Conduction modeling of multi-track raster scanning of heat over a rectangular surface region

An analytical conduction model is implemented for the multi-track raster scanning of a heat spot over sequential neighboring tracks upon a half-space surface, focusing on maximum temperature rise as a function of subsurface position as its primary example of a thermal response of interest, with simple rectangular surface regions scanned and the non-dimensionalization of the analysis (to heat spot size and power as well as half-space thermal properties) intended to broaden applicability of model output. As baseline, first the case of isolated single passes is studied, as would also approximate cases of long times between sequential passes, with no heating from a prior pass available to affect that during a following pass. Asymptotic behavior is found describing all slow scanning speed conditions as described by low Peclet number Pe ≤ 0.1, where isotherms of maximum temperature rise may be plotted as a function of position through the subsurface cross-section. Similar description for fast-scanning high Peclet conditions Pe ≥ 40 is also achieved, although temperature rise and depth must both additionally be multiplied by Pe before attaining isotherms asymptotically pertinent for all high Pe. Such high Pe cases are those for which the effect of shorter finite times between neighboring passes is further studied here, since for any given practical stroke length the correspondingly higher scanning speed results in reduced time between passes and thus more opportunity for a heat accumulation effect. Indeed, at a fixed Pe, for a given point within the body cross-section located in proximity to a track several passes into the region being scanned, its temperature rise is greater than for a single isolated pass, and that temperature rise increases further with decreasing time between neighboring passes, as would result from a decreased period of the cyclic scanning motion. This heat accumulation effect builds for passes further into the rectangular surface region from the first track scanned at the region’s edge, and stabilizes after some number of tracks in, a number found to decrease with increased track spacing. Likewise, if instead holding scanning period and in turn time between passes constant, maximum temperature rises at any given position are also found to increase with decreasing Pe, when maximum temperature rise and depth position are expressed in their modified high Peclet asymptotic forms as multiplied by Pe. The analysis further suggests that if maximum temperature rises and depths are both expressed in such modified forms, any further effect of Peclet number may instead be combined with that of non-dimensionalized scanning period as a function only of their product L* = Peτ*, the non-dimensional track length. Finally, while maximum temperature rise behavior around the mid-stroke location will be similar regardless of whether scanning is via back&forth or one-way pass motions and furthermore in one-way motions these maximum temperature rises will remain fairly uniform with position along the track length as each possesses the same elapsed time since proximity during the previous adjacent pass, in back&forth motion the time since proximity during the previous pass before turn-around and in turn maximum temperature rise will instead vary with position along the track.