Extended surfaces have fins attached to the primary surface on one side of a two-fluid or a multifluid heat exchanger. Fins can be of a variety of geometry—plain, wavy or interrupted—and can be attached to the inside, outside or to both sides of circular, flat or oval tubes, or parting sheets. Pins are primarily used to increase the surface area (when the heat transfer coefficient on that fluid side is relatively low) and consequently to increase the total rate of heat transfer. In addition, enhanced fin geometries also increase the heat transfer coefficient compared to that for a plain fin. Fins may also be used on the high heat transfer coefficient fluid side in a heat exchanger primarily for structural strength (for example, for high pressure water flow through a flat tube) or to provide a thorough mixing of a highly-viscous liquid (such as for laminar oil flow in a flat or a round tube). Fins are attached to the primary surface by brazing, soldering, welding, adhesive bonding or mechanical expansion, or extruded or integrally connected to tubes. Major categories of extended surface heat exchangers are Tube-fin*Tube-fin* (Figure 1), and *Tube-fin* (Figure 2, individually finned tubes – Figure 2a and flat fins on an array of tubes – Figure 2b) exchangers. Note that shell-and-tube exchangers sometimes employ individually finned tubes—low finned tubing (similar to Figure 2a but with low height fins) [Shah (1985)].

Basic heat transfer and pressure drop analysis methods for extended and other heat exchangers have been described by Shah (1985). An overall design methodology for heat exchangers has also been presented by Shah (1992). Detailed step-by-step procedures for designing extended surface plate-fin and tube-fin type counterflow, crossflow, parallelflow and two-pass cross-counterflow heat exchangers have been outlined by Shah (1988).

In this entry, the theoretical and experimental/analytical nondimensional heat transfer coefficients (Nusselt Number, **Nu**, or *Colburn factor*, j) and the Fanning Friction Factor for some important extended surface geometries are summarized and a table of fin efficiencies for some important extended surfaces is provided.

The concept of fin efficiency accounts for the reduction in temperature potential between the fin and the ambient fluid due to conduction along the fin and convection from or to the fin surface, depending on fin cooling or heating situation. The *fin temperature effectiveness* or *fin efficiency* is defined as the ratio of the actual heat transfer rate through the fin base divided by the maximum possible heat transfer rate through the fin base, which can be obtained if the entire fin is at base temperature (i.e., its material thermal conductivity is infinite). Since most real fins are “thin,” they are treated as one-dimensional (1-D), with standard idealizations used for analysis [Huang and Shah (1992)]. This 1-D fin efficiency is a function of fin geometry, fin material thermal conductivity, heat transfer coefficient at the fin surface and fin tip boundary condition; it is not a function of the fin base or fin tip temperature, ambient temperature or heat flux at the fin base or fin tip. Fin efficiency formulas for some common plate-fin and tube-fin geometries of uniform fin thickness are presented in Table 1 [Shah (1985)]. These results are not valid when the fin is thick or is subject to variable heat transfer coefficients or variable ambient fluid temperature, nor for fins with temperature depression at the base [see Huang and Shah (1992) for specific modifications to the basic formula or for specific results]. In an extended surface heat exchanger, heat transfer takes place from both the fins (η_{f} < 100%) and the primary surface (η_{f} = 100%). In this case, the total heat transfer rate is evaluated through a concept of *total surface effectiveness or surface efficiency* η_{o} defined as:

where A_{f} is the fin surface area, A_{p} is the primary surface area and A = A_{f} + A_{p}. In Eq. (1), the heat transfer coefficients of finned and unfinned surfaces are idealized to be equal. Note that η_{o} is always required for the determination of thermal resistances for heat exchanger analysis [Shah (1985)].

Accurate and reliable surface heat transfer and flow friction characteristics are key input for exchanger heat transfer and pressure drop analyses, or the rating and sizing problems [Shah (1985), (1992)]. The nondimensional surface heat transfer characteristic is usually presented in terms of the Nusselt Number, Stanton Number or *Colburn factor* vs. Reynolds Number.

