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### Computation of heat conduction in welding of thick wall reactor vessels

The welding of reactors in refinery is complex due to various phase formation during and after welding. Each phase was unique with success of welding lies in control of phase structure during weld solidification.

**S Chidambaram**

Numaligarh Refinery LimitedNumaligarh Refinery Limited

**Viewed :**2026

**Article Summary**

The mathematical modelling was developed with moving heat source constitutive equations to obtain temperature prediction at heat affected zone. The maximum temperature of 737°C obtained at 0.5 cm away from weld fusion zone with favourable microstructure of austenitic stainless steel weldments of reactor.

The heat transfer during welding is analysed for reactor wall both circumferentially and radially. The heat transfer in circumferential direction analysed using three dimensional constitutive heat conduction and heat transfer in radial direction analysed using Rosenthal equation.

Mathematical modeling of heat transfer in circumferential direction or welding direction:

The heat dissipation of weld joint is analysed to measure peak temperature, heating rate, cooling rate and temperature profiles of various thermal cycles’ experienced during welding. The transient constitutive heat conduction equation is written as:

(1)

α is thermal diffusivity in sq.metre per second and T is temperature

x, y, z – axial coordinates

The weld deposition of skirt to reactor wall joint welded by shielded metal arc welding of electrode E347 electrode travels in X direction. The boundary conditions defined as rate of change of travel movement with respect to time is defined as velocity of electrode Vx. Also assuming electrode supplies all heat from electrode to work piece, the rate of change of temperature with respect to time is zero at any spatial point during welding, ∂T/∂t=0. By using constant steady state conditions and applying boundary conditions first order heat equation is solved as:

(2)

Q is rate of heat supplied during welding in watt

r, is longitudinal distance from welding zone in m

k, thermal conductivity in watt per meter Kelvin

x is distance travelled in n seconds

Thermal conductivity of SS347 is k = 22.8 watt per meter Kelvin

Thermal diffusivity of SS347 is α = 4.2 sq.metre per second

The rate of heat supply is defined as product of maximum heat input and travel speed. Maximum heat input is given as 1.6 kJ/mm and travel speed is 165 mm/min, rate of heat supply is calculated as 4400 watt. The velocity of electrode Vx is 0.00233 m/s and distance travelled from welding zone r = 0.5 cm with velocity of electrode moving within ten second x = “V”x X 10. By substituting above data in equation 2 yields result of temperature as 34°C at 0.5 cm distance from weld fusion zone circumferentially. In X direction (circumferentially), moving electrode with velocity of 140 mm/min peak temperature not attained and thermal cycles were not experienced. At distance 1 cm from weld fusion zone, temperature is calculated as 30°C and remains constant at various distances away from weld fusion zone beyond 1 cm. From above temperature calculations it was confirmed that metallurgical structure remains stabilized austenite with no significant changes circumferentially at any point of distance near welding.

Mathematical modeling of heat transfer in radial direction or perpendicular to welding direction:

The heat flow in direction perpendicular or normal to welding is analysed to measure peak temperature, heating rate, cooling rate and temperature profiles of various thermal cycles’ experiences during welding.

The temperature distribution of semi infinite plate derived by Rosenthal equation which follows as

(3)

Where r2= y2+ z2

n, efficiency of welding SMAW process = 0.75,

E voltage in volts, I current in ampere, r is radial distance normal to welding direction,

T0 - room temperature in Kelvin.

The set of temperature calculations were performed with respect to distance of interest from weld fusion zone which represented in figure2 where temperature is function of time according to equation 3. The peak temperature is marked with locus point in graph and at 0.5 cm distance from weld fusion zone peak temperature reached 737°C whereas at 1 cm distance from weld fusion zone peak temperature reached 203°C. The peak temperature decreases with increase in distance from weld fusion zone and also rate of heating and cooling decreases with increase in distance from weld fusion zone. The time required to reach peak temperature increases with increase in distance from weld fusion zone. Beyond 1 cm peak temperature is lower and no detrimental effect occurs in microstructure due to welding.

The maximum peak temperature attained is near about 737°C from 0.5 cm from weld fusion zone. The solidification pattern during welding from weld fusion zone clearly indicates austenite grain nucleates and grown with very small amount of ferrite phases nucleate around 800°C. The ferrite phases decreases beyond 400°C further with major austenite phases along with little or no ferrite phase microstructure depends on base material composition as shown in Figure 3. The maximum peak temperature attained is near about 203°C from 0.5 cm from weld fusion zone. The solidification pattern during welding from weld fusion zone clearly indicates austenite grain heated around 203°C and again cools with no effect on microstructure as shown in Figure 3. The above results predicted based on neither microstructure susceptible to residual hydrogen nor high temperature hydrogen attack phenomena.

