Equation 1.21 is Bernoulli’s equation with negligible gravity forces
2.2 Hydrostatic Pumps
2.2.1 Basic equations
The main parameter for a hydraulic pump is the stroke displacement, D, defined as the displacement pr. revolution of the pump. It relates to the pump flow as follows:
n D
QtP = ⋅ (2.1)
where
QtP theoretical pump flow, [volume/time]
D stroke displacement, [volume/revolution]
n rotational speed of the pump, [revolutions/time]
The rotational speed relates to angular velocity as follows:
π ω
= ⋅
n 2 (2.2)
where
n rotational speed, [revolutions/time]
ω angular velocity, [radians/time]
Defining the unit displacement, Dω, as the displacement pr. radian leads to:
π Dω
2
D= ⋅ ⋅ (2.3)
ω⋅ω
= D
QtP (2.4)
where
QtP theoretical pump flow, [volume/time]
D stroke displacement, [volume/revolution]
Dω unit displacement, [volume/radian]
ω angular velocity, [radians/time]
Some pumps have adjustable displacements. In that case the displacements may be determined from:
max ,
max D D
D
D=α⋅ ω =α⋅ ω (2.5)
where
D stroke displacement, [volume/revolution]
α displacement control parameter, 0.0≤ α≤1.0 Dmax maximum stroke displacement, [volume/revolution]
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The power supplied to the fluid by the pump may be determined based on the general expression for hydrokinetic power:
p Q
P= ⋅ (2.6)
where
P power, [power]
Q flow, [volume/time]
p pressure, [pressure]
Accordingly, the theoretical power put into the hydraulic system, e.g. the fluid, by the pump may be determined as:
(
P S)
tP PtP F P ,
t Q p p Q p
P → = ⋅ − = ⋅Δ (2.7)
where
Pt,P→F theoretical power supplied by the pump to the fluid, [power]
QtP theoretical pump flow, [volume/time]
pP pressure at the pressure side of the pump, [pressure]
pS pressure at the suction side of the pump, [pressure]
ΔpP pressure rise across the pump, [pressure]
The pressure and flow variables of Equation (2.7) are shown in Figure 2.6.
pP
Δ
pP
pS
M QtP
Figure 2.6 Diagram illustration of motor driven pump. The pressure and flow variables associated with the pump are shown.
The theoretical mechanical power delivered to the pump from some power source is given as:
ω
⋅
→P = tP PS
,
t M
P (2.8)
where
Pt,PS→P theoretical power supplied to the pump by the power supply, [power]
MtP theoretical input torque to the pump from the power supply, [torque]
ω angular velocity, [radians/time]
Without any losses, the power delivered to the pump by the power supply equals the power supplied by the pump to the fluid, i.e., combining Equation (2.7) and (2.8) gives:
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ω
Δ = ⋅
⋅ P tP
tP p M
Q (2.9)
Inserting Equation (2.4) and (2.3) into Equation (2.9) leads to:
π
The expressions developed for the pump flow and pump torque in the previous section correspond to an ideal pump without power losses of any kind. In reality the pump will produce less flow than the theoretical value and the pump will require more torque than the theoretical value. Hence, only a part of the power supplied to the pump will end up as hydrokinetic power, whereas the power loss will heat up the pump and its surroundings including the hydraulic fluid.
The fact that the pump delivers less flow than theoretically expected is expressed by means of a volumetric efficiency:
tP
QtP theoretical flow of the pump, [volume/time]
The volumetric loss is mainly due to leakage in the form of laminar clearance flow from the high pressure chambers to the low pressure chambers within the pump. This leakage flow is mainly laminar (although some models also include turbulent leakage) and therefore proportional to the pressure rise and inverse proportional to the viscosity:
μ
QlP leakage flow within the pump, [volume/time]
KlP leakage constant for the pump, [volume]
ΔpP pressure rise across the pump, [pressure]
μ dynamic viscosity, [pressureּtime]
The volumetric constant tends to vary with the displacement for the same type of pump (larger pumps will have larger dimensions including the leakage clearances).
