Heat accumulation during pulsed laser materials processing: erratum

Rudolf Weber; Thomas Graf; Peter Berger; Volkher Onuseit; Margit Wiedenmann; Christian Freitag; Anne Feuer

doi:10.1364/OE.22.028232

Abstract

With this erratum we aim to correct a transcription error that occurred in our previous paper: In Eq. (3)a)-(3c), Eq. (5), and Eq. (6) the Greek characters were not converted correctly. The properly formatted formulae are listed below. All other contents, calculations and conclusions of the original paper remain unchanged.

Corrected formula

In the formulae (3a)-(3c) in [1], the roman characters r (just the first letter in the denominator), p, and k must be replaced by the Greek characters ρ, π, and κ, respectively. The formulae correctly read:

T_{1 D} - T_{0} = Δ T_{1 D} = \frac{Q_{1 D}}{ρ \cdot c_{p} \cdot \sqrt{4 \cdot π \cdot κ \cdot t}} e^{- \frac{r^{2}}{4 \cdot κ \cdot t}} (r^{2} = z^{2})

T_{2 D} - T_{0} = Δ T_{2 D} = \frac{Q_{2 D}}{ρ \cdot c_{p} \cdot \sqrt{{(4 \cdot π \cdot κ \cdot t)}^{2}}} e^{- \frac{r^{2}}{4 \cdot κ \cdot t}} (r^{2} = x^{2} + y^{2})

T_{3 D} - T_{0} = Δ T_{3 D} = \frac{Q_{3 D}}{ρ \cdot c_{p} \cdot \sqrt{{(4 \cdot π \cdot κ \cdot t)}^{3}}} e^{- \frac{r^{2}}{4 \cdot κ \cdot t}} (r^{2} = x^{2} + y^{2} + z^{2})

where ΔT_nD is the temperature increase with respect to the initial temperature T₀, _{$n D \in [1, 2, 3]$}, ρ is the mass density of the solid or liquid material, c_p its specific heat capacity, κ = λ_th /(ρ c_p) the temperature conductivity, λ_th the heat conductivity, t is time, and x,y,z are the spatial coordinates.

The same corrections apply to Eq. (5)

T_{n D} - T_{0} = Δ T_{n D} = \frac{Q_{n D}}{ρ \cdot c_{p} \cdot \sqrt{{(4 \cdot π \cdot κ \cdot t)}^{n D}}} e^{- \frac{1}{t} \cdot \frac{r_{n D}^{2}}{4 \cdot κ}}

and Eq. (6)

Δ T_{n D} (t, N) = \frac{Q_{n D} \cdot Θ (t - \frac{N - 1}{f_{L}})}{ρ \cdot c_{p} \cdot \sqrt{{(4 \cdot π \cdot κ \cdot (t - \frac{N - 1}{f_{L}}))}^{n D}}} e^{- \frac{1}{(t - \frac{N - 1}{f_{L}})} \cdot \frac{r_{n D}^{2}}{4 \cdot κ}}

The Heaviside function Θ is equal to zero for arguments <0 and equal to one for arguments ≥0.

All other contents, calculations and conclusions of the original paper are correct and remain unchanged. The description of the variables was not changed and is kept in the text for easier reading of the formulae.

References and Links

1. R. Weber, T. Graf, P. Berger, V. Onuseit, M. Wiedenmann, C. Freitag, and A. Feuer, “Heat accumulation during pulsed laser materials processing,” Opt. Express 22(9), 11312–11324 (2014). [CrossRef] [PubMed]

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Equations (5)

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T_{1 D} - T_{0} = Δ T_{1 D} = \frac{Q_{1 D}}{ρ \cdot c_{p} \cdot \sqrt{4 \cdot π \cdot κ \cdot t}} e^{- \frac{r^{2}}{4 \cdot κ \cdot t}} (r^{2} = z^{2})

T_{2 D} - T_{0} = Δ T_{2 D} = \frac{Q_{2 D}}{ρ \cdot c_{p} \cdot \sqrt{{(4 \cdot π \cdot κ \cdot t)}^{2}}} e^{- \frac{r^{2}}{4 \cdot κ \cdot t}} (r^{2} = x^{2} + y^{2})

T_{3 D} - T_{0} = Δ T_{3 D} = \frac{Q_{3 D}}{ρ \cdot c_{p} \cdot \sqrt{{(4 \cdot π \cdot κ \cdot t)}^{3}}} e^{- \frac{r^{2}}{4 \cdot κ \cdot t}} (r^{2} = x^{2} + y^{2} + z^{2})

T_{n D} - T_{0} = Δ T_{n D} = \frac{Q_{n D}}{ρ \cdot c_{p} \cdot \sqrt{{(4 \cdot π \cdot κ \cdot t)}^{n D}}} e^{- \frac{1}{t} \cdot \frac{r_{n D}^{2}}{4 \cdot κ}}

Δ T_{n D} (t, N) = \frac{Q_{n D} \cdot Θ (t - \frac{N - 1}{f_{L}})}{ρ \cdot c_{p} \cdot \sqrt{{(4 \cdot π \cdot κ \cdot (t - \frac{N - 1}{f_{L}}))}^{n D}}} e^{- \frac{1}{(t - \frac{N - 1}{f_{L}})} \cdot \frac{r_{n D}^{2}}{4 \cdot κ}}

Abstract

Corrected formula

References and Links

Cited By

Equations (5)

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