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Radius-normal property of the representation ellipsoid

As previously described, the heat flux, J, in an anisotropic material is related to the temperature gradient, gradT, by:

J = k(gradT)

The components of the temperature gradient parallel to the principal axes (x, y and z) will be:

gradT_x = (gradT)l,   gradT_y = (gradT)m,   gradT_z = (gradT)n

where l, m and n are the direction cosines specifying the direction of gradT, and for example, l is the cosine of the angle between the x-axis and the thermal gradient vector.

Therefore, the components of the heat flow are:

j_x = k₁(gradT)l,   j_y = k₂(gradT)m,   j_z = k₃(gradT)n

where k₁, k₂ and k₃ are the values of thermal conductivity along the principal axes, x, y and z, and are called the principal values.

The direction cosines of J are therefore proportional to k₁l, k₂m, k₃n.

We have already shown that the equation for the representation ellipsoid is

k₁x² + k₂y² + k₃z² = 1

Suppose P is a point on the surface of the ellipsoid such that OP is parallel to the temperature gradient. The coordinates of P are therefore (rl, rm, rn), where r is the distance OP.

r² = k₁x² + k₂y² + k₃z²

The components of the gradient at (x₁ y₁ z₁) are obtained by partial differentiation of this equation, which gives components in the directions of the x, y, and z axes of:

,   ,   

The components are therefore equal to

But since

x₁ = rl, y₁ = rm, z₁ = rn

the normal at P has direction cosines proportional to k₁l, k₂m, k₃n. Hence the normal at P is parallel to the heat flux.

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