The nondimensional surface pressure drop characteristic is usually presented in terms of the *Fanning friction factor* vs. Reynolds Number. Some important analytical solutions and empirical correlations for some important extended surfaces are summarized below.

Analytical solutions for developed and developing velocity/temperature profiles in constant cross-section noncircular flow passages are important for extended surface (plate-fin) heat exchangers. Fully developed laminar flow solutions are applicable to highly compact plate-fin exchangers with plain uninterrupted fins, developing laminar flow solutions to interrupted-fin geometries, and turbulent flow solutions to not-so-compact extended surfaces.

Fully developed laminar flow analytical solutions are presented in Table 2 for specified ducts for three important thermal boundary conditions [Shah and London (1978); Shah and Bhatti (1987)]. The following observations may be made from this table:

There is a strong influence of flow passage geometry on Nu and fRe. Rectangular passages approaching a small aspect ratio exhibit the highest Nu and fRe.

Three thermal boundary conditions (denoted by the subscripts H1, H2, and T) have a strong influence on the Nusselt numbers. Here, H1 denotes constant axial wall heat flux with constant peripheral wall temperature, H2 denotes constant axial and peripheral wall heat flux and T denotes constant wall temperature.

As Nu = αD

_{h}/λ, a constant Nu implies the convective heat transfer coefficient α is independent of the flow velocity and fluid Prandtl Number.An increase in α can best be achieved either by reducing D

_{h}or by selecting a geometry with a low aspect ratio, rectangular flow passage. Reducing the hydraulic diameter is an obvious way to increase exchanger compactness and heat transfer, or D_{h}can be optimized using well-known heat transfer correlations based on design problem specifications.Since fRe = constant, f ∞ 1/Re ∞ 1/um. In this case, it can be shown that Δp ∞ u

_{m}

Many additional analytical results for fully developed laminar flow (Re ≤ 2,000) are presented in Shah and London (1978) and in Shah and Bhatti (1987). The entrance effects, flow maldistribution, free convection, fluid property variation, fouling and surface roughness all affect fully developed analytical solutions. In order to account for these effects in real plate-fin plain fin geometries having fully developed flows, it is best to reduce the magnitude of the analytical Nu by at least 10% and to increase the value of the analytical fRe by 10% for design purposes.

The transition regime (2,000 < Re < 10,000) correlations for f and Nu can be found in the work of Bhatti and Shah (1987). Fully-developed, turbulent flow Fanning friction factors are given by Bhatti and Shah (1987) as

where

Equation (2) is accurate within ±2% [Bhatti and Shah (1987)]. The fully developed, turbulent flow Nusselt number correlation for a circular tube is given by Gnielinski, as reported in Bhatti and Shah (1987), as:

which is accurate within about ±10% with experimental data for 2,300 ≤ Re ≤ 5 × 10^{6} and 0.5 ≤ Pr ≤ 2,000. For higher accuracies in turbulent flow, refer to the correlations by Petukhov et al. reported by Bhatti and Shah (1987).

A careful observation of accurate experimental friction factors for all noncircular smooth ducts reveals that ducts with laminar fRe < 16 have turbulent f factors lower than those for a circular tube, whereas ducts with laminar fRe > 16 have turbulent f factors higher than those for a circular tube [see Shah and Bhatti (1988)]. Similar trends have been observed for the Nusselt numbers. If one is satisfied within ±15% accuracy, Eqs. (2) and (3) for f and Nu can be used for noncircular passages, with the hydraulic diameter as the characteristic length in f, Nu and Re; otherwise refer to Bhatti and Shah (1987) for more accurate results.

For hydrodynamically and thermally developing flows, the analytical solutions are boundary condition dependent (for laminar flow heat transfer only) and geometry-dependent [see Shah and London (1978), Shah and Bhatti (1987) and Bhatti and Shah (1987) for specific solutions].

Analytical results presented in the preceding section are useful for well-defined constant cross-sectional extended surfaces with essentially unidirectional flows. Flows encountered in enhanced extended surfaces are generally very complex, having flow separation, reattachment, recirculation and vortices. Such flows significantly affect Nu and f for specific exchanger surfaces. Since no analytical or accurate numerical solutions are available, the information is derived experimentally. Kays and London (1984) and Webb (1994) have compiled most of the experimental results reported in open literature. Empirical correlations for some important extended surfaces are summarized.