Combining Equations (2.1), (2.11) and (2.12) gives:
n
It is seen that the volumetric efficiency will depend on pressure rise, rotational speed and the viscosity (mainly temperature), i.e., change with the working conditions. The
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variation of the volumetric efficiency may be viewed graphically for 2 different situations, see Figure 2.7. The curves are based on Equation (2.13).
ηvP
ηvP
pP
Δ 1.0
n 1.0
. const .
const
n= μ= ΔpP =const. μ=const.
Figure 2.7 The volumetric efficiency for a hydraulic pump as function of the pressure rise across the pump and the rotational speed of the pump, respectively.
As may be seen the volumetric efficiency decreases linearly with increasing pressure rise when rotational speed and viscosity is held constant. When varying the rotational speed for fixed viscosity and pressure rise it is seen that the volumetric efficiency goes to zero and below. At zero volumetric efficiency the pump is just capable of producing enough theoretical flow to make up for its own internal leakage and the pump is on the border line of acting as a hydraulic motor driven by the leakage flow.
It should be noted that in case of insufficient suction pressure, the volumetric efficiency will decrease with increasing pump speed because there is not enough time for the suction chambers to be properly filled with fluid.
The fact that the pump requires more input torque than theoretically expected is expressed by means of a hydro-mechanical efficiency:
P hmP tP
M
= M
η (2.14)
where
ηhmP hydro-mechanical efficiency
MtP theoretical input torque to the pump, [torque]
MP (actual) input torque to the pump, [torque]
In general, 4 different types of hydro-mechanical losses may be experienced:
1. Mechanical friction due to mechanical contact between parts of the pump moving relative to each other. This loss is proportional to the pressure rise.
2. Viscous (laminar) friction due to shearing of fluid films between parts of the pump moving relative to each other. This loss is proportional to the speed of the moving parts and the viscosity.
3. Hydrokinetic (turbulent) friction due to turbulent pump flow around restrictions, bends, etc. within the pump. This loss is proportional to the square of the flow.
4. Static friction due mainly to sealings. This loss is constant.
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The different hydro-mechanical losses may be expressed as either pressure drops, or more conveniently, as extra input torques to the pump:
P
MmP input torque required to overcome mechanical friction, [torque]
MνP input torque required to overcome viscous friction, [torque]
MhP input torque required to overcome turbulent friction, [torque]
MsP input torque required to overcome static friction, [torque]
KmP, KνP, KhP pump dependant constants
ΔpP pressure rise across the pump, [pressure]
n rotational speed, [revolutions/time]
μ dynamic viscosity, [pressureּtime]
Inserting Equations (2.15), (2.16), (2.17) and (2.18) into Equation (2.14) gives:
sP
Introducing Equation (2.10) in the above and rearranging leads to an expression that shows the dependency of the hydro-mechanical efficiency on pressure, speed and viscosity.
C0...3 pump dependent constants
In Figure 2.8 the variation of the hydro-mechanical efficiency for constant pump speed and constant pressure rise, respectively, may be viewed graphically. The curves are based on Equation (2.20).
As seen in Figure 2.8 the hydro-mechanical efficiency goes to zero as the pressure rise goes to zero for fixed pump speed. This corresponds to the pump meeting no resistance (no demand to pressurize the fluid), but still demanding some input torque from the power supply in order to overcome the viscous, turbulent and constant friction losses.
The (total) efficiency of a pump is defined as follows:
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P
PPS→P (actual) power delivered by the power supply to the pump, [power]
Figure 2.8 The hydro-mechanical efficiency for a hydraulic pump as function of the pressure rise across the pump and the rotational speed of the pump, respectively The power delivered by the pump to the fluid is found by combining Equation (2.7) and (2.11)
Similarly, the power delivered by the power supply to the pump is found by combining Equation (2.8) and (2.14)
hmP
Inserting Equation (2.22) and (2.23) into Equation (2.21) yields the following expression for the pump efficiency
hmP
ηhmP hydro-mechanical efficiency of the pump
The dependency of the total efficiency on the pressure rise across the pump, the speed of revolution and the viscosity is complex, especially as the volumetric efficiency is best at low pressure and high speed, whereas the hydro-mechanical efficiency is best at high pressure and low speed.
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