146 5 Non-electric Hybrid Propulsion Systems
referred to as “full hybrid,” since larger degrees of hybridization are used, with larger improvements (70% and more) in reducing fuel consumption than with parallel “mild” concepts.
All types of hybrid-hydraulic propulsion systems include a high-pressure accumulator and a low-pressure reservoir. The accumulator contains the hy- draulic fluid and a gas such as nitrogen (N2) or methane (CH4), separated by a membrane (see Fig. 5.15). When the hydraulic fluid flows in, the gas is compressed. During the discharge phase, the fluid flows out through the motor and then into the reservoir. The state of charge is defined as the ratio of instantaneous gas volume in the accumulator,Vg, to its maximum capacity.
Typical data of an accumulator designed for a truck are listed in Table 5.2.
The reservoir can be regarded as an accumulator working at a much lower pressure, e.g., 8.5–12.5 bar.
d/2 w mm
mg
mo
membrane
oil Vg
Fig. 5.15.Schematic of a hydraulic accumulator.
Table 5.2.Typical data of hydraulic accumulators.
fluid capacity 50 l
maximum gas volume 100 l
minimum gas volume 50 l
pre-charge pressure (at 320 K) 125 bar
maximum pressure 360 bar
5.4.1 Quasistatic Modeling of Hydraulic Accumulators
The causality diagram of a simple hydraulic accumulator used for quasistatic simulations is sketched in Fig. 5.16. The input variable is the mechanical
5.4 Hydraulic Accumulators 147 powerP2(t) =p2(t)·Q2(t) required at the shaft. Herep2(t) is the pressure in andQ2(t) is the volumetric flow from the accumulator. The output variable is the state of charge of the accumulator, i.e., the volume occupied by the gas, Vg(t).
P 2 HA Vg
Fig. 5.16.Hydraulic accumulators: causality representation for quasistatic model- ing.
Thermodynamic Model
A basic physical model of a hydraulic accumulator can be derived from the mass and energy conservation laws and the ideal gas state equation for the charge gas [197, 79]. The resulting set of nonlinear equations includes an energy balance
mg·cv· d
dtϑg(t) =−pg(t)· d
dtVg(t)−h·Aw·(ϑg(t)−ϑw) , (5.16) a mass balance
d
dtVg(t) =Q2(t), (5.17)
and the ideal gas law
pg(t) =mg·Rg·ϑg(t)
Vg(t) , (5.18)
whereAw is the effective accumulator wall area for heat convection,his the effective heat transfer coefficient,mgis the gas mass,cvis the constant-volume specific heat of the gas, ϑg(t) is the gas temperature,pg(t) the gas pressure, Vg(t) the gas volume, ϑw the wall temperature, and Rg is the gas constant.
More detailed models [197] may describe the charge gas with a nonlinear state equation (Benedict–Webb–Rubin equation of state) and may take into account the heat exchange between the charge gas and the elastomeric foam that is usually inserted on the gas side of the accumulator to reduce the thermal loss to the accumulator walls.
In a first approximation the gas pressure pg(t) equals the fluid pressure at the accumulator inlet, p2(t). Frictional losses caused by flow entrance ef- fects, viscous shear, and piston-seal friction or bladder hysteresis can hardly be accounted for in an analytical fashion. Usually the pressure loss term is evaluated as a fraction ofp2(t) (e.g., 2%) [197].
A relationship between fluid flow rate, charge volume, and power can be derived by solving (5.16)–(5.18) at steady state. Assuming thatpg(t) =p2(t),
148 5 Non-electric Hybrid Propulsion Systems
i.e., no pressure losses at the accumulator inlet, the resulting equation for fluid pressure is written as
p2(t) = h·Aw·ϑw·mg·Rg
Vg(t)·h·Aw+mg·Rg·Q2(t) . (5.19) Combining (5.19) with the definition of output power, the fluid flow rate is obtained,
Q2(t) =Vg(t) mg
· h·Aw·P2(t)
Rg·ϑw·h·Aw−Rg·P2(t) . (5.20) The state of charge is thus given by integrating (5.17). With only one state variable, this quasi-stationary model has a complexity that is equivalent to that of most battery models. Thus it can easily be embedded, for instance, in a dynamic programming algorithm [278] and also generally in quasistatic algorithms.
