# Convection (38867)

### Convective Thermal Flow Problems (38867)

Christian Moosmann, moosmann@imtek.uni-freiburg.de

Andreas Greiner, greiner(at)imtek.uni-freiburg.de

Many thermal problems require simulation of heat exchange between a solid body and a fluid flow. The most elaborate approach to this problem is computational fluid dynamics (CFD). However, CFD is computationally expensive. A popular solution is to exclude the flow completely from the computational domain and to use convection boundary conditions for the solid model. However, caution has to be taken to select the film coefficient.

An intermediate level is to include a flow region with a given velocity profile, that adds convective transport to the model. Compared to convection boundary conditions this approach has the advantage that the film coefficient has not to be specified and that information about the heat profile in the flow can be obtained. A drawback of the method is the greatly increased number of elements needed to perform a physically valid simulation, because the solution accuracy when employing upwind finite element schemes depends on the element size. While this problem still is linear, due to the forced convection, the conductivity matrix changes from a symmetric matrix to an un-symmetric one. So this problem type can be used as a benchmark for problems containing un-symmetric matrices.

Fig. 1. Convective heat flow examples: 2D anemometer model (left), 3D cooling structure (right)

Two different designs are tested: a 2D model of an anemometer-like structure mainly consisting of a tube and a small heat source (Fig 1 left) [1]. The solid model has been generated and meshed in ANSYS. Triangular PLANE55 elements have been used for meshing and discretizing by the finite element method, resulting in 19 282 elements and 9710 nodes. The second design is a 3D model of a chip cooled by forced convection (Fig 1 right) [2]. In this case the tetrahedral element type SOLID70 was used, resulting in 107 989 elements and 20542 nodes. Since the implementation of the convective term in ANSYS does not allow for definition of the fluid speed on a per element, but on a per region basis, the flow profile has to be approximated by piece-wise step functions. The approximation used for this benchmarks is shown in figure 1.

The Dirichlet boundary conditions are applied to the original system. In both models the reference temperature is set to 300 K, Dirichlet boundary conditions as well as initial conditions are set to 0 with respect to the reference. The specified Dirichlet boundary conditions are in both cases the inlet of the fluid and the outer faces of the solids. Matrices are supplied for the symmetric case (fluid speed is zero; no convection), and the unsymmetric case (with forced convection). Table 1 shows the output nodes specified for the two benchmarks, table 2 links the filenames according to the different cases.

Matrices are in the Matrix Market format. The matrix name is used as an extension of the matrix file. *.C.names contains a list of ouput names written consecutively. The system matrices have been extracted from ANSYS models by means of mor4fem.

Model |
Number | Code | Comment |

Flow Meter | 1 | out1 | outlet position |

2 | out2 | outlet position | |

3 | SenL | left sensor position | |

4 | Heater | within the heater | |

5 | SenR | right sensor position | |

cooling Structure | 1 | out1 | outlet position |

2 | out2 | outlet position | |

3 | out3 | outlet position | |

4 | out4 | outlet position | |

5 | Heater | within the heater |

Model |
fluid speed (m/s) | Link | File Size, Bytes |

Flow Meter | 0 |
(File 1) |
664950 |

0.5 | (File 2) |
775969 | |

cooling Structure | 0 | (File 3) |
4055692 |

0.1 | (File 4) |
4159963 |

Further information on the models can be found in [3] where model reduction by means of the Arnoldi algorithm is also presented.

## Bibliography

1
H. Ernst : *High-Resolution Thermal
Measurements in Fluids*, PhD thesis, University of Freiburg, Germany(2001).

2
C. A. Harper : *Electronic packaging and
interconnection handbook*, New York McGraw- Hill, USA (1997)

3
C. Moosmann, E. B. Rudnyi, A. Greiner, J. G. Korvink:
*Model Order Reduction for Linear Convective Thermal Flow*,
Proceedings of 10th International Workshops on THERMal
INvestigations of ICs and Systems, THERMINIC2004, 29 Sept - 1 Oct ,
2004, Sophia Antipolis, France.