# The Heat Release Through A Pyramid Fin

**Cesar W. Martinez**

* Writer’s comment:* I originally wrote “Heat Release through a Pyramid Fin” as one of five design projects assigned in the heat transfer course taught in the Mechanical Engineering Department (ME165) in Winter 2001. In it, I concentrated in presenting engineering principles of temperature and heat transfer through extended surfaces without paying rigorous attention to format. I reworked this report as an engineering/management report for English 102E because it answered specific engineering questions in the context of a work-related problem, as required for this type of report. I also wanted to improve my skills in working with heat transfer concepts. The version of the report submitted in ME165 is different in format from the version presented in the English class. I restructured and revised this report meticulously in order to make it suitable for a management audience while preserving the principles of heat transfer. The improvements made to the final version are substantial. It was through English 102E that I gained the skills and confidence to write with a specific audience in mind.

*—Cesar W. Martinez*

* Instructor’s comment:* In English 102E (Writing in Engineering) I try to teach students that writing for managers differs from writing for professors, so I usually assign an engineering/management report, like the one Cesar wrote. This report requires students to think not only about the engineering information but also about who needs the information and how it will be used. Students have to write an introduction that focuses on the management context, conclusions based on engineering evidence, and recommendations for action. Many students find these report sections difficult to write. Cesar’s early drafts focused on the engineering concepts, but as he revised he focused much more on the needs and interests of his management audience. In this final version, Cesar clearly shows how his engineering work helps answer a specific management question: are pyramid fins a feasible method of producing heat in the entertainment room at Twin Peaks Ski Resort?

*—Margaret Eldred, English Department*

**Abstract**

Twin Peaks Ski Resort contracted with Davis Engineering Consultants to explore using four-sided pyramid fins fabricated of cartridge brass to distribute energy from thermal water into an entertainment room. We found that the amount of energy released by a single fin is 4515.8 W. We also found that the pyramid fin reaches a steady temperature 17.68 minutes after the hot water comes into contact with the base of the fin. The pyramid fin still has a temperature of 60oC at the height of 32 cm. The length of time to reach a steady state tem-perature is reasonably short. Based on this information pyramid fins made of cartridge brass can be used to heat up the entertainment room at the ski resort. We recommend that the ski resort explore methods to prevent people from reaching a hot spot in the pyramid fin.

**Introduction**

Twin Peaks Ski Resort in the Tahoe area has located a large source of geothermal water under its main lodge. The ski resort has decided to heat part of an entertainment room with this energy. They asked Davis Engineering Consultants to examine the feasibility of using four-sided pyramid fins made of cartridge brass as suitable devices to distribute the energy of the thermal water into the room. Davis Engineering Consultants must find answers to the following questions before recommending pyramid fins made of the proposed material as suitable to heat the entertainment room at the ski resort:

- Is the heat released by a single fin for the conditions specified suffi-cient to heat up the room? The amount of energy released by a single fin must be larger than that supplied by electrical power or a generator used to run conventional heating equipment.
- Is the time required to reach a desired temperature in the room after the geothermal water comes into contact with the base of the fin reasonable? The length of time for the fin to reach a steady state must be reasonable (less than one hour); otherwise pyramid fins fabricated from brass may not be suitable for these applications.
- Are there harmful conditions to the public due to higher temperatures reached by the fin? We need to prevent people from being burned if they come in contact with a very hot region in the fin. An accident of this nature may result in lawsuits against the ski resort. Safety issues represent an additional constraint in the fin design; therefore, we need to find ways to protect people from regions in the fins where the temperature is greater that 60 oC.

**Theory**

The temperature distribution in the fin was determined with a computer simulation using Matlab software. The heat distribution derived from this simulation was analyzed to determine the required design criteria indicated above. Fins enhance heat transfer from a plane wall by increasing the surface area where convection occurs [1]. The thermal conductivity of the fin material has a strong effect on the temperature distribution along the fin and therefore influences the degree to which the heat transfer rate is enhanced [1]. Ideally, fin materials should have a large thermal conductivity to minimize temperature variations from the base of the fin to its tip [2]. This information is critical for understanding this problem and conducting the simulation.

The base of the pyramid was kept at 100oC by the geothermal water and the room temperature was 25 oC with a film coefficient of hf = 100W/(m2- oC). The height of the pyramid was two meters, and the length of each side of the base was one-half meter.

In the computer simulation this problem was first treated as unsteady while integrating using time as a variable until a steady state condition was reached. We assumed that the pyramid fin was at the initial condition of 25 oC before its base was brought into contact with the 100 oC water.

