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25.1
SECTION 25
HYDRAULIC AND PNEUMATIC
SYSTEMS DESIGN
Determining Response Time of Pilot-
Operated Solenoid-Energized Spool
Valves in Hydraulic Systems 25.1
Hydraulic-System Reservoir and Heat
Exchanger Selection and Sizing 25.12
Choosing Gaskets for Industrial
Hydraulic Piping Systems 25.19
Computing Friction Loss in Industrial
Hydraulic System Piping 25.26
Hydraulic-Cylinder Clearance for
Damping End-of-Stroke Forces 25.29
Hydraulic System Pump and Driver
Selection 25.32
Hydraulic Piston Acceleration,
Deceleration, Force, Flow, and Size
Determination 25.36
Hydropneumatic Accumulator Design
for High Force Levels 25.39
Membrane Vibration in Hydraulic
Pressure-Measuring Devices 25.41
Power Savings Achievable in Industrial
Hydraulic Systems 25.42
Pneumatic-Circuit Analysis Using
Various Equations and Coefficients
25.44
Air Flow Through Close-Clearance
Orifices in Pneumatic Systems 25.49
Labyrinth Shaft Seal Leakage
Determination 25.58
Pipe-Wall Thickness for Hydraulic
Systems without Fluid Shock 25.67
Pipe-Wall Thickness for Hydraulic
Systems with Fluid Shock 25.68
Allowable Stress in Hydraulic System
Piping 25.68
Hydraulic Fluid Compressibility and
System Bulk Modulus 25.69
Selection of Fluids for Oil Hydraulic and
Control Systems 25.69
Effect of Trapped Air on Hydraulic
System Bulk Modulus 25.71
Surge Pressure in Hydraulic Cylinders
25.72
Sizing a Hydraulic System Fluid
Reservoir 25.72
Required Volume of Bladder-Type
Accumulator 25.73
Determining Hydraulic Accumulator
Demand Volume 25.74
Effective Force Developed by a Double-
Acting Hydraulic Cylinder 25.74
Hydraulic Cylinder Oil Consumption and
Extension Time 25.75
Hydraulic Cylinder Power Output 25.76
Selecting Hydraulic Motors and Pumps
by Using Manufacturer’s Size Tables
25.76
DETERMINING RESPONSE TIME OF PILOT-
OPERATED SOLENOID-ENERGIZED SPOOL
VALVES IN HYDRAULIC SYSTEMS
A pilot-operated solenoid-energized spool control valve in a hydraulic system has
the dimensions, operating pressures, and performance given in Table 1. Pilot supply
pressure is 100 lb/in
2
(689 kPa); main supply pressure is 500 lb/in
2
(3445 kPa).
Find the maximum velocity of this valve, its acceleration time, and the total re-
sponse time. Next, using the same dimensions and main operating pressure, find
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Source: HANDBOOK OF MECHANICAL ENGINEERING CALCULATIONS
25.2 DESIGN ENGINEERING
TABLE 1 Dimensions and Operating Conditions*
Pilot Spool Main spool
Diameter, in d ϭ 0.25 D ϭ 2.5
Mass, lb-sec
2
/in m ϭ 0.0002 M ϭ 0.05
Stroke, in s
ϭ 0.375 S ϭ 1.5
Land length, in — L
ϭ 6.0
Radial clearance, in — C
ϭ 0.0003
Coefficient of friction — F
R
ϭ 0.04
Solenoid force, lb (initial; final) F
ϭ 1; 8.5
SOL
—
Back pressure, lb / in
2
— p
B
ϭ 20
Supply pressure, lb/in
2
p ϭ 100 P ϭ 500
Differential pressure, lb/in
2
⌬p ϭ 70 (approx) ⌬P ϭ 450 (approx)
Port area, in
2
a
0
ϭ 0.05 A
M
ϭ 1.2
Flow coefficient ƒ
ϭ 0.6 ƒ ϭ 0.6
Viscosity, CP
ϭ 80
ϭ 80
Density, lb-sec
2
/in
4
0.000,085 0.000,085
*SI values given in calculation procedure.
the same unknowns when the pilot pressure is made equal to the main operating
pressure i.e., 500 lb /in
2
(3445 kPa). As a further modification, a small actuating
piston is placed at each end of the main spool, Fig. 3, to increase the longitudinal
velocity for a given pilot-fluid flow rate. Trial and error would normally be used
to calculate the most effective diameter for the actuating piston. In this procedure
we will use a diameter d
x
ϭ 1.4 in (3.56 cm) for this small actuating piston. If the
dimensions and operating pressures are unchanged, analyze the valve for the same
unknowns as above.
