Chapter 2: Heat flow in welding

 

Chapter 2: Heat flow in welding

Temperature Distribution in Welding

Temperature distribution in welding depends upon the nature of the welding process used, type of the heat source employed, energy input per unit time, configuration of the joint (linear or circular), type of joint (butt, fillet, et.c.),physical properties of the metal being welded, and the nature of the surrounding medium i.e. ordinary atmospheric conditions or underwater. Although it is beyond the scope of this book to analyse all these aspects of heat flow in detail but brief descriptions of the following cases are

(A) Arc Welding

(i) Linear butt welds,

(ii) Circular butt welds,

(iii) Fillet welds.

B) Resistance .Welding

(i) Upset butt welding,

(ii) Spot welding.

(C) Electroslag Welding,

(D) Underwater Welding.

Temperature Distribution in Arc Welding

Nearly 90% of welding in world is carried out by one or the other arc welding process, therefore it is imperative to discuss the problem of temperature distribution in arc welding in the maximum possible detail to arrive at the best possible understanding of the problem. Because linear butt welds are perhaps the most used type of welds in welded fabrication therefore this type of joint will be detailed the most.

Temperature Distribution in Linear Butt Welds

Heat flow in welding is mainly due to the heat input by wetding source in a limited zone, and its subsequent flow into the body of the workpiece by conduction. Alimited amount of heat loss is by way of convection and radiation as well but that can be accounted for by allotting heat transfer efficiency factor at the accounting of heat input. So, the problem of temperature distribution can be seen as a case of heat flow by conduction when the heat input is by a moving heat source. This case can be further simplified by assuming workpiece of large dimensions to approach the infinity concept i.e. the temperature at the farthest end of the workpiece in all directions remains unchanged', This leads to a condition of quasi-stationary state which can be defined as a condition in which an observer at the arc will see a fixed temperature field all around the arc at all times. In other words under a condition of quasi-stationary state of heat flow isotherms representing, different temperatures remain at a fixed distance with respect to the heat source. Mathematically stated it means dT/dt = 0 

where T is the temperature at any time and t is the time unit. Fig. 2.1 represents the condition of quasi-stationary state with observation points at A and B for welding along centre line of the plate. To determine the temperature at any point, in a workpiece, during welding, the problem can be solved by considering from the basic Fourier's Equation of heat flow by conduction,

If one dimensional form of heat conduction is considered as shown in Fig. 2.2, Fourier's Law states that the rate of heat transfer per unit time, q, is a product of,

(a) area, A, normal to heat flow path,

(b)the temperature gradient at the section i.e. the rate of change of temperature with reference to the distance in the direction of heat flow,

(c) k, thermal conductivity of the material of the body.

Mathematically stated,

where, dQ = quantity of heat conducted in time, dt .

Note: (i) Since the heat flow always occurs in the direction of decreasing temperature, the temperature gradient (dt/dx) will,therefore, always be -ve, hence the negative sign on the right hand side of the equation (2.1).

(ii) The following assumptions have been made in the above equation,

(a) the heat flow in y and z directions is zero,

(b) the temperature in any plane perpendicular to the x-axis is uniform throughout the plane.

The equation (2.1) represents the fundamental heat conduction law for uni-directional flow of heat. As normally the heat will flow in all the three directions in a given body, therefore a comprehensive equation must deal with 3-dimensional heat flow. Three Dimensional Heat Flow Equation

Consider an infinitely small solid cubic body as shown in Fig.2.3.

Let the edges parallel to the three axes be respectivelydx, dy and dz. The volume ofthe cubicelement is, therefore, given by the relation,

Let dQx represent the total quantity of heat entering the face area dy . dz in time dt as shown in Fig. 2.3. So, on the basis of equation (2.1), we have,

Note: The gradient in equation (2.3) is expressed as a partIal derivative of the temperature T as it is a function of x, y, z and t. Now,a corresponding quantity of heat will be leaving the cubic element (dx . dy . dz). Let it be dQx + dx, the value of which may be obtained by Taylor Expansion of dQx + dx' whereby,

Ignoring the higher terms of Taylor expansion in equation (2.4), the net heat gained by the element (dx· dy . dz) due to conduction in x-direction will, thus, be,

