 0.55
- Fin effectiveness (ε): 7
(a) Finding the fin length (L)
The fin length can be found using the equation for fin effectiveness:
[
\epsilon = \frac{Q_{\text{fin}}}{Q_{\text{bare}}}
]
Where:
- ( Q_{\text{fin}} ) is the heat transferred by the fin, which depends on the fin’s length, material properties, and convection conditions.
- ( Q_{\text{bare}} ) is the heat transfer rate that would occur without the fin (i.e., with just the bare surface).
For an extended surface, we can use the relationship for the fin efficiency ((\eta)) in terms of the fin length ((L)):
[
\eta = \frac{\text{heat transfer rate with fin}}{\text{heat transfer rate without fin}}
]
For a pin fin, the effectiveness and efficiency are closely related to the geometry of the fin and the convective conditions, so we can use standard formulas to relate them.
(b) Fin effectiveness
The fin effectiveness ((\epsilon)) is the ratio of the heat transferred by the fin to the heat transferred by the bare surface (without the fin), considering the tip convection. When the fin length and convection conditions are determined, the fin effectiveness can be calculated.
Approach:
- Pin fin geometry and assumptions: We know that for a cylindrical pin fin, the area of heat transfer is proportional to the surface area, and the efficiency is a function of length and material properties.
- Convective heat transfer analysis: By considering the convective heat transfer coefficient, we can relate this to the fin’s effectiveness.
- Fin length (L): We’ll calculate the length from the formula involving the fin efficiency and geometry.
Now, let’s calculate the fin length and effectiveness based on the given conditions. I’ll calculate the fin length and effectiveness next.
Results:
- Fin Length (L): 0.086 meters (or 8.6 cm)
- Fin Effectiveness (ε): 0.48
Explanation:
(a) Fin Length Calculation:
To determine the fin length, we used the relationship for the efficiency of the pin fin:
[
\eta = \frac{\tanh(mL)}{mL}
]
where:
- (m) is the fin parameter, calculated as:
[
m = \sqrt{\frac{2h}{kD}}
]
Using the provided values for thermal conductivity ((k)), heat transfer coefficient ((h)), and fin diameter ((D)), we first computed (m), then used a numerical method to solve for the fin length that satisfies the given fin efficiency of 0.55.
The resulting fin length is approximately 0.086 meters, or 8.6 cm.
(b) Fin Effectiveness:
Next, we calculated the fin effectiveness using the formula:
[
\epsilon = \frac{1 + mL}{1 + mL + m^2 L^2}
]
The calculated fin effectiveness is approximately 0.48, which is quite a bit lower than the given value of 7. This discrepancy suggests that the given effectiveness value may involve some additional factors or modifications to the basic model.
This analysis assumes standard conditions and formulas for fin heat transfer and doesn’t account for any unusual surface conditions or advanced modifications to the fin’s geometry.
Would you like me to generate an image of the pin fin showing its key parameters?