2. METAL 2007, HRADEC nad MORAVICI, CZECH REPUBLIC
APPLICATION OF THE MATHEMATICAL MODELING TO THE DYNAMIC DESIGN OF THE THERMAL SPACE OF THE FURNACES FOR ROLLING MILLS
Dan CONSTANTINESCU
University Politehnica of Bucharest, faculty Material’s Science and Engineering

4. 1. HEAT TRANSFER INSIDE THE BILLETS
Using the Bessel functions, it was deduced for a cylindrical billet:
temperature for the surface:
average final temperature:
Temperature’s equation in the section of the cylinder may be expressed by:
CONDITION:

5. 1. HEAT TRANSFER INSIDE THE BILLETS
Figure 1: Temperatures in the cylindrical billet (140mm)
Figure 2: Analyze of the temperature in the cylindrical billet applying equation (3)

6. 1. HEAT TRANSFER INSIDE THE BILLETS
A study of the mechanism of thermal stresses and the establishing of the critical thermal values was imposed in order to include it in the mathematical remodeling of the thermal space
In order to obtain the variation of thermal stresses in the round billet it is used the Bessel function
If “R” is the cylinder radius and “r” the current radius
r=R (surface)
r=0 (axis)
β: coefficient of dilatation E: Young module ν: Poisson coefficient
ν1, φ1: series of Bessel function Δθ: temperature gradient

7. 1. HEAT TRANSFER INSIDE THE BILLETS
For whole section of the billet (cylinder), if the heating is symmetrical (1=2):
In the case of billets with rectangular section with the thickness X, the admitted thermal stresses reported to (x) axis:
x=0; axial and tangential stress
C: furnace’s temperature
mi: temperature in the centre of the billet
m0: initial temperature of the billet
ms: temperature at the surface of the billet
x=X; axial and tangential stresses

8. 1. HEAT TRANSFER INSIDE THE BILLETS
In order not to exceed the level of the admitted resistance of the specific category of the steel it is necessary to apply the above-mentioned equation and to obtain a model of the thermal regime of the furnace.
5. Temperatures in case of a 70mm rectangular billet of carbon steel sample

9. 1. HEAT TRANSFER INSIDE THE BILLETS
The value of σ0 represents the residual stress in the material due to a precedent thermo-technological process. The value of σ0 do not represents the cracking resistance of the material, but the true value of the thermal stress for specific heating conditions, at the beginning of heating process
The model which connects the thermal stress and the thermal field of the aggregate is obtained by simulation on the computer.
The values of the thermal stress can be expressed function of the temperature by a polynomial equation with the general form
(there are used to calculate the real values in the centre of the sample not for the average value in the section):
The results concerning the modeling of thermal fields in the furnace are used as a component included to the re-modeling of the thermal space of the aggregate. The graph depicted in the figure 6 explain the relation between the heating conditions necessary for a specific category of steels and the geometry of the vault.

10. Establishing of the admitted temperature of the furnace, depending on Δθadmis (maxim) and the billet’s dimensions (Φ=300mm)
Θc=850C
Θc=1300C

11. But… 2. THE PROBLEME OF ENERGY AND MASS TRANSFER
The theoretical output t, give the energetic efficiency and is strongly connected to the energy saving problem
The oxidation process of the metal can be controlled by the air excess coefficient.
If it is defined “the factor of the fuel” :
The equations above are used to choose the thermal source in correlation with the preheating degree of the fuel and the combustion air in order to control the oxidation process.
But…

12. …It is also necessary to correlate the temperature of the flue gases, temperature of the thermal isolation and the temperature of the steel. The establishing of the values of the heat exchange coefficients is yet difficult.
gpm: heat exchange coefficient from the gases to the metallic material, if it is considerate that the temperature of the gases is the same with the temperature of the thermal isolation, kJ.m-2.h-1.K-1
pm: heat exchange coefficient by radiation between the thermal isolation and the metal, kJ.m-2.h-1.K-1
gp: radiation heat exchange coefficient between the gases and the thermal isolation, kJ.m-2.h-1.K-1
c : convection heat exchange coefficient between the gases and the thermal isolation, kJ.m-2.h-1.K-1
g: temperature of the flue gases, oC
p: temperature of the thermal isolation, inside the furnace, oC
: thermal emisivity coefficients
σ=s/S
s: heated surface of the billets, m2
S: surface of the thermal isolation, m2
From the above equation, the following relations will be used for modeling:
- thermal isolation-steel
- flue gases-steel

