Electric batteries and fuel cells modeled by Bondgraphs

Electric batteries and fuel cells modeled by Bondgraphs

Simulation Practice and Theory 7 (1999) 613±622 www.elsevier.nl/locate/simpra Electric batteries and fuel cells modeled by Bondgraphs Jean Thoma Tho...

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Simulation Practice and Theory 7 (1999) 613±622

www.elsevier.nl/locate/simpra

Electric batteries and fuel cells modeled by Bondgraphs Jean Thoma Thoma Consulting, Bellevueweg 23, CH 6300 Zug, Switzerland Received 16 March 1999; received in revised form 1 September 1999

Abstract In an electric battery, electric charge ¯ows against the electric ®eld, driven by the concentration gradient or chemical tension. Outside it ¯ows with the electric ®eld through the load resistor to which it supplies energy. The whole is well represented by a Bondgraph (BG) and we develop the associated equations, especially for the element SPAC (see Section 2), which a€ords the coupling of chemical and electric ¯ows. So it is a case of coupled reactions, driven by the concentration gradients between the two battery compartments. The electric charge is taken in ions against its potential gradient, driven by the chemical tension or potential. The BG has an electrical and a chemical part, connected by two elements SPAC. There is also a ¯ow source in the chemical part, which is driven when an external current ¯ows. The reaction proceeds between two multiport C which represent chemical e€ort sources and entrains the electric charge. The whole is programmed and simulated by the 20SIM program and shows the switching on and o€ of electric current and the gradual equalization of concentrations with depletion of the voltage: the battery is discharged. Essential is the selective membrane, that divides two compartments with di€erent concentrations, and lets one species of ions run through. Fuel cells are similar but have two constituents, hydrogen and oxygen, and one product, water. Other substances can be used. Ó 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Batteries; Chemical tension (potential); Electrolyte; Ions; Fuel cells; Bondgraphs

1. Introduction In a common electric battery, electricity or electric charge ¯ows conventionally from the plus pole to the minus pole under the in¯uence of an electric ®eld, where

E-mail address: [email protected] (J. Thoma) 0928-4869/99/$ - see front matter Ó 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 8 - 4 8 6 9 ( 9 9 ) 0 0 0 2 5 - 7

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it goes through an external electric load or resistor and delivers the useful power. Since charge is conserved, the same charge must ¯ow in the battery from the minus to the plus pole against the action of the electric ®eld. This is under the action of chemical tension, usually called chemical potential. These are the ideas of Falk [1]. We take here chemical potential according to him as a fundamental quantity of physics, and not as the free enthalpy per mass, which is usual but only valid for special conditions. We shall henceforth, as said, call it chemical tension to emphasize the analogy with electrical tension or voltage. We propose here to cast this in the Bondgraph (BG) language, which, with its power signs, causalities and signal bonds (activation), and even multibonds, is very suitable for such problems. Chemical tension is an e€ort variable in the sense of BGs, with the molar mass ¯ow as complementary ¯ow [2]. So in the battery we have coupled ¯ows, where the chemical ¯ow goes from higher to lower concentration and drags the electric charge along against the electric ®eld. The electrolyte, usually a liquid will dissociate into positive and negative ions and we have selective walls which prevent the migration of one ion between the two compartments. The migration of ions follows the di€erence of concentrations. Chemical tension in the concentrated part is larger and drives the migration to low concentration. Thereby it takes the electric charge along against the potential increases. This is in a nutshell the operation of an electric battery. For details, see Ref. [7], p. 133. This article will establish a BG along these ideas, really a multi-BG with voltage and chemical tension as e€orts and current and molar ¯ow as ¯ows. Naturally, the concentration is connected with the molar ¯ow by a simple integration; this establishes the concentrations. As always with BGs, we bring out the main physical e€ects, and we bring a simulation by the 20SIM program [3].