**Offset Strip Fins**. This is one of the most widely used, enhanced fin geometries (Figure 3) in aircraft, cryogenics and many other industries that do not require mass production. This surface has one of the highest heat transfer performance relative to the. friction factor. Extensive analytical, numerical and experimental investigations have been conducted over the last 50 years. The most comprehensive correlations for j and f factors for offset strip-fin geometry are provided by Manglik and Bergles (1995), as follows:

where

Geometrical symbols in Eq. (6) are shown in Figure 3.

These correlations predict the experimental data of 18 test cores within ±20% for 120 ≤ Re ≤ 10^{4}. Although all the experimental data for these correlations have been obtained for air, the j factor takes into consideration minor variations in the Prandtl number, and the above correlations should be valid for 0.5 < Pr < 15.

**Louver Fins**. Louver or multilouver fins are extensively used in the auto industry due to their mass production manufacturability and hence, lower cost. It has generally higher j and f factors than those of the offset strip-fin geometry; also, the increase in the friction factors is usually higher than the increase in the j factors. However, the exchanger can be designed for higher heat transfer and the same pressure drop compared with offset strip-fins by a proper selection of exchanger frontal area, depth and fin density. Published literature on and correlations for louver fins have been summarized by Webb (1994) and Cowell (1995) while flow and heat transfer phenomena have been discussed by Cowell (1995). Because of the lack of systematic studies on modern louver fin geometries in the open literature, no correlation can be recommended for design purposes.

Two major types of tube-fin extended surfaces are: a) individually-finned tubes, and b) flat fins (also sometimes referred to as plate fins) with or without enhancements/ interruptions on an array of tubes as shown in Figure 2. An extensive coverage of published literature on and correlations for these extended surfaces are provided by Webb (1994), Kays and London (1984) and Rozenman (1976). Empirical correlations for some important geometries are summarized below.

**Individually-Finned Tubes**. This fin geometry, helically-wrapped (or extruded) circular fins on a circular tube as shown in Figure 2a, is commonly used in process and waste heat recovery industries. The following correlation for j factors is recommended by Briggs and Young [see Webb (1994)], for individually-finned tubes on staggered tube banks.

where l_{f} is the radial height of the fin, δ_{f} is the fin thickness, p_{f} is the fin pitch and s = p_{f} – δ_{f} is the distance between adjacent fins. Equation (7) is valid for the following ranges: 1100 ≤ Re_{d} ≤ 18,000, 0.13 ≤ s/l_{f} ≤ 0.63, 1.01 ≤ s/δ_{f} ≤ 6.62, 0.09 ≤ l_{f}/d_{o} ≤ 0.69, 0.011 ≤ δ/d_{o} ≤ 0.15, 1.54 ≤ X_{t}/d_{o} ≤ 8.23; fin root diameter do between 11.1 and 40.9 mm; and fin density N_{f} (= 1/p_{f}) between 246 and 768 fins per meter. The standard deviation of Eq. (7) from experimental results has been computed at 5.1%.

For friction factors, Robinson and Briggs [see Webb (1994)] recommended the following correlation:

Here, is the diagonal pitch and X_{t}
> and X_{l} are the transverse and longitudinal tube pitches, respectively. The correlation is valid for the following ranges: 2000 ≤ Re_{d} ≤ 50,000, 0.15 ≤ s/l_{f} ≤ 0.19, 3.75 ≤ s/δ_{f} ≤ 6.03, 0.35 ≤ l_{f}/d_{o} ≤ 0.56, 0.011 ≤ δ_{f}/d_{o} ≤ 0.025, 1.86 ≤ X_{t}/d_{o} ≤ 4.60, 18.6 ≤ d_{o} ≤ 40.9 mm, and 311 ≤ N_{f} ≤ 431 fins per meter. The standard deviation of Eq. (8) from correlated data was 7.8%.

