PAYNTER S VERIDICAL STATE EQUATION IN INTEGRAL CAUSAL FORM

Transcription

1 PAYNTER S VERIDICAL STATE EQUATION IN INTEGRAL CAUSAL FORM PETER BREEDVELD University of Twente, Control Laboratory, EL/TN 85 P.O Box 7, 75 AE Enschede, Netherlands phone: , fax: , P.C.Breedveld@el.utwente.nl Keywords: van-der-waals gas, thermodynamics, -based description, initialization, integral causality Abstract Compared to the ideal gas law, the van-der-waals state equation considerably improves the thermodynamic description of a substance at a qualitative level. However, the quantitative accuracy is in many cases not sufficient to obtain reasonably realistic simulation results. Many state equations with a more satisfying numerical accuracy have been proposed. However, most do not satisfy the conditions derived in previous work [Breedveld, ] for symbolically deriving the preferred integral causal form of the three- storage element that describes the energy storage in a gas. Paynter [985] has introduced a modification of the van-der- Waals equation that may be considered the necessary quantitative improvement in order to obtain realistic simulation results, which he called the veridical state equation. However, like the common form of the van-der-waals equation, his state equation is not in the full integral causal form required for constitutive relations of power s of an energy-based dynamic submodel and thus not optimally suited for simulation. Consequently, it has been hardly used in -based, plug-&-play modeling and simulation. This paper shows that the integral causal form of the corresponding constitutive relations can be symbolically derived and introduces a suitable initialization strategy. Some examples of its use in simulation are shown. INTRODUCTION The van-der-waals state equation considerably improved the thermodynamic description of a substance at a qualitative level with respect to the ideal gas law, as it shows phase transition (intrinsically unstable region, cf. [Breedveld, 99; ]) and a critical point. Although many close quantitative approximations to the measurement data for the state equation p(v,t) of pure substances exist (e.g. [Çengel and Boles, 989] or [Gyftopoulos and Beretta, 99]), these relations commonly require specification of quite a few parameters and, more imantly, they require the temperature as a dynamic input variable, i.e. differential causality of the thermal in bond graph terminology. Although specifying the state of matter by its temperature makes much more sense than specifying its initial entropy, one should only conclude from this that initialization of storage elements in dynamic simulations is better done in terms of the temperature, not the entropy. However, the dynamic state itself should be the entropy in order to prevent numerical differentiation. Previous work demonstrated that if cv is assumed constant, the preferred integral form ( causality ) of the thermal can be found in case of the ideal and the van-der-waals gas, but not for the more accurate curve fits of the measurement data commonly used in simulation, as they do not depend linearly on the temperature. Karnopp and Rosenberg [975] solved this causality problem for the ideal gas law by formulating the constitutive relations of the two- C describing a fixed amount of gas in integral causal form. The author showed how this can be extended to a three- C describing a variable amount of gas [Breedveld, 98] and later how a substance characterized by the van-der-waals equation can be described by a three- C element with constitutive relations in integral causal form [Breedveld, 99, ]. This paper takes Paynter s veridical (literally: truth-speaking ) state equation as a starting point to find the constitutive relations of a three- C- type storage element in integral causal form that describes a pure substance also quantitatively in a reasonably accurate manner, even though Paynter considered his results as an intermediate step, predicting that further modifications along the lines of his extension with a non-analytical term would give even more accurate results. If such other extensions are found, it will have to be tested in each case whether or not these forms satisfy the conditions for derivation of the integral causal form. Naturally, an alternative path would be to try to improve the numerical accuracy of the integral causal form. However, this would go beyond the scope of this paper. Some typical simulation results are shown and compared with the results based on the ideal gas law and the van-der-waals equation, in order to demonstrate that the results lead to a functional implementation. Although the essence of Paynter s extension is that it contains a non-analytic term, it is still such that the pressure depends linearly on the temperature and that it leads to analytically integrable expressions. These are two crucial conditions on a state equation in order to be able to find an integral causal form [Breedveld, ]. The models will not include the dynamics of phase transitions, i.e. the (saturated) vapor state is assumed below the critical temperature and the results for the region of co-existing phases will not be