Accumulator Efficiency
The efficiency of a hydraulic accumulator may be defined as the ratio of the total energy delivered during a complete discharge to the energy that is nec- essary to charge up the device. This definition is conceptually identical to the
“global efficiency” introduced in Chap. 4 for electrochemical batteries. The en- ergy spent to charge the accumulator depends on the type of process assumed for the accumulator charge/discharge cycle. Similarly to electrochemical bat- teries, constant flow rate or constant power processes may be assumed. For hydraulic accumulators, however, the usual definition of efficiency is based on a reference cycle consisting of isentropic compression and expansion from the maximum to the minimum volume and vice versa.
The thermodynamic transformations followed by the gas in the reference cycle are represented in the temperature/entropy (ϑ–s) diagram of Fig. 5.17.
The transformation AB is an isentropic compression from an initial state A, which is characterized by a maximum volume and a temperature that equals that of the surrounding ambient (ϑA = ϑw). The transformation BC is an isochoric cooling of the gas, which loses thermal energy5to the ambient until ϑC=ϑA. The transformation CD is an isentropic expansion that ends when pD = pA. The transformation DE is a further expansion that ends when VE=VA.
By definition, the isentropic transformation yields the following equations pB
pA
= VA
VB
γ
=r−γ, ϑB ϑA
= pB
pA
γ−1γ
=r1−γ, (5.21)
5 This corresponds to a worst-case scenario in which the device is left to cool for a long time.
5.4 Hydraulic Accumulators 149
A B
C
D
E s
ϑA=ϑC
pA VA VB ϑ
Fig. 5.17.Temperature–entropy (ϑ–s) diagram of the accumulation process.
where r is the expansion ratio and γ is the ratio of the specific heats. The compression work is calculated along the path AB. For a closed, adiabatic system, the work exchanged equals the variations of the internal energy. Thus the workWAB is calculated as
WAB=mg·cv,g·ϑA· ϑB
ϑA −1
=mg·cv,g·ϑA· r1−γ−1
. (5.22) The transformation BC is isochoric, i.e., the volume VC is equal to VB. Moreover, sinceϑC=ϑA, the pressure can be calculated as
pC= mg·Rg·ϑC
VC = mg·Rg·ϑA
VB = pA·VA
VB =pA
r . (5.23) For the reference cycle ABCDA, the expansion ends at the state D, which is characterized by a pressurepD=pA, and a temperatureϑDevaluated as
ϑD
ϑC
= pD
pC
γ−1γ
= pA
pA
r γ−1γ
=rγ−1γ . (5.24) The expansion work is thus calculated as
WCD =mg·cv,g·ϑC· ϑD
ϑC
−1
=mg·cv,g·ϑA·
rγ−1γ −1
. (5.25) For the reference cycle ABCEA, the expansion ends at the state E, which is characterized by a volumeVE=VA, and a temperature evaluated as
ϑE
ϑC = VE
VC 1−γ
=rγ−1. (5.26)
The expansion work is thus
150 5 Non-electric Hybrid Propulsion Systems WCE=mg·cv,g·ϑC·
ϑE ϑC
−1
=mg·cv,g·ϑA· rγ−1−1
. (5.27) The accumulator efficiency is the ratio between the discharge energy and the charge energy. For the isochoric reference cycle ABCEA, the efficiency is evaluated as
ηha,V = −WCE
WAB = 1−rγ−1
r1−γ−1 . (5.28)
For the “isobaric” reference cycle ABCDA, the efficiency is evaluated as ηha,P =−WCD
WAB
=1−rγ−1γ
r1−γ−1 . (5.29)
The variations ofWAB,WCD,WDEandηhawith the expansion ratiorare shown in Fig. 5.18. The plots clearly show that the efficiency of the isobaric cycle is always lower than the efficiency of the isochoric cycle which reaches the 100% value for r = 1. In both cases, the efficiency is a monotonically increasing function ofr.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 1 2 3 4
W/(mg cv,g!A) [!]
r [!]