Calculation Procedure:
1. Compute the axial force on the main spool of this valve
The forces acting on the main spool at maximum velocity are: Pilot backpressure,
p
B
; viscous damping force, D
V
; and radial jet force P , Fig. 2. From the equation,
rad
P ϭ 2F ƒA ⌬P
as r M
where the symbols are as given, Table 2. Then, P
ax
ϭ 2(0.04)(0.6)(1.2)(450) ϭ 26
lb (115.6 N). converting to pressure by dividing by the area of the main spool valve
end, we have 26/4.9
ϭ 5.3 lb / in
2
(36.5 kPa).
2. Compute the combined hydrodynamic resistance of the valve
Provisionally, estimate that D
V
is equivalent to 3.2 lb/in
2
(22 kPa) and P
B
ϭ pilot-
valve backpressure
ϭ 20 lb/in
2
(138.8 kPa). The combined hydrodynamic resis-
tance is then the sum of: Radial pressure, lb/in
2
(kPa) ϩ Viscous drag, lb/in
2
(kPa)
ϩ Pilot-valve backpressure, lb/in
2
(kPa). Or combined hydrodynamic resistance ϭ
5.3 ϩ 3.2 ϩ 20 ϭ 28.5 lb / in
2
(196.4 kPa).
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HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN
HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.3
TABLE 2 Valve Symbology*
Pilot
Actuating
piston
Main
valve
Spool
Dimensions
and Mass
Diameter, in
Cross-sectional area, in
2
Mass, lb-sec
2
/in
Stroke: Intermediate
Full
Engagement (length in contact), in
Land length, in (total)
Spool-to-bore radial clearance, in
d
—
m
x
s
—
—
C
d
x
a
p
—
—
S
a
l
p
—
C
D
A
s
M
—
S
—
L
C
Solenoid
Forces
Initial, lb
Gradient, lb/in
Final, lb
Ratio, A/B
A
B
F
SOL
r
—
—
—
—
—
—
—
—
Drag
Forces
Back pressure, psi
Viscous drag, lb (or psi)
Radial jet, lb
Coefficient of friction
Axial jet, lb
Acceleration force, lb
p
B
—
—
—
—
F ϭ ma
p
B
d
V
P
rad
F
R
P
ax
—
p
B
D
V
P
rad
F
R
P
zx
F ϭ Ma
Oil
pressure,
flow, and
port size
Pressure:
Supply, psi
Pilot downstream, psi
Differential, psi
Port area, in
2
(effective orifice)
Flow coefficient (0.55 to 0.70)
Viscosity, centipoise
Oil density, lb-sec
2
/in
4
Flow rate, in
3
/sec
Oil velocity, in/sec (through port)
Oil mass flow, lb-sec/in
Oil-jet deflection angle, deg
p
p
1
⌬p
a
0
ƒ
q
—
—
—
p
p
1
⌬p
—
ƒ
q
—
—
—
P
p
1
⌬P
A
M
ƒ
—
V
0
M
ƒ
a
Valve
response
Acceleration time, sec
Shifting velocity, in/sec
Shifting time, sec
(after energization)
t
a
v
p
t
T
a
v
T
T
a
V
T
*SI values given in calculation procedure.
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HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN
25.4 DESIGN ENGINEERING
3. Calculate the pilot-valve flow rate
The pilot-valve pressure differential, delta P
ϭ 100 Ϫ 28.5 ϭ 71.5 lb/in
2
(492.6
kPa). Hence, the valve flow rate is, using the equation below
2⌬p
q ϭ ƒa
o
Ί
where q ϭ flow, in
3
/s (mL / s); ƒ ϭ flow factor, dimensionless, ranging from 0.55
to 0.70 depending on valve type; a
o
ϭ cross-sectional area, in
2
(cm
2
); of the min-
imum port opening—usually the drilled port hole;
⌬ p ϭ p Ϫ p
1
ϭ differential
pressure, lb/in
2
(kPa) measured across the pilot inlet and outlet ports; p ϭ fluid
mass density, lb-s
2
/in
4
, normally 0.000085 for oil. Substituting, q ϭ 0.6
(0.5)[(2)(71.5)/0.000085)]
ϭ 40 in
3
/s (656 mL /s), using a value of ƒ ϭ 0.6 for
0.5
this valve.
