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:

Defining Equation for Heat Transfer Coefficient
Values of h are determined from experimental data. Various forms (hi, ho) 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:

Resistance Form
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:

Inside Layer
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
Outside Layer

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 (qi=qw=qo, Twi, and Two).

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:

Combining Resistances
In this example, the resistances are in series, so the heat transfer problem becomes:
Layer Problem
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

Overall 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
Overall Heat Transfer Coefficient
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 Ui and Uo 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:

Dimensional Analysis
and the dimensionless groups involved: