4.2.
Heat Transfer with High-Finned Trufin Tubes
4.2.1. Fin Temperature Distribution and Fin Efficiency
1. Temperature Distribution in Fins. The
temperature in a fin is not constant, due to the
resistance to conductive heat transfer in the fin
metal. A typical temperature profile in a fin is
shown in Fig. 4.12.
The details of calculating the temperature
distribution are quite complex and will not be
given here; the most comprehensive reference
on this subject is the book, "Extended Surface
Heat Transfer" by Kern and Kraus (3). The
results depend upon a number of parameters,
including fin geometry (shape, height, and
thickness), fin material, and outside fluid
temperature and heat transfer coefficient. It is
also necessary to make a number of assump-
tions; for example, most analyses assume that
the outside fluid has a constant bulk
temperature and a constant heat transfer
coefficient at all points on the fin surface. This
is known not to be true, but the real state of
affairs is not well understood and would
introduce great complexity into the analysis if
one tried to be completely rigorous. As a practical matter, the results obtained from the simplified analysis
seem to be consistent with experience and lead to acceptable designs.
The subject of fin efficiency was discussed in Section Iand curves for fin efficiency and fin resistance were
given for low-finned Trufin. Since the values for high fin were not given, the method of obtaining the values
will be repeated.
2. Fin Efficiency and Resistance. The fin efficiency, Φ, is the ratio of the total heat transferred from the real
fin in a given situation to the total heat that would be transferred if the fin were isothermal at its base
temperature. For the kinds of fins that are considered here, a good equation to use over the range of
interest is:
1
Φ= (4.1)
m2 do
1+
3 dr
where
216
2
m=H (4.2)
æ 1 ö
çç + R fo ÷÷k wY
è ho ø
Equation (4.1) is actually based upon fins of uniform thickness, whereas the fins on Wolverine high-finned
trufin are actually slightly thicker at the base and thinner at the tips. The error is small and in fact the
Wolverine fins are slightly more efficient than this equation indicates.
The geometrical variables are defined in Fig. 4.2 and ho and Rfo are respectively the actual convective heat
transfer coefficient and the actual fouling resistance on the fin side, based on the fin area. To gain an
appreciation of the probable magnitude of Φ in a typical problem, consider the following example:
Type H/R tube, 3003 aluminum:
dr = 1.00 in.
do = 1.875 in.
H = 0.437 in.
s = 0.076 in.
Y = 0.015 in.
110 Btu/hr ft °F
2
kw =
10 Btu/hr ft °F
2
ho =
R = 0.0
Then:
æ 0.437 ö 2
m=ç ft ÷ = 0.439
è 12 ø æ 1 hrft ° F öæ
2
Btu öæ 0.015 ö
çç ÷÷çç110 ÷ç ft ÷
è 10 Btu øè hrft 2 ° F ÷øè 12 ø
1
Φ= = 0.919 , i.e., 91.9% fin efficiency
(0.439) 2 1.875
1+
3 1.000
There are small differences between the nominal dimensions and the actual dimensions , and some
variation from lot to lot in the latter. See Section 6 for details. Nominal dimensions will be used in the
examples in this Section.
As we will observe later, this efficiency is, if anything, biased towards the low side of most applications.
Cop per fins have a higher thermal conductivity and would give a higher Φ. (Coppernickel (90/10) would
give Φ = 0.730 under otherwise identical conditions, but is not commonly used for high-finned tubes.)
Thicker fins (our example used the thinnest available) would give higher efficiencies. The film heat
transfer coefficient was typical of atmospheric air under nominal operating conditions; an extreme value of
20 Btu/hr ft°F would give Φ = 0.862.
A quantity somewhat more directly useful in design calculations is the "Fin Resistance", Rfin, defined as:
217
é 1 − Φ ùé 1 ù
A fin = ê Aroot ú ê + R fo ú (4.3)
êë A fin + Φ úû ë ho û
where Aroot is the surface area of a unit length of plain(unfinned) tube between the fins and Afin is the heat
transfer area of all of the fins on a unit length of tube. Continuing with the example above, we can
compute the value of Rfin as follows:
Aroot = π ç ft ÷(11 fins per inch )(12 inches / ft )ç
æ1 ö æ 0.076 ö
÷ = 0.219 ft / ft of length
2
è 12 ø è 12 ø
æ 1 ö 2 ü ìïæ fins öæç 2 sides ö÷æ 12in ö üï
ìπ
[
A fin = í (1.875) 2 − (1) 2 ç ] ÷ ft ý × íç11 ÷ç ÷ çç ÷÷ ý = 3.62 ft 2 / ft of length
î4 è 144 ø þ ïîè in. øè fin øè ft ø ïþ
é 1 − 0.919 ù é 1 ù hr ft 2 ° F
R fin = ê 0.219 úê + 0ú = 0.00827
ë 3.62 + 0.919 û ëê10 Btu / hr ft ° F
2
úû Btu
which corresponds to an effective heat transfer coefficient for the fins only of 121 Btu/hr ft °F. This may be
2
compared to a typical value of h,, for air-cooled exchangers of 10 Btu/hr ft °F, which indicates that the fin
2
resistance is only a small part of the total resistance to heat transfer.
