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Hydraulic accumulator with gas as compressible medium

Accumulators

This block models a gas-charged accumulator. The accumulator consists of a precharged gas chamber and a fluid chamber. The fluid chamber is connected to a hydraulic system. The chambers are separated by a bladder, a piston, or any kind of a diaphragm.

As the fluid pressure at the accumulator inlet becomes greater than the precharge pressure, fluid enters the accumulator and compresses the gas, storing hydraulic energy. A decrease in the fluid pressure causes the gas to decompress and discharge the stored fluid into the system.

During typical operations, the pressure in the gas chamber is equal to the pressure in the fluid chamber. However, if the pressure at the accumulator inlet drops below the precharge pressure, the gas chamber becomes isolated from the system. In this situation, the fluid chamber is empty and the pressure in the gas chamber remains constant and equal to the precharge pressure. The pressure at the accumulator inlet depends on the hydraulic system to which the accumulator is connected. If the pressure at the accumulator inlet builds up to the precharge pressure or higher, fluid enters the accumulator again.

The motion of the separator between the fluid chamber and the gas chamber is restricted by two hard stops that limit the expansion and contraction of the fluid volume. The fluid volume is limited when the fluid chamber is at capacity and when the fluid chamber is empty. The hard stops are modeled with finite stiffness and damping. This means that it is possible for the fluid volume to become negative or greater than the fluid chamber capacity, depending on the values of the hard-stop stiffness coefficient and the accumulator inlet pressure.

The diagram represents a gas-charged accumulator. The total
accumulator volume (*V*_{T})
is divided into the fluid chamber on the left and the gas chamber
on the right by the vertical separator. The distance between the left
side and the separator defines the fluid volume (*V*_{F}).
The distance between the right side and the separator defines the
gas volume (*V*_{T} – *V*_{F}).
The fluid chamber capacity (*V*_{C})
is less than the total accumulator volume (*V*_{T})
so that the gas volume never becomes zero.

The hard stop contact pressure is modeled with a stiffness term and a damping term. The relationship of the gas pressure and gas volume between the current state and the precharge state is given by the polytropic relation, with pressure balanced at the separator:

$$\left({p}_{G}+{p}_{A}\right){\left({V}_{T}-{V}_{F}\right)}^{k}=\left({p}_{pr}+{p}_{A}\right){V}_{T}^{k}$$

$${p}_{F}={p}_{G}+{p}_{HS}$$

$${V}_{C}={V}_{T}-{V}_{dead}$$

$${p}_{HS}=\{\begin{array}{ll}{K}_{S}\left({V}_{F}-{V}_{C}\right)+{K}_{d}{q}_{F}^{+}\left({V}_{F}-{V}_{C}\right)\hfill & \text{if}{V}_{F}\ge {V}_{C}\hfill \\ {K}_{S}{V}_{F}-{K}_{d}{q}_{F}^{-}{V}_{F}\hfill & \text{if}{V}_{F}\le 0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$

$${q}_{F}^{+}=\{\begin{array}{ll}{q}_{F}\hfill & \text{if}{q}_{F}\ge 0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$

$${q}_{F}^{-}=\{\begin{array}{ll}{q}_{F}\hfill & \text{if}{q}_{F}\le 0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$

where

V_{T} | Total volume of the accumulator, including the fluid chamber and the gas chamber |

V_{F} | Volume of fluid in the accumulator |

V_{init} | Initial volume of fluid in the accumulator |

V_{C} | Fluid chamber capacity, the difference between total accumulator volume and the gas chamber dead volume |

V_{dead} | Gas chamber dead volume, a small portion of the gas chamber that remains filled with gas when the fluid chamber is at capacity |

p_{F} | Fluid pressure (gauge) in the fluid chamber, which is equal to the pressure at the accumulator inlet |

p_{pr} | Pressure (gauge) in the gas chamber when the fluid chamber is empty |

p_{A} | Atmospheric pressure |

p_{G} | Gas pressure (gauge) in the gas chamber |

p_{HS} | Hard-stop contact pressure |

K_{s} | Hard-stop stiffness coefficient |

K_{d} | Hard-stop damping coefficient |

k | Specific heat ratio (adiabatic index) |

q_{F} | Fluid flow rate into the accumulator, which is positive if fluid flows into the accumulator |

The flow rate into the accumulator is the rate of change of the fluid volume:

$${q}_{F}=\frac{d{V}_{F}}{dt}$$

At *t* = 0,
the initial condition is *V*_{F} = *V*_{init},
where *V*_{init} is the value
you assign to the **Initial fluid volume** parameter.

The Gas-Charged Accumulator block does not consider loading on the separator. To model additional effects, such as the separator inertia and friction, you can construct a gas-charged accumulator as a subsystem or a composite component, similar to the block diagram below.

The process in the gas chamber is assumed to be polytropic.

Loading on the separator, such as inertia or friction, is not considered.

Inlet hydraulic resistance is not considered.

Fluid compressibility is not considered.

**Total accumulator volume**Total volume of the accumulator including the fluid chamber and the gas chamber. It is the sum of the fluid chamber capacity and the minimum gas volume. The default value is

`8e-3`

m^3.**Minimum gas volume**Gas chamber dead volume, a small portion of the gas chamber that remains filled with gas when the fluid chamber is at capacity. A nonzero volume is necessary so that the gas pressure does not become infinite when the fluid chamber is at capacity. The default value is

`4e-5`

m^3.**Precharge pressure (gauge)**Pressure (gauge) in the gas chamber when the fluid chamber is empty. The default value is

`10e5`

Pa.**Specific heat ratio**Specific heat ratio (adiabatic index). To account for heat exchange, you can set it to a value normally between 1 and 2, depending on the properties of the gas in the gas chamber. For dry air at 20°C, this value is 1 for an isothermal process and 1.4 for an adiabatic (and isentropic) process. The default value is

`1.4`

.**Hard-stop stiffness coefficient**Proportionality constant of the hard-stop contact pressure with respect to the fluid volume penetrated into the hard stop. The hard stops are used to restrict the fluid volume between zero and fluid chamber capacity. The default value is

`1e10`

Pa/m^3.**Hard-stop damping coefficient**Proportionality constant of the hard-stop contact pressure with respect to the flow rate and the fluid volume penetrated into the hard stop. The hard stops are used to restrict the fluid volume between zero and fluid chamber capacity. The default value is

`1e10`

Pa*s/m^6.

**Accumulator flow rate**Volumetric flow rate through the accumulator port at time zero. Simscape™ software uses this parameter to guide the initial configuration of the component and model. Initial variables that conflict with each other or are incompatible with the model may be ignored. Set the

**Priority**column to`High`

to prioritize this variable over other, low-priority, variables.**Volume of fluid**Volume of fluid in the accumulator at time zero. Simscape software uses this parameter to guide the initial configuration of the component and model. Initial variables that conflict with each other or are incompatible with the model may be ignored. Set the

**Priority**column to`High`

to prioritize this variable over other, low-priority, variables.**Pressure of liquid volume**Gauge pressure in the accumulator at time zero. Simscape software uses this parameter to guide the initial configuration of the component and model. Initial variables that conflict with each other or are incompatible with the model may be ignored. Set the

**Priority**column to`High`

to prioritize this variable over other, low-priority, variables.

**Atmospheric pressure**Absolute pressure of the environment. The default value is

`101325`

Pa.

The block has one hydraulic conserving port associated with the accumulator inlet.

The flow rate is positive if fluid flows into the accumulator.