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Wärmeleitung in anisotropen zusammengesetzten Körpern beliebiger Gestalt

Heat conduction in anisotropic composites of arbitrary shape (a numerical analysis)

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Abstract

To investigate the thermal response of composed anisotropic structures, a numerical study has been carried out. By the use of a numerical mapping technique it was possible to handle complex multi-body geometries very efficiently. Due to the transformation method the restriction of the well known finite difference method to simple solution regions is removed. For the iterative solution of the corresponding finite difference equations the Strongly Implicit Procedure (SIP) has been employed. Based on the solution methodology, several numerical examples revealing the effects of anisotropy in thermal conductivity and composite structures are presented.

Zusammenfassung

Um das thermische Verhalten von zusammengesetzten anisotropen Körpern zu untersuchen, wurden numerische Rechnungen durchgeführt. Unter Verwendung einer numerischen Transformationsmethode war es möglich, auch komplexe Geometrien, die aus mehreren Körpern bestehen, effektiv zu behandeln. Die Beschränkung der bekannten Finiten Differenzen Methode auf relativ einfache Lösungsgebiete, wird durch diese Transformationsmethode beseitigt. Zur iterativen Lösung des entsprechenden Gleichungssystems wurde ein implizites Verfahren (SIP) angewandt. Mit Hilfe dieser Lösungsmethode wurden anisotrope Wärmeleitvorgänge in Körpern komplexer und zusammengesetzter Geometrie berechnet und der Einfluß der Anisotropie auf zusammengesetzte Gebiete aufgezeigt.

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Abbreviations

D:

domain

J:

Jacobian of transformation (=xξyη−xηyξ)

k n11 , k n12 , k n22 :

dimensionless thermal conductivities in macro element n (k nlk =λ nlk /λ 111 )

m:

total number of grid points

N:

total number of macro elements

P,Q:

coordinate control functions

q:

heat flux per unit area

Res:

residuum

x, y:

cartesian coordinate directions in physical plane

α,β,γ:

transformation factors (α=x 2η +y 2η ;β=xξxη+yξyη;γ=x 2ξ +y 2ξ )

Γ:

bounding surface of geometry

λ:

thermal conductivity

θ:

dimensionless temperature

ξ,η:

coordinate directions in transformed plane

―:

matrix notation

x:

derivative with respect to x

y:

derivative with respect to y

n:

normal

ij:

nodal point i,j

ξ:

derivative with respect to ξ

η:

derivative with respect to η

→:

vector notation

n:

number of macro element

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Authors and Affiliations

  1. Institut für Technische Thermodynamik, Technische Hochschule Darmstadt, Petersenstraße 30, 6100, Darmstadt, Federal Republic of Germany

    U. Projahn,  H. Rieger &  H. Beer

Authors
  1. U. Projahn
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  2. H. Rieger
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  3. H. Beer
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Projahn, U., Rieger, H. & Beer, H. Wärmeleitung in anisotropen zusammengesetzten Körpern beliebiger Gestalt. Wärme- und Stoffübertragung 15, 223–232 (1981). https://doi.org/10.1007/BF01003642

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  • DOI: https://doi.org/10.1007/BF01003642

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