Heat conduction in a planar slab with power-law and polynomial heat flux input

• We find temperature history in insulated slab due to heat flux increasing with fourth-power of time.
• We find the response of the slab to power-law heat fluxes applied over a finite time interval.
• Superposition facilitates solutions for a quadratic heat flux pulse and a quartic heat flux pulse.
• Solutions suitable for short and long time evaluation are validated with intrinsic verification.
• Exact results are useful test problems for numerical and inverse problem code validation.

A new solution is presented for one-dimensional heat conduction with boundary heat flux increasing as fourth power of time. Green's functions (GFs) are used and resulting slowly-converging summations in the solution are replaced with equivalent closed-form expressions. Solutions to time power-law heat flux variations that stop suddenly after a specified time are included which are especially useful as “building blocks” for superposition. An example superposition of power-law heat flux solutions to determine the response of a composite heat flux is given. Finite heating power-law solutions are combined using superposition to produce special quadratic and quartic heat flux pulse solutions which are higher-order analogs of the classic triangular heat flux pulse. The pulse solutions provide test cases for solution verification. The quartic flux approximates Gaussian heating with zero value and gradient at beginning and end of heating. An example is presented illustrating the use of the quartic heat flux solution to guide grid refinement in a numerical solution. Intrinsic verification concept is demonstrated in several ways.