Experiments involving the interaction of a high-power laser beam with metal targets demonstrate that combustion plays an important role. This process depends on reactions within an oxide layer, together with oxygenation and removal of this layer by the wind. We present an analytical model of laser-induced combustion. The model predicts the threshold for initiation of combustion, the growth of the combustion layer with time, and the threshold for self-supported combustion. Solutions are compared with detailed numerical modeling as benchmarked by laboratory experiments

Boley, C D

Rubenchik, A M

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LLNL-CONF-401854Modeling of Laser-Induced MetalCombustionC. D. Boley, A. M. RubenchikFebruary 29, 20085th Annual High Energy Laser Lethality ConferenceMonterey, CA, United StatesMarch 3, 2008 through March 7, 2008Disclaimer  This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.  This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.Modeling of Laser-Induced Metal Combustion*C. D. Boley1 and A. M. Rubenchik2Lawrence Livermore National LaboratoryLivermore, CA 94551 Experiments involving the interaction of a high-power laser beam with metal targets demonstrate that combustion plays an important role. This process depends on reactions within an oxide layer, together with oxygenation and removal of this layer by the wind. We present an analytical model of laser-induced  combustion. The model predicts the threshold for initiation of combustion, the growth of the combustion layer with time, and the threshold for self-supported combustion. Solutions are compared with detailed numerical modeling as benchmarked by laboratory experiments.Keywords: solid-state laser, high average power, lethality, combustion, analytical modelNomenclaturea spot radiusC specific heatD thermal diffusivityI   laser intensityQ combustion power fluxT temperatureTc combustion initiation temperaturea optical absorptivityk thermal conductivityr density1. IntroductionDuring the last decade, our laboratory at LLNL has been developing solid-state lasers with high average power, for applications to tactical defense. Our most current laser containsfour diode-pumped ceramic Nd:YAG slabs12-15. It produces approximately 25 kW of average power for 10 seconds, at a wavelength of 1.053 mm. Upon addition of another slab and an    *Work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.1E-mail address: boley1@llnl.gov; telephone: (925) 423-7365; fax: (925) 423-6195.2E-mail address: rubenchik1@llnl.gov; telephone: (925) 422-6131. 2 increased diode duty factor, the laser has yielded 67 kW for 0.25 s. The waveform consists of pulses of length 0.5 ms, at 200 Hz, corresponding to a duty factor of 10%. During lasing operations, waste heat is stored in the slabs. In actual applications, the hot slabs would be rapidly cycled out of the aperture and cooled, while cold slabs would simultaneously be cycled into the aperture.We have studied the material interactions produced by such lasers, including high-explosive initiation, in some detail1-9. Because the thermal conduction length between pulses is small compared to typical target dimensions, the macroscopic heat distribution is generally determined by the time-average power rather than the instantaneous power. At a 10% duty factor, the latter is of course 10 times the former. We have constructed a computational model (THALES2-9) which yields detailed predictions. At a power near 100 kW, the laser has been projected to be effective in defense against mortars and rockets2,8,9. A lower power suffices for neutralization of threats such as landmines1 and improvised explosive devices.We have previously shown that combustion plays an important role in facilitating material heating2,3,8. We have incorporated a model of combustion into THALES and found parameter values which seem consistent with experiment. While this detailed treatment is useful, it does not readily illuminate some general features and trends. Among these are the threshold for combustion and the possibility of self-supported combustion. The latter occurs when combustion reactions continue after the laser has been turned off. This has been observed in LLNLexperiments on titanium3 but not on iron2,3,8. An analytical approach can add to our understanding of such issues.In this paper, we present an analytical model of laser-induced combustion in metals. The model is based on the phenomenology of combustion11, together with solutions of the heat  3 conduction equation in solids10. It predicts the threshold for initiation of the combustion process, the relative role of laser heating and energy deposition due to combustion affects, and the threshold for self-supported combustion. We compare analytical results with evidence from experiments. The model appears to be consistent with this evidence.2. ModelThe equation governing heat conduction within a solid isTtTC 2Ñ=¶¶ kr ,    (1)whereT is the local temperature, Cr is the heat capacity per volume, and k is the thermal conductivity. We shall also use CD rk /= , the thermal diffusivity. We assume that the material properties can be treated as constants. We work in cylindrical coordinates ( rz, ), so that the temperature has the dependence ),,( trzT . Energy is supplied to the system by both laser heating and combustion. We assume that the beam is turned on at 0=t and thereafter is constant. The