VVT14 - Hollow cylinder with convective boundary conditions

Description

The purpose of this test case is to determine the temperature distribution through the thickness of a hollow cylinder by convective heat transfer.

Geometry

The hollow cylinder is defined with an inner diameter of 600 mm and an outer diameter of 780 mm. The axisymmetric modeling technique is used, which requires the sketch on the XZ plane. Cylinder thickness is 50 mm, therefore a plate is sketched with the length of 50 mm and height of 10 mm, starting from the inner diameter of the cylinder. A bounded plane was attached to this plate.

Simulation model

This model uses the Axisymmetric Thermal solution type.

A 2D mesh of quadrilateral thin shell elements is generated on the surface of the plate, using 30 mm element size.

The meshed elements have the following material and physical properties:

• Thermal conductivity: k = 40 W/m·°C
• Mass density: ρ = 1kg/m³
• Thickness: not required for the axisymmetric thermal analysis

The following boundary conditions are applied:

• Convection to Environment constraint on the inside edge of the cylinder with a heat transfer coefficient h_i = 150 W/m²·°C and an environment temperature T_i = 500 ºC.
• Convection to Environment constraint on the outside edge of the cylinder with a heat transfer coefficient h_o = 142 W/m²·°C and an environment temperature T_o = 20 ºC.

The following figure shows the mesh and the location of the convective boundary conditions.

The following solution options are set:

• Solution Type = Steady State
• Results Options: Select the heat flux and temperature at nodes and elements

The default solver parameters are selected.

Theory

The heat transfer rates by conduction through the cylinder thickness and by convection along the inside surface of the hollow cylinder is determined [4], [5], [6] using the following equations:

From these results, the inner surface temperature is determined using the following equations:

From these results, the outer surface temperature is determined using the following equations:

Results

The following table compares the node temperatures computed by the thermal solver with theoretical values.

The following table shows the heat transfer rates by conduction through the cylinder thickness.