Background

The concept of thermal resistances R (m².K)/W is invaluable for studying the thermal behaviour of a series of elements (layers) with interfaces, the elements being represented by their thermal conductivity and the interfaces by their heat transfer coefficient. The total thermal resistance of the layered structure, R, is given by the sum of the thermal resistances, r_i, of all the elements and interfaces. (From this point on, an upper case R is used to denote directly measurable quantities (using Equation 7) whereas lower case r is used to denote component thermal resistances.) The thermal resistance of an element or interface is given by:

Equation 1

where δτ is the temperature difference and q the heat flux. For a specimen of thickness x, with temperatures T_h and T_c at the bottom (hot) and top (cold) surfaces respectively, and heat flux q (W/m2) through the specimen, the thermal conductivity λ (W/(m. K)) is given by:

Equation 2

Similarly, the heat transfer coefficient h, W/(m^2K) at the interface of two surfaces in contact is given by:

Equation 3

where T_h is the temperature at the 'hot' interface side and T_c is the temperature at the 'cold' interface side. For an element (layer) of thickness x_l then using Equations 1 and 2:

Equation 4

and for an interface with heat transfer coefficient h_i using Equations 1 and 3:

Equation 5

Thus the total thermal resistance of a layered structure of elements and interfaces can be expressed as:

Equation 6

where h_i is the heat transfer coefficient of the i^th interface, and λ_l the thermal conductivity of the l^thlayer of thickness x_l.

For the instrument the measured thermal resistance R_i is given by:

Equation 7

where T_h is the lower 'hot' plate temperature, T_c is the upper 'cold' plate temperature and q is the heat flux.

Analysis for heat transfer coefficients across polymer-steel interfaces (page 4)