intensity distribution is taken to be Gaussian. Thus the absorbed laser flux has the form)/exp()( 220 arIrP -= a , (2)where a is the surface absorptivity and a is the spot radius.Combustion occurs within a thin oxide layer on the surface11. The combustion products are rapidly removed by the wind. Making simple assumptions regarding the turbulent boundary layer3, one finds that the energy flux due to combustion takes the form)/exp( TTQQ cc -= ,  (3)with cQ proportional to the wind speed. Modeling the laser irradiation of iron at Mach 0.3, we found3 that the temperature response was consistent with =cQ 0.8 kW/cm2 and 2000=cT K. To get a qualitative understanding of the combustion effect, we approximate this flux as a step  4 function in temperature. We say that the combustion flux equals cQ for temperatures above a particular “on-off” temperature 0T , while it vanishes for temperatures below this,   0TTQc > ,   »Q     (4) 00 TT < .The quantity 0T is of the same order as cT but the two are not necessarily equal. Under the reasonable assumption that the temperature decreases monotonically away from the center of the beam, the combustion flux is a flat-top with radius )(tR , defined by the condition0]),(,0[ TttRrzT === . For qualitative purposes, we neglect the modification of the surface profile by material removal. This is most appropriate for large spot sizes.These considerations regarding the energy sources are summarized in the boundary condition:QPzTz+=÷øöçèæ¶¶=0k .  (5)Thus we have a temperature distribution produced by two sources – a Gaussian function in radius produced by the laser and a step function in radius due to combustion. Since the heat conduction equation is linear, the solution is the sum of the solutions due to each source separately. The solution at the surface will suffice for our purposes.The temperature distribution due to a given heat flux is a familiar problem for which many explicit solutions are available10. For our laser source, the surface temperature can be written as )]1/()/(exp[)1(2),,0( 202/12/10~xarxxdxIatrzTtP +-+== òkpa, (6) 5 where 2~/4 atDt = . Similarly, the surface temperature due to a heat flux 0Q delivered within a radius 0r has the formkdktDkkrJkrJQRtrzTQ ])([erf)()(),,0(2/101000 ò¥==k, (7)in terms of Bessel functions and the error function. If 0r changes slowly in time compared to the average surface temperature, this equation will remain approximately valid. In our model, we assume that this condition is satisfied.Putting these results together, we see that the combustion radius )(tR is determined by the condition]),(,0[]),(,0[0 ttRTttRTT QP += .  (8)It will be convenient to arrange this in the final dimensionless formlllll drtJJrqxrxxdxIt]2/)([erf)()()]1/()(exp[)1(1~2/1~100~~2~02/1~.~òò¥++-+= , (9)where the dimensionless radius of combustion, the dimensionless intensity, and the dimensionless combustion flux are defined by, respectively,atRr /)(~= , 02/10~2/ TIaI pa= , 0~/ TQaq c k= .   (10)The dimensionless time was defined after Eq. (6). Equation (9) is the basic statement of our model. This is a nonlinear condition giving the radius of combustion as a function of time, for particular values of the laser intensity and combustion flux. In the regime of interest, the solutions of Eq. (9) have a characteristic behavior. Before a particular time 00 >t , there is no solution. At this time, there occurs a single solution which develops two branches for larger times. One of these has a radius increasing with time, while the  6 Fig. 1. Evolution of dimensionless combustion radius with dimensionless time, for two choices of the dimensionless intensity~I . Here the dimensionless combustion flux is .1.1~=qother has a radius decreasing with time, as illustrated in Fig. 1. Only the former branch is physical. A natural interpretation of this behavior is that, before the threshold time 0t , the radius is zero. The system then develops an instability which causes the solution to jump to the stable branch. The entire solution follows the arrows shown in Fig. 1. Here two typical solutions, corresponding to two choices of the intensity, are given. As the intensity decreases, the threshold time increases and the curves move to the right.  7 Fig. 2. Evolution of radius with temperature 14000 =T C. Red line: Present model; blue line: detailed THALES calculation; black line: THALES calculation without combustion.3. ApplicationsWe now turn to practical applications, beginning with the laser irradiation of an iron slab.We work with a laser power of 6.3 kW and a spot radius of 1 cm, corresponding to a central intensity of 2 kW/cm2. We use 8.0=cQ kW/cm2, as noted after Eq. (3), and try =0T 1400 C, remembering that this need not be the same as cT . The other assumed material parameters are 2.5=Cr J/(cm3 K), 4.0=k W/(cm K), and 5.0=a . These represent reasonable values within the temperature range of interest.Figure 2 shows the time dependence of the predicted combustion radius, where 0TT = .This radius becomes nonzero at a threshold time somewhat greater than 8 s, at which it jumps to THALESPresent ModelTHALES(no combustion) 8 about 0.4 cm and then rises steadily. The figure also shows the corresponding result given by detailed THALES modeling with the combustion flux of Eq. (3) and the same material parameters as above. This indicates a gentler initial rise, as would be expected. The two predictions become comparable at longer times. Finally, the location of this radius is shown for a THALES calculation without combustion. This clearly has a milder behavior than either of the results involving combustion. From these features, we infer that the analytical model is qualitatively realistic. Next, we turn to the issue of self-supported combustion (ignition). We can obtain a relevant estimate by considering the model, as defined by Eq. (9), in the limit of zero intensity and long time. We havelllldJJrq )()(1 100~~ò¥= ,                 (11)which implies that the (dimensional) radius of combustion is cQTR 2/0kp= . We see that in order for this situation to be reached, the beam radius must be sufficiently large:cQTa 2/0kp³ .                    (12)This is also consistent with dimensional analysis, within a factor of order unity. However, dimensional analysis gives no assurance that ignition will actually occur, in contrast with our model.We would not expect Eq. (12) to be satisfied in iron, where self-supported combustion was not observed in experiments2,3,8. However, it turns out to be nominally satisfied, since7.1=a cm (the circular equivalent of the experimental 3x3 cm2 beam) and the right-hand side is 1.3 cm. We conjecture that this is due to inaccuracies in our nominal model parameters. It may  9 be that the experiments would have reached ignition for a somewhat larger spot size, at fixed central intensity and thus greater power.In titanium, however, the situation is much more definite. The ignition condition (12) is more likely to be satisfied for three reasons: (1) the thermal conductivity is approximately one-third that of iron, depending on the temperature; (2) titanium burns at a much lower temperature than iron (~1000 K versus ~2000 K); and (3) the heat of combustion is at least comparable to that of iron11. These considerations appear to explain why titanium ignited in our experiments.4. ConclusionsWe have presented an analytical model of laser-induced combustion in metals. The model incorporates simple principles of combustion together with the heat conduction equation. Evaluating solutions of the model, we have found that it predicts a threshold for the combustion process, followed by an increase of the combustion radius with time. It also gives a threshold for self-supported combustion. To the extent that experiments are available in this regime, the model appears to be consistent with reality.AcknowledgmentsWe are happy to acknowledge our debt to R. M. Yamamoto, project manager of the LLNL high average power laser program. 10 References1Boley, C. D., C. B. Dane, S. N. Fochs, T. J. McGrann, and A. M. Rubenchik, “On Excavation and Destruction of Landmines by a High-Energy Pulsed Laser,” Sixth Annual Directed Energy Symposium, Albuquerque, NM, Oct. 20-24, 2003 (UCRL-PRES-153899).2Boley, C. D., S. N. Fochs, and A. M. Rubenchik, “Lethality of a High-Power Solid-State Laser,” Ninth Annual Directed Energy Symposium, Albuquerque, NM, Oct. 30 – Nov. 3, 2006 (UCRL-PRES-225460).3Boley, C. D., S. N. Fochs, and A. M. Rubenchik, “Lethality Effects of a High-Power Solid-State Laser.” Fourth High Energy Laser Lethality Conference, Monterey, CA, March 19-23, 2007 (UCRL-PRES-228666); UCRL-JRNL-234510, Sept. 11, 2007.4Boley, C. D., and A. M. Rubenchik, “Modeling of Material Removal by Solid State Heat Capacity Lasers,” Proc. 15th Annual Solid State and Diode Laser Technology Review, Albuquerque, NM, June 3-6, 2002.5Boley, C. D., and A. M. Rubenchik, “Modeling of High-Energy Pulsed Laser Interactions with Coupons,” University of California, UCRL-ID-151857, Feb. 6, 2003.6Boley, C. D., and A. M. Rubenchik, “Simulations of Target Interactions with Pulsed, High-Energy Laser,” Proc. 17th Annual Solid State and Diode Laser Technology Review, Albuquerque, NM, June 8-10, 2004.7Boley, C. D., and A. M. Rubenchik, “Modeling of the Initiation of High-Explosive Targets by Solid-State Heat Capacity Lasers,” Seventh Annual Directed Energy Symposium, Rockville, MD, Oct. 18-21, 2004 (UCRL-PRES-207230).8Boley, C. D., and A. M. Rubenchik, “Lethality of High-Power Solid-State Lasers on High-Explosive Targets,” Eighth Annual Directed Energy Symposium, Lihue, HI, Nov. 14-18, 2005 (UCRL-PRES-216504).9Boley, C. D., and A. M. Rubenchik, “Modeling of Antimortar Lethality by a Solid-State Heat-Capacity Laser,” J. Directed Energy 2, 97-106 (2006).10Carslaw, H. S., and J. C. Jaeger, “Conduction of Heat in Solids,” 2nd Edition, Clarendon Press, Oxford, 1959.11Prokhorov, A. M., V. I. Konov, I. Ursu, and I. N. Mihailescu, “Laser Heating of Metals,” Adam Hilger, 1990. 12Soules, T. F., et al., “Ceramic Nd:YAG – Current Material of Choice for SSHCL,” Eighth Annual Directed Energy Symposium, Lihue, HI, Nov. 14-18, 2005.13Yamamoto, R. M., et al., “Laser Performance of the Solid-State Heat-Capacity Laser (SSHCL),” Ninth Annual Directed Energy Symposium, Albuquerque, NM, Oct. 30 – Nov. 3, 2006.14Yamamoto, R. M., et al., “Evolution of a Solid State Laser,” Defense and Security Symposium, SPIE, Orlando, FL, April 9-13, 2007 (UCRL-ABS-229142).15Yamamoto, R. M., et al., “The Use of Large Transparent Ceramics in a High Powered , Diode Pumped Solid State Laser,” ASSP 2008, Nara, Japan, Jan. 27-30, 2008 (UCRL-CONF-235413).

Modeling of Laser-Induced Metal Combustion

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Modeling of Laser-Induced Metal Combustion

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