Appendix 2:
Aids and Supplemental Material
This appendix contains a collection of material that is relevant to the subject matter of this book. It is intended as a working aid and supplement to the information in the body of the book. The usual lists of the chemical elements, periodic table, electromotive series, etc., that are found in many scientific texts have not been provided because they are readily available in many sources.
A2.1 Properties of Homogeneous Membranes
A series of experiments were performed with five samples of ion transport membranes. Ion exchange membranes have proven to be the best means of reducing diffusion between two compartments of a dual liquid electrolyte cell. These tests were directed at determining the diffusion rate of molecular bromine in a NaBr solution. The relative electrical (ionic) conductance of these membranes was also measured.
A2.1.1 Diffusion Tests
Two 250 ml glass flasks with side tubes were employed per test setup. A membrane sample was placed over the flat ground mouths of the flasks with an RTV sealant. Two flasks were brought together, mouth-to-mouth, with the side tubes facing upwards. Both flasks were then filled with 250 ml of 2 molar NaBr solution. Free bromine was introduced into one of the flasks to make a 1/8 molar solution of Br2. Five such setups were made, one for each of the membranes.
The start times of the tests were recorded. A titration solution of N sodium thiosulfate was prepared to measure the bromine in the dilute side periodically. A 1 ml sample was extracted from the dilute side flask and diluted with a 0.01 N solution of H2SO4 starch and iodine indicator solutions were employed to provide good end-point indications.
The following membranes were tested:
- NAFION N324, untreated, not pre-wetted
- NAFION N324, pre-treated in hot water, cathode side away from bromine
- NAFION N324, pre-treated in hot water, cathode side facing bromine
- RAI R-4010-5, special 6 mils thick experimental membrane
- RAI 4010, standard 3 mil thick membrane.
The diffusion data presented in Table A2.1 are in terms of ml of 0.001 N of titrant needed per ml of sample, after normalizing for the background factors.
Table A2.1 Diffusion rates of bromine across various membranes.
| Elapsed time, hrs | Membrane #1 | Membrane #2 | Membrane #3 | Membrane #4 | Membrane #5 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 24 | 0 | 0 | 0 | 0 | ? |
| 49 | 0 | 1.3 | 1.3 | 0.7 | 2.6 |
| 93 | 0.4 | 5.1 | 4.8 | 2.1 | 7.1 |
The provided data shows that the diffusion rate of bromine through unpre-treated NAFION 324 is 1/5 of that of the R-4010-5 RAI, which is the membrane provided by TRL in the cells delivered to NP.
Now, consider the significance of these numbers if they, indeed, represent the diffusion process accurately. Taking the data from membrane #4, the RAI membrane, about 2.1 × 10–3 N of bromine had diffused across a membrane with an area of about 0.8 in2 after 93 hours. This corresponds to about 1/4 × 2.1 × 10–3 gram equivalent weight of bromine since the solution volume is 250 ml, or 1/4 liter. At 26AH per equivalent weight, we get 26 × 1/4 × 2.1 × 10–3 AH over 93 hours. Or, we have as the current equivalent of the diffusion a figure of 0.014AH over 93 hours. This is roughly equal to 0.10 ma per square inch of membrane area if the concentration difference is 1/8 molar Br2. That is a fairly small number.
This means that a fully charged cell with 2 molar bromine in solution would be discharging at 16 times this rate per in2 of membrane area, or about 1.6 ma/in2 of area. If the cell normally discharges at 200 ma per in2, then the self discharge rate would be less than 1% of the current flow.
A2.2 The van der Waals Equation and its Relevance to Concentration Cells
These types of forces are frequently important to the processes that transpire in any electrochemical system. Where interfaces appear in an electrochemical cell, such as that between the electrolyte and an electrode, we are apt to encounter some form of a van der Waals type of effect and probably mechanisms of adsorption as well. These physical effects are particularly important in the operation of a concentration cell where the storage of a component within an electrode is designed to take place to a very high degree.
It is important to explore the subjects of forces other than the classical, macroscopic ones that are so familiar in physical mechanics because we are seeking an explanation for observed conditions that arise in the experimental concentration cells regarding capacity, electrical, and population densities of species.
