RAJAN ARAVINDH (US)
WO2019066651A1 | 2019-04-04 |
CN102753806A | 2012-10-24 |
LI ET AL.: "Performance analysis of a thermally regenerative electrochemical refrigerator", ENERGY, 17 June 2016 (2016-06-17), pages 43 - 51, XP029743095, DOI: 10.1016/j.energy.2016.06.045
ANONYMOUS: "Cool Solutions: Science and Engineering Help Address the Impacts of Climate Change", TECHNOLOGY ORG, 30 September 2019 (2019-09-30), pages 1 - 18, XP055978266, Retrieved from the Internet
CLAIMS What is claimed is: 1. A system for electrochemical cooling, the system comprising: a cold electrochemical cell comprising a first ion exchange membrane, wherein a first ion flux between a first working fluid and a second working fluid across the first ion exchange membrane absorbs heat from the first working fluid and the second working fluid, lowering a temperature of the first working fluid and the second working fluid exiting the cold electrochemical cell; a heat source; a cold side heat exchanger, wherein the cold side heat exchanger absorbs heat from the heat source to increase the temperature of the first working fluid and the second working fluid flowing through the cold side heat exchanger; a hot electrochemical cell comprising a second ion exchange membrane, wherein a second ion flux between the first working fluid and the second working fluid across the second ion exchange membrane releases heat into the first working fluid and the second working fluid, raising the temperature of the first working fluid and the second working fluid exiting the hot electrochemical cell; a heat sink; and a hot side heat exchanger, wherein the hot side heat exchanger rejects absorbed heat into the heat sink to decrease the temperature of the first working fluid and the second working fluid; and wherein the first working fluid and the second working fluid circulate continuously through a first fluid path and a second fluid path that are independent of one another without physically mixing together. 2. The system of claim 1, wherein the cold electrochemical cell, the hot electrochemical cell, or both further comprise a housing comprising a chamber separated into a first side and a second side by the first ion exchange membrane or second ion exchange membrane, respectively, and wherein the first side and the second side independently comprise one or more of a porous electrode, a flow channel plate, or a combination thereof. 3. The system of claim 2, wherein the porous electrode comprises carbon felt, a metal foam, conducting polymer foam, conducting ceramic foam, or a combination thereof. 4. The system of claim 1, wherein the system comprises adiabatic walls, wherein the adiabatic walls are configured to be removable to allow the system to absorb heat from an environment surrounding the system, wherein heat absorption results in a temperature drop in the surrounding environment. 5. The system of claim 1, further comprising a first circulation pump for circulating the first redox active electrolyte and a second circulation pump for circulating the second redox active electrolyte. 6. The system of claim 1, wherein the first working fluid and the second working fluid do not circulate through the heat source, the heat sink, or both the heat source and the heat sink. 7. The system of claim 1, wherein the first ion exchange membrane is a first cation exchange membrane, wherein the second ion exchange membrane is a second cation exchange membrane, and the ion flux is an anionic flux. 8. The system of claim 7, wherein the first cation exchange membrane and the second cation exchange membrane are independently selected from a hydrocarbon membrane, a per- fluorinated sulfonic acid membranes, or a solid ion-conducting electrolyte. 9. The system of claim 1, wherein the first ion exchange membrane is a first anion exchange membrane, wherein the second ion exchange membrane is a second anion exchange membrane, and the ion flux is a cationic flux. 10. The system of claim 9, wherein the first anion exchange membrane and the second anion exchange membrane are independently selected from an ionomer membrane or a solid-ion conducting electrolyte membrane. 11. The system of claim 1, wherein the first working fluid is a first redox active electrolyte and the second working fluid is a second redox active electrolyte. 12. The system of claim 1, further comprising an external power supply that drives current through the cold electrochemical cell and the hot electrochemical cell. 13. The system of claim 1, wherein the system produces a temperature drop of from about 0.05K to about 50K. 14. An apparatus comprising one or more systems according to claim 1. 15. The apparatus of claim 14, wherein the apparatus comprises a refrigerator, an air conditioner, or a heat pump. 16. A method for electrochemical cooling, the method comprising: circulating a first working fluid in a first closed circuit comprising a first half of a cold electrochemical cell, a first portion passing through a cold side heat exchanger, a first half of a hot electrochemical cell, and a first portion passing through a hot side heat exchanger; circulating a second working fluid in a second closed circuit comprising a second half of a cold electrochemical cell, a second portion passing through a cold side heat exchanger, a second half of a hot electrochemical cell, and a second portion passing through hot side heat exchanger; wherein the first half of the cold electrochemical cell is separated from the second half of the cold electrochemical cell by a first ion exchange membrane; wherein the first half of the hot electrochemical cell is separated from the second half of the hot electrochemical cell by a second ion exchange membrane; and driving current from an external power supply through the cold electrochemical cell and the hot electrochemical cell, wherein the current drives a first redox reaction in the cold electrochemical cell, where the first redox reaction has an entropy change greater than 0, and wherein the current drives a second redox reaction in the hot cell, where the second redox reaction is a reverse of the first redox reaction, and wherein the second redox reaction has an entropy change less than 0; wherein the system produces a temperature drop. 17. The method of claim 16, wherein the temperature drop is about 0.05K to about 50K. 18. The method of claim 16, further comprising a first circulation pump for circulating the first working fluid and a second circulation pump for circulating the second working fluid. 19. The method of claim 16, wherein the first working fluid and second working fluid do not circulate through a heat source, a heat sink, or both the heat source and the heat sink. 20. The method of claim 16, wherein the first ion exchange membrane is a first cation exchange membrane and the second ion exchange membrane is a second cation exchange membrane. 21. The method of claim 20, wherein the first cation exchange membrane and the second cation exchange membrane are independently selected from a hydrocarbon membrane, a per- fluorinated sulfonic acid membranes, or a solid ion-conducting electrolyte. 22. The method of claim 16, wherein the first ion exchange membrane is a first anion exchange membrane and wherein the second ion exchange membrane is a second anion exchange membrane. 23. The method of claim 22, wherein the first anion exchange membrane and the second anion exchange membrane are independently selected from an ionomer membranes or a solid- ion conducting electrolyte membrane. 24. The method of claim 16, wherein the current from the external power supply applied to the cold electrochemical cell and the hot electrochemical cell have the same numerical potential with the opposite charge. 25. The method of claim 16, further comprising removing an adiabatic barrier surrounding the first and second closed circuits and allowing the first working fluid, the second working fluid, or both to absorb environmental heat. |
[0113] While the temperature coefficient represents the entropic driving force that leads to cooling, the heat capacity of the electrolyte represents the thermal inertia. The volumetric heat capacities of the electrolytes were measured at the 50% SOC configuration using a calorimeter to be 2.49 ± 0.38 MJ m -3 K -1 for aqueous I 3 -/I- and 3.58 ± 0.78 MJ m -3 K -1 for aqueous FCN 3- /FCN 4- . These values are used as constants for the included thermal calculations, since it is not expected the heat capacities of the electrolytes to change significantly due the small intra-electrolyte species conversion occurring with electron transfer. The SOC dependent temperature coefficients and heat capacities can be used to predict the maximum isentropic temperature difference as a function of reaction extent as depicted in FIG.1.2B, achievable in the absence of thermodynamic irreversibilities (i.e., inter-electrolyte and exo-cell heat transfer, and Joule heating), within a closed electrochemical cell. FIG. 1.2B was generated by the method described below. In general, the temperature difference may be increased by alternative half-cell reactions with larger temperature coefficients, lower specific heat capacity, and higher solubility of the limiting reagent. In order to obtain the maximum isentropic drop in temperature for a given flowrate, electrode volume, and operating current in an open system, the electrochemical cell must adhere to the tenants of a plug flow reactor (i.e., without any heat or mass dispersion). In such an architecture, for a given pair of electrolytes, the isentropic drop in temperature increases with higher electrolyte residence time and operating current. The reader is referred to the theoretical work presented elsewhere herein for additional thermodynamic cycle details, the BECR figure of merit, and the predicted performance. Additionally, it was confirmed that the I 3 -/I-//Nafion//FCN 3- /FCN 4- system possessed the required lifetime for the proof-of-concept experiment by cycling it at ~16 mA/cm 2 within a potential window far away from water-splitting regimes ([-600 mV, 600 mV]) for 100 cycles over 200 hours. For perspective, the proof-of-concept refrigerator was operated for only 50 cycles and 20 hours, observing steady, continuous cooling. BECR Implementation [0114] The implementation experiment (FIG.1.3A) and the schematic in FIG.1.1B are alike, save for the heat exchangers. In the prototype, one heat exchanger regulates the temperature of each half-cell electrolyte at the inlets of the hot and cold cells to 27 °C, thereby, maintaining constant inlets temperature to clearly demonstrate that cooling is occurring from only the electrochemical reactions and is not an artifact of ambient (room) temperature fluctuations (i.e. T H = T C = 27 °C). The electrochemical cells have acrylic housing to increase thermal resistance to the surroundings and a graphite serpentine flow field to increase the residence time, and therefore the degree of electrochemical conversion of the electrolyte. Electrolyte flowrates were measured using optical tachometers. Eight different electric current inputs (up to 4 A) were sourced with a galvanostat to the electrochemical cells for 20 minutes to achieve steady-state (with 1 minute of open circuit data measurement on either side of the sourced current domain, FIG. 1.3B). When the current is applied, the electrolytes exiting the cold cell is driven to a lower SOC, decreasing the outlet temperatures of both electrolytes using the entropy of the reactions. Conversely, the electrochemical reaction in the hot cell increases the SOC (the direction of I is reverse in hot cell relative to cold cell), in turn increasing the outlet temperature over time. Since the electrochemical cells are in series and flowrates of one electrolyte stream in both cells are equal, the SOC of the electrolytes entering both cells remain invariant after steady-state has been achieved (when the V HOT and V COLD saturate). The cells reach thermal steady-state (dT i /dt = 0) at ~7 minutes and electric steady-state (dV i /dt = 0) at ~2 minutes. At thermal steady-state, the difference between the electrolyte outlet and inlet temperatures is equal to the net heat generation within the corresponding electrode. The difference of the values of this temperature differential was used at steady-state (t = 21 minutes) and at the onset of the current (t = 1 minute) as the steady-state temperature response of the electrolyte to the heat process occurring in the electrode (FIG.1.3C). [0115] The steady-state temperature response was used to evaluate the electrolyte specific cooling load (FIG. 1.4A and FIG. 1.4B) and rejected heat (FIG. 1.4C and FIG.1.4D) using the heat capacity and flowrate of the electrolytes. The FCN 3- /FCN 4- electrolyte outperforms its counterpart because |α FCN | > |α I |, and because it has a lower overpotential at 4A, where the entropic cooling effects of the triiodide reduction are negated by Joule heating. The larger Joule heating associated with the I 3 -/I- reaction arises from its higher activation overpotential, attributed to the large solvent reorganization energy of the covalent bond cleavage that accompanies the triiodide reduction. The electrolyte specific cooling loads were combined to obtain the total cooling load, Q C ̇ (FIG.1.4E). The coefficient of performance (COP), β, was evaluated using (FIG.1.4E), where P IN – I(V HOT +V COLD ). The circulating pumping power was neglected in these calculations; an estimate of the pumping power based on Darcy’s law is provided below. BECR can achieve a target cooling capacity by