1. The air pressure inside the teadbox of a paper machine is adjusted using a dual valve system (see schematic below). The control valve at the inlet is used to regulate the flow of air entering the headbox whereas the manual valve at the outlet acts as a constant resistance to the air leaving the headbox. The mass Eoverate across either valve may be calculated using the following equation
W(kg /hr) = 360 Cy NIC-'b
where Pt valve inlet pressure, bar
P2 valve cutlet pressure, bar
T gas temperature, K
gas specific gravity (density of the gas divided by the density of air with both gases at standard conditions)
The control valve has a linear inherent characteristic, is normally 50% open and has a maximum
of 3. For the purpose of this problem, the volume of air in the headbox may be assumed to remain constant at 5 m3. The normal air pr sure is the headbox is 1.5 bar and the air supply reservoir is at 2 bar and 25°C.(5)
a)What is the C, of the valve in the outlet line?
b)Derive the dynamic model (differential equation) for the air pressure P inside the headbox. The valve stem position x of the inlet valve is the main input variable. You may assume isothermal conditions.
c)Linemize the model obtained in part (b) about normal conditions. Show that the linearized model is that of a first order lag:
