Example 1
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.
|
Pipe (Inner Cylinder) |
Insulation (Asbestos) |
|
10mm Inner Radius |
20mm Inner Radius |
|
20mm Outer Radius |
30mm Thick |
|
k = 19.0 |
k = 0.2 |
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.
Learn More about Validation
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. Symmetry can be used since the temperatures are isothermal along the direction of curvature.
A mild steel bar 100mm long is initially heated to 100°C steady state. At time>0 the ends of the bar are changed to a constant 20°C. This problem is a one-dimensional dynamic conduction problem.
The fin is created as a flat plate. Boundary conditions were applied by holding strips of elements at the two ends at constant temperatures. The solution was converged its maximum. The theoretical and TAI results are very closely matched.
We have a simple model of 2 concentric cylinders. The model contains three thermal nodes, one of which is a constant temperature boundary node. The TAI-obtained temperature of the outer cylinder was 442.72°C, which is extremely close to the 442.71°C analytical.
Solar energy is applied to two parallel plates of glass separated by a small distance. The goal is to determine the fraction of heat transferred to the plates and to the environment. The relative error between RadTherm and the analytical solution is insignificant.
This problem is similar to the previous validation, except there is one plate of glass above another surface with a given absorptivity. We determine the fraction of energy absorbed by the second glass surface ("collector"). The relative error between RadTherm and the analytical solution is insignificant: 0.00004%.
