**1. Introduction**

Electrical discharge machining (EDM) is a non-traditional machining process in which material removal takes place by thermal energy generated from sparking between the workpiece and tool [1]. To predict EDM performance measures, researchers have developed various models such as the mathematical model, numerical model, regression model, etc. based on this material removal theory. Each model has a different level of prediction accuracy due to their assumptions related to flushing efficiency, material properties of the workpiece and tool, the shape of the heat source, crater shape, etc. [2]. A mathematical model for MRR was developed by DiBitonto et al. [3] considering a point heat source and constant fraction of energy (Fc = 0.183) transferred to the cathode. Singh and Ghosh [4] estimated the electrostatic force responsible for material removal in a short pulse interval (<5 μs) using a thermo-electric model and calculated crater depth. For the estimation of the

*Modeling of Material Removal Rate in Electrical Discharge Machining by a Novel Approach… DOI: http://dx.doi.org/10.5772/intechopen.81083* 

 geometrical dimensions of a micro-crater, Yeo et al. [5] proposed analytical models of the anode and cathode based on electro-thermal theory. Salonitis et al. [6] also developed models of the MRR and the average surface roughness by introducing the new concept of erosion front velocity. Madhu et al. [7] used the finite element method to predict MRR and the depth of damaged layer during EDM. The result showed that the spark-radius and the power intensity affect the geometrical dimensions of the crater. During thermal analysis of EDM in FlexPDE software, Lasagni et al. [8] observed that the material removal can be controlled by the melting enthalpy and the melting point. A 2D axisymmetric model of single spark EDM was developed by Joshi and Pande [9] considering realistic boundary conditions like Gaussian heat distribution, variable spark radius, latent heat of melting, etc. for prediction of the crater shape and MRR. Tao et al. [10] used FLUENT software to present a numerical model considering the plasma heating phase and the bubble collapsing phase for the material removal process. To simulate the crater formation due to a single discharge, Assarzadeh and Ghoreishi [11] developed an electro-thermal based model using ABAQUS software. The DFLUX subroutine was used to program the non-uniform heat flux. The maximum errors in predicting the crater radius and depth were 18.1% and 14.1%, respectively. Ming et al. [12] developed a hybrid model for EDM based on the finite-element method (FEM) and Gaussian process regression (GPR) to predict MRR and surface roughness. Puertas et al. [13] used response surface methodology to develop multiple regression equations for material removal rate, electrode wear and surface roughness. Using the nonlinear regression method with logarithmic data transformation, Chattopadhyay et al. [14] developed the empirical models for prediction of output responses. To investigate the effect of input parameters on material removal rate, electrode wear rate and surface roughness, Sanchez et al. [15] developed an inversion model based on the least squares theory. In the present work, a mathematical model for MRR is developed based on the new approach of using basic material removal theory and the regression method.

#### **2. Experimentation**

 The experimental results presented in DiBitonto et al. [3] are used in this modeling approach, as it is one of the founding research works in the field of EDM quite often cited in EDM literature. Those data were obtained by electrical discharge machining of steel (iron) as the cathodic workpiece with copper as the anode. The average thermophysical properties of the workpiece are shown in **Table 1**.


#### **Table 1.**  *Thermophysical properties of workpiece.*

## **3. Different models for prediction of MRR**

#### **3.1 Novel approach**

In this work, a mathematical model for MRR is developed using the theory of melting and vaporization of material from the workpiece as well as experimental data shown in **Table 1**. The experimental results of MRR are used to calculate the volumetric rate of material removal from the workpiece.

The energy responsible for material removal from workpiece [16] is

$$E\_{\rm MRR} = \rho V\_W [C\_p (T\_v - T\_o) + L\_m + L\_v] \tag{1}$$

where ρ - Density (kg/m3 ), VW - Volumetric material removal rate from workpiece (m3 /s), Cp - Specific heat (J/kg K), Tv - Vaporization temperature (K), To - Room temperature (K), Lm - Latent heat of melting (J/kg), Lv - Latent heat of vaporization (J/kg).

Energy released during single discharge per unit time [17] is

$$E \quad = \text{ VI}\left(\frac{T\_{ou}}{T\_{ou} + T\_{off}}\right) \tag{2}$$

where V - Voltage (V), I - Discharge current (A), Ton - Pulse on time (μs), Toff - Pulse off time (μs).

