The application of hydropneumatic accumulators in the field of energy storage is leading to the use of higher pressures (well above 200 bar). The performances and predictions of the restored volume cannot be obtained with the use of the classical equation of perfect gas and with the adiabatic coefficient equal to 1.4 (Boyle-Mariotte's Law). In practice, for applications above 250 bar, this results in over sizing of accumulators capacity, to an error of more than 10%.

For this reason, a computer model has been developed in order to predict the performances. Theoretical and experimental results are presented to show the effect of real gas, the effect of charging and discharging, and evolution of stabilisation overtime. Thus, the efficiency with which the accumulator stores energy can be estimated more accurately when calculating the overall efficiency of the system in which they are designed to operate.

Introduction

The performances calculations for hydraulic accumulators are difficult to carry out in applications where the pressures is greater than 210 bar, with a sufficient accuracy if based on the assumptions of a perfect gas.

Furthermore, in practice neither the storage of energy nor the complete discharge correspond to the two ideal states :
- isothermal, where the thermal exchange between the gas and the oil is such that the gas temperature remains constant.
-adiabatic, where no exchange of heat takes place between the gas and oil in the case of very rapid cycles.
The calculated levels of stores energy or released energy for these states are very different.

In practice, the operating model of the accumulator can be considered as polytropic and it is necessary to provide, in the calculation, a method of defining the energy transfer flow between the gas and the oil.

This is why we have developed a method of calculation which takes into account the characteristics of a real gas (generally nitrogen) and the thermal exchange between gas and oil.

Before describing the principles of the calculation method and the equations used we will identify the important differences that exist between the characteristics of perfect and real gases.

Notation

P - gas pressure
V - gas volume
T - gas temperature
m- mass of gas
h - heat exchange coefficient
S- exchange surface
N - polytropic coefficient
- adiabatic coefficient
Cv - specific heat (constant volume)
Cp - specific heat (constant pressure)

Indices

0 - precharge condition
1 - initial state
2 - final state
g - gas
o - oil

Function

Perfect gas and real gas

a) Equation of state:
The equation of state for a perfect gas (Boyle-Mariotte's Law) is defined by the relation

When applied to the physical characteristics of nitrogen, this law gives a deviation greater than 10% for pressures around 250 bar.

Such differences have to be taken into account at higher pressures and a number of authors have proposed relationships to permit a description of the effect of real gases in certain conditions, of pressure and temperature.(Van der Waals, Beattie Bridgeman 1927, Benedict, Webb and Rubbin 1940).

We have chosen the Beattie Bridgeman relationship BB. This gives satisfactory results for pressures of 0 to 600 bar and for which the numerical solution does not noticeably extend the time of calculation of the simulation model. The Benedict, Webb & Rubbin Law (BWR) is nevertheless more precise, especially at the higher pressures, but is very much more difficult to resolve numerically. As a general rule, the factors of compressibility obtained with BB Law will be lower than the real values (5% at 500 bar).

 

 

b) Transformation gas law:

The adiabatic transformation of a perfect gas is generally described by the relation

The adiabatic coefficient is defined by the ratio of the specific heats and is very sensitive to pressure and temperature, and can reach values much greater than 1.4.

for example

= 1.71 at 400 Bar and 20ºC(nitrogen)

This is why we must consider using energy conservation equation as
- first, the internal energy of the gas, as the specific heat is not constant.
- secondly, the transfer of energy between oil and gas (and vice versa),

The results obtained on the calculated performances from the assumption of a perfect gas (Boyle's law and coefficient of 1.4) and a real gas (BB law and energy conservation equation) has been presented in the case of a very rapid discharge which can be considered as adiabatic.

 

 

Method Principle

From the known initial state P1, V1, T1, we calculate as functions of time the evolution of each of the characteristic parameters of the gas P(t), V(t) and T(t) until the final instant of P2, V2 and T2 determined from the criteria of volume or pressure at the end of the cycle.
These three unknowns are obtained by solving the three equations below

- Equation of state for a real gas (Beattie Bridgeman)

P.V. = R.T.f (T,V) + g(T,V)
and Cv, P = K(P,T)

- Energy conservation of the gas-oil system

- Law of discharge or charging flow

The solution of these equations for P, V, and T, integrated at a constant step time between the initial instant t1, (where P1, V1 and T1 are known) and t1 (where P2 or V2 are only known).
The simulation programme is written in FORTRAN and used on a micro-computer.

