Finite Width Slab and Hollow Cylinder Under an Arbitrary Temperature Transient on a Growing or Receding Boundary: Forward and Inverse Formulations

Semi-analytical solutions based on Duhamel's and Laplace convolution theorems along with a Zakian series representation of the inverse Laplace transform were derived to solve forward, unsteady heat-conduction problems of a single phase, homogeneous, and finite-width slab and hollow cylinder. Both had a constant-velocity growing or receding boundary under a time-dependent, arbitrary thermal load on the moving boundary with convection on the static surface. Additionally, the inverse thermal problem was solved by modeling an arbitrary surface loading using a polynomial and temperatures measured at the opposite surface with convection. In order to assure the accuracy and versatility of the derived semi-analytical solutions, results were compared with finite element solutions with excellent agreement using a test case of an asymptotic exponential thermal excitation. In practice, the resulting direct solutions can be used to determine transient temperature during machining, wear, erosion, corrosion, and/or additive manufacturing, especially for lower temperature solid-state methods such as cold-spray. Inverse solutions can be used to remotely assess surface temperature and/or erosion/wear and/or oxidation/growth rates in severe conditions where direct measurements are not feasible.