--%>

Dynamic model for the paper machine headbox

Explain and derive the Dynamic model for the paper machine headbox?

E

Expert

Verified

We'll first develop a dynamic model for the paper machine headbox.

A stock balance around the headbox. A suffix hb refers to the head box.

Accumulation = Input – Output

dmhb/dt = ρqin – ρqout

Now ρ, is the stock density, but since the consistancy is 0.5% only,  ρ = ρw = Water density

d(ρVhb)/dt = ρqin – ρqout

dVhb/dt = qin – qout

Ahbdh/dt = qin – qout

where, Ahb, is the c/s area of header and assumed to be constant.

qout, is the flow out of the header, and is only through slice, and can be written as, CAs(2gh)1/2, where As, is the cross sectional area of slice perpendicular to the flow, and C is characteristic constant coefficient for the slice.

Hence,
Ahbdh/dt = qin – CAs(2gh)1/2,

To find, we can write Bernoulli's equation between Vacuum Degasser and Headbox. Suffix vd refers to vacuum degasser.

Pvd/ρ + W = (Phb + ρgh)/ρ + V2/2
 
V = {2[Pvd - (Phb + ρgh)]/ρ + 2W}1/2

qin = AinV =  Ain{2[Pvd - (Phb + ρgh)]/ρ + 2W}1/2

Hence the dynamic model is,

Ahbdh/dt = Ain{2[Pvd - (Phb + ρgh)]/ρ + 2W}1/2 – CAs(2gh)1/2,

where the rate of stock height change in head box is related to the pressure in vacuum degasser and height.

We have to linearize the non linear dynamic model.

So that the effective model will be,

τdh'/dt = KpP'hb  + Khh',

So as we see the the response of the height of stock to variations in vacuum degasser pressure is first order lag. We don't know the dynamics of level sensor or transmitter, but we'll assume it's also first order lag.

Hence the effective system will be a second order. And hence it'll be oscillatory, and we propose the PID controller for level control by manipulating the speed of fan pump.

Using MATLAB control toobox and given values of the parameters in the problem, we approximately find the following controller parameter settings.

    Kc = 9.6, τI = 2.3 min, τD = 3.5 min.

Similarly, a propotional integral controller is proposed for pressure control in head box.

The pressure in the head box is related to in flow of air, which is controlled by PI controller.
The air is available at pressure of 300 kPa.

dPhb/dt = f(Qin), where is the inlet flow rate of air.

This will be pure capacitive system, hence we propose PI controller.

We find the controller parameters for this,

Kc = 14.5, τI = 4.5 min.

   Related Questions in Mechanical Engineering

  • Q : Problem on motion of the system (a) A

    (a) A plane moving at a constant velocity, V, crashes into a building as shown in figure below. Due to the design against plane crash of that building, neither major damage nor explosion occurs but the building vibrates after the crash. Assuming that the building can

  • Q : Define spinning jenny Spinning jenny:

    Spinning jenny: It is a multi-spool spinning wheel. It was invented in the year 1764 by James Hargreaves in Stanhill, close to Blackburn, Lancashire in the north west of England (though Thomas Highs is another candidate recognized as inventor).

  • Q : Problem on discharge head loss Water is

    Water is draining from the tank A to tank B. The elevation difference among the two tanks is 10 m. The pipe joining the two tanks has a sudden-expansion section as shown below. The cross-sectional area of the pipe from A is 8 cm2, and the area of the pipe f

  • Q : Bearing number ti Diameter of inner and

    Specify how the Bearing number ti Diameter of the inner and outer can be calculated?

  • Q : Change of power in flow of kinetic

    Air at 20 m/s, 260 K, 75 kPa with 5 kg/s flows into a jet engine and it flows out at 500 m/s, 800 K, 75 kPa. What is the change (power) in flow of kinetic energy?

  • Q : Powder Technology Explain the term

    Explain the term Powder Technology?

  • Q : Value Engineering The value engineering

    The value engineering method focuses on functional values, and aims to increase the difference between the cost and value of a product: by lowering cost and/or adding value. In many cases, the emphasis is simply on reducing costs, and the design effor

  • Q : Welding rod 7018 State abbreviation of

    State abbreviation of the welding rod 7018?

  • Q : Quantity in Product design specification

    Quantity: The total quantity of the product predicted and, more importantly, the production rates and batch sizes needed, should be specified. This will have implications for the types of manufacturing equipment and work organization necessary. It wil

  • Q : Mechatronics Define the tem

    Define the tem Mechatronics and what are its elements?