Formulario AMP for Written Exam

Waterjet calculations and principles

Theoretical velocity and compressibility

WATERJETV_TH = √(2P/ρ₀) Theoretical velocity from Bernoulli.

V_∞ = constant

V_TH,C = √(2L/(1-m)ρ₀) [(1+ ρ₀P/L)^(1-m)/m -1] Theoretical velocity considering compressibility.

Ψ = V_TH,C / V_TH = √(L/P(1-m)) [(1+ ρ₀P/L)^(1-m)/m -1] Compressibility coefficient Ψ < 1.

Jet velocity and coefficients

V_J = C_V⋅V_TH,C⋅Ψ⋅√(2P/ρ₀⋅0.001) Velocity of the jet [hundreds of m/s]

C_V = V_J / V_H,C Velocity coefficient.

S₀ = π d_N² / 4 Nozzle cross section.

S_J = π d_J² / 4 Jet cross section.

C_C = S_J / S₀ Contraction coefficient ≈ 0.62.

C_D = Q_W / Q_TH Overall coefficient of discharge.

Q_TH = S₀⋅V_TH = π d_N² / 4⋅ √(2⋅P/ρ₀).

Q_W = S_J⋅V_J = C_C⋅C_V⋅S₀√(2P/ρ₀) = C_D⋅S₀⋅ √(2P/ρ₀⋅1000).

ṁ_W = β⋅Q_W Water flow rate [L/min] β = 1000 g/ℓ.

Balance with abrasive

Q_air + Q_W + Q_abkc = Q_TOT.

V_air⋅V_air + ṁ_W⋅V_W + ṁ_abkc⋅V_abkc = (ṁ_air + ṁ_w + ṁ_abkc)⋅V_avg.

H_L = ṁ_abkc/ṁ_W = V_W/V_J + 1 Abrasive loading ratio.

R = ṁ_W / ṁ_abkc Abrasive flow rate.

Waterjet calculations (continued)

WATERJETV_TH = √(2P / ρ₀) Theoretical velocity from Bernoulli.

μ = constant

V_TH,c = √[2L / (1-m)ρ₀] [(1+ ρ/L)^1-m -1] Theoretical velocity considering compressibility.

Ψ = V_TH,c / V_TH = √[L / P(1-m)] [(1+ ρ/L)^1-m -1] Compressibility coefficient Ψ < 1.

Velocity of the jet

V_J = C_V · V_TH,c = C_V · Ψ · √(2P / ρ₀) Velocity of the jet [m/s] hundreds.

Coefficients and calculations

L = 300 MPa M = 0.1368 at 25ºC

C_V = V_S / V_TH,c Velocity coefficient.

S₀ = π d_n² / 4 Nozzle cross section.

S_J = π d_J² / 4 Jet cross section.

C_C = S_J / S₀ Contraction coefficient ≈ 0.62.

C_D = Q_W / Q_TH = Ψ · C_V · C_C Overall coefficient of discharge.

Q_TH = S₀· V_TH = πd_n² / 4 · √(2P / ρ₀).

Q_W = S_J· V_J = C_C C_V S₀ √(2P / ρ₀) = C_D· S₀ · √(2P / ρ₀) · 0.60 [ℓ / min] Water flow rate.

ṁ = β · Q_W ρ = 1000 g/ℓ

Balance with abrasive (continued)

Q_air + Q_W + Q_ab = Q_TOT.

ṁ_air V_air + ṁ_W V_W + ṁ_ab V_ab = (ṁ_air + ṁ_W + ṁ_ab) · V_avg.

H_a = ṁ_ab / ṁ_w Abrasive loading ratio.

R = ṁ_ab / ṁ_w + ṁ_air Abrasive load rate → V_out = V_W = V_j / H_t + 1 → V_out = R · V_S Conservation of momentum.

Thrust forces

THRUST FORCES normed = a_avg ∙ |ΔV₁| = (ṁ_in + m_store) ∙ |Δ₁| [N] - [kg ∙ m/s²]

|ΔV₁| = |V_in - V_out|

V_x = V₁ ∙ sinΘ

V_y = V₁ ∙ cosΘ - (v - v_o ∙ sinΘ) = 2V ∙ cosΘ(V_in - V_out) = 2 ∙ V_out

dCobra/dt = (V_obr - V_rar) / (V_obr - V_r)

Δt = V_rar (l_out / q_out)

Jet reactive force

JET REACTIVE FORCE Basically the same as above, without abrasive.

