Mathematical Models of Gas in Hydropneumatic Accumulators Used in Numerical Tests of Drive Systems with Energy Recovery

1. Introduction

In recent years, new-generation energy-saving drive systems with energy recovery have been discussed in numerous articles, and hydropneumatic accumulators have been chosen as secondary sources of energy in many of them [1,2,3]. Although the initial study on the efficiency of these drive systems is usually carried out by computer simulation, various models have been used to describe the behavior of gas in the hydraulic accumulator in the studies discussed, for example, in [2,3,4]. It is not clear whether the results obtained with various simulation models of hydropneumatic accumulators can be compared to each other. The magnitude of errors resulting from simplifications in these models remains unclear. Despite multiple studies on this problem, for example, presented in [5,6,7,8], the knowledge of this topic has not yet been clearly summarized. In many studies, various mathematical models of gas in the hydropneumatic accumulator have been compared based on the pressure–volume curves recorded during the accumulator testing [5,6]. However, in numerous drive systems, efficiency is of primary importance and is also a key parameter to assess the energy recovery performance.

The objective of this paper is to solve the issues mentioned above by reviewing the state-of-the-art models of gas in a hydropneumatic accumulator in terms of how these models affect the error of a numerical simulation of the efficiency of energy recovery in a drive system. This review will result in practical recommendations on the model most suitable for simulation testing of drive systems with energy recovery. This study is limited to the mathematical models most commonly used to simulate drive systems, including models delivered by engineering software. The energy losses related to the resistance of hydraulic oil to flow are excluded from this study.

2. Energy Balance of a Hydraulic Accumulator

According to the literature, the equation for the energy balance for a hydraulic accumulator is usually derived from the first law of thermodynamics given by Equation (1)

where U [J] is the internal energy of gas, Q [J] is heat, and W [J] is work.

The increment of work performed by the gas in the hydraulic accumulator is given by Equation (2)

where P [Pa] is gas pressure and V January is gas volume.

Equations (3)–(5) can be used to describe the increment of internal energy of the gas.

$d U = {(\frac{δ U}{δ T})}_{V} \cdot d T + {(\frac{δ U}{δ V})}_{T} \cdot d V$

(3)

$d U = {(\frac{δ U}{δ T})}_{P} \cdot d T + {(\frac{δ U}{δ P})}_{T} \cdot d P$

(4)

$d U = {(\frac{δ U}{δ P})}_{V} \cdot d P + {(\frac{δ U}{δ V})}_{P} \cdot d V$

(5)

The most commonly used equation is Equation (3). When Equation (3) is used, the final energy balance formula does not involve the time derivative of pressure. On the other hand, the time derivative of volume is eliminated using Equation (4).

The energy balance is represented by Equation (6) when Equations (2) and (3) are substituted into Equation (1), as well as some common state-of-the-art thermodynamic equations.

$\frac{d Q}{d t} = c_{V} \cdot m_{g} \cdot \frac{d T}{d t} + T \cdot {(\frac{δ P}{δ T})}_{V} \cdot \frac{d V}{d t}$

(6)

where c_V [J∙kg⁻¹∙K⁻¹] is the specific heat capacity at a constant volume, m_g [kg] is the mass of gas, T [K] is gas temperature, and t [s] is time.

The energy balance can also be expressed with Equation (7) when Equations (2) and (4) are substituted into Equation (1) together with the common thermodynamic equations.

$\frac{d Q}{d t} = c_{P} \cdot m_{g} \cdot \frac{d T}{d t} - T \cdot {(\frac{δ V}{δ T})}_{P} \cdot \frac{d P}{d t}$

(7)

where c_P [J∙kg⁻¹∙K⁻¹] is the specific heat capacity at a constant pressure.

In this study, Equation (6) is selected for further simulation.

Equations (6) and (7) are correct if the thermodynamic processes running in the hydraulic accumulator are reversible. The reversibility of the process depends primarily on the ratio of the velocity of the accumulator piston to the mean square velocity of the gas particles. The so-called Process Reversibility Factor is a measure of reversibility and can be evaluated with a formula derived in [9] for the ideal gas and adiabatic process. For example, for an accumulator filled with ideal nitrogen at a temperature of 23 °C, whose piston moves the velocity of 0.5 m/s, the Process Reversibility Factor is 1.0047. Since this is close to 1, the process is almost fully reversible. Therefore, it can be assumed that the processes running in some hydraulic accumulators are reversible.

3. Models of Gas State Used in Modeling of Hydraulic Accumulators

The temperature derivative of pressure and the temperature derivative of volume are involved in Equations (6) and (7), respectively. These derivatives must be calculated with some gas equation of state. Although more than a hundred real gas equations of state have already been developed [10,11,12], only a few of them are commonly used to model hydraulic accumulators. The equations most commonly used in practice, including the generally available scientific studies and engineering software, are listed below in (a) through (e). Therefore, these will be evaluated herein in terms of their effect on the results of simulation computations.

