This experiment determined the temperature profile of a vertical steel fin both experimentally and theoretically. A steel fin was placed on a heating element set to 365 K and allowed to reach steady state. Thermocouples measured the decreasing temperature profile from the base to tip of the fin. The experimental results varied slightly from the theoretical profile mainly due to errors in temperature measurements at the fin surface. Comparing the experimental and theoretical profiles showed good agreement between the two methods of analyzing the fin's temperature change.

1. Temperature Profile Analysis of a Vertical Steel Fin
At the completion of this lab, the temperature profile of a vertical
fin was both experimentally and theoretically determined. A
heating element was set to 365 K heated a 1018 steel fin was
placed on top. After reaching steady state, the fin cooled as
expected with the base temperature identical to the heating
element, and the fin tip nearly ambient temperature with a
decreasing profile along the length of the fin. The experimental
temperature profile varied slightly from the theoretical expected
results due mainly to errors in the temperature measurements.
ABSTRACT
The rate of heat transfer is determined by 𝑄 = ℎ𝐴∆𝑇 where h is the heat transfer
coefficient, and A is the total surface area of the object. In order to create a larger rate
of cooling, either h, A, or the temperature differential must be increased. However, in
most applications, both the heat transfer coefficient and temperature differential are
constrained and will remain relatively constant. Therefore, increasing the surface area
is the most common way to increase the rate of cooling. The most efficient way of
increasing surface area is adding a fin, or multiple fins to the object. Figure 2 shows
some examples of objects with cooling fins including a heat sink (a), an air cooled
engine (b), and extremities on the human body (c). The objectives for this experiment
were to experimentally and theoretically determine the temperature profile for a 90°
steel angle fin and then compare the two.
PROCEDURE
This lab utilized a Benchmark brand hot-plate and a 1018 steel 90° angle fin with
a thermal conductivity (k) of 51.9 W/m·K. The fin was placed on top of the heating
element. The hot-plate was set to a temperature of approximately 365 Kelvin and
thermocouples were attached to the fin at varying positions along the length of the fin
in order to measure the temperature. When the temperature readings stabilized, the
temperature was recorded with reference to its position on the fin. This was repeated
3 times at the same height measurements but differing positions on the fin.
The theoretical temperature profile was determined using Eq.1. This equation
was used because there was convection from the fin on all sides including the tip of the
fin.
𝑇(𝑥) =
cosh 𝑚 𝐿−𝑥 +
ℎ
𝑚𝑘
sinh 𝑚 𝐿−𝑥
cosh 𝑚𝐿 +
ℎ
𝑚𝑘
sinh 𝑚𝐿
𝑇𝑏𝑎𝑠𝑒 − 𝑇∞ + 𝑇∞ Eq.1
Although the fin was actually 90° steel angle, the vertical plate formulas for m (Eq.2)
and L (Eq.3) were used because the angle could be thought of as a plate if it were
flattened out.
𝑚 =
2ℎ
𝑘𝑡
Eq.2
𝐿 𝑐 = 𝐿 +
𝑡
2
Eq.3
The convective heat transfer coefficient (h) was determined using the Rayleigh
and Nusselt numbers to determine natural convection. The temperature of the film
was determined as the difference between the surface temp of the plate and the temp
of the surroundings. The kinematic viscosity (v) and the Prandtl number (Pr) were
determined from the temperature of the film and used in Eq.4 to determine the
Rayleigh number.
𝑅𝑎 𝐿 =
𝑔β(𝑇𝑠−𝑇∞)𝐿3
𝑣2 𝑃𝑟 Eq.4
The Nusselt number was used in order to determine the heat transfer coefficient, h.
𝑁𝑢 = 0.825 +
0.387𝑅𝑎1/6
1+
.492
𝑃𝑟
9/16 8/27
2
=
ℎ𝐿
𝑘
Eq.5
OBJECTIVES AND THEORY RESULTS AND DISCUSSION
Figure 1: Experimental Temperature Profile for 1018 Steel Fin
The experimental temperature profile shown in Figure 1 displays the
temperature drop from the base of the fin to the tip. The further away from the
base, the cooler the temperature became until the temperature at the tip was
nearing ambient temperature. The experimental values are compared to the
experimental values in Figure 3 which shows that the experimental values were
accurate to the expected values.
Figure 3: Experimental and Theoretical Temperature Results
The error associated with the experimental values stems largely from the
fact that each temperature was taken on the surface of the fin, rather than
inside the metal. The thermocouple was not embedded into the material and
therefore picked up some interference from the ambient temperature, and
surface temperature was cooler than the internal temperature. Also, the steel
itself had minor oxidation on the surface and was not polished to a smooth
finish which also created some error in the temperature measurement. Another
factor contributing to a small error could be that the fin was placed directly on
top of the heat plate and it was not permanently attached to the base, and it
occasionally was bumped onto cooler parts of the hot-plate during the
experiment.
CONCLUSION
KEVIN LESTER
BRAD WHEELER
Figure 2: Various Examples of Cooling Fins
This experiment determined that the theoretical method for finding
temperatures at a given point along a fin is viable and accurate compared to the
experimental results. However, if this experiment were to be repeated, in order
to minimize error, the fin should be welded perpendicular to a horizontal plate
and then placed on the heating element. Also the thermocouples should be
insulated from the ambient air and the fin should be a smooth finish with no
oxidation to avoid any interference to the heat transfer.