Fin (extended surface)
Encyclopedia
In the study of heat transfer
, a fin is a surface that extends from an object to increase the rate of heat transfer to or from the environment by increasing convection
. The amount of conduction
, convection
, or radiation
of an object determines the amount of heat it transfers. Increasing the temperature
difference between the object and the environment
, increasing the convection heat transfer coefficient, or increasing the surface area
of the object increases the heat transfer. Sometimes it is not economical or it is not feasible to change the first two options. Adding a fin to an object, however, increases the surface area and can sometimes be an economical solution to heat transfer problems.
With these assumptions, the conservation of energy can be used to create an energy balance for a differential cross section of the fin.
Fourier’s law states that
,
where
is the cross-sectional area of the differential element. Therefore the conduction rate at x+dx can be expressed as
Hence, it can also be expressed as
.
Since the equation for heat flux is
then
is equal to
where
is the surface area of the differential element. By substitution it is found that
This is the general equation for convection from extended surfaces. Applying certain boundary conditions will allow this equation to simplify.
where P is the perimeter of the cross-sectional area. Thus, the general equation for convection from extended surfaces with constant cross-sectional area simplifies to
.
The solution to the simplified equation is
where
and
The constants
and 
can be found by applying the proper boundary conditions. All four cases have the boundary condition 
for the temperature at the base. The boundary condition at 
, however, is different for all of them, where L is the length of the fin.
For the first case, the second boundary condition is that there is free convection at the tip. Therefore,
which simplifies to
Knowing that
,
the equations can be combined to produce
and 
can be solved to produce the temperature distribution, which is in the table below. Then applying Fourier’s law at the base of the fin, the heat transfer rate can be found.
Similar mathematical methods can be used to find the temperature distributions and heat transfer rates for other cases. For the second case, the tip is assumed to be adiabatic or completely insulated. Therefore at x=L,
because heat flux is 0 at an adiabatic tip. For the third case, the temperature at the tip is held constant. Therefore the boundary condition is:
For the fourth and final case, the fin is assumed to be infinitely long. Therefore the boundary condition is:
The temperature distributions and heat transfer rates can then be found for each case.
,
where
is the fin cross-sectional area at the base. Fin performance can also be characterized by fin efficiency. This is the ratio of the fin heat transfer rate to the heat transfer rate of the fin if the entire fin were at the base temperature.
The fin efficiency is defined as
in this equation is equal to the surface area of the fin. Fin efficiency will always be less than one. This is because assuming the temperature throughout the fin is at the base temperature would increase the heat transfer rate.
The third way fin performance can be described is with overall surface efficiency.
,
where
is the total area and 
is the sum of the heat transfer rates of all the fins. This is the efficiency for an array of fins.
s in cars and heat exchanger
s in power plants. They are also used in newer technology such as hydrogen fuel cells. Nature has also taken advantage of the phenomena of fins. The ears of jackrabbits and Fennec Foxes act as fins to release heat from the blood that flows through them.
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...
, a fin is a surface that extends from an object to increase the rate of heat transfer to or from the environment by increasing convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....
. The amount of conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....
, convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....
, or radiation
Radiation
In physics, radiation is a process in which energetic particles or energetic waves travel through a medium or space. There are two distinct types of radiation; ionizing and non-ionizing...
of an object determines the amount of heat it transfers. Increasing the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
difference between the object and the environment
Natural environment
The natural environment encompasses all living and non-living things occurring naturally on Earth or some region thereof. It is an environment that encompasses the interaction of all living species....
, increasing the convection heat transfer coefficient, or increasing the surface area
Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...
of the object increases the heat transfer. Sometimes it is not economical or it is not feasible to change the first two options. Adding a fin to an object, however, increases the surface area and can sometimes be an economical solution to heat transfer problems.
Simplified case
To create a simplified equation for the heat transfer of a fin, many assumptions need to be made:- Steady stateSteady stateA system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...
- Constant material properties (independent of temperature)
- No internal heat generation
- One-dimensional conduction
- Uniform cross-sectional area
- Uniform convection across the surface area
With these assumptions, the conservation of energy can be used to create an energy balance for a differential cross section of the fin.
Fourier’s law states that
,where
is the cross-sectional area of the differential element. Therefore the conduction rate at x+dx can be expressed asHence, it can also be expressed as
.Since the equation for heat flux is
then
is equal towhere
is the surface area of the differential element. By substitution it is found thatThis is the general equation for convection from extended surfaces. Applying certain boundary conditions will allow this equation to simplify.
Uniform cross-sectional area
For all four cases, the above equation will simplify because the area is constant andwhere P is the perimeter of the cross-sectional area. Thus, the general equation for convection from extended surfaces with constant cross-sectional area simplifies to
.The solution to the simplified equation is
where
and
The constants
and can be found by applying the proper boundary conditions. All four cases have the boundary condition for the temperature at the base. The boundary condition at , however, is different for all of them, where L is the length of the fin.For the first case, the second boundary condition is that there is free convection at the tip. Therefore,
which simplifies to
Knowing that
,the equations can be combined to produce
and can be solved to produce the temperature distribution, which is in the table below. Then applying Fourier’s law at the base of the fin, the heat transfer rate can be found.Similar mathematical methods can be used to find the temperature distributions and heat transfer rates for other cases. For the second case, the tip is assumed to be adiabatic or completely insulated. Therefore at x=L,
because heat flux is 0 at an adiabatic tip. For the third case, the temperature at the tip is held constant. Therefore the boundary condition is:
For the fourth and final case, the fin is assumed to be infinitely long. Therefore the boundary condition is:
The temperature distributions and heat transfer rates can then be found for each case.
| Case | Tip condition (x=L) | Temperature distribution | Fin heat transfer rate |
|---|---|---|---|
| A | Convection heat transfer |  |
 |
| B | Adiabatic |  |
 |
| C | Constant Temperature |  |
 |
| D | Infinite Fin Length |  |
 |
Fin performance
Fin performance can be described in three different ways. The first is fin effectiveness. It is the ratio of the fin heat transfer rate to the heat transfer rate of the object if it had no fin. The formula for this is,where
is the fin cross-sectional area at the base. Fin performance can also be characterized by fin efficiency. This is the ratio of the fin heat transfer rate to the heat transfer rate of the fin if the entire fin were at the base temperature.The fin efficiency is defined as
in this equation is equal to the surface area of the fin. Fin efficiency will always be less than one. This is because assuming the temperature throughout the fin is at the base temperature would increase the heat transfer rate.The third way fin performance can be described is with overall surface efficiency.
,where
is the total area and is the sum of the heat transfer rates of all the fins. This is the efficiency for an array of fins.Fin uses
Fins are most commonly used in heat exchanging devices such as radiatorRadiator
Radiators are heat exchangers used to transfer thermal energy from one medium to another for the purpose of cooling and heating. The majority of radiators are constructed to function in automobiles, buildings, and electronics...
s in cars and heat exchanger
Heat exchanger
A heat exchanger is a piece of equipment built for efficient heat transfer from one medium to another. The media may be separated by a solid wall, so that they never mix, or they may be in direct contact...
s in power plants. They are also used in newer technology such as hydrogen fuel cells. Nature has also taken advantage of the phenomena of fins. The ears of jackrabbits and Fennec Foxes act as fins to release heat from the blood that flows through them.