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Application of a Real Gas Model by Van Der Waals for a Hydrogen Tank Filling Process

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Application of a Real Gas Model by Van Der Waals for a Hydrogen Tank Filling Process Application of a Real Gas Model by Van-der-Waals for a Hydrogen Tank Filling Process Application of a Real Gas Model by Van-der-Waals for a Hydrogen Tank Filling ApplicationProcess of a Real Gas Model by van der Waals for a Hydrogen Kormann, Maximilian and Krüger, ImkeTank Lisa Filling Process 665 Maximilian Kormann1 Imke Lisa Krüger2 1Dassault Systèmes, Germany, [email protected] 2Dassault Systèmes, Germany, [email protected] Abstract should take less than 200 s for 5 kilogram (US DOE et al., 2009). For safety reasons, the temperature inside the tank Hydrogen fuel tanks operate at high system pressure lev- may not exceed 85 ◦C. els. In these regions effects occur, which cannot be han- Due to its small molecules, hydrogen shows a dif- dled by an ideal gas model. One of these is the Joule- ferent behavior when compressed or expanded than e.g. Thomson effect. It describes an adiabatic throttling with- air. When air is expanded isenthalpic, temperature sinks out change in enthalpy, but a change in temperature. The where as for hydrogen the temperature rises. This is due tank filling process can be simplified to a throttling valve, to the different signs of the Joule-Thomson coefficient, so the effect is of interest. In this investigation the van der that describes the temperature evolution during a pressure Waals equations are implemented in a real gas model for change. the Hydrogen Library and the Pneumatic Systems Library To design the tank filling processes and identify cooling (PSL) by Dassault Systèmes and the model is applied to needs during the injection, system simulation is the best a hydrogen tank filling process. Performance and accu- way. We will demonstrate in this paper that the van der racy are compared to the CoolProp fluid properties library, Waals approach can be used for hydrogen to evaluate tank which is imported with the ExternalMedia library. filling processes at an early stage of development. Keywords: Hydrogen, Pneumatic Systems Library, Tank Filling Process, Medium Models, Real Gas, van der 2 Thermodynamic background Waals, Joule-Thomson-Effect, CoolProp To describe the temperature change at an adiabatic throt- 1 Introduction tling some thermodynamic basics have to be given. 2.1 Ideal Gas The calculation of thermodynamic properties plays an im- portant role in simulating thermohydraulic systems. The The ideal gas model assumes the fluid to consist of dimen- specific behaviour we want to study as well as the pressure sionless particles which can move freely in the given vol- and temperature range define the exactness needed of the ume until they hit another particle or the wall. The thermal medium properties model. The ideal gas approach is only equation of state between the pressure p and temperature valid for low pressures, complex approaches that cover the T is defined according to (Adkins, 1983; Tschoegl, 1983) liquid phase as well slow down the simulation speed. with the specific gas constant R: The van der Waals model gives a simplified analytical approach and it assumes the deviation to be acceptable for RT = pv (1) e.g. early stages of design or if only the qualitative be- and the specific volume havior should be shown. The Joule-Thomson effect that is to be examined in this paper can be analytically modeled V 1 v = = (2) and studied with the van der Waals approach. It results in m r simple equations for e.g. the internal pressure. The model has been implemented for the Hydrogen and 2.2 Real gas equation by van der Waals Pneumatic Systems Library by Dassault Systèmes. New gases can be easily added, the only parameters needed are The ideal gas model lacks two effects: Interactive forces the molar mass of the gas, the critical pressure and temper- between the fluid molecules and their dimension. For the ature, the sutherland constants and the table-based specific first effect, the pressure in the van der Waals model (der internal energy. Waals, 1967) is increased by the internal pressure of the gas. The influence of the dimension of the molecules is A recent application where this effect occurs is the hy- considered by reducing the specific volume by the volume drogen tank filling processes for fuel cell vehicles. Hy- of the fluid molecules: drogen has to be stored at high pressures e.g. 200 bar due a to the low energy density at low pressures. In order to RT = p + (v − b) (3) be