a) Energy conservation

The variation of internal energy, obtained during expansion or compression of the gas, is written in the way below

where:

The term 1 is work provided externally to the accumulator system
The term 2 is energy variation due to transfer of heat between gas and oil
The term 3 expressed for a real gas in the way below

which reduces to :

 

b) Energy exchange flow

In effect, one has to consider the gas-oil system below:

The calorific capacity of the oil is such that we can assume that its temperature will not change during the phases of compression or expansion and that thermal conductivity of the elastomeric vessel (bladder), of negligible thickness, is so great that the internal temperature of the surface is little different to that of the oil.
The transfer of energy, resulting from the effect of natural convection and conduction, can then be described in the following way.

Calculation of the exchange coefficient can be deduced from using the Nusselt Number Nu

Nu = h. L/K

where :

L = characteristic thickness of the exchange

K = thermal conductivity coefficient of the gas

This number is non-dimensional, being an experimental characteristic, in the case of natural convection, by Polhausen. the viscosity and the thermal conductivity of the gas have important features in the estimation of h, as well as the height of the gas enclosure (bladder).

The only important parameter that is likely to affect the results of the performance calculations of the accumulator is therefore the coefficient of thermal exchange h, between oil and gas.

This overall method of calculation allows the determination of the development of pressure, volume and temperature as a function of time, and of the energy actually stored or released during that period of the cycle.

The next important phase of the cycle is the calculation of the development of pressure, volume and temperature as a function of time, and of the energy actually stored or released during that period of the cycle.

The next important phase of the cycle is the calculation of the development of P, V and T during the stabilization of the accumulator where no work is done on the gas. This aspect of the calculation is very much more important as it allows the definition of the initial conditions for the next phase of the cycle. The whole calculation, then, permits the estimation of the overall performance of the accumulator.

Furthermore, we can take account of the heat exchanged between the shell of the accumulator and the external environment. These effects are not significant in the case of rapid cycling. However, if the pressure plateaux are long, these differences become significant, in particular where the temperature difference between shell and ambient air is important.

Accumulator Optimisation

This method of calculation makes it possible to apply the characteristics of real gases to the practical operation conditions of an accumulator and of its own characteristics. It permits the determination of the levels of energy stored or discharged an of the overall efficiency of the accumulator. Furthermore, it is possible to estimate the value of the polytropic coefficients N for the cycle under consideration, whether for charge or discharge, and to calculate the performances of these given accumulators in the prescribed conditions using the classical relationship.
This study on the behavior of accumulators in energy storage suggests the following conclusions :

- the performances of accumulators depend heavily on the value of the polytropic coefficient N(generally somewhere between 1 and ), it is important to select the most favorable conditions to reduce as far as possible this coefficient and thus to increase the amount of energy which can be stored per unit of volume.

- we can influence the value of N in three ways :

1st - choice of gas
2nd - accumulator construction
3rd - optimisation of working conditions (oil temperature, precharge pressure...)
The last way is relatively easy to arrange but the first two are more difficult and fundamental.

a) Choice of gas

The choice of gas can be seen in two different ways :
- the use of a gas which has physical properties such that the adiabatic coefficient is inferior to that of nitrogen at the working pressures and temperature usually encountered, which permits an improved performance in adiabatic conditions.
- the choice of a gas for which the compressibility characteristics are nearer to the perfect gas permits an improvement of the results in the case of isothermal conditions or for a small compression ratio.
- a compromise between these two aspects depends on the type of application, pressure an temperature, but other considerations relative to the phenomenon of permeability across the bladder wall or the risks of combustion with the oil need to be taken into account and so complicate the selection.

b) Accumulator construction

The heat exchange surface between gas and oil is an important factor for lowering the value of N. It is for this reason that the surface area and the height should be as large as possible in relation to the volume of the accumulator.

Conclusion

The effects of the real gas and the thermal exchange between the gas and the oil in hydraulic accumulators, can according to the conditions of use, lead to important differences in their performances, based on a theory more "real", permits a better understanding of their efficiency of energy storage and their inclusion more effectively in the calculation of the overall performances of the system for which they are intended.

This allows for quick answers for any sort of applications encountered, and gives, with a reasonable accuracy, most of the information required to determine the accumulator's size. OLAER Fawcett Christie have collaborated on a number of different projects for which information has been given to complete the Computer Numeric model of the overall hydraulic system using accumulators. These have been, mostly, in the sector of energy saving for mobile applications.