Semi-analytical unsteady heat conduction in large plane walls with heat convection exchange: The transversal method of lines (TMOL) in the “small time” sub-domain

Purpose: The purpose of this study is to address one-dimensional, unsteady heat conduction in a large plane wall exchanging heat convection with a nearby fluid under “small time” conditions. Design/methodology/approach: The Transversal Method of Lines (TMOL) was used to reformulate the unsteady, one-dimensional heat conduction equation in the space coordinate and time into a transformed “quasi-steady”, one-dimensional heat conduction equation in the space coordinate housing the time as an embedded parameter. The resulting ordinary differential equation of second order with heat convection boundary conditions is solved analytically with the method of undetermined coefficients. Findings: Semi-analytical TMOL dimensionless temperature profiles of compact form with/without regressed terms are obtained for the whole spectrum of Biot number (0 < Bi < ∞) in the “small time” sub-domain. In addition, a new “large time” sub-domain is redefined, that is, setting a smaller critical dimensionless time or critical Fourier number τ_cr = 0.18. Originality/value: The computed dimensionless center, surface and mean temperature profiles in the large plane wall accounting for all Biot number (0 < Bi < ∞) in the “small time” sub-domain τ < τ_cr = 0.18 exhibit excellent quality while carrying reasonable relative errors for engineering applications. The exemplary level of accuracy indicates that the traditional evaluation of the center, surface and mean temperatures with the standard infinite series retaining a large number of terms is no longer necessary.