Annular Fins: How Circular Fins Move Heat Off Pipes, Tubes and Cylinders

Look closely at the cylinder of an old air-cooled motorcycle engine, the tubes of a refinery air cooler, the coil behind a window air conditioner or the heat pipe inside a computer cooler, and you will often see the same shape: thin rings or a continuous helix of metal standing out from a round tube. These are annular fins, also called circumferential, radial or circular fins. Their job is to give a hot (or cold) cylindrical surface more area in contact with the surrounding fluid, usually air, so that it can exchange heat faster than the bare tube could.

The idea is simple, but the details are not. A fin only helps if heat can conduct out along it faster than it is lost from its faces, and in an annular fin the area grows with radius, so the usual straight-fin formulas do not apply. The governing equation leads to modified Bessel functions, the efficiency has to be read from charts or computed numerically, and the real-world performance depends on things the simple theory ignores, such as how well the fin is bonded to the tube and how unevenly air flows over it.

This article covers what annular fins are, how their temperature distribution and efficiency are calculated, how they compare with straight fins and pins, where they are used, what limits their performance and what people commonly get wrong about them. It includes a worked example computed with the standard textbook solution, clearly labeled as hypothetical.

What an annular fin is

An annular fin is a flat ring attached around the outside of a tube or cylinder. In the simplest geometry, the fin has a constant thickness t, an inner radius r1 equal to the outer radius of the tube, and an outer radius r2. Heat conducts radially outward from the tube wall through the fin, and at the same time it is lost by convection from both flat faces of the ring (and, to a small extent, from its outer edge) into the surrounding fluid.

Wikipedia's article describes the defining feature this way: an annular fin is a fin whose cross-sectional area varies with radius. As you move outward from the tube, the circumference of the ring grows, so the area available for heat to conduct through grows, and the face area available for convection grows too. That geometric fact has two consequences. It makes the mathematics harder than for a straight fin of constant cross-section. And, as the Wikipedia article points out, it means that for the same radial length an annular fin offers more surface than a pin fin, which is why it can transfer more heat.

Annular fins come in several practical forms:

Strictly, only the first few are true annular fins, but the theory of the annular fin is the starting point for analyzing all of them.

Why fins work at all

The convection bottleneck

Heat leaving a solid surface into a gas is usually limited by convection. Newton's law of cooling says the heat flow per unit area is h times the temperature difference between the surface and the fluid, where h is the convection coefficient. For air moving over a surface, h is small compared with the ability of a metal to conduct heat. If the metal could conduct heat to a larger surface without much temperature drop, the total heat transfer would rise roughly in proportion to the added area.

That is what a fin does. It is an extended surface. The design trade-off is that as the heat travels out along the fin, the fin's temperature drops, so the outer parts of the fin are cooler than the base and exchange less heat per unit area. A fin made of a good conductor, kept thin enough to be light but thick enough to conduct, and not extended beyond the point of diminishing returns, can multiply the heat transfer of a bare surface many times.

Efficiency and effectiveness

Two numbers summarize fin performance, and they are defined differently enough that they are often confused.

Fin efficiency, η_f, is the ratio of the heat actually transferred by the fin to the heat that would be transferred if the entire fin were at the base temperature. An ideal fin made of a material with infinite conductivity would have an efficiency of 1. Real fins always have efficiency below one.

Fin effectiveness, ε_f, is the ratio of the heat transferred by the fin to the heat that would have been transferred from the base area the fin occupies, had the fin not been there. Effectiveness tells you whether adding the fin was worthwhile.

Overall surface efficiency, η_o, applies to an array: it is the total heat from all the fins and the bare tube between them, divided by the heat that would be transferred if the entire area (fins plus bare tube) were at the base temperature. Heat exchanger designers work mostly with η_o, because it combines fin efficiency with the fraction of area that is fin.

