Page 623 - The Mechatronics Handbook

0066_Frame_C20 Page 93 Wednesday, January 9, 2002 5:49 PM

in discharge. This hypothesis is physically acceptable; hypothesizing b = 0.3, P s = 10 bar, P amb = 1 bar, we
get that P 1 can vary between 3.33 bar and 10 bar. It is the same for P 2 .
Linearizing the subsonic feed flow rate curve with a secant passing through the points P 1 = P s , G 1 = 0
∗
and P 1 = b 1 P s , G 1 = G 1 sonic , of angular coefficient K L1 , we get

G 1 = K L1 P s – P 1 ) (20.69)
(

where

K L1 = -------------------- = ----------------------- = ------------- (20.70)
G 1sonic
r n C 1
r n C 1 P s
∗
∗
(
–
P s – b 1 P s P s 1 b 1 ) 1 b 1 ∗
–
or
(
r n P s – P 1 )
G 1 = --------------------------C 1 (20.71)
∗
( 1 b 1 )
–
In the neighborhood P 1 = P 1r and C 1 = C 1r = 0 (the subscript r indicates of reference), we get
(
----------------------------C 1 =
G 1 = r n P s – P 1r ) K 11 C 1 (20.72)
∗
( 1 b 1 )
–
Equation (20.72) is valid for V ref1 > 0, to which C 1 > 0 corresponds analytically.
The flow rate discharge is expressed by

G 1 = r n C 1 P 1 (20.73)

which in the neighborhood P 1 = P 1r becomes

G 1 = r n P 1r C 1 = K 12 C 1 (20.74)

Equation (20.74) is valid for V ref1 < 0, to which C 1 < 0 corresponds analytically.
Calculating a mean of slopes K 11 and K 12 we get a mean slope K mean given by

∗
(
K 11 + r n P s – b 1 P 1r )
K mean = --------------------- = --------------------------------- (20.75)
K 12
∗
(
2 21 b 1 )
–
The linearized flow rate as a function of C 1 therefore becomes
∗
(
---------------------------------C 1 =
G 1 = r n P s – b 1 P 1r ) K 1 C 1 (20.76)
∗
21 b 1 )
(
–
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