(a): Ideal gas law by Clapeyron (1834)
(b): van der Waals real gas equation of state (1873)
$(P + \frac{a_{b} \cdot n^{2}}{V^{2}}) \cdot (V - n \cdot b_{b}) = n \cdot R \cdot T$
(9)
(c): Benedict–Webb–Rubin (BWR) real gas equation of state (1940) [13]
$P = \frac{n \cdot R \cdot T}{V} + \frac{n^{2} \cdot (B_{c} \cdot R \cdot T - A_{c} - \frac{C_{c}}{T^{2}})}{V^{2}} + \frac{n^{3} \cdot (b_{c} \cdot R \cdot T - a_{c})}{V^{3}} + \frac{n^{6} \cdot a_{c} \cdot α_{c}}{V^{6}} + \frac{n^{3} \cdot c_{c} \cdot (1 + \frac{n^{2} \cdot γ_{c}}{V^{2}}) \cdot e^{- \frac{n^{2} \cdot γ_{c}}{V^{2}}}}{V^{3} \cdot T^{2}}$
(10)
(d): Redlich–Kwong–Soave (RKS) real gas equation of state (1972)
$(P + \frac{n^{2} \cdot a_{d} \cdot a (T)}{V \cdot (V + n \cdot b_{d})}) \cdot (V - n \cdot b_{d}) = n \cdot R \cdot T$
(11)
$a (T) = {[1 + k_{d} \cdot (1 - \sqrt{\frac{T}{T_{C}}})]}^{2}$
(12)
(e): RKS equation with the Ghanbari–Check amendment (2012) [14]
$a (T) = {[1 + {D_{e} \cdot k}_{d} \cdot (1 - \sqrt{\frac{T}{T_{C}}})]}^{2}$
(13)
$D_{e} = \{\begin{matrix} 1 w h i l e \frac{T}{T_{c}} \leq 1 \\ \frac{11}{15} w h i l e \frac{T}{T_{c}} > 1 \end{matrix}$
(14)

In Equations (8)–(14), the following nomenclature is used: n [mol] is the amount of gas, R [J∙mol⁻¹∙K⁻¹] is the ideal gas constant, T_c [K] is the critical temperature of nitrogen, and a_b, b_b, a_c, b_c, c_c, A_c, B_c, C_c, α_c, a_d, b_d, k_d, and γ_c are the model parameters that are determined by experiments for nitrogen and can be found in the reference tables for chemistry; in this study, parameters a_b through γ_c have been evaluated based on [15,16]. The Ghanbari–Check amendment to the RKS real gas equation of state is related to the master formula given by Equation (11).

In many studies identified by the author, the BWR and RKS real gas equations of state are claimed sufficient for typical engineering calculations. However, no specific numerical results or theoretical considerations have proved this recommendation or clarified its range of applicability.

Apart from the so-called limited applicability gas equations of state outlined above, the so-called universal gas equations of state are available in the literature, including Equation (15) [17]. Equation (15) derives from the results of experimental tests and includes thirty-three parameters: A1 through A32 and γ. Equations of this type are called the virial equations. They are hardly used for engineering because of their high complexity.

$P = \frac{n \cdot M \cdot R \cdot T}{V} + {(\frac{n \cdot M}{V})}^{2} \cdot [A_{1} \cdot T + A_{2} \cdot T^{0.5} + \sum_{i = 3}^{5} (A_{i} \cdot T^{3 - i})] + {(\frac{n \cdot M}{V})}^{3} \cdot \sum_{i = 6}^{9} (A_{i} \cdot T^{7 - i}) + {(\frac{n \cdot M}{V})}^{4} \cdot \sum_{i = 10}^{12} (A_{i} \cdot T^{11 - i}) + {(\frac{n \cdot M}{V})}^{5} \cdot A_{13} + {(\frac{n \cdot M}{V})}^{6} \cdot (A_{14} \cdot T^{- 1} + A_{15} \cdot T^{- 2}) + {(\frac{n \cdot M}{V})}^{7} \cdot A_{16} \cdot T^{- 1} {+ (\frac{n \cdot M}{V})}^{8} \cdot (A_{17} \cdot T^{- 1} + A_{18} \cdot T^{- 2}) + {(\frac{n \cdot M}{V})}^{9} \cdot A_{19} \cdot T^{- 2} + [{(\frac{n \cdot M}{V})}^{3} \cdot (A_{20} \cdot T^{- 2} + A_{21} \cdot T^{- 3}) + {(\frac{n \cdot M}{V})}^{5} \cdot (A_{22} \cdot T^{- 2} + A_{23} \cdot T^{- 4}) {+ (\frac{n \cdot M}{V})}^{7} \cdot (A_{24} \cdot T^{- 2} + A_{25} \cdot T^{- 3}) {+ (\frac{n \cdot M}{V})}^{9} \cdot (A_{26} \cdot T^{- 2} + A_{27} \cdot T^{- 4}) {+ (\frac{n \cdot M}{V})}^{11} \cdot (A_{28} \cdot T^{- 2} + A_{29} \cdot T^{- 3}) {+ (\frac{n \cdot M}{V})}^{13} \cdot (A_{30} \cdot T^{- 2} + A_{31} \cdot T^{- 3} + A_{32} \cdot T^{- 4})] \cdot e^{- γ \cdot {(\frac{n \cdot M}{V})}^{2}}$

(15)

4. Specific Heat Capacity at a Constant Volume

The specific heat capacity is used in the energy balance equation for a hydraulic accumulator, where c_p and c_V denote the specific heat capacity at a constant pressure and constant volume, respectively. Both of these parameters are constant for ideal gas; however, in real gas, they depend on the temperature and pressure. In this paper, the energy balance equation is given by Equation (6), which only involves the specific heat capacity at a constant volume. Consequently, the methods for modeling the specific heat capacity at a constant volume are only analyzed below.

