Extended surfaces have fins attached to the primary surface on one side of a twofluid 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 highlyviscous 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 TubefinTubefin (Figure 1), and Tubefin (Figure 2, individually finned tubes – Figure 2a and flat fins on an array of tubes – Figure 2b) exchangers. Note that shellandtube 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 stepbystep procedures for designing extended surface platefin and tubefin type counterflow, crossflow, parallelflow and twopass crosscounterflow 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 onedimensional (1D), with standard idealizations used for analysis [Huang and Shah (1992)]. This 1D 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 platefin and tubefin 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 crosssection noncircular flow passages are important for extended surface (platefin) heat exchangers. Fully developed laminar flow solutions are applicable to highly compact platefin exchangers with plain uninterrupted fins, developing laminar flow solutions to interruptedfin geometries, and turbulent flow solutions to notsocompact 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 wellknown 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 platefin 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). Fullydeveloped, 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 geometrydependent [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 welldefined constant crosssectional 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 stripfin 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 stripfin 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 stripfins 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 tubefin extended surfaces are: a) individuallyfinned 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.
IndividuallyFinned Tubes. This fin geometry, helicallywrapped (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 individuallyfinned 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 lowfinned 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 airconditioning/refrigeration industry as well as in applications where the pressure drop on the fin side prohibits the use of enhanced flat fins. An inline tube bank is generally not used unless very low finside 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 Singlephase 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 lowfinned tube crossflow heat exchangers. Int. Commun. Heat Mass Transfer. 15: 95–101. DOI: 10.1016/07351933(88)900103
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: 10.1016/08941777(94)00093N
Ganguli, A. and Yilmaz, S. B. (1987) New heat transfer and pressure drop correlations for crossflow over lowfinned 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: 10.1016/0142727X(92)900428
Kays, W. M. and London, A. L. (1984) Compact Heat Exchangers. 3rd edn. McGrawHill, 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: 10.1016/08941777(94)00096Q
Rabas, T. J. and Taborek, J. (1987) Survey of turbulent forcedconvection heat transfer and pressure drop characteristics of lowfinned 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 BNWLPPR 7100; II: Data Analysis and Correlation. Report BNWLPFR 7102. 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. McGrawHill, New York.
Shah, R. K. (1988) Platefin and tubefin 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. JM. 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 Singlephase 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 singlephase heat exchangers. TwoPhase Flow Heat Exchangers: ThermalHydraulic 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.
Les références

Bhatti, M. S. and Shah, R. K. (1987) Turbulent and transition convective heat transfer in ducts. Handbook of Singlephase Convective Heat Transfer. Ed. by S. KakaÃ§ R. K. Shah and W. Aung, 4: 166 pages. John Wiley. New York.
 Chai, H. C. (1988) A simple pressure drop correlation equation for lowfinned tube crossflow heat exchangers. Int. Commun. Heat Mass Transfer. 15: 95â€“101. DOI: 10.1016/07351933(88)900103
 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: 192â€“199. DOI: 10.1016/08941777(94)00093N
 Ganguli, A. and Yilmaz, S. B. (1987) New heat transfer and pressure drop correlations for crossflow over lowfinned tube banks. AIChE Symp. Ser. 257(83): 9â€“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: 282â€“293. DOI: 10.1016/0142727X(92)900428
 Kays, W. M. and London, A. L. (1984) Compact Heat Exchangers. 3rd edn. McGrawHill, 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: 171â€“180. DOI: 10.1016/08941777(94)00096Q
 Rabas, T. J. and Taborek, J. (1987) Survey of turbulent forcedconvection heat transfer and pressure drop characteristics of lowfinned tube banks in crossflow. Heat Transfer Eng. 8(2): 49â€“62. DOI: 10.1080/01457638708962793
 Rozenman, T. (1976) Heat transfer and pressure drop characteristics of dry cooling tower extended surfaces. I: Heat Transfer and Pressure Drop Data. Report BNWLPPR 7100; II: Data Analysis and Correlation. Report BNWLPFR 7102. 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. GaniÄ‡. 4. III: 4â€“174 to 4â€“311. McGrawHill, New York.
 Shah, R. K. (1988) Platefin and tubefin heat exchanger design procedures. in Heat Transfer Equipment Design. Eds. R. K. Shah, E. C. Subbarao and R. A. Mashelkar. 255â€“266. Hemisphere Publishing Corp., Washington, DC.
 Shah, R. K. (1992) Multidisciplinary approach to heat exchanger design. Industrial Heat Exchangers, Ed. JM. Buchlin. Lecture Series No. 1991: 04. von KÃ¡rmÃ¡n Institute for Fluid Dynamics. Belgium.
 Shah, R. K. and Bhatti, M. S. (1987) Laminar convective heat transfer in ducts. Handbook of Singlephase Convective Heal Transfer. Eds. S. KakaÃ§ 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 singlephase heat exchangers. TwoPhase Flow Heat Exchangers: ThermalHydraulic Fundamentals and Design. Eds. S. KakaÃ§ A. E. Bergles, and E. O. Fernandes. 81â€“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.