Fixed-Displacement Pump (2P)

Mechanical-hydraulic power conversion device

Since R2022b

Libraries:
Simscape / Fluids / Two-Phase Fluid / Fluid Machines

Description

The Fixed-Displacement Pump (2P) block represents a pump that extracts power from a mechanical rotational network and delivers it to a two-phase fluid network. The pump displacement is fixed at a constant value that you specify through the Displacement parameter.

Ports A and B represent the pump inlets. Ports R and C represent the drive shaft and case. During normal operation, the pressure gain from port A to port B is positive if the angular velocity at port R relative to port C is positive.

The Fixed-Displacement Pump (2P) block assumes that liquid enters the pump inlet. You can use the Report when fluid is not fully liquid parameter to choose what the block does when the fluid does not meet liquid conditions. To move vapor in a two-phase fluid network, use a compressor.

Operating Modes

The block has four modes of operation, as shown by this image.

The working mode depends on the pressure gain from port A to port B, Δp = p_B – p_A and the angular velocity, ω = ω_R – ω_C:

The quadrant labeled 1 represents the forward pump mode. In this mode, the positive shaft angular velocity causes a pressure increase from port A to port B and flow from port A to port B
The quadrant labeled 2 represents the reverse motor mode. In this mode, the flow from port B to port A causes a pressure decrease from B to A and negative shaft angular velocity.
The quadrant labeled 3 represents the reverse pump mode. In this mode, the negative shaft angular velocity causes a pressure increase from port B to port A and flow from port B to port A
The quadrant labeled 4 represents the forward motor mode. In this mode, the flow from port A to port B causes a pressure decrease from A to B and positive shaft angular velocity.

The response time of the pump is negligible in comparison with the system response time. The pump reaches steady state nearly instantaneously and is treated as a quasi-steady component.

Energy Balance

The block associates the mechanical work done by the pump with an energy exchange. The governing energy balance equation is

$ϕ_{A} + ϕ_{B} + P_{h y d r o} = 0,$

where:

Φ_A and Φ_B are the energy flow rates at ports A and B, respectively.
P_hydro is the pump hydraulic power, which is a function of the pressure difference between the pump ports: $P_{h y d r o} = Δ p \frac{\dot{m}}{ρ}$ .

The block generates mechanical power due to torque, τ, and angular velocity, ω:

$P_{m e c h} = τ ω .$

Flow Rate and Driving Torque

The mass flow rate generated at the pump is

$\dot{m} = {\dot{m}}_{Ideal} - {\dot{m}}_{Leak},$

where:

$\dot{m}$ is the actual mass flow rate.
${\dot{m}}_{Ideal}$ is the ideal mass flow rate.
${\dot{m}}_{Leak}$ is the internal leakage mas flow rate.

The driving torque required to power the pump is

$τ = τ_{Ideal} + τ_{Friction},$

where:

τ is the actual driving torque.
τ_Ideal is the ideal driving torque.
τ_Friction is the friction torque.

Ideal Flow Rate and Ideal Torque

The ideal mass flow rate is

${\dot{m}}_{Ideal} = ρ D ω,$

and the ideal required torque is

$τ_{Ideal} = D Δ p,$

where:

ρ is the average of the fluid densities at ports A and B.
D is the Displacement parameter.
ω is the shaft angular velocity.
Δp is the pressure gain from inlet to outlet.

Leakage and Friction Parameterization

You can parameterize leakage and friction analytically, by using tabulated efficiencies or losses, or by using input efficiencies or input losses.

Analytical

When you set the Leakage and friction parameterization parameter to Analytical, the leakage flow rate is

${\dot{m}}_{Leak} = \frac{K_{HP} ρ_{Avg} Δ p}{μ_{Avg}},$

and the friction torque is

$τ_{Friction} = (τ_{0} + K_{TP} | Δ p | tanh \frac{4 ω}{(5 \cdot 10^{- 5}) ω_{Nom}}),$

where:

K_HP is the Hagen-Poiseuille coefficient for laminar pipe flows. The block computes this coefficient from the specified nominal parameters.
μ is the dynamic viscosity of the fluid, which is the average of the values at the ports.
K_TP is the friction torque vs. pressure gain coefficient at nominal displacement, which the block determines from the Mechanical efficiency at nominal conditions parameter, η_m:

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

$τ_{f r, n o m} = (\frac{1 - η_{m, n o m}}{η_{m, n o m}}) D Δ p_{n o m} .$
Δp_Nom is the value of the Nominal pressure gain parameter. This value is the pressure gain at which the nominal volumetric efficiency is specified.
τ₀ is the value of the No-load torque parameter.
ω_Nom is the value of the Nominal shaft angular velocity parameter.

