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Hydraulics, open channels and fluid power
Usable oil volume from a gas-charged accumulator
Oil delivered as the pressure falls from ph to pl in an accumulator whose gas volume is V₀ at precharge p₀ (all absolute), for an ideal gas with polytropic exponent n and negligible losses.
- Ref
- HYD-50
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- ΔV [m³]
A gas-charged accumulator holds gas of volume V₀ at the precharge pressure p₀ when no oil is inside. As the oil pressure changes, the gas follows p Vⁿ = constant, so at pressure p the gas volume is V₀(p₀/p)^(1/n). The usable oil delivered as pressure falls from ph to pl is the difference of the two gas volumes. Use absolute pressures, with p₀ ≤ pl < ph. n = 1 is isothermal (slow) and n ≈ 1.4 adiabatic (fast).
Oil delivered as the pressure falls from ph to pl in an accumulator whose gas volume is V₀ at precharge p₀ (all absolute), for an ideal gas with polytropic exponent n and negligible losses.
Calculator
- Gas volume at the lower pressure, V₀(p₀/pl)^(1/n)0.008333 m³
- Gas volume at the upper pressure, V₀(p₀/ph)^(1/n)0.005 m³
- Usable oil as a share of V₀0.3333
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The accumulator delivers 0.003333 m³ (3.333 litres), 33.3 % of its gas volume, as the pressure falls from 100 to 60 bar.
- A fast discharge (n = 1.4) gives less oil than a slow one (n = 1), so size the accumulator with the exponent suited to the duty time.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| V0 Gas volume at precharge | 1e-4 to 1 m³ | 3.333e-5 | 0.3333 | 0.3333 | +10% | +10 % |
| pl Lower working pressure (absolute) | 5,000,000 to 9,000,000 Pa | 5.556e-4 | 0.005 | 0.004444 | +10% | −22.7 % |
| ph Upper working pressure (absolute) | 1e7 to 5e7 Pa | 0.003333 | 0.007333 | 0.004 | +10% | +13.6 % |
| p0 Precharge pressure (absolute) | 100,000 to 5,000,000 Pa | 6.667e-5 | 0.003333 | 0.003267 | +10% | +10 % |
| n Polytropic exponent | 1 to 1.4 | 0.002684 | 0.003333 | 6.495e-4 | +10% | −5.58 % |
Sweep of Lower working pressure (absolute)
| Lower working pressure (absolute) (Pa) | Usable oil volume (m³) |
|---|---|
| 5,000,000 | 0.005 |
| 5,500,000 | 0.004091 |
| 6,000,000 | 0.003333 |
| 6,500,000 | 0.002692 |
| 7,000,000 | 0.002143 |
| 7,500,000 | 0.001667 |
| 8,000,000 | 0.00125 |
| 8,500,000 | 8.824e-4 |
| 9,000,000 | 5.556e-4 |
Sweep one variable
Variables and how each one contributes
Usable oil volume
What the equation gives you. Everything else feeds it.
Gas volume at precharge
The gas volume at the precharge pressure, in m³. Raises the usable volume in direct proportion. Raising it by 10 % (to 0.011) moves the result by +10 %, from 0.003333 to 0.003667 m³.
Precharge pressure (absolute)
The precharge pressure (absolute), in Pa. A higher precharge raises the usable volume for the same working range. Raising it by 10 % (to 5,500,000) moves the result by +10 %, from 0.003333 to 0.003667 m³.
Lower working pressure (absolute)
The lower working pressure (absolute), in Pa. A higher lower limit gives less usable oil. Raising it by 10 % (to 6,600,000) moves the result by −22.7 %, from 0.003333 to 0.002576 m³.
Upper working pressure (absolute)
The upper working pressure (absolute), in Pa. A higher upper limit gives more usable oil. Raising it by 10 % (to 1.1×107) moves the result by +13.6 %, from 0.003333 to 0.003788 m³.
Polytropic exponent
The polytropic exponent: 1 for slow (isothermal), up to 1.4 for fast (adiabatic). A larger exponent means a stiffer gas, so less oil is delivered. Raising it by 10 % (to 1.1) moves the result by −5.58 %, from 0.003333 to 0.003147 m³.
Worked example with the values above
- Gas at pl: 0.01 × (5,000,000/6,000,000)^(1/1) = 0.0083333 m³.
- Gas at ph: 0.01 × (5,000,000/1×107)^(1/1) = 0.005 m³.
- ΔV = 0.0083333 − 0.005 = 0.0033333 m³.
Assumptions and limits
- A single-phase liquid in a uniform line, with a linear elastic wall and the wave travelling without loss.
- Rapid closure, with the valve moving in less than the wave return time.
Common mistakes
- Using gauge pressures in a gas-law formula.
- Using a precharge above the lower working pressure.
- Using the isothermal exponent for a fast discharge.
Statistics and uncertainty
Gas behaviour sits between isothermal and adiabatic, so the exponent matters: the usable volume from a quick discharge can be 20 % to 30 % lower than from a slow one.
Where it is used
- Checking pipelines and pump stations for water hammer.
- Sizing surge tanks and gas-charged accumulators.
Maths you need
- Water hammer: a pressure wave from a sudden velocity change
- Bulk modulus: B, the stiffness of the liquid
- Polytropic process: p Vⁿ = constant