--%>

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 : Life expectancy in Product design

    Life expectancy: This part of the specification will state how long the product should remain in working order provided the customer gives reasonable care and maintenance. Also take into account technological advances and ongoing improvements that wou

  • Q : Sources of energy what are different

    what are different alternate sources of energy

  • Q : Benefits of gear drive Benefits of gear

    Benefits of gear drive: In general, gear drive is helpful for power transmission among two shafts, which are close to each other (at most at 1m distance). Additionally, it has maximum efficiency while transmitting power. It is resilient as compare to

  • Q : What is wrap-around Wrap-around : The

    Wrap-around: The main aim of a wrap-a-round is to make a straight line about a pipe to help in cutting the pipe to its appropriate length. It is employed mostly as a straight edge or a template.   

  • Q : Modal Combination Rules What are the

    What are the Modal Combination Rules in order to determine the peak value of the total response?

  • Q : What is machine shop Machine shop : A

    Machine shop: A facility which employs machines to fabricate devices from stock raw materials or to change mechanisms based on given specifications. Also termed as "Back" Shops. The general machines in a machine sh

  • 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 : Problem on turbulent flow The

    The approximate equation for the velocity distribution in a rectangular channel with the turbulent flow is 655_turbulent flow.jpg

    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 : 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.