   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 (W m K ), 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 s m . 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