In the literature, one of the following approaches is usually adopted to model the specific heat capacity in the thermodynamic simulation models of hydraulic accumulators; the specific heat capacity can be (a) represented by a constant value that usually corresponds to a specified gas temperature and pressure, (b) calculated with a formula fitted to experimental data [18,19], or (c) calculated using a gas equation of state.

The approach referred to in (a) can be used when there are minor fluctuations in gas temperature and pressure in the hydraulic accumulator.

The method outlined in (b) does not significantly improve the accuracy. In practice, the influence of the gas temperature and pressure on the specific heat capacity of the gas is governed by a complex relationship that cannot be precisely captured by a typical mathematical function fitted to the experimental data with a generally available numerical algorithm. Figure 1 illustrates a discrepancy between the results of experimental testing and the predictions made with various mathematical models fitted to the experimental data.

The approach mentioned in (c) is the preferred one for high-precision simulation. It has been implemented, for example, by Siemens in Simcenter Amesim 2021.1 software [20]. In general terms, the instantaneous specific heat capacity at a constant volume can be calculated on the basis of a gas equation of state, as shown in Equation (16)

$c_{V} = c_{V, P = 0} + T \cdot \int_{0}^{P} ({{(\frac{δ V}{δ P})}_{T} \cdot (\frac{δ^{2} P}{δ T^{2}})}_{V}) d P$

(16)

where c_V,P₌₀ is the specific heat capacity at a constant volume and zero pressure.

The accuracy in estimating the specific heat capacity depends considerably on what gas equation of state is adopted for simulation. For example, when the Van der Waals equation or the RKS equation is chosen, Equation (16) is simplified with Equation (17) or Equation (18), respectively.

$c_{V} = c_{V, P = 0}$

(17)

$c_{V} = c_{V, P = 0} + T \cdot \frac{a_{d} \cdot m \cdot k_{d}}{2 \cdot T_{c}^{2} \cdot b_{d}} \cdot {(\frac{T}{T_{c}})}^{- \frac{3}{2}} \cdot (k_{d} + 1) \cdot l n |\frac{V + m \cdot b_{d}}{V}|$

(18)

Figure 1 shows c_V predicted for various pressure and temperature values using different methods. For clarity, the gas pressure is only displayed on the horizontal axis in Figure 1, which is related to the gas temperature as presented in Equation (19).

$T = T_{0} \cdot {(\frac{P}{P_{l}})}^{\frac{2}{7}}$

(19)

The changes in c_V presented in Figure 1 roughly correspond to adiabatic gas compression and expansion, assuming the initial gas temperature T₀ and pressure P_l of 296.15 K and 5 MPa, respectively.

Figure 1.
Specific heat capacity at constant volume predicted with various methods; curve 2 is based on data from [18,19]; curve 6 is based on data from [21].

When the c_V predicted with Equation (16) and the 33-parameter virial gas equation of state is adopted as the reference, the following can be concluded:

Equation (16) gives the best predictions of c_V when the RKS equation of gas state is substituted into Equation (16).
Bilinear interpolation provides an even more accurate prediction of c_v. The accuracy of this method strongly depends on the amount of experimental data available. For the pressure and temperature range considered herein, a large amount of experimental data is available (Figure 2). The tests carried out herein show that this method is fast and can be used successfully to update the estimate of c_V while a computer simulation of hydraulic accumulator operation is running.
The changes in c_V that occur during the operation of a hydraulic accumulator are relatively small. Therefore, the final result of a computer simulation of the hydraulic accumulator is unlikely to improve significantly due to changing the method to update the specific heat capacity during the simulation.

5. Modeling of Heat Transfer Between the Environment and the Gas in the Hydropneumatic Accumulator

In practice, heat transfer between the environment and the gas in the hydraulic accumulator is usually described by Equation (20)

$\frac{d Q}{d t} = h \cdot A \cdot (T_{w} - T)$

(20)

where h [W/(m²·K)] is the overall heat transfer coefficient and T_w [K] is the temperature of accumulator walls.

The overall heat transfer coefficient is determined based on the so-called Thermal Time Constant using the following given by Equation (21)

$h = \frac{m \cdot c_{V}}{τ \cdot A}$

(21)

where τ is the Thermal Time Constant and A is the surface of heat transfer between gas and accumulator walls.