2. Development of the theory As shown in Fig. 1, there is a tank ®lled with a conducting liquid, as example with diluted HCl. This forms the so-called electrolyte, where the liquid HCl dissociates into positive and negative ions. These are, here, the positive hydrogen ions and the negative chlorine ions. Both ions carry charge and want to migrate, but the hydrogen ions are blocked by the selective membrane. The chlorine ions can migrate with negative charge, perhaps under a certain migration resistance. In the tank there are two electrodes of metal, which are connected outside the tank over the electric load resistance. The essential part is, that the electric charge for the ions must by supplied from the electrodes, where it comes from the outside, through the electrical branch. So on the positive electrode electrons are supplied, migrate against electrical tension, as said under the in¯uence of the concentration gradient, and are given up again at the negative electrode to form neutral HCl at right. Since the charge of electrons is negative, the conventional current travels from the negative to the positive electrode.

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Fig. 1. Electrolytic tank with two electrodes, connected by an electric resistor on top. The membrane in the tank separates electrolyte from high to low concentrations. Chlorine ions can migrate through the membrane and take electric charge along against the electric ®eld.

The electrons cannot travel trough the electrolyte and must be supplied to the electrode by the electric wire. Within the electrolyte, we have the so-called electrochemical potential lcl ‡ eu;

…1†

where e is the elementary charge and u is the voltage. The transport of the chlorine ions is coupled to the electric current due to charge conservation i …2† n_ cl ˆ : e Essential is now, that the concentration of chlorine is much higher left than right, which gives rise to a di€erence of chemical tension by l1cl ÿ l2cl ˆ RT

ln n1cl : ln n2cl

…3†

It drives ions from left to right and takes the electric charge along. Near to the membrane we have a depletion layer, that is the chemical tension decreases slowly, not abruptly at the membrane. This generates a voltage as follows: u ˆ e…qH ÿ qcl †:

…4†

The net charge generates a voltage by a basic equation of electricity (5) d2 u qnet qH ÿ qCl ˆ : ˆ e e dx2

…5†

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In the right compartment, the chlorine density is positive compared to the smaller hydrogen mass density and we have the curves of u bending upwards. This gives the indicated form of the curves. At the membrane itself there is half the di€erence of electric tension, in accordance with the fact that the number of uncompensated charges is the same in both compartments due to electro-neutrality. This is almost a series (or 1-) junction, combined with a transformer for the elementary charge e. It has been called IONI before [4], because an unknown resistance can appear. Here we use another element, which we call SPAC. This is a non-standard BG element, but convenient for battery work. It has four letters to distinguish it from standard BG elements, which have 1±3 letters [2]. We could give any name, but we use the four letter combination SPAC, which reminds one vaguely of space charge. The idea is then to treat it as a power conserving element, that involves the electric and chemical strands of a multiple BG. In the example of Fig. 1, electrodes are made from the metal platinum, which allow the hydrogen ions to penetrate the surface. They are surrounded by free hydrogen supplies by external tubes, where it can bubble up the electrodes. The liquid is here HCl which dissociates into positive hydrogen. Let us repeat, the selective (or semipermeable) membrane in the middle allows chlorine ions to pass freely but blocks the hydrogen ions. Such membranes are also found in thermodynamics for explaining mixtures [4]. They can be a thin layer, as usually associated with the word membrane, but the main action is the selectivity. In nature there are many more or less perfect examples of them. Even electrons can run through the electrodes and the electric wire, but not through the electrolyte; so this can be considered a selective membrane too (see Fig. 2).

Fig. 2. Details with the electrodes, the membrane and the development of chemical tension of chlorine and the electric tension due to the space charge.

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Important is to note, that on the membrane itself, there is exactly one half of the di€erence of chemical tension and voltage. This comes from electro-neutrality, that is the negative charge at left must balance exactly the positive charges at right. Hence on the membrane itself there is drop of chemical tension l1b ÿ l2b ˆ 1=2 

ln n1cl ln n2cl

…6†

and a corresponding increase of voltage u1b ÿ u1 ˆ …l1d ÿ l2d †=2e:

…7†

Important is that the di€erences of voltage and chemical tension depend only on the di€erence of HCl concentrations in both compartments according to Eqs. (6) and (7), and not on the space coordinate. These are needed to calculate the thickness of the depletion layer according Eq. (5), but not on the global functioning of the battery. For the constitutive equation of the multiport C, we calculate ®rstly the molar masses by integration of the molar ¯ows, and have then the chemical tension, and also an intermediate quantity a as follows: Z n1 _ dt n; l1 ˆ prop n1 ; a ˆ K=2  log : …8† n1 ˆ n2 This will be used for the BG, see Section 4. More complex dependencies on the molar masses can of course be introduced.