WAB
!WCD
!WCE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.2 0.4 0.6 0.8 1
"ha [!]
r [!]
"ha,V
"ha,P
Fig. 5.18. Compression workWAB, expansion works WCD, WCE (top), and effi- ciencyηha(bottom) as a function of the expansion ratior.
Specific Energy of Accumulators
The results of the analysis developed in the previous section allow for the evaluation of the specific energy of a hydraulic accumulator. The compression
5.4 Hydraulic Accumulators 151 work given by (5.22) must be divided by the accumulator mass. This is given by the sum of three terms, viz. the mass of the charge gas, the mass of the hydraulic fluid, and the mass of the housing. The latter term may be evaluated as the mass of a spherical shell designed to bear a pressurepB, the maximum gas pressure. The thickness of the housing (see Fig. 5.15) may be evaluated using the formula:
w=pB·d
4·σ , (5.30)
wheredis the diameter of the accumulator shell andσis the maximum tensile stress. The housing mass is thus
mm=ρm·π·d2·w= 3
2·VA·pB·ρm
σ , (5.31)
since VA, the maximum volume occupied by the gas, by definition is the accumulator volume. The fluid mass to be considered in the calculation is the mass that occupies the maximum volume left by the gas, i.e., the volume VA−VB. The mass of the charge gas is the constant mg already introduced in the previous section. The orders of magnitude of such masses are rather different. Evaluating the gas mass as mg =VA·pA/(Rg·ϑA), the ratios of fluid to gas mass and of housing to gas mass are
mo
mg
=ρo·Rg·ϑA
pA
·(1−r), mm
mg
= 3
2 ·Rg·ϑA·r−γ·ρm
σ . (5.32) Using typical data for the materials, such as the ones listed in Table 5.3, clearly shows thatmgcan be neglected when compared withmoandmm. For higher pressures, evenmo can be neglected when compared withmm. Table 5.3. Typical parameter values for hydraulic accumulator models (methane as compressed gas).
ρm/σ 10−5kg/J ρo 900 kg/m3
ϑA 300 K
Rg 520 J/(kg K) pB 400–800 bar
In the latter case, the expression for the specific energy is written as Eha
mha
≈WAB
mm
=α·rγ· r1−γ−1
, (5.33)
with α = 2·cv,g·σ/(3·ρm·Rg). The variation of the specific energy as a function of the compression ratioris shown in Fig. 5.19.
152 5 Non-electric Hybrid Propulsion Systems
The specific energy of (5.33) can be maximized with respect torby setting the relevant derivative to zero. This condition yields the optimal compression ratio
ropt= 1
γ γ−11
. (5.34)
Withγ = 1.31 (methane), the optimal compression ratio isropt= 0.42, and the corresponding efficiency isηha,P = 0.60.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
WAB/(mm!) [!], "ha,P [!]
r [!]
ropt = 0.42
Fig. 5.19.Specific energy of a hydraulic accumulator as a function of the compres- sion ratio with methane as a hydraulic fluid.
5.4.2 Dynamic Modeling of Hydraulic Accumulators
The physical causality representation of a hydraulic accumulator is sketched in Fig. 5.20. The model input variable is the flow rate of the hydraulic fluid Q2(t). A positiveQ2(t) discharges the accumulator, a negativeQ2(t) charges it. The model output variables are the hydraulic fluid pressurep2(t) and the state of chargeVg(t).
The state of charge is calculated by directly integrating (5.17), while in a first approximation the fluid pressure equals the gas pressure pg(t), which may be integrated using (5.16) and (5.18).