An exact definition of the Thermal Time Constant is given, for example, in [5,22,23,24,25]. The Thermal Time Constant τ depends on many factors, including the type and volumetric capacity of the hydropneumatic accumulator, and should be determined by experiments. In simulation computations, the Thermal Time Constant for a given accumulator is estimated at a specified constant value. This assumption does not capture the actual properties of real accumulators; however, according to [22], it is acceptable when simulation computations do not need to be carried out with exceptional precision. The Thermal Time Constants for typical hydraulic accumulators can also be roughly estimated using empirical relationships, for example, Equations (22) or (23), given in [23]

$τ [s] = 1.338 \cdot P_{0} [b a r] \cdot {V_{0} [m^{3}]}^{0.313}$

(22)

$τ [s] = 0.300 \cdot P_{0} [b a r] \cdot {V_{0} [m^{3}]}^{0.220} + 86.2 \cdot {V_{0} [m^{3}]}^{0.490}$

(23)

where P₀ [bar] is the initial gas pressure at a temperature of 20 °C.

Equations (22) and (23) refer to bladder- and piston-type accumulators, respectively. Figure 3 shows sample values of τ predicted for a number of sample accumulators using Equations (22) and (23).

It must be emphasized that the Thermal Time Constants plotted in Figure 3 refer only to hydropneumatic accumulators with steel housings. In recent years, alternative materials have been increasingly used to manufacture accumulators. In the up-to-date vehicles, entire accumulator housings, or some of their sections, are made from aluminum alloys or composite materials, including carbon fiber-reinforced plastics (CFRPs) [8,26,27,28].

CFRPs have many advantages over steel, including less weight and thermal expansion combined with superior rust resistance and flexural rigidity. However, CFRPs are significantly less resistant to heat; in practice, an operation temperature of less than 80 °C is advisable for CFRPs.

The material used to manufacture the housing of a hydropneumatic accumulator with a specified volume and pre-charge pressure will affect the Thermal Time Constant of the accumulator because of a significant difference in thermal conductivity of steel, aluminum, and CFRPs. The thermal conductivity of steel is 80 times greater than that manifested by CFRPs in the direction transverse to reinforcing inserts. This will result in the greatest Thermal Time Constants exhibited by the CFRP accumulators. The Thermal Time Constant will also be affected by the thermal insulation of the accumulator.

In general, the greater the Thermal Time Constant of an accumulator, the better the efficiency of the accumulator. To maximize efficiency, the gas chambers of some accumulators are filled with elastomer foam [26,28] or nitrogen, which when used in conventional accumulators is replaced with another gas or gas mixture [7].

More accurate models of heat transfer between the gas held in the accumulator, the environment, and the hydraulic oil have been discussed in the literature, including [8,28,29,30]. Although they are likely to improve the accuracy of numerical studies on certain drive systems, they are not discussed herein. Because of outstanding complexity, they are not suitable for extensive simulation models that must allow for a hydropneumatic accumulator included in a sophisticated, multi-component system.

6. Models of Gas in Hydropneumatic Accumulators Used in Selected Commercial Software

A number of commercial software is available for the simulation of systems with hydropneumatic accumulators. In this paper, all of them could not be reviewed in detail. Therefore, the review presented herein presents the three most popular computation engines.

6.1. Matlab, Simulink, v. 2016b (Simscape Tool)

The Matlab Simulink software includes two models of a hydropneumatic accumulator called a “Gas Charged Accumulator” and a “Gas Charged Accumulator (TL)” [31]. In both of these models, the behavior of gas is governed by the ideal gas law, a polytropic process describes the thermodynamic cycle that the gas undergoes during the accumulator operation, and a constant polytropic index is defined by the user for the entire simulation. This approach to modeling gas in the hydropneumatic accumulator has been the most commonly used in commercially available software. For example, it is adopted by the software Easy5 v. 2020 [32] and was applied in DADS v. 8.0 developed by CADSI (Coralville, IA, USA) in 1995 [33]. It must be noted that a modified version of the latter has become a module of Simcenter 3D by Siemens (Munich, Germany).

If a polytropic process is used in a simulation to describe the thermodynamic gas transformation, the simulation will capture the heat transfer between the gas and the environment to some extent. Unfortunately, if a constant polytropic index is specified for the entire simulation, the results of the simulation will only be relatively accurate for a specified gas pressure and for some constant rate of accumulator charging and discharging. This assumption is not acceptable for simulations of systems with energy recovery, as each individual cycle of operation of these systems may differ from the other cycles in terms of the rate at which the accumulator is being charged or discharged and the intervals between the consecutive operation stages when the accumulator is undercharging or discharging. The assumption that the polytropic process is the only gas transformation occurring in the accumulator and the polytropic index is constant over the entire simulation means the efficiency of energy recovery from the gas in the accumulator is 100%.

In [34,35,36], recommendations are given on what polytropic index to assume for simulation computations to properly model a hydraulic accumulator being charged or discharged at a specified rate and average gas pressure.

