Analyzing an array of fins is a little more complex than that of a single fin.
However, once you understand the terms that are involved as well as the geometry, solving
problems involving arrays are considerably less difficult.
So let's start by looking at a picture of a fin array.
These fins actually go back into the wall like that.
If we look at this fin, let's start by identifying the different lengths.
So from this out of the base is the length of the fin, how far it goes back is the width
of the fin, and this part right here is the thickness of the fin.
So these are considered rectangular fins that are attached to a base that is at some temperature,
so our base here is at some temperature T_b.
And as always, we want to find the heat transfer from the array as well as the efficiency.
So in this screencast, we're going to calculate the heat transfer, and here we have to take
into account the heat transfer from the fins themselves, as well as that from the base.
However, it's not just as easy as separating the two, adding the heat transfer of the base,
and adding that to the number of fins times their heat transfer.
First of all, each fin has an efficiency, we call that eta_f, that has to be taken into
account, and secondly, the heat transfer from the base only occurs where the fins aren't,
and that's what we call the exposed base or the prime base.
So when you see in any equation this A_b, that refers to as I said the exposed base.
Let's look at those two parts of the heat transfer.
So the heat transfer for all the fins, so we'll call this q of the fins and this is
total, is equal to the number of fins that we'll call N times the efficiency of each
fin, times the area of each fin times our theta_b, which is the temperature of the base,
minus T infinity.
And T infinity is the temperature of the air or whatever fluid flowing over the base, and
we also call that the bulk temperature.
So it's easy to count the fins, and we can figure out the area, but let's start with
the efficiency, and the efficiency is based on what type of condition you have.
The efficiency of a fin is equal to the heat transfer from that fin divided by the maximum
heat transfer, and this maximum heat transfer is that which would occur if the entire fin
were at the temperature of the base, creating the maximum driving potential for convection.
In other words, this q_max is equal to our convective heat transfer coefficient times
the area of the fin times Tb minus T infinity.
The efficiency can then be calculated using the appropriate expression for the tip condition.
However, for rectangular fins, you can use a corrected length, and use the equation for
an adiabatic tip, and in fact the area is considered negligible if either h times the
thickness divided by the thermal conductivity of the fin, or if it's a pin fin, h times
the diameter of the pin fin divided by 2k, if either of those is less than 0.0625.
So why would we want to assume an adiabatic tip?
Well, the different expressions for the efficiencies of different tip conditions can be quite complex,
and the adiabatic tip is one of the easiest ones.
So let's look at this corrected length.
So our Lc, if you have this rectangular fin, is the length of the fin plus the thickness
divided by two.
Or, if you have a pin fin, its corrected length is its length plus its diameter divided by
4.
And so this corrected length is used in the calculation of the area of the fin, so the
area of the fin is equal to 2 times that w times Lc, or it can be equal to pi times D
times Lc.
So what is the efficiency of an adiabatic tip?
So the efficiency is equal to the hyperbolic tangent of this expression, which I'll define
m, times this corrected length, and that entire expression is divided by m times the corrected
length.
And our m is equal to 2 times the convective heat transfer coefficient divided by the thermal
conductivity times the thickness of the fin to the 1/2, or if it's a pin fin, it's equal
to 4 times h divided by k times D all raised to the 1/2.
So now, we have our heat transfer from the fins, which is just equal to this N times
this area of the fin, which we just found, times this efficiency which we just found,
times again our Tb minus T infinity.
So now what we have to do is find the heat transfer from the base.
So since that heat transfer mode is convection, we use Newton's Law of Cooling, which is the
convective heat transfer coefficient, times the exposed area of the base, and again Tb
minus T infinity.
And one of the things that we assume when calculating heat transfer from an array of
fins is that the h for the fin is the same as the h for the base.
So all we have to do now is calculate the area of the base, and what that's equal to
is the total area of the base, subtracting from that is the number of the fins times
the area of the fin that is attached to the base.
So for example, it's the thickness times the width for a rectangular fin, or it's pi times
D for a pin fin.
So now all we do is add the total heat transfer from the fins to that heat transfer from the
exposed base, and we end up with the heat transfer from an array of fins.