By substituting the value of dQx from (2.3) in the right hand side of equation (2.5), we get,

From (2.6) and (2.2), we get,

Similarly, the net heat gained by the cubic element by conduction in y and z directions may be obtained as follows,

The total heat gained (dQ) by the cubic element (dx . dy . dz) is the sum of the heat gained by conduction in x, y, and z direction. Thus, from equations (2.7), (2.8) and (2.9), we get,

Now, the heat gained by the cubic element (dx . dy . dz) can also be expressed in terms of increase in internal energy dE, expressed as,

Assuming k uniform in all directions of the equation (2.12) can be written as,

where (α) is known as the thermal diffusivity of the material of cubic element and its unit is m2/sec.

From equations (2.13) and (2.14), we get,

This is known as Fourier's Equation of three dimensional heat conduction in solids. To apply equation (2.15) to welding, let us consider the situation in arc welding as expressed in Fig. 2.4.

Let 0 be the origin (i.e. the starting point for welding) for the cartesian coordinates (x, y, z). Suppose, we are interested in finding temperature at any point A (x, y, z). Because of moving heat source the distance of point Ais changing every moment with respect to the arc or the tip of the electrode. If the origin of coordinate system is shifted from 0 to the tip of the electrode (which is assumed to lie in the plane of top surface of the plate) then the temperature at any point A located at a fixed distancE)from the tip of the electrode remains fixed because of the establishment of a quasi-stationary state and can thus be determined mathematically.

Let point A in the new coordinate system with respect to the tip of the electrode have the coordinates

where, v = the welding speed,

t = time taken by welding, starting from the origin, O.

Now, to determine the temperature distribution in the plate with respect to the tip of the electrode, we are required to change equation (2.15) from cartesian coordinate system (x, y, z) to a new coordinate system

Differentiating (2.16), we get,

Equation (2.32) is a more convenient form of heat flow equation for a quasi-stationary state of welding. It can be used for determining temperature distribution in specific cases for example, in a semi-infinite plate which is representative of welding a large thick plate.

Temperature Distribution in a Semi-Infinite Plate (3- dimensional case)

Considering a case of laying a single weld bead, using a point heat source, on the surface of a very large and thick plate (workpiece), as shown in Fig. 2.5. Let us assume that the Z-axis is placed in the direction of thickness of the plate downwards. For determining the temperature distribution the solution of equation (2.32) must satisfy the following conditions.

(1) Since welding is done by a point heat source, the heat flux through the surface of the hemisphere drawn around the source must tend to the value of the total heat,Qp delivered to the plate, as the radius of the sphere tends to zero. If R is the radius of the sphere, then the total heat flowing through the hemispherical surface of heat source as given by Fourier's Equation will be,

(2) Heat losses through the surface of the plate (workpiece) being negligible, there is no heat ransmission from the plate to the surroundings

(3) The temperature of the plate remains unchanged at a great distance from the heat source

Now, condition (2) i.e. equation (2.36) assigns a semi-circular form to the isothermf: located at sections parallel to plane YZ (Fig. 2.5), thus they are dependent only on radial distance 'l' from the heat source. Keeping in view the semi-circular form of isotherms, equation (2.32) can be written more conveniently in cylindrical coordinates, as shown in Fig. 2.6.

Now, temperature <l>is a function of y and z while y and z are functions of Z and", therefore equation (2.32) can be converted into polar coordinates by determining the double differential of  Φ w.r.t. and z in terms of Z and 

Equations (2.48) and (2.49) are important relationships for determining temperature distribution in a semi-infinite plate.

2.1.1.3  temperature distribution in large (infinite) plate of finite thicknes

Equation (2.48) can help in determining temperature distribution in semi-infinite plate but the normal cases encountered are those of large plates (in ship-building and pressure vessel fabrication, etc.) of finite thickness. Thus, equation (2.48) must be modified to account for the limited range of Z dimension.

Therefore, boundary condition of equation (2.51) is satisfied. However, it. remains to be proved that conditions of equations (2.33) and (2.35), viz.,

Equation (2.62) satisfies equations (2.33) and (2.35).

To sum up it can be said that the following two equations can be used for determining temperature distribution during butt weLding of large (semi-infinite) plates of infinite thickness and finite thickness respectively .

