# Heat Transfer Between Concentric Cylinders with Heat Source

In this application, AcuSolve is used to simulate a 2D cable problem to demonstrate one way Flux-AcuSolve coupling. The 2D cable is used to demonstrate how the volumetric heat load conducts through a solid. AcuSolve results are compared with analytical results as described in Incropera (2006). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to model cases with volumetric heat source applied on a solid volume.

## Problem Description

The problem consists of two solid cylinders in contact with each other wherein the inner solid cylinder is provided with a volumetric heat source of 1.46686 W and is in contact with the outer solid cylinder, the outer surface of which is maintained at a temperature of 20 degrees C (293K). The radius of the inner cylinder is 0.002 m and the radius of the outer cylinder is 0.003 m. This problem forms the basis of a simple conduction analysis between two concentric cylinders. The only difference from the basic problem is that the heat source is calculated using another software called Flux and is provided using AcuConsole’s EMag (Electromagnetics) Manager to account for volumetric losses from Flux to AcuSolve.

The simulation was performed as a two dimensional problem by constructing a volume mesh that contains a single layer of elements normal to the radial and circumferential directions.

## AcuSolve Results

The AcuSolve solution converges to a steady state and the results reflect the mean temperature distribution. The heated volume (inner cylinder) is at the highest temperature due to the heat source provided and conducts through the outer cylinder volume and forms a temperature gradient along the radial direction between both the inner and outer solid volume. The temperature within the medium increases logarithmically as a function of the radius. The following images show the temperature with the medium as a function of radius from the edge of the inner cylinder. The temperature is compared with the analytical solution as described in Incropera (2006). The image below shows black circles representing the analytical solution and a solid red line for the AcuSolve results.

## Summary

The AcuSolve solution compares well with analytical results for the 2D solid conduction model case with volumetric heat source applied on a solid volume. In this application, the temperature within the cylinders is driven by a gradient along the outer and inner solid volumes, due to the volumetric heat source and fixed temperature. The AcuSolve solution for the temperature as a function of radius matches well compared to the analytical solution.

## Heat Transfer Between Concentric Cylinders with Heat Source

AcuConsole database file: <your working directory>\annulus_heat_source\annulus_heat_source.acs

Global

• Problem Description
• Analysis type - Steady State
• Turbulence equation - Advective Diffusive
• Absolute temperature offset - 273.15
• Auto Solution Strategy
• Convergence tolerance - 0.0001
• Relaxation Factor - 0.4
• Flow - off
• Temperature - on
• Material Model
• Insulation
• Density
• Type - Constant
• Conductivity - 2702.0
• Specific Heat
• Type - Constant
• Conductivity - 908.0
• Conductivity
• Type - Constant
• Conductivity - 0.8

Model

• Volumes
• Solid
• Element Set
• Medium - Solid
• Material model - Insulation
• SolidHeated
• Element Set
• Medium - Solid
• Material model - Insulation
• Total heat source - 1.46686 (applied through EMag Manager)
• Surfaces
• +ZInner
• Simple Boundary Condition - (disabled, no BC needed)
• +ZOuter
• Simple Boundary Condition - (disabled, no BC needed)
• -ZInner
• Simple Boundary Condition - (disabled, no BC needed)
• -ZOuter
• Simple Boundary Condition - (disabled, no BC needed)
• InterfaceAll
• Simple Boundary Condition - (disabled, no BC needed)
• InterfaceInner
• Simple Boundary Condition - (disabled, no BC needed)
• InterfaceOuter
• Simple Boundary Condition - (disabled, no BC needed)
• OuterWall
• Simple Boundary Condition
• Type - Wall
• Temperature BC type - Value
• Temperature - 20.0 K

## References

F. P. Incropera and D. P. DeWitt. "Fundamentals of Heat Transfer – Sixth Edition". John Wiley & Sons. New York. 2006.