Next: Infinite Body, 1-D, Steady-Periodic Up: Library of Green's Functions Previous: Plate, steady 1-D Helmholtz.

K. D. Cole

In this section, steady-periodic heat conduction is treated. Also called time-harmonic or thermal-wave behavior, this special case is important whenever the causal effect is harmonic in time and has continued long enough for any start-up transients to die out.

Consider a one-dimensional region in which the temperature is sought. The transient temperature distribution satisfies

 (1)

Here is the thermal diffusivity (m s), is the thermal conductivity (WmK), is the volume heating (W m), and is a specified boundary condition. Index represents the boundaries at the limiting values of coordinate . The boundary condition may be one of three types at each boundary: boundary type 1 is specified temperature ( and ); boundary type 2 is specified heat flux (); and, boundary type 3 is specified convection where is a constant-with-time heat transfer coefficient (or contact conductance).

Since in this section the applications of interest involve steady-periodic heating, the solution is sought in Fourier-transform space, and the solution is interpreted as the steady-periodic response at a single frequency . For further discussion of this point see Mandelis (2001, page 2-3). Consider the Fourier transform of the above temperature equations:

 (2)

Here is the steady-periodic temperature, is the steady-periodic volume heating, is the steady-periodic specified boundary condition, and .

The temperature will be found with the Fourier-space Green's function, defined by the following equations:

 (3) (4)

Here and is the Dirac delta function. The coefficient preceding the delta function in Eq. (3) provides the 1-D frequency-domain Green's function with units of sm. This is consistent with earlier work with time-domain Green's functions.

If the steady-periodic Green's function is known (given below), then the steady-periodic temperature is given by the following integral equation:

 (5)

For a derivation of this equation see Beck et al. (1992, pp. 40-43). Next the steady-periodic Green's functions are given for 1-D bodies for cases XIJ.

Next: Infinite Body, 1-D, Steady-Periodic Up: Library of Green's Functions Previous: Plate, steady 1-D Helmholtz.

2004-08-10