## Carburization; Theory and Numerical Methods

The computational weld mechanics CWM analysis of the process of welding on pressurized pipelines involves several difficult problems that make this a particularly challenging problem.

The greatest challenge is to develop the capability to deal with creep at temperatures in the range 900 °С (1652 °F, 1173 °K) to 1500 °С (2732 °F, 1773 °K). For sufficiently short times at temperatures below 900 °С (1652 °F,1173 °K), the creep rate is sufficiently slow that during the rather short times at high temperatures during welding creep can be ignored in most problems and rate independent plasticity is the appropriate material behavior. However, it is exactly at temperatures above 900 °С (1652 °F, 1173 °K) that internal pressure can cause bum-through. It is exactly in this temperature range that the steel deforms primarily by viscous flow, which is rate dependent plastic or visco-plastic or creep. In this temperature range, a realistic analysis must deal with viscous flow and use realistic values of the viscosity, the material parameter that relates the deviatoric strain-rate to stress.

A second challenge is the need to couple the thermal analysis with the stress analysis more strongly than is usually done in CWM. In particular, the thermal analysis must be done on the geometry deformed by the stress analysis.

A third challenge is to deal with the length scales that range from a fraction of a mm near the weld pool to a 3.66m (12 ft.) long pressure vessel length. This large range of length scales can make the FEM problem numerically ill-conditioned. Special solvers that were sufficiently robust were needed to deal with this problem.

A fourth challenge is to manage the complexity of the software and the complexity of input data required for analyses. The addition of filler metal is desirable to provide a more complete and realistic simulation.

While welding on the outer wall of a pressurized natural gas pipeline, the heat from the weld can cause localized carburization on the inner wall of steel pipe. The carbon levels can exceed the eutectic composition of approximately 4.3 weight percent carbon in iron. This phenomenon can produce a thin layer of liquid cast iron on the internal wall of the pipe directly under the welding arc.

The diffusion of carbon is assumed to depend primarily on the spatial gradient of carbon composition and the temperature through the temperature dependence of the diffusivity of carbon. Local equilibrium is assumed at the liquid-solid interface.

Natural gas pipelines are welded under pressure for two reasons; to make a new connection to an existing pipeline or to repair damage such as that caused by corrosion [1]. If the pipeline is sufficiently thin, the part of the inner wall under the arc can reach temperatures approaching the melting point, 1530 °С for a typical HSLA pipeline steel. At these temperatures natural gas, which is mostly methane CH4, can decompose by the reaction CHA —>C + 2 H2 .The resulting carbon atoms react with the surface of the pipe. We conjecture that at temperatures above the eutectic temperature for a typical HSLA pipeline steel, 1147 °С, a very thin Fe-C film forms, perhaps only 1 to 2 atoms thick at first, and then the film grows by diffusion of carbon. The carbon diffuses into this film from the gas atmosphere, through the liquid film across the liquid-solid interface and into the solid. In this very thin film of liquid, it is assumed that advection, i. e., stirring or fluid flow effects, can be neglected. A film of such a brittle material increases the risk of cracking in the pipe. The addition of hydrogen from the decomposition of natural gas adds to the concern.

In this chapter, this phenomenon is simulated by the following method. The 3D transient energy equation is solved to compute the transient temperature field. While the temperature at a point on the internal pipe wall exceeds the eutectic temperature, it is assumed a thin layer of liquid exists. Further it is assumed this layer grows by diffusion of carbon into the underlying solid. The limiting process is assumed to be the temperature dependent diffusion of carbon. A thermal-stress analysis computes distortion which can change the wall thickness and thus the temperature computed in the thermal analysis and the carburization analysis. The addition of hydrogen from the decomposition of natural gas adds to the concern. Modeling the hydrogen behavior will be presented in section 6-3.

The temperature curves obtained at the points on the internal surface of the pipe were used to simulate the growth of the carburized layer. The carburized layer on the internal surface of the pipe was assumed to result from the absorption of carbon by the pipe material from the hydrocarbon rich environment inside the pipe. The increase in the carbon concentration results in melting starting at the internal surface of the pipe at temperatures (~ 1147°C) significantly lower than solidus temperature (~1500°C) corresponding to the nominal concentration of carbon in steel, c0 =0.1 %. They also assumed that the rate of carbon absorption is controlled by the diffusion of carbon through the growing liquid layer and that concentrations at the liquid-solid interface, ey ande/ ;/, and liquid-

gas phase interface, cL(), are determined by the equilibrium liquidus

and solidus concentrations of Fe - Fe}C diagram, Figure 6-3.

Therefore they neglected any effects of the melting kinetics and kinetics of the reaction between the gas phase and liquid material on the growth rate.