competitive to conventionally fueled cars, tank filling v2 DOI Proceedings of the 13th International Modelica Conference 665 10.3384/ecp19157665 March 4-6, 2019, Regensburg, Germany Application of a Real Gas Model by Van-der-Waals for a Hydrogen Tank Filling Process The two new parameters can be derived from the critical 2.4 Caloric equation of state state of the fluid (Weigand et al., 2008). The internal pres- sure is defined using the cohension pressure parameter: Assuming a constant amount of mass the total differential of the internal energy is (Weigand et al., 2008): 27(RT )2 a = c (4) 64pc du du du = dT + dv = cvdT + pT dv (16) The volume occupied by the molecules can be written di- dT v dv T rectly: RTc For an ideal gas, the internal pressure is zero and thus the b = (5) 8pc internal energy only depends on the temperature with the specific heat capacity at constant volume c . 2.3 Internal Pressure v The internal pressure of a fluid is defined as the partial 2.5 Specific Enthalpy derivation of the specific internal energy u by the specific volume at constant temperature (Moore, 1986): The enthalpy describes the energy of a system. The en- thalpy can only change when energy is transferred over du the system border. The specific enthalpy of a fluid is de- pT = (6) dv T fined as (Weigand et al., 2008): It can be derived from the fundamental thermodynamic relation with the specific entropy s under assumption of h = u + pv (17) a constant amount of mass (Callen, 1985) by deriving the internal energy partially by the volume at constant temper- For an ideal gas the expression simplifies to ature: du = Tds − pdv (7) h = u + RT (18) du ds = T − p (8) dv T dv T As described according to equation (16), for an ideal gas With the Maxwell-Relation (Weigand et al., 2008) the internal energy only depends on the temperature. In this case, also the enthalpy only depends on the inner en- ds d p = (9) ergy. This is not the case for a real gas. dv T dT v the expression for the internal pressure becomes: 2.6 The Joule-Thomson effect du d p When a gas is expanded from a high pressure over an isen- pT = = T − p (10) dv T dT v thalpic valve, a change in temperature can be observed. For an ideal gas we get from equation (1): This behaviour cannot be described by an ideal gas model as it does not model a pressure dependency of the specific d p R = (11) enthalpy as mentioned in section 2.5. The effect is de- dT v v scribed by the Joule-Thomson coefficient (Greiner et al., d p R 1993): pT = T − p = T − p = 0 (12) dT v v dT As the ideal gas equation does not model forces between mJT = (19) the fluid molecules except elastic collisions, the internal d p h pressure is zero. Thus the Joule-Thomson effect cannot be covered by an ideal gas law. In the real gas equation the To derive its equation, the total differential of the spe- van der Waals forces between the molecules are covered cific entropy under assumption of a constant amount of by the internal pressure term. From equation (3) we get: mass will be set into the fundamental thermodynamic re- d p R lation from equation (7). It reduces the dependency of the = (13) change in enthalpy to pressure and temperature: dT v v − b d p R pT = T − p = T − p (14) ds ds dT v v − b ds = dT + dp (20) dT p d p T R RT a a = T − − = (15) dh = du + pdv + vdp (21) v − b v − b v2 v2 ds ds The internal pressure for a real gas modeled with the van dh = T dT + T dp +Vdp (22) der Waals equation is always positive. dT p d p T 666 Proceedings of the 13th International Modelica Conference DOI March 4-6, 2019, Regensburg, Germany 10.3384/ecp19157665 Application of a Real Gas Model by Van-der-Waals for a Hydrogen Tank Filling Process With the definition of the specific heat capacity at con- As there is no easy-to-handle analytical expression for stant pressure cp and the Maxwell-Relation for the Gibbs cv and to avoid the use of polynomials, the temperature- energy we get: dependent part of the specific internal energy and the inter- nal pressure correction of the reference point (marked with ds an under-brace in equation (31)) is provided as a table- T = cp (23) dT p interpolation uT (T). Then we get the following equation for the specific enthalpy: ds dv = − (24) d p T dT p a h = uT (T) + pv + (33) dv v dh = cpdT − T dp +Vdp (25) dT p 3.3 Parametrization Now we can set dh = 0 and rearrange: Compared to ideal gas models, the additional parameters for creating a new gas model are pressure and temperature ! dT 1 dv in the critical state as well as the tables for the specific = = T − v mJT (26) internal energy and heat capacity (see table1).
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