achieving a high degree of electrochemical conversion at low electrolyte flowrate. Therefore, unlike flow batteries, the BECR is not bound by the tradeoff between high pumping power and high porous electrode specific surface area. The precipitous drop in COP with increasing current (shown in FIG.1.4E) has thermodynamic and kinetic origins. Even though the cells operate in the ohmic regime (FIG.1.21) at the higher current values, the Joule heating scales with I 2 , increasing the irreversibility. Furthermore, T 2 and increasing T 4 decrease and increase respectively with increasing current. This leads to heat transfer processes that are increasingly deviant from the isothermal heat transfer characteristic of the ideal Carnot cycle. The measured cooling load and COP (evaluated above) are in good agreement with the predicted values, calculated using an energy balance. [0116] Finally, it is emphasized that while this proof-of-concept demonstration of continuous electrochemical refrigeration is made possible by appreciable electrolyte redox conversion, significant improvement can still be made by increasing the residence time of the electrolytes. The first Damköhler number, given by the ratio of the reaction rate and the advective molar transport rate, for this work is ~ 0.08, which amounts to ~8% conversion for a plug flow reaction with first-order kinetics. There is substantial scope for improvement by operating at higher currents (increased rate of reaction) or decreasing the volumetric flowrate of the electrolytes (decreased molar influx). Of course, the latter must be done keeping in mind that decreasing the electrolyte flowrate to arbitrarily low values will decrease the cooling load, and hence there exists an optimum electrolyte flowrate for a given current that provides maximum electrolyte specific cooling load. As future motivation, Kim et al. reported a two-fold increase in the temperature coefficient of FCN 3- /FCN 4- by employing 15 wt% methanol-water solution as the solvent. Yamada et al. increased the temperature coefficient of I 3 -/I- to 2 mV/K using a supramolecular scheme. These works could be immediately adopted into the BECR cycle to improve the cooling load several fold. Discussion and Conclusions [0117] A continuous electrochemical refrigerator that is inspired by the reverse Brayton cycle has been demonstrated herein. It achieves a peak COP of 8.09 with a temperature drop of 0.07 K, and a peak cooling load of 0.934 W (at 14.6 mW/cm 2 , normalized by the cross-sectional area of the porous electrodes). It is acknowledged that only a modest temperature drop of 0.15 K at peak cooling load was observed (FIG.1.3C) but theoretically it could have achieved a temperature drop of 2-7 K (FIG. 1.2B) if a greater degree of electrochemical conversion could be achieved. It demonstrates cooling for a total of ~18 hours without any sign of deterioration of the system performance. Additionally, the inverse gravimetric heat capacities and the gravimetric maximum possible entropy change of existing refrigerants have been compared with those of the disclosed electrochemical refrigerants (FIG.1.5A). The maximum possible entropy change and the specific heat represent the entropic driving force and the thermal inertia towards cooling. With existing aqueous redox chemistries, it was found that electrochemical refrigeration compares well against vapor compression, suggesting that it could someday be an appreciable zero-GWP refrigeration technology. A more compelling argument for electrochemical refrigeration can be made upon comparing the entropy change per unit carrier, ∆S CARRIER , for the technologies (FIG. 1.5B). Through this lens, the entropic driving force for electrochemical refrigeration is more than double that of vapor compression. Table 2.2 details the device COP and temperature drop achieved by experimental works corresponding to the previously mentioned refrigeration technologies. It must be noted that popular refrigerants used in other refrigeration technologies were chosen in FIGs. 1.5A-1.5B and Table 2.2 simply to provide context to this work. Research within each of these technologies may have led to other refrigerants that demonstrate superior performance. For example, a rotary magnetocaloric refrigerator that employs LaFeSiH as the refrigerant has shown to a temperature lift of 11 K, and a COP of 1.9. Table 2.2 shows that significant work must be done to improve the temperature drop of the disclosed technology, while maintaining the COP, to improve its commercial viability. It is believed that this can be done through additional research into high temperature coefficient half-cell reactions with high solubility, low freezing point solvents with low specific heat capacities, and electrochemical cell architectures that can drive isentropic reactions to completion. Higher temperature coefficient half-cell reactions with high solubility have larger entropy changes per unit volume of electrolyte, and thereby increase an electrolyte’s capacity for cooling, in addition to delaying the onset of temperature induced precipitation. Low vapor pressure (e.g., ionic liquids) and low freezing point solvents (e.g., alcohols) allow for the BECR cycle to operate in a larger parameter space (operating current and electrolyte flowrate) with little threat of excessive internal pressurization and/or the electrolyte freezing. Lower electrolyte specific heat elicits a greater temperature response per unit electrochemical reaction. Electrochemical cell architectures that drive the reactions toward completion (higher first Damköhler number) isentropically (minimal overpotential) can generate higher electrolyte temperature drops. The operating current and electrolyte flowrate influence the COP and cooling load, and they must be rationally optimized according to the thermodynamic, kinetic, and physical properties of the electrolytes. [0118] Electrochemical Thermodynamic Cycles. The following nomenclature is used below: [0119] The entropic heat absorbed during an electrochemical reaction may be utilized via two different schemes to achieve refrigeration (FIG. 1.6). If the electrochemical system under consideration has diathermal walls, and an electrochemical reaction with I > 0 and ΔS rxn > 0 is driven, heat will flow into the system from the surroundings. If this process is done slow enough, the heat transfer may be done under near isothermal conditions. This concept is represented as scheme [P→I] in FIG.1.6. Conversely, if the electrochemical system has adiabatic walls, driving an electrochemical reaction with I > 0 and ΔS rxn > 0 will result in drop in temperature of the system. [0120] Once the required amount to heat has been absorbed from the system, the adiabatic walls may be removed to allow the electrochemical system to absorb heat from the surroundings. This concept is represented as scheme [P→A] in FIG. 1.6. The isothermal and isentropic electrochemical processes can be used as drivers in the electrochemical analogue of the Stirling and Brayton cycles, respectively. These schemes are highlighted in the context of their respective thermodynamic cycles in FIG.1.7. Methods [0121] Materials. All chemicals were purchased from commercial suppliers (Sigma-Aldrich, VWR International, The Science Company), and were of at least reagent grade and were used without purification. MilliQ grade (19.7 MΩ cm) ultrapure deionized water was used to prepare all the solutions. Non-aqueous solvents were used without drying. [0122] Electrochemical cells. The electrochemical cells were designed to maintain an adiabatic environment and increase the residence time of the electrolytes within the cell. A 0.5 inch-thick acrylic endplates and a serpentine channel made of resin impregnated graphite (Graphite Store, MW001204) were used. FIG.1.18 shows an exploded view of the cold and hot electrochemical cells. The acrylic housing sandwiches the graphite serpentine, porous electrodes (AvCarb, G600A), and cation exchange membrane (Nafion 211). Eight M6 bolts were used to hold he assembly together. Electrolytes were introduced into the cells using PVDF adapters (McMaster- Carr, 5533K411). The current collectors were thin copper strips that were placed in between the acrylic housing and graphite serpentine. The size of the porous electrodes (and therefore the rest of the assembly) was chosen using a conservative estimation. A simple energy balance affords the following equation for small temperature changes; VĊ V ΔT = IαT ∞ -I 2 R. Here, it is assumed that the majority of the overpotential occurs due to the ohmic overpotential from the membrane resistance. Some conservative values are used for the flowrate, V̇ (1 mL/s), heat capacity C V (3.5 MJ/m 3 K), temperature coefficient, α (1.5 mV/K), and membrane resistance (1 Ω cm 2 ). If a 0.5 K drop in temperature at 10 A is desired, the cross-sectional area of the cell should be ~36 cm 2 .8 cm × 8 cm cross-sectional area porous electrodes were selected to further decrease the effect of overpotentials. [0123] Heat exchangers. An active heat exchanger was chosen (FIGs.1.19A-1.19B) rather than a passive heat exchanger to eliminate ambient temperature fluctuation and maintain a constant electrochemical inlet cell temperature; this allows us to unequivocally demonstrate electrochemical cooling decoupled from ambient temperature fluctuations of the heat exchangers. The active heat exchange was performed using a thermoelectric module attached to a cold plate, which had a tube for the electrolyte to flow. Since the electrolytes must always be kept from mixing with each other, two active heat exchanger modules were used (TE technology, CP- 061HT). 1/8” (3.175 mm) OD stainless steel tube was embedded (chosen for its corrosion resistance to the triiodide/iodide, and ferricyanide/ferrocyanide redox couples) into two 1/4" (6.35 mm) thick copper block, which were mounted to the thermoelectric heat exchanger. 1/4" (6.35 mm) thick neoprene rubber was use to insulate the thermoelectric and copper block from the environment. The parts of the stainless steel tubing that were in contact with the ambient air were liberally covered in a polymer as insulation. The temperature was set and regulated by souring power to the thermoelectric modules using a programmable temperature controller (SRS, PTC10) that employed two PTC440 TEC drivers. The temperature feedback to the closed PID loop was provided by thermistors (OMEGA, Item# 44031) that monitored the temperature of the electrolytes exiting the heat exchangers. The programmable temperature controller used a PTC320 card to acquire inputs from the thermistors located at the outlets. [0124] Porous electrode and ion exchange membrane preparation. Prior to all experiments, the porous electrodes were immersed in an acid bath containing three parts 68% nitric acid and one part 18 M sulfuric acid. They were heated to 80 °C and refluxed overnight. The electrodes were washed in DI water until pH neutral and then heated in an oven at 400 °C for three hours. The electrodes were assembled into the electrochemical cells and flooded with electrolytes as soon as possible. The as purchased Nafion membrane was first heated in a 2 M KOH solution at 60 °C overnight. It was then washed thoroughly with DI water and then stored in a 4 M KCl solution. After each experiment, the membrane was washed thoroughly with DI water and stored in the KCl solution until the next experiment. [0125] Electrolytes for the experiment. Of all the redox active species employed in the experiment, the species with the limiting solubility was the ferrocyanide with a room temperature solubility of ~0.6 M. Therefore, with reference to the cold electrochemical cell reaction, the cell is at 100% state of charge when the ferrocyanide ion is at that value. Conversely it is at 0% state of charge when all of the ferrocyanide has been converted to ferricyanide at 0.6 M. The quantity of the iodide ions and triiodide ions are determined using the solubility constraint of the ferrocyanide species and by the requirement, albeit arbitrary, that at 50 % state of charge, both the electrolyte had the same concentration of potassium ions. Potassium ions were used for the iodide, ferricyanide, and ferrocyanide species. The triiodide ion is achieved by dissolving molecular iodine into an iodide solution which causes the iodine to abstract an iodide to form the triiodide. This reaction has an equilibrium constant that strongly favors the triiodide ion and therefore it is assumed that no iodine is present in the solution. The experiment was primed with electrolyte at 80% state of charge and the details are provided in Table 2.5. [0126] Galvanic-Electrolytic Cycling. Once the chosen half-cell reactions were confirmed to have a high isothermal entropy change and that the system possessed no temperature hysteresis, galvanic and electrolytic cycling was performed to ensure that the system did not change appreciably over the course of the experiment. This was done using the electrochemical cell illustrated in FIGs. 1.9A-1.9D. The electrolytes were prepared in the 10% state-of-charge configuration as given in Table 2.5 below. The iodide/triiodide was connected to the positive terminal of the galvanostat (BioLogic, VSP 300). This cycling experiment is inherently different from what one may perform for a battery. In the case of a battery power source, one is interested in the how much electrical work the battery can output in galvanic mode after it has been charged in electrolytic mode. The potential therefore cycles between 0 V and some