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# Fixed-Displacement Pump

Fixed-displacement hydraulic pump

Pumps and Motors

## Description

The Fixed-Displacement Pump block represents a positive, fixed-displacement pump of any type as a data-sheet-based model. The key parameters required for this block are pump displacement, volumetric and total efficiencies, nominal pressure, and angular velocity. All these parameters are generally provided in the data sheets or catalogs. The fixed-displacement pump is represented with the following equations:

$q=D\cdot \omega -{k}_{leak}\cdot p$

$T=D\cdot p/{\eta }_{mech}$

${k}_{leak}=\frac{{k}_{HP}}{\nu \cdot \rho }$

${k}_{HP}=\frac{D\cdot {\omega }_{nom}\left(1-{\eta }_{V}\right)\cdot {\nu }_{nom}\cdot {\rho }_{nom}}{{p}_{nom}}$

$p={p}_{P}-{p}_{T}$

where

 q Pump delivery p Pressure differential across the pump pP,pT Gauge pressures at the block terminals T Torque at the pump driving shaft D Pump displacement ω Pump angular velocity kleak Leakage coefficient kHP Hagen-Poiseuille coefficient ηV Pump volumetric efficiency ηmech Pump mechanical efficiency ν Fluid kinematic viscosity ρ Fluid density ρnom Nominal fluid density pnom Pump nominal pressure ωnom Pump nominal angular velocity νnom Nominal fluid kinematic viscosity

The leakage flow is determined based on the assumption that it is linearly proportional to the pressure differential across the pump and can be computed by using the Hagen-Poiseuille formula

$p=\frac{128\mu l}{\pi {d}^{4}}{q}_{leak}=\frac{\mu }{{k}_{HP}}{q}_{leak}$

where

 qleak Leakage flow d, l Geometric parameters of the leakage path μ Fluid dynamic viscosity, μ = ν.ρ

The leakage flow at p = pnom and ν = νnom can be determined from the catalog data

${q}_{leak}=D{\omega }_{nom}\left(1-{\eta }_{V}\right)$

which provides the formula to determine the Hagen-Poiseuille coefficient

${k}_{HP}=\frac{D\cdot {\omega }_{nom}\left(1-{\eta }_{V}\right)\cdot {\nu }_{nom}\cdot {\rho }_{nom}}{{p}_{nom}}$

The pump mechanical efficiency is not usually available in data sheets, therefore it is determined from the total and volumetric efficiencies by assuming that the hydraulic efficiency is negligibly small

${\eta }_{mech}={\eta }_{total}/{\eta }_{V}$

The block positive direction is from port T to port P. This means that the pump transfers fluid from T to P provided that the shaft S rotates in the positive direction. The pressure differential across the pump is determined as $p={p}_{P}-{p}_{T}$.

## Basic Assumptions and Limitations

• Fluid compressibility is neglected.

• No loading on the pump shaft, such as inertia, friction, spring, and so on, is considered.

• Leakage inside the pump is assumed to be linearly proportional to its pressure differential.

## Dialog Box and Parameters

Pump displacement

Pump displacement. The default value is 5e-6 m^3/rad.

Volumetric efficiency

Pump volumetric efficiency specified at nominal pressure, angular velocity, and fluid viscosity. The default value is 0.92.

Total efficiency

Pump total efficiency, which is determined as a ratio between the hydraulic power at the pump outlet and mechanical power at the driving shaft at nominal pressure, angular velocity, and fluid viscosity. The default value is 0.8.

Nominal pressure

Pressure differential across the pump, at which both the volumetric and total efficiencies are specified. The default value is 1e7 Pa.

Nominal angular velocity

Angular velocity of the driving shaft, at which both the volumetric and total efficiencies are specified. The default value is 188 rad/s.

Nominal kinematic viscosity

Working fluid kinematic viscosity, at which both the volumetric and total efficiencies are specified. The default value is 18 cSt.

Nominal fluid density

Working fluid density, at which both the volumetric and total efficiencies are specified. The default value is 900 kg/m^3.

## Global Parameters

Parameter determined by the type of working fluid:

• Fluid kinematic viscosity

Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

## Ports

The block has the following ports:

T

Hydraulic conserving port associated with the pump suction, or inlet.

P

Hydraulic conserving port associated with the pump outlet.

S

Mechanical rotational conserving port associated with the pump driving shaft.

## Examples

The Power Unit with Fixed-Displacement Pump example contains a fixed-displacement pump, which is driven by a motor through a compliant transmission, a pressure-relief valve, and a variable orifice, which simulates system fluid consumption. The motor model is represented as an Ideal Angular Velocity Source block, which rotates the shaft at 188 rad/s at zero torque. The load on the shaft decreases the velocity with a slip coefficient of 1.2 (rad/s)/Nm. The load on the driving shaft is measured with the torque sensor. The shaft between the motor and the pump is assumed to be compliant and simulated with rotational spring and damper.

The simulation starts with the variable orifice open, which results in a low system pressure and the maximum flow rate going to the system. The orifice starts closing at 0.5 s, and is closed completely at 3 s. The output pressure builds up until it reaches the pressure setting of the relief valve (75e5 Pa), and is maintained at this level by the valve. At 3 s, the variable orifice starts opening, thus returning the system to its initial state.