Dynamic model for the paper machine headbox

Question

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

Answer 1

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

dm_hb/dt = ρq_in – ρq_out

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

d(ρV_hb)/dt = ρq_in – ρq_out

dV_hb/dt = q_in – q_out

A_hbdh/dt = q_in – q_out

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

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

Hence,
A_hbdh/dt = q_in – CAs(2gh)^1/2,

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

P_vd/ρ + W = (P_hb + ρ_gh)/ρ + V²/2
 
V = {2[P_vd - (P_hb + ρgh)]/ρ + 2W}^1/2

q_in = A_inV =  A_in{2[P_vd - (P_hb + ρgh)]/ρ + 2W}^1/2

Hence the dynamic model is,

A_hbdh/dt = Ain{2[P_vd - (P_hb + 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 = K_pP'_hb  + K_hh',

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.

    K_c = 9.6, τI = 2.3 min, τ_D = 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.

dP_hb/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.