3. Writing the Bondgraph These ideas are displayed by the BG of Fig. 3, with the electrical side or branch on top and the chemical side below. The electrical branch has a load resistor in the top middle, modulated by a consumer signal. Next there are the two coupling capacitors, which are necessary to establish the correct causalities. As most coupling capacitors, they have no real physical equivalent, but some capacity near to the electrodes can always be found as an excuse. The chemical branch starts from Clef with the HCl density n1 at left and goes to the Crig with HCl density n2 at right, where n1 is much larger than n2, so that we have a chemical tension by Eq. (3). The arrow on the R-elements on top indicates that it can vary: that is the variation of electrical load of the battery. Electric and chemical branches combine in the elements SPAC as described by Eqs. (6) and (7). Between them we have a multibond, with a chemical and an electric branch, corresponding to the migration of the charged ions in the electrolyte. The electric strand (part bond) has a dissipation resistor and drives a ¯ow source on a series junction. It drives in turn the molar (chemical) ¯ow from left to right. This source and the resistor could be combined in a multiport R, but in this way one sees the operation better. What remain are the elements SPAC, which are power conserving structures. Power is taken from the chemical branch and transferred to the electric side. This is really the source of electrical power. As said, this goes by transfer of voltage

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Fig. 3. BG of the battery, with the chemical tanks Clef and Crig on bottom and electric load resistor on top. The electric tension is generated in the elements SPAC against a corresponding drop in chemical tension.

and chemical tension, and both SPACs depend on the densities n1 and n2, as indicated by the full arrows. The operation of elements SPAC is shown by a kind of BG or graphical way in Fig. 4. The ®gure is strictly not necessary, all the information being given by the equations of SPAC but we ®nd it pedagogically useful. It uses Block-BGs, as introduced in [2], where on one side, e€ort is interrupted to bring energy by addition. The curved line brings this energy from the lower (chemical) bond to the higher (electrical) one, and the hole structure is power conserving as said. The little vertical line on the curved line reminds that in the electrical bond we have another physical dimension, voltage. They are connected by the elementary charge or by FaradayÕs constant. Operation of the battery is then as follows: with small load resistance, really ®nite load conductance, electric current ¯ows through the upper branch and through the middle between the SPACs with an R-element and an SF (¯ow source). This SF drives the chemical branch, it makes current but no power. The power comes from the SPACs, ultimately from the di€erence of concentrations, that is absorbed from the chemical branch and drives the electric branch. If the load conductance is zero, both molar ¯ow and current stop, the battery is at rest. The concentration di€erence becomes less with the ion ¯ow from left to right and the voltage diminishes. When the concentrations are equal, the battery is discharged.

4. Simulation by TWENTESIM We use now the 20sim program [3], where Fig. 5 gives the BG drawn with it. It is similar to our Fig. 3 but has the following special points:

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Fig. 4. Graphical representation of SPAC by block-BG. The little vertical line reminds, that the dimension of voltage is di€erent from chemical potential.

1. The elements SPAC are realized as a set of equations as mentioned by Eq. (8). There is the intermediate variable a calculated in the block CHETEN below. 2. In the block CHETEN, we take the molar densities n1 and n2 from the C-elements and calculate the variable a only once. The code has logarithm function, but with the small quantities b1 to avoid an argument of zero. 3. It is necessary to put a resistance R_2 in parallel with MSF_1, which makes a slight inclination of the source characteristics. 4. The multiplier and the step generator allow, with a ˆ 0, to settle down for about 1 s, and then to start. The load resistor is taken as a conductance modulated by a square wave; this simulates the switching on and o€. Fig. 6 shows a sample run of this simulation, with switching on after the settling time, The load is switched on and o€ by sqrwave_1 and controls all currents and the molar mass ¯ow. One sees also that as the concentrations equalize, the molar mass and the electric ¯ow get weaker: the battery is discharged.

5. Interpretation and conclusion Other chemical reactions can be used, if there is dissociation, as with water in a tank. Here the current generation is reversible: A tank or cell works below a voltage

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Fig. 5. BG for simulation by Twentesim, similar to Fig. 3, but with a block CHETEN on bottom. The block MOG stands for modulated conductance.

