This article describes heat and mass transfer from spheres. An immersed sphere is commonly understood to be a sphere made of solid material. But as far as heat transfer is concerned, fuel drops in an internal combustion engine or in the front of a nozzle of a furnace, or liquid drops in a spray drying oven, are considered as spheres of small size. In such cases, mass transfer takes place simultaneously with heat transfer. An understanding of heat transfer from single spheres is needed before predicting the thermal performance of clouds of spheres which are heated or cooled in a stream of fluid.

The distributions of local convective heat transfer coefficient around a sphere immersed in an air stream are shown in Figure 1.

The heat transfer coefficient decreases gradually from the front stagnation point to the minimum value at about 105° from the stagnation point. This decrease is due to the growth of the thermal boundary layer, which increases thermal resistance. Between 105° and 120°, a sharp increase in heat transfer coefficient occurs, after which the increase becomes less. The sudden increase of the local convective heat transfer coefficient following the minimum point may be explained by the fact that flow in the boundary layer undergoes a transition from Laminar to Turbulent Flow and that an intensive turbulent state exists on the downstream side of the sphere.

By using a series expansion technique for solving the laminar boundary layer equations, Frössling succeeded in obtaining the following relation for the local heat transfer coefficient

where Nusselt Number, Nu = αD/λ (D sphere diameter, λ thermal conductivity of the fluid, α the heat transfer coefficient), Reynolds Number, Re = U_{∞}D/ν (U_{∞} is the free stream velocity and ν, the kinematic viscosity of the fluid) and x is the distance along the perimeter measured from the stagnation point. The coefficients in Equation (1) are valid for naphtalene with a Prandtl Number equal to 2.53 arid an experimental pressure distribution has been used in deriving it.

From the engineering point of view, the average heat transfer coefficient is most important. This must be sought from experiments because of the difficulty of performing a theoretical analysis in the separated flow region on the downstream side of the sphere. Several experiments have been carried out and a variety of empirical formulas are available. A few will be presented here.

For air flowing in the Reynolds number interval 20 < Re_{D} < 150,000, one formula reads

which, for other gases, is modified to

In Eqs. (2) and (3), the thermophysical properties should be evaluated at the so-called film temperature (average of free stream temperature and surface temperature).

A common formula recommended for Reynolds numbers in the range of 3.5 ≤ Re_{D} ≤ 7.6·10^{4} and Prandtl numbers in the range of 0.6 ≤ Pr ≤ 380 is:

where η_{∞} and η_{w} are the dynamic viscosities at the free stream temperature and sphere surface temperature, respectively.

Freely-falling liquid drops present a special case of convective heat and mass transfer. In the Reynolds number range 1 ≤ Re_{D} ≤ 70,000, a formula reads

As is obvious, many formulas reduce to Nu_{D} = 2 as Re_{D} → 0. This is the value for heat transfer by conduction in a spherical shell and can easily be found analytically.

For air flow across a sphere, recent measurements have been correlated as:

which is valid in the range 100 < Re_{D} < 2·10^{5}. At higher Reynolds number in the range of 4·10^{5} < Re_{D} < 5·10^{6}, the following formula has been suggested:

For heat transfer from a sphere to a liquid metal, the following expression has been suggested:

The range of application is 3.6·10^{4} < Re_{D} < 2·10^{5}.

The analogy between equations describing heat transfer and equations for isothermal mass transfer implies that there exists for every heat transfer situation a corresponding mass transfer situation, with analogous boundary conditions and with the same geometry.

The mass transfer coefficient is expressed by the dimensionless Sherwood Number, defined as:

where β is the mass transfer coefficient, L is the characteristic length and δ is the mass diffusivity.

If the Schmidt number, defined as

is introduced, the formulas given previously can be used for mass transfer problems if Nu is replaced by Sh and Pr is replaced by Sc. (See Analogy Between Heat, Mass and Momentum Transfer.)

#### REFERENCES

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

Kreith, F. and Bohn, M. S. (1993) *Principles of Heat Transfer*. 5th edn. West Publ. Comp.

Schlichting, H. (1979) *Boundary Layer Theory*. 7th edn. McGraw-Hill.