114 Chapter 4 / Mathematical Modeling of Fluid Systems and Thermal SystemsAssuming that the relationship between the variation in the nozzle back pressure and
the variation in the nozzle–flapper distance is linear, we have
(4–13)
where is a positive constant. For the diaphragm valve,
(4–14)
whereK 2 is a positive constant. The position of the diaphragm valve determines the
control pressure. If the diaphragm valve is such that the relationship between and z
is linear, then
(4–15)
whereK 3 is a positive constant. From Equations (4–13), (4–14), and (4–15), we obtain
(4–16)
whereK=K 1 K 3 /K 2 is a positive constant. For the flapper, since there are two small
movements (eandy) in opposite directions, we can consider such movements separately
and add up the results of two movements into one displacement x. See Figure 4–8(d).
Thus, for the flapper movement, we have
(4–17)
The bellows acts like a spring, and the following equation holds true:
(4–18)
whereAis the effective area of the bellows and ksis the equivalent spring constant—
that is, the stiffness due to the action of the corrugated side of the bellows.
Assuming that all variations in the variables are within a linear range, we can obtain
a block diagram for this system from Equations (4–16), (4–17), and (4–18) as shown in
Figure 4–8(e). From Figure 4–8(e), it can be clearly seen that the pneumatic controller
shown in Figure 4–8(a) itself is a feedback system. The transfer function between and
eis given by
(4–19)
A simplified block diagram is shown in Figure 4–8(f). Since and eare proportional,
the pneumatic controller shown in Figure 4–8(a) is a pneumatic proportional controller.
As seen from Equation (4–19), the gain of the pneumatic proportional controller can be
widely varied by adjusting the flapper connecting linkage. [The flapper connecting link-
age is not shown in Figure 4–8(a).] In most commercial proportional controllers an ad-
justing knob or other mechanism is provided for varying the gain by adjusting this linkage.
As noted earlier, the actuating error signal moved the flapper in one direction, and
the feedback bellows moved the flapper in the opposite direction, but to a smaller degree.
pc
Pc(s)
E(s)
=
b
a+b
K
1 +K
a
a+b
A
ks
=Kp
pc
Apc=ks y
x=
b
a+b
e-
a
a+b
y
pc=
K 3
K 2
pb=
K 1 K 3
K 2
x=Kx
pc=K 3 z
pc
pb=K 2 z
K 1
pb=K 1 x
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