A highly efficient fin is not necessarily a useful one. A very short fin is nearly isothermal and so has an efficiency close to 1, but it adds little area. A very long fin has low efficiency because its outer portion is nearly at the fluid temperature and does nothing. The designer is looking for the size at which further increases in fin length or thickness stop paying for themselves in weight, cost or air-side pressure drop.

The governing equation

Assumptions

The classical analysis of an annular fin, as summarized in Wikipedia from Anthony Mills's textbook Heat Transfer, makes four main assumptions:

• the fin's thermal conductivity and other properties are constant
• there is no heat generation inside the fin
• conduction is one-dimensional, in the radial direction only, so the temperature does not vary across the fin's thickness
• the fin is at steady state

Two more assumptions are implicit in most textbook treatments and are worth stating, because they are the ones most often violated in practice: the convection coefficient h is the same over the whole fin surface, and the base of the fin is at a single uniform temperature equal to the tube's outer surface temperature.

The energy balance

Consider a thin ring of the fin between radius r and r + Δr. Heat conducts into the ring at its inner edge and out at its outer edge, and the difference is lost by convection from the ring's two faces. Using Fourier's law for the conducted heat, q = −k·dT/dr, and Newton's law of cooling for the convected heat, and letting Δr shrink to zero, gives a differential equation for the temperature T(r). For a fin of thickness t that loses heat from both faces, it can be written as

d/dr (r · dT/dr) − (2h/(k·t)) · r · (T − T∞) = 0

where T∞ is the fluid temperature. Some textbooks, and the Wikipedia article, write the coefficient using half-thickness or a single face; the physics is the same, but you must be consistent about whether t means full thickness. The combination

m = √(2h / (k·t))

is the fin parameter, with units of inverse length. A large m means the fin loses heat to the fluid quickly compared with how well it can conduct heat along itself: high h, low k or a thin fin. A small m means the fin conducts well relative to its losses.

Turning it into Bessel's equation

If you define a dimensionless temperature θ = (T − T∞)/(Tb − T∞), where Tb is the base temperature, and a dimensionless radius z = m·r, the equation becomes

z² · d²θ/dz² + z · dθ/dz − z² · θ = 0

This is the modified Bessel equation of order zero, as the Wikipedia article states. Its general solution is a combination of two standard functions:

θ = C1 · I0(m·r) + C2 · K0(m·r)

I0 is the modified Bessel function of the first kind of order zero, which grows with its argument. K0 is the modified Bessel function of the second kind of order zero, which decays. Their derivatives involve the first-order functions I1 and K1. These functions are tabulated in mathematical handbooks and are built into scientific software, so the "Bessel function" part of the problem is no longer the obstacle it once was.

The constants C1 and C2 come from the boundary conditions. At the base, θ = 1. At the tip, the usual simple assumption is that the outer edge is insulated (adiabatic), so dθ/dr = 0 at r = r2.

The efficiency formula

With an adiabatic tip, the fin efficiency works out to

η_f = [2·r1 / (m·(r2² − r1²))] × [K1(m·r1)·I1(m·r2) − I1(m·r1)·K1(m·r2)] / [I0(m·r1)·K1(m·r2) + K0(m·r1)·I1(m·r2)]

This is the exact expression, within the assumptions, for a constant-thickness annular fin. A homework set on annular fins by Enrico Nobile at the University of Trieste presents the same Bessel-function solution with m = √(2h/kt). One caution about the Wikipedia version of this formula: as rendered, the grouping in its denominator is easy to misread, so it is worth checking against a textbook before using it in calculations.