positive value. In this work however, the near complete chemical conversion and the thermal effects that accompany that conversion are of interest. Therefore, depending on the activity of the redox active species, the voltage required to drive the reaction in a specific direction can go from positive to negative as ΔG rxn crosses over the point of chemical equilibrium. Regardless of the points of chemical equilibrium, the potentiostat was used to enforce maximum chemical conversion until mass transfer effects become prominent. Through iterations, the voltage window was chosen to be [600 mV, -600 mV] and the cell was cycled at ±200 mA, at ~ 16 mA/cm 2 (FIGs. 1.15A-1.15B). This experiment shows that the redox system is indeed reversible within the potential and time windows if the proof-of-concept experiment. This is to be expected because both the redox reactions occur at neutral pH which suppresses hydrogen evolution. The window is intentionally kept away from the water splitting regimes. The source of the iodide, ferricyanide and ferrocyanide ions were potassium salts. Therefore, Nafion, a cation exchange membrane, ensured that only potassium ions shuttled from one electrode to the other to preserve charge balance. Any solvent flux due to osmotic pressure was suppressed by maintaining an air-tight seal. [0127] Note that I 2 is sparingly soluble in water. The equilibrium constant for the I 3 - from I- and I 2 heavily favors the product, and so the amount of I 2 and KI added to the electrolyte reflects the required I 3 - concentration. [0128] Proof-of-concept setup. The tubing used to connect the components together were 1/4” (6.25 mm) OD, 1/8” (3.175 mm) ID. The tubing used to contain the ferricyanide/ferrocyanide electrolyte was a flexible PVC tubing, and rigid PFA tubing was used to contain the triiodide/iodide electrolyte due to the electrolyte’s tendency to engage in SN2 nucleophilic attack with non- fluorinated polymers. All adapters and fittings that were made of PVDF. At the inlet and outlet electrolyte ports of the electrochemical cells, RTDs were introduced into the electrolyte streams using PVDF tee-adapters. Previous experiments showed that it was necessary for these RTDs to have a metal shield to prevent capacitive coupling to the voltage applied by the galvanostat. In addition to the components shown in FIG.1.3A, two sealed Erlenmeyer flasks were used for each electrolyte stream as a reservoir source (and sink) for the inlets (and outlets) of the electrochemical cells, which served three purposes: (i) they acted as gas traps that removed any air bubbles from the system, (ii) allowed one pump to generate two flow streams, and (iii) by placing the outlet (or inlet) tubing above (or below) the liquid level, they ensured the flow of electrolytes in only one direction (eliminating any possibility of unintended back flow). K-type thermocouples were placed close to the electrochemical cells to provide evidence that the temperature changes caused by sourcing the electrochemical current are not correlated to changes in environmental temperature. A photograph of the completed setup is shown in FIG. 2.20. Peristaltic pumps (Control Company, 3385) circulate the electrolyte. The hot and cold electrochemical cells are connected electrically in series to the galvanostat (Biologic, VSP-300). Eight stainless-steel shielded RTDs (Adafruit Industries LLC, 3290) are introduced into the electrolyte flow paths at the inlets and outlets of the electrochemical cells. The setup was assembled inside a fume hood whose temperature was monitored using three thermocouples (FIG.1.22). [0129] Calculation of equilibrium time cell resistance. The equilibrium time cell resistance during the proof-of-concept were calculated using the expression below where i denotes the values corresponding to the hot or cold cell. [0130] Calculation of measured and predicted cooling loads. The measured total cooling load was calculated by the sum of the electrolyte specific products of the volumetric specific heat (C V,e ), volumetric flowrate (V e ̇), and equilibrium time drop in temperature (ΔT ij ). [0131] Subscript e denotes electrolyte and subscripts /" and j denote the RTDs located at the electrolyte outlet and inlet on the cold cell respectively. The predicted total cooling load was calculated using an energy balance.
[0132] T is the volume averaged temperature within the cold cell. Since the maximum drop in temperature in this work is ~0.2 K, it is assumed that the cell is isothermal at 27 °C.
[0133] Non-lsothermal Temperature Coefficient Measurements. Half-cell reactions were screened using a non-isothermal temperature coefficient measurement. To contain a variety of redox active couples and solvent combinations, the cell housing was made of a 7 cm long PTFE cylinder with a 1.5 cm bore across its length house the electrolyte solution. The electrodes were made of resin impregnated graphite (Graphite Store, MW001204). After the electrolyte was loaded into the cell housing, the PTFE annulus was sandwiched by the graphite electrodes using neoprene gaskets and two M16 bolts that thread into the bottom electrodes. Fiber glass washers are used to keep the graphite electrode electrically isolated via zinc plated bolts. The assembled cell is then sandwiched between two thermoelectric modules (TE technology, CP-061 HT) using a lead screw mechanism that minimizes interfacial thermal resistance by applying pressure. When the experiment is started, the programmed thermoelectric modules apply varying thermal gradients as shown in FIGs. 1.8A-1.8D. The top thermoelectric module is always maintained at a higher temperature than the bottom one. The temperatures of the electrodes are continuously monitored using K-type thermocouples and the voltage across the two electrodes is measured using a data acquisition unit (Agilent, 34970A). The hot electrode and cold electrode are always connected to the positive and negative terminals, respectively. The non-isothermal temperature coefficient is evaluated by the slope of the voltage change and the difference in temperature between the two electrodes. For these measurements, electrolytes were prepared with equal concentrations of the reduced and oxidized species.
[0134] Isothermal Temperature Coefficient Measurements. Once two promising half-cell reactions with the appropriate temperature coefficients were shortlisted, they were assembled in an electrochemical cell with an ion-exchange membrane (IEM) of choice (FIGs. 1.9A-1.9D). This was done for two reasons: (i) to confirm that the temperature coefficients added up in a complete electrochemical cell setup and that there were no mechanisms that diminished the entropy of the half-cell reactions, and (ii) to evaluate the lifetime of the electrochemical system under varying temperature under the open-circuit condition. To test a variety of redox systems, the tenants of maximum inertness were maintained. The cell housing was made of PTFE and 2 cm through holes were bored into the middle to house the porous electrodes. Carbon felt (AvCarb, G600A) were used as the cathode and anode material. They sandwiched the ion-exchange membrane of choice. The current collectors were made of resin impregnated graphite (Graphite Store, MW001204) and zinc plated screws were screwed into them to facilitate robust electrical connections. The cell housing was sealed using M6 bolts and neoprene gaskets. The electrolytes were first introduced into the cell using syringes and then air-tight sealed using redundant stop cocks and one-way valves. This restricted any solvent motion due to osmotic pressure. After the electrochemical cell was assembled and checked for any leakage, it was placed in between the same two thermoelectric modules in the previous section. This time however, the modules were programmed to establish the same temperature. The temperature feedback that informed the PID controller were disconnected from the modules and firmly affixed on the cathode and anode current collectors. Thermocouples were used to measure the temperature of both current collectors and the open circuit voltage of the cell was monitored continuously. Keeping in line with convention, the half-cell reaction with the positive temperature coefficient was connected to the positive terminal. When the experiment is started, the thermoelectric modules draw power to drive the feedback temperatures (located on the current collectors) to the set value. However, thermal equilibrium is achieved when the entire cell (specifically the porous electrodes) attains the bespoke temperature. At this point, there should ideally be no change in the open circuit voltage with time. This was taken to be the point of thermal equilibrium and the open circuit voltage and electrochemical cell temperature were recorded for data processing. This was repeated for multiple set values and the rate of change of the voltage with respect to temperature afforded the temperature coefficient of the half-cell reactions. See FIGs.1.10A-1.10B for generated data and processing. [0135] Quantifying Electrolyte Specific Heat Capacity. The measurement of electrolyte specific heats was mildly complicated by the tendency of the iodide/triiodide electrolyte to liberate corrosive I 2 vapor upon heating. This could be detrimental costly laboratory equipment (e.g., automated differential scanning calorimeters), therefore, simpler “styrofoam cup” calorimetry was performed. Consider the “styrofoam cup” calorimeter assembly as shown in FIG.1.11. It is filled with some liquid whose specific heat capacity is to be measured. A resistive heater is immersed into the liquid and generates heat at the rate P. A stir bar keep the liquid well mixed so that the temperature is homogenous. Employing a simple energy balance for the liquid that begins at thermal equilibrium (T=T amb ), the following expression is obtained under the assumption of high thermal resistance of the Styrofoam cup. [0136] Using the above equation, the specific heat capacity of the liquid may be evaluated using a linear regression. For the resistive heater, a 1Ω thin-film resistor (Riedon PF1262-1RF1) capable of generating up to 20 W of heat was chosen. It was screwed into a 2 cm × 2 cm × 3 mm graphite plate with some heat sink compound to improve its thermal conductance with the liquid. The temperature of the liquid was measured using four RTDs (Adafruit Industries LLC, 3290) placed in random locations within the fluid (FIGs. 1.12A-1.12B). Prior to the experiment, the resistive heater and RTDs were immersed in the liquid of interest, contained in the styrofoam cup, with fast stirring using a stir bar (700 RPM). During the experiment, a power source (Keithley 2230G-30-3) was used to source 1.5 A through the heater, and the temperatures and voltage across the resistor was continuously monitored using a data acquisition unit (NI, cDAQ). The experimental time was short (<50s) such that the temperature-time slopes were linear as shown in FIG.1.13. The temperature of the liquid was taken to be the average of the four temperature traces and some of the data was visually truncated to ignore the noise floor regime. The experiment was first validated using deionized water. This set-up was then used to measure the specific heat capacities of the electrolytes (FIG.1.14). The y-intercept in the plot below is different for the different liquids because of the polymer adhesive that had to be applied to the resistive heater for corrosion resistance. [0137] Electrolyte Temperature as a Function of Reaction Extent. For an arbitrary electrochemical system, the first law of thermodynamics may be written as the following expression. [0138] In the above expression, H is the total enthalpy, N is the number of particles, H ഥ is the partial molar enthalpy, T is temperature, S is entropy, μ is chemical potential, and bscript i represents all species participating in the electrochemical system. Recalling that ((δH ഥ i/δ ഥ where the partial molar heat capacity and employing the concept o at capa CP, the above expression may be written as the following. TdS ^ ^ ^ [0139] Finally, it is recognized that by fixing the initial and final states of the reaction, a reaction coordinate can be defined. One point on the reaction coordinate will then map to relative concentrations of all species in the electrochemical system. It is also recognized that the changes in the partial molar enthalpies and chemical potential are simply the enthalpy and Gibbs free energy of the reaction. The above equation can now be written as the following for the special case of an isentropic reaction. d S ^ [0140] The reaction coordinate, ξ, is interchangeable with the state of charge. Finally, the linear fit from the trends in FIG.1.2A, and specific heat capacities may be used to integrate both sides of the above equation to find the final temperature as a function of reaction extent. [0141] Quantifying Electrolyte Flowrate. Electrolyte flow was generated using two identical peristaltic pumps (Control company, 3385). The peristaltic pumps use a potentiometer to control the speed of the peristaltic pump. Once connected to a closed-flow loop, the speed of the pump decreases and, therefore, the pump flowrate needs to be calibrated against the rotational speed that is set by the potentiometer’s position when connected to the flow loop. To measure the speed of the pump, an optical tachometry technique was used as shown in FIGs. 1.16A-1.16D. A reflective piece of foil was first attached to the circumference of the pump head. Directly above the pump