The rate of heat transfer in forced convection depends on properties of
both the fluid (density, heat capacity, etc.) and of the flow (geometry,
turbulence, etc.). The calculation is generally complex, and may
involve boundary layer theory and tricky mathematics, so we typically
use empirical correlations based on masses of data. These enable us to
determine *heat transfer coefficients* for use in calculations.

A heat transfer coefficient, *h*, is the proportionality factor
between the heat flux and an overall temperature difference driving
force:

The defining equation can be rearranged into "resistance form", relating
heat flow to the temperature difference and a *resistance*:

Consider the case where heat is transferred from a fluid through a wall to another fluid. The heat first transfers from the bulk fluid to the inside of the wall. Transfer is primarily convective, and we usually assume that all of the resistance can be "lumped" into a "film" adjoining the wall:

Since is is a "no accumulation, no generation" case, the heat flow must
be constant and continuous through each of the layers. Consequently,
the problem becomes a system of three equations with three unknowns
(*q _{i}=q_{w}=q_{o}, T_{wi}, and
T_{wo}*).

If the resistance form is used, a single equation can be developed. To do this, recall from previous studies (in transport phenomena, electric circuits, etc.) that resistances combine according to the connection pattern:

The method of combining resistances suggests an "overall" approach might
be useful. This produces the idea of an *overall heat transfer
coefficient*

Real problems may not be as simple as three layers in series. It is usually wise to include resistances for scaling or fouling of the wall, contact resistances between two solid layers, etc.

The heat transfer coefficient depends on fluid properties (heat capacity, viscosity, thermal conductivity, density) and flow properties (pipe diameter, velocity). Dimensional analysis shows the relation between the variables:

- the
*Reynolds Number*, the ratio of convective to molecular transport; - the
*Prandtl Number*, the ratio of momentum to heat transfer; and - the
*Nusselt Number*. Most of the correlations will thus take the form:

**References:**- Brodkey, R.S. and H.C. Hershey, Transport Phenomena: A Unified Approach, McGraw-Hill, 1988, pp. 500-04.
- Levenspiel, O., Engineering Flow and Heat Exchange, Revised Edition, Plenum Press, 1998, pp. 173-74, 197-201.
- McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering (5th Edition), McGraw-Hill, 1993, pp. 319-24.
- McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering (6th Edition), McGraw-Hill, 2001, pp. 325-30.

R.M. Price

Original: 12/8/99

Modified: 1/4/2002, 2/4/2003

Copyright 1999, 2002, 2003 by R.M. Price -- All Rights Reserved