Dealing with the tip

Real fin tips are not insulated; their thin outer edge also loses heat. The standard engineering approximation, attributed to Harper and Brown, is to pretend the tip is insulated but extend the fin slightly to make up for the lost edge area. For a thin rectangular fin, lecture notes from Florida A&M and Florida State University's engineering college give the corrected length as Lc = L + t/2, which follows from Lc = L + Ac/P with cross-sectional area Ac = W·t and perimeter P ≈ 2W. For an annular fin the equivalent is a corrected outer radius, r2c = r2 + t/2, which a 2019 paper in the journal Energies lists among its standard nomenclature. A Gonzaga University paper argues that an appropriately defined corrected length makes the insulated-tip solution reproduce the convecting-tip heat transfer exactly. A Brazilian master's thesis, by contrast, argues that Harper and Brown's original goal was simply to reduce laborious computation and that for typical fin proportions the correction has little practical meaning. Both views agree that for thin fins the effect is small.

Why not just use a straight-fin formula?

For a straight rectangular fin with an insulated tip, the efficiency is simply tanh(m·Lc)/(m·Lc). It is tempting to apply that to an annular fin of the same radial length. The error depends on the radius ratio r2/r1. When the fin is short compared with the tube radius, the circumference barely changes from base to tip and the straight-fin result is close. When the fin extends to several times the tube radius, the area at the tip is much larger than at the base, more of the fin's area sits in its cooler outer region, and the annular fin's efficiency is noticeably lower than the straight-fin formula suggests. That is why annular fin charts are plotted against both a fin parameter and the radius ratio.

Charts, approximations and their history

Gardner's charts

Before computers, engineers read annular fin efficiencies from charts. The best known came from Karl A. Gardner's paper "Efficiency of Extended Surface," published in the November 1945 Transactions of the American Society of Mechanical Engineers (volume 67, number 8, beginning on page 621). Gardner solved the fin equations with Bessel functions for practically any extended surface whose thickness varies as a power of the distance along the fin, and presented curves of fin efficiency for straight fins, annular fins and spines. Versions of his annular fin chart, plotting efficiency against a combination of fin length, thickness and h/k, with separate curves for different radius ratios, appear in many heat transfer textbooks.

Schmidt's approximation

Because the Bessel-function expression was tedious to evaluate, approximate formulas became standard in heat exchanger design. The best known is due to Schmidt, published in 1949, who expressed the annular fin efficiency in terms of a hyperbolic tangent, like the straight-fin formula, but with an adjusted equivalent length that depends on the radius ratio. A source discussing it states that the Schmidt approximation gives fin efficiency within 0.01 of the exact value for efficiencies greater than 0.5. Hong and Webb proposed a modification in 1996 to improve accuracy. The same approach, the "equivalent circular fin" method, is used to estimate efficiencies of non-circular fins, such as the hexagonal or rectangular region around each tube in a plate-fin coil. Marcinkowski and Taler, for example, compared exact analytical methods, the equivalent circular fin method (Schmidt's method) and the sector method for complex-shaped fins.

Industrial methods

Heat exchanger design organizations developed their own methods. Heat Transfer Research, Inc. (HTRI) published report ESG-1, "Efficiency of Extended Surfaces," by T. Rozenman and J. Taborek in 1971. According to HTRI's abstract, it reviews the fin efficiency concept for straight longitudinal fins, provides simple analytical expressions for round and rectangular cross fins that can be coded instead of read from graphs, and notes that the least realistic of the usual assumptions is a uniform heat transfer coefficient over the fin surface. Nonuniform coefficients lower the efficiency below the theoretical value, and the report provides a correction factor.

A short timeline

The milestones mentioned in this article, gathered in one place:

The pattern is typical of engineering science. A clean analytical solution arrives early, practical approximations follow so that designers can use it quickly, and decades later numerical simulation is used to measure how far reality departs from the idealized model.

A worked example (hypothetical)

To see how the numbers behave, consider a fin that might be found on a small tube in an air-cooled device. All values are illustrative, not taken from any specific product.

