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Hydraulic accumulator with spring used for energy storage
This block represents a spring-loaded accumulator, where fluid entering the accumulator compresses the spring, thus storing hydraulic energy. Since the spring compression increases as fluid enters the chamber and decreases as the accumulator is discharged, the pressure is not constant. The spring is preloaded. If the fluid pressure at the accumulator inlet becomes higher than the preload pressure, fluid enters the accumulator chamber and compresses the spring, thus storing hydraulic energy. A decrease in the fluid pressure at the inlet forces the stored fluid back into the system.
Generally, the pressure in the fluid chamber is equal to that of the system. But if the pressure at the accumulator inlet drops below the accumulator's preload pressure, the fluid chamber gets isolated from the system with the inlet valve. In this case, the spring is stopped against the hard stop and no longer affects the system behavior. If the pressure at the inlet builds up to the preload value or higher, the chamber starts being filled 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, represents a spring-loaded piston chamber. The piston motion is restricted by the hard stops, which limit the bladder expansion and contraction. 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:
$${q}_{F}=\frac{d{V}_{F}}{dt}$$
$${K}_{spr}=\frac{{V}_{0}}{{p}_{\mathrm{max}}-{p}_{pr}}$$
$${p}_{spr}={p}_{pr}+\frac{{V}_{F}}{{K}_{spr}}$$
$${p}_{HS}=\{\begin{array}{ll}\left({V}_{F}-{V}_{0}\right){q}_{F}{K}_{HS}\hfill & \text{for}{V}_{F}{V}_{0}\text{,}{q}_{F}0\hfill \\ -{V}_{F}{q}_{F}{K}_{HS}\hfill & \text{for}{V}_{F}0\text{,}{q}_{F}0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$
$${p}_{F}={p}_{spr}+{p}_{HS}$$
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 |
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_{max} | Pressure needed to fully fill the accumulator |
p_{spr} | Pressure developed by the spring |
K_{spr} | Spring gain coefficient |
p_{HS} | Pressure developed by hard stop in the piston-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 fluid volume parameter (V_{init}):
$${V}_{F}={V}_{init}$$
$${p}_{init}={p}_{pr}+\frac{{V}_{init}}{{K}_{spr}}$$
The Spring-Loaded 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 spring has linear characteristics.
No loading on the separator, such as inertia, friction, and so on, is considered.
Fluid compressibility is not taken into account.
Accumulator volumetric capacity. The default value is 8e3 m^3.
Pressure at which fluid starts entering the chamber. The default value is 10e5 Pa.
Pressure at which the accumulator is fully charged. The default value is 30e5 Pa.
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.