6.2. Simcenter Amesim, v. 2021.1

Multiple models of hydropneumatic accumulators are available in Simcenter Amesim, v 2021.1 [20]. In the most sophisticated of these models, gas is described using the RKS equation, and an arbitrarily selected gas equation of state is used to update the specific heat capacity at constant volume during the course of numerical simulation. The model provides a very time-efficient numerical simulation. Unfortunately, at the beginning of the simulation, the amount of gas to be considered, expressed in moles, is determined so roughly that the actual initial conditions of the simulation, including the initial gas temperature, pressure, and volume, are slightly different from the initial conditions requested by the user.

The heat transfer between the gas and the environment is defined using a single time constant that does not change during the entire simulation. The review of the state-of-the-art literature reveals that Simcenter Amesim is the software most used for the simulation testing of drive systems with energy recovery.

6.3. MSC Adams, v. 2005.0.0

Various models of hydropneumatic accumulators are available in MSC Adams v. 2005.0.0 software [37]. The most complex of these models comprises a virial gas equation of state whose exact form has not been given in the software user manual. Moreover, the manual does not explain how the specific heat capacities c_v and c_P are estimated during the simulation.

The heat transfer between the gas and the environment is governed by a heat transfer coefficient h_p·A [W/K], which is defined at the beginning of the simulation and does not change until the simulation ends. This coefficient can be calculated based on the Thermal Time Constant.

Unfortunately, the software was unstable when the model of the hydropneumatic accumulator was involved in the simulation. A very short integration step was used to successfully complete the simulation, resulting in a very long total simulation time. Furthermore, the most recent versions of MSC Adams no longer comprise the module for modeling hydropneumatic accumulators. In return, a co-simulation recommended that hydraulic systems be solved using, for example, EASY5 v. 2020 software.

7. Materials and Methods

In this study, the results calculated with the five limited applicability gas equations of state presented in Section 3 have been compared with their respective predictions derived from the reference formula, Equation (15), which is assumed to capture the behavior of nitrogen with a negligible error. Furthermore, the following assumptions have been made for each of the simulations related to this study. An initial temperature of gas in the accumulator of 23 °C has been assumed. Each simulation has also been carried out for identical initial gas pressure and volume. To meet the initial conditions in terms of volume, temperature, and pressure of gas, an appropriate amount of gas, expressed in moles, is set. As a result of this approach, the quantity of gas considered in a specified simulation depends on the gas equation of state chosen for the simulation.

7.1. Stage I—Prediction Errors Related to the Gas Equation of State and the Specific Heat Capacity Model

The objective of the first stage is to study the maximum errors caused using different gas equations of state to solve a specified engineering problem by numerical simulation. It has been assumed that these errors increase with an increase in the difference between the temperatures at which gas is compressed and expanded in the operation cycle of a hydropneumatic accumulator. Since the so-called reverse Otto cycle provides the greatest temperature difference possible, a hydraulic accumulator operating according to the reverse Otto cycle is analyzed herein.

Figure 4 shows an example of the reverse Otto cycle, consisting of adiabatic compression, isochoric cooling, adiabatic expansion, and isochoric heating. The reverse Otto cycle is used to model the hydraulic accumulator operation when both the compression and the expansion of the gas in the accumulator run very quickly, while the intervals between the compression and expansion cycles are long [38]. The efficiency of the hydraulic accumulator operation predicted by assuming the reverse Otto cycle usually underestimates the actual efficiency that the accumulator exhibits in real operating conditions.

The gas equations of state considered in the first stage of this study have been evaluated in terms of relative errors Δ in predicting pressure P₁ and P₃, temperature T₁ and T₃ (for nomenclature, see Figure 4), and efficiency of energy recovery η. The above-mentioned operation parameters have been predicted with each of the analyzed gas equations of state, assuming four different values of pre-charge pressure of the accumulator (P_l) and four different volumetric ratios F_max given by Equation (24)

$F_{m a x} = \frac{V_{n} - V_{m i n}}{V_{n}} \cdot 100 %$

(24)

where V_n is the reference volume of the accumulator; in this paper, the initial volume of gas V₀ is adopted as the reference volume, and V_min is the minimum gas volume during the operation cycle of the accumulator.

According to the general guidelines given in the literature, some gas equations of state, including Equations (8) and (9), can only be used to describe the behavior of the gas at a low pressure and under conditions far from the gas critical point. In Figure 4, the gas state represented by V₁, P₁, and T₁ is the farthest away from the critical point, while the state that corresponds the best with the critical point is distinguished by V₃, P₃, and T₃. Therefore, these two points have been used to evaluate the gas equations of state analyzed herein.

In the first stage of this study, the influence of the method to describe the gas in terms of specific heat capacity at a constant volume on the accuracy of the simulation has also been studied. The first of the investigated approaches was to calculate the specific heat capacity with Equation (16), involving the instantaneous gas temperature and pressure. In the second approach, the specific heat capacity was simply evaluated at some constant value, regardless of the gas state.

To compare various gas equations of state in a synthetic way, the mean of the normalized relative errors given by Equation (25) has been used as an additional assessment parameter.