Fig. 6. A simulation with rapid switching on and o€ of the load by the block MOG, and gradual equalization of the chemical tensions in Clef and Crig.

of 1.23 V as electric battery as described here. Above, the current reverses and the cell absorbs power. Then water is separated, hydrogen migrates to one electrode, oxygen to the other. The process is used in industrial scale for the production of hydro-

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gen. Another process is the production of aluminum from aluminum oxide. The stoichiometric coecients can, as with all chemical reactions, be taken as BG transformers. All those processes depend on temperature and pressure, but normally one thinks of 298 K and 100 kPa. Also there is always the coupling with entropy ¯ow, which can be expressed by the elements COUPL [4] neglected. They are responsible for the temperature generation which is often important. According to the old adage ``if I can program it, I have understood it'' we have shown an exceptionally simple case to show the basic mechanism of a battery. Real batteries have more complex charge and discharge curves, which can be accommodated by nonlinear C-elements for the constituents. Also the fact that in real batteries the capacity decreases with a decrease of temperature is probably a surface phenomena of the electrodes. It can be represented by BGs that have modulated R-elements between the reaction capacitors. The same holds for the decrease of capacity with discharge current, the so-called Peukert e€ect. It is probably due to cavities in the electrodes, which are blocked by discharged remains, as easily shown by BGs. For a modern view on batteries, from a small razor to large automobile traction batteries, with many interesting designs, cross-sections and production process, see [5]. We have calculated in Volts and Ampere, but for chemical tension we use milli-mole and kilo Gibbs (kG) to get convenient numerical values, following a proposition of Job [6]. 1 The Gibbs (G) has been proposed by [6] and is simply Joules per mole. The constant K of Eq. (8) contains the quantities RT of Eq. (3). In the signal block in the center, we have a quantity K, which calculates the change from electric current to molar ¯ow and therefore equals the reciprocal FaradayÕs constant, 0.01033 millimole/As. A slight modi®cation leads to the fuel cells with one bottle or C-element with hydrogen and two bottles, one for oxygen and the other for water, they are connected as all chemical reactions by another chemical resistance behind a series junction. The operation is as follows: hydrogen comes from the left, is ionized, goes through the membrane dragging along the electric charge. At right it is de-ionized and combines with the oxygen to form water. In practice one electrode is surrounded by hydrogen, the other by oxygen and the resulting water appears in the tank. Adding a little acid facilitates the ionization. We have developed the electric batteries as a BG. The reticulation works ®ne, somewhat similar to chemical reactions which are known in biochemistry since a long time [8]. As we have seen, batteries work essentially because of the coupling of electric charge with chemical ¯ow, that goes through a selective membrane. 1 Jean Thomas will send a copy of this interesting publication, which is dicult to access and an English translation of the preface to interested people.

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Electric batteries are with us since about 200 years and have not much improved since about 100 years. Yet they would be so much needed for electric automobiles and solar electric power to bridge the gap of day and night. Perhaps the BGs can make them better.

References [1] G. Falk, Energy and Entropy, Springer, Berlin, 1976. [2] J. Thoma, Simulation by Bondgraphs, Springer, Berlin, 1991. [3] J.F. Broenink, P. Weustink, PC Version on the Bondgraph Modeling, Analysis and Simulation Tool CAMAS, ICBGM; SCS publishing, San Diego, CA, USA, 1995, p 203. [4] J. Thoma, B. Bouamama, Modeling and Simulation of Thermal and Chemical Processes, Editions PPUR, Lausanne, Switzerland in French and English, Springer, Berlin (to appear in 1999). [5] L. Trueb, P. Ruetschi, Batterien and Akkumulatoren, Springer, Berlin, 1999. [6] G. Job, H. Schroedel, The values of the Chemical Potential, Verlag, Hannover, Germany, 1981. [7] E. Wiberg, Die Chemische Anit at, Verlag Walter de Gruyter, Berlin, 1972. [8] OKK (Oster, Perelson, Katchalsky) Network Thermodynamics: Dynamic Modeling of Biophysical Systems, Quarterly Rev. of Biophysics 6 (1973).