accurate. Hence this approximation is mainly intended to get good results for temperatures close to, but above, the critical temperature, or for low densities at temperatures below the critical point. Before a simulation is possible, initial conditions should be set. At this point an additional reason for the commonly used differential causality of the thermal becomes clear: the reference value for the entropy just plays an arbitrary role in the relations, but a non-zero reference temperature should be provided, due to the nature of the constitutive form. This means that relations should be found between common sets of initial values like (T, p, V ), (T, p, N ) or even (T, V, N ), and the required initial states (S, V, N ). These relations are not trivial and will be given

2 special attention, in particular (T, p, N ), which will be used in the simulations. PAYNTER S MODIFICATION Paynter [985] proposed a much closer approximation of the state equation of pure substances than the one by van-der-waals by making a non-analytic extension. However, his equation is not in optimal form for numerical simulation either, as the thermal of the multi capacitor characterized by this equation is still in differential causal form. This severely restricts dynamic simulation to implicit numerical integration schemes and it often leads to models in which thermal damping is ignored without justification, as demonstrated by Pourmovahed and Otis, 98]. In this paper the integral causal form is derived assuming cv to be constant, viz. cv = ( n+ ) R, where R is the universal gas constant and n the number of atoms per molecule (e.g. n = for nitrogen (N), used in the example simulations). The relations are written such that they characterize a proper three- C-element, by allowing variation of the amount of gas (addition of a material ) in principle. Paynter s structural extension consists of adding a term ρr to the van-der-waals equation, where ρ r is the reduced density. He also adapted the reduced van-der-waals parameters a and b from their common values, viz. 3 and /3 respectively, to 7 and / respectively, although he clearly states that these are parameters to be fitted to actual data. In order to sup that, a and b are expressed herein in terms of the critical values of the temperature Tc, pressure pc and the compressibility factor Zc, thus allowing the user to obtain an even closer approximation for the pure substance being simulated (note that the subscript c will be used to indicate the critical point). The example simulations will be performed with the values used by Paynter, i.e. 7 and / respectively, although some multiple run tests were performed varying a and b, which showed that a=6 gives a result that much closer approximates figure. in [Gyftopoulos and Beretta, 99]. The relation given by Paynter is [985] using the same symbols: 3 ( ) v x pr = + ρr M Tr Aρr ( Tr) () v with M = M = ( bρr ) for the van-der-waals form, v M = M = ρr M = ρr ( bρr) for Paynter s veridical x form and A= a = a pcvc where a is the common van-der-waals parameter. The other common van-der-waals parameter b is related to b by b = bv. For the reduced specific volume holds c = ρr, such that ( ) sgn( ) apc p = p ( ) 3 ctr T r = ( b) a ξ( ) apc = + pt 3 c r ( b) using the non-analytical term ξ = sgn( ) = ( ). Also note that Paynter used reduced variables throughout his 985 () paper, even though he omitted the common subscript r. These reduced, i.e. dimensionless, variables are not the required variables for -based submodels. Herein, in order to keep notation compact, we will take the following strategy. We first assume that after integration of the flows of the s of the submodel into the extensive states S (entropy), V (volume) and N (amount of moles), first the intensive states s and v are computed by dividing by N and next is computed by dividing by vc. It then suffices to derive the temperature T(s, ), the pressure p(s, ) and the total material potential µ tot(s, ) in order to obtain the efforts of the corresponding s in integral causality. It is assumed that cv is independent of T, such that UVSN (,, ) = NcTVSN v (,, ) = NcTsv v (, r) resulting in the required expression for the effort of the material (cf. [Breedveld, ]): tot µ (, s ) = u+ pv Ts= ( cv st ) + pv= (3) = ( cv s) T( s, ) + vcp( s, ) Note that the relation for µ tot may be omitted if the amount of moles N is kept fixed at all times which means that the C-element reduces to a two-. Equation () already provides the desired relation p (T, ), however for later use it is already written in separately integrable terms as follows (Note that > b always holds as b would mean the presence of a solid phase for which this relation is not adequate): a ab T p = pc ξω+ + ap c () ( b) ( b) T c where 6 ( b) Ω= ( b) ( b) ( b) ( b ) The compressibility factor Z, which is a measure for the deviation from the ideal gas law for which Z is always equal to, is defined as Z = pv/ RT. Its critical value Zc = pv c c/ RTc is often used as one of the parameters to specify pure substances. The van-der- Waals equation in reduced form can thus be written: pr = Tr ( Zc( b)) a (5) For large values of, Paynter s relation should converge to the van-der-waals form, hence Zc = ( b) and consequently b is found in terms of Zc: b = Zc. The Riedel parameter at the p critical point ac becomes [Paynter, 985]: a r c = = + a. Tr v c, T c Hence a = a c, such that the reduced van der Waals constants a and b can be expressed in terms of the parameters ac and Zc commonly known for real substances. In the example simulations we will follow Paynter and choose ac = 8 and Zc = /5 =.7, corresponding to a = 7 and b =.5, to obtain values close to the average for many substances. Gyftopoulos and Beretta [99] show in their Table. that most of the listed pure substances have values that lie within % of this value for Zc. However, the plug & play -based submodel that is proposed will require only