The extensive work on low-finned tubes has been assessed by Rabas and Taborek (1987), Ganguli and Yilmaz (1987) and Chai (1988). A simple but accurate correlation for heat transfer, given by Ganguli and Yilmaz (1987), is

A more accurate correlation for heat transfer is given by Rabas and Taborek (1987). By comparing existing data in literature and various correlations, Chai (1988) has arrived at the best correlation for friction factors:

This correlation is valid for 895 ≤ Re_{d} ≤ 713,000, 20 ≤ θ ≤ 40, X_{t}/d_{o} < 4, N_{r} ≥ 4, and θ is the tube layout angle in degrees. It predicts 89 literature data points within a mean absolute error of 6%; the range of actual error is from –16.7 to +19.9%.

**Flat Plain Fins on a Staggered Tube Bank**. This geometry, as shown in Figure 2b, is used in the air-conditioning/refrigeration industry as well as in applications where the pressure drop on the fin side prohibits the use of enhanced flat fins. An in-line tube bank is generally not used unless very low fin-side pressure drop is the essential requirement. Heat transfer correlation for Figure 2b flat plain fins on staggered tube banks with four or more tube rows has been provided by Gray and Webb [see Webb (1994)] as follows:

For the number of tube rows N from 1 to 3, the j factor is lower and is given by:

Gray and Webb hypothesized that the friction factor consists of two components: one associated with the fins and the other associated with the tubes, as follows:

where

and f_{t} is the friction factor associated with the tube defined the same as ft. In equation form, it was expressed in terms of the Euler Number, Eu = 4 f_{tb} = f_{t}πd_{o}/[N(X_{t} – d_{o})], by Zukauskas and Ulinskas (1983). Equation (13) correlated with 90% of the data for 19 heat exchangers within ±20%. The range of dimensionless variables of Equations (13) and (14) are: 500 ≤ Re ≤ 24,700, 1.97 ≤ X_{t}/d_{o} ≤ 2.55, 1.7 ≤ X_{l}/d_{o} ≤ 2.58, and 0.08 ≤ s/d_{o} ≤ 0.64.

The subject of extended surface heat transfer is very extensive and is difficult to condense in a few pages. This attempt to summarize some important typical results, both analytical and experimental, is but an introduction to the subject. Key references are provided below for further exploration of the subject.

### Nomenclature

A total heat transfer area (primary + fin) on one fluid side of a heat exchanger, **A**_{p−} primary surface area, **A**_{f−} fin surface area, m^{2}

A_{o} minimum free flow area on one fluid side of a heat exchanger, m^{2}

b plate spacing, h + δ_{f}, m

D_{h} hydraulic diameter of flow passages, m

d_{o} tube outside diameter, m

f Fanning friction factor, , dimensionless

f_{th} Fanning friction factor, , dimensionless

h height of the offset strip fin (see Fig. 3), m

j Colburn factor, NuPr^{–1/3}/Re, dimensionless

l_{f} offset strip fin length or fin height for individually finned tubes, m

mass velocity, kg/m^{2}s

N number of tube rows

N_{f} number of fins per meter, 1/m

Nu Nusselt number, αD_{h}/λ, dimensionless

Pr fluid Prandtl number, dimensionless

p_{f} Fin pitch, m

Re Reynolds number, D_{h}/η, dimensionless

Re_{d} Reynolds number, ρu_{m} d_{o}/η, dimensionless

s distance between adjacent fins, **p**_{f−}δ_{f} , m

u_{m} mean axial velocity in the minimum free flow area, m/s

X_{d} diagonal tube pitch, m

X_{l} longitudinal tube pitch, m

X_{t} transverse tube pitch, m

α heat transfer coefficient, W/m^{2}K

δ_{f} fin thickness, m

η_{f} fin efficiency, dimensionless

η_{o} extended surface efficiency, dimensionless

λ fluid thermal conductivity, W/mK

η fluid dynamic viscosity, Pa · s

ρ fluid density, kg/m^{3}

#### REFERENCES

Bhatti, M. S. and Shah, R. K. (1987) Turbulent and transition convective heat transfer in ducts. *Handbook of Single-phase Convective Heat Transfer*. Ed. by S. Kakaç R. K. Shah and W. Aung, 4: 166 pages. John Wiley. New York.