head, an LED and a photodiode (PD) was mounted within a recession cut into a piece of foam. The assembly was arranged such that during the rotation of the pump head, the LED and PD would be directly over the foil. At this instance, the light emanating from the LED is reflected to the PD. The time-varying current generated from the PD passes through a 909 kΩ resistor and the time-varying voltage is recorded by an oscilloscope (Tektronix TBS1202B). This voltage trace comprised of moments of high intensity when the foil was directly underneath the LED and PD, and of low intensity in the absence of the foil during the rotation. The periodicity of this trace informed the pump speed. A calibration experiment was performed to map the flowrate of each pump to the pump speed. For a specific setting on the pump, the speed was first measured using the optical tachometry technique above. Then, the flowrate for that setting was measured using a graduated cylinder and a stopwatch. This was done for three different setting and a linear regression was performed to evaluate the calibration (FIG.1.17). This was done for both pumps. During the proof-of-concept experiment, the speeds of both pumps were recorded, and the calibration curves were used to evaluate the flowrate of the electrolytes. Raw data acquired is shown in Table 2.6. a All measurements were repeated five times to acquire statistics. [0142] Estimation of Pumping Power. The pressure drop ∆P experienced by a fluid as it flows through a porous medium is given by Darcy’s law which is given below. ^ [0143] where ν is fluid velocity, path length, and K is permeability. Converting the fluid velocity to volumetric flowrate and multiplying both sides of the above equation with the volumetric flowrate V̇ to obtain the pumping power P PUMP . ^ P PUMP [0144] The permeability for a porous electrode is given by the Kozeny-Carman equation given below as a function of pore radius r p , electrode porosity ε, and Kozeny-Carman constant. [0145] Using a pore radius rman constant of 5.55, the above permeability is calculated to be 3.92×10 -10 m 2 . Assuming that the electrolyte takes the convoluted path of the graphite serpentine where each serpentine flow-field has a total of 20 channels. The channels are 3 mm wide and 3 mm deep. It was conservatively estimated that the entire depth is occupied by the porous electrode due to the compressive stress put into the electrochemical cell assembly. In reality, there will be parallel pathways that reduce the pumping demand e.g. graphite serpentine channel area unobstructed by the electrode, bypass electrolyte routes within the porous electrode with a very short path length from the inlet to the outlet. The volumetric flowrate of the electrolyte is taken to be 1 mL/s, and the dynamic viscosity of the electrolyte is taken to be 8.9×10 -4 Pa s. Using these parameters, the pumping power required to drive the electrolyte through one porous electrode is estimated to be 0.4 W. Since one pump drives the electrolyte through two porous electrodes in parallel, the total power consumed by one pump is 0.2 W. Two pumps in were used to drive electrolyte through two sets of porous electrodes each in parallel; therefore, the entire pumping power is estimated to be 0.4 W. [0146] Uncertainty and Error Propagation. The electrolyte specific cooling load QC, at thermal equilibrium (acquired at time t eq ) is given by the following expression. [0147] where V̇ is volume e between the electrolyte inlet and outlet temperature of the cold electrochemical cell, and subscript e represents the electrolyte. The uncertainty in the electrolyte specific cooling load is then given by the following expression. ^ Q C Q C , [0148] The uncertainty in the equilibrium time difference between the outlet and inlet temperatures is given by the standard deviation of such differences acquired across all repetitions and is given by the expression below. ^ ^ T ij [0149] In the above equation, subscript rep represents each of N repetitions. The uncertainty in the total cooling load, which is simply the sum of the electrolyte contributions, is given by, ^ Q C ^ [0150] A similar approach is taken for rejected heat Q H . The COP β is given by the ratio of the total cooling load Q C and the electrical work input W IN . The uncertainty of the COP is then given by the corresponding formula for error propagation. ^ ^ ^ ^ [0151] The uncertainty of the work input is given by the product of the current (uncertainty of current regulated by the galvanostat is effectively zero) and the uncertainty of the cell voltages. Example 3: System Dynamics and Metrics of an Electrochemical Refrigerator Based on the Brayton Cycle [0152] The global cooling demand is projected to triple by 2050 due to rising regional temperatures, rapid urbanization, and regional population growth. Currently, this demand is poised to be fulfilled almost completely by vapor compression. While efficient and scalable, vapor compression uses working fluids, hydrofluorocarbons, that have a global warming potential orders of magnitude larger than CO 2 . Studies have shown that if the status quo is left unchallenged, HFC emissions will be 9 GtCO 2 -eq/yr or 15% of the total CO 2 -eq emissions. [0153] One potential solution to mitigating this problem is the discovery and development of technologies that have zero global warming potential, called not-in-kind technologies. Inspired by the vast improvements made in flow battery chemistry, herein is disclosed a novel, continuous electrochemical refrigeration cycle. Though electrochemically driven refrigeration has been modelled by a few works previously, these previous works have based their analyses around the Stirling cycle. This cycle necessitates isothermal electrochemical reactions, which are practically difficult to realize as this implies that the electrolytes must be well-mixed and fins must be incorporated onto the electrochemical reaction vessel. Additionally, electrochemical refrigeration based on the Stirling cycle inherently divides the entropic heat absorbed between the moving electrolyte and heat transfer across the electrochemical cell boundaries. Therefore, despite tenuous academic interest, continuous electrochemical refrigeration based on the Stirling cycle has never been experimentally demonstrated. [0154] This work introduces the Brayton Electrochemical Refrigeration (BECR) cycle, which employs the electrochemical analogue to the Brayton refrigeration cycle. Herein is demonstrated an experimental proof-of-concept of the BECR cycle which demonstrates its technological superiority over the ones that rely on the Stirling cycle. The analytic framework which quantitatively describes the performance of a BECR system using thermodynamic and kinetic parameters is also presented. First, the ideal BECR cycle is analyzed (perfect heat exchangers and zero electrochemical overpotential) that reveals the performance of the technology in the limit of internal reversibility. Though unrealistic, this analysis demonstrates the fullest extent of the BECR performance, and through the dimensionless figure-of-merit χ, provide intuition for the operation of the BECR cycle and simple expressions for the cooling load and coefficient of performance (COP). Two main practical considerations that limit the operation of the BECR cycle are then identified. Finally, the non-ideal BECR cycle with real heat exchanger efficiencies and electrochemical overpotentials are analyzed, subject to the practical considerations previously mentioned. This analysis explores the thermodynamic, kinetic, and operational requirements of a BECR system that generates useful cooling, and how it may be scaled up to generate higher cooling loads. This framework will guide the concerted efforts that are required of engineers to advance the competency of this technology and introduce it into the refrigeration and heat pumping market. BECR Cycle Operating Principle [0155] The BECR cycle relies on the thermogalvanic effect, the metric for which is the temperature coefficient α. For a general half-cell reaction, O + ne- → R the temperature coefficient is given by where U is the equilibrium potential of the half-cell reaction measured against a reference, T is temperature, ∆S rxn is the entropy of the half-cell reaction, n is the number of electrons transferred, and F is Faraday’s constant. The highest performing BECR cycle will comprise two half-cell reactions with opposite signs of α, as the magnitude of the subsequent cooling effect is additive, similar in manifestation to using both n- and p-type semiconductors with thermoelectric legs. This maximizes the ∆S rxn of the net electrochemical reaction, and therefore, the refrigeration performance. Overall, the BECR cycle is comprised of four processes that are undertaken by a generic electrochemical system consisting of two redox active species, electrodes, and an appropriate ion-exchange membrane (IEM). These processes are best described on a temperature-entropy diagram as shown in FIG.2.1A. In the first process, the electrochemical system is driven from state 1 to state 2 via an isentropic electrochemical reaction. The direction of the applied voltage bias is such that the entropy of the reaction, ∆S rxn , is greater than zero. The total derivative of the entropy of an electrochemical system undergoing an isentropic reaction is given by where C P is the averaged specific heat capacity of the system, and ξ is the reaction coordinate (see below for derivation). Equation 3.2 shows that the temperature of the electrolytes must decrease for an isentropic (i.e., dS = 0) electrochemical reaction. The extent of the electrochemical reaction causes the volume averaged temperature of the electrochemical system to cool down to some temperature T 2 below the cold source temperature, T C . In the next process, the system is electrically disconnected and placed in thermal contact with the cold heat source and is driven from state 2 to state 3 via isomolar heat absorption. This process is described as isomolar because the number of participating species do not change in the absence of an electrochemical reaction. The process is complete when the temperature of the system reaches that of T C . In the third process, the reverse electrochemical reaction occurs isentropically, driving the system from state 3 to state 4. The voltage bias is reversed from that of the isentropic reaction in the first process. The entropy of the reaction, ∆S rxn is less than zero and, therefore, the volume averaged temperature of the system increases. The extent of the electrochemical reaction is such that the net charge transferred between states 3 and 4 is equal to that of the charge transferred between states 1 and 2, i.e. the chemical configuration of the system at state 4 is the same as that of state 1. The extent of the electrochemical reaction causes the volume averaged temperature of the electrochemical system to heat up to some temperature T 4 above the hot sink temperature, T H . In the fourth and final process, the system is electrically disconnected, placed in thermal contact with the hot heat sink, and driven from state 4 to state 1 via isomolar heat rejection. The process is complete when the temperature of the system reaches that of T H . Upon completion, the temperature and chemical configuration of the system is restored to state 1. [0156] The overall electrical work input required to complete the cycle is the difference in Gibbs free energy of the closed-circuit electrochemical reactions and has a positive value as necessitated by the Clausius statement (see below). This sequential cycle can be made continuous by employing all soluble redox couples, flow battery architectures for the electrochemical cells, and heat exchangers that allow both electrolytes to separately exchange heat with their respective environments (FIG.2.1B). FIG.2.1A is a depiction of the ideal BECR cycle that operates with isentropic electrochemical reactions and perfect heat exchanger efficiency (ε HX = 1). These assumptions are relaxed for the non-ideal BECR cycle, which features entropy generation from Joule heating within the electrochemical cell, electrolyte mixing, internal heat transfer, and a realistic heat exchanger efficiency. For ideal and non-ideal BECR cycles operating with the same cold source and hot sink, these mechanisms have a net effect of reducing the cooling load Qc and coefficient of performance β.
[0157] A critical point that must be addressed at this juncture is about the nature of an electrochemical cell capable of driving an isentropic electrochemical reaction. To this end, it is sufficient for the electrochemical cell to be adiabatic and operate without irreversibilities. However, since the purpose of the isentropic reaction is to obtain the maximum drop (or rise) in temperature using a temperature dependent heat generation term, the electrochemical cell must be built as per the tenets of a plug flow reactor. The ideal electrochemical plug flow reactor that achieves a high isentropic change in temperature prevents exocellular heat transfer, operates with a high electrolyte residence time and high operating current, and mitigates thermal dispersion to support the spatial temperature evolution of the electrolyte. See below for further discussion.
Total Derivative of Entropy Change for an Electrochemical System
[0158] For an arbitrary electrochemical system, the first law of thermodynamics may be written as
[0159] Applying the chain rule to Equation 3.3, the following equation is derived.
