DOWNLOAD FINE and COARSE MESH MODELS w/results (132k)

Problem Description

The cross-sectional heat flow of an insulated pipe provides an excellent evaluation of the TAI conduction solver.  The cross section can be represented as a two-dimensional symmetric model as shown in the figure below.   Symmetry can be used since the temperatures are isothermal along the direction of curvature.

Two different meshes were tested: a coarse mesh (left - 210 elements) and a fine mesh (right - 3584 elements).  The expected outward temperature gradient is logarithmic—not a trivial linear solution.

Insulated Pipe Setup

Boundary Conditions

• The pipe inner temperature is 150°C.

• The insulation outer temperature is 20°C.

Assumptions

• Emissivity is set to zero (no radiation); the minimal software emissivity is slightly larger than zero.

• No convection

• No conduction along the pipe length

Objective

Predict the steady-state temperature gradient.

Analytical Solution

TAI Results

The convergence criteria was set to provide the maximum possible convergence.  The maximum temperature change at convergence was a numerical zero and the residual was 5.24557E-6 watts (course mesh) and 5.60893E-6 watts (fine mesh).  There is some modeling error due to boundary temperature application; conduction is computed from the centroids of elements.  The centroid of the inner and outer element rows are slightly different than the analytical radii.  Also radiation can not be completely eliminated in the solver.  There is a small amount of radiation which cannot be completely eliminated in the solver. Nonetheless, very good correlation was obtained.

Element #	Radius	Analytical	Model	RadTherm 5.0	WinTherm 5.0	MusesPro 5.0
154	30.3	90.7	Coarse	89.7	89.7	89.7
156	36.1	65.8	Coarse	64.6	64.6	64.6
2762	42.7	42.2	Fine	41.0	41.0	41.0
1047	26.2	110.9	Fine	110.7	110.7	110.7

Note: Test run by RES 3-2000 using versions 5.0.0.