--%>

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 : Radial bearings Explain the importance

    Explain the importance to remember about the radial bearings?

  • 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 : Petroleum technology theories for the

    theories for the origin of petroleum, methods for detection of petroleum deposits, fractional distillation

  • 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 : 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 : Preventing the excess discharge pressure

    How we can prevent the excess discharge pressure?

  • Q : Describe Process Flow Diagram Process

    Process Flow Diagram: A Process Flow Diagram or System Flow Diagram exhibits the relationships among the main components in the system. It too has fundamental information regarding the material balance for the process.

  • Q : Explain steam turbine diaphragm Steam

    Steam turbine diaphragm: Steam turbine includes of phases, number and size of the phases depends on the break horse-power of the turbine. The phase has set of moving and fixed blades. Moving blades are joined to th

  • 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 : Energy in the home-personal energy

    Energy in the home personal energy use and home energy efficiency. Estimate your personal Annual Energy Usage for the following - list all data in kWh (KiloWatt hours):