An interesting phenomenon was observed in the mid-nineteenth century when various gases were compressed at different constant temperature conditions. Attempts were made to come up with a modified version of the simple gas equation (Boyle’s Law) that would conform to the new experimental data. A gas was seen to depart from the standard relationship for an ideal gas, viz,
(A2.1)
At the low-pressure ranges and lower temperatures, it appeared that within certain ranges, at constant temperature, the pressure remained quite constant while the volume changed significantly. In 1879, van der Waals proposed a relationship of the form
(A2.2)
which seemed to fit the empirical data quite well. If the process involves the adsorption of a gas, then the mathematics and assumptions for a model to describe the process can be structured in a more simple fashion. Hence, much of the early work focused on the circumstances of a gas being adhered to a solid absorbent. It is much easier to treat this sort of situation than that of solids or ions being captured by solid adsorbents. The pressure or force term is replaced in this case by P’ in order to differentiate from that of a pure gas. Obviously, a new term appears for the force to condense the gas or compress the gas to a very small volume or, perhaps, the liquid state.
A2.3 Derivation of Electrolyte Interconnectivity Losses
The efficiency of operation of a multi-cell, full-flow electrolyte array is affected by the dissipative losses through the electrolyte common to all cells that are connected electrically in series. Unless appropriate design compensations are incorporated, these coulombic losses can be significant. Multiple cell arrays considered here are arranged as a series of bipolar electrodes connected together by electrolyte electrically in parallel via manifolding and associated hydraulic (fluid) circuits. A single electrolyte is supplied from a hydraulic circuit in parallel connection as shown in Figure A2.1.
Figure A2.1 Hydraulic circuit in parallel.
The equivalent electric circuit represents half of what one would observe for a two-electrolyte array. A simple circuit is illustrated in Figure A2.2.
Figure A2.2 Equivalent circuit.
R equals the resistance of the feeder channels from the manifold into the cell compartment. The manifold cross sectional area is so large that it contributes very little resistance to the above circuit. An analysis proceeds as follows.
Referring to the circuit in Figure A2.2, for a single loop, or single cell N = 1, where N is the number of series cells, the electric current, i1, in this loop is found by Kirchoff’s law.
For a circuit where N = 2, we have
(A2.3)
as is shown in Figure A2.3 below.
Figure A2.3 A two loop circuit.
The currents i1 and i2 are those in the first and second loops, and they are equal in magnitude.
For a three-loop circuit, the solution of the first loop current is found in similar fashion and is
(A2.4)
Next, as is illustrated in Figure A2.4, i1 = i3.
Figure A2.4 A three loop circuit.
A four-loop configuration (not shown) gives the following as a first loop current,
(A2.5)
where i1 = i4, and i2 = i3.
A five-loop current is represented by
(A2.6)
where i1 = i5, and i2 = i4, and so the progression proceeds.
It is obvious that a symmetrical pattern is developing here. A six-loop current is as follows:
(A2.7)
where i1 = i6, i2 = i6, and i3 = i5.
The general expression for the first and last loop of a circuit in an array with N loops is
It is apparent that the electric current in any one of the series’ coupled loops is not necessarily equal and that its value depends on the number of cells in series. A rapid examination of the relationship to find a repetitive symmetry is shown here.
We will now derive the expression for the current in the nth loop (cell) of an N cell array in terms of E, R, and i1.
As was learned previously, for n = 2, i2 = 2i1 –E/R, and for n = 3, i3 = 3i1 – 3E/R.
By carrying out these steps in a manner similar to the preceding, we can see a pattern developing. The series of steps that must be performed in order to see the format develop are not shown here. However, these are the results of the process.
The general form of the relationship for the nth loop current, In, is
(A2.9)
For those circuits with four or more loops, the above relationship is valid. Examination of the iterative pattern shows that the coefficients a and b are
(A2.10)
The resultant expression is then
(A2.11)
The dissipative currents through interior cells in an array are greater than those through the end cells, necessitating periodic equilibrating or rebalancing of such batteries if they depend on electro-deposition and the removal of materials such as metal plating during charging and discharging.
By substituting the expression for i1, or the first loop current, which is also the last loop current in an array (see equation (A2.8)), into the above equation and solving for n = N/2, the innermost and highest self-discharging rate cells gives
(A2.12)
In the design of modules, the above dissipation factors should be taken into account, and electrolyte channels or feeder tubes should be configured to give the maximum electrical isolation without either unduly increasing hydraulic impedance to the electrolyte flow or increasing risks of passage blocking.
A2.4 Efficiency Calculations
The total energy efficiency of a secondary battery system is exceedingly pertinent when evaluating its importance to applications such as wind, solar, and load leveling uses. In these applications, increasing the efficiency and effectiveness is among the main considerations for selecting a storage mechanism.