3 specification of pc, Tc, ac and Zc from which vc, a and b are derived in an initial section of the model (only computed at the first time step), such that the results can be easily adapted to the pure substance at hand. Naturally, the plug & play submodel can be easily modified to allow specification of a substance by any desired parameter set. DERIVATION OF THE INTEGRAL CAUSAL FORM FOR THE THERMAL PORT Expressions for p (T, ) = p(t(s, ), ), viz. () and µ(t(s, ), p(t, ), s, ), viz. (3), have been found already, which means that, if T is known, the mechanical and material are in integral causality. What rests is to find T(s, ), which is the key issue if the thermal has to be in integral causality too. In previous work [Breedveld, 987, ] it has been shown that if p(v, T) depends linearly on T, and if the resulting expression for ds is integrable, the required T(v, s) can be found. It is obvious that Paynter s relation satisfies the first condition and consequently p cv ds = dv dt T + or dv= T ds ZcR p r = d + dt (6) cv cv Tr T d = needs to be integrated (cf. [Breedveld, ]). The result of the integration depends on the two possible integration intervals, which are most easily handled if the arbitrary reference is chosen in the critical point: v = vc so =, T = Tc, s = sc. Note that this reference is not necessarily equal to the initial equilibrium state from which a simulation can be started. From () follows pr = Tr d = 6 b = ξ + + (7) 3 ( b) ( b) ( b) ( b) ab a ( b) ( b) and thus pr ( b) ξ d = λ( ) = b( b ) ln + T r v d r b = (8) 7 + ( b) ln + + v ( ) r + ξ + a b b b such that an expression for s is found that can be used for initialization p r T T s = ZcR d + cvln = ξλ+ cvln (9) T r T d c T = c v ( ) with ( ) ( ) ln r b Λ= ZcRλ = ZcRb b + Rln + b ZRv c ( r ) v ( ) r + ξ + a b b b and finally the required form for T is obtained: β γ ( b) T = T c expξ () b where RZc( ) 7 Ξ= + v ( ) r + ξ + a + s () cv b b b 3 with γ = bb ( ) Zc = ( 3 Zc) (Zc ) and β = ξrc v. INITIALIZATION As the choice for the initial value of the entropy is arbitrary, some attention should be paid to the proper initialization of the 3- C-element characterized by the expressions derived above. The available volume V, the initial temperature T, and the initial pressure p commonly specify the initial state of a substance in a container. It is thus essential to derive the initial amount of moles N and the initial entropy S before a simulation is started, e.g. in an initial section of the submodel. The initial amount of moles N can be found from the initial (reduced) density, which is a function of pressure and temperature. However, in case of the van-der- Waals equation this is a third order polynomial and in case of Paynter s extension a fourth order polynomial in the density: ( ρr ) + ξ( bρr ) ( Tr ) aρr + prtr = () Apart from using iteration in the initial section of the model, which is numerically somewhat costly, the ideal gas law can be used to approximate the initial amount of moles: N = pv RT. This initial guess can be further improved by using the compressibility factor for initial states close to the critical point, e.g.: N = pv RTZc for Tr <.,.5 < pr < 3. Another solution, especially if the simulation tool does not allow initial iteration, is to set the volume flow to zero, connect sources with the desired initial temperature and total material potential to the thermal and material via relatively small resistors and next to run a dynamic simulation in order to find the equilibrium states that are the initial states to be computed. As soon as N is known, S can be found using (taking sc again as the reference) S = N ξλ + cvln( TTc ) (3) v ( ) ln r b Λ = Rγ + Rln + b with ZRv c ( r ) v r + ξ( + a) b b b or approximated by the much simpler expression for the van-der- Waals case: S = N Rln (3v r ) + cvln( TTc ). Now that we have expressions for all efforts as a function of the integrated flows (extensive states), viz. equations, (3), and () and a way to initialize the storage element ((3)or ()), it is ready for simulation. SIMULATIONS RESULTS A test-bed model in a plug & play modeling environment [- sim, ] in which the three C (van-der-waals or Paynter) can be easily replaced is used to generate some simple example

