An apparatus for processing an edge of a substrate sheet. The apparatus includes a finishing member, a shroud, and a tubular member. The finishing member is rotatably maintained within a chamber of the shroud. The shroud includes a wall segment terminating at a major edge that defines a portion of a slot. The slot is configured to receive the edge of the substrate sheet, facilitating interface of the edge with the finishing member. The wall segment further defines a nozzle passageway that terminates at an opening in the major edge. The tubular member projects from the major edge and defines a passage in fluid communication with the nozzle passageway. Cooling agent delivered to the nozzle passageway is injected onto the finishing member via the tubular member even in the presence of vacuum induced cross-flow. In some embodiments, a spatial arrangement of the tubular member is adjustable.