Cell’s Thermochemistry and I²R Formula

Abstract

All electrochemical power sources of practical use emit heat during discharge. This heat primarily warms the battery itself. The amount of heat, as well as the heat flow, depend on the discharging rate. In case of the lithium-ion battery, high-rate (fast) process can overheat the battery up to unsafe conditions, causing sometimes fire and explosion. Thus, understanding and quantifying the character of heat releasing processes is important for batteries’ usage, maintenance, modeling, and developing.All of the following reasoning relates to the situation when the battery operates under projected use conditions.Direct calorimetric measurements that provide immediate answer need special equipment, such as, for example, the ARC instrument. However, both entities, that is, total heat and heat flow, can be derived relatively easily employing the general ideas of thermochemistry and thermodynamics. The approach is based on comparing the enthalpy of the redox reaction of the cell and the useful energy produced by the cell during discharge at a certain rate. The latter is measured in a typical battery test; the enthalpy ΔH being the sum of Gibbs’ (free) energy ΔG and the entropy component T ΔS (T is absolute temperature, and ΔS is entropy change, ΔH = ΔG + T ΔS) can be easily determined through additional though simple experiment. Electrochemical objects give a unique opportunity of obtaining both enthalpy components directly by measuring the open circuit voltage (OCV) of the cell at various temperatures and state of charge. Indeed, by definition, the cells’ OCV is in fact the free energy expressed in electric units U = -ΔG/zF, where U is the OCV, F is the Faraday constant, and z is the number of electrons involved in the redox reaction. Further, since dG = -SdT + Vdp (V and p are volume and pressure, respectively), the temperature coefficient of the OCV (∂U/∂ T)p = -ΔS/zF. The talk presents the results of examining of three lithium-ion batteries of extensively used chemistries: lithium cobalt oxide-graphite, lithium iron phosphate-graphite, and lithium nickel cobalt aluminum oxide-graphite. For performing the comparison and analyzing the results, uncomplicated but special computational procedures were developed. Particular attention is given to the effect of the entropy of the reactions on batteries’ thermal behavior. As it was mentioned above, the enthalpy is comprised of Gibbs energy and the entropy member. Since there is a solid-state reaction in the batteries, a good assumption is that entropy should not change much, and the entropy term might be neglected. Of course, such an assumption needs verification. Our investigation showed in which cases this hypothesis is correct, and if it is incorrect – to what degree.Total heat and heat flow depend upon the current of discharge I. In battery engineering practice, it is commonly assumed that this dependence can be expressed in terms of the Joule-Lenz law where the voltage is substituted with resistance and current according to Ohm’s equation. However, the electrochemical electrodes are essentially non-Ohmic elements meaning that there is a non-linear connection between voltage and current. In the present work, empirical relationships have been suggested: Total Heat is proportional to I1/2 , and Heat Flow – to I3/2 . The proportionality factor evidently cannot be interpreted as resistance – first of all, because its unit of measure is different. The explanation of the phenomenon is based on the non-Ohmic nature of the battery behavior.In some experimental series, we measured the temperature of the cells during cycling. The experiments revealed impressive matching of the temperature and entropy profiles.