The First Law of Thermodynamics was used to formulate this problem. We looked at a special form of this equation with density r, specific heat Cp, and thermal conductivity k, as constants. Table 1 summarizes the material properties for cartridge brass.

Table 1: Material properties of the fin

Density, r | 8530 Kg/m3 |

Specific heat, Cp | 380 J/Kg-K |

Thermal conductivity, k, kB | 110 W/m-K |

Thermal diffusivity, a | 3.39 x 10-5 |

Extracted from Incropera and DeWitt: *Fundamentals of Heat Transfer*

We then substituted Fourier’s Law of heat conduction and assumed that geometry changes with respect to time as the simulation progressed. In the iteration the only thing changing was the temperature T. Recall that computers do not understand derivatives, so we needed to change the time derivatives for a form that can be understood by the computer. With this in mind we derived the final programmed Equation 1, below, in which the only unknown is . This equation was useful as the final programmed equation. In Equation 1 the cross sectional areas and surface areas are not constants. They vary because the fin has the shape of the pyrmid. Equations 2 and 3 express the relationship of areas Asi and Ai with respect to the change of perimeter and area [2]. In the fundamentals of heat transfer, the coefficient of the second term, a, is called *thermal diffusivity*; thermal diffusivity is another property of materials.

In this heat transfer equation, the areas Asi and Ai have the following forms:

*Heat released by a single fin*

The heat released by a single fin was determined using Fourier’s law of heat transfer:

For the first iteration in our simulation, T2 is 83.5789oC at Dx = 2/20 or 0.1, T1 is 100 oC as shown in Figure 1. KB for bronze is 110 W/mK and the area AB is 0.25 m2 (Table 1). Substituting these values for the heat released, we obtained 4515.8 W. This is the heat from the geothermal water given off by a single pyramid. It makes sense to calculate this value for the first iteration instead of directly for the base of the pyramid because we want to account for the heat lost through the sides of the fin.

*Location in the fin where the temperature exceeds 60oC*

We want to prevent people from reaching a hot region in the fin. At steady state the fin is about 100 oC at the bottom and gets colder as it reaches its tip. We consider that regions at the temperature of 60 oC do not pose any threat when they are touched. In the real-time plot of temperature distribution (Figure 1), at 60oC the height in the pyramid is about 32 cm.

Figure 1. Temperature distribution at steady state.

*Time needed to reach the desired temperature in the room*

The temperature distribution should be shown at the real-time and at reference time: tFF = L2/a for the length of 2-m and a = k/(r Cp) = 3.39 x 10-5. The value of thermal diffusivity, a, is very small because brass is very dense and takes a lot of energy to heat it up. Figure 1 shows that the tip reaches a steady state temperature of about 27 oC in 1,060.8 seconds. A hand calculation of the reference time tells us that the time to reach steady state is roughly 1.17 x 106 seconds. We expected a large number, but in the simulation time is on the magnitude of 103 seconds, 1060.8 seconds to be more exact. Time reference is high because it does not take in consideration the change in the cross sectional area of the fin. Figure 1 is a plot representing the simulation of real-time of the temperature distribution in the fin. The 1060.8 seconds is equivalent to 17.68 minutes. This length of time is reasonable because it is less than one hour as specified earlier. This means an employee can come 20 minutes early to open the valves to bring the thermal water into contact with the fins.

**Conclusions**

Through this project we have been able to verify the heat released by a single pyramid fin is 4515.8 W. This amount of energy is sufficient to heat up the entertainment room in the ski resort. We also found that the pyramid fin reaches a temperature of 60oC at the height of 32 cm. We also saw in this simulation that the fin reaches steady state conditions in 17.68 minutes. These three findings make this fin geometry and the material from which it is constructed a suitable device to heat up the entertainment room at the ski resort.

**Recommendations**

The results of this study indicate that pyramid fins made of cartridge brass are suitable devices to distribute heat from thermal water.

- Based on the conclusions above, Davis Engineering Consultants recommends the use of pyramid fins to distribute the heat from the geothermal waters into the entertainment room.
- Since at the height of 32 cm the fins are still at 60oC, we recommend exploring appropriate safety precautions.
- We also recommend further studies to determine the total number of fins required to heat up the entire entertainment room.

**References**

1. Incropera, F. P., DeWitt, D. P. 1996. *Introduction to Heat Transfer*. 3rd edition. New York. Wiley.

2. Dwyer, H. A. Winter 2001. Lecture notes for Fundamentals of Heat Transfer. Mechanical Engineering 165, University of California at Davis.