• Tube outer radius r1 = 12.5 mm (a 25 mm diameter tube)
• Fin outer radius r2 = 25 mm (so the fin stands 12.5 mm proud of the tube)
• Fin thickness t = 1 mm
• Convection coefficient h = 50 W/(m²·K), a plausible value for air in moderate forced flow
• Base temperature 100 °C, air temperature 25 °C, so a 75 K difference

The corrected outer radius is r2c = 25.5 mm. The fin's convecting area, counting both faces, is 2π(r2c² − r1²), which is about 0.00310 m². The base area the fin occupies on the tube, 2π·r1·t, is about 0.0000785 m².

Using the exact Bessel-function efficiency for four common materials (conductivities chosen within the ranges listed in standard tables: pure copper about 401, pure aluminium about 237, a carbon steel value of 50 within its usual 36 to 54 range, and a stainless steel value of 16.3):

Several useful lessons fall out of this table.

First, for a short fin in air, even steel performs respectably. The convection coefficient of air is so low that the fin does not need a superb conductor to stay near its base temperature. Copper beats aluminium by barely 1 percent here, which, given aluminium's lower density and cost, is a large part of why aluminium dominates air-side fins.

Second, effectiveness is very high. Each fin transfers dozens of times more heat than the patch of tube it covers would have on its own. That is the normal situation in gas-side heat transfer, and it is why finned tubes are standard in air-cooled equipment.

Third, conductivity matters more as conditions get harsher. Holding the aluminium fin's geometry fixed and raising h shows how efficiency drops:

At h = 500, a level more typical of liquids or very high-velocity gas flows, a quarter of the fin's potential is lost to its own temperature drop.

Fourth, making the fin taller gives diminishing returns. Keeping the aluminium fin at 1 mm thick and h at 50:

The heat per fin keeps rising, because area grows roughly with the square of radius, but efficiency falls steadily. Whether the larger fin is worthwhile depends on the material cost, the weight, the space and, in a bank of tubes, the extra air-side pressure drop and fan power.

Finally, for comparison, treating this aluminium fin as a straight rectangular fin of the same 12.5 mm length and thickness gives tanh(mLc)/(mLc) of about 0.977, a little higher than the annular value of 0.967. The difference is small here because the fin is short relative to the tube radius; it grows for taller fins.

Annular fins compared with other fin shapes

The Wikipedia article's key comparison is that an annular fin transfers more heat than a pin fin of the same length because its surface area grows with distance from the base. The cost is that a disc blocks flow in a way a pin does not, so annular fins are best suited to flow that passes across the tube, between the fins, rather than along it.

Where annular fins are used

Air-cooled heat exchangers

The largest industrial use of annular and helical fins is in air-cooled heat exchangers, sometimes called fin-fan coolers, used in refineries, gas plants, power stations and chemical plants to cool process fluids with ambient air instead of water. A bank of finned tubes carries the hot fluid, and large fans push or pull air across the fins. Because the air-side coefficient is low and the tube-side coefficient (liquid or condensing vapor) is high, nearly all the thermal resistance is on the air side, and fins multiply the air-side area.

Published rules of thumb and manufacturers' literature give a sense of typical proportions, though actual designs vary and the governing American Petroleum Institute standard, API 661 (also ISO 13706), should be consulted for requirements. Typical fin spacing is about 7 to 11 fins per inch. A Hudson Products brochure describes fins as commonly about 5/16 to 1 inch high. An excerpt from Stephen Hall's Rules of Thumb for Chemical Engineers cites fin heights of 12.7 to 15.9 mm for 1-inch carbon steel tubes with aluminium fins, and suggests denser fins for steam condensers and water coolers and sparser fins for gas coolers and viscous hydrocarbon coolers. Helical, L-footed aluminium fins are the most common type, with steel fins reserved for very high temperatures.

