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Hydraulic accumulator with gas as compressible medium
This block models a gas-charged accumulator. The accumulator consists of a precharged gas chamber and a fluid chamber connected to a hydraulic system. The chambers are separated by a bladder, piston, or another kind of elastic diaphragm.
If the fluid pressure at the accumulator inlet becomes higher than the preload pressure, fluid enters the accumulator chamber and compresses the gas, thus storing hydraulic energy. A decrease in the fluid pressure at the inlet forces the stored fluid back into the system.
Normally, pressure in the gas chamber is equal to that of the fluid chamber. But if pressure at the accumulator inlet (p) drops below the accumulator's preload pressure (p_{pr}), the gas chamber gets isolated from the system with the inlet valve. In this case, pressure in the gas chamber remains constant and equal to the preload value, while pressure at the inlet depends on pressure in the system to which the accumulator is connected. If pressure at the inlet builds up to the preload value or higher, the chambers start interacting again.
The fluid compressibility, inlet hydraulic resistance, and diaphragm mechanical properties, such as inertia and damping, are not accounted for in the model. The calculation diagram of the model, shown in the preceding figure, contains two rigidly connected chambers with equal effective area pistons. The left chamber represents the fluid chamber, while the chamber on the right side corresponds to the gaseous chamber. The pistons represent the separator between chambers, such as a bladder, diaphragm, or a piston. The piston motion is restricted by the hard stops, which limit the bladder expansion and contraction. The bladder expansion is limited when the fluid chamber gets emptied. The contraction limitation takes effect when the chamber is completely full. The distance from the left stop in terms of fluid volume equals to V_{F}, and the distance to the right stop is V_{0} – V_{F}, where V_{0} is the accumulator capacity and V_{F} is the volume of fluid in the accumulator. The hard stops are considered absolutely plastic.
The accumulator is described with the following equations:
$${q}_{F}=\frac{d{V}_{F}}{dt}$$
$${p}_{G}=\left({p}_{init}+{p}_{A}\right){\left(\frac{{V}_{0}-{V}_{init}}{{V}_{0}-{V}_{F}}\right)}^{k}-{p}_{A}$$
$${p}_{F}={p}_{G}+{p}_{HS}$$
$${p}_{HS}=\{\begin{array}{ll}\left({V}_{F}-{V}_{0}-{V}_{dead}\right){q}_{F}{K}_{HS}\hfill & \text{for}{V}_{F}{V}_{0}\text{,}{q}_{F}0\hfill \\ -{V}_{F}{q}_{F}{K}_{HS}\hfill & \text{for}{V}_{F}0\text{,}{q}_{F}0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$
where
q_{F} | Flow rate at the accumulator inlet |
V_{F} | Volume of fluid in the accumulator |
V_{0} | Accumulator capacity |
V_{init} | Initial fluid volume |
V_{dead} | Accumulator gaseous dead volume, a small portion of the gaseous chamber that remains filled with gas even if fluid volume is close to capacity |
p_{F} | Pressure at the accumulator inlet (gauge) |
p_{pr} | Preload pressure (gauge) |
p_{init} | Initial pressure (gauge), that is, pressure in the accumulator after the initial volume is added to the preloaded accumulator |
p_{A} | Atmospheric pressure |
p_{G} | Pressure in the gaseous chamber (gauge) |
p_{HS} | Pressure developed by hard stop in the bladder-hard stop interaction |
K_{HS} | Proportionality coefficient in the absolutely plastic hard stop characterization. With this model, the bladder can penetrate into the stop and the fluid volume can theoretically exceed the capacity at the top and become negative at the bottom. |
k | Specific heat ratio |
t | Time |
The block calculates the initial conditions based on the value you assign to the Initial volume parameter (V_{init}):
$${V}_{F}={V}_{init}$$
$${p}_{init}=\left({p}_{pr}+{p}_{A}\right){\left(\frac{{V}_{0}}{{V}_{0}-{V}_{init}}\right)}^{k}-{p}_{A}$$
The Gas-Charged Accumulator block represents the accumulator as a data-sheet-based model and uses parameters that are generally available in the catalogs or manufacturer data sheets. If a model with a higher degree of idealization is desirable, you can build it as a subsystem or a composite component, similar to the following block diagram:
The block positive direction is from the inlet into the accumulator. This means that the flow rate is positive if fluid flows into the accumulator.
The process in the gaseous chamber is represented with the ideal gas model.
The process is assumed to be polytropic.
No loading on the separator, such as inertia, friction, and so on, is considered.
Fluid compressibility is not taken into account.
Accumulator capacity. The default value is 8e-3 m^3.
Accumulator gaseous dead volume, that is, a small portion of the gaseous chamber that remains filled with gas even if fluid volume is close to capacity. The purpose of this parameter is to prevent computational failure if for some reasons the fluid volume becomes greater than the accumulator capacity. The default value is 4e-5 m^3.
Preload gauge pressure. The default value is 10e5 Pa.
Specific heat ratio (adiabatic index). No units. The default value is 1.4. To account for heat exchange, you can set it to a value between 1 and 2, depending on the properties of the gas being used in the accumulator. For example, for dry air at 20 degrees C, this value will be within a range between 1 (isothermal process) and 1.4 (adiabatic process).
Initial volume of fluid in the accumulator. This parameter specifies the initial condition for use in computing the block's initial state at the beginning of a simulation run, according to the equations listed in the block description. The default value is 0.
Proportionality coefficient in the absolutely plastic hard stop characterization. The default value is 1e15 Pa*s/m^6.
The block has one hydraulic conserving port associated with the accumulator inlet.
The flow rate is positive if fluid flows into the accumulator.