Numerical Uncertainty in Heat Conduction Calculations for a Tube Immersed in a Fluidized Bed

Abstract

A numerical code has been evaluated with regard to numerical uncertainties involved in calculating heat flux through the wall of a horizontal tube in a bubbling bed of sand. The two-dimensional unsteady heat conduction equation is solved numerically with a non-linearly varying temperature boundary condition prescribed according to measurements. The finite difference method used is an implicit method with a second-order accurate discretization scheme both in temporal and spatial domains. Previous literature dealing with numerical calculations in heat conduction usually reports any detailed study about numerical errors. In the present analysis, a rigorous grid dependence test is applied, and it is shown that the results, in particular heat flux, are very sensitive to the grid size and distribution. Therefore, to achieve better grid convergence when heat flux is sought, the discretization error in the heat flux rather than in the temperature calculations should be considered. This should be done even in cases where temperature is the primary unknown, because it is usually the derivative of temperature which is of any physical importance. The errors are also strongly dependent on the number of iterations which need to be increased as the grid is refined. The present application showed that a non-uniform grid refinement throughout the calculation domain gives a more efficient (less expensive) solution than uniform grid refinement. Furthermore, for calculation of the temperature gradient at the wall, a parabolic profile assumption gives a faster grid convergence compared to a linear profile assumption. The present study shows that the previously published results concerning calculated heat transfer coefficients should be interpreted with caution, unless the authors have provided some measure of grid dependency of their results.