Air-cooled engines

Engines that are cooled by air rather than by liquid coolant rely on fins cast or machined into their cylinders and heads, and those fins are annular or nearly so around the cylinder barrel. The National Advisory Committee for Aeronautics (NACA), the predecessor of NASA, studied cylinder fins intensively in the 1930s and 1940s because, in the words of one report, the power output of air-cooled engines is limited in part by the quantity of waste heat that can be removed. NACA analyses compared the fin proportions of aluminium, copper, magnesium and steel needed to dissipate maximum heat for a given width, weight and airflow, and wind-tunnel tests on electrically heated finned steel cylinders covered a range of fin pitches, thicknesses and widths. One later study found that an optimum fin width existed, beyond which further increase reduced heat transfer on the front of the cylinder. The ideal fin shapes were hard to cast or machine, which led to experiments with preformed fins that one report credited with practically unlimited fin dimensions and spacing and better conductivity than the usual integral steel fin. NACA also studied the baffles that guide air around and between the fins, with a 1943 report on improved baffle designs. This work fed into the cowlings developed by NACA from 1928 to 1929 for radial aircraft engines, which were designed to cut drag while still cooling the cylinders as well as an uncovered engine.

HVAC and refrigeration coils

The evaporator and condenser coils in air conditioners, heat pumps and refrigerators usually use continuous plate fins pierced by round tubes rather than individual annular fins. The region of plate around each tube behaves approximately like an annular fin, and the equivalent circular fin method lets designers estimate its efficiency with the annular fin theory. When such a coil operates below the dew point, moisture condenses on the fins and the effective efficiency changes. A 2006 ASME paper notes that wet-surface efficiency can differ from dry-surface efficiency by as much as 35 percent.

Heat sinks and heat pipes in electronics

Many tower-style computer coolers and LED lamp heat sinks stack thin circular or rectangular fins on heat pipes or a central post. Where the fins are round discs on a cylinder, annular fin theory applies directly. A Python textbook problem by Christian Hill, in Learning Scientific Programming with Python, uses SciPy's modified Bessel functions to compute the efficiency of an annular fin heat sink with very thin (0.1 mm) aluminium fins, reporting an efficiency of about 0.994 for its test case, a reminder that in low-h natural or light forced convection, thin aluminium fins can be nearly isothermal.

Other uses

Finned tubes also appear in boiler economizers, waste heat recovery units, oil coolers, gas-fired water heaters, transformer coolers and space heaters. A study of a finned-tube oil cooler by Karczewski applied Schmidt's approximation to an annular fin with an inner radius of 15 mm and reported efficiencies of about 0.93 from both Schmidt's method and an alternative correlation.

Manufacturing and materials

Choice of material

The worked example above shows why aluminium dominates air-side fins: its conductivity is high enough that fin efficiency stays near 1 in typical air flows, and it is light, inexpensive and easy to form. Copper is used where its higher conductivity matters, where corrosion resistance in certain environments is needed, or in combination with copper tubes. Carbon and stainless steel fins are used at high temperatures or in corrosive or erosive conditions, accepting lower efficiency. The NACA studies of the 1930s compared aluminium, copper, magnesium and steel cylinder fins on exactly these grounds.

The bond between fin and tube

A fin is only as good as its thermal connection to the tube. The classical theory assumes the base of the fin is at the tube's outer temperature, but in a wound or pressed fin there is a contact resistance at the joint. If the fin loosens over time through thermal cycling or corrosion, that resistance grows and performance falls. Air cooler fin types are largely distinguished by how they attach:

HTRI's ESG-1 abstract lists "fin bond" as one of the topics engineers must treat, alongside the efficiency itself.

Fin spacing and fouling

Packing fins more closely adds area, but each additional fin narrows the air passage, raises the pressure drop and makes the fins more likely to trap dirt, insects, lint and ice. Dense fins are appropriate where the air is clean and the tube-side coefficient is high; sparser fins are preferred where fouling is a concern or the tube-side coefficient is low. The rule-of-thumb preference for sparser fins on viscous hydrocarbon coolers reflects the latter: when the tube side is the bottleneck, extra air-side area buys little.