[0160] Recalling that Equation 3.4. may be written as given below under the assumption of thermal equilibrium.
[0161] For pedagogical purposes, the concept of a state-averaged heat capacity is employed. This may be done by fixing the initial and final states of the reaction, averaging the number of particles and partial molar heat capacity for each species. Under this treatment, the following is obtained:
[0162] Finally, it is recognized that by fixing the initial and final states of the reaction, a reaction coordinate can be defined. One point on the reaction coordinate will then map to relative concentrations of all species in the electrochemical system. It is also recognized that the changes in the partial molar enthalpies and chemical potential are simply the enthalpy and Gibbs free energy of the reaction. Equation 3.6. can now be written as
Electronic Work Required for the BECR Cycle
[0163] For a generic redox reaction A + ne B , the temperature coefficient of the redox reaction is given by where the equilibrium potential of the redox reaction is measured relative to some reference potential. For a redox reaction with a positive temperature coefficient, a cathodic current results in a positive entropy of reaction, and an anodic current leads to a negative entropy of reaction. Therefore, the entropy of the net electrochemical reaction may be maximized by pairing half-cell reactions with both positive and negative temperature coefficients. In the cold cell, the voltage bias is applied such that the half-cell reaction with α > 0 undergoes a cathodic current, and that with α < 0 undergoes an anodic current. For a complete electrochemical system with a cathode and anode redox reaction, the equilibrium potential of the cell can be written as [0164] In the above equation, the temperatures of the half-cell reactions are constrained to be different in order to ignore entropy generation from internal heat transfer. This may be realized by an arbitrarily high thermal resistance of the separator. The infinitesimal reversible electrical work done on/by any electrochemical system is given by dW ^ ^ U CELL ( T , q ) dq , where dq is positi in the direction of a cathodic current. In the absen ons and 100% Faradaic efficiency, the charge transferred is coupled to the chemical composition of the electrochemical system. Therefore, the equilibrium potential of the cell is a function of the charge transferred from the chemical composition. The total electrical work input into the BECR cycle then is given by where the limits of the integrals represent the states on the BECR cycle. It is recognized that the chemical constituency is the same in states 2 and 3, and in states 4 and 1. For clarity, the chemical constituency is labeled in states 1 and 4 as q i and in states 2 and 4 as q f [0165] Furthermore, the half-cell reaction with α > 0 is labeled as A, and that with α < 0 as B. Adopting these limits and substituting for the equilibrium potentials the following equation is arrive at: [0166] In Equation 3.11, by disallowing for any internal heat transfer, the temperature of the half- cell reaction ensemble can be mapped to the chemical composition and therefore, the charge transferred (Equation 3.7). As shown in FIG.2.8, for every state between the initial and final states, the equilibrium potential in the hot cell is greater than that of the cold cell. This may be explicitly derived by integrating Equation 3.7 to express the temperature of a half-cell reaction as a function of the reaction coordinate. W IN > 0 must be greater than zero because the change from q i to q f represents a positive change along the reaction coordinate (the direction of charge transferred in the cold cell is chosen to be positive as a reference), thus proving the need for an external power supply from electrochemical thermodynamics. Adiabatic Plug Flow Electrochemical Cells for High Isentropic Temperature Change [0167] An electrolyte passing through its corresponding electrodes is subject to the heat generation or consumption according to Equation 3.7. However, the heat generation term is directly proportional to the electrolyte temperature and therefore, the temperature field of the electrolyte dictates the outlet temperature of the electrolyte from the field. Temperature inhomogeneity is supported by high electrolyte residence time, high reaction rate, and low thermal dispersion. Two sources of thermal dispersion are (i) porous electrode tortuosity (ii) thermal back conduction. Tortuous electrodes can mix electrolyte streams of different temperature leading to reduction in temperature inhomogeneity. Thermal back conduction along the electrode fibers or the current collector can also reduce thermal gradients within the electrolyte. This spectrum of temperature homogeneity and the resulting outlet temperature are shown in the three cases featured in FIG.2.9. [0168] In all three cases, the electrolyte enters the electrode at flowrate V ^ and temperature T I leaves it at the same flowrate but at different temperatures that is a fun tion of the temperature field of the electrolyte, and the current distribution along the one-dimensional electrolyte flow is uniform. In case A, there is no thermal dispersion whatsoever and the electrode may be likened to a plug flow reactor. The electrolyte enters the electrode at temperature T IN and gradually drops to T A due to entropic heat absorption associated with the half-cell reaction. In case C, the electrolyte is well-mixed and the electrolyte is at a homogenous temperature T C . Case B represents some intermediately mixed case the architecture supports a smaller temperature gradient relative to case A. [0169] Applying energy balance for case A on a differential volume at some position in the x-wise direction (FIG. 2.10), the following equation is derived.
[0170] Upon integration, the outlet temperature of the electrolyte is given by
[0171] Applying energy balance to case C, the following equation is obtained.
[0172] Rearranging the equation affords the expression for the outlet temperature for case C.
[0173] The ratio of the outlet temperature in both cases is then given by where The right-hand side of the above equation is always less than 1 for
This implies that keeping all things constant, a lower outlet temperature can be achieved in a plug flow reactor temperature.
Energy Balance in Electrochemical Cells
[0174] The heat associated with electrochemical reactions was first addressed by Sherfey and Brenner and has since then been developed in several works for specific electrochemical systems. Bernardi et. al developed a more general framework for the heat evolved in an electrochemical system that accounted for phase changes, changing heat capacity, and electrolyte mixing. This work will use the framework developed by Bernardi et al. to describe electrochemical systems that employ soluble redox species and where electrolyte mixing is not appreciable. The following analysis was performed on a low order one-dimensional adiabatic plug flow electrochemical cell to elucidate the interplay of the thermodynamics, kinetics, and operating parameters. Due to symmetry, the analysis was focused on to a porous electrode and one-half of the IEM as shown in FIGs.2.2A-2.2B. Applying the first law of thermodynamics to a control volume in an open system, it is assumed (i) a constant current density within the porous electrode, (ii) one-dimensional electrolyte flow with high heat transfer and mass transfer Péclet numbers with all-soluble redox reaction species, (iii) negligible electrolyte mixing, (iv) negligible edge effects, and (v) adiabatic boundary conditions on either side of the domain in the z-direction; this energy balance leads to the expression below. where V̇ and C v are the volumetric flowrate of the electrolyte and volumetric specific heat of the electrolyte, T is the local temperature, I⃗ is the current vector, n̂ is the unit vector in the direction of a cathodic current, L e is the length of the electrode in the x-direction, and η is the total local overpotential. The details for the derivation of Equation 3.17 may be found below. The outlet temperature of the electrolyte may now be predicted once the Joule heating due to the overpotentials is detailed. The total local overpotential is the summation of the local activation, local concentration, and volume normalized ohmic overpotentials and these expressions are derived from Butler-Volmer kinetics as shown below. The expressions for all the heat generation terms involved in Equation 3.17 are summarized in Table 3.4. [0175] The overpotential terms are derived from Butler-Volmer kinetics in the linear (activation overpotential) and Tafel (concentration overpotential) regimes for a generic half-cell reaction O + ne- → R. Details are provided below. Derivation of a Low-Order Model for an Adiabatic Plug Flow Reactor [0176] The Brayton Electrochemical Refrigeration (BECR) cycle is realized using an adiabatic plug flow electrochemical cell. A low order 1D model was developed as follows for one half of the adiabatic plug flow electrochemical cell to project the performance of this technology. As depicted in FIG.2.11, the domain of interest features a current collector, a porous electrode, and one half of the ion exchange membrane. [0177] The local energy balance in an elemental volume comprising of the porous electrode and electrolyte is given by the following equation. [0178] Upon deta substituting the equation of continuity in the absence of a fluid source, sim ompressible fluid, ignoring variation in potential energy, considering flow to be perpendicular to gravity, and neglecting viscous dissipation and viscous work, Equation. 3.18 may be simplified to the following. [0179] The first term in the equati from the surroundings. [0180] Employing effective medium the porosity weighted sum of the liquid and solid phase thermal conductivities [0181] Considering standard values for the terms on the right hand side of the above equation ( the approximation is made going forward.
Furthermore, g in equation 3.17 represents the irreversible heat generation within the control volume resulting from finite electrochemical kinetics. The second term in Equation 3.19 is the reversible work done by the control volume which, in this context, is the reversible coulombic energy associated with the electrochemical reactions at the electrode-electrolyte interface.
[0182] At this point, the following simplifying assumptions are made; (i) the applied current is uniformly distributed in the bulk of the porous electrode, and (ii) the state of the electrolyte (temperature and bulk species concentration) only varies in the x direction. It is acknowledged that a uniform current density is antithetic to complete electrochemical conversion of the reactants within the flowing electrolyte. However, it is certainly achievable by potential control of segmented current collectors. In the domain of interest, the outer wall of the current collector and the midplane of the ion exchange membrane are adiabatic. The former is enforced via insulative housing and the latter via symmetry. These conditions, along with the uniform rate of reaction enforced by the uniform current density, ensure that the temperature and bulk species concentration only vary in the x-direction. Taking the above into consideration, Equation 3.19 may be written as the following.
[0183] In the above equation, the total local enthalpy per unit mass may be written as the summation of the individual components comprising the electrolyte leading to the following.
The first term on the right-hand side may be simplified using a lumped gravimetric specific heat. The second term on the same side may be simplified by employing the Nernst-Planck flux equation and the continuity equation. Ion-migration and thermophoresis are considered negligible due to a supporting electrolyte and a small local thermal gradient, respectively. The non- dimensional Nernst-Planck flux equation then becomes the equation below. [0184] In the equation above, the mass transfer Peclet number and the non- dimensional length . In this work, the velocity of the electrolyte is on the order of 1 mm/s.
The Peclet number for proton diffusion cm) is much greater than 1.
Therefore, the gradient of the bulk species concentration may be approximated by
Plugging the above assumptions into Equation 3.22, the following result is obtained.
[0185] Once again, it is recognized that from the Nernst equation. With these assumptions, the above equation may be rearranged to the form below.
[0186] We note that the coefficient of the term on the left-hand side of the above equation maybe be given by the inverse of the thermal Peclet number Considering values for each parameter representative to this work cm), the thermal Peclet number is much greater than 1. Therefore, the Laplacian of the temperature field is negligible, and the above analysis may be summarized by the equation below after regrouping a couple of terms.
Derivation of Local Activation, Concentration, and Ohmic Overpotentials
[0187] The modified Butler-Volmer equation for a generic half-cell reaction O + ne -> R with 0 =0.5 occurring in a porous electrode is given by the following expression
[0188] At low current density, the surface concentration is equal to the bulk concentration for both species. In this limit, the overpotential is merely the activation overpotential. Under this assumption Equation 3.28 may be written as In this work, the local temperature dependent rate constant is solved for using and assuming that E a 50 kJ/mol. Applying this simplification and rearranging the above expression, the local activation overpotential may be expressed as
[0190] Now, in order to develop an expression for the concentration overpotential, the Tafel regimes are employed. At high cathodic current densities (T -n > 0), the first exponential term in the square bracket is much greater than the second, i.e. ' ' n this regime, the concentration overpotential is much larger than the activation overpotential. Taking these into consideration, Equation 3.28 may be written as the following for high cathodic current density.
[0191] Rearranging the terms in Equation 3.31 above, the concentration overpotential may be expressed as the following.