The block determines the Hagen-Poiseuille coefficient from the nominal fluid and component parameters

$K_{HP} = \frac{D ω_{Nom} μ_{Nom} (1 - η_{v,Nom})}{Δ p_{Nom}},$

where:

ω_Nom is the value of the Nominal shaft angular velocity parameter. This value is the angular velocity at which the block specifies the nominal volumetric efficiency.
μ_Nom is the value of the Nominal Dynamic viscosity parameter. This value is the dynamic viscosity at which the block specifies the nominal volumetric efficiency.
η_v,Nom is the value of the Volumetric efficiency at nominal conditions parameter. This is the volumetric efficiency that corresponds to the specified nominal conditions.

Tabulated Efficiencies

When you set the Leakage and friction parameterization parameter to Tabulated data - volumetric and mechanical efficiencies, the leakage flow rate is

${\dot{m}}_{Leak} = {\dot{m}}_{Leak,Pump} \frac{(1 + α)}{2} + {\dot{m}}_{Leak,Motor} \frac{(1 - α)}{2},$

and the friction torque is

$τ_{Friction} = τ_{Friction,Pump} \frac{1 + α}{2} + τ_{Friction,Motor} \frac{1 - α}{2},$

where:

α is a numerical smoothing parameter for the motor-pump transition.
${\dot{m}}_{Leak,Motor}$ is the leakage flow rate in motor mode.
${\dot{m}}_{Leak,Pump}$ is the leakage flow rate in pump mode.
τ_{Friction,Motor} is the friction torque in motor mode.
τ_{Friction,Pump} is the friction torque in pump mode.

This hyperbolic function describes the smoothing parameter α

$α = tanh (\frac{4 Δ p}{Δ p_{Threshold}}) \cdot tanh (\frac{4 ω}{ω_{Threshold}}),$

where:

Δp_Threshold is the value of the Pressure gain threshold for motor-pump transition parameter.
ω_Threshold is the value of the Angular velocity threshold for motor-pump transition parameter.

The block calculates the leakage flow rate from the volumetric efficiency, which the Volumetric efficiency table parameter specifies over the Δp–ɷ domain. When operating in pump mode, the leakage flow rate is:

${\dot{m}}_{Leak,Pump} = (1 - η_{v}) {\dot{m}}_{Ideal},$

where η_v is the volumetric efficiency, which the block obtains either by interpolation or extrapolation of the tabulated data. Similarly, when operating in motor mode, the leakage flow rate is:

${\dot{m}}_{Leak,Motor} = - (1 - η_{v}) \dot{m} .$

The block calculates the friction torque from the mechanical efficiency, which the Mechanical efficiency table parameter specifies over the Δp–ɷ domain. When operating in pump mode, the friction torque is

$τ_{Friction,Pump} = (1 - η_{m}) τ,$

where η_m is the mechanical efficiency, which the block obtains either by interpolation or extrapolation of the tabulated data. Similarly, when operating in motor mode, the friction torque is

$τ_{Friction,Motor} = - (1 - η_{m}) τ_{Ideal} .$

Tabulated Losses

When you set the Leakage and friction parameterization parameter to Tabulated data - volumetric and mechanical losses, the block specifies the leakage volumetric flow rate in tabulated form over the Δp–ɷ domain:

$q_{Leak} = q_{Leak} (Δ p, ω) .$

The block calculates the mass flow rate due to leakage from the volumetric flow rate:

${\dot{m}}_{Leak} = ρ q_{Leak} .$

The block calculates the friction torque in tabulated form:

$τ_{Friction} = τ_{Loss} (Δ p, ω) \tanh (\frac{4 ω}{ω_{t h r e s h o l d}}),$

where q_Leak(Δp,ω) and τ_Loss(Δp,ω) are the volumetric and mechanical losses, specified through interpolation or extrapolation of the tabulated data in the Volumetric loss table and Mechanical loss table parameters.

Input Efficiencies

When you set the Leakage and friction parameterization parameter to Input signal - volumetric and mechanical efficiencies, the leakage flow rate and friction torque calculations are identical to the Tabulated data - volumetric and mechanical efficiencies setting. The block replaces the volumetric and mechanical efficiency lookup tables with the physical signal input ports EV and EM.

The efficiencies are positive quantities with values between 0 and 1.The block sets input values outside of these bounds to 0 for inputs smaller than 0 or 1 for inputs greater than 1. The block saturates the efficiency signals at the value of the Minimum volumetric efficiency or Minimum mechanical efficiency parameter and the Maximum volumetric efficiency or Maximum mechanical efficiency parameter.

Input Losses

When you set the Leakage and friction parameterization parameter to Input signal - volumetric and mechanical losses, the leakage flow rate and friction torque calculations are identical to the Tabulated data - volumetric and mechanical efficiencies setting. The block replaces the volumetric and mechanical loss lookup tables with the physical signal input ports LV and LM.

The block expects the inputs to be positive and sets the signs automatically from the Δp–ɷ quadrant where the component is operating. If you provide a negative signal, the block returns zero losses.

Faults

To model a fault, in the Faults section, click the Add fault hyperlink next to the fault that you want to model. Use the fault parameters to specify the fault properties. For more information about fault modeling, see Introduction to Simscape Faults.

You can model a displacement fault, leakage, or a shaft friction torque fault.

