Derive a general analytical model $($equation$)$ for the volume build rate of a direct, vector$-$scan, single$-$laser SL process in terms of the following parameters: Effective laser spot size or cured line width on the surface $($Lw$)$; Cured layer thickness $($Zr$)$; Laser scan speed $($Vs$)$; and time per layer to configure the next layer of liquid resin, the recoating or wiper time $($tw$).$ Assume neighboring $($cured$)$ tracks in a layer are just touching $($not overlapping$)$ at the surface, but overlap about $10\%$ maximum with the layer $($tracks$)$ below. Assume the uncured volume of resin between tracks is part of the build volume. Note that linear shrinkage $($SL$)$ is already included in the track cross$-$section dimensions, but not in the length direction.

For Zr $=$ Lw $=0\mathrm{.}1$ mm$,$ Vs $=2000$ mm$/$s$,$ and a volume curing shrinkage of $15\%,$ calculate the following:

$($a$)$ The volume build rate $($cm$3/$h$),$ neglecting the wiper time.

$($b$)$ The time to build a single $10$ x $10$ cm layer and a $10-$cm cube $(=1$ L$).$

$($c$)$ For what wiper cycle time, does the wiper time add $10\%$ to the total time in $($b$).$

$($d$)$ For what $($smaller$)$ layer area does the wiper time in $($c$)$ double the total time.

$($e$)$ For a $10\mathrm{.}0$ x $10\mathrm{.}0$ cm cured layer, if the first track defines the outer edges of the layer, what must be the length of the edges scanned by the laser.

$($f$)$ Repeat part $($b$)$ for the laser set at the fill or $$hatch$$ mode, where Lw $=0\mathrm{.}5$ mm and Zr $=0\mathrm{.}1$ mm$.$