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Analysis of transient nonlinear heat conduction in wood using finite-difference solutions

Beschreibung des nichtlinearen Wärmeüberganges in Holz mit Hilfe von Finite-Differenz-Methoden

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Abstract

This paper describes three-dimensional, transient heat conduction in a rectangular piece of wood with crossgrain, and in an orthotropic wooden cylinder. Computerized solutions of a generalized, nonlinear heat equation are derived by discretizing the space and the time domains, using explicit and implicit finite-difference techniques. A simplified example of linear heat conduction in cylindrical coordinates illustrates how to apply the finite-difference solutions. The accuracy of the solutions for this special case is evaluated via comparing them with a well-known exact solution.

Zusammenfassung

Der Verlauf des dreidimensionalen Wärmeübergangs innerhalb eines Holzquaders mit quer verlaufender Faserrichtung und in einem orthotropen Holzzylinder werden beschrieben. Numerische Lösungen der allgemeinen, nicht-linearen Wärmegleichung werden abgeleitet durch Unterteilen in Raum-und Zeitintervalle mit Hilfe der expliziten und impliziten Finite-Differenz-Methode. Ein vereinfachtes Beispiel einer linearen Wärmeleitung in Zylinder-Koordinaten beleuchtet die Anwendungsmöglichkeiten dieses Lösungsansatzes. Die Genauigkeit der Lösungen für diesen Spezialfall wird durch Vergleich mit der bekannten exakten Lösung abgeschätzt.

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Abbreviations

A :

axial direction

a :

coordinate in the axial direction (m)

C :

circumferential direction

c :

coordinate in the cirumferential direction (m)

c.. :

directional cosine

Cp :

specific heat at constant pressure (J/kg · °C)

D :

distance between mesh points (m)

d :

ordinary differential operator

Fo :

Fourier number

{G} :

gradient of temperature

J :

Bessel function

[K] :

thermal conductivity tensor

k :

thermal conductivity (W/m · °C)

kcr :

“critical” thermal conductivity (W/m · °C) in the stability criterion

[I] :

identity matrix

L :

length (m)

[M] :

matrix of directional cosines

MC :

moisture content based on dry mass (percent)

Q :

heat flux (W/m2)

R :

radial direction

r :

coordinate in the radial direction (m)

SG :

specific gravity based on dry mass and green volume (−)

T :

temperature (°C)

V :

volume (m3)

X :

X-axis

x :

coordinate in theX-direction (m)

Y :

Y-axis

Y :

coordinate in theY-direction (m)

Z :

Z-axis

z :

coordinate in theZ-direction (m)

δ:

interval (−)

α:

diffusivity coefficient in radial direction (m2/s)

β:

fiber grain angle (rad)

Γ:

growth ring angle (rad)

ŗ:

time (s)

Ω:

0,0.5, 1

δ:

partial differential operator

φ:

angle in cylindrical coordinates (rad)

θ:

density (kg/m3)

a :

axial direction

avg:

average

c :

circumferential direction

e :

exterior

i :

nodal point 0, 1, 2, 3… inY-orC-direction

m :

nodal point 0, 1, 2, 3… inX-orR-direction

max:

maximum

min:

minimum

n :

nodal point 0, 1, 2, 3… inZ-orA-direction

r :

radial direction

t :

total

x :

X-direction

y :

Y-direction

z :

Z-direction

o :

initial

∞:

ambient

p :

time level 0, 1, 2, 3…

T :

transpose

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Author notes

  1. P. Steinhagen

    Present address: Engineering Faculty, University of Biobio, Concepcion, Chile

Authors and Affiliations

  1. Forestry Research and Experiment Station, B. P. 763, Agdal-Rabat, Morocco

    A. Khattabi

  2. College of Forestry, Wildlife and Range Sciences, University of Idaho, 83843, Moscow, ID, USA

    P. Steinhagen

Authors
  1. A. Khattabi
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  2. P. Steinhagen
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Khattabi, A., Steinhagen, P. Analysis of transient nonlinear heat conduction in wood using finite-difference solutions. Holz als Roh-und Werkstoff 51, 272–278 (1993). https://doi.org/10.1007/BF02629373

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