Variable-Displacement Pump (IL)

Variable-displacement pump in an isothermal liquid network

Since R2020a

Libraries:
Simscape / Fluids / Isothermal Liquid / Pumps & Motors

Description

The Variable-Displacement Pump (IL) block models a pump with variable-volume displacement. The fluid may move from port A to port B, called forward mode, or from port B to port A, called reverse mode. Pump mode operation occurs when there is a pressure gain in the direction of the flow. Motor mode operation occurs when there is a pressure drop in the direction of the flow.

The shaft rotation corresponds to the sign of the fluid volume moving through the pump, which is received as a physical signal at port D. Positive fluid displacement at D corresponds to positive shaft rotation in forward mode. Negative fluid displacement at D corresponds to negative shaft angular velocity in forward mode.

Operation Modes

The block has eight modes of operation. The working mode depends on the pressure gain from port A to port B, Δp = p_B – p_A; the angular velocity, ω = ω_R – ω_C; and the fluid volumetric displacement at port D. The figure above maps these modes to the octants of a Δp-ω-D chart:

Mode 1, Forward Pump: Positive shaft angular velocity causes a pressure increase from port A to port B and flow from port A to port B.
Mode 2, Reverse Motor: Flow from port B to port A causes a pressure decrease from B to A and negative shaft angular velocity.
Mode 3, Reverse Pump: Negative shaft angular velocity causes a pressure increase from port B to port A and flow from B to A.
Mode 4, Forward Motor: Flow from port A to B causes a pressure decrease from A to B and positive shaft angular velocity.
Mode 5, Reverse Motor: Flow from port B to port A causes a pressure decrease from B to A and positive shaft angular velocity.
Mode 6, Forward Pump: Negative shaft angular velocity causes pressure increase from A to B and flow from A to B.
Mode 7, Forward Motor: Flow from port A to B causes a pressure decrease from A to B and negative shaft angular velocity.
Mode 8, Reverse Pump: Positive shaft angular velocity causes a pressure increase from port B to port A and flow from B to A.

The pump block has analytical, lookup table, and physical signal parameterizations. When using tabulated data or an input signal for parameterization, you can choose to characterize pump operation based on efficiency or losses.

The threshold parameters Pressure gain threshold for pump-motor transition, Angular velocity threshold for pump-motor transition, and Displacement threshold for pump-motor transition identify regions where numerically smoothed flow transition between the pump operational modes can occur. For the pressure and angular velocity thresholds, choose a transition region that provides some margin for the transition term, but which is small enough relative to the typical pump pressure gain and angular velocity so that it will not impact calculation results. For the displacement threshold, choose a threshold value that is smaller than the typical displacement volume during normal operation.

Analytical Leakage and Friction Parameterization

If you set Leakage and friction parameterization to Analytical, the block calculates internal leakage and shaft friction from constant nominal values of shaft velocity, pressure gain, volumetric displacement, and volumetric efficiency. The leakage flow rate, which is correlated with the pressure differential over the pump, is calculated as:

${\dot{m}}_{l e a k} = K ρ_{a v g} Δ p,$

where:

Δp is p_B – p_A.
ρ_avg is the average fluid density.
K is the Hagen-Poiseuille coefficient for analytical loss,

$K = \frac{D_{n o m} ω_{n o m} (1 - η_{v, n o m})}{Δ p_{n o m}},$
where:
- D_nom is the Nominal displacement.
- ω_nom is the Nominal shaft angular velocity.
- η_nom is the Volumetric efficiency at nominal conditions.
- Δp_nom is the Nominal pressure gain.

The friction torque, which is related to the pump pressure differential, is calculated as:

$τ_{f r} = (τ_{0} + k | Δ p \frac{D}{D_{n o m}} |) \tanh (\frac{4 ω}{5 \times 10^{- 5} ω_{n o m}}),$

where:

τ₀ is the No-load torque.
k is the friction torque vs. pressure gain coefficient at nominal displacement, which is determined from the Mechanical efficiency at nominal conditions, η_m,nom:

$k = \frac{τ_{f r, n o m} - τ_{0}}{Δ p_{n o m}} .$
τ_fr,nom is the friction torque at nominal conditions:

$τ_{f r, n o m} = (\frac{1 - η_{m, n o m}}{η_{m, n o m}}) D_{n o m} Δ p_{n o m} .$
ω is the relative shaft angular velocity, or $ω_{R} - ω_{C}$ .