$\bar{‖Δ_{i}‖} = \frac{\sum_{k = 1}^{k = z} (\sum_{j = 1}^{j = x} (\frac{|∆_{i, j, k}| - \min_{i} |∆_{i, j, k}|}{\max_{i} |∆_{i, j, k}|}))}{x \cdot z}$

(25)

where i is the ID index of a given gas state equation, j is the ID index of a given value of accumulator pre-charge pressure, k is the ID index of a given value of the volumetric ratio, x is the number of considered individual values of accumulator pre-charge pressure, and z is the number of considered values of the volumetric ratio.

The measure given by Equation (25) is not sensitive to the increase in dispersion of the relative errors with an increase in both the pre-charge pressure P_l and the volumetric ratio F_max. If a standard arithmetic mean were used, the average error would be mostly affected by the largest error, which is usually noted for the extreme values of P_l and F_max. The measure defined by Equation (25) does not suffer from this problem. This approach is inspired by the methods developed in the field of operations research to solve a problem whose preferred solution must satisfy multiple contradictory quality factors.

7.2. Stage II—Prediction Errors Caused by Neglecting Heat Transfer

In real operation conditions of hydropneumatic accumulators, the differences between the temperatures at which the gas undergoes compression or expansion are far smaller than those assumed in the first stage of the research. This is related to the heat transfer between the gas and the environment that occurs while the gas is being compressed or expanded, as well as the time intervals between compressing and expanding the gas being so short that the gas temperature hardly reaches the ambient temperature. This stage of the research is executed to answer the following question: to what extent can the calculation errors revealed in the first stage be reduced by making the assumed thermodynamic cycle more realistic?

To perform this, simulation models of a hydropneumatic accumulator have been implemented in Matlab Simulink using various gas equations of state. In each of these models, heat exchange between the gas in the hydropneumatic accumulator and the environment has also been captured using the approach described in Section 5. Each of the models has been used to simulate the accumulator operating cycle shown in Figure 5.

This operating cycle consists of four stages with duration times t₁ through t₄. During the cycle, the accumulator is charged and discharged at a constant rate (Q₁ = const., Q₂ = 0, Q₃ = const., Q₄ = 0). The parameter F_max is the same for each consecutive cycle. Since duration times t₁ through t₄ significantly affect the temperature of the gas in the accumulator, different values for the duration times were assumed during the tests. However, such duration times were chosen for simulation so that they corresponded well with duration times noted for real systems with energy recovery.

In this research stage, the models are only compared with each other in terms of the accuracy in predicting the efficiency of recovering the energy supplied to the accumulator. The calculations correspond with a 10 L accumulator with steel housing. Furthermore, since accumulators are also made from materials that translate into a greater Thermal Time Constant, this research stage includes computations related to a hypothetical accumulator whose Thermal Time Constant is three times greater than that of the steel accumulator.

7.3. Stage III—Prediction Errors of the Hydropneumatic Accumulator Models Delivered by Commercial Engineering Software

In simulation studies of sophisticated drive systems, researchers usually rely on component models predefined in various commercial software. Unfortunately, these models are discussed so roughly in their respective user manuals that users are often forced to trust the software vendor. For this reason, in the third stage of the research, the accuracy of two hydraulic accumulator models will be verified, including a model available in MSC Adams 2005.0.0 and a model delivered by the Simcenter Amesim 2021.1 package. These are the only hydropneumatic accumulator models among those described in Section 7 that capture the energy losses resulting directly from heat exchange between the environment and the gas in the accumulator. To examine the accuracy of these models, numerical computations will be carried out with these models, and the results of these computations will be compared with the results of calculations made using a reference model implemented in the Matlab Simulink environment for the purpose of the second stage of the research.

8. Results

The lower section of each of the tables presents the reference value of the analyzed quantity calculated with the reference model and marked with the symbol F. In the columns on the left, and in the upper section of the table, conditions captured by the simulation are specified, including the accumulator pre-charge pressure and the parameter F_max. The main part of the table presents the absolute errors exhibited by the tested gas models when predicting the analyzed quantity. The absolute error typical for a specified gas model is calculated as the prediction made with that model minus the value brought by the reference model. The prediction quality of each model is also rated in the diagram; the lower the rating, the better the prediction quality.

For a more convenient interpretation of the results summarized in the tables, Figure 11 and Figure 12 show the absolute errors of pressure predictions related to each of the gas equations of state tested herein. Each of the figures is related to a specified gas temperature and the parameter F_max changing from 0 to 80%. For each of the models, the amount of gas (expressed in moles) is chosen, and a pressure of 5 MPa is predicted by the model at a temperature of 23 °C and F_max = 0%.

