Convection and Heat Transfer Coefficients 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: Values of h are determined from experimental data. Various forms (h i , h o ) are used depending on the particular application. The defining equation can be rearranged into "resistance form", relating heat flow to the temperature difference and a resistance: This form is especially useful in the many applications where it is necessary to combine heat transfer coefficients for a number of "layers." 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: The heat then transfers by conduction through the wall: and then through another film layer on the outside of the wall to the surrounding fluid 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: In this example, the resistances are in series, so the heat transfer problem becomes: which can be solved directly for the heat transfer rate. If the intermediate temperatures are needed, the rate can be plugged back into the equations for the individual layers. The method of combining resistances suggests an "overall" approach might be useful. This produces the idea of an overall heat transfer coefficient Each overall heat transfer coefficient is determined for a specific mean area and temperature difference. For the layer problem being discussed, the overall coefficient is given by The overall coefficient U can be defined in terms of the inside or the outside wall area. Both values work the same, but the numbers for U i and U o will be different. 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. Correlating Heat Transfer Coefficients 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: and the dimensionless groups involved: 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: The correction factors can be used to adjust for thermal variation in properties, tube curvature, entrance and exit effects, etc. Choice of a correlation depends on the flow regime, geometry, and whether or not a phase change occurs. References: 1. Brodkey, R.S. and H.C. Hershey, Transport Phenomena: A Unified Approach, McGraw-Hill, 1988, pp. 500-04. 2. Levenspiel, O., Engineering Flow and Heat Exchange, Revised Edition, Plenum Press, 1998, pp. 173-74, 197-201. 3. McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering (5th Edition), McGraw-Hill, 1993, pp. 319-24. 4. 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 RMP Lecture Notes