(Solved): Q3/ The conceptual design of a pneumatic system for temperature controlled of a room is shown in f ...

Q3/ The conceptual design of a pneumatic system for temperature controlled of a room is shown in fig. (3). The control system is a regulator, whose objective is to maintain a constant room temperature \( T_{1} \) corresponding to the set point, in the presence of disturbance caused by variation of the outside temperature \( T_{a} \). The displacement \( x_{f} \) of the flapper increases when the set point is increased by \( T_{r} \). Also, the bimetal is mounted such that \( x_{f} \) decreases when the room temperature increases by \( T_{1} \). For the pneumatic thermostat, the constant supply pressure is \( P_{s} \) and the pressure \( P_{1} \) is controlled by the displacement \( x_{f} \) in order to balance the pressure \( P_{2} \) in the pneumatic diaphragm actuator through the resistance \( R \). 1. State the mathematical equations for all controlled variables 2. Determine the overall block diagram for this system for output \( T_{1} \) and input \( T_{r} \). (50 Marks) Figure (3)