In a laser cutting process of aluminum plates of 1mm thick, a through hole is to be drilled, followed by melting-dominated cutting. If the focused laser beam is 0.25 mm in diameter (assuming it is the same of the cut width w), (a) In order to drill a hole through the plate in 1 ms, determine the laser power required. (b) In the cutting, the laser power is adjusted to be 1,500W. Determine the cutting velocity V achievable.

energy is used to heat the aluminum, we should calculate the energy necessary to bring the solid aluminum to the melting temperature and add the energy to carry out the change of solid state to liquid,

let's use the calorimeter equation

Q₁ = m c_e ΔT

and the energy of change of these solid to liquid (fusion process)

Q₂ = m L

the energy required to create the hole is

Q_ {total} = Q₁ + Q₂

if there are no losses this is the laser energy

E = Q_ {total}

The aluminun data c_e =9000 J/kgC, L = 322 103 J/kg, ρ = 2.7 103 kg/m3 , T₂ = 660C, T₀= 25C

Let's find the mass of the hole, which we approximate by a cylinder of diameter d = 0.25 mm = 0.25 10⁻³ m and a thickness of e = 1 mm = 1 10⁻³ m

let's use the concept of density

ρ = m / V

the volume of a cylinder is

V = π r² e = π (d²/4) e

we substitute

m = [tex]\rho \pi \frac{d^2 e}{4}[/tex]

let's calculate

m = π/4 2.7 10³ (0.25 10⁻³)² 1 10⁻³

m = 1,325 10⁻⁷ kg

we calculate the energy

E = 1,325 10⁻⁷ 900 (660 - 25) + 1,325 10⁻⁷ 322 10³

E = 7.57 10⁻² + 4.27 10⁻²

E = 1.184 10⁻¹ J

Let's use the power ratio

P = E / t

P = 0.1184 /1 10⁻³

P = 118.4 W

b) In this part they indicate that the laser power is P = 1500 W, find the time to deposit the energy to melt the aluminum

P = E / t

t = E / P

t = 0.1184 / 1500

t = 7.9 10⁻⁵ s