[0192] The same simplification may be done for high anodic current densities (/ -n < 0), and in this case, the concentration overpotential may be expressed as the following.
[0193] We have ignored the potential drop within the electrolyte and the corresponding ohmic overpotential under the treatment of a supporting electrolyte. The Joule heating due to the ohmic overpotential from the ion-exchange membrane is simply assumed to be split among the electrodes and therefore the ohmic overpotential is expressed as
[0194] The values for the activation and concentration overpotentials may be solved for by evaluating the local bulk and surface concentrations for both species in the porous electrode. In order to derive the local bulk concentration, the Nernst-Planck flux equation is used to start; this states that the flux of an ionic species is proportional to the gradient of its electrochemical potential. In the limit of no thermophoresis and negligible migration, the flux of an ionic species i may be expressed as [0195] We recall that the continuity equation relates the flux of a species to its local source. Assuming unity Faradaic efficiency, for each redox active species, the local source term is related to the local current density through its stoichiometry. Applying Equation 3.35 to the continuity equation the following expression is obtained. [0196] Similar to the treatment of Equation 3.24, the change in the bulk concentration is only dependent on the advection term in the limit of high Pe M , i . Integrating the resulting equation, the local bulk concentration of a species i may be expressed as the following. [0197] The local surface concentration may by now solved for using a more microscopic species balance at the electrode-electrolyte interface. Under this treatment, the surface concentration of species I is given as [0198] Now, through Equation 3.37 and Equation 3.38, the local bulk and surface concentrations are known. This allows for the calculation of the local activation and concentration overpotentials. [0199] Finally, the temperature coefficient of a half-cell reaction O+ne-→R is given by , where U is the equilibrium potential of the half-cell reactions with respect to some arbitrary reference. The Nernst equation details the equilibrium (no net current) potential for the half-cell reaction as given below. [0200] In the absence of any currents however, the bulk concentrations and the surface concentrations are equal. Correcting for the presence of concentration gradients, and applying the partial derivative operator, the temperature coefficient may now be expressed as the summation of the standard temperature coefficient and the Nernst entropic terms as given below. Ideal BECR Cycle Performance [0201] The ideal BECR cycle is internally reversible (η = 0) and has perfect heat exchangers (ε HX = 1). Under these assumptions, Equation 3.17 is integrable and the performance of the ideal BECR cycle is governed by the dimensionless figure of merit, χ. ^ ^ where N is the number of cold and hot electrochemical cells in their respective stacks, and I is the cathodic operating current. As a physical interpretation, χ is the ratio of the entropic driving force (NIα = χ E ) and the thermal inertia of the electrolyte (VĊ V = χ I ). It is also a combination of operating parameters (N, I, and V)̇ and thermodynamic properties (α and C V ). Therefore, a dearth of favorable material properties may be compensated for by the operating parameters to meet the cooling load and COP demands. The COP, β, and cooling load, Q C , for the ideal BECR cycle are given in Equations 3.42 and 3.43 below, respectively. [0202] In the above equations, T R = T H /T C The details of the derivation are given below. [0203] FIG.2.3A shows the v β relative to the Carnot COP, given by β Carnot = 1/(T R – 1), as a function of χ for three different temperature ratios. β monotonically decreases with increasing χ because of the nature of the BECR cycle. Increasing values of χ implies that the heat transfer processes occur less isothermally, and the BECR cycle deviates further from the Carnot cycle leading to lower efficiency. For a given value of T R , χ must be greater than or equal to ln(T R ) in order to provide practical cooling. In the limit of the equality, Q C = 0, and (as expected) the Carnot efficiency is recovered. As shown in Equation 3.43, Q C cannot be parametrized by χ and T R alone, and it additionally requires the specification of T C and χ E . The cooling load and COP achievable by various values of χ E for a modest space cooling application (T H = 313 K, T C = 295 K) is shown in FIG.2.3B. The cooling load, Q C , first increases rapidly as χ increases beyond χ = ln(T R ) until χ = 0.38 and then starts decreasing relatively gradually. At χ > ln(T R ), the temperature of the electrolytes exiting the cold cell are increasingly lower than T C and subsequently, Q C first increases. Beyond χ = 0.38, the thermal inertia starts to decrease and although temperature of the electrolytes exiting the cold cell is increasingly lower than T C as χ increases; for a fixed value of χ E , there is not enough electrolyte volume undergoing this drop in temperature. Therefore, Q C decreases beyond this value. The crossover value of χ is a transcendental function of T R and its details are provided below. [0204] One method of relaxing the requirements imposed on χ by the source and sink temperatures, and the COP and cooling load required by the application, is by employing a regenerator (see below). A high regeneration (internal exchange of heat between states 1 and 4), given by f reg , ensures that most of the isentropic reaction occurring in the cold electrochemical cell is used to create useful cooling load, rather than lowering the electrolyte temperature from T H to T C . Therefore, a higher f reg leads to higher values of β and Q C for a given value of χ as shown in FIGs.2.3C-2.3D. Derivation of Cooling Load and COP Expressions for the Ideal BECR Cycle [0205] The ideal BECR cycle is internally reversible ( η =0) and has 100% efficient heat exchangers mit of the former, Equation 3.22 takes the following form. The equation w integrable if it were not for the position dependence of the temperature coefficient. [0206] In order to simplify the following analysis and to keep the interplay of the main parameters obvious, the domain in which the temperature coefficient remains a constant is explored. It is noted that or a generic half-cell reaction the temperature coefficien he following using the Nernst equation. [0207] Therefore, the temperature coefficient is the summation of the standard temperature coefficient, and what will be here on out referred to as the compositional entropy. The former is a measure of the inherent difference in entropy between the product and reactant species in some reference concentration. The latter originates from the concept of fugacity and is therefore a contributing factor in all systems under a chemical equilibrium. Therefore, the half-cell reactions that are of interest in this work are those with redox active species that have a large inherent difference in entropy that originates from their structure and interactions with their immediate environment. In other words, half-cell reactions with a high α 0 value. By applying the dilute solution approximation, the reaction quotient may be expressed using the species concentrations. FIG. 2.12 plots the compositional entropy term as a function of the ratio of the surface concentrations of the reduced and oxidized species. This work is only concerned with redox couples with | ^ | ^ 0.5 mV/K and as shown in FIG.2.12, it requires high degrees of electrochemical conversion f th sitional entropy to approach that value. Therefore, to analyze the ideal BECR cycle performance as analytically as possible, the compositional entropy term is neglected and treat it as a constant value throughout the reaction coordinate . Equation 3.44 may be integrated across the length of the electrode shown in FIG 211 to the following form. [0208] The argument of the exponent in Equation 3.46 is dimensionless and is the figure of merit where a cathodic current is chosen to be positive as a reference. For the ideal BECR cycle, since the heat exchangers are 100% efficient, the inlet temperatures of the electrolytes in the cold cell are equal to that of heat sink T H . Similarly, the inlet temperatures of the electrolytes in the hot cell are equal to that of the cold source 7 C . The exit temperatures of both electrolyte (T 2 ) may be given as follows.
[0209] Furthermore, to simplify the analysis of the ideal BECR cycle and maintain the optics of the interplay between the governing parameters clear, it is assumed that
The same treatment can be applied to the electrolyte exit temperatures in the hot cell. It is also recognized that a high degree of chemical conversion may be achieved with the same amount of current by stacking electrochemical cells in series electrically and in parallel hydraulically. This is shown in FIGs. 2.4A-2.4B. By splitting an electrolyte flow among N electrochemical cells and connecting the cells in electrical series, the thermal inertia is decreased by a factor of N for each cell. This kind of architecture is useful for a fully functioning refrigerator that wants to achieve a high degree of electrochemical conversion at high flowrates, but is current limited by a fuse. In this case, the exit temperatures in Equation 3.37 are modified by simply considering . Under these assumptions, the expressions for the cooling load and rejected heat are given by the following equations respectively.
[0210] Rearranging some terms and using the above equations respectively become the following. [0211] The COP of the ideal BE Ideal BECR Cycle with Reg [0212] The temperature constraints that the source and sink impose on the ideal BECR cycle may be mitigated by the inclusion of a regenerator. The regenerator is a heat exchanger that allows the electrolytes exiting the cold side and hot side heat exchangers to undergo an iso-molar heat transfer process between themselves. The result is that the inlet temperature of the cold EC cell electrolytes are lower than T H , and that of the hot EC cell electrolytes are higher than T C . Therefore, the inclusion of refrigeration allows for lower values of T 2i for a given value of ^ . In this work, regeneration is quantified using the metric given below. [0213] A similar energy all the states are known for, the cooling load and COP may now be written as given below. ^ ^ [0214] The minimum is given by, ^ ^ l ^T χ Corresponding to Maximum C [0215] For a given value of T R and f reg , the maximum COP is given by setting ^ ^ ^ ^ ^ 0. The resulting quadratic equation is solved and the root outside of the domain given i E tion 3.49 is discarded. The value of χ ^ that generates the maximum cooling ( ^ max, ^ ) load is given by the equation below. [0216 ] g C , , E , g y g value of ^ that gene es the maximum cooling ( ^ max,Q ) load is given b C the transcendental equation given below. e x 1 [0217] The equation give requirements of a scaled up BECR cycle that is looking to generate high cooling loads. Both the values in Equation 3.57 and Equation 3.58 are plotted in FIGs.2.15A-2.15B. Practical Material Considerations [0218] Before analyzing the non-ideal BECR cycle with realistic parameters, it is important to consider practical material constraints. Two constraints on χ are recognized, arising from (i) the freezing or evaporation of the solvent, and (ii) insufficient solubility. It is again noted that the continuous nature of the BECR cycle is made possible by all-soluble redox active half-cell reactions that are dissolved in a solvent. These solvents are most susceptible to freezing and evaporation at states 2 and 4, respectively. In order to operate the BECR cycle continuously, T 2 must be greater than the freezing point of the electrolytes, and the solvent’s saturation pressure at T 4 must be lower than the maximum pressure that the system can withstand. FIG.2.4A shows the maximum possible χ, denoted by χ PHASE , that limits the BECR cycle performance via freezing or pressurization for five common solvents. The vapor pressure and freezing points of the pure solvents are used and therefore, the reported data is conservative. The freezing point and Antoine equation coefficients were obtained from the NIST database. Water and benzene are freezing point limited due to hydrogen bonding and larger molecular weight, respectively. The alcohols