Diaphragm Accumulators - Airline Hydraulics

FORMULAS FOR SIZING
ACCUMULATORS
The compression and expansion
processes taking place in
hydropneumatic accumulator are
governed by the general gas laws.
The following applies for ideal gases:
p 0 x V n 0 = p 1 x V n 1 = p 2 x V n 2 ,
where the time related change of
state is represented by the polytropic
exponent “n”. For slow expansion
and compression processes which
occur almost isothermically, the
polytropic exponent can be set at
n=1. For rapid processes, the
diabetic change of state can be
calculated using n = k = 1.4 (for
nitrogen as a diatomic gas) (1 .
For pressures above 3000 psi the real
gas behavior deviates considerably
from the ideal one, which reduces the
effective fluid volume ∆V. In such
cases a correction is made which
takes into account a change in the
adiabatic exponent (k).
By using the following formulas, the
required gas volume V 0 can be
calculated for various calculations.
Pressures of up to 150 psi must
always be used as absolute
pressures in the formulas.
Calculation Formulas ∆V
V0
polytropic:
=
isothermal:
(n=1)
adiabatic:
(n = k = 1.4)
V0 =
P0
P1
V0 =
( ) ( )
P0
P1
∆V
( ) ( )
1/n 1/n
P0
P2
P0
P2
∆V
( ) ( )
P0
P1
0.714
P0
0.714
P2
Correction factors to take into
account the real gas behavior (2
V 0,real = C i x V 0,ideal or
∆V real = ∆ V ideal
C i
for adiabatic change of condition:
V 0,real = C a x V 0,ideal or
∆V real = ∆ V ideal
C a
1 An estimate of the accumulator size and a selection of
precharge pressure can be calculated similar to the
sample shown. For more accurate sizing and design
assistance, please contact HYDAC.
2 The correction factors can be taken from the graphs in
the next column, depending on the pressure ratio p 2 /p 1
and the maximum working pressure p 2 , which is given
as a parameter, for an isothermal or adiabatic
change of condition.
correction factor Ci
correction factor Ca
Correction factor for isothermal
change of condition
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
1
max working pressure p2 = 400 bar (5800 psi)
300 bar (4350 psi)
200 bar (2900 psi)
max working pressure p2 = 400 bar (5800 psi)
300 bar (4350 psi)
200 bar (2900 psi)
SIZING EXAMPLE
2 3 4 5
pressure ratio p 2 /p 1
Correction factor for adiabatic
change of condition
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
1
2 3 4 5
pressure ratio p 2 /p 1
A brake cylinder has to be rapidly
operated by means of an
accumulator. The system must
operate between 3000 psi and 1500
psi. The required fluid volume to
operate properly is 15 in 3 . The
maximum operating temperature is
140°F while the minimum is 50°F.
Given:
max. working pressure
p 2 = 3000 psi
min. working pressure
p 1 = 1500 psi
effective fluid volume
∆V = 15 in 3
max. operating temperature
T 2 = 140°F
min. operating temperature
T 1 = 50°F
Required:
1 necessary accumulator size,
taking into account the real gas
behavior
2 gas precharge pressure p 0 at 68°F
(T 0 )
Solution:
Since it is a rapid process, the
change of condition of the gas can
be assumed to be adiabatic.
1 Determination of gas precharge
pressure
p 0,T2 = 0.9 x p 1
= 0.9 x 1500
= 1350 psi.
In order to maintain the minimum
working pressure (p 1 ), one must
calculate the precharge pressure
at T 1 .
T 1 + 460
p 0
,T 1
= p 0
,T 2 x (
T 2 + 460
)
50 + 460
= 1350 x
140 + 460
= 1148 psi ≈ 1150 psi
Determination of the required
gas volume:
V 0, ideal =
1150 =
= 46.5 in 3
( )
P0, (T1)
P1
∆V
( ) ( )
1150
1500
0.714
P0, (T1)
0.714
P2
15
( ) ( )
0.714 0.714
1150
3000
Taking into account the real gas
p 0
= 2 - Ca ≈ 1.16
p 1
behavior:
V 0,real = C a x V 0, ideal
= 53.9 in 3
Selected:
Diaphragm accumulator
SBO 200-1 (60 in 3 )
b) Determination of the gas
precharge pressure
p 0 at 68°F:
T0 + p 0,T 0 = p 0,T2 x ( T2 + 460)
= 1350 x
68 + ( 140 + 460)
= 1188 psi
Selected:
Gas precharge pressure
p 0,T 0 ≈ 1200 psi
7