**1. Selection Method**

When selecting an accumulator, a number of parameters are involved, and the most important ones are listed below.

a. Working Pressure: P1 and P2

P1 means gas pressure at the minimum hydraulic pressure, while P2 means gas pressure at the maximum hydraulic pressure. For safety considerations, P2 must be lower than or equal to the maximum working pressure of the accumulator.

b. Work Fluid Quantity: ΔV

This parameter indicates the fluid quantity that could be stored or released, and it is needed to correctly determine the accumulator size.

c. Application Field

It is very necessary to make sure what kind of state the gas is in during operation, isothermal or adiabatic.

If the compression or expansion process is conducted slowly, say 3 minutes or more, the gas will keep its temperature at a relatively constant value, and this is isothermal state. Typical process includes pressure stabilization, volume compensation, counter balancing, lubrication circuit, etc.

Under other conditions, such as energy storage, pulsation dampening, shock absorbance and more, heat exchange is negligible as a result of the high transfer speed, so the gas is in adiabatic state. In addition, adiabatic process is the dominating one when compression or expansion process is less than 3 minutes.

d. Operating Temperature

The operating temperature determines what kind of materials should be applied to make bladders and steel shells, and it also has an influence over the pre-loading pressure as well as the accumulator volume.

e. Fluid Type

Different fluids requires different raw materials.

f. Maximum Required Flow Rate

The volume V0 and the size of the connection are related to response speed.

g. Working Location

It is important to know the eventual destination of the accumulator, and this is helpful to design an accumulator that can fully meet technical requirements for this location.

Based on the foregoing, it is possible to select an accumulator that best fits for the actual application.

**2. Gas Precharge Pressure: P0**

In order to obtain the optimum efficiency and maximum service life of accumulator and its components, the gas precharge pressure must be correctly selected. Theoretically speaking, when the gas precharge pressure P0 is as close as possible to the minimum working pressure P1, maximum fluid storage or release quantity is gained.

For actual applications, a safety factor should be specified, and its value should be in accordance with the following equation unless otherwise stated to avoid close of the hydraulic orifice during working: P0=0.9P1.

The limit values of P0 are: 0.25×P2≤P0≤0.9×P1.

Please be noted that there are some special values for different applications as listed below.

a. Piston accumulator: P0=0.95-0.97 P1 or P0=P1-(2~5bar)

b. Pulsation damper and shock absorber: P0=0.6-0.75 Pm or P0=0.8P1 where Pm stands for average working pressure

c. Hydraulic line shock damper: P0=0.6-0.9Pm where Pm means average working pressure with free flow

d. Accumulator + additional gas bottle: P0=0.95-0.97P1

P0 is valid for maximum operating temperature required by the user.

Usually, the checking or precharging process for an accumulator is carried out at a temperature that is different from the operating temperature T2, so P0 at checking temperature Tc becomes:

For example, for Tc=20°C, we have P0(20°C)=Po

Note: The accumulator is precharged with nitrogen at 20°C directly in the factory.

**3. Calculation Principle**

The compression and expansion of gas inside the accumulator are in accordance with the Boyle-Mariotte law, which can be expressed as:

P0×V0n=P1×V1n= P2×V2n

Fig. 12 illustrates the P-V relationship for the gas, where V0 refers to the nitrogen precharge volume in L at pressure P0. It is the maximum gas volume that could be stored in the accumulator, and it is equal to, or slightly lower than, the nominal capacity.

The meanings of the symbols in Fig. 12 are demonstrated below.

V1: Nitrogen volume in L at pressure P1

V2: Nitrogen volume in L at pressure P2

ΔV: Volume of discharged or stored liquid in L

P0: Precharge pressure in bar

P1: Minimum working pressure in bar

P2: Maximum working pressure in bar

n: Polytropic exponent

As a pressure-related function, the P-V curve is influenced by the polytropic exponent n, and typical values for n are listed below.

a. n=1: If the compression and expansion of nitrogen gas are conducted in a slow way, the entire heat exchange process will be carried out between the gas and the environment, indicating that there is no temperature change. So, this is isothermal process.

b. n=1.4: If the above behaviors are so quick that there is no heat exchange, this is adiabatic process.

These are theoretical but not practical conditions. However, it has been precisely and reasonably proven that when the accumulator works as volume compensator, leakage compensator or pressure compensator, this process is isothermal type, while for other applications, such as energy storage unit, pulsation damper, emergency energy reserve, water hammer absorber, shock absorber, hydraulic spring and more, it is adiabatic type.

If a much more accurate calculation is needed, it is possible to use intermediate values of n, and n is a function of t as shown in Fig. 13, where t means the expansion or compression time.

Note: In all calculations, pressures are in bar while temperatures are in K.

P-V Curve n-t Curve

**4. Volume Calculation (Isothermal)**

When n=1, the Boyle-Mariotte law could be simplified as:

P0×V0=P1×V1= P2×V2

So, V1=

, and the volume difference between the minimum and maximum working pressures represents the quantity of stored fluid.

