An insert piece for a blood tubing set , comprising at least, respectively: - a first connection site for connecting a first tubing portion of the blood tubing set to the insert piece ; - a second connection site for connecting a second tubing portion of the blood tubing set to the insert piece ; - a third connection site for connecting a third tubing portion of the blood tubing set to the insert piece ; - a main line for conducting a first liquid, preferably blood, through the insert piece , the main line being in fluid communication with the first connection site and with the second connection site ; - a secondary line for conducting a second liquid, preferably an infusion, into the main line , the secondary line being in fluid communication with the third connection site , and the secondary line being also in fluid communication with the main line in an intermediate portion of the insert piece being arranged between the first connection site and the second connection site ; wherein - the first connection site comprises a flow-through lumen having a first cross section area; - the second connection site comprises a flow-through lumen having a second cross section area; - the intermediate portion comprises a flow-through lumen having a third cross section area; characterized in that : - the secondary line comprises a lumen portion which protrudes into an interior of the intermediate portion and which comprises at least an outlet or outlet opening for the second fluid; and - no flow obstacles are present inside the insert piece, with the exception of the secondary line and/or of an optional baffle or deflector element arranged in the area of the intermediate portion to limit the second liquid flowing out of the outlet opening or of the outlet of the secondary line in its radial movement or in its movement in the outlet direction. - a first connection site for connecting a first tubing portion of the blood tubing set to the insert piece ; - a second connection site for connecting a second tubing portion of the blood tubing set to the insert piece ; - a third connection site for connecting a third tubing portion of the blood tubing set to the insert piece ; - a main line for conducting a first liquid, preferably blood, through the insert piece , the main line being in fluid communication with the first connection site and with the second connection site ; - a secondary line for conducting a second liquid, preferably an infusion, into the main line , the secondary line being in fluid communication with the third connection site , and the secondary line being also in fluid communication with the main line in an intermediate portion of the insert piece being arranged between the first connection site and the second connection site ; wherein the secondary line being in fluid communication with the third connection site , and the secondary line being also in fluid communication with the main line in an intermediate portion of the insert piece being arranged between the first connection site and the second connection site ; - the first connection site comprises a flow-through lumen having a first cross section area; - the second connection site comprises a flow-through lumen having a second cross section area; - the intermediate portion comprises a flow-through lumen having a third cross section area; - the secondary line comprises a lumen portion which protrudes into an interior of the intermediate portion and which comprises at least an outlet or outlet opening for the second fluid; and - no flow obstacles are present inside the insert piece, with the exception of the secondary line and/or of an optional baffle or deflector element arranged in the area of the intermediate portion to limit the second liquid flowing out of the outlet opening or of the outlet of the secondary line in its radial movement or in its movement in the outlet direction. <EOS>