--%>

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 : Calculate the surface area available

    You are working in a company that has developed a process to produce DNA vaccines from E.coli. After the fermentation step a lysis step is used to release the DNA from the cells resulting in a sticky mixture. You find that an aggressive cleaning solution heated to 80°C is required to

  • Q : Static and Dynamic load What are Static

    What are Static and Dynamic load and what are their significant dynamic effects?

  • Q : Problem on work of compression A diesel

    A diesel engine operates devoid of a spark plug by using the high-temperature gas produced throughout the compression stage to ignite the fuel. During a typical compression, pure air that is originally at 21 °C and 0.95 bar is reversibly and adiabatically compress

  • Q : Product performance in Product design

    Product performance: Depending on the product, this may take many forms. Speed, loads to be withstood, number of work cycles, and intermittent or continuous working are some examples of considerations. 

  • Q : Arena Are you able to assist with these

    Are you able to assist with these two assignments in Arena simulation below? You can use the Basic Process instead of Blocks and Elements. An office of state license bureau has two types of arrivals. Individuals interested in purchasing new plates are characterized to have inter-arrival times dis

  • Q : Synchronous generator & Generating units

    Question 1: Three generating units operating in parallel at 60 Hz have rating of 300, 500, and 600    MW and have speed-droop characteristics of 5, 4, and 3%, respectively. Due to a change in load, a

  • 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 : Problem on discharge of water In the

    In the below system, d = 6 in., D = 12 in., Δz1 = 6 ft, and Δz2 = 12 ft. The discharge of water in the given system is 10 cfs. Is the machine a turbine or a pump ? Determine the pressures at points A and B? Neglect head losses. Suppos

  • Q : Define Mechanism Mechanism : The

    Mechanism: The mechanism is a system of moving parts which changes an input motion and force into the desired output force and motion.

  • 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