Limits of the simple theory

How large are the errors?

A 2019 open-access study in the journal Energies by Mladen Bošnjaković, Simon Muhič, Ante Čikić and Marija Živić, titled "How Big Is an Error in the Analytical Calculation of Annular Fin Efficiency?", used computational fluid dynamics to evaluate fin efficiency according to its original definition and compared the result with the analytical formula. They found that the numerically determined efficiency was greater than the analytical value by up to 12.3 percent. The two assumptions that contributed most to the error were that the fin base is at the same temperature as the outer tube surface and that the heat transfer coefficient is the same everywhere on the fin.

Nonuniform heat transfer coefficient

In a real finned tube in cross-flow, air strikes the front of the tube and fin roots at high velocity, a horseshoe vortex forms at the junction of fin and tube, and a wake develops behind the tube. The local h on a fin can therefore vary considerably from front to back and from root to tip. Since the classical efficiency formula uses a single average h, it cannot capture these effects exactly. HTRI's report singled this assumption out as the least realistic and provided a correction factor.

Two-dimensional conduction

For thick fins, or fins with large h, the temperature varies across the fin's thickness as well as along its radius. The one-dimensional model then overestimates efficiency. The usual rule is that the one-dimensional model is acceptable when the fin's Biot number based on half-thickness, h·(t/2)/k, is small. For aluminium fins in air this is virtually always the case.

Variable properties and radiation

At high temperatures, as in engine cylinders, furnace tubes and some waste heat units, the fin's conductivity changes with temperature and radiation from the fin surface adds to convection. Both effects require numerical treatment beyond the simple Bessel solution.

Wet and frosted fins

As noted, condensing moisture changes the heat and mass transfer on the fin. Frost formation on refrigeration coils both insulates the fin and blocks the air passage, and requires periodic defrosting.

Design considerations

Pulling these threads together, an engineer specifying annular fins typically weighs:

1. Where the thermal resistance is. If the fluid inside the tube is a gas too, fins outside help less. If the inside is a condensing vapor or liquid, nearly all the resistance is on the outside and fins pay off well.
2. Fin material and thickness. Aim for a fin parameter m that keeps efficiency high at the design h. Aluminium in air at modest h can be very thin; steel at high h must be thicker or shorter.
3. Fin height. Taller fins add area roughly as the square of radius but lower efficiency; diminishing returns set in as m·(r2 − r1) grows.
4. Fin spacing. Denser fins add area but increase pressure drop, fan power and fouling.
5. Bond type. Choose an attachment that will keep contact resistance low over the equipment's life at its operating temperatures.
6. Overall surface efficiency. Combine fin efficiency with the area fraction of fins to get η_o, the number that actually enters the heat exchanger's overall heat transfer coefficient.
7. Margins. Analytical efficiencies are approximations. Studies such as the 2019 Energies paper and the wet-surface comparisons show that real performance can differ substantially, in either direction depending on conditions.

Common misconceptions

"Higher fin efficiency is always better"

Efficiency only tells you how close the fin is to being isothermal. A tiny fin can have an efficiency of 0.99 and still add little heat transfer. What matters is the total heat transferred per unit cost, weight or pressure drop.

"Copper fins are always worth it"

In air at ordinary convection coefficients, aluminium fins are already close to isothermal, so copper's higher conductivity gains little, as the worked example shows. Copper earns its place where h is high, fins are tall, or other requirements favor it.

"A fin always increases heat transfer"

Not necessarily. If the fin is made of a poor conductor, or the convection coefficient is very high, a fin can conduct heat so poorly that it adds little, and in extreme cases its root can actually insulate the surface it covers. The effectiveness, the ratio of heat with the fin to heat from the base area without it, should be well above 1 for a fin to be worthwhile. In gas-side applications with metal fins it usually is.

