Rolling Stock Heat Conduction With State-Dependent Density
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Solution Overview
Problem
Existing methods for modeling the thermal state of flat metal rolling stock during production, such as in casting, roughing, and cooling, face inaccuracies due to assuming constant density, which is not applicable when the metal has already solidified and dimensions change, leading to inaccuracies in heat conduction equation solutions.
Innovation Solution
Incorporating a factor that accounts for the dependence of density on the thermal state of the rolling stock in the heat conduction equation, using coefficients that adjust for changes in length, width, and thickness, allowing for accurate modeling of state-dependent density and improved thermal behavior prediction.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If constant density is assumed in the heat conduction equation, then the calculation is simpler, but the accuracy of thermal state modeling deteriorates when dimensions change
Solution Approach 1:
The patent applies the dynamics principle by transitioning from a static constant density assumption to a dynamic state-dependent density model. The density is expressed as a function of the thermal state (temperature) and dimensional changes: ρ(T,L,W,H) = ρ₀(T₀,L₀,W₀,H₀) × (L₀/L) × (W₀/W) × (H₀/H). This allows the density to adapt dynamically as the rolling stock cools and contracts, maintaining calculation accuracy while preserving computational feasibility through the functional relationship.
Solution Approach 2:
The patent implements parameter changes by making the density parameter variable rather than constant. The density is updated based on changes in temperature and dimensional parameters (length, width, height) during the cooling process. This is achieved by incorporating the relationship ρ = ρ₀ × (L₀/L) × (W₀/W) × (H₀/H) into the heat conduction equation, allowing the system to account for thermal contraction effects on density without requiring complex real-time measurements.
2Measurement precision
If state-dependent density is incorporated in the heat conduction equation, then the accuracy of thermal state modeling is improved, but the calculation complexity increases
Solution Approach 1:
The patent applies preliminary action by pre-establishing the functional relationship between density and dimensional/thermal parameters before solving the heat conduction equation. The density function ρ(T,L,W,H) is defined in advance based on the initial state (T₀, L₀, W₀, H₀) and the expected thermal contraction behavior. This pre-characterization allows the density to be updated systematically during the cooling process without requiring complex iterative calculations or real-time measurements, thus improving accuracy while managing computational complexity.
3Device complexity
If dimensional changes are not considered, then the model is simpler, but the precision of heat conduction solutions deteriorates during cooling
Solution Approach 1:
The patent implements parameter changes by incorporating dimensional parameters (L, W, H) into the density expression to account for thermal contraction. The relationship ρ = ρ₀ × (L₀/L) × (W₀/W) × (H₀/H) explicitly links density changes to dimensional changes during cooling. This allows the model to maintain precision in heat conduction solutions by considering the coupled effects of temperature and dimension changes, while avoiding the need for fully three-dimensional moving boundary calculations through the use of this functional relationship.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
This approach enables accurate modeling of the thermal state and behavior of rolling stock, improving the precision of heat conduction equation solutions and reducing the need for shifting support points, thus enhancing the control and treatment processes in metal rolling stock production.
Implementation Method 1
the temporal development of a thermal state of the rolling stock is modeled by means of a model of the rolling stock by iteratively solving at least one heat conduction equation
Data Source
AI summary
At a treatment time, a treatment device is intended to act, at least substantially in the thickness direction, on a planar, hot item of metal rolling stock. At least for a period before the treatment time, the development over time of a thermal state (Z) of the rolling stock is modelled by means of a model of the rolling stock by iteratively solving at least one thermal conductivity equation. The treatment device is controlled on the basis of the particular thermal state (Z) determined by the model for the rolling stock for the treatment time.