4. Determine the maximum velocity of the main spool and the viscous damping
force
The maximum velocity of the main spool, Fig. 1, V
ϭ (flow rate, in
3
/s/(area of
spool end, in
2
) ϭ 40 / 4.9 ϭ 8.2 in/s (20.8 cm/s). Knowing the velocity, we can
find the damping force, D
V
, from
D
LV
D ϭ
V
6
C ϫ 6.9 ϫ 10
where D
ϭ spool diameter, in (cm); L ϭ length of spool lands, in (cm); V ϭ main
spool velocity, in / s (cm/s); mu
ϭ absolute viscosity, centipoise; C ϭ spool-to-bore
radial clearance, in (cm). If the temperature varies more than 30 to 50 degrees, it
is nearly impossible to compute the viscous resistance. Substituting, D
V
ϭ
2.5
(6)(8.2)(80)/(0.0003)(6.9 ϫ 10
6
) ϭ 3.05 lb/in
2
(21 kPa). Thus, the provisional
estimate of D
V
ϭ 3.2 was close enough (within 4.9 percent) and recalculation is
not necessary.
5. Find the accelerating pressure and acceleration time of the spool
The forces acting upon the spool during acceleration are: p
R
, P
ax
, D
V
, and F, where
F
ϭ Ma. Assuming a mean value for initial port opening A
M
ϭ 0.4 in
2
(2.58 cm
2
),
then from
P
ϭ 2ƒA ⌬P cos
␣
as M
where
␣
normally varies from 70 degrees at initial opening to 90 degrees at full
opening. In calculations, use the axial jet pressure during initial opening, and the
axial component of radial pressure during the remainder of travel. Substituting,
P
ax
ϭ 2(0.6)(0.4)(450)(0.26) ϭ 56 lb (248.1 N). Then 56 / 4.9 ϭ 11.4 lb / in
2
(78.5
kPa),
␣
ϭ 75 deg; cos
␣
ϭ 0.26.
Viscous drag will be the average: D
V
ϭ 3.2/2 ϭ 1.6 lb/in
2
(11 kPa). Backpres-
sure is still p
B
ϭ 20 lb / in
2
(137.8 kPa). So the total is 11.4 ϩ 1.6 ϩ 20 ϭ 33 lb /
in
2
(227.4 kPa).
Therefore, accelerating pressure
ϭ 100 Ϫ 33 ϭ 67 lb / in
2
(461.6 kPa). Con-
verting to force, we have 67 (4.9)
ϭ 328 lb (1441.2 N). The acceleration time, t
a
s ϭ MV /F ϭ 0.05 (8.2)/328 ϭ 0.0013 s.
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HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN
25.5
FIGURE 1 Typical solenoid-energized pilot-operated spool valve.
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HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN
25.6 DESIGN ENGINEERING
FIGURE 2 Jet-force drag in pilot-operated spool valves.
6. Determine the main spool displacement and the energization time interval
The displacement of the main spool during the acceleration period is negligible,
being less than 1 percent of the total stroke. Time for the total stroke of 1.5 in (3.8
cm) is 1.5/8.2
ϭ 0.182 s, and the time interval from energization of the solenoid
to completion of the main valve stroke, T
ϭ 0.190 s.
7. Analyze the valve with the higher pilot pressure
Much larger flow will pass through the pilot valve because of the higher pressure.
Maximum velocity period: P
ax
ϭ 5.3 lb/in
2
(36.5 kPa), the same as before; D
V
ϭ
7.7 lb/in
2
(53.1 kPa)— a higher estimate, proportional to the anticipated velocity;
p
B
ϭ 20.0 lb/in
2
(137.8 kPa), the same as before. The total is 33.0 lb / in
2
(227.5
kPa).
The new
⌬ P ϭ 467 lb/in
2
(3217.6 kPa), and Q ϭ 4.7 (467) ϭ 102 in
3
/s
0.5
(1671.5 mL/s); V ϭ 102/4.9 ϭ 20.8 in / s (52.1 cm/s); D
V
ϭ 1.82 (20.8) ϭ 37.8
lb (168.1 N)
ϭ 7.75 lb / in
2
(53.4 kPa), which proves out the assumption of 7.7 lb
/in
2
(53.1 kPa).
