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4.2 Fluid compressibility and its effect on flow rate
In reality a fluid power system has moving mass and the combination of this with fluid compressibility results in system dynamics that usually cannot be neglected. Also a servovalve spool cannot move immediately to its commanded position and it therefore contributes to dynamic behaviour. Whether or not the servovalve dynamics should be included depends usually on the magnitude of the moving mass for a linear actuator or the rotary inertia for a motor combined with actuator size. In practice damping is fortuitously provided by losses due to viscous and leakage effects but a closed-loop control system will almost certainly become unstable as the servoamplifier gain is increased due to dynamic effects. Instability can lead to disastrous consequences if severe pressure oscillations occur, for example in motor servodrives.
Consider first the effect of fluid compressibility for an arbitrary control volume which has a flow rate 4L
into it and a flow rate 4R out of it as shown in figure 4.1.9ROXPH9 3UHVVXUH3 4L
4R
Figure 4.1 Flow into and out of a control volume
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Considering mass flow continuity including fluid compressibility and the equation of state it can be shown that:
GW G3 ȕ 9 GW R G9 L 4
4
The fluid bulk modulus is β and varies with pressure, temperature, and fluid type and it is very important to refer to manufacturers data sheets when considering fluid properties. Most of the author’s work with mineral oil uses ISO 32 mineral oil, that is, a mineral oil with a viscosity of 32cSt at a temperature of 400C (1centiStoke =PV). At typical servodrive pressures around 100→240bar the fluid bulk modulus is β ≈ [1P. Considering a very small inherent air content in practice reduces this value to typically β ≈ [1P.
If flexible hose is used then the effective bulk modulus of the fluid/hose combination is drastically reduced. Measurements by the author has indicated a value typically half that for a rigid steel line of the same length.
(4.1)
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For a combination of different lines and volumes connected together, 9, 9, 9 etc. and each having a bulk modulus ȕ, ȕ, ȕ etc, the following equation may be used to calculate the effective bulk modulus ȕH:
−
−
−
− + + +
=
−
−
−
− + +
+
=
3 2 t 1
t 3 3 t 2
2 t 1
1
V V V V
β 1 V V β
1 V V β
1 V V βe
1
Returning to equation (4.1), the difference between the input flow rate 4L
and the output flow rate 4R is equal to the rate of change of the boundary volume plus an additional flow rate due to fluid compressibility.It is crucial to appreciate here that the dynamic fluid compressibility effect can only occur if pressure is varying with time. For steady state conditions the compressibility flow rate term is zero but there could still be a moving boundary such as a piston moving with a constant velocity. Considering the actuators shown in figure 4.2 the flow rate continuity equation may be applied to the appropriate control volume:
3
$
3
$
8 4
4Z 43
43
Figure 4.2 Flow rates into and out of hydraulic actuators
GW G3 ȕ 9 GW 4 G9
4L R
/LQHDUDFWXDWRU0RWRU
GW G3 ȕ Ȧ 9 ' 4 GW RXW
G3 ȕ 8 9
$ 4 RXW
GW G3 ȕ Ȧ 9 ' 4 GW LQ
G3 ȕ 8 9
$ 4 LQ
P
P
9 and 9 are the total fluid volumes on either side of the actuator. Volumes are a combination of line and actuator volumes, the latter also varying with time for a linear actuator during motion. Usually pressure transients occur very rapidly and within a small movement for a linear actuator. For this reason, particularly when using transfer functions as introduced in Chapter 5, it is common to assume a constant actuator volume on either side and at the particular steady-state condition selected.
(4.2)
(4.3)
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Considering the electrical analogy where pressure is analogous to voltage P→V and flow rate is analogous to current Q→ i then fluid compressibility is sometimes referred to as fluid capacitance:
Capacitance ȕ
& 9