Numerical analysis of the cooling of thin-walled pipes from a trip steel.

Behulova, Maria

1. INTRODUCTION

TRIP steels belong to multiphase steels widely used in the last decades in automotive industry especially because of their good formability, high strength and ductility (Chatterjee & Bhadeshia, 2007). The final microstructure and properties of TRIP steels depend on not only on the chemical composition but also on applied method of heat or thermo-mechanical treatment (Masek et al., 2009). In this reason, it is very important by manufacturing of thin-walled pipes from MnSi TRIP steel to comply very exactly the designed parameters of technological process.

2. PROBLEM DESCRIPTION

In the production process, pipes from the TRIP steel with the chemical composition given in the Table 1 are heated in a box or continuous furnace to the temperatures from 800[degrees]C to 900[degrees]C. The outer diameter of pipes d varies from 25 mm to 100 mm while the wall thickness is from 2 mm to 10 mm (Fig. 1). The distance of pipes axes can be from 170 mm to 250 mm. During the pipes heating in a continuous furnace, the rate of conveyer belt v is considered to be from the interval from 10 mm.[s.sup.-1] to 50 mm.[s.sup.-1.] After the heating, pipes are cooled down by the mechanisms of free convection and radiation to the surrounding air.

The main aim of numerical simulation was to determine the cooling times for pipes with various dimensions from the initial temperatures at the furnace output to the temperatures of 400[degrees]C and 100[degrees]C. The finite element program code ANSYS was exploited to perform numerical analyses (Ansys, 2005).

[FIGURE 1 OMITTED]

3. SIMULATION MODEL FOR PIPE COOLING

The process of transient heat conduction during cooling of thin-walled pipes can be mathematically described by Fourier-Kirchhoff's partial differential equation (Incropera&DeWitt, 1996). For heat conduction in solid isotropic material, it takes the following form

[partial derivative]T/[partial derivative]t = a ([[nabla].sup.2] T + [q.sub.v]/[lambda]) (1)

in which a = [lambda]/([rho].c) is the thermal diffusivity, [rho] is the density, c the specific heat capacity, [lambda] the thermal conductivity, [q.sub.v] the heat generated in unit volume per second and [[nabla].sup.2]T = [DELTA]T is the Laplace operator of temperature.

For the explicit solution of the Fourier-Kirchhoff's differential equation of heat conduction, the following conditions must be defined:

* geometrical conditions (shape and dimensions of a body),

* physical conditions (material--thermal properties),

* initial conditions (temperature distribution at the beginning of the process, T = T(x, y, z, t = 0) and

* boundary conditions (conditions at the interface of a body and surroundings).

Simulation model for the cooling of thin-walled pipes was developed for the optional parameters important from the technological point of view:

* geometrical parameters--outer pipe diameter d, wall thickness s and distance of pipe axes l (Fig. 1),

* processing parameters--rate of conveyer belt, initial pipe temperature [T.sub.0] and cooling air temperature (surrounding temperature) [T.sub.r].

Geometrical and finite element models (Fig. 2) were created as 2D in order to take into account non-axisymmetric conditions of convection and radiation cooling. In this reason, the whole middle pipe and one half of neighbouring pipes were modeled.

Applied temperature dependent thermal properties of the TRIP steel on the base of MnSi are plotted in Fig. 3.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Thin-walled pipes are cooled from the initial temperature of [T.sub.0] by air with the temperature of [T.sub.r] by free convection and radiation. Heat transfer coefficient h by free convection was computed using the criterial equation (Incropera & DeWitt, 1996)

[NU.sub.L, m] = C [[Ry.sup.n].sub.L, m] (2)

in which Nu = [h.sub.r] d/[[lambda].sub.r] is the Nusselt number, Ry = [gd.sup.3] [[alpha].sub.v] [DELTA]T/[a.sub.r][v.sub.r] is the Rayleigh number, [[alpha].sub.v] is the thermal expansion coefficient, [DELTA]T = [T.sub.s] - [T.sub.r] is the temperature difference between the pipe surface temperature [T.sub.s] and surrounding temperature [T.sub.r], g is the gravity acceleration, [v.sub.r] is the kinematic viscosity, [[lambda].sub.r] is the thermal conductivity and [a.sub.r] is the thermal diffusivity of air. The constant values of C and n are dependent on Rayleigh number.

