1. Selection Method

Many parameters need to be considered when selecting an energy accumulator, the most important of which are as follows:

1) Minimum pressure P1 and maximum pressure P2 – working pressure.

For safety reasons, the P2 value must be lower than or equal to the maximum operating pressure specified by the selected accumulator.

2) The volume of liquid that can be stored or utilized, ΔV.

In order to accurately determine the size of the accumulator, this data is needed in addition to the maximum and minimum pressures.

3) Methods and Applications

It is important to determine whether the gas in operation is in an isothermal or adiabatic state.

For example, if compression (or expansion) occurs slowly (approximately 3 minutes), the gas maintains a roughly constant temperature; this state is isothermal. (Examples: pressure stability, volume compensation, counterweights, lubrication lines). In other cases (energy storage, pulsation buffering, shock buffering, etc.), due to the high transmission speed, heat exchange is negligible; therefore, this state is adiabatic. When the compression or expansion time is less than 3 minutes, this adiabatic condition will exist as a precursor.

4) Operating temperature

The operating temperature determines the choice of capsule material and steel shell material, and also affects the initial load pressure, as well as the accumulator volume.

5) Types of liquids

This will determine the choice of materials.

6) Requires a large flow rate

The volume Vo and the connector specifications are related to the reaction rate.

7) Location of use

It is important to understand where the energy storage device will be used in order to ensure that the design meets the requirements of local design and testing parameters.

Based on the above information, a suitable energy storage device can be selected for the required specific purpose.

2. Pre-inflation pressure

Choosing the correct pre-charge pressure is fundamental to achieving optimal efficiency and maximum service life for the accumulator and its components. Theoretically, the maximum liquid storage (or release) can be achieved when the pre-charge pressure Po is as close as possible to the minimum operating pressure.

A safety factor should be provided in practical applications. To prevent the valve from closing during operation, this value (unless otherwise specified) is:

The limiting values ​​of Po are: Po = 0.9P1, Po min ≥ 0.25 × P2, Po max ≤ 0.9P1

Special values ​​are used for:

Ø Piston accumulator Po=0.95-0.97 P1

Or Po = P1 - (2 to 5 bar)

Ø Pulsation buffering and vibration reduction Po=0.6-0.75 Pm or Po=0.8P1

Where: Pm = average working pressure

Ø Liquid Buffer

Po=0.6-0.9Pm

Where: Pm = Average working pressure in free flow state

Ø Accumulator + Auxiliary Gas Cylinder

Po=0.95-0.97P1

The Po value is applicable to the maximum operating temperature required by the user.

The accumulator is typically tested or pre-charged at a temperature different from the operating temperature T2. Thus, the Po value at the test temperature Tc becomes:

Poc=After 

If Tc = 20℃, then: Po(20℃) = Po

Note: The pre-charge pressure of the accumulator is directly achieved by the factory at a temperature of 20°C. The charging gas is nitrogen.

3. Calculation Principle

The compression and expansion of the gas inside the accumulator are based on Boyle-Mariotte's law of state changes in an ideal gas:

Po×Von=P1×V1n= P2×V2n

The PV diagram in Figure 12 illustrates the "pressure-volume" relationship of the accumulator.

In the figure: Volume of pre-charged nitrogen (liters) when Vo = pressure Po

This is the maximum gas volume that the accumulator can store, which is equal to or slightly lower than its rated capacity.

V1 = Volume of nitrogen gas at pressure P1 (liters)

V2 = Volume of nitrogen gas at pressure P2 (liters)

△V = Volume of liquid discharged or stored (liters)

Po = Pre-charge nitrogen pressure (bars)

P1 = Minimum working pressure (bars)

P2 = Maximum working pressure (bars)

n = polytropic index

As a function of pressure, the volume change curve is related to powers of n, with the powers of nitrogen falling between the following limits:

If the compression and expansion of nitrogen occur very slowly, such that the entire heat exchange process takes place between the gas and the surrounding medium, this is called isothermal; the state is called isothermal.

n = 1.4 If the operating speed is so fast that heat exchange does not occur, the state is adiabatic.

These are theoretical states, not actual states.

However, it can be reasonably and accurately determined that when the accumulator is used as a volume compensator, leakage compensator, or lubrication compensator and pressure compensator, the state is isothermal.

In other applications, such as energy storage devices, pulsation dampers, emergency power sources, dynamic pressure compensators, shock absorbers, hydraulic springs, etc., the condition should be determined to be adiabatic.

For more precise calculations, intermediate values ​​of n can be used, where n is a function of t. As shown in the curve in Figure 13, n is a function of the expansion or contraction time.

Note: In the calculations, pressure is expressed in absolute pressure "bar"; temperature is expressed in "Kelvin".

4. Volume Calculation (Isothermal)

When n = 1, the Boyle-Mariotte law becomes:

Po×Vo=P1×V1= P2×V2  

Therefore, V1 = Vo × ; V2 = Vo × ;

The difference between the volumes V1 (at minimum working pressure) and V2 (at maximum working pressure) gives the amount of liquid stored.

△V=V1－V2=Vo －Vo

Therefore, △V = Vo( )

The accumulator volume will be:

In= ＝

This means that when ΔV increases, Po decreases, and the difference between the two working pressures P1 and P2 decreases, the volume of the accumulator increases.

 5. Volumetric compensation (isothermal)

Using an accumulator for volume compensation is a typical example of calculations performed under isothermal conditions.