Tabulated Data Parameterizations

When using tabulated data for pump efficiencies or losses, you can provide data for one or more of the pump operational modes. The signs of the tabulated data determine the operational regime of the block. When data is provided for less than eight operational modes, the block calculates the complementing data for the other modes by extending the given data into the remaining octants.

The Tabulated data - volumetric and mechanical efficiencies parameterization

The leakage flow rate is calculated as:

${\dot{m}}_{l e a k} = {\dot{m}}_{l e a k, p u m p} (\frac{1 + α}{2}) + {\dot{m}}_{l e a k, m o t o r} (\frac{1 - α}{2}),$

where:

${\dot{m}}_{l e a k, p u m p} = (1 - η_{υ}) {\dot{m}}_{i d e a l}$
${\dot{m}}_{l e a k, m o t o r} = (η_{v} - 1) \dot{m}$

and η_v is the volumetric efficiency, which is interpolated from the user-provided tabulated data. The transition term, α, is

$α = \tanh (\frac{4 Δ p}{Δ p_{t h r e s h o l d}}) \tanh (\frac{4 ω}{ω_{t h r e s h o l d}}) \tanh (\frac{4 D}{D_{t h r e s h o l d}}),$

where:

Δp is p_B – p_A.
Δp_threshold is the Pressure gain threshold for pump-motor transition.
ω is ω_R – ω_C.
ω_threshold is the Angular velocity threshold for pump-motor transition.

The friction torque is calculated as:

$τ_{f r} = τ_{f r, p u m p} (\frac{1 + α}{2}) + τ_{f r, m o t o r} (\frac{1 - α}{2}),$

where:

$τ_{f r, p u m p} = (1 - η_{m}) τ$
$τ_{f r, m o t o r} = (η_{m} - 1) τ_{i d e a l}$

and η_m is the mechanical efficiency, which is interpolated from the user-provided tabulated data.

The Tabulated data - volumetric and mechanical losses parameterization

The leakage flow rate is calculated as:

${\dot{m}}_{l e a k} = ρ_{a v g} q_{l o s s} (Δ p, ω, D),$

where q_loss is interpolated from the Volumetric loss table, q_loss(dp,w,D) parameter, which is based on user-supplied data for pressure gain, shaft angular velocity, and fluid volumetric displacement.

The shaft friction torque is calculated as:

$τ_{f r} = τ_{l o s s} (Δ p, ω, D),$

where τ_loss is interpolated from the Mechanical loss table, torque_loss(dp,w,D) parameter, which is based on user-supplied data for pressure gain, shaft angular velocity, and fluid volumetric displacement.

Input Signal Parameterization

When you select Input signal - volumetric and mechanical efficiencies, ports EV and EM are enabled. The internal leakage and shaft friction are calculated in the same way as the Tabulated data - volumetric and mechanical efficiencies parameterization, except that η_v and η_m are received directly at ports EV and EM, respectively.

When you select Input signal - volumetric and mechanical losses, ports LV and LM are enabled. These ports receive leakage flow and friction torque as positive physical signals. The leakage flow rate is calculated as:

${\dot{m}}_{l e a k} = ρ_{a v g} q_{L V} \tanh (\frac{4 Δ p}{p_{t h r e s h}}),$

where:

q_LV is the leakage flow received at port LV.
p_thresh is the Pressure gain threshold for pump-motor transition parameter.

The friction torque is calculated as:

$τ_{f r} = τ_{L M} \tanh (\frac{4 ω}{ω_{t h r e s h}}),$

where

τ_LM is the friction torque received at port LM.
ω_thresh is the Angular velocity threshold for pump-motor transition parameter.

The volumetric and mechanical efficiencies range between the user-defined specified minimum and maximum values. Any values lower or higher than this range will take on the minimum and maximum specified values, respectively.