4 env. temp. Se env. temp. Se heat conduction RS thermal heat conduction thermal MSf mech. Directsum material C Paynters gas MSf mech. Directsum material C Paynters gas WaveGenerator Figure : Test bed model for the three C representing energystorage in a Paynter gas (direct graphical model input; signals only for initialization) Sf Sf p WaveGenerator Figure : Alternative test bed model with thermal in differential causality Sf Sf p Z log Z pr Figure 3: Z- pr-plot (log-scales), critical isotherm for nitrogen, using integral causality, explicit integration and high heat conduction (computed points are shown) simulation runs (Figure ). A zero flow source (Sf) is attached to the material in order to restrict the example experiments to a fixed number of moles. The thermal is connected to the environmental temperature via a Fourier type heat conduction represented by an RS two- element. The container is assumed to have a piston to be able to vary the available volume, driven by a velocity source at the mechanical which imposes either a sinusoidal signal to demonstrate loop behavior and thermal damping or a constant negative velocity representing compression, followed by a constant positive velocity representing expansion. Combined with a high thermal conduction to the environment to guarantee isothermal behavior, the latter mechanical input is used to create the commonly used log Z log pr plots, which may be used to judge the accuracy of the model. A version of the three with differential causality at the causal provides the... log pr Figure : like figure 5, but using the test bed in Figure (thermal in differential causality) and implicit numerical integration (computed points are shown) possibility to eliminate the heat conduction (RS) completely to provide exact isothermal results (Figure ), but requires use of implicit integration schemes like the BDF-method (DASSL). The log Z log pr plots in figures 3 and demonstrate that the simulation results are identical. Although this may seem to make this whole exercise of deriving a symbolic integration superfluous, the reader should keep in mind that implicit numerical integration is not always a solution, even if it is available, for instance when the submodel is embedded in a model with certain discontinuities or nonlinearities that do not match well with implicit integration. Figure 5 shows cycles in the p-v-plane and T-v-plane near the critical point to demonstrate that the behavior is as expected. Figure 6 demonstrates in a multiple run for n =,, 3 that the thermal damping (area of the cycle) decreases when the heat capacity and thus the time constant increase, i.e. for a higher

5 pr Tr pr Pntr Figure 5: pr-- (o) and Tr--cycles (x) for nitrogen showing thermal damping for a much lower heat conduction than in Figure Figure 6: pr-- cycles for n =,, 3, showing a decreasing enclosed area (thermal damping) for increasing n (Paynter gas) 3.5 a=6 pr compressibility factor Z Figure 7: pr--plot for nitrogen: isotherms for various temperatures (9 39 K in steps of K) below and above the critical point (7 K) number of atoms per molecule. Figure 7 gives the results of isotherms in the pr--plot for nitrogen below and above the critical point (7 K). In Figure 8 the value for a was reduced from 7 to 6 in order to match the resulting Z-pr-plot (log-scales) for nitrogen with isotherms for T =,7 (Tc),65,, 55, 3K, pc=3.3 Mpa with the experimental version in figure. in [Gyftopoulos and Beretta, 99]. Next some comparisons are made between a van der Waals gas and Paynter s modification. Figure 9 shows that the compressibility factor Z takes off to infinity earlier for increasing density ρ. Finally Figure shows pr--cycles for the van-der-waals and Paynter cases. CONCLUSION In this paper the preferred integral causal form of Paynter s veridical state equation [985] has been derived in order to make it available for -based modeling and simulation, in which storage elements in integral causality are preferred and sometimes even required. This is not only done in order to be able to use explicit.. log pr Figure 8: Z-pr-plot (log-scale), for nitrogen with a=6; isotherms for T =,7 (Tc),65,, 55, 3K, pc=3.3 MPa numerical simulation schemes (other nonlinear elements may obstruct the use of implicit integration schemes for instance), but also in order to prevent omission of the thermal time constant due to heat exchange with the environment, as the differential causality of the may mislead the modeler to just choose the environmental temperature. Pourmovahead and Otis [98] have shown that this thermal damping often has a considerable dynamic effect, such that it should not be omitted from a model a priori. As Paynter s extension is non-analytic, the expression was first separated in analytical expressions on separate intervals, substituted in an expression for the total differential of the entropy after which integration could be performed term by term as to obtain the desired expression for the temperature. After rewriting the expression could be collected again using the non-analytic ξ = sgn in order to compensate sign changes and terms ( v r ) ( ξ ) in order to allow additional terms. The resulting expressions were written in terms of the critical pressure, temperature, compressibility factor and Riedel parameter to provide for a generic submodel.