Chai, H. C. (1988) A simple pressure drop correlation equation for low-finned tube crossflow heat exchangers. *Int. Commun. Heat Mass Transfer*. 15: 95–101. DOI: 10.1016/0735-1933(88)90010-3

Cowell, T. A., Heikal, M. R., and Achaichia, A. (1995) Flow and heat transfer in compact louvered fin surfaces, *Exp. Thermal and Fluid Sci.* 10: 192–199. DOI: 10.1016/0894-1777(94)00093-N

Ganguli, A. and Yilmaz, S. B. (1987) New heat transfer and pressure drop correlations for crossflow over low-finned tube banks. *AIChE Symp. Ser.* 257(83): 9–14.

Huang, L. J. and Shah, R. K. (1992) Assessment of calculation methods for efficiency of straight fins of rectangular profiles. *Int. J. Heal and Fluid Flow*. 13: 282–293. DOI: 10.1016/0142-727X(92)90042-8

Kays, W. M. and London, A. L. (1984) Compact Heat Exchangers. 3rd edn. McGraw-Hill, New York.

Manglik, R. M. and Bergles, A. E. (1995) Heat transfer and pressure drop correlations for the rectangular offset strip fin compact heat exchanger. *Exp. Thermal and Fluid Sci.* 10: 171–180. DOI: 10.1016/0894-1777(94)00096-Q

Rabas, T. J. and Taborek, J. (1987) Survey of turbulent forced-convection heat transfer and pressure drop characteristics of low-finned tube banks in crossflow. *Heat Transfer Eng.* 8(2): 49–62.

Rozenman, T. (1976) Heat transfer and pressure drop characteristics of dry cooling tower extended surfaces. I: *Heat Transfer and Pressure Drop Data*. Report BNWL-PPR 7-100; II: *Data Analysis and Correlation. Report BNWL-PFR 7-102.* Battelle Pacific Northwest Laboratories. Richland, WA.

Shah, R. K. (1985) Compact heat exchangers. in *Handbook of Heat Transfer Applications*. 2^{nd} edn., Eds. W. M. Rohsenow, J. P. Hartnett, and E. N. Ganić. 4. III: 4–174 to 4–311. McGraw-Hill, New York.

Shah, R. K. (1988) Plate-fin and tube-fin heat exchanger design procedures. in *Heat Transfer Equipment Design*. Eds. R. K. Shah, E. C. Subbarao and R. A. Mashelkar. 255–266. Hemisphere Publishing Corp., Washington, DC.

Shah, R. K. (1992) Multidisciplinary approach to heat exchanger design. *Industrial Heat Exchangers*, Ed. J-M. Buchlin. Lecture Series No. 1991: 04. von Kármán Institute for Fluid Dynamics. Belgium.

Shah, R. K. and Bhatti, M. S. (1987) Laminar convective heat transfer in ducts. *Handbook of Single-phase Convective Heal Transfer*. Eds. S. Kakaç R. K. Shah and W. Aung, pages, 3: 137 pages, John Wiley, New York.

Shah, R. K. and Bhatti, M. S. (1988) Assessment of correlations for single-phase heat exchangers. *Two-Phase Flow Heat Exchangers: Thermal-Hydraulic Fundamentals and Design.* Eds. S. Kakaç A. E. Bergles, and E. O. Fernandes. 81–122. Kluwer Academic Publishers, Dordrecht. The Netherlands.

Shah, R. K. and London, A. L. (1978) Laminar flow forced convection in ducts, Supp. 1 to *Advances in Heat Transfer*. Academic Press. New York.

Webb, R. L. (1994) *Principles of Enhanced Heat Transfer*. John Wiley. New York.

Zukauskas, A. and Ulinskas, R. (1983) Banks of plain and finned tubes. *Heal Exchanger Design Handbook*. 2: 2.2.4. Hemisphere. New York.