The complex flow pattern around a tube or circular cylinder in crossflow influences the temperature field in the fluid created when the tube surface is kept a temperature different from that of the approaching stream. A description of the flow phenomena as a fluid flows across a tube have been presented elsewhere (see Tubes, Crossflow Over). Figures 1 and 2 show temperature fields at Re_{0} = 5 and Re_{D} = 40, respectively.

As the Peclet Number, Re_{D}Pr, is increased the importance of the convective transport relative to the diffusive transport is increased. The isotherms come closer to each other and the temperature gradients become greater. Also the thermal wake becomes narrower and more elongated in the main flow direction as the Peclet number is increased. The heat transfer between the tube and fluid is enlarged accordingly. The heat transfer coefficient is usually expressed by a dimensionless number, the Nusselt Number defined as

where α is the heat transfer coefficient, D the tube diameter and λ the thermal conductivity of the fluid.

Figure 3 presents the local Nusselt number for a single tube in crossflow for Reynolds numbers 5 to 40.

For Re_{D} = 5 the distribution is quite even over the tube surface. This is a reflection of the importance of the molecular diffusion. As the Reynolds number is increased the macroscopic convective transport becomes more and more important and the distributions vary considerably over the tube surface. It is also evident that the upstream side dominates the heat transfer process. The low heat transfer on the rear side is due to the low intensity in the recireulating flow region at that side and thus the heat exchange is not promoted.

The flowing medium is also important for the heat transfer process. If the Prandtl Number is low, as for liquid metals, the Peclet number becomes low and the Nusselt number is decreased since the molecular diffusion dominates. The distribution over the tube surface becomes more uniform. If instead the Prandtl number is large, the convective contribution is enlarged and the distribution is more uneven. Also the level in the Nusselt numbers is increased.

In Figure 4, local Nusselt numbers at high Reynolds numbers are provided.

**Figure 4. Local Nusselt number over the tube surface. Pr = 0.72. From Lienhard, J. H. (1981), A Heat Transfer Text Book, Prentice Hall Inc.**

Compared to the Nusselt numbers in Figure 3, the magnitudes are much higher. Also the distributions are remarkably different. These distributions reflect the various flow phenomena discussed in Tubes, Crossflow Over. At Re_{D} = 70800, the Nusselt number is largest at the forward stagnation point. It decreases along the tube surface as the thermal boundary layer thickness is increased and reaches a minimum at the separation point. On the rear side, the heat transfer coefficient is increased due to the intensive motion of *vortices*. At the highest Reynolds numbers two minima appear. The first one is assumed to be related to a transition from laminar to turbulent flow in the boundary layer while the latter one is regarded to closely coincide with the *separation* of the turbulent *boundary layer*. The region between the first minimum and the following maximum is believed to be a transition region. The details of this transition (separation followed by reattaehment or attached flow) is not evident. From Figure 4 it is found that the influence of the Reynolds number is large and as turbulent flow prevails the heat transfer coefficient is considerably enlarged.

On the forward side of the tube in crossflow, where a laminar boundary layer exists, the heat transfer coefficient can be determined by semi-analytical series expansion techniques or numerical solution of the boundary layer equations. However, for accurate predictions the surface pressure distribution or the velocity distribution outside the boundary layer has to be known. By using a velocity distribution based on measurements by Hiemenz, Frössling was able to obtain the following relation (by series expansion technique) for the local Nusselt number:

where x is the arc distance on the circumference from the front stagnation point. Equation (2) is regarded to be valid up to the separation point [Frössling (1940)].

For engineering calculations correlations of the average heat transfer coefficient are necessary. The following are examples.

For Reynolds numbers Re_{D} > 44 a formula by Collis and Williams is sometimes used. It reads

where t_{∞} is the fluid temperature and tf the film temperature defined as

where t_{w} is the surface temperature of the tube [Hinze (1975)].

A popular formula, based on experimental data for gases and liquids, reads

where the constants C and m are given in Table 1.

As Equation (5) is used, the physical properties should be evaluated at the film temperature according to Equation (4).

Another correlation is the one suggested by Whitaker which is

which is recommended for use in the intervals 0.67 < Pr < 300, 10 < Re_{D} < 10^{5},0.25 < η_{∞}/η_{w} < 5.2 [Incropera and Dewitt (1981)].

Žukauskas and Ziugzda (1985) have suggested the correlation

In this correlation the physical properties in the Nusselt and Reynolds numbers are to be evaluated at the freestream temperature. The values of C and m are given in Table 2.

Other correlations are available, more or less complicated.

The reader should remember that each correlation is reasonable over a certain range of conditions but for most engineering calculations one should not expect accuracy to much better than 25 percent.

Investigations have revealed that the average and local heat transfer from tubes or circular cylinders is augmented by freestream turbulence. To predict the heat transfer, several characteristics of the turbulence field have to be considered. The turbulence intensity has been found to be the most important one.

At low Reynolds numbers, the heat transfer is influenced by free convection. The motion due to free or natural convection is caused by the buoyancy force which is described by the Grashof Number

Sometimes the ratio of the Grashof number to the square of the Reynolds number is used to characterize the flow and heat transfer. If Gr/Re^{2} << 1, forced convection prevails while if Gr/Re^{2} >> natural or free convection occurs. When Gr/Re^{2} is of the order of unity, mixed convection, that is combined forced and free convection, is maintained.

Surface roughness also affects the heat transfer from tubes or cylinders. In air flow the heat transfer can be enhanced if the *relative roughness* (roughness height to tube diameter) is increased. Similar to the influence of freestream turbulence, the surface roughness may cause an onset to the critical flow regime at a lower Reynolds number. In viscous liquids having a high Prandtl number, the thermal resistance is concentrated to the viscous sublayer. For *enhancement of the heat transfer* under such circumstances small surface roughness elements should be applied.

#### REFERENCES

Eckert, E. R. G. and Drake, R. M. (1972) *Analysis of Heat and Mass Transfer*, McGraw-Hill, New York.

Frossling, N. (1940) Verdunstung, Wärmeübertragung und Geschwindigkeitverteilung bei Zweidimensionaler und Rotationssymmetrischen Laminarer Grenzschichtstromung, Lunds Universitets Arsskrift, N.F., Avd 2, Bd. 36, Nr 4.

Hinze, J. O. (1975) *Turbulence*, McGraw-Hill, New York.

Incropera, F. P. and DeWitt, D. P. (1981) *Fundamentals of Heat Transfer*, J. Wiley and Sons, New York.

Lienhard, J. H. (1981) A Heat Transfer Text Book, Prentice Hall Inc.

Morgan, V. T. (1975) The overall convective heat transfer from smooth circular cylinders, T. F. Irvine, Jr. and J. P. Hartnett (Eds.) Vol. 11, *Advances in Heat Transfer*, Academic Press, New York.

Žukauskas, A. A. and Ziugzda, J. (1985) *Heat Transfer of a Cylinder in Cross Flow*, Hemisphere Publishing Corporation, New York.