When a fluid flows across a solid object or ensemble of solids at a different temperature, crossflow heat transfer results. Heat transfer is a function of Reynolds Number,
and Prandtl Number,
where ρ is the density, u is a bulk velocity, η is viscosity, λ is the fluid conductivity, and cp is specific heat. D is a characteristic length, such as a diameter. Another popular choice for crossflow heat exchangers is a hydraulic diameter, Dh.
At very low flow rates, heat transfer is by Conduction alone. If the objects are inclined to the vertical, natural or mixed-convection may be important. Under these circumstances, the engineer may also be interested in maintaining thermal stratification. As Re is increased, forced-convection becomes the dominant mode of heat transfer. The local heat transfer coefficient, α, is defined by,
where is the local wall heat flux, Tw is the wall temperature, and Tb is a reference bulk temperature. For single objects, Tb is chosen as the free-stream temperature. For banks of objects, a variety of references are used. α is nondimensionalized in terms of a local Nusselt Number, Nu,
Figure 1(a) and (b) show Nu distributions around a circular cylinder in the Re ranges 4 × 103 to 5 × 104 and 3.98 × 104 to 4.26 × 105, respectively. It can be seen that at low Re, Nu is a maximum at φ = 0°, where skin-friction is zero. In this region, the flow is similar to that occurring at the stagnation point for flow normal to a flat-plate. Nu decreases to a minimum near the separation point, at the side of the cylinder, and then increases in the wake. Most heat transfer occurs through the front half of the cylinder. As Re increases, heat transfer in the latter half increases due to wake-turbulence. At Re = 2 × 105, a turbulent thermal boundary layer is established and the Nu distribution is quite complex, with twin minima occurring at around 90° and 140° probably due to laminar-to-turbulent transition in the boundary-layer and boundary-layer separation, respectively. At high Re, there is a large Nu maximum in the turbulent boundary-layer around φ = 110°. Heat transfer at the rear stagnation-point is by now as much as at the front stagnation point.
Figure 1. Distribution of local heat transfer coefficient around a circular cylinder for flow of air. Sources: a) Lohrisch (1929) and b) Schmidt and Wenner (1941), by permission of V. D. I. Verlag.
Overall heat transfer is expressed in terms of an average heat transfer coefficient, , defined by means of a rate equation,
where is the total rate of heat transfer, A is the total heat transfer area and ΔTM is an average or effective temperature difference between the solid-wall and the bulk of the fluid. For situations involving the use of extended surfaces or fins, a modified rate equation is employed. (See Tube banks, crossflow over.) is nondimensionalized either as an average Nusselt number, , according to Eq. (4) or as an average Stanton Number,
Re and Pr are often evaluated at the bulk temperature, Tb, though some authors advocate the use of a mean film temperature (Tw + Tb)/2. and are frequently correlated according to a power-law relationship,
where c and m are Re-dependent and a accounts for natural convection, if present. Use of n = 1/3, based on the Colburn-Chilton analogy, is widespread, though empirical correlations may involve different values of m. (See Analogy Between Heat, Mass and Momentum Transfer) Figure 2 shows and as a function of Re and Pr. A Pr-independent heat transfer factor, j, is defined as,
and may be considered the heat transfer analogue of the friction factor, f/2.
Figure 2. Average heat transfer for flow around a circular cylinder expressed in the form of Nusselt and Stanton numbers. Source: Žukauskas and Žiugžda. (1985).
Temperature variation of fluid properties affects both f and j. This may be accounted for by writing,
where j' denotes a temperature-independent (adiabatic) value and Prw is evaluated at the wall temperature, Tw. f is treated in a similar manner. Temperature-independent values of f' and j' appear in the literature for a large number of geometries. These measures of overall performance are of much interest to the heat transfer engineer.
The geometry of the heat transfer surface will substantially alter the mechanisms of fluid flow and heat transfer in crossflow. Other influencing factors include: the effect of thermal boundary conditions (constant vs. Tw), the influence of containing ducts or other blockage, free-stream turbulence, as well as the use of roughened surfaces or fins to enhance heat transfer.
Lohrische, W. (1929) Forschungsarbeiten auf dem Gebeite des Ingenieurwesens. 322, 46 (In German).
Schmidt, E. and Wenner, K. (1941) Forschung auf dem Gebeite des Ingenieurwesens. 12, 2, 65-73 (In German).
Žukauskas, A. and Žiugžda, J. (1985) Heat Transfer of a Cylinder in Crossflow. Hemisphere, New York. (Translator E. Bagdonaite, Editor G. F. Hewitt.)