PWARX Model Identification Based on Clustering Approach - Complex System Modelling and Control Through Intelligent Soft Computations - page 179

Knowing that

log H þ

pH m ¼ ð ½ Þ ð 30 Þ

H þ OH

K w ¼ ½ ½ ; ð 31 Þ

Equation ( 29 ) can be then rewritten as:

210 pH m pK a2 1 þ ð Þ

10 pH m 14 10 pH m

W a4 þ þ W b4

10 pH m pK a2

¼ 0 ð 32 Þ

10 pH m pK a1

1 þ þ

￿ The mass balance yields to:

A dh

dt ¼

q 1 þ q 2 þ q 3 q 4 ð 33 Þ

h 0 : 5 , Eq. ( 33 ) becomes:

Taking into account that the exit fl ow rate q 4 ¼ C v :

A dh h 0 : 5

dt ¼

q 1 þ q 2 þ q 3 C v ð 34 Þ

where C v is the constant valve coef cient.

The differential equations of the ef fl uent reaction invariants ð W a4 ; W b4 Þ can be

￿

determined as follows:

Ah dW a4

dt ¼

q 1 ð W a1 W a4 Þþ q 2 ð W a2 W a4 Þþ q 3 ð W a3 W a4 Þ ð 35 Þ

Ah dW b4

dt ¼

q 1 ð W b1 W b4 Þþ q 2 ð W b2 W b4 Þþ q 3 ð W b3 W b4 Þ ð 36 Þ

Nominal model parameters and operating conditions (Xiao et al. 2014 ) are given
in Table 6 .
The static nonlinearity of this process can be represented by the titration curve
shown in Fig. 6 with a beginning pH of 2.7 and an ending pH of 10.7. A brief glance
at the curve indicates that the process of pH neutralization is highly nonlinear.

Table 6 Operation
parameters of the pH
neutralization process

q 1 ¼ 16 : 6ml/s W a1 ¼ 3 10 3 mol/l

q 2 ¼ 0 : 55ml/s

10 2 mol/l W a2 ¼ 3

q3 ¼ 15 : 6 ml/s

10 3 mol/l W a3 ¼ 3 : 05

h ¼ 14 : 0cm W b1 ¼ 0

A ¼ 207 cm 2 10 2 mol/l W b2 ¼ 3

C v ¼ 8 : 75 ml/cm/s

10 5 mol/l W b3 ¼ 5

pK a1 ¼ 6 : 35

pH 4 ¼ 7

pK a2 ¼ 10 : 25 s ¼ 0 : 5