Accelerating time, t
a
ϭ (0.05)(20.8)/(2280) ϭ 0.0005 s. The 1.5-in (3.81-cm)
stroke takes 1.5/20.8
ϭ 0.072 s. Total time ϭ 0.081 s.
The flow rate of the pilot oil is more important than pressure intensity in ob-
taining a fast-acting valve. A slightly larger pilot valve and enlarged porting have
a marked effect on the operational speed of the main valve.
Note that increasing the pilot pressure fivefold, from 100 lb/in
2
to 500 lb / in
2
(689 kPa to 3445 kPa) only doubles the speed of response, from 0.19 s to 0.08 s.
Increasing the port area can result in an nearly proportional gain in speed, and no
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HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN
HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.7
additional pressure is necessary, saving on pumping costs. Costs of producing a
0.375-in (9.52-mm) pilot spool are not much greater than those for a 0.25-in (6.35-
mm) spool. The increase in capacity is 50 percent without the additional heat losses
entailed by an increase in pressure.
8. Analyze the valve fitted with actuating pistons
For the valve, Fig. 3, with the small actuating pistons, taking the summation of the
viscous drag,
͚D
V
, and inserting the known optimum values in parentheses after
the computed values, we have:
8.2
ϫ 4.9 ϫ 4 ϫ 80
͚D ϭ
V
6
0.0003 ϫ 6l9 ϫ 10 ϫ 1.4
2.5
ϫ 6
ϫϩ2 ϫ 1.5
ͩͪ
1.4
12800 15
ϭϩ3
ͩͪ
2075 ϫ 1.4 1.4
ϭ 60.5 lb (269.1 N) [59.0 lb optimum; 262.4 N]
Introducing the value of
͚ D
V
in the equation,
3k 3(P
ϩ ͚D )
3 ax V
a ϭϭ
p
2k 2(p Ϫ p )
2 B
we have
3(26
ϩ 60.5)
22
a ϭϭ1.62 in (10.45 sq cm) [1.59 in ; 10.26 cm ]
p
2(100 Ϫ 20)
d
ϭ 1.44 in (3.66 cm) [1.425 in, 3.62 cm].
With optimum a
p
ϭ 1.59 cu in (26.06 mL), piston velocity using
2⌬p
q ϭ ƒa
o
Ί
kk
23
v ϭ k Ϫ
1
23
Ί
aa
pp
22
kk kk
12 13
ϭϪ
23
Ί
aa
pp
80 (26 ϩ 59)
v ϭ 4.6 Ϫis
p
Ί
2.53 4.0
ϭ 15.0 in/s (38.1 cm / s)
The total time, T
ϭ 0.100 ϩ 0.009 ϭ 0.109 s. Using a pilot pressure of 500 pst
(3445 kPa), V
ϭ 20.8 in / s (52.8 cm / s) and d ϭ 1.06 in (2.69 cm). Then:
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HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN
25.8
FIGURE 3 Piston-operated solenoid-energized spool valve.
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HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN
HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.9
TABLE 3 Effect of Adding Actuating Pistons*
Pilot
pressure p,
psi
Main valve
diameter,
in
Maximum
valve
velocity,
without
piston
in/sec
Total
shift
time
T,
sec
Maximum
valve
velocity,
with
piston in
sec
Piston
diameter,
in
Total
shift
time
T,
sec
100 2.50 8.2 0.190 15.0 1.425 0.109
500 2.50 20.8 0.081 55.0 1.06 0.036
*SI values given in calculation procedure.
20.8 ϫ 4.9 ϫ 4.80
͚DV ϭ
6
0.0003 ϫ 6.9 ϫ 10 ϫ 1.06
15
ϫϩ3 ϭ 258 lb
ͩͪ
1.06
3(26
ϩ 258)
2
a ϭϭ0.886 in
p
2(500 Ϫ 20)
d
ϭ 1.06 in
z
480 284
v ϭ 4.6 Ϫ
p
Ί
0.784 0.61
ϭ 55 in/s
T
ϭ 0.036 s
Table 3 and Fig. 4 depict the effect of actuating-piston area upon the spool shifting
velocity and shifting time.