△V= VT×（T2－T1）×（β－3α）

In the formula:

VT = Pipe volume (liters)

T2 = Maximum temperature (°C)

T1 = Minimum temperature (°C)

β = Coefficient of volumetric expansion of the fluid (1/℃)

α = Linear expansion coefficient of the pipeline (1/℃)

P1 = Minimum allowable working pressure (bar)

P2 = Maximum permissible working pressure (bar)

The necessary gas volume is:

In=

6. Leakage compensation (isothermal)

Calculation of accumulator volume:

ΔV=Q1×t

Po=0.9×P1

P1 = Minimum allowable working pressure (bar)

P2 = Maximum permissible working pressure (bar)

In =

7. Volume Calculation (Insulation)

Start with the basic formulas

Po×Von=P1×V1n= P2×V2n

Similar to isothermal calculations, we obtain the following formula for calculating the volume in an adiabatic state:

ΔV = Vo =0.7143

In =

This formula is valid in adiabatic conditions during either the expansion or compression phase.

The volume of an energy storage device is affected not only by pressure but also by operating temperature.

The effect of temperature

The operating temperature can vary significantly during operation, and this should be taken into account when calculating the volume.

The relationship between temperature and volume is

   VOT=VO  ；   

In the formula

T2 = t2 (°C) + 273 = Maximum operating temperature (°K)

T1 = t1(°C) + 273 = Minimum operating temperature (°K)

VO = Volume (liters) calculated ignoring temperature changes.

VOT = Actual volume (liters) after taking temperature changes into account

High voltage correction factor

Under high pressure, the industrial nitrogen used in the accumulator is far from ideal, and the formula involves ideal gases. Therefore, this fact must be taken into account when the working pressure P2≥200bar, whether in an adiabatic or isothermal state.

The actual volume VOT value becomes:

VOT = VO × Ci ×  (isothermal)

VOT = VO × Ca ×  (adiabatic)

Isothermal condition

8 Emergency Energy Storage

Typically, storage occurs very slowly (isothermal) while venting is very fast (adiabatic). The gas volume is derived from this formula:

In =

Storage capacity is derived from this formula:

ΔV = Vo

In the formula:

n = 1.4 Adiabatic coefficient (rapid emission stage)

nc = 1-1.4 polyvariate coefficient (slow emission stage)

This value is a time function, which can be derived from the curve in Figure 13.

In most cases, we can assume nc = 1, which simplifies the calculation without affecting the result.

 9 Absorption Pulsations

Typical calculations are performed under adiabatic conditions of high-speed storage and discharge.

When calculating, the liquid volume ΔV to be considered is related to the pump model and the pump's working volume.

The volume is:

In =

In the formula:

q = Pump operating volume (liters)

=A×C(piston area×stroke)=Q/N(flow rate/stroke)

P = Average operating pressure of the pump (bar)

P1=P-X(bar)

P2=P+X(bar)

α = Residual pulsation ± (%)

K = This is a coefficient that is related to the number of pistons and the pump's operating method.

       Pump model K

One piston, single-acting, 0.69    

One piston, double-acting, 0.29   

Two pistons, single-acting, 0.29   

Two pistons, double-acting, 0.17   

3 pistons, single-acting, 0.12   

3 pistons, double-acting, 0.07   

Four pistons, single-acting, 0.13   

4 pistons, double-acting, 0.07   

5 pistons, single-acting, 0.07   

5 pistons, double-acting, 0.023  

Six pistons, single-acting, 0.07   

7 pistons, double-acting, 0.023  

10. Hydraulic pipelines absorb shock.

A rapid increase or decrease in fluid volume causing a rapid increase in pressure is called water hammer.

The degree of overpressure is represented by ΔPmax, which occurs when a valve in a pipeline is closed and is affected by the pipeline length, flow velocity, fluid density, and valve closing time. It is calculated using the following formula:

ΔPmax(bar) =

Within a specified range of ΔP values, the required accumulator volume to reduce the impact pressure is given by the following formula:

In=

In the formula:

Vo = Volume of the accumulator (liters)

Q = Flow rate in the pipeline (m³/h)

L = Total length of the pipeline (m)

γ = Specific gravity of the liquid (kg/m3)

V = Flow velocity (m/s) = 1000Q / 3.6S

S = Internal cross-sectional area of ​​the pipe (mm²) = 0.25 × πd²

d = Pipe inner diameter (mm)

ΔP = Permissible overpressure (bar)

P1 = Free-flowing working pressure (absolute pressure, bar)

P2 = Maximum allowable pressure (absolute pressure, bar) = P1 + ΔP

t = deceleration time (valve closing time) (s)

11. Accumulator + Additional Gas Cylinder (Transfer)

If the pressure difference between P1 and P2 is small but a large amount of liquid is required, the synthesized volume Vo should be greater than ΔV.

In this situation, increasing the amount of nitrogen in the gas cylinder is a very convenient thing to do.

The calculation of volume depends on the application method and should take into account whether it isothermal or adiabatic conditions.

And the effect of temperature. To obtain maximum efficiency, a higher pre-charge pressure is used. This is determined by the following formula: P0 = 0.95 - 0.97P1   

This means that the volume of the liquid plus the volume change caused by temperature must be less than 0.75 of the accumulator volume. 

The volume of the gas cylinder is determined based on the difference.

VOB = VOT － VOA

In the formula:

VOA = Volume of accumulator

VOB = Volume of the additional gas cylinder

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