Evaluation of a Mathematical Model Based on Lubanska Equation to Predict Particle Size for Close-Coupled Gas Atomization of 316L Stainless Steel

Silva, Flávia Costa da; Lima, Moysés Leite de; Colombo, Giovanna Fiocco

doi:10.1590/1980-5373-MR-2021-0364

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Articles • Mat. Res. 25 • 2022 • https://doi.org/10.1590/1980-5373-MR-2021-0364 copy

Evaluation of a Mathematical Model Based on Lubanska Equation to Predict Particle Size for Close-Coupled Gas Atomization of 316L Stainless Steel

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

Close-Coupled Gas Atomization (CCGA) is often used to produce spherical metal powders with a wider Particle Size Distribution (PSD) (10 – 500 µm) compared to that required by the main Additive Manufacturing processes (10 – 105 µm). This work presents an accuracy evaluation of a mathematical model based on the Lubanska equation to predict the d50 for CCGA. Atomization experiments of 316L steel were conducted to evaluate the tip diameter and atomization gas pressure effects on PSD and, the d50 experimental results were used as the reference to the mathematical model evaluation. The mathematical model accuracy could be improved by: (i) considering the backpressure phenomenon for the metal flow rate calculation, since it was an important inaccuracy source; (ii) reviewing the tip diameter effect, which had a lower impact on d50 than that predicted by the Lubanska equation. The atomization gas pressure was the most influential parameter on d50 and d90 and the increase of the gas pressure led to a significant reduction in PSD and, consequently, increased yield.

Keywords:
close-coupled gas atomization; additive manufacturing; 316L powder; Lubanska equation; process parameters

Equations
1. $d 50 = K \sqrt{[\frac{η_{m} σ_{m} d_{0}}{η_{g} U_{g}^{2} ρ_{m}} (1 + \frac{\dot{M}}{\dot{G}})]}$	d50=median particle size (μm) K = a constant that depends on the type of atomizer η_m = kinematic viscosity of the metal (m²/s) ν_m = dynamic viscosity (N.s/m²) ρ_m = density of the metal (kg/m³) ρ_g = density of the gas (kg/m³) η_g = kinematic viscosity of the gas (m²/s) σ_m = surface tension of the metal (N/m) T = temperature of the metal (K) d₀ = tip diameter (m) U_g = gas velocity (m/s) R = constant of the gases (8.314 m³.Pa/K.mol) M = molar mass of the gas (kg/mol) k = coefficient of heat capacities at constants pressure and volume of a gas Cp/Cv P₁ = atomization pressure (Pa) P₂ = atmospheric pressure in the atomization chamber (Pa) T₁ = gas temperature inside the nozzle (298.15 K) P_g = gas pressure (bar) $\dot{M} =$ metal flow (kg/s) φ = coefficient of friction A₀ = tip area (m²) P_f = pressure in the fusion chamber (bar) g = gravity acceleration (m/s²) h_m = height of liquid metal (m) Re = Reynolds number V_m = exit velocity of the metal (m/s) $\dot{G} =$ gas flow (kg/s) A_g = gas exit area (m²)
2. $η_{m} = \frac{ν_{m}}{ρ_{m}}; \ln ν_{m} = (- 2.396 + 7950 / T)$
3. $σ_{m} = 1840 - 0.4 (T - 1823)$
4. $\begin{array}{l} ρ_{m} = (69.4 % F e) + (66.3 % C r) + (71.4 % N i) + \\ (57.2 % M n) + (51.5 % M o) + (49.3 % S i) - \\ 0.86 (T - 1823) \end{array}$
5. $U_{g} = \sqrt{\frac{R . T_{1}}{M} . \frac{2. k}{k - 1} . {[1 - (\frac{P_{2}}{P_{1}})]}^{\frac{k - 1}{k}}}$
6. $\dot{M} = φ A_{0} ρ_{m} \sqrt{2 [\frac{P_{f}}{ρ_{m}} + g h_{m}]}$
7. $φ = 1 - \frac{7.96}{\sqrt{R e}}$
8. $R e = \frac{ρ_{m} V_{m} d_{0}}{μ_{m}}$
9. $\dot{G} = A_{g} \sqrt{(k ρ_{g} P_{g}) {[\frac{2}{k + 1}]}^{(k + (\frac{1}{k}) - 1)}}$
10. $ρ_{g} = 4.5 (1.6317 * 10^{- 6} P_{g} + 1.0585)$

Experiment Code	P_g(bar)	d₀(mm)	P_f(bar)	h_m(mm)	T_s(K)
A	30	1.5	0.3	90	200
B	40
C	50
D	30	2.0
E	40
F	50
G	30	2.5	0.2
H	40		0.2
I	50		0.3

Experiment Code	Lubanska (μm)	Lubanska with $\dot{M}$ exp (μm)	Experimental (μm)
A	84	85	103
B	78	72	76
C	74	59	62
D	118	102	111
E	108	77	73
F	100	70	65
G	149	110	104
H	135	77	71
I	131	80	63
Mean Error (μm)	9.0	5.3	6.6
Mean Absolute Error (μm)	68	18	-

Experiment Code	P_g (bar)	d₀ (mm)	P_f (bar)	$\dot{M}$ _cal (10^-3kg/s)	$\dot{M}$ _exp (10^-3kg/s)
A	30	1.5	0.3	37.5	39.5
B	40	1.5	0.3	37.5	27.3
C	50	1.5	0.3	37.5	10.5
D	30	2.0	0.3	67.7	43.5
E	40	2.0	0.3	67.7	18.9
F	50	2.0	0.3	67.7	12.6
G	30	2.5	0.2	92.7	39.4
H	40	2.5	0.2	92.7	9.1
I	50	2.5	0.3	106.3	14.7

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