"The straight-fin formula is close enough"

It is close for short fins on large tubes. For fins that extend several times the tube radius, the straight-fin formula overestimates efficiency because it ignores how much of the annular fin's area lies near its cool outer edge.

"Analytical fin efficiency is exact"

It is exact only within its assumptions. Real tubes have contact resistance, nonuniform air flow, two-dimensional conduction near the root and, sometimes, condensation or frost.

Frequently asked questions

What is the difference between an annular fin and a circular fin?

They are the same thing. Annular, circular, circumferential and radial fin all refer to a ring-shaped fin around a tube.

Why do annular fin formulas need Bessel functions?

Because the fin's cross-sectional area and face area both grow in proportion to radius, the temperature equation has a radius-dependent coefficient. Its solution is not an exponential or hyperbolic function, as for a straight fin, but a combination of modified Bessel functions of order zero.

What is a typical annular fin efficiency?

It depends entirely on geometry, material and conditions. Thin aluminium fins in low-velocity air can exceed 0.95; steel fins in high-h service can fall well below 0.7. The worked example above shows how the same geometry ranges from 0.68 to 0.98 across common metals.

What does the corrected radius do?

It accounts for heat lost through the fin's outer edge by pretending the fin is slightly longer, by half its thickness, and that its edge is insulated. For thin fins the correction is small.

Are helical fins analyzed as annular fins?

Yes, as a close approximation. A helical fin wound at a small pitch looks locally like a stack of annular fins, and the annular fin efficiency is the standard way to estimate its performance.

How do you calculate fin efficiency without charts?

Any scientific computing environment with modified Bessel functions (I0, I1, K0, K1) can evaluate the exact formula directly. SciPy's special functions are a common choice. Approximations such as Schmidt's are also widely used in design codes.

Why are fins on the air side, not the water side?

Because water and condensing vapors have convection coefficients far higher than air, so the air side is usually the bottleneck. Adding area where the resistance is highest has the largest effect.

Summary

An annular fin is a ring of metal around a tube that increases the area available for exchanging heat with a fluid. Because its cross-section grows with radius, its temperature distribution is governed by the modified Bessel equation of order zero, and its efficiency is a ratio of combinations of the functions I0, I1, K0 and K1. Engineers have used Gardner's 1945 charts, Schmidt's 1949 approximation and later methods to apply that theory to everything from air-cooled engine cylinders to refinery air coolers, HVAC coils and electronic heat sinks. The theory captures the main trade-offs well: material, thickness and height against convection coefficient, with diminishing returns as fins grow. It also rests on assumptions, such as a uniform convection coefficient and a perfect bond to the tube, that real equipment violates, with computational studies finding discrepancies of up to about 12 percent. Used with those limits in mind, annular fin analysis remains one of the most practical tools in thermal engineering.

Sources

• Annular fin (Wikipedia)
• Fin (extended surface) (Wikipedia))
• Bošnjaković, M. et al. (2019). How Big Is an Error in the Analytical Calculation of Annular Fin Efficiency? Energies 12(9):1787 (RePEc record)
• Gardner, K.A. (1945). Efficiency of Extended Surface. Transactions of the ASME 67(8) (record)
• HTRI ESG-1: Efficiency of Extended Surfaces (Rozenman & Taborek, 1971)
• Nobile, E. Annular fin homework, University of Trieste
• FAMU-FSU College of Engineering lecture notes: Correction Length
• Gonzaga University: Exact insulated tip fin length correction for tip convection
• Karczewski: finned-tube oil cooler fin efficiency study (BazTech)
• Calculating the efficiency of complex-shaped fins (Marcinkowski & Taler)
• NASA Technical Reports Server: NACA air-cooled cylinder fin research
• U.S. Centennial of Flight Commission: NACA cowling
• Learning Scientific Programming with Python: heat dissipation of an annular fin heatsink
• Air Coolers excerpt from Rules of Thumb for Chemical Engineers (Hall)
• List of thermal conductivities (Wikipedia)