F = ṁ_in ∙ ΔV₁ = β ∙ C_C ∙ ∙ 2V₃ = 2β C_C C_v y_{ϕ o} √ (2p / _o) = √β C_C C_v y_ϕ² d_n² √ (2p / _s)= C_C C_v^{2 2} d_n P

Power calculations

POWER P_HYDR = ¹⁄₂ ṁ_in v₃² = ¹⁄_{2 o 3} v₃² = ₄ β C_C C_v d_n² √ (2p / _o) ²= ^V₂/₀ C_C C_v ³ d_n² Pv/P = P_HYDR [KW]

P_e = Neg. - P_HYDR

Taper and trail back

If you have to compensate it: angle of entrance of the jet T = (w_top - w_bottom) / 2

T_a = arctan (T / h_a) compensation angle.

Production considerations

t_{cut & figure} t_{cut aqn} t_{extra = tcut + taper + tsetup}

C = C_stor ∙ C_{machine on hour} C₁ per C_c = on m / piece

Other issues

RAPIDS must be counted twice if you are doing a shape like that: 2 Turns due at the head.

AVAILABILITY is multiplied to the time of work at week Tmax = 1000 ℓ There I = Tmax

Control if the pump is able to supply all the Q (or all the heads) Usually you find then you express it in LASER

Laser sources specifications

• CO₂ λ = 10.604 μm
• DIODE λ = 0.81; 0.98 μm
• Nd:YAG λ = 1.064 μm
• Yb: λ = 1.090 μm

Optics and focus

If plate thickness < 3 mm - Δz = 0, focus on the surface; > 3 mm - Δz = ^z/₂ focus in the middle at the peak.

FREE OPTICS (CO₂) M² = ^{BPP π}/_{λn μm} d₀ = ⁴/_π M² λ f_l [μm]

I₀ = ^P/_{50 cm²} [Kw/cm²] Θ = ^{d_foc [rad]}/_ff or Θ = ^{4 BPP}/_d0

Real beam and Gaussian beam

REAL BEAM d₀ = d_0g M² = ⁴/_π λ f_l [μm]

d²() = d₀² + (z-₀)² ^mrad Θ² = ^d_foc/_ff

d²() = d₀² + Δz² e²

GAUSSIAN BEAM d₀ = d_0g = ⁴/_π λ f_l

Steen's law and fiber optics

STEEN'S LAW A . P = V . w . t . ρ (c (Tm - Ta) + Lf + m (Lv)) W = d_weld

FIBER NA = √n_core² - n_clad² α = θ_accept = arcsin(NA)

Sinθ_i = ^n₂/_Sinθt

d₀ = ^f/_focm . d_f0 [μm] Θ = ^{4 BPP}/_d0 [mrad]

Caustic equation and irradiance

CAUSTIC EQUATION d²() = d₀² + Δz² e² d(2max-) = √ ^P/_I . ⁴/_π

Irradiance I = ^{A . P}/_S > 10⁶ W/cm² to make process feasible

Piercing a hole and processing time

Energy required E = Volume . C [J] t_p = ^E/_J [s] J = ^{A . P} [W . m^-2]

Pulsed Power and one application d_s² = d_o² + Δz² . θ²

In this case, P_MAX ≠ P_AVG because the beam source is pulsed. P_peak = ^P_AVG/_{fp . τ}

I_cut = ^P_peak/_Scut

D_f = √^P_peak/_Icut . ⁴/_π

tan(α'/2) = D_f/2f   f_f = D_f/2 . tan α'/2

EDM and material removal rate

Usually you have:

• n_o holes
• d_hole
• d_tool

You can choose two t_on:

1. Maximize MRR_piece, but you have to go high MRR_tool
2. Minimize MRR_tool (so the number of tools)

∆MRR = MRR_piece - MRR_tool → if you have to choose, choose the one with the HIGHEST ∆MRR

Holes and tool considerations

V_hole = πd_h² . ℓ_h [mm³]

V_tot = n_o holes . V_hole

Processing time = _Vtot/^MRR_piece

TOOL V_tool = πd_tool² . ℓ_tool or T_{tool eff} = _Vhole/^MRR_tool

n_o tools = _{Processing time}/^{T_{tool eff}}

Roughness

Has the minimum time R_a = 3 . (log₁₀ t_on)² - 10 log₁₀ t_on + 9

With EDM → R_{a, max} = 10 μm