Then, the volume difference could be further expressed as:

And the accumulator volume will be illustrated as:

Seen from this equation, we can find that the accumulator volume increases when ΔV is increasing, when P0 is decreasing, and when the difference between two operation temperature (P1 and P2) is decreasing.

**5. Volume Compensation (Isothermal)**

A typical example of volume calculation in isothermal condition is when an accumulator works as a volume compensator, and here is the equation.

ΔV=VT×(T2-T1)×(β－3α), where

VT: Piping volume in L

T2: Max. temperature in °C

T1: Min. temperature in °C

β: Cubic expansion coefficient of fluid in 1/°C

α: Linear expansion coefficient of piping in 1/°C

P1: Min. permissible operating pressure in bar

P2: Max. permissible operating pressure in bar

The necessary gas volume could be expressed like this:

**6. Leakage Compensation (Isothermal)**

The accumulator volume is specified as:

ΔV=Q1×t

P0=0.9×P1

P1: Min. permissible operating pressure in bar

P2: Max. permissible operating pressure in bar

**7. Volume Calculation (Adiabatic)**

Let,s start from the basic formula:

P0×V0n=P1×V1n= P2×V2n

As illustrated for isothermal condition, we can deduce the following equation that is valid for both compression and expansion in adiabatic condition :

The accumulator volume is affected by pressure as well as operating temperature.

Part 1: Temperature

During working, the operating temperature will change greatly, and this must be taken into consideration when calculating the volume. Here is their relationship.

Where

T2: t2(°C)+273=max. working temperature in K

T1: t1(°C)+273=min. working temperature in K

VO: Volume calculated neglecting thermal variation in L

VOT: Actual volume for thermal variation in L

Part 2: Pressure (High Pressure Correction Factor)

Under high pressure, the properties of nitrogen gas are far from those under ideal state. However, previous formulas are for ideal gas, so this fact must be taken in account when working pressure is equal to, or higher than, 200bar, no matter it is isothermal or adiabatic state.

The actual volume value could be expressed as:

VOT=VO×Ci× （isothermal）

VOT=VO×Ca× （adiabatic）

Correction Factor in Isothermal Condition Correction Factor in Adiabatic Condition

**8. Emergency Energy Storage**

Typical occasion is when storage is slow (isothermal) and discharge is fast (adiabatic), and gas volume is given by:

While stored volume is given by:

Where

n=1.4 (adiabactic coefficient for quick discharge phase)

nc=1-1.4 (polytropic coefficient for slow storage phase)

This value is a function of time, and it could be derived from Fig. 13. In most cases, nc could be assumed to be 1, and the calculation is simplified without affecting the result.

**9. Pulsation Compensation**

A typical calculation is in adiabatic condition due to the high speed storage and discharge. During calculation, please be noted that ΔV is related to the type and capacity of pump.

And the volume could be calculated like this:

Where

q: pump displacement (L) = A×C (piston area × stroke) = Q/N (flow rate/strokes)

P: Average working pressure of pump (bar)

P1=P-X (bar)

P2=P+X (bar)

α: Remaining pulsation (±%)

K: A coefficient that is related to the number of piston and the acting way of pump, and please refer to the following data.

Pump type K

1 piston, single acting 0.69

1 piston, double acting 0.29

2 pistons, single acting 0.29

2 pistons, double acting 0.17

3 pistons, single acting 0.12

3 pistons, double acting 0.07

4 pistons, single acting 0.13

4 pistons, double acting 0.07

5 pistons, single acting 0.07

5 pistons, double acting 0.023

6 pistons, single acting 0.07

7 pistons, double acting 0.023

**10. Shock Absorbance in Hydraulic Line**

Water hammer is a pressure surge caused when a fluid in motion is forced to stop or change direction suddenly. The level of overpressure is indicated by ΔPmax, and it is generated when valve is closed. Moreover, it is affected by the pipe length, flow velocity, fluid density, and the valve shut down time as shown in the formula below:

The accumulator volume required to reduce and limit shock pressure within allowed ΔP is:

Where

V0: Accumulator volume (litre)

Q: Flow rate in the pipe (m3/h)

L: Total length of pipeline (m)

γ: Specific gravity of liquid (kg/m3)

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

S: Internal sectional area of pipe (mm2)=0.25×πd2

d: Internal diameter of pipe (mm)

ΔP: Allowable overpressure (bar)

P1: Operating pressure by free flow (absolute bar)

P2: Max. allowable pressure (absolute bar)= P1+ΔP

t = Deceleration time (s) (valve shut down time)

**11. Accumulator + Additional Gas Bottle (Transmission)**

If the pressure difference between P1 and P2 is small but a lot of fluid is needed, V0 should be higher than ΔV. Under this condition, it is very convenient to use additional gas bottle to increase nitrogen quantity.

Volume calculation is related to practical application, and whether it is isothermal or adiabatic process should be taken into consideration. Also, the influence of temperature can,t be neglected. To get maximum efficiency, the precharge pressure should be of a higher value.

This means the fluid volume and the volume change caused by temperature must be less than 75 percent of the accumulator volume.

The volume of gas bottle is determined by this:

VOB =VOT －VOA

Where

VOA: Accumulator volume

VOB: Additional cylinder capacity