Fraction of energy (%) responsible for material removal

$$F\_{MRR} = \frac{E\_{MRR}}{E} \times 100\tag{3}$$

The regression equation for fraction of energy (%) responsible for material removal is developed using statistical software MINITAB 16 and is given by

$$F\_{\rm MRR} = \ 2.4 \Re \times I^{0.9472} \times T\_{\rm on} \, ^{-0.1855} \tag{4}$$

Energy responsible for material removal from workpiece

$$\begin{aligned} E\_{MRR} &= \quad F\_{MRR} \times E\\ &= \quad 0.0249 \times I^{0.9472} \times T\_{on} \, ^{-0.1855} \times VI \left(\frac{T\_{ou}}{T\_{on} + T\_{gf}}\right) \\ &= \quad \frac{0.0249 \times V \times I^{1.9472} \times T\_{on} \, ^{0.8145}}{T\_{ou} + T\_{gf}} \end{aligned}$$

$$\text{But, } E\_{MRR} = \rho \, V\_W \Big[ C\_p \left( T\_v - T\_o \right) + L\_m + L\_v \Big].$$

$$\frac{0.0249 \times V \times I^{1.9472} \times T\_{on}^{-0.8145}}{T\_{on} + T\_{off}} = \rho \left[ V\_W \left[ C\_p (T\_v - T\_o) + L\_m + L\_v \right] \right]$$

*Modeling of Material Removal Rate in Electrical Discharge Machining by a Novel Approach… DOI: http://dx.doi.org/10.5772/intechopen.81083* 

By putting the value of material properties of workpiece from **Table 1**,

$$
\Delta V\_W = \frac{4.1419 \times 10^{-4} \times V \times I^{1.9472} \times T\_{on}^{0.8145}}{T\_{on} + T\_{off}} \, mm^3/\text{sec}
$$

$$
\text{MRR} = \frac{0.0248 \times V \times I^{1.9472} \times T\_{on}^{0.8145}}{T\_{on} + T\_{off}} \, mm^3/\text{min} \tag{5}
$$

The material removal rate can be obtained using Eq. (5) at any set of process parameters for a given workpiece and tool combination.

#### **3.2 Other models**

#### *3.2.1 DiBitonto's model*

DiBitonto et al. [3] developed a simple mathematical model of material removal from steel (iron) in EDM. An analytical solution of the model is presented using a point heat source and power as the boundary condition at the interface of plasma and workpiece. They assumed the plasma radius to be negligibly small and a constant fraction of energy transfer to the workpiece as 0.183 for a large range of discharge current. The maximum erosion rate is determined from

$$\left(\left(V\_w\right)\_{max}\right)\_{max} = \frac{16\pi X^3 \pi\_0 {}^{3/2} F\_c U I}{\Im\left(\pi\_0 + \delta^\*\right) \left(T\_m - T\_0\right) \rho \, C\_p} \tag{6}$$

where X is given by

$$X^2 = \ln\left(\frac{1 \star 3\delta^\*/\pi\_0}{4\pi^{3/2}\sqrt{\pi\_0}}\right) \tag{7}$$

τ0 - dimensionless optimum pulse time (μs), Fc - Fraction of power transfer, U voltage (V), I - current (A), δ \* - dimensionless pause time, Tm - melting temperature (K), T0 - room temperature (K), ρ - density (kg/m3 ), Cp - specific heat (W/m K).

#### *3.2.2 Joshi's model*

A thermophysical model for the EDM process has been developed by Joshi and Pande [9] using finite element analysis. It has been carried out to predict the material removal rate by considering more realistic boundary conditions like Gaussian distribution of heat flux, spark radius based on discharge current and pulse on time, latent heat of melting, energy distribution factor for workpiece as 0.183, and neglecting convection heat loss from workpiece.

#### *3.2.3 Assarzadeh's model*

A comprehensive electro-thermal based single spark EDM model has been developed by Assarzadeh and Ghoreishi [11]. This model has been developed based on temperature dependent materials properties, Gaussian distribution of heat flux, convection, latent heat of fusion and expanding plasma channel with current and pulse on time. They have also assumed constant energy dispersing into the workpiece as 18% of total energy for all of the process parameters.
