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Numerical modeling example

Numerical modeling example. A simple s teel reheat furnace model – pg. 99-116. Hot steel slab. Rolling mill. Reheat furnace. Final product. The implicit solution. Same equations with a minor change. The slab …. Dimensions: 10 m long 1 m wide 0.3 m thick. Steel Properties:

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Numerical modeling example

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  1. Numerical modeling example A simple steel reheat furnace model – pg. 99-116 Hot steel slab Rolling mill Reheat furnace Final product

  2. The implicit solution Same equations with a minor change

  3. The slab … • Dimensions: • 10 m long • 1 m wide • 0.3 m thick • Steel Properties: • k = 35 W/m-K • CP = 473.3 J/kg-K • r = 7820 kg/m3 • Model heat transfer through the 0.3 m dimension • “Convective” heat transfer based on a furnace gas temperature of 1400 K • Heat transfer symmetry on both sides of the slab; h = 200 W/m2-K • Centreline of the slab has a zero gradient boundary condition (q = 0)

  4. Some computational considerations: No stability problems with this method. Time steps, t, and spatial steps, x, can be arbitrarily chosen by the user. Precision of the solution can be affected by the gird size (time and/or space) and this can be checked. We want to cover the slab region 0 < x < 0.15 m With x = 0.01 m we will require 15 cells

  5. Same problem formulation and set up: Three different types of cells (implicit form): • Boundary next to the hot combustion gases – radiation modeled as a convective process. One cell only. • An array of interior cells. Many cells. • The centre plane boundary condition – plane of symmetry with q = 0. One cell only.

  6. Cell adjacent to furnace gases: q = hA(T - TP) Accumulation within cell P = Input to cell P = TP evaluated at time step n+1 Output from cell P = TP and TE evaluated at time step n+1 Generation within cell P = 0 Accumulation = Input – Output + Generation

  7. General interior cells: Accumulation within cell P = Input to cell P = TW and TP evaluated at time step n+1 Output from cell P = TP and TE evaluated at time step n+1 Generation in cell P = 0

  8. Centreline Cell: Accumulation within cell P = TW and TP evaluated at time step n+1 Input to cell P = Output from cell P = 0 Generation in cell P = 0

  9. The set of equations involve 15 cells with cell 1 adjacent to the furnace gases through to cell 15 on the centre symmetry plane of the slab Equation for cell 1: Equations for cell 2 – 14: Equation for cell 15:

  10. Leads to a set of linear equations: A tridiagonal matrix

  11. The vector of knowns (usually true for linear problems) The vector of unknowns

  12. Diagonal elements of [A], Elements “banding” the diagonal coefficients in [A],

  13. For ease of writing a computer code, define dimensionless groups: Diagonal elements of [A]: Banding elements of [A]:

  14. Spreadsheet solution VBA (Excel add-in) format: 'Invoke TDMA to evaluate current temperatures ' Delta(1) = D(1) Y(1) = B(1) / Delta(1) For I = 2 To Number_of_Cells Delta(I) = D(I) - C(I) * E(I - 1) / Y(I - 1) Y(I) = (B(I) - C(I) * Y(I - 1)) / Delta(I) Next I X(Number_of_Cells) = Y(Number_of_Cells) For II = 2 To Number_of_Cells I = Number_of_Cells - II + 1 X(I) = Y(I) - E(I) * X(I + 1) / Delta(I) Next II ' 'Output from TDMA procedure is the solution vector X(I) Solver based on Tridiagonal Matrix Algorithm

  15. Consider the system of equations: [A] x = b With 5 equations and 5 unknowns:

  16. Decompose the original problem, [A] x = b To one of the form: [L][U] x = b Where[L] = and[U] = Lower triangular matrix Upper triangular matrix  Need to find matrices [L] and [U] whose product equals original [A] matrix

  17. With [A] x = band[L][U] x = b Let [U]x = ythen [L]y = b And b is known [L] = Solve for y by forward substitution in the system [L]y = b etc. i.e.

  18. With a solution for y from [L]y = b Then solve for x from[U]x = y and y known [U] = Solve for x by backward substitution in the system [U]x = y i.e. x5 = y5 etc. And x is the solution to the original problem!!

  19. A simpler case arises when the coefficient matrix [A] is tridiagonal: i.e. if [A] = [L] =

  20. [U] = With backward substitution solution

  21. 'Invoke TDMA to evaluate current temperatures ‘Forward Substitution Code Delta(1) = D(1) Y(1) = B(1) / Delta(1) For I = 2 To Number_of_Cells Delta(I) = D(I) - C(I) * E(I - 1) / Y(I - 1) Y(I) = (B(I) - C(I) * Y(I - 1)) / Delta(I) Next I ‘Backward Substitution Code X(Number_of_Cells) = Y(Number_of_Cells) For II = 2 To Number_of_Cells I = Number_of_Cells - II + 1 X(I) = Y(I) - E(I) * X(I + 1) / Delta(I) Next II ' 'Output from TDMA procedure is the solution vector X(I)

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