Problem Statement:
This report examines the effects of fin geometry on the total rate of heat dissipation from identical pipes with two distinct fin arrangements. Also examined is the optimal design for maximizing heat dissipation and for minimizing net weight in each case. Each pipe has the same diameter and surface temperature, one with circular radial fins and the other with rectangular axial fins. Figure 1 shows the two finned pipes, and listed below are material constants as well as dimensional constraints.
Figure 1: Fin Arrangement 1 (left), and 2 (right)
- Material: Aluminum (k = 180 W/mK , ρ = 2700 kg/m3 )
- Pipe Surface Temperature: Tb = 120 °C
- Environment Temperature: Tꝏ = 25 °C
- Combined Heat Transfer Coefficient: h = 60 W/m2 K
- Pipe Length: L = 1 m
- Maximum Diameter (including fins): D = 6 cm
- Minimum Rate of Heat Dissipation: Q = 5 kW
- Maximum Weight of Assembly: Mmax = 3 kg
- Minimum Fin Thickness: t = 2 mm
- Minimum Fin Spacing: s = 2 mm (for Arrangement 1, this distance represents
Assumptions:
- The fin tips are considered to be adiabatic, and corrected fin length is used to determine heat transfer.
- Outer diameter of pipe (not fins) is assumed to be 15 mm in all cases, router = 13 m
- Thickness of pipe itself is assumed to be 2 mm, rinner = 13 m
- Heat dissipation from unfinned surface of pipe considered negligible
Results and Discussion:
Part 1: Characterize Effects of Varied Parameters
For the annular fins in Arrangement 1 , heat dissipation rate is determined by the product of fin efficiency (Equation 1 & 2) and maximum heat dissipation (Equation 3).The efficiency equation involves solutions of the modified Bessel function. Arrangement 1 has straight fins with insulated tips and constant thickness, and rate of heat dissipation is determined directly from Equation 3. L is corrected length, k is thermal conductivity, A is cross section area area, p is fin perimeter, Tb is temperature at the pipe, Tꝏ is environment temperature, and tanh is the hyperbolic tan function. For both arrangements, m = (2h/kt)0.5 , where h is the combined heat transfer coefficient, and t is fin thickness.
Equation 1: Efficiency of Annular Fins of Rectangular Profile
Equation 2: Heat Dissipation from Each Fin
Equation 3: Heat Dissipation from Fin with Adiabatic Tip
Using these equations, the effects of varying fin thickness, fin spacing, and fin length on total rate of heat dissipation will be investigated. In each case, the other two parameters will be held constant. Figure 2 shows the control dimensions for both fin arrangements. A simplified cross-sectional view of Arrangement 2 with a single fin is shown, and total number of fins will be determined from the pipe’s circumference by assuming a 2 mm gap between the base of each.
Figure 2: Control Dimensions for Arrangement 1 (left) and Arrangement 2 (cross section)
Figure 3: Total Heat Dissipation vs Fin Thickness, Arrangement 1
Figure 4: Total Heat Dissipation vs Fin Spacing, Arrangement 1
Figure 5: Total Heat Dissipation vs Fin Length, Arrangement 1
Figure 6: Total Heat Dissipation vs Fin Thickness, Arrangement 2
Figure 7: Total Heat Dissipation vs Fin Spacing, Arrangement 2
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Figure 8: Total Heat Dissipation vs Fin Length, Arrangement 2
Figures 3 – 8 show the effects of varying thickness, spacing, and fin length for each of the two arrangements. For both arrangements, increasing thickness decreases total heat dissipation. This result may seem counterintuitive since increased thickness raises heat transfer from individual fins, but it also means that there are less total fins.
The effect of increased spacing between fins is decreased total heat dissipation for both arrangements. This result is logical since larger gaps between fins means that there are less total fins along the 1 m pipe. When the length of fins (outer radius) is increases, the heat dissipation rate also increases, for both arrangements. In both cases, longer fins result in greater fin area which is directly proportional to heat transfer rate. In summary, varying any of these three parameters has the same effect on heat dissipation rate for both arrangements.
Figure 9: Total Weight vs Fin Thickness, Arrangement 1
Figure 10: Total Weight vs Fin Spacing, Arrangement 1
Figure 11: Total Weight vs Fin Length, Arrangement 1
Figure 12: Total Weight vs Fin Thickness, Arrangement 2
Figure 13: Total Weight vs Fin Spacing, Arrangement 2
Figure 14: Total Weight vs Fin Length, Arrangement 2
Figures 9 – 14 show the effects of varying these three parameters on the total weight of the assemblies. These results are intuitive; increasing thickness and fin length results in greater total weight, while increasing spacing results in lower total weight.
Part 2a: Optimize for Heat Dissipation
Arrangement 1:
Now the ideal thickness, spacing, and length values will be determined for maximizing heat dissipation rate. Since it is known how Q is affected by each, the natural starting point is to consider the design configuration that has minimum thickness, minimum spacing, and maximum length (2 mm, 2 mm, and 15 mm respectively). This gives a net weight of 2.21 kg, and total heat dissipation rate of 6.59 kW. The weight threshold is not reached, and this is the maximum possible heat that can be achieved under these spacing length and thickness constraints.
Arrangement 2:
Determining the ideal specifications for maximizing Q from this assembly is similarly straightforward, as net weight also does not approach the 3 kg as Q is maxed. For fins of 2 mm thickness, 15 mm long, and spaced 2 mm apart, net weight is only 1.75 kg. However, this maximum rate of heat dissipation is calculated to be only 4.18 kW, which means that under no configurations will Arrangement 2 meet the 5 kW threshold.
Part 2b: Optimize for Minimum Weight
Arrangement 1:
A configuration with fin thickness 2 mm, spacing 2 mm, and outer radius 30 mm has a total Q of 6.59 kW. This means that some heat transfer can be sacrificed for reduced weight.
To determine which parameter(s) should be adjusted to make Q = 5 kW and reduce weight, the approximate slopes of Figures #, #, and # are considered. It can be seen that relative to spac1ing and thickness, fin length clearly has the least effect on the magnitude of heat dissipation. Therefore fin length will be reduced to make the assembly lighter. Again using Excel, it is found that with minimum thickness and spacing and an outer radius of 27 mm, Arrangement 1’s rate of heat dissipation will be 5 kW.
Arrangement 2:
Since this assembly cannot meet Q requirements, it is inadvisable that it should be configured for optimizing weight. But if this is still the goal, then fin length is again the parameter that should be reduced for lower weight. As explained, this is because shortening fins is the most efficient way to decrease mass while suffering minimal heat losses. For example, the weight of configuration with 2 mm thickness and 2 mm spacing is reduced by 35 % if the outer radius is decreased from 30 mm to 22.5 mm.
All calculations for Part 2 were performed in the appended Excel sheet using the same equations/cells as in Part 1, but are not explicitly shown.
| Arrangement 1 | Maximize Q | Minimize M |
| Outer Radius (mm) | 30 | 27 |
| Fin Thickness (mm) | 2 | 2 |
| Fin Spacing (mm) | 2 | 2 |
Table 1: Optimal design dimensions for Arrangement 1
| Arrangement 2 | Maximize Heat Dissipation | Minimize Weight |
| Outer Radius (mm) | 30 | 22.5 |
| Fin Thickness (mm) | 2 | 2 |
| Fin Spacing (mm) | 2 | 2 |
Table 2: Optimal design dimensions for Arrangement 2
Appendix:
Jacob's Portfolio 