13. (qex is the conduction thermal flow) Particularity of the furnace
In the case when using natural gases for rolling mill furnaces, the particular equations are:
For the computation of the flue gases temperature it is obtained:
(qex is the conduction thermal flow)
Particularity of the furnace
For example, in the case of a rotary type furnace for circular billets it can be established the dependence of the flue gases at the exit from the furnace (g), furnace’s productivity (P) and the disposal mode of the burners . It was deuced:
D,d: dimensions of the circular furnace
1, 2: final and initial temperature of the billets,

14. Main sectors of the dynamic of the gazes in a continuous furnace
3. DYNAMIC OF THE FLUE GASES
1. In the case of the walking beam furnaces it is necessary that the value of the flue gazes speed is near to the critical level (wcr), it means in the sector of the turbulent regime. It is not enough to obtain this value at the exit from the burners, but to maintain it in the most of the sectors of the thermal space.
2. In order to obtain the most uniform temperature, it is necessary to assure an advanced recirculation of the gazes.
Main sectors of the dynamic of the gazes in a continuous furnace

15. the walking beam furnace
3. DYNAMIC OF THE FLUE GASES
The experimental physical model for the recirculation of the flue gases in the case of
the walking beam furnace

16. 3. DYNAMIC OF THE FLUE GASES
Simulation of the jets in the thermal space of the walking beam furnace

17. Speed distribution in the tap holes
3. DYNAMIC OF THE FLUE GASES
WHAT TO ANALYSE BY MODELING ?
By experiments and mathematical modeling it was remarked that, the abstraction of the flue gazes do not have a uniform distribution; the maximum value of the speed is reached in the central tap holes, simultaneously with low values of the speed thru the lateral tap holes
Influence of the temperature and of the jet of the gazes on the dynamic in the furnace depending on the heating zone
Speed distribution in the tap holes
of the furnace

18. 3. DYNAMIC OF THE FLUE GASES
Establishing of the temperature ta function of the recycling coefficient (K=1+Dr/Dt) and the temperatures t1 and t2.
t1: temperature of the primary jet
t2: temperature of the recycled jet
ta: temperature of the mixed jets

19. 4. GEOMETRY OF THE THERMAL SPACE
Main monitoring points recommended for the walking-beam furnace
Temperature monitoring in the walking beam furnace
CO2 monitoring in the walking beam furnace

20. 4. GEOMETRY OF THE THERMAL SPACE
A basic schema for modeling the geometry of the continuous furnace (walking beam furnace)

21. group A1: using air from the heat recovery R1 which can assure a higher temperature at the superior level of the furnace
group A3: used only in special conditions, in connection with the geometry of the thermal space in the first heating zone
in order to assure dynamic and thermal regime, the geometry of the vault (especially in the first zone) is decisive
group A2: burners using air from R2 in order to assure a lower temperature at the inferior levels of the furnace
group ARFP of special burners
the low oxidation of the steel is also highly influenced by the profile of the vault in the firs zone

22. 5. Discussions
The results from the studies regarding the thermal stresses contribute at the re-modeling of the geometry of the thermal space of the furnace. The aspects regarding the design of the vault are essential in determining the dynamic and the thermal solutions.
Starting from the diagrams that show the variation of maximum admitted temperature of the furnace it is possible the modeling of the vault of the continuous working furnace.
In order to reduce the oxidation and the decarburizing process an important action regards the control of chemical composition of the flue gases. This is possible by the control of the air combustion excess coefficient and the designee of the heating furnace.
Using the proposed general solutions for the remodeling of the thermal regime it can be obtained a better control of the temperatures in each heating zone of the furnace and to correlate it with the necessary temperatures of the billets. It is also possible to control the temperature of the thermal isolation, and by this to save thermal energy.
Using the results of the modeling, it is possible to control the flue gases temperature in each heating zone of the furnace in connection with the temperature of the steel.
The basics of the general solution to modeling the thermal regime allowed establishing the disposal mode of the burners in connection with the design of the furnace and the necessary output. The design of the furnace can be also changed having in view the thermal and the dynamic particularities of the flow gases.