That's explanation of how your database works. I was looking for a database-free explanation of the real-world system/process it models.
I went back and reread your initial question and you mentioned these new tasks are unrelated to this database, but somehow need to be included. But with the structure you posted it does seem related to this database (T_area, T_equipment, etc.).
Why wouldn't the existing task table work?
Also, and back to your relationship issue, this new table wouldn't need to be related to T_department, T_area and T_equipment since those 3 tables are in relation to each other. You would be storing redundant data if you had all three of those pieces of information in that table.
Suppose you were making a database for professionl sports teams. You would have a League table, a Team table, and a Players table. Those 3 tables would not all be related. Players would relate to a team and teams would relate to a League. You could still find out that Wayne Rooney played in the Premier League, but that specific piece of information wouldn't be stored in the Players table (or the Leauge Table). You would have to associate Rooney with a Team and a Team to a League to get there, but you still could.
Your tables are breaking that idea. If a T_equipment is related to a T_area which is related to a T_department, then don't need to relate T_department to T_NewTasks. You simply relate a T_equipment to a T_NewTask and you can follow the path from T_equipment to T_area to T_department to find out what T_department the T_NEwTask relates to. You don't do it directly.
I went back and reread your initial question and you mentioned these new tasks are unrelated to this database, but somehow need to be included. But with the structure you posted it does seem related to this database (T_area, T_equipment, etc.).
Why wouldn't the existing task table work?
Also, and back to your relationship issue, this new table wouldn't need to be related to T_department, T_area and T_equipment since those 3 tables are in relation to each other. You would be storing redundant data if you had all three of those pieces of information in that table.
Suppose you were making a database for professionl sports teams. You would have a League table, a Team table, and a Players table. Those 3 tables would not all be related. Players would relate to a team and teams would relate to a League. You could still find out that Wayne Rooney played in the Premier League, but that specific piece of information wouldn't be stored in the Players table (or the Leauge Table). You would have to associate Rooney with a Team and a Team to a League to get there, but you still could.
Your tables are breaking that idea. If a T_equipment is related to a T_area which is related to a T_department, then don't need to relate T_department to T_NewTasks. You simply relate a T_equipment to a T_NewTask and you can follow the path from T_equipment to T_area to T_department to find out what T_department the T_NEwTask relates to. You don't do it directly.