Costs associated with the capital equipment and its operation are greatly affected by the overall, or energy turnaround, efficiency. Losses are incurred during charging as well as during the discharge mode. The following can summarize energy loss mechanisms for the battery:
Voltage Losses
- Ohmic potential drops through the electrolyte
- Polarization voltage loss at electrode interfaces
- Electrolyte concentration potentials in the immediate vicinity of the electrodes
- Electrical connections and electrode ohmic resistance.
Coulombic Losses
- Dissipation through electrolyte interconnections via manifolding
- Diffusion of reactant species away from electrodes or into opposite membrane sides
- Plating fall off from electrodes such as metallic zinc or iron
- Gas evolution
- Hydrolysis and irreversible formation of acid or base
- Electrolyte pump power.
All of the above and additional, perhaps somewhat more subtle, losses such as circulating parasitic currents contribute to the turn-around efficiency of redox systems. As an example of the many sources of energy loss, consider the case of a zinc/bromine system. Some of the more predominant loss mechanisms, with some estimate of the magnitude of their contributions to the total, are listed in Table A2.2.
Table A2.2 Energy loss mechanisms in metal/halogen batteries.
| Coulombic losses | Voltaic factor |
| Br2 diffusion 2–3% | Polarization 0.05–0.10 volts |
| H2 evolution ~ 1–5% | Internal resistance 0.05–0.10 volts |
| Zinc dendrites ~ 1% | |
| HBr formation 2–3% | |
| Manifolds < 1% | |
| Pump power ~ 1% |
These values are based on many designs and operating data with numerous large-scale zinc/bromine systems. In all of these instances, the bromine was stored in the adsorbed state in microporous carbon positive electrodes. Such methods do promote the formation of HBr and consequent coulombic losses due to hydrogen evolution and zinc consumption.
Certainly, the energy efficiency of a battery system is of paramount importance when estimating operating power costs. It is also reflected in the net energy storing capacity of the power source as well as in determining the capital equipment costs.
Total turn-around energy efficiency, η, is defined as the ratio of output energy to the electrical load from the battery divided by the total energy expended elsewhere and required to charge the battery. Let us now define coulombic efficiency, hc, and the voltage efficiency, hv. If we let ic be the charging current and id the discharging current, then the coulombic inputs to outputs from the cell are the following:
(A2.13)
and
(A2.14)
The coulombic efficiency then is simply Qd/Qc.
The voltage efficiency factor is another matter. There actually is no direct way to measure that parameter, but we may find it interesting to invent such a function and define it in a manner similar to that for hc. It can be considered the ratio of the driving potential over the charging potential over the entire range of the two modes. Unless one were to average the voltages and call their quotient the voltage efficiency, or unless the voltages were constant, then we could ascribe a physical significance to the term without too much difficulty.
However, in real situations both current and voltage will vary during the charge mode as well as during discharge. We can define the voltage efficiency term, which is usually referred to as
(A2.15)
This brings us to the main issue at hand – the total efficiency of battery operation.
Power at any instant in time is
(A2.16)
The energy, dE, over a period in time, dt, is
(A2.17)
The total energy efficiency then becomes the quotient of input energy divided into output energy over the entire span of time for charge and discharge, or
(A2.18)
Depending on the module design, the number of cells in a series array, the type of electrolytes employed, rates of charging and discharging, and many other factors, efficiencies ranging between 60% and 75% for total turn-around can be expected.
In certain applications where convenience, emergency availability, etc., are the main application values, efficiency may not be as important. However, in load leveling and solar and wind power applications, energy efficiency is very important and second perhaps only to cost considerations.
A2.5 Specific Resistivity and Specific Gravity of Some Reagents
The graphs below are plots of Na2S electrolyte resistance and specific gravity as it depends on its concentration in water. Figure A2.5 shows the change of resistivity of NaOH, depending on its concentration in water. The importance of such information is realized when it is employed to control the pH of the electrolyte in a sodium sulfide concentration cell. The pH is important because it minimizes the formation of hydrogen gas during charging at the negative electrode, especially when the availability of sodium ions is at a minimum. Also, adding NaOH to the solution will increase electrolyte conductivity.
Figure A2.5 Resistivity of NaOH solutions.
In cells that make use of the ferric/ferrous system, salt solution weights of these heavy salts are important in determining cell power and cell energy to weight delivery capabilities, i.e., PD and ED. Figure A2.6 shows specific gravity versus concentration.
Figure A2.6 Specific gravity of FeCl3 solutions.
Similarly, we are interested in the specific gravity versus water concentration of the salt Na2S as shown in Figure A2.7.
Figure A2.7 Sodium monosulfide solution specific gravity.