Radiative heat flux [[PHI].sub.12] between surfaces [S.sub.1] and [S.sub.2] with the temperatures of [T.sub.s1] and [T.sub.s2] and emissivities [[epsilon].sub.1] and [[epsilon].sub.2] can be calculated from the equation (Incropera & DeWitt, 1996)

[[PHI].sub.12] = [[epsilon].sub.1] [[epsilon].sub.0] [S.sub.r] ([[T.sub.s1.sup.4] - [[T.sub.s2.sup.4])[[phi].sub.12] (3)

where [[sigma].sub.0] is the Stefan-Boltzmann constant, [S.sub.r] is the reference surface ([S.sub.1] or [S.sub.2]) and [[phi].sub.12] is the view factor. For the arrangement of pipes according to Fig. 1, it can be evaluated from the relationship

[[phi].sub.12] = 1/[pi] [arcsin d/l + [square root of [(l/d).sup.2] - 1 - l/d]] (4)

The view factor between defined surfaces and from the surface to the surroundings is calculated in the program system ANSYS automatically. Evaluation of the convection heat transfer coefficient in the dependence on the pipe surface temperature was implemented to the ANSYS using a user defined subroutine.

The described simulation model was verified by experimental temperature measurement in the nodes 1 to 4 (Fig. 1) during pipe cooling (Behulova, 2007). As it follows from comparison of measured and numerical results, mean relative errors of computed temperatures are from 0.42 % (node 2) to 1.38% (node 3).

4. RESULTS OF NUMERICAL SIMULATION

Numerical analysis of pipes cooling was carried out for pipes with outer diameter from 25 mm to 100 mm and the wall thickness from 2 mm to 10 mm. Thin-walled pipes were cooled down from the temperature of 800[degrees]C on the air with the temperature of 20[degrees]C.

The time histories of temperature in the node 2 for chosen outer diameters and wall thickness of pipes are illustrated in Fig. 4. The average cooling rates to the temperature of 400[degrees]C vary from 0.46 K.[s.sup.-1] for a pipe with diameter of 100 mm and wall thickness of 10 mm to 2.65 K.[s.sup.-1] for a pipe with smallest dimensions (d = 25 mm, s = 2 mm).

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The results of numerical simulations were processed to the diagrams for easy and fast determination of the pipe cooling time from the initial temperature of 800[degrees]C to the temperatures of 400[degrees]C (Fig. 5a) and 100[degrees]C (Fig. 5b) in the dependence on the outer diameter and wall thickness of a pipe.

5. CONCLUSION

The obtained results in the form of diagrams will be used directly in the manufacturing process of thin-walled pipes for efficient specification of process parameters.

6. ACKNOWLEDGEMENTS

The research has been supported by the project VEGA MS and SAV of the Slovak Republic No. 1/0837/08.

7. REFERENCES

Ansys Theoretical Manual, Release 10.0, SAS IP, Inc., (2005) Behulova, M. (2007). Simulation model for cooling process of thin-walled pipes. Materials Science and Technology [online]. ISSN 1335-9053

Chatterjee, S. & Bhadeshia, H. K. D. H. (2007). Transformation induced plasticity assisted steels. Mater. Sci Tech, Vol. 23, No. 9, pp. 1101-1104, ISSN 1743-2847

Incropera, F., P. & DeWitt, D. P. (1996). Findamentals of Heat and Mass Transfer. New York, J. Wiley&Sons, ISBN 0-471-30460-3

Masek, B. et al. (2009). The Influence of Thermomechanical Treatment of TRIP Steel on its Final Microstructure. J. Mater EngPerform, Vol. 18, No. 4, pp. 385-389, ISSN 1544-1024

Tab. 1. Chemical composition of the TRIP steel

C Mn Si P S

0,19 1,45 1,9 0,02 0,07

 Cr Ni Cu Al Nb

0,07 0,03 0,04 0,02 0.003