The overall heat transfer coefficient is employed in calculating the rate of heat transfer from one fluid at an average bulk temperature T_{1} through a solid surface to a second fluid at an average bulk temperature T_{2} (where T_{1} > T_{2}). The defining equation is generally only applicable to an incremental element of heat transfer surface dA for which the heat transfer rate is d , and the equation is strictly valid only at steady state conditions and negligible lateral heat transfer in the solid surface, conditions generally true enough in most practical applications. The defining equation is

where U is referenced to a specific surface (see below).

In the particular situation of heat transfer across a plane wall of uniform thickness, U is related to the individual film heat transfer coefficients, α_{1} and α_{2}, of the two fluids by the equation

where δ_{w} is the thickness of the wall and λ_{w} is the thermal conductivity of the wall.

If there are fouling deposits on the wall, they have a resistance to heat transfer, R_{1} and R_{2}, in units of m^{2} K/W, and these resistances must be added in (see Fouling and Fouling Factors)

For the special but very important case of heat transfer through the wall of a plain round tube, the different heat transfer areas on the inside and outside surfaces of the tube need to be considered. Let dA_{i} be the inside incremental area and dA_{o} be the outside. Then (including fouling resistances R_{fi} and R_{fo} inside and out):

where U_{i} is termed the "overall heat transfer coefficient referenced to (or based on) the inside tube heat transfer area", and r_{i} and r_{o} the inside and outside radii of the tube.

Alternatively, the overall coefficient may be based on the outside heat transfer area, giving

where U_{o} is termed the "overall heat transfer coefficient based on the outside tube heat transfer area." Note that

These ideas may be extended to more complicated surfaces such as finned or composite tubes, but it is then necessary to add further resistance terms (and the area ratio corrections) for the fins or imperfect metal-to-metal contact.

Generally, in order to use these equations in heat transfer applications, the basic equation must be integrated:

where A_{T} is the total area required to transfer T and T_{1}, T_{2} and sometimes U must be expressed as functions of the heat already transferred from one end up to a given point in the heat transfer device. This is the basic design equation for most heat exchangers.

#### REFERENCES

Hewitt, G. F., Shires, G. L., and Bott, T. R. (1994) *Process Heat Transfer*, CRC Press, Boca Raton, Florida.

Incropera, F. P. and DeWitt, D. P. (1990) *Introduction to Heat Transfer, 2nd ed.*, John Wiley & Sons, New York.