Suitability of the Method Of Lines for rendering analytic/numeric solutions of the unsteady heat conduction equation in a large plane wall with asymmetric convective boundary conditions

• Unsteady, one-dimensional, heat conduction in a large plane wall with asymmetric convective boundary conditions.
• Implementation of the Method Of Lines (MOL).
• The adjoint system of first order ordinary differential equations is solved analytically with the eigenvalue method.
• Piecewise temperatures at the lines are expressed in terms of linear combinations of exponential functions of time.
• A practical example deals with the temperature–time evolution in a large single-pane, window.

The primary objective of the present study is to implement the Method Of Lines (MOL) for the analysis of the unsteady, one-dimensional, heat conduction equation in a large plane wall with different convective boundary conditions at the two exposed surfaces. In the equation, MOL discretizes the space derivative while leaving the time derivative continuous. By way of MOL, the adjoint system of linear, first order ordinary differential equations will be solved analytically (not numerically) with the eigenvalue method. The outcome of the computational procedure provides a discrete sequence of piecewise temperatures-time variations at each line, which is expressed in terms of linear combinations of exponential functions of time containing the eigenvalues and eigenvectors. A practical example dealing with the temperature evolution in a large single-pane window is tackled with two meshes, one having three and the other having five lines. The collection of analytic/numeric temperature–time solutions provided by MOL and the eigenvalue method exhibits excellent quality at all time.