2.2. Efficiency of Heat Sources

It is evident from the solution of Problem 2.1 that to solve the temperature distribution problems we require to know the efficiency η  of the heat source used for welding ; where η defined as,

Thus, if the efficiency η of the heat source is known, the energy (Q) transferred from it to the workpiece, can be determined. In arc, electroslag, and electron beam welding,

where V and I are the arc voltage and welding current respectively. In gas tungsten arc welding (GTAW)with dcen (direct current, electrode negative) the majority of the heat is produced by electrons bombarding the workpiece (anode) that is as a result of the release ofthe work function and the conversion oftheir kinetic energy into heat at the workpiece. In GTAWwith a.c., however, electrons bombard the workpiece only during straight polarity half cycle i.e. for a period when electrode is negative thus resulting in significantly lower arc efficiency. Also, the heat loss to the surrounding can be rather high; particularly so for long arc lengths. Apart of the heat generated is taken away by the cooling water employed to keep the electrode cool. In consumable electrode welding processes like SMAW,SAW, GMAWand FCAW,using dcep or a.c. the heat going to both the electrode and the workpiece finally lands up on the workpiece through transfer of molten meta1. Thus, the heat transfer

efficiencies of these processes are high. In SAW process, the heat transfer 11 is further increased because the arc remains under a blanket of flux, the heat loss to the surroundings is, thus, minimised.

The efficiency of heat transfer in ESW is lower than that in SAW, mainly owing to heat loss to the water-cooled copper shoes and, to lesser extent, by radiation and convection from the surface of the molten slag.

the molten slag.

In EBW (electron beam welding) process the welds are produced by the phenomenon of keyholing. These keyholes act like bIack bodies to the heat source and trap most of its energy; leading to very high efficiency of heat transfer in EBW process.

In laser welding the heat transfer efficiency can be strongly affected by the wavelength and energy density of the laser beam, the workpiece material and its surface condition, and the joint design. For example, with well poli>Shed AI or Cu, the surface reflectivity can be around 99%, i.e. the efficiency can be only about 1% for a 10.6  μm.continuous wave CO2 laser. For steels, especially when coated with thin layers of materials that enhance absorption of the beam energy (for example, graphite and zinc phosphate), quite reasonable efficiencies can be obtained. When keyholes are established during laser welding, the efficiency of the process can rather be impressive.

In Oxy-acetylence welding the energy transferred is given by,

Both these values refer to standard state of 1atm, and 25°C tem perature.The heat source efficiency in Oxy-acetylene gas welding varies over a rather wide range as the efficiency decreases significantly

with increasing fuel consumption rate VC2H2 , because of incomplete 2 ·2 combustion. Efficiency of oxy-acetylene gas welding is also found to depend on the torch nozzle diameter, welding speed, material thickness, and thermal conductivity of the workpiece. In Table 2.1 are listed the efficiencies of most of arc. beam and flame welding

2.3. Further Modifications of Temperature Distribution Equtions

Different researchers have tried to modify Rosenthal's equations to determine more accurately the temperature distribution under different sets of welding conditions. The ones put forward by Adams, and Wells have received wide recognition and are included in the following sub-sections.

2.3.1. Adams Modification

In the earlier treatment of problem based on Rosenthal's solution the heat source has been assumed to be a point heat source which is obviously not true, particularly for small sized workpieces. Thus, recognising the existence of finite sized weld pools, Adams used the fusion line as the boundary condition and modified Rosenthal's equations viz., equations (2.48) and (2.62). The following equations were derived by Adams for the peak temperature, Tp, at a distance y from the fusion boundary at the workpiece surface.

2.3.2. Wells Modification

The relationship between the heat flow and weld bead dimensions is given by the Wells simplification of Rosenthal's Equation by the following relationship.

The main difficulty in the use of heat flow equations is the variation of physical constants (like k, p, C, etc.) with temperature. Energy absorption within the weld pool by latent heat and its subsequent release at the tail end of the weld pool on solidification is one reason why actual isotherms around a moving weld pool are more elongated than indicated by calculations. Wells equations (2.65) to (2.67) give good corr~lation with experimental data for low carbon steel and can be adjusted to apply more accurately to

metals having high latent heats by using.a specific heat value corrected by the following relationship.