formulas hawe

8/13/2019 Formulas HAWE

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Appendix

Formulas and units

Hydraulic system and circuit design is limited only by the creativity of the application

engineer. All basic circuit design begins with the ultimate actuator functions in mind how-

ever. The most critical parameters are the required rates of travel, feeds, speeds, accel-

eration, deceleration requirements as well as ambient conditions and duty cycles. Addi-

tional consideration must be given to load induced as well as external forces.

Thermal efficiencies and noise abatement considerations are another major factor in

todays fluid power system design.

The following formulas and tables are intended to serve as guideline only and should help

with the planning of your hydraulic system.

A-18 04/01

Equipment Formulas and description Symbol

General information

Hydraulic cylinders

Single acting

F: force

p: pressure

A: area

Q: flow

v: speed

V: volume

t: time

s: travel (stroke)

M: torque

Basic equations (static, without any loss)

Q V

t

V A s

=

=

F p A

p F

A

Q A v

M V p

v st

=

=

=

=

=

2

vm

si

d

i

2

4

vm

si

d: piston diameter (mm)

A: piston area (mm2)

Fs: force (N)

pB: operating pressure (bar)

v: piston speed

Qin: inflow (lpm)

s: stroke (mm)

t: time (S)

Fs [N] = - 0.1 pB [bar] A [mm2]

=

pB [bar] =

Qin [lpm] = 0.06 A [mm2]

A [mm2] = [mm]

A1 [mm2

]

-10 Fs [N]

s [mm]

1000 t [s]

Qin

m

s


2/6

Formulas and units

A-1904/01


Double acting

Hydraulic pumps /

hydraulic motors

Extending

Basic equations (balance of forces):

Retracting

Basic equations (balance of forces):

Q A v

Q A v

in

out

1

3

=

=

( )

p A p A F

p A p A F

Q A v

Q A v

1 1 3 3

33

1 1

3

1

1

= +

=

=

=

in

out

p3 is the result of back pressure from pipes and valves for Qout

Attention: Pressure intensification could occur!

p1 result of back pressure from pipes and valves for Qout

Basic equations: Simplified:

Flow:

Middle torque:

Power:

Geometric volume per

revolution (piston pumps): V A h=

Q V n=

M V p

=

2

P p Qhydr =

V: displacement (cm3)

A: effective piston area (mm2)

h: double stroke (mm)

n: rev. rating (rpm)

M: middle torque (Nm)

p: pressure (bar)

|p: effective pressure (bar)

Q: flow (lpm)Phydr: hydraulic performance (kW)

Pmech: mechanical performance (kW)

T : total efficiency (including volumetric and

mechanical losses)

Pdrive: drive performance (kW)

Pout: output hydraulic motor (kW)

Simplified:

Simplified:

A1: piston area (mm2)

A3: rod side area (mm2)

d1: piston # (mm)

d2: rod # (mm)

F: force (N)

Qin: inflow (lpm)

Oout: outflow (lpm)

p1: pressure, piston side (bar)

p3: pressure, rod side (bar)

s: stroke, travel (mm)

Guideline: A power rating of 1 kW for the drive is neces-

sary to achieve a delivery flow of Q = 1 lpm

with operating pressure p = 500 bar!

1)

1) po result of back pressure from pipes and valves2) incl. degree of efficiency T , 0.82

p p po= 1

Power consumption (pump):

Power rating (motor):

Hydraulic motor

Hydraulic pump

TTmech

n2MQpP

=

=

TTmech n2MQpP ==2)

2)

p3

[bar] A3

[mm2] - 10F [N]p1 [bar] =

p1 [bar] =

F [N] =

F [N] =

A1 [mm2]

A [mm2] h [mm]

318V [cm3] ,

Q [lpm] ,

M [Nm] ,

Phyd [kW] ,

Pdrive [kW] ,

Pout [kW] ,

,

V [m3] n [min-1]

1000

M [Nm] n [min-1]

12000

V [cm3] |p [bar]

62

|p [bar] Q [lpm]

612

|p [bar] Q [lpm]

500

|p [bar] Q [lpm]

740

- p1 [bar] A1 [mm2] + p3 [bar] A3 [mm

2]

10

p1 [bar] A1 [mm2] - 10F [N]

A3 [mm2]

p1 [bar] A1 [mm2] - p3 [bar] A3 [mm

2]

10

( )

( )FApA

1p

FApAp

dd4

A

d78.0d4

A

331

1

3311

22213

21

211

=

=

=

=

Qin

Qin

Qin

Qout

Qout


3/6

Appendix

A-20 04/01


Orifices

(ideally, sharp edged)

e.g. orifice inserts type

EB; by-pass check

valves type BC, BE

Pipes /

hoses

Basic equation: Simplified:

Basic equations:

The diameter of pipes and/or hoses should be selected in such a way that back

pressure is minimized.

Q: flow (lpm)

|p: back pressure between A and B (bar)

d: orifice diameter (mm)

: density (approx. 0.9 g/cm3)

a: flow coefficient (approx. 0.78)

R : pipe back pressure coefficient

|p: back pressure (bar)

l: pipe length (m)

d: pipe diameter (mm)

n: cinematic viscosity (mm2/s)

Q: flow (lpm)

Re: Reynolds No. (< 2300) : density (approx. 0.9 g/cm3)

v: flow velocity

Valves

Directional valves

Pressure valves

Flow valves

Check valves

Losses of pressure by streaming fluid

The loss of pressure in hydraulic systems is caused by:

Back pressure of valves

Back pressure of pipes

Back pressure due to geometric shape (elbows etc.)

