1. Numerical modeling example A simple s teel reheat furnace model – pg. 102-107 Reheat furnace Hot steel slabRolling mill Final product

2. The problem: Steel companies use models to simulate the batch heating of steel slabs (~4 hours, reaching 1450-1550K) prior to the rolling mill. This software is built into their process control systems. Two-dimensional models often suffice for this purpose (a balance between computing speed and model detail). The reheat furnace may contain many steel slabs and their heating progress must be monitored and paced through the furnace consistent with downstream operations in the rolling mill. Different zones of the furnace provide different heating strategies over the heating cycle.

3. A full model would involve: Implicit finite volume formulation. Variable properties – thermal conductivity and heat capacity are a function of temperature and steel composition. The phase change in steel structure from BCC to FCC at 910  C. Surface oxidation of the steel to the various oxides – occurs more rapidly above 570  C. FeO 0.95 (dominant oxide) has a heat of formation (a generation term) and low thermal conductivity (good insulator).

4. Our treatment will involve: Heat a steel slab from ambient temperature to a relatively uniform ~1400 K over a 4 hour period. A one-dimensional model will be used. Constant properties, no phase change and no scale growth. Explicit formulation – spreadsheet demonstration. Implicit formulation – can be coded in matlab or VBA (Excel add-in feature).

5. The slab … Dimensions: 10 m long 1 m wide 0.3 m thick Steel Properties: k = 35 W/m-K C P = 473.3 J/kg-K  7820 kg/m 3 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/m 2 -K Centreline of the slab has a zero gradient boundary condition (q = 0)

6. Problem formulation and set up: Three different types of cells (explicit form): 1.Boundary next to the hot combustion gases – radiation modeled as a convective process. One cell only. 2.An array of interior cells. Many cells. 3.The centre plane boundary condition – plane of symmetry with q = 0. One cell only.

7. Cell adjacent to furnace gases: q = hA(T  - T P ) Accumulation within cell P = Input to cell P = Output from cell P = Generation within cell P = 0 Accumulation = Input – Output + Generation

8. Alternative surface cell formulation: Evaluate the cell P temperature as “true” surface value Place the P node at the surface Adjust the cell volume (optional) Accumulation term is now half the original value used previously

9. General interior cells: Accumulation within cell P = Input to cell P = Output from cell P = Generation in cell P = 0

10. Centreline Cell: Accumulation within cell P = Input to cell P = Output from cell P = 0 Generation in cell P = 0

11. Some computational considerations: We want to cover the slab region 0 < x < 0.15 m With  x = 0.01 m we will require 15 cells Explicit scheme stability criteria: Time steps of  t = 2.5 – 5 s will meet this criteria To cover the 3-4 h time required for the batch heating process we will need ~3000 – 4000 time steps!

12. 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:

13. Spreadsheet solution