Pump Operation

The pump flow rate is:

$\dot{m} = {\dot{m}}_{i d e a l} - {\dot{m}}_{l e a k},$

where ${\dot{m}}_{i d e a l} = ρ_{a v g} D \cdot ω .$

The pump torque is:

$τ = τ_{i d e a l} + τ_{f r},$

where $τ_{i d e a l} = D \cdot Δ p .$

The mechanical power delivered by the pump shaft is:

$φ_{m e c h} = τ ω,$

and the pump hydraulic power is:

$φ_{h y d} = \frac{Δ p \dot{m}}{ρ_{a v g}} .$

To be notified if the block is operating beyond the supplied tabulated data, you can set Check if operating beyond the range of supplied tabulated data to Warning to receive a warning if this occurs, or Error to stop the simulation when this occurs. when using input signals for volumetric or mechanical losses, you can be notified if the simulation surpasses operating modes with the Check if operating beyond pump mode parameter.

You can also monitor pump functionality. Set Check if pressures are less than pump minimum pressure to Warning to receive a warning if this occurs, or Error to stop the simulation when this occurs.

Examples

Hydrostatic Transmission

A hydraulic transmission system built of a variable-displacement pump and a fixed-displacement motor. The pipes between the pump and the motor are connected by two replenishing valves (check valves) and a charge pump on the pump side and two pressure relief valves on the motor side. The motor drives a mechanical load consisting of an inertia, viscous friction, and time-varying torque. The system is tested with both positive and negative pump flow rate by varying the pump displacement.

Open Model

Ports

Conserving

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A — Liquid port
isothermal liquid

Liquid entry or exit port to the pump.

B — Liquid port
isothermal liquid

Liquid entry or exit port to the pump.

R — Mechanical port
mechanical rotational

Rotating shaft angular velocity and torque.

C — Mechanical port
mechanical rotational

Pump casing reference angular velocity and torque.

Input

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D — Volumetric displacement, cm^3/rad
physical signal

Volumetric displacement of the pump, in m^3/rad, specified as a physical signal.

EV — Volumetric efficiency
physical signal

Pump efficiency for fluid displacement, specified as a physical signal. The value must be between 0 and 1.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical efficiencies.

EM — Mechanical efficiency
physical signal

Pump efficiency for the mechanical delivery of power, specified as a physical signal. The value must be between 0 and 1.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical efficiencies.

LV — Leakage flow rate, m^3/s
physical signal

Pump volumetric losses, in m^3/s, specified as a physical signal.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical losses.

LM — Friction torque, N*m
physical signal

Pump mechanical losses in N*m, specified as a physical signal.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical losses.

Parameters

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Leakage and friction parameterization — Method of calculating leakage flow rate and friction torque
`Analytical` (default) | `Tabulated data - volumetric and mechanical efficiencies` | `Tabulated data - volumetric and mechanical losses` | `Input signal - volumetric and mechanical efficiencies` | `Input signal - volumetric and mechanical losses`

Parameterization of the leakage and friction characteristics of the pump.

In the Analytical parameterization, the leakage flow rate and the friction torque are calculated by analytical equations.
In the Tabulated data - volumetric and mechanical efficiencies parameterization, the volumetric and mechanical efficiencies are calculated from the user-supplied Pressure gain vector, dp, Shaft angular velocity vector, w, and Displacement vector, D parameters and interpolated from the 3-D dependent Volumetric efficiency table, e_v(dp,w,D) and Mechanical efficiency table, e_m(dp,w,D) tables.
In the Tabulated data - volumetric and mechanical loss parameterization, the leakage flow rate and friction torque are calculated from the user-supplied Pressure gain vector, dp; Shaft angular velocity vector, w; and Displacement vector, D parameters and interpolated from the 3-D dependent Volumetric loss table, q_loss(dp,w,D) and Mechanical loss table, torque_loss(dp,w,D) tables.
In the Input signal - volumetric and mechanical efficiencies parameterization, the volumetric and mechanical efficiencies are received as physical signals at ports EV and EM, respectively.
In the Input signal - volumetric and mechanical loss parameterization, the leakage flow rate and friction torque are received as physical signals at ports LV and LM, respectively.