# Nonlinear heat transfer (38883)

### Nonlinear heat transfer modelling (38883)

Jan Lienemann <lieneman (at) imtek (dot) de>, A. Yousefi <yousefi (at) iat (dot) uni-bremen (dot) de>, and Jan G. Korvink <korvink (at) imtek (dot) de>

## Introduction

The simulation of heat transport for a single device is easily tackled by current computational resources, even for a complex, finely structured geometry; however, the calculation of a multi-scale system consisting of a large number of those devices, e.g., assembled printed circuit boards, is still a challenge. A further problem is the large change in heat conductivity of many semiconductor materials with temperature. We model the heat transfer along a 1D beam that has a nonlinear heat capacity which is represented by a polynomial of arbitrary degree as a function of the temperature state. For accurate modelling of the temperature distribution, the resulting model requires many state variables to be described adequately. The resulting complexity, i.e., number of first order differential equations and nonlinear parts, is such that a simplification or model reduction is needed in order to perform a simulation in an acceptable amount of time for the applications at hand. Thus the need for model order reduction emerges.

## Model description

We model the heat transfer along a
1D beam with lengtha *L*, cross
sectional area *A* and nonlinear heat conductivity
represented by a polynomial in temperature *T(x,t)*
of arbitrary degree *n*:

*kappa(T)=a _{0}+a_{1}T+a_{2}T^{2}+...*

We output of the model is the temperature *T(x,t)*, the
degrees of freedom are the temperature from left to right.

*x=L*) is fixed at ambient temperature 0; this node does not occur in the model any more. The model features two inputs: The first one is a time-dependent uniform heat flux

*f*[W/m

^{2}] flowing in from the left end (at

*x=0*). The second one is a time dependant heat source

*Q*[W/m

^{3}] in the beam volume, e.g. from an electric current.

## Benchmark examples

An interactive matrix generator
has been created using Wolfram
Research's *webMathematica*. Models
produced by this generator are in the DSIF format, which allows
for nonlinear terms.

### Linear example (heat conductivity not temperature dependant)

Property | Value | |
---|---|---|

Number of nodes | 15 | |

Beam length (l) [m] | 0.1 | |

Cross-sectional area (A) [m^{2}] |
1e-4 | |

Material density (rho) [kg/m^{3}] |
3970 | |

Heat capacity
(C_{p}) [J/kg K] |
766 | |

Heat conductivity a_{0} [W/m K] |
36 |

### Nonlinear examples (heat conductivity temperature dependant)

Property | Value | |
---|---|---|

Number of nodes | 15 | |

Beam length (l) [m] | 0.1 | |

Cross-sectional area (A) [m^{2}] |
1e-4 | |

Material density (rho) [kg/m^{3}] |
3970 | |

Heat capacity
(C_{p}) [J/kg K] |
766 | |

Heat conductivity a_{0} [W/m K] |
36 | |

Heat conductivity a_{1} [W/m K^{2}] |
-0.1116 | |

Heat conductivity a_{2} [W/m
K^{3}] |
0.00017298 | |

Heat conductivity a_{3} [W/m
K^{4}] |
-1.78746e-7 | |

Heat conductivity a_{4} [W/m
K^{5}] |
1.3852815e-10 |

Property | Value | |
---|---|---|

Number of nodes | 410 | |

Beam length (l) [m] | 0.1 | |

Cross-sectional area (A) [m^{2}] |
1e-4 | |

Material density (rho) [kg/m^{3}] |
3970 | |

Heat capacity
(C_{p}) [J/kg K] |
766 | |

Heat conductivity a_{0} [W/m K] |
36 | |

Heat conductivity a_{1} [W/m K^{2}] |
-0.1116 | |

Heat conductivity a_{2} [W/m
K^{3}] |
0.00017298 | |

Heat conductivity a_{3} [W/m
K^{4}] |
-1.78746e-7 | |

Heat conductivity a_{4} [W/m
K^{5}] |
1.3852815e-10 |

The .m files contain matrices E, A, B, F and C and the vector f for the following system of equations:

E x' + A x | = B u + F f(x) |

y | = C x |

The two outputs are on the left end and in the middle of the beam.

Details of the implementation are available in a separate
report (File 4, 182 kB)
(A full paper appeared in the
*Proceedings of the 12th Mediterranean Conference on
Control and Automation*, June 6-9, 2004, Kusadasi, Aydin,
Turkey). A typical input to
this system is a step response; periodic on/off switching is also
possible. The reduced model should thus both represent the step
response as well as the possible influence of higher order
harmonics.

Last modified: Tue Jan 11 16:51:40 CET 2005