Based on the results obtained, the following conclusions can be drawn:

The greatest error has been noted for the model based on Clapeyron’s equation, which meets initial expectations. This model is expected to show an increase in prediction error with an increase in gas pressure, as well as when the gas approaches the phase transition curve because theoretical derivations leading to this model neglect a finite volume of gas particles, the possible attraction and repulsion of gas particles, and the possible formation of a new phase in the gas.
The computation errors caused by the model based on Clapeyron’s equation are so large that this model cannot be used to predict the behavior of gas under conditions similar to those assumed in this study. The efficiency in recovering energy from a conventional accumulator predicted with this model suffers from an error of approx. 10%.
The van der Waals equation reveals satisfactory accuracies for the pressures and temperatures meeting typical operation conditions of real hydropneumatic accumulators. At extremely high pressures, this equation leads to unacceptable errors. This simple equation is a quick remedy to find the initial conditions of a simulation and carry out the simulation.
The behavior of the gas has been captured the best by the BWR equation. Because of the relatively large number of coefficients involved in this equation, it can successfully describe the behavior of the gas over a relatively large range of temperatures and pressures (see Figure 11 and Figure 12). Unfortunately, this equation cannot provide an accurate prediction for the instantaneous specific heat at constant volume (see Figure 1). Probably, for this reason, the BWR equation is not the best option, which has been revealed in the rankings posted under the tables with the test results. Therefore, in simulations based on the BWR equation, the instantaneous specific heat capacity should be predicted with a method different from the approach adopted in this article.
The RKS equation with the Ghanbari–Check amendment, labeled EV herein, exhibited a performance almost as effective as the BWR equation. According to Figure 11 and Figure 12, the RKS equation is less accurate in modeling gas behavior; however, it gives more reliable predictions of the instantaneous specific heat at a constant volume.
The models featuring the dynamic update of the specific heat at a constant volume have generally produced better results, except for a few exceptions. For example, the BC model delivers the best prediction for pressures P₁; however, this has been achieved at the expense of inaccurate estimates of temperature T₁.
A comparison of the results plotted in Figure 11 and Figure 12 leads to a conclusion that for some models, the error related to the gas equation of state can be either a positive or negative value, depending on the gas temperature. Furthermore, the temperature can affect the rate at which the error caused by an increase in pressure grows.
The results confirm the general statement quoted in the literature that all of the analyzed real gas equations of state provide satisfactory accuracy in predicting the nitrogen pressure and temperature when the typical operating conditions of hydropneumatic accumulators are considered using these equations.

The computations carried out in the first stage prove that all of the tested real gas equations are suitable for simulation studies. Thus, they do not need to be investigated in the second stage. The only issue that needs to be tested is to find out whether the ideal gas law could also be successfully applied for simulation under certain conditions.

Figure 13 and Figure 14 present the example results related to the second stage of this study. Namely, the efficiency in energy recovery averaged for ten cycles of the hydropneumatic accumulator operation is plotted. Averaging is justified by the differences shown in Figure 15 between the consecutive cycles, resulting from the modeling of heat exchange between the gas and the environment.

The following conclusions can be drawn from the second stage of this study:

The model based on the ideal gas law gives accurate results if duration times t₁ through t₄ are short and F_max is small. It should be noted that in a number of studies, including [28], F_max is estimated at approx. 40%.
In many real applications of hydropneumatic accumulators, each of the duration times t_i is from 6 s to 12 s [2]. If these are investigated with the model based on the ideal gas law, and F_max = 60% is assumed, the energy recovery efficiency is overestimated by approx. 6% (for the steel body accumulator).
When duration times t_i exceed 60 s, the overestimation may be even larger; it may be as large as the errors obtained in the first stage.
The higher the efficiency in recovering the energy from the real hydropneumatic accumulator, the more accurate the predictions of the efficiency delivered by the model based on the ideal gas law. This is because the higher the efficiency of the real process, the smaller the difference between the pressures recorded at any specified gas volume during gas compression and expansion. When the pressures during the real process are similar, their model predictions will suffer from almost identical error magnitudes. If the errors are identical, they will cancel out each other when the efficiency of energy recovery is calculated.
Thermal Time Constant and the time the operation cycle of an accumulator lasts are decisive for the efficiency of the accumulator. The efficiency will increase by increasing the Thermal Time Constant and decreasing the operation cycle duration.
The results in Figure 13 are related to a conventional design accumulator with steel housing, while Figure 14 refers to an accumulator whose Thermal Time Constant is 3 times greater than that of the conventional design steel accumulator. The increase in the Thermal Time Constant causes both the efficiency predictions and the errors in predicting the efficiency to change slower as duration times t₂ and t₄ increase.
When duration times t₁ through t₄ do not exceed 5 s, and an accumulator with an increased Thermal Time Constant is considered, the errors in predicting the efficiency of the accumulator in recovering energy do not significantly exceed 1%. Under these conditions, the model based on the ideal gas law proves acceptable accuracy.

In Table 7, the results obtained using the commercial software MSC Adams v. 2005.0.0 and Simecenter Amesim v. 2021.1 are compared with the results related to the model of the hydropneumatic accumulator developed herein. All of the results are similar, which meets the initial expectations. The calculations carried out with Simcenter Amesim software exhibit a slightly lower initial gas temperature because the software algorithm is unable to precisely adjust the number of gas moles in the accumulator to meet the requested initial conditions.

