Gas-Charged Accumulator

Hydraulic accumulator with gas as compressible medium

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Accumulators

Description

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:

$(p_{G} + p_{A}) {(V_{T} - V_{F})}^{k} = (p_{p r} + p_{A}) V_{T}^{k}$

$p_{F} = p_{G} + p_{H S}$

$V_{C} = V_{T} - V_{d e a d}$

$p_{H S} = {\begin{array}{l} K_{S} (V_{F} - V_{C}) + K_{d} q_{F}^{+} (V_{F} - V_{C}) & if V_{F} \geq V_{C} \\ K_{S} V_{F} - K_{d} q_{F}^{-} V_{F} & if V_{F} \leq 0 \\ 0 & otherwise \end{array}$

$q_{F}^{+} = {\begin{array}{l} q_{F} & if q_{F} \geq 0 \\ 0 & otherwise \end{array}$

$q_{F}^{-} = {\begin{array}{l} q_{F} & if q_{F} \leq 0 \\ 0 & otherwise \end{array}$

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:

$q_{F} = \frac{d V_{F}}{d t}$

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.

Basic Assumptions and Limitations

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.

Parameters Tab

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.

Variables Tab

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.