Three-dimensional thermo-mechanical analysis of molybdenum wire electrode during wire EDM process
3.2 Generation of spark plasma on the wire surface
3.3.2 Governing equation and boundary conditions A. Thermal analysis
Governing equation
Heat flow through the wire is governed by a three-dimensional transient heat conduction equation given by the partial differential equation in cylindrical polar coordinates (Carslaw and Jaeger, (1959)).
2 2 2
2 2 2 2
1 1 1
T T T T T T
r r r r z n z t
(3. 1)
where T is the temperature, α is thermal diffusivity (m2/s), t is time (s) and n = ρcv/k where ρ is density of the material (kg/m3), c is specific heat (J/kg K), v is wire velocity (m/s) and k is thermal conductivity (W/mK).
Boundary conditions
The heat flux is assumed to be distributed over boundary 1 (Figure 3.2). On the remaining portion of the wire (boundary 2), convection between the wire surface and the dielectric is considered as the boundary condition. Mathematically, these boundary conditions can be described as follows:
For boundary region 1:
fo
( ) r w
k T Q r r
r R
(3.2a) where Q(r) is the heat flux applied and Rw is the radius of the wire.
For boundary region 2:
for r w
k h
r R
T
(3.2b)
where h is the convective heat transfer coefficient (W/m2K).
The initial temperature of the wire at time t = 0 is assumed to be at room temperature of 300 K.
Spark radius
For a single pulse discharge process in wire EDM, it is very difficult to measure the spark radius experimentally due to very short duration of the pulses (in the range of microseconds).
A number of researches have been reported to determine the spark radius during a single discharge. DiBitonto et al. (1989) approximated spark as a point for cathode erosion and calculated the plasma radius, Ra at anode erosion(Patel et al. (1989)) as a function of time, which is given as
' n'
Rak t (3.3)
where n’ and k’ were considered to have values of 0.75 and 0.788e-6 respectively, based on experimental results.
Pandey and Jilani (1986) suggested an equation for the discharge radius, R in the EDM process based on the boiling point temperature Tb of work material, energy density E0 and thermal diffusivity α of workpiece material. The equation has limited applicability for steel- copper pair, given as
0.5
0 1
0.5 2
tan 4
b
E r t
T K r
(3.4) A semi-empirical equation of spark radius, R (µm) termed as “equivalent heat input radius”
was derived by Ikai and Hashiguchi (1995) as a function of discharge current, I (A) and pulse on-time, ton (μs), given as
3 0.43 0.44
(2.04 10 ) on ( )
R I t m (3.5)
In the present numerical model, this approach (Equation 3.5) has been used to calculate the spark radius as it gives more realistic results compared to the above mentioned approaches.
Heat flux on the wire electrode
Different approaches of heat flux were used by several researchers in the numerical modeling of electrical discharge machining process. Literature suggests that two forms of heat input sources have been considered in the thermal models developed initially viz. point heat source (DiBitonto et al. (1989)) and uniformly distributed heat flux (Beck (1981), Jilani and Pandey (1982), Pandey and Jilani (1986)). However, these assumptions were found to be not realistic in nature. In this work, the approach of Gaussian distribution of heat flux as suggested by Patel et al. (1989) is used because the results predicted were in higher agreement with the experimentally obtained values. The Gaussian heat flux equation is,
2
2 2
( ) 4.57 F V Ic exp 4.5 r
Q r R R
(3.6)
where Fc is the fraction of total EDM spark power going to the wire electrode (cathode); V is discharge voltage (V); I is discharge current (A) and R is spark radius (µm). The energy distribution factor (Fc) is a critical factor in the heat flux equation because it gives the fraction of total energy absorbed by the wire electrode during the discharge phenomenon.
The discharge energy is distributed between the wire electrode and workpiece, while the rest of the heat is lost into the dielectric. Researchers employed various values of Fc in the literature. Previous reports suggested that 50-60 % of the total heat generated during spark discharges is absorbed by the cathode and anode (Yeo et al. (2008)). However, this assumption was not able to produce realistic results when compared to the experimental data.
DiBitonto et al. (1989) collected data over a long period at various process conditions.
Comparing the analytical and experimental data, it was recommended that the energy distribution factor for cathode should be chosen as 0.183 for better agreement between the analytical and experimental results. In the present model, we have chosen this value of energy distribution factor, Fc to calculate the heat flux and to estimate the temperature increment on the wire as the wire acts as the cathode for a wire EDM process.
Incorporation of the concept of moving heat source
The wire is constantly moving during the WEDM cutting operation. The effect of moving wire tool is incorporated in the numerical model by using the concept of moving heat source characteristics, which gives a more realistic approach to the model. The temperature
distribution in the wire is evaluated at the end of a single pulse by using moving heat source characteristics in the developed model. The wire moves at a constant velocity during the discharge phenomenon. Thus, the Gaussian heat flux is assumed to move along the length of wire at a constant velocity, which gives the temperature generated at the end of a single pulse. For a single pulse cycle, the wire moves a distance (L), which is given by
(on off) L v t t
(3.7) It is assumed that the Gaussian heat flux traverses a distance over the wire surface that is traveled by the wire during a single pulse at a constant velocity. Thus, the Gaussian heat flux moves a distance L over the wire surface during a single discharge cycle. The incorporation of moving heat source characteristics gives a realistic approach to the process model. In the present model, a user subroutine DFLUX has been written and employed in ABAQUSTM to incorporate the effect of moving wire electrode during wire EDM machining. The DFLUX subroutine is provided in Appendix A3.
Convective heat transfer coefficient
The heat transfer phenomenon in wire EDM process is quite complex; therefore, estimating the convective heat transfer coefficient (h) experimentally is quite complicated. The convective heat transfer coefficient is assumed to have a constant value of 10,000 W/m2K referring to Dekeyser et al. (1985b). Banerjee et al. (1993) has also used this value of h and the results were quite satisfactory.