Figure 6-3: The iron-iron carbide phase diagram with interface concentrations shown for liquid layer growing at temperature T] |

9> з Ч-- о |

6,70 |

It is possible to neglect the absorption of carbon by a solid material at a temperature below the eutectic temperature because the diffusivity of carbon in the solid phase is more than two orders of magnitude lower than the diffusivity of carbon in the liquid. As a result the overall contribution of gas to the solid phase transport into carburization is negligibly small compared to the contribution from the fluxing effect. In this model of the fluxing process the changes in the concentrations and phase state are simulated only at temperatures above the eutectic temperature. It is assumed that an infinitesimally thin layer of the liquid phase is formed at the pipe internal surface at the moment when its temperature becomes higher than the eutectic temperature. After that the layer grows with the rate limited by the diffusion of carbon from the liquid gas interface to the liquid solid interface. The boundary conditions for a corresponding diffusion problem and typical concentration distribution in the system during this process are illustrated in Figure 6-4.

Distance, х |

Figure 6-4: The distribution of the carbon concentration in the system with a liquid layer growing from the surface (x=0). The liquid-solid interface is located atx = x,. |

с о |

с CD О С О о с О -Q о и |

An analytical solution exists for the diffusion problem with a constant growth temperature (constant values of c/0, cLy and ), [4].

Under these conditions the thickness of the growing layer at time t can be found as:

xs(t) = 2P^D~t (6-1)

where the coefficient p is determined by the transcendental equation:

-PD,S

(с -с )4кВ - (с - с )exP(-l2) I C)№exp A (6‘2)

yc-A, cLrHKp (cLr cL0) о +{cyL cQ) J—

erjKP) erfiP

where D; and D() are diffusion coefficients of the liquid and solid

phases, respectively (D, = 2.0xl0~*m/s2 and

D0 =.0xl0~9m/s2 [5]). Equations (6-1) and (6-2) are used to

estimate the total thickness of the liquid layer which can be obtained under different welding conditions and to determine the size of mesh cells and number of cells in a mesh for the numerical solution of the diffusion problem with a varying growth temperature.

The numerical algorithm that has been used for the solution of the diffusion problem is based on the cellular model of the solidification process [6 and 7]. In this model space is divided into cells and every cell is characterized by a fraction of liquid, f, and fraction of

solid, 1 - ft, phases and carbon concentration in the liquid, cl and/or in solid cs, phase. The size, Ax, and the number of cells in the 1-D mesh were estimated using (6-1) and (6-2) so that the liquid layer at the end of the growth process would consist of at least 10 cells and the total length of the mesh would be 2-3 times larger than the final thickness of the liquid layer. The diffusion transport is described in this model by an explicit finite difference scheme implemented separately for liquid and solid parts of the system. The solution for the concentration in liquid is evaluated only in the cells with a nonzero fraction of liquid, and the solution for the concentration in solid is evaluated in the rest of the system. The liquid and solid concentration fields overlap in one cell in which melting occurs and the fraction of liquid gradually increases. The length of time step, At, was limited by the stability criterion and by the solidification limit allowing not more than 0,1 of a cell to be solidified in one step.

Different boundary conditions were implemented at the external boundaries of the mesh and at the internal (liquid-solid) interface. The external boundaries had temperature (time) dependant Dirichlet type conditions cL0 and c0 on liquid and solid sides of the mesh, respectively. The Neumann type condition was implemented in the melting cell with a zero transport from the liquid to solid phase and vice versa.

The evolution of a melting cell is determined by both diffusion and melting processes. At the beginning of the time step the concentrations in the liquid and solid phases in the melting cell are equal to c^andcy, respectively. At every time step we calculate

first the changes in the concentration in the solid and liquid phases in the melting cell using the diffusion equation. The zero liquid-solid diffusion exchange condition imposed on the melting cell (cell number i, cells with numbers < і are liquid) results in the following

changes in solid, Acs, and liquid, Ac,, concentrations produced by diffusion, adopted from Artemev et al. [11]: дс - ^ A-(с, [* + !]-с, [Q)

' ax2-о-./;)

Ac, = 7l2 ;j /l j; (6-3) |

At-Ds -(сДг-1]-с;[г])

Ax2 - fi

where cv[z] and c;[z] are concentrations in the solid and liquid phases in the cell number i. When these increments are added to the concentrations, the updated values cl[i] and c [/] differ from the

equilibrium conditions at the liquid-solid interface. c^[/] and

c/ [/] are used then to calculate the increase in fraction of liquid, Af,

using the condition that at the end of a time step equilibrium should be restored in the melting cell. The equation adopted from [11]:

Ar _ (c,+ И-cL )-f,+ (c[[/]-CyL)(1 - ft) ^ ^

AJ,----------------------------------------------- (6-4)

CLy CyL

At the end of a time step the fraction of liquid is updated using the calculated value of Af,, and equilibrium values are assigned to concentrations in liquid and solid. If Af, is such that an estimated fraction of liquid becomes larger than 1.0 then f is assigned the value 1.0 and melting is transferred into the next cell. The fraction of liquid in the next cell is calculated so that the equilibrium interface conditions are established there. At the very beginning of melting we need to initialize melting in the first cell in the system when the temperature becomes just higher than the eutectic temperature. We do this by using equations (6-1) and (6-2) to estimate the thickness of liquid produced in such a first melting step. After that the described combination of diffusion steps and melting steps is used. The simulation is effectively terminated when the temperature curve drops below the eutectic temperature after passing through the maximum.