The back pressure |p of valves caused by streaming fluid, may be

found in the |p - Q curves of the corresponding pamphlets. They will

be approx. 30% as a rough estimation when calculating the

performance loss of a complete circuit

Examples:

Directional valve

Pressure limiting valve

Flow control valve

Releasable

check valve

Q d p

422 [ ] [ ]

[ ]

[ ][ ]

[ ]

2

2

2

mmd

lpmQ81.1p

barp

lpmQ37.1d

barpmmd55.0Q

Re = v d

R =

64

Rep

l

dvR=

2

2

mm

s

2

[ ]p

l

bar

m

mms

Q l

d mm

,min

61

2

4

mm

s

2

Simplified:

m

s

Q [lpm] $ 0.108 d [mm]

d4 [mm]

9.2 Q [lpm]d [mm] %

6.1 pm


4/6

Formulas and units

A-2104/01


Back pressure due to

geometric shape

(elbows etc.)

Leakage losses

(by concentric (e = 0)

and eccentric gaps)

Volumetric losses

(due to pressure increase)

Volumetric losses

(due to temperature rise)

Basic equations:

Basic equation:

Basic equation:

Basic equation:

90 elbow = 0.15

straight pipe fitting = 0.5

elbow fitting = 1.0


: back pressure coefficient

: cinematic viscosity (mm2/s)

d: pipe diameter (mm)


e: eccentricity (mm)

|r: gap (mm)


d: diameter (mm)

: cinematic viscosity (mm2/s)

l: gap length (mm)


QL: leakage losses

V V p

withbar

o

P

=

0 7 10

0 7 10 1

4

4

,

,

V V

withK

o

T

=

0 7 10

0 7 10 1

3

3

,

,

p1: pressure, start (bar)

p2: pressure, end (bar)

Vo: initial volume (l )

|}: volume alternation (l)

P: compressibil ity

}1: temperature, start (C)

}2: temperature, end (C)

|}: temperature, difference (K)

Vo: initial volume (l)

|V: volume alternation (l)

T: expansion coefficient

V p= = 0 7 10 0 7 104 3, ,

p v=

2

2v

Q

A

Q

d= =

42

V V p

with p p p

P o= =

2 1

V V

with

T o

=

2 1

Pressure increase caused

by temperature rise

(without volumetriccompensation)

Attention: A temperature rise of trapped oil volume will cause a pressure increase!(i.e. a pressure limiting valve will be required sometimes)

Guideline: The pressure will rise by approx. 10 bar for 1 K of temperature increase.

F p A=

Simplified:

Simplified:

Simplified:

i.e. |} , 1K & |p , 10 bar

0.7

0.7

0.7

0.7

0.7 0.7

]mm[d

min]/l[Q2,2]bar[p

4

2

=

( )r

e5,11

l

prd1848Q 2

3

L

=+

= .

( )23L 5,11l

p

p12

rdQ +

=

.

Simplified:

[lpm]


5/6

Appendix

A-22 04/01


Hydraulic accumulators

Cavitation

Thermal level

Dissipation power

and oil temperature

Hydraulic accumulators are intended for the supply of pressurized fluid during

sudden demands (quick, adiabatic pressure alternations), compensation of

leakage losses or to dampen oscillations (slow, isotherm pressure alternations).

Pressure alternations,

isotherm (slow)

adiabatic (quick)

Basic equations:o1 p1.1p =

isotherm (slow)

adiabatic (quick)

V V p

p=

1

1

2

1

V V p

p=

1

1

2

10 71,

po: filling pressure for the gas (bar)

p1: lower operating pressure (bar)

p2: upper operating pressure (bar)

V1

: initial volume (l)

|V: volume alternation (l)

Approx. 9 % (volumetric) air are solved in oil at atmospheric pressure. There is the danger of bubble cavitation during

atmospheric pressure below 0,2 bar. These situations can occur, accompanied by sudden noise, during suction process

of pumps and cylinders as well as at extreme throttle sections. The hydraulic components where this occurs will show

increased wear.

The hydraulic power losses in a hydraulic system result in a temperature rise of the fluid and the equipment which is

partly radiated to the surroundings via the surface of the system. They roughly amount 20 - 30% of the induced perfor-

mance. The induced and the radiated heat will balance at some point after the warm-up of the system.

Basic equations: P Pv hydr= 0 3, + ambvc

P

A

Surface with unhindered circulation c , 75

Surface with bad circulation c , 120

with fan (v , 2 m/s) c , 40

Oil/water radiator c , 5

Pv: performance loss, transformed in heat (kW)

Phydr: hydraulic performance (kW)

}oil max: max. fluid temperature (C)

}amb: ambient temperature (C)

A: surface of the system (tank, pipes etc.) (m2)

Simplified:

}oil max

][mA

[kW]P3,0c

2

hydrUmgmaxl

+ }amb}oil max

]m[A 2

0.71

0.3

0.3


6/6

Formulas and units

A-2304/01

Conversion table

Nomenclature

Pressure

Force

Length, travel,

stroke

Torque

Performance

Area

Volume

Temperature

Mass

Cinematic

viscosity

Codings

p

Unit Factor

10

10

1

0.07

Unit

bar

bar

bar

bar

N

N

mm

mm

Nm

kW

mm2

mm2

l

l

l

l

C

kg

,

,

,

,

,

=

,

,

,

=

,

,

,

,

,

,

,

,

,

=

12

N

mm

12

kgf

cm

12

kg m

s

12

2

kg m

s

21064.1

x

1 MPa

1 psi

1 lbf

1 in

1 ft

1 PS, 1 hp

1 UK gal

1 US gal

5 (F-32)/9

1 lb

1 cSt

1 ft2

1 in2

1 ft3

1 in3

F

l, s, h

M

P

A

V

T, }

1

4.45

25.4

304.8

1

0.74

92903

645.16

28.92

m

4.55

3.79

1

0.45

1mm

s

2

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