Related Calculations. Pilot-operated flow-control valves are probably the most
common valves used in industrial hydraulic systems. Speed of response of these
valves is important during the design and operation of any hydraulic system. The
procedure given here analyzes the speed of response of a typical valve in terms of
the fluid flow rate, characteristic force-vs-airgap curve of the solenoid; shape, size,
clearance, and displacement of each spool; and the fluid viscosity.
The method given in this procedure relates the above parameters for the valve
in Fig. 1. and can be applied to any other pilot-operated spool valve. And the
procedure includes a special technique for a large spool valve, Fig. 3, actuated by
a small auxiliary piston.
In the sequence of operation of solenoid-energized pilot-operated spool valve,
here is what happens. The solenoid is energized, the pilot spool moves quickly to
the full open position, Fig. 5, and the main spool is shifted at a rate determined by
the amount of fluid that can move through the pilot ports against these five resisting
forces: (1) pilot system backpressure, lb/in
2
(kPa); (2) viscous damping force, lb
(N); (3) radial jet force, lb (N); (4) axial jet force, lb (N); (5) acceleration force, lb
(N).
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HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN
25.10 DESIGN ENGINEERING
SI Values
in./sec cm/sec in.
2
cm
2
5 12.7 1 6.45
10 25.4 2 12.9
15 38.1 3 19.4
20 50.8 4 25.8
25 63.5 5 32.3
FIGURE 4 Effect of varying piston diameter.
FIGURE 5 Before energization and after full stroke of a solenoid-energized spool valve.
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HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN
[...]... representing the value of the friction factor, ƒ, as a function of the Reynolds number, R, is often called the Stanton chart, after its developer, who was the first to employ this representation of the friction factor A chart taking advantage of the functional relationships established by research was drawn up by Lewis F Moody, and is reproduced in Fig 12 in a form convenient for the user of this handbook In Fig... predetermined control In this procedure we assumed that at the start of dashpot action inertia forces alone are dissipated through the ejection of dashpot oil Hence, the kinetic energy of the moving parts equals the work done during penetration of the dashpot The assumed value of the coefficient of discharge, CD, may be checked against the Reynolds number of the calculated flow and adjusted if it deviates too much... maximum line pull of 20,000 lb (88,964.4 N) at a maximum linear speed of 280 ft / min (1.4 m / s) with a maximum drum torque of 200,000 lb ⅐ in (22,597.0 N ⅐ m) at a drum speed of 53.5 r / min Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at... s (0.34 m / s) having a mass, M, of 64 lb-s2 / ft (95.3 kg-s2 / m), a length, L ϭ 4 in (10.16 cm), a dashpot radius of R ϭ 1 in (2.54 cm), a dashpot capacity of 12.5 cu in (204.8 cu cm), a coefficient of discharge, CD ϭ 0.62, and a pressure differential, ⌬P, of 30 lb / in2 (206.7 kPa) What annular clearance is needed when handling hydraulic fluid with a specific gravity of 0.85? Calculation Procedure:... Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.20 DESIGN ENGINEERING From Table 5, the stress area for 3 / 3-10 NC bolt is 0.3340 sq in (2.15 sq cm) The bolt material specified can easily take a stress of 30,000... (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.22 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as... PNEUMATIC SYSTEMS DESIGN Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 25.23 A S B E S T O S M E T A L M E T A L ⁄ only Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The... Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 25.25 *Or acceptable substitute Spiral-wound Profile Corrugated and corded* Interlocked plies of preformed metal strip are spiral-wound with an interleaving cushion of asbestos or fluorocarbon... diameter, in (mm); v ϭ kinematic viscosity of hydraulic fluid, centistokes Substituting, R ϭ (7740)(20)(1) / 110 ϭ 1407.3 2 Determine the relative roughness of the piping Since the Reynolds number for this piping is less than 2000, roughness of the pipe does not enter into the calculation See Fig 12 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006... reserved Any use is subject to the Terms of Use as given at the website FIGURE 12 Stanton diagram is useful in hydraulic-system calculations HYDRAULIC AND PNEUMATIC SYSTEMS DESIGN 25.27 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2006 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website HYDRAULIC . use is subject to the Terms of Use as given at the website.
Source: HANDBOOK OF MECHANICAL ENGINEERING CALCULATIONS
25.2 DESIGN ENGINEERING
TABLE 1 Dimensions. Speed of response of these
valves is important during the design and operation of any hydraulic system. The
procedure given here analyzes the speed of response
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