A stator assembly for surrounding a rotor, the stator assembly comprising: a pair of adjacent turbine vane or blade segments (10,28); a gap between inner bands of the pair of adjacent turbine vane or blade segments (10,28); a seal extending between the pair of adjacent vane or blade segments (10,28) and defining a radial depth of said gap; each of the pair of turbine vane or blade segments (10,28) comprising: at least one airfoil (16,18) extending radially outwardly from a radially inner band, said radially inner band formed with substantially axially-extending side edges; said at least one airfoil (16,18) formed with a leading edge, a trailing edge, a pressure side and a suction side; said pressure side facing one of said substantially axially-extending side edges (32,34) and said leading edge located proximate to said one of said substantially axially-extending side edges; and a plurality of cooling passages (46,48,50,52,54) formed in said radially inner band in fluid communication with a source of cooling air and exiting said inner band via a plurality of exit holes in said one of said substantially axially-extending side edges (32,34), said plurality of exit holes confined to a region where static pressure Ps of hot combustion gases flowing along said one of said substantially axially-extending side edges (32,34) lies in a pressure range substantially between a stage inlet total pressure P T and a pressure that is substantially about 1.5 times a dynamic pressure ΔP D below the stage inlet total pressure, wherein the dynamic pressure ΔP D is P T -P S at the stage inlet. a pair of adjacent turbine vane or blade segments (10,28); a gap between inner bands of the pair of adjacent turbine vane or blade segments (10,28); a seal extending between the pair of adjacent vane or blade segments (10,28) and defining a radial depth of said gap; each of the pair of turbine vane or blade segments (10,28) comprising: at least one airfoil (16,18) extending radially outwardly from a radially inner band, said radially inner band formed with substantially axially-extending side edges; said at least one airfoil (16,18) formed with a leading edge, a trailing edge, a pressure side and a suction side; said pressure side facing one of said substantially axially-extending side edges (32,34) and said leading edge located proximate to said one of said substantially axially-extending side edges; and a plurality of cooling passages (46,48,50,52,54) formed in said radially inner band in fluid communication with a source of cooling air and exiting said inner band via a plurality of exit holes in said one of said substantially axially-extending side edges (32,34), said plurality of exit holes confined to a region where static pressure Ps of hot combustion gases flowing along said one of said substantially axially-extending side edges (32,34) lies in a pressure range substantially between a stage inlet total pressure P T and a pressure that is substantially about 1.5 times a dynamic pressure ΔP D below the stage inlet total pressure, wherein the dynamic pressure ΔP D is P T -P S at the stage inlet. at least one airfoil (16,18) extending radially outwardly from a radially inner band, said radially inner band formed with substantially axially-extending side edges; said at least one airfoil (16,18) formed with a leading edge, a trailing edge, a pressure side and a suction side; said pressure side facing one of said substantially axially-extending side edges (32,34) and said leading edge located proximate to said one of said substantially axially-extending side edges; and a plurality of cooling passages (46,48,50,52,54) formed in said radially inner band in fluid communication with a source of cooling air and exiting said inner band via a plurality of exit holes in said one of said substantially axially-extending side edges (32,34), said plurality of exit holes confined to a region where static pressure Ps of hot combustion gases flowing along said one of said substantially axially-extending side edges (32,34) lies in a pressure range substantially between a stage inlet total pressure P T and a pressure that is substantially about 1.5 times a dynamic pressure ΔP D below the stage inlet total pressure, wherein the dynamic pressure ΔP D is P T -P S at the stage inlet.