During the tests with MSC Adams v. 2005.0.0 software, the hydropneumatic accumulator was charged with a slightly greater amount of hydraulic oil (F_max = 60.06). Consequently, the prediction for pressure P₂ resulting from this simulation slightly exceeded its respective prediction derived from the author’s model.

It can be concluded that all of the three computation models considered in Table 7 are suitable for studies on hydropneumatic systems with energy recovery.

9. Summary

This paper presents a discussion on the effect of the gas state equation on the accuracy of the numerical model of a hydropneumatic accumulator. The ideal gas law by Clapeyron, and the equations of van der Waals, BWR, and RKS, including the genuine RKS equation and the Ghanbari–Check amendment, have been evaluated in terms of the accuracy in predicting the efficiency of energy recovery from the gas in a hydropneumatic accumulator. A model based on the so-called virial gas equation of state has been used as a reference model. The evaluation has been carried out using two models of heat transfer between the environment and the gas in the accumulator; according to the first model, the heat transfer is governed by the so-called reverse Otto cycle, while the second one is governed by the Thermal Time Constant.

The results obtained herein show that in the typical operating conditions of hydropneumatic accumulators, all of the investigated real gas models can predict the efficiency of energy recovery from the gas in the accumulator with an error not exceeding ±1%, which is satisfactory in the author’s opinion. According to this study, in extreme operating conditions, which are unlikely for real accumulators, the prediction error manifests an increase.

The analyses presented herein clearly prove that the results of simulation computations of a hydropneumatic accumulator are likely to suffer from unacceptable errors when the simulation model of the accumulator is derived from the ideal gas law by Clapeyron. However, this refers only to the hydrostatic drive systems with energy recovery that include a conventional accumulator design and exhibit relatively long periods when the accumulator is undercharging or discharging, long intervals between these periods, and/or irregular and varying cycles of the accumulator operation. Unfortunately, the conditions mentioned above are typical for hydrostatic drive systems with energy recovery, including lifting systems of forklifts, manipulators of earthmoving machinery, and drive systems of mobile machines.

Even greater prediction errors may occur when a polytropic process and a constant polytropic index are assumed to simulate the thermodynamic process that occurs to the gas in a hydropneumatic accumulator. Unfortunately, these are assumed by most of the models of hydropneumatic accumulators predefined in commercially available software for the simulation testing of hydrostatic drive systems. Few exceptions comprise MSC Adams v. 2005.0.0 and various versions of Simcenter Amesim software.

The efficiency of hydropneumatic accumulators predicted in this paper does not correspond to the overall efficiency, which is even worse. The overall efficiency of energy recovery from the gas held in hydropneumatic accumulators has been found by experiments, for example, in [40,41].

It is expected that this paper will let the readers understand the process of modeling gas in hydropneumatic accumulators. Furthermore, this paper provides a number of guidelines, supported by the results of numerical calculations, on how to select a gas model reasonably suited for simulation testing of a specific drive system with a hydropneumatic accumulator.

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Andrzej Kosiara www.mdpi.com

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Mathematical Models of Gas in Hydropneumatic Accumulators Used in Numerical Tests of Drive Systems with Energy Recovery

1. Introduction

2. Energy Balance of a Hydraulic Accumulator

3. Models of Gas State Used in Modeling of Hydraulic Accumulators

4. Specific Heat Capacity at a Constant Volume

5. Modeling of Heat Transfer Between the Environment and the Gas in the Hydropneumatic Accumulator

6. Models of Gas in Hydropneumatic Accumulators Used in Selected Commercial Software

6.1. Matlab, Simulink, v. 2016b (Simscape Tool)

6.2. Simcenter Amesim, v. 2021.1

6.3. MSC Adams, v. 2005.0.0

7. Materials and Methods

7.1. Stage I—Prediction Errors Related to the Gas Equation of State and the Specific Heat Capacity Model

7.2. Stage II—Prediction Errors Caused by Neglecting Heat Transfer

7.3. Stage III—Prediction Errors of the Hydropneumatic Accumulator Models Delivered by Commercial Engineering Software

8. Results

9. Summary

Greenberg

1. Introduction

2. Energy Balance of a Hydraulic Accumulator

3. Models of Gas State Used in Modeling of Hydraulic Accumulators

4. Specific Heat Capacity at a Constant Volume

5. Modeling of Heat Transfer Between the Environment and the Gas in the Hydropneumatic Accumulator

6. Models of Gas in Hydropneumatic Accumulators Used in Selected Commercial Software

6.1. Matlab, Simulink, v. 2016b (Simscape Tool)

6.2. Simcenter Amesim, v. 2021.1

6.3. MSC Adams, v. 2005.0.0

7. Materials and Methods

7.1. Stage I—Prediction Errors Related to the Gas Equation of State and the Specific Heat Capacity Model

7.2. Stage II—Prediction Errors Caused by Neglecting Heat Transfer

7.3. Stage III—Prediction Errors of the Hydropneumatic Accumulator Models Delivered by Commercial Engineering Software

8. Results

9. Summary

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