Johnston. The Noble-Abel Equation of State: thermodynamic derivations for ballistics modeling

28 Pages·2016·1.09 MB·English

by  Ian A.

·e Noble-Abel Equation of State: ·ermodynamic
Derivations for Ballistics Modelling
Ian A. Johnston
WeaponsSystems Division
Defence ScienceandTechnologyOrganisation
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ABSTRACT
Accuratemodellingofguninteriorballisticspromotesmoree(cid:14)cientgunand
propelling charge design. In order to simulate interior ballistic (cid:8)ow(cid:12)elds,
such models require a description of the thermodynamic behaviour of the
propellant gas. ·e Noble-Abel equation provides a simple and reasonably
accurate equation ofstate forpropellant gases at the highdensities and tem-
peraturesexperiencedinguns.Mostcomputational(cid:8)uiddynamics-basedbal-
listics models, however, require additional thermodynamic functions which
mustbederivedfromtheequationofstate. ·isnotepresents thederivation
ofsuchthermodynamicfunctionsforNoble-Abelgases. Althoughthederiva-
tionsaregearedtowardthefunctionalrequirementsofthecommercialFluent
code, the results are equally applicable to all computational (cid:8)uid dynamics
solvers. Alsopresented isabriefnumericalexampleforatypical propellant,
highlighting the di(cid:11)erent thermodynamics of the Noble-Abel and ideal gas
equations.
APPROVEDFORPUBLICRELEASE
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Publishedby
DefenceScienceandTechnologyOrganisation
POBox1500
Edinburgh,SouthAustralia5111,Australia
Telephone: [phone]
Facsimile: [phone]
'CommonwealthofAustralia2005
ARNo. 013-525
November,2005
APPROVED FORPUBLICRELEASE
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·e Noble-Abel Equation of State: ·ermodynamic Derivations
for Ballistics Modelling
EXECUTIVESUMMARY
Accurate modelling of gun interior ballistics promotes more e(cid:14)cient gun and pro-
pelling charge design. In order to simulate interior ballistic (cid:8)ow(cid:12)elds, such models re-
quire a description of the thermodynamic behaviour of the propellant gas. ·e Noble-
Abelequationprovidesasimpleandreasonablyaccurateequation-of-stateforpropellant
gasesatthehighdensitiesandtemperaturesexperiencedinguns.
Most computational (cid:8)uid dynamics-based ballistics models, however, require addi-
tionalthermodynamicfunctionswhichmustbederivedfromtheequation-of-state. ·is
notepresentsthederivationofarangeofthermodynamicfunctionsforNoble-Abelgases.
·eyinclude:
(cid:15)
Entropy,
(cid:15)
Speedofsound,
(cid:15)
·efunctionalformofthespeci(cid:12)cheats,
(cid:15)
·erelationshipbetweenthespeci(cid:12)cheatsandgasconstant,
(cid:15)
·eisentropicprocess,and
(cid:15)
Variouspartialderivativesofdensityandenthalpy.
Althoughthederivationsaregearedtowardthefunctionalrequirementsofthecommer-
cial Fluent code, the results are equally applicable to all computational (cid:8)uid dynamics
solvers.
Also presented is a brief numerical example for a high-energy tank gun propellant
(JA2),demonstratingthedi(cid:11)erencebetweentheNoble-Abelandidealgasequations-of-
stateandtwoofthederivedfunctions.
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Contents
Nomenclature ix
1 Introduction 1
2 ·ermodynamicFunctions 3
2.1 EquationofState . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Speci(cid:12)cHeats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 ·eSpeci(cid:12)cGasConstant . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.5 ·eIsentropicProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.6 SoundSpeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.7 PartialDerivativesofStateVariables . . . . . . . . . . . . . . . . . . . . . 7
3 Example 8
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Figures
1 PressureasafunctionofdensityandtemperatureforJA2 . . . . . . . . . . . 9
2 SoundspeedasafunctionofdensityandtemperatureforJA2 . . . . . . . . 10
3 EntropyasafunctionofdensityandtemperatureforJA2 . . . . . . . . . . . 10
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Tables
1 PropertiesofJA2PropellantGas . . . . . . . . . . . . . . . . . . . . . . . . . 9
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Nomenclature
a
SpeedofSound[m/s]
b
Co-volume[m3/kg]
c
p Speci(cid:12)cHeatatConstantPressure[J/(kgK)]
c
v Speci(cid:12)cHeatatConstantVolume[J/(kgK)]
E
IntensiveTotalEnergy[J/kg]
e
InternalEnergy[J/kg]
F
VectorofFluxes
h
Enthalpy[J/kg]
n(cid:136)
NormalUnitVector
P
Pressure[Pa]
R
Speci(cid:12)cGasConstant[J/(kgK)]
s
Entropy[J/(kgK)]
S
SurfaceArea[m2]
t
Time[s]
T
Temperature[K]
u
VelocityVector[m/s]
U
VectorofConservedVariables
v
Speci(cid:12)cVolume[m3/kg]
V
Volume[m3]
a
Coe(cid:14)cientforIntermolecularA(cid:145)raction[m5/(kgs2)]
g
RatioofSpeci(cid:12)cHeats
r
Density[kg/m3]
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