Making an allowance for state-dependent density when solving a heat conduction equation

A treatment method for an item of metal rolling stock, wherein the rolling stock is a flat hot item of rolling stock which extends in a longitudinal direction, in a width direction and in a thickness direction,

• wherein at a treatment time, the rolling stock is acted on at least substantially in the thickness direction by means of a treatment device,
• wherein at least for a period before the treatment time, the temporal development of a thermal state (Z) of the rolling stock is modelled by means of a model of the rolling stock (1) by iteratively solving at least one heat conduction equation,
• wherein control of the treatment device, taken as a basis for the treatment device to act on the rolling stock, is effected depending on that thermal state (Z) which is determined by means of the model for the rolling stock for the treatment time,
• wherein the density (ρ) of the rolling stock influences the heat conduction equation,
• wherein the density (ρ) is dependent on the respective thermal state (Z) of the rolling stock, wherein
• an allowance is made for the dependence of the density (ρ) of the rolling stock on the respective thermal state (Z) of the rolling stock in the heat conduction equation by a factor (a, a′) of the form
• a
• =
• xL
• ·
• xB
• xD
• ·
• ρ
• 0
• ⁢
• or
• ⁢
• a
• ′
• =
• 1
• x
• ⁢
• D
• 2
• ·
• ρ
• ,
• wherein ρ and ρ0 are densities (ρ, ρ0) of the rolling stock related to a current and a predetermined thermal state (Z, Z0) of the rolling stock and the coefficients xL, xB and xD arise as a quotient of an extent (L, B, D, L0, B0, D0) of the rolling stock in the longitudinal direction, in the width direction and in the thickness direction in the case of the respective and the predetermined thermal state (Z, Z0) of the rolling stock.