The figure above shows a pneumatic pressure system. Before time t=0, the system is at a steady state with a certain pressure throughout. Both bellows in the system are the same. At t=0, the input pressure changes and this causes the pressures in bellows 1 and 2 to change as well. Each bellow has a volume of 5104m3 and can handle a pressure difference of between 0.5105N/m2 and 0.5105N/m2. The figure also shows how the mass flow rate changes through the valves. The bellows can expand or contract in proportion to the air pressure and they are considered to have a spring constant of k=1105N/m. The area of each bellow is A=15104m2.(5+2+8) (a) Given, 2 P= gas pressure in the vessel at steady state (before changes in the pressure have occurred), lbf/ft2 pi= small change in inflow gas pressure, lbf/ft2 po= small change in gas pressure in the vessel, lbf/ft2 V= Volume of the vessel, ft3 m= mass of the gas in the vessel, lb q= gas flow rate, lb/sec = density of gas, lb/ft3 R=qpip0,Cdpo=qdt Show, Pi(s)P0(s)=RCs+11 (b) Let's define the movement of the center point of the rod connecting the two bellows as " x ". The force acting on Bellows 1 in the " x " direction is A(P+p1), while the force acting on Bellows 2 in the opposite " x " direction is A(P+p2). Show that these two forces are balanced by kX(s)=A[P1(s)P2(s)] (c) Based on the results from previous parts, determine the relationship between the system's input Pi(s) and output X(s) by finding the transfer function X(s)/Pi(s). It is assumed that the expansion process is isothermal and the temperature of the entire system remains constant at 30C, and that the polytropic exponent " n " is equal to 1 . The gas flow resistance R is given by the relation: R=dqd(PT, The capacitance of the pressure vessel is given by the relation: C=nRoirTV