When you enable the Displacement fault parameter, the block scales the displacement by the value of the Faulted displacement factor parameter when the fault triggers,

$D_{F a u l t} = f_{D} D,$

where f_D is the value of the Faulted displacement factor parameter. When the Leakage and friction parameterization parameter is Analytical, the block does not use the faulted displacement value to calculate the Hagen-Poiseuille coefficient or the friction torque at nominal conditions.

When you enable the Leakage fault parameter and Leakage and friction parameterization is Analytical, Tabulated data - volumetric and mechanical efficiencies, or Input signal - volumetric and mechanical efficiencies, the faulted volumetric efficiency is

$η_{v, F a u l t} = \frac{η_{v}}{f_{L e a k}},$

where f_Leak is the value of the Faulted leakage factor parameter and η_v is the volumetric efficiency. When Leakage and friction parameterization is Analytical, the block uses the faulted volumetric efficiency to calculate the Hagen-Poiseuille coefficient.

When Leakage and friction parameterization is Tabulated data - volumetric and mechanical losses or Input signal - volumetric and mechanical losses, the faulted leakage volumetric flow rate is

$q_{L e a k, F a u l t} = f_{L e a k} q_{L e a k} .$

When you enable the Shaft friction torque fault parameter and Leakage and friction parameterization is Analytical, Tabulated data - volumetric and mechanical efficiencies, or Input signal - volumetric and mechanical efficiencies, the faulted mechanical efficiency is

$η_{m, F a u l t} = \frac{η_{m}}{f_{F r i c t i o n}},$

where f_Friction is the value of the Faulted shaft friction torque factor parameter and η_m is the mechanical efficiency. When Leakage and friction parameterization is Analytical, the block uses the faulted mechanical efficiency to calculate the friction torque at nominal conditions.

When Leakage and friction parameterization is Tabulated data - volumetric and mechanical losses or Input signal - volumetric and mechanical losses, the faulted friction torque is

$τ_{F r i c t i o n, F a u l t} = f_{F r i c t i o n} τ_{F r i c t i o n} .$

Assumptions and Limitations

The block treats the pump as a quasi-steady component.
The block ignores the effects of fluid inertia and elevation.
The pump wall is rigid.
The block ignores external leakage.

Ports

Input

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EV — Volumetric efficiency, unitless
physical signal

Physical signal input port for the volumetric efficiency coefficient. The input signal has an upper bound at the Maximum volumetric efficiency parameter value and a lower bound at the Minimum volumetric efficiency parameter value.

Dependencies

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

EM — Mechanical efficiency, unitless
physical signal

Physical signal input port for the mechanical efficiency coefficient. The input signal has an upper bound at the Maximum mechanical efficiency parameter value and a lower bound at the Minimum mechanical efficiency parameter value.

Dependencies

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

LV — Volumetric loss, m^3/s
physical signal

Physical signal input port for the volumetric loss which is the internal leakage flow rate between the pump inlets.

Dependencies

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

LM — Mechanical loss, N*m
physical signal

Physical signal input port for the mechanical loss which is the friction torque on the rotating pump shaft.

Dependencies

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

Conserving

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A — Pump inlet
two-phase fluid

Two-phase fluid conserving port associated with the pump inlet.

B — Pump outlet
two-phase fluid

Two-phase fluid conserving port associated with the pump outlet.

C — Pump case
mechanical rotational

Mechanical rotational conserving port associated with the pump case.

R — Pump shaft
mechanical rotational

Mechanical rotational conserving port associated with the rotational pump shaft.

Parameters

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Parameters

Leakage and friction parameterization — Parameterization of 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`

Method to compute the flow-rate and torque losses due to internal leaks and friction. Use the Analytical if the block parameters are generally available from component data sheets. When you select Tabulated data - volumetric and mechanical efficiencies or Tabulated data - volumetric and mechanical losses, the block uses lookup tables to map pressure gains and shaft angular velocities to component efficiencies or losses.

When you select Input signal - volumetric and mechanical efficiencies or Input signal - volumetric and mechanical losses, the block performs the leakage flow rate and friction torque calculations the same as the Tabulated data - volumetric and mechanical efficiencies or Tabulated data - volumetric and mechanical losses settings, respectively. When you select Input signal - volumetric and mechanical efficiencies, the block enables the EV and EM ports. You use these ports to specify the volumetric and mechanical efficiency. When you select Input signal - volumetric and mechanical losses, the block enables the LV and LM ports. You use these ports to specify the volumetric and mechanical losses.

Displacement — Fluid volume displaced per shaft rotation
`30` `cm^3/rev` (default) | positive scalar

Fluid volume displaced per shaft rotation. The block maintains this value throughout the simulation.

Nominal shaft angular velocity — Shaft angular velocity for given volumetric efficiency
`1800` `rpm` (default) | scalar

Angular velocity of the rotary shaft that corresponds to the given volumetric efficiency. The standard operating conditions in the manufacturer data sheet typically list these values. The block uses this parameter to calculate the leakage flow rate and friction torque.