From mathematical calculations and from above discussion it was clearly reveals that welding produces delta ferrite nucleation from which austenite grain grows over weld metal and delta ferrite at grain boundaries avoid solidification cracking during welding of reactors. Heat Affected Zone of 0.5 cm away from weld fusion boundary were experiences 737°C whereas 1cm experiences 203°C shows mathematically the weld metal, heat affected zone and base metal were not affected by welding with above ideal conditions.

**Introduction**The heat transfer during welding is analysed for reactor wall both circumferentially and radially. The heat transfer in circumferential direction analysed using three dimensional constitutive heat conduction and heat transfer in radial direction analysed using Rosenthal equation.

**Mathematical modeling of reactor welded joint**Mathematical modeling of heat transfer in circumferential direction or welding direction:

The heat dissipation of weld joint is analysed to measure peak temperature, heating rate, cooling rate and temperature profiles of various thermal cycles’ experienced during welding. The transient constitutive heat conduction equation is written as:

(1)

α is thermal diffusivity in sq.metre per second and T is temperature

x, y, z – axial coordinates

The weld deposition of skirt to reactor wall joint welded by shielded metal arc welding of electrode E347 electrode travels in X direction. The boundary conditions defined as rate of change of travel movement with respect to time is defined as velocity of electrode Vx. Also assuming electrode supplies all heat from electrode to work piece, the rate of change of temperature with respect to time is zero at any spatial point during welding, ∂T/∂t=0. By using constant steady state conditions and applying boundary conditions first order heat equation is solved as:

(2)

Q is rate of heat supplied during welding in watt

r, is longitudinal distance from welding zone in m

k, thermal conductivity in watt per meter Kelvin

x is distance travelled in n seconds

Thermal conductivity of SS347 is k = 22.8 watt per meter Kelvin

Thermal diffusivity of SS347 is α = 4.2 sq.metre per second

The rate of heat supply is defined as product of maximum heat input and travel speed. Maximum heat input is given as 1.6 kJ/mm and travel speed is 165 mm/min, rate of heat supply is calculated as 4400 watt. The velocity of electrode Vx is 0.00233 m/s and distance travelled from welding zone r = 0.5 cm with velocity of electrode moving within ten second x = “V”x X 10. By substituting above data in equation 2 yields result of temperature as 34°C at 0.5 cm distance from weld fusion zone circumferentially. In X direction (circumferentially), moving electrode with velocity of 140 mm/min peak temperature not attained and thermal cycles were not experienced. At distance 1 cm from weld fusion zone, temperature is calculated as 30°C and remains constant at various distances away from weld fusion zone beyond 1 cm. From above temperature calculations it was confirmed that metallurgical structure remains stabilized austenite with no significant changes circumferentially at any point of distance near welding.

Mathematical modeling of heat transfer in radial direction or perpendicular to welding direction:

The heat flow in direction perpendicular or normal to welding is analysed to measure peak temperature, heating rate, cooling rate and temperature profiles of various thermal cycles’ experiences during welding.

The temperature distribution of semi infinite plate derived by Rosenthal equation which follows as

(3)

Where r2= y2+ z2

n, efficiency of welding SMAW process = 0.75,

E voltage in volts, I current in ampere, r is radial distance normal to welding direction,

T0 - room temperature in Kelvin.

The set of temperature calculations were performed with respect to distance of interest from weld fusion zone which represented in figure2 where temperature is function of time according to equation 3. The peak temperature is marked with locus point in graph and at 0.5 cm distance from weld fusion zone peak temperature reached 737°C whereas at 1 cm distance from weld fusion zone peak temperature reached 203°C. The peak temperature decreases with increase in distance from weld fusion zone and also rate of heating and cooling decreases with increase in distance from weld fusion zone. The time required to reach peak temperature increases with increase in distance from weld fusion zone. Beyond 1 cm peak temperature is lower and no detrimental effect occurs in microstructure due to welding.

**Results and discussions**The maximum peak temperature attained is near about 737°C from 0.5 cm from weld fusion zone. The solidification pattern during welding from weld fusion zone clearly indicates austenite grain nucleates and grown with very small amount of ferrite phases nucleate around 800°C. The ferrite phases decreases beyond 400°C further with major austenite phases along with little or no ferrite phase microstructure depends on base material composition as shown in Figure 3. The maximum peak temperature attained is near about 203°C from 0.5 cm from weld fusion zone. The solidification pattern during welding from weld fusion zone clearly indicates austenite grain heated around 203°C and again cools with no effect on microstructure as shown in Figure 3. The above results predicted based on neither microstructure susceptible to residual hydrogen nor high temperature hydrogen attack phenomena.

**Conclusion**From mathematical calculations and from above discussion it was clearly reveals that welding produces delta ferrite nucleation from which austenite grain grows over weld metal and delta ferrite at grain boundaries avoid solidification cracking during welding of reactors. Heat Affected Zone of 0.5 cm away from weld fusion boundary were experiences 737°C whereas 1cm experiences 203°C shows mathematically the weld metal, heat affected zone and base metal were not affected by welding with above ideal conditions.

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