An analytical study of unsteady heat transfer in the rock surrounding a deep airway

The heat transfer between an airway and the air flowing though it is an unsteady problem. The governing equation of unsteady heat transfer was solved using the method of separation of variables. The solution is an infinite series including Bessel functions. The theoretical solution was analyzed by solving for the positive roots of the transcendental equation by iteration. The dimensionless surface temperature of the surrounding rock is only affected by the Bi number but not by the thermo-physical coefficients of the rock. The dimensionless coefficient of heat transfer, k, decreases with the Fo number similarly to the influence of the Bi number on k. A formula for determining the fully developed stage (FDS) suitable for unsteady heat transfer in the airway is proposed. The FDS from theoretical analysis occurs with Fo from 1.6 to 2. The ratio of excess temperature in the surrounding rock is independent of the initial conditions and only dependent on the Bi number and the relative position in the airway, at the FDS. The calculation error is large when using just the first term from the complete series when Fo is from 2 to 12. Five terms give a solution approximately equal to that found using the complete series. The first term could replace the complete series only when Fo is greater than 12. The FDS plays an important role in predication of the temperature field of the surrounding rock and in simplified calculations.