The fin resistance then can be directly incorporated into the equation for the overall heat transfer
coefficient as follows:
1 1 ∆x Ao æA ö 1æA ö
= + R fo + R fin + + R fi çç o ÷÷ + çç o ÷÷ (4.4)
U o ho k w Am è Ai ø hi è Ai ø
The value of Rfin may be calculated for any desired case by using Eqns. 4.1, 4.2 and 4.3.
4.2.2. Effect of Fouling on High-Finned Trufin
As a matter of consistency and principle, the analysis to this point has steadfastly incorporated the term
Rfo, the resistance due to fouling on the finned surface. As a matter of fact, fouling on high-finned Trufin
with air on tlie fins is seldom a serious problem, unless there is extensive deposition of material as from
massive corrosion (indicating a poor material choice) or a heavy dust storm or ingestion of debris. In the
latter cases, continued operation is out of the question, and there is no alternative but to shut down and
remove the obstructions. Under normal conditions, the continuous movement of air past the surface tends
to minimize deposition of sand and dust, and such deposits as may form can usually be removed by
occasionally running a compressed air jet over the sur face. Accordingly, Rfo is usually taken as zero for
high finned Trufin applications.
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4.2.3. Contact Resistance in Bimetallic Tubes
In Type L/C Trufin, there is an internal liner of a metal other than the 3003 aluminum of the outer tube and
fins. The two metals will sometimes be in imperfect contact with one another, leading to an additional
resistance to the flow of heat. Generally at low temperatures of the metal-to-metal interface, the liner is
exerting a positive pressure upon the aluminum finned tube. But as the tube temperature rises, the
aluminum expands more rapidly than the liner and a definite gap develops. The gap is filled with air,
introducing a substantial additional resistance to the flow of heat.
There have been several studies, both experimental and analytical, made of this problem and the results
have been surveyed by Kulkarni and Young (4). This paper and its references should be consulted for
details and predictive methods, but it is desirable to summarize here the main findings:
1. At the fabrication temperature of approximately 70°F, there is a positive contact pressure of about
400 psi for a stainless steel liner inside aluminum. Presumably a similar value would exist for
other liner metals.
2. This results in a contact resistance of about 0.00005 hrft °F/Btu, based upon the contact surface.
2
This is negligible for any practical application.
3. At the point of zero contact pressure (which occurs at a bond temperature of about 200-215°F in
the steel/aIuminurn case), the bond resistance has been measured to be about 0.0002
hrft °F/Btu. This is still negligible for most applications.
2
4. At tube side fluid temperatures of 1000°F and air side temperatures of 200°F, the bond resistance
is computed to increase to values as high as 0.003 hrft °F/Btu (based on contact area) at air side
2
coefficients of 5 Btu/hrft °F (based on fin area) and 0.002 hrft °F/Btu for air side coefficients of 10
2 2
Btu/hrft °F. When the corresponding area ratios (say between 1: 10 and 1:20) are taken into
2
account, bond resistance is seen to be about 10-25 percent of the total resistance to heat transfer
and definitely needs to be considered in the design. However, it would not seem that a very
detailed calculation of the effect is in order unless many such high temperature cases are to be
handled.
The complete formulation of the overall heat transfer coefficient calculation for the bimetallic tube
with contact resistance is then:
1
U0 = (4.5)
1 æ ∆x A ö æ A ö æ ∆x A ö æA ö 1 æ Ao ö
+ R fo + R fin + çç w o ÷÷ + Rb çç o ÷÷ + çç w o ÷÷ + R fi çç o ÷÷ + çç ÷÷
ho è k w Am ø è Ab ø è k w Am ø 2 è Ai ø hi è Ai ø
æ ∆x w Ao ö
where çç ÷÷ is the wall resistance for the fin metal root, Rb is the bond resistance based
è k w Am ø1
æ ∆x w Ao ö
upon the bond contact area Ab, çç ÷÷ is the wall resistance for the liner tube, and the other
è k w Am ø 2
terms have their usual meaning.
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