--%>

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 : What is carnot engine Explain the term

    Explain the term Carnot engine?

  • Q : Problem on head loss The pump

    The pump illustrated in the figure adds 20 kW of power to the flowing water. The only vital loss is that which takes place across the filter at the inlet of the pump. Find out the head loss for this filter. Note that the gage pressure upstream of the filter is negativ

  • Q : Measuring the Temperature in Wet Bulb

    Explain phenomenon to measure the Temperature in Wet Bulb Thermometer?

  • Q : Problem related to the value of the

    Water flows via a pipe of about 300 mm diameter at the rate of 66 L/s. Evaluate the value of the manometer reading, ‘h’. The Specific gravity of manometer liquid is taken as 0.8.  

  • Q : Conformance to standards in product

    Conformance to standards and specifications: These are standards laid down by national and international authorities. For instance, in Canada there is the Standards Council of Canada (SCC). The United States has many standards bodies including MIL (US

  • Q : Problem on basic process of Arena

    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 chara

  • Q : Problem on degree of freedom Draw a

    Draw a frequency-response curves for a damped single degree of freedom system subjected to a harmonic excitation under three different damping ratios. System has a natural frequency of ωn as the forcing frequency of the excitation is ω. Describe

  • Q : Bernoulli's equation From Bernoulli's

    From Bernoulli's equation we know that presure head + velocity head at inlet and outlet header is same. If so  what is ' W' then in the equation ?

  • Q : Efficient use of waste heat and

    Efficient use of waste heat and renewable heat sources 1. Describe how you might recover heat from (a) a process exhaust gas stream (e.g. from an oven) and (b) a process warm water stream (e.

  • Q : Define feasibility study Feasibility

    Feasibility study: In order to take wise investments in a market-place experiencing rising stages of risk, companies are turning to feasibility studies to find out if they must propose new products, services or commence a new business endeavor. The ma