6 5 log Z Pntr log Z vdw pr Pntr pr vdw 3 log rho Figure 9: compressibility factor versus density for Paynter (o) and van der Waals gas (x; van der Waals curve taking off to infinity earlier for increasing ρ) An imant aspect that is often not well understood is that the fact that initial conditions have to be specified in terms of the temperature, due to the nature of the relations, does not imply that the temperature should be the independent variable during simulation. As soon as the initial temperature (and pressure) can be used to find the proper initial conditions for the extensive states, simulation can take place in an optimal way, viz. in integral causality, thus allowing robust explicit integration schemes like Runge-Kutta th order. Herein, expressions were derived and strategies proposed to find the initial extensive states in an initial section of the model (only computed at the first time step) Figure : pr--cycles for van der Waals (x) and Paynter (o) case REFERENCES -sim,, -based modeling and simulation software, see: for more information and free demo (no implicit integration); contact author for models used herein. Breedveld, P.C., 98, Physical systems theory in terms of bond graphs, Ph.D. Thesis, Electrical Engineering, University of Twente, Netherlands, ISBN Breedveld, P.C., 99, An alternative formulation of the state equations of a gas, Entropie, énergétique et dynamique des systèmes complexes, Vol. 6/65, pp , ISSN Breedveld, P.C.,, Constitutive relations of energy storage in a gas in preferred integral causality, to be published in Proceedings IEEE IECON, Nagoya, Japan, October -7,. Çengel, Y.A. and Boles, 989, M.A., Thermodynamics, an engineering approach, McGraw-Hill, N.Y. Gyftopoulos, E.P. and Berretta, 99, G.P., Thermodynamics Foundations and Applications, MacMillan, N.Y. Karnopp, D.C. and Rosenberg, R.C., 97, System Dynamics: A Unified Approach, Wiley, N.Y. Paynter, H.M., 985 Simple Veridical State Equations for Thermofluid Simulation: Generalization and Improvements Upon Van der Waals, ASME Journal of Dynamic Systems, Meas. & Control, Vol. 7, No., pp Pourmovahed, A. and Otis, D.R., 98, Effects of thermal damping on the dynamic response of a hydraulic motoraccumulator system, ASME Journal of Dynamic Systems, Meas. & Control, Vol. 6, No., pp. -6. About the author Peter Breedveld is an associate professor with tenure at the University of Twente, Netherlands, where he received a B.Sc. in 976, an M.Sc. in 979 and a Ph.D. in 98. He has been a visiting professor at the University of Texas at Austin in 985 and at the Massachusetts Institute of Technology in He is or has been an industrial consultant. He initiated the development of the modeling and simulation tool that is now commercially available under the name -sim. In 99 he received a Ford Research grant for his work in the area of physical system modeling and the design of computer aids for this purpose. He is an associate editor of the Journal of the Franklin Institute, SCS Simulation and Mathematical and Computer Modeling of Dynamical Systems. His scientific interests are: Integrated modeling, control and design of physical systems; graphical model representations (bond graphs); generalized thermodynamics; computer-aided modeling, simulation, analysis and design; dynamics of spatial mechanisms; mechatronics; generalized networks; numerical methods; applied fluid mechanics; applied electromagnetism; qualitative physics; surface acoustic waves in piezo-electric sensors and actuators.