and pentane are relatively more volatile and are vapor pressure limited. The pressure limit is taken to be 10 atm for the present considerations (consistent with limits on conventional refrigerators appliances), but higher pressures are possible. This material constraint may be mitigated by using solvent blends, which have depressed freezing points and lower vapor pressures, or through employing room temperature ionic liquids as solvents, which inherently possess low vapor pressures. Details on the derivation of χ PHASE are provided below. It is emphasized herein that χ PHASE is just one way of screening solvents based on the source and sink temperatures. Solvents must also be rationally screened to facilitate reversible electrochemical reactions (i.e., they must not participate in side half-cell reactions, they must be stable in the pH required for the chosen half-cell reactions, etc.), and they must not degrade the chosen ion exchange membrane. [0219] In the electrochemical cells, the molar influx of the reactant species must be able to support the rate of the reaction dictated by the operating current. As per Faraday’s law of electrolysis for an open system, there is a maximum possible operating current for a given volumetric flowrate determined by the solubility limit of the reactant species. FIG.2.4B shows the maximum possible χ, denoted by χ SOLUBILITY , that limits the BECR cycle performance because of insufficient reactant solubility as a function of the solubility limit, and parametrized by the solvent specific heat and temperature coefficient. As a point of reference, the standard for high temperature coefficient half-cell reactions is the ferricyanide/ferrocyanide redox couple (α = -1.4 mV/K). However, the solubility limit of potassium ferrocyanide is ~0.6 M at room temperature. As shown in FIG.2.4B, for use in a space cooling application, a half-cell reaction with α = 1 mV/K must have a minimum solubility of ~3 M in water. This material constraint may be mitigated in general with the discovery of new half-cell reactions with high temperature coefficients, low specific heats, and high half-cell species solubility. For instance, Kim et al. reported over a factor of 2 increase in the temperature coefficient of the ferricyanide/ferrocyanide redox couple in 20% v/v aqueous solution of methanol. This result is even more relevant to this work, because the specific heat of aqueous methanol solution is lower than that of water. Additionally, a solubility 1.5 M has recently been achieved for the ferricyanide/ferrocyanide redox couple using hydrophilicity of ammonium counter-ions. Details on the derivation of χ SOLUBILITY are provided below. χ is Limited by Solvent Freezing and Vapor Pressure [0220] At state 2 the electrolytes are at the lowest temperature. Since this works considers liquid states electrolytes that is movable using a pump, the electrolytes can potentially freeze or vaporize beyond a critical value. Therefore, for an ideal BECR cycle with regeneration, the following inequality must hold. T 2 ^ T 1 exp( [0221] T f.p is the freezing point of t and the following is subsequently derived. ^ ^ ^ ln ^ [0222] Similarly, at state 4, th pressure must not exceed a critical value that the system is capable of handling. T 4 ^ T 3 exp [0223] T v.p is the saturation temp Replacing T 3 in the equation below the following is obtained. ^ ^ ^ ln ^ [0224] Since the BECR cycle the maximum possible value of ^ is the les of the two right hand sides. This value is given as χ is Limite [0225] The electrochemical reactions occurring within the EC cells are sustained by the concentration of the reactant species in the corresponding electrodes. Therefore, the limiting solubility of the reactants dictates the rate of electrochemical conversion (determined by the operating current) relative to the rate of species influx (determined by the volumetric flowrate). A simple balance is employed that is centered on Faraday’s law of electrolysis to reflect this conservation. For the electrolyte contains the solubility limited reactant, the following equality can be written. [0226] The greatest possib of maximum possible conversion, the outlet concentration is zero. Remembering that for the BECR cycles the electrolyte flow rate is split between N EC cells, the following equality holds in the limit of maximum possible reactant conversion. n FC SOL ^ [0227] The value of ^ in the above ^ SOLUB Non-Ideal BECR Cycle Perfo [0228] The previous analysis informs the asymptotic performance of the BECR cycle with idealized components. However, Joule heating and incomplete heat exchange are unavoidable, and they must be considered to predict the performance of a real world BECR cycle. The local overpotential terms detailed in Table 3.4 are substituted into Equation 3.17 and solved to evaluate the electrochemical cell exit temperature of the electrolytes as a function of the inlet temperatures, χ, and electrochemically relevant parameters such as the standard rate constant, mass transfer coefficient, reactant solubility limit, and membrane resistance. The heat exchanger exit temperature of the electrolytes are related to the corresponding inlet temperature through the heat exchanger efficiency, ε HX . These relations are used in an iterative procedure to evaluate the non- ideal BECR cycle states, and therefore, its performance. First, the effect of increasing operating current on BECR cycle operating in a space cooling application with N=1 is illustrated. Electrochemical cells are chosen that are 8 cm × 8 cm × 5 mm. The catholyte and anolyte are hypothetical redox couples with standard temperature coefficients of 2 mV/K and -2 mV/K, respectively. The limiting solubility of the in both electrolytes is 3 M. The solvent is chosen to be water and the specific heat is that of 3 M brine solution (3.56 MJ/m 3 K). For both half-cell reactions the standard rate constant k 0 = 10 -7 m/s, mass transfer coefficient k M = 10 -5 m/s, and area specific resistance of the IEM ASR = 1 Ω cm 2 . Keeping in mind the material constraints in FIGs.2.4A- 2.4B, χ = 0.12 and ε HX = 0.8 were fixed, and an iterative solution procedure was executed to evaluate how varying current densities, and, therefore, overpotential, affect the non-ideal BECR cycle performance. FIG.2.5A shows the non-ideal BECR cycles for three different current values (1 A, 8 A, 10 A). At 1 A, the overpotentials are low enough such that the electrochemical reaction processes (1 nA to 2 nA and 3 nA to 4 nA ) are nearly isentropic shown by the nearly vertical transitions. Conversely, at 10 A, the excessive overpotential leads to large entropy generation from Joule heating in the electrochemical reaction processes (1 nC to 2 nC and 3 nC to 4 nC ). The non-ideal cooling loads and COPs are shown alongside their corresponding ideal cycle trends for comparison in FIG. 2.5B. The cooling load first increases with the operating current due to a larger rate of reaction but starts to decrease at ~5 A due to the overpotentials that scale rapidly with current density. At ~11 A, the cooling load is 0 W because the Joule heating is high enough such that the exit temperatures of the electrolytes are nearly equal to T C . The COP monotonically decreases with increasing operating current due to the entropy generation from Joule heating. The operating current dependent contributions of the terms in Table 3.4 corresponding to the cold cell and hot cell are detailed in FIGs.2.5C-2.5D, respectively. Although the total heat generation in the cold cell is endothermic across all current values, the electrolyte drop in temperature is insufficient to generate cooling in the cold side heat exchanger at T C . In both the cells, ohmic overpotential is the dominant source of Joule heating and it can be reduced by using electrodes with larger cross- sectional areas or improved membranes with lower area specific resistance. [0229] A more general analysis was then performed of the non-ideal BECR cycle by varying the operating current density χ E and operating current density simultaneously (FIG.2.6), keeping all other parameters the same as in the results of FIGs.2.5A-2.5D. As expected, the highest cooling is achieved at high values of χ E (as shown in FIG. 2.3B) and low values of current density. Increasing the entropic driving force χ E by increasing the number of cell stacks N, operating current I, or electrolyte temperature coefficient α, increases the degree of entropic heat generation and, therefore, increases the amount of useful cooling. Increasing the operating current density for a fixed electrochemical cell, however, generates an increasing amount of electrochemical overpotential and, therefore, decreases the amount of useful cooling. At ~0.3 A/cm 2 , the Joule heating is high enough such that the temperature of the electrolytes exiting the cold EC cell is nearly equal to T C . No practical cooling is achievable past this value of current density. This cutoff point can be increased by lowering the overpotentials: (i) choosing half-cell reactions with higher rate constants, (ii) increasing solubility of the limiting reagent species, (iii) improving electrodes and flow fields to generate higher mass transfer coefficients, (iv) and developing ion exchange membranes with low area specific resistance. Finally, to quantify the terms in Table 3.4 relative to the standard temperature coefficient, three dimensionless parameters, X, A, and M were derived. [0230] Parameter X represents he limiting reagent species with a solubility limit C SOL . The difference between the inlet concentration C IN and outlet concentration C OUT is derived using Faraday’s law within an open electrochemical system, making X a proxy for the ratio of the operating current and electrolyte flowrate. [0231] Parameter A represen urrent from the local exchange current density, which is governed by the standard rate constant of the half-cell reaction, k 0 , in addition to the other terms. [0232] Parameter M represen urrent to the local limiting current density, which is governed by the mass transfer coefficient k M , in addition to the other terms. It can be extrapolated from the above formulations that parameter A influences the activation overpotential that depends on the local bulk concentrations and rate constant, and parameter M (for mass) governs the concentration overpotential and the Nernst entropic terms that depend on the local surface concentrations. Since parameter X dictates the local bulk conversion, it affects the activation (FIG. 2.7A) and concentration (FIG.2.7C) overpotentials, as well as the Nernst entropic terms (FIG. 2.7D). The ohmic overpotential is only dependent on the area specific resistance of the membrane and current density and it is shown for completion in FIG 2.7B. The activation and concentration overpotentials, and the Nernst entropic terms can naturally be normalized by temperature as per the terms in Table 3.4 and the ohmic overpotential is normalized by T amb =298 K to facilitate comparison with the standard temperature coefficient of the half-cell reaction of interest. The activation overpotential increases with increasing X due to lower bulk concentrations, and with increasing A due to a greater ratio of the operating current density and exchange current density. The concentration overpotential and the Nernst entropic terms both increase with increasing X due to lower bulk concentrations, and with increasing M as the concentration boundary layer thickens. In both the above terms, there is a maximum possible M for a given X, at which the surface concentration of the limiting reagent species reaches zero. Beyond this point, the concentration overpotential increases precipitously generating large amounts of Joule heat. It is believed that dimensionless parameters X, A, and M, along with the operating parameter χ can help screen half-cell reactions, estimate the operating parameters of the BECR cycle, and project its performance. [0233] In summary, the Brayton Electrochemical Refrigerator (BECR) cycle has been introduced; this can generate continuous refrigeration by using all-soluble redox active species and open flow cell architectures. The electrochemical cells must be adiabatic and support electrolyte plug flow that mitigates internal electrolyte mixing and heat transfer. A low order one dimensional model was developed for such electrochemical cells to predict the performance of this technology. The model considers all sources of heat generation (i.e., entropic heat, activation, ohmic, and concentration) in an electrochemical system with liquid electrolytes. It makes assumptions of high thermal and mass transfer Peclet numbers and uniform current density throughout the porous electrode. The assumptions simplify the analysis of the BECR cycle and predict the BECR performance to the first order. In the asymptotic limit of internal reversibility and complete heat exchange, the analysis recovers the figure-of-merit χ. Low χ values afford high COP and the cooling load scales with the entropic driving force, χ E . The inclusion of regeneration will allow for a higher COP and cooling load for a given χ. Two material constraints have been identified that restrict the values available for χ. In general, the BECR cycle benefits from low electrolyte freezing point, low electrolyte vapor pressure, high temperature coefficient, low electrolyte specific heat, and high reactant solubility. The performance of the non-ideal BECR cycle was reported using realistic values for thermodynamic and kinetic parameters. Finally, dimensionless parameters X, A, and M were introduced; these will quantify detrimental heat generation. In conjunction with χ, these parameters will help screen half-cell reactions, and estimate the performance of a real BECR cycle. In two other works, it has been demonstrated the BECR proof-of-concept and identified four material level challenges that must be solved to help increase this technology’s market readiness. Dimensionless Parameters X, A, and M [0234] The expressions for the activation overpotential, concentration overpotential, and Nernst entropic terms are given in Equation 3.30, Equation 3.32, and Equation 3.40, respectively. These terms are dependent on the bulk and/or surface concentration of the participating species given in Equation 3.37 and Equation 3.38. First, the concentration terms are modified to be constrained by Faraday’s law of electrolysis. For an open electrochemical electrode, Faraday’s law affords the following balance for the limiting reagent species. [0235] The limiting case for the above species balance is when the species enters at the solubility limit and is absent in the effluent stream. In this case, the ratio of the right-hand side of Equation 3.70 takes on the maximum possible value as shown below. [0236] We now define the dimensionless parameter X to be the ratio between the left-hand sides of Equation 3.70 and Equation 3.72. [0237] Equation 3.37 for the bulk concentration may then be written as the following. [0238] Additionally, 100% faradaic efficiency ensures that the summation of the local concentrations of the participating species is equal to the solubility limit. Employing this with the Equation 3.73, the following expression may be written. [0239] Equation 3.74 may be used in conjunction with Equation 3.38 to describe the surface concentration using parameter X. [0240] We use Equation 3.74 in the expression for the activation overpotential in Equation 3.30 to generate the following equation. [0241] We now can define the dimensionless parameter A as given in Equation 3.77. [0242] Finally, to compare the activation overpotential with the standard temperature coefficient which has units of V/K, the length averaged and temperature normalized activation overpotential as given below is reported. [0243] For the concentration overpotential and Nernst entropic terms, the parameter M is defined as given below. [0244] The volume averaged and temperature normalized concentration overpotential and Nernst entropic terms are then derived from Equation 3.32 and Equation 3.40 to generate the equations below. [0245] It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above- described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims. REFERENCES 1. Abraham, T. J. et al. (2013) High Seebeck coefficient redox ionic liquid electrolytes for thermal energy harvesting. Energy Environ. Sci.6, 2639–2645 2. Ambrose, D., et al. (1975). Thermodynamic properties of organic oxygen compounds XXXVII. Vapour pressures of methanol, ethanol, pentan-1-ol, and octan-1-ol from the normal boiling temperature to the critical temperature. The Journal of Chemical Thermodynamics 7, 185–190. 10.1016/0021-9614(75)90267-0. 3. Aschenbrenner, O., et al. (2009). Measurement of vapour pressures of ionic liquids and other low vapour pressure solvents. Green Chem.11, 1217–1221.10.1039/B904407H. 4. Bard, A. J. (2001) Electrochemical methods: fundamentals and applications (2 nd ed). Wiley 5. Bernardi, D. (1985). A General Energy Balance for Battery Systems. Journal of The Electrochemical Society 132, 5.10.1149/1.2113792. 6. Bonetti, M. et al. (2011) Huge Seebeck coefficients in nonaqueous electrolytes. J. Chem. Phys. 134, 114513 7. Bridgeman, O.C., et al. (1964). Vapor Pressure Tables for Water. Journal of Heat Transfer 86, 279–286.10.1115/1.3687121. 8. Çengel, Y.A. (2011). Heat and mass transfer^: fundamentals & applications / Yunus A. Çengel, Afshin J. Ghajar.4th ed.. (New York^: McGraw-Hill). 9. Chum, H. et al. Review of Thermally Regenerative Electrochemical Systems. (1981). 10. Davis, R. The Heat of Adsorption of Hydrogen Gas on Lanthanum Pentanickel. 11. Dittman, G.L. (1977). Calculation of brine properties. http://www.osti.gov/servlets/purl/7111583/. 12. Dittmar, L., et al New Concept of an Electrochemical Heat Pump System: Theoretical Consideration and Experimental Results. in Electrochemical Engineering and Energy (eds. Lapicque, F., et al.) 57–65 (Springer US, 1995). doi:10.1007/978-1-4615-2514-1_6. 13. Duan, J. et al. Aqueous thermogalvanic cells with a high Seebeck coefficient for low-grade heat harvest. Nat. Commun.9, 5146 (2018). 14. Duan, Z. N., et al. Thermodynamic and electrochemical performance analysis for an electrochemical refrigeration system based on iron/vanadium redox couples. Electrochimica Acta 389, 138675 (2021). 15. Efficient and Climate-Friendly Cooling (2020). (United Nations Environment Programme). https://wedocs.unep.org/bitstream/handle/20.500.11822/31587/ ECFC.pdf?sequence=1&isAll owed=y. 16. Fogler, H. Essentials of Chemical Reaction Engineering, 2nd Edition. (Pearson, 2017). 17. Gerlach, D. W. et al. Basic modelling of direct electrochemical cooling. Int. J. Energy Res. 31, 439–454 (2007). 18. Hammond, R. H. et al. An electrochemical heat engine for direct solar energy conversion. Sol. Energy 23, 443–449 (1979). 19. Hansen, J. et al. Global temperature change. Proc. Natl. Acad. Sci.103, 14288 (2006). 20. Henry, A. A New Take on Electrochemical Heat Engines. Joule 2, 1660–1661 (2018). 21. Hu, R. et al. (2010) Harvesting waste thermal energy using a carbon-nanotube-based thermo-electrochemical cell. Nano Lett.10, 838–846 22. Hupp, J. T. et al. (1984) Solvent, Ligand, and Ionic Charge Effects on Reaction Entropies for Simple Transition-Metal Redox Couples. Inorg. Chem.28, 3639–3644 23. Jacobs, S. et al. The performance of a large-scale rotary magnetic refrigerator. New Dev. Magn. Refrig.37, 84–91 (2014). 24. Jiang, B., et al. (2016). A comparative study of Nafion series membranes for vanadium redox flow batteries. Journal of Membrane Science 510, 18–26. 10.1016/j.memsci.2016.03.007. 25. Kim, J. H. et al. Iron (II/III) perchlorate electrolytes for electrochemically harvesting low- grade thermal energy. Sci. Rep.9, 8706 (2019). 26. Kim, T., et al. (2017). High thermopower of ferri/ferrocyanide redox couple in organic-water solutions. Nano Energy 31, 160–167.10.1016/j.nanoen.2016.11.014. 27. Kreysa, G. et al. Theoretical consideration of electrochemical heat pump systems. Electrochimica Acta 35, 1283–1289 (1990). 28. Lee, S. W. et al. An electrochemical system for efficiently harvesting low-grade heat energy. Nat. Commun.5, 3942–3942 (2014). 29. Li, B., et al. (2016). Performance analysis of a thermally regenerative electrochemical refrigerator. Energy 112, 43–51.10.1016/j.energy.2016.06.045. 30. Luo, J., et al. (2019). Unprecedented Capacity and Stability of Ammonium Ferrocyanide Catholyte in pH Neutral Aqueous Redox Flow Batteries. Joule 3, 149–163. 10.1016/j.joule.2018.10.010. 31. Makarov, D. M. et al. Density and volumetric properties of the aqueous solutions of urea at temperatures from T=(278 to 333) K and pressures up to 100^MPa. J. Chem. Thermodyn. 120, 164–173 (2018). 32. Marcus, R. A. et al. Solvent dynamics and vibrational effects in electron transfer reactions. J. Electroanal. Chem. Interfacial Electrochem.204, 59–67 (1986). 33. Marcus, Y. et al. (2006) Ion Pairing. Chem. Rev.106, 4585–4621 34. McKay, I. S. et al. (2019) Electrochemical Redox Refrigeration. Sci. Rep.9, 13945 35. Metz, B., et al. Safeguarding the Ozone Layer and the Global Climate System (IPCC/TEAP). https://www.ipcc.ch/report/safeguarding-the-ozone-layer-and- the-global- climate-system/. 36. Newell, Ty. A. Thermodynamic analysis of an electrochemical refrigeration cycle. Int. J. Energy Res.24, 443–453 (2000). 37. Newman, J., et al. (1975). Porous-electrode theory with battery applications. AIChE Journal 21, 25–41.10.1002/aic.690210103. 38. Oh, K., et al. (2015). Three-dimensional, transient, nonisothermal model of all-vanadium redox flow batteries. Energy 81, 3–14.10.1016/j.energy.2014.05.020. 39. Osborn, A.G., et al. (1974). Vapor-pressure relations for 15 hydrocarbons. J. Chem. Eng. Data 19, 114–117.10.1021/je60061a022. 40. Pecharsky, V. K. et al. Giant Magnetocaloric Effect in Gd5(Si2Ge2). Phys Rev Lett 78, 4494–4497 (1997). 41. Poletayev, A. D. et al. (2018) Continuous electrochemical heat engines. Energy Environ. Sci.11, 2964–2971 42. Pollard, R., et al. (1981). Mathematical Modeling of the Lithium-Aluminum, Iron Sulfide Battery: I . Galvanostatic Discharge Behavior. Journal of The Electrochemical Society 128, 491–502.10.1149/1.2127445. 43. Rajan, A., et al. Electrolyte engineering can improve electrochemical heat engine and refrigeration efficiency. Trends in Chemistry.10.1016/j.trechm.2021.12.006. 44. Ritchie, H. et al. Urbanization. https://ourworldindata.org/urbanization (2018). 45. Schafner, K., et al. (2019). Membrane resistance of different separator materials in a vanadium redox flow battery. Journal of Membrane Science 586, 106–114. 10.1016/j.memsci.2019.05.054. 46. Sherfey, J.M., et al. (1958). Electrochemical Calorimetry. Journal of The Electrochemical Society 105, 665.10.1149/1.2428687. 47. The Core Writing Team, R. K. Pachauri, and L. A. Meyers (2014). Climate Change 2014: Synthesis Report. Contribution of Working Groups I, II and III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC). 48. The Future of Cooling (2018). (International Energy Agency). https://www.iea.org/futureofcooling/. 49. Thermodynamic Properties of DuPont Suva 410A Refrigerant. < https://www.cantas.com/urunpdf/20.09.008_h64423_Suva410A_the rmo_prop_si.pdf> accessed March 16, 2022. 50. Thermodynamics Research Center (2021) Thermodynamics Source Database in NIST Chemistry WebBook, NIST Standard Reference Database Number 69, DOI: 10.18434/T4D303 51. Velders, G. J. M. et al. Preserving Montreal Protocol Climate Benefits by Limiting HFCs. Science 335, 922–923 (2012). 52. Velders, G. J. M., et al. The large contribution of projected HFC emissions to future climate forcing. Proc. Natl. Acad. Sci.106, 10949–10954 (2009). 53. Willingham, C.B., et al. (1945). Vapor pressures and boiling points of some paraffin, alkylcyclopentane, alkylcyclohexane, and alkylbenzene hydrocarbons. Journal of Research of the National Bureau of Standards 35, 219.10.6028/jres.035.009. 54. Worswick, R. D., et al. The enthalpy of solution of ammonia in water and in aqueous solutions of ammonium chloride and ammonium bromide. J. Chem. Thermodyn.6, 565–570 (1974). 55. Wu, M.S., et al. (1998). Thermal behaviour of nickel/metal hydride batteries during charge and discharge. Journal of Power Sources 74, 202–210.10.1016/S0378-7753(98)00064-0. 56. Yamato, Y. et. al. (2013) Effects of the Interaction between Ionic Liquids and Redox Couples on Their Reaction Entropies. J. Electrochem. Soc.160, H309–H314 57. Yang, H., et al. (2004). Determination of the Reversible and Irreversible Heats of a LiNi[sub 0.8]Co[sub 0.15]Al[sub 0.05]O[sub 2]/Natural Graphite Cell Using Electrochemical- Calorimetric Technique. Journal of The Electrochemical Society 151, A1222. 10.1149/1.1765771. 58. Yang, Y. et al. Charging-free electrochemical system for harvesting low-grade thermal energy. Proc. Natl. Acad. Sci.111, 17011–17016 (2014). 59. Zhou, H., et al. Supramolecular Thermo-Electrochemical Cells: Enhanced Thermoelectric Performance by Host–Guest Complexation and Salt-Induced Crystallization. J. Am. Chem. Soc.138, 10502–10507 (2016).