Direct measurement of the genuine efficiency of thermogalvanic heat-to-electricity conversion in thermocells

Harvesting wasted thermal energy could make important contributions to global energy sustainability. Thermogalvanic devices are simple, chemistry-based devices which can convert heat to electricity, through facile redox chemistry. The efficiency of this process is the ratio of electrical energy generated by the cell (in Watts) to the quantity of thermal energy that passes through the cell (also in Watts). Prior work estimated the quantity of thermal energy passed through a thermocell by applying a conductive heat transfer model to the electrolyte. Here, we employ a heat flux sensor to unambiguously quantify both heat flux and electrical power. By evaluating the effect of electrode separation, temperature difference and gelation of the electrolyte, we found significant discrepancy between the estimated model and the quantified reality. For electrode separation, the trend between estimated and measured efficiency went in opposite directions; as a function of temperature difference, they demonstrated the same trend, but estimated values were significantly higher. This was due to significant additional convection and radiation contributions to the heat flux. Conversely, gelled electrolytes were able to suppress heat flux mechanisms and achieve experimentally determined efficiency values in excess of the estimated values (at small electrode separations), with partially gelled systems being particularly effective. This study provides the ability to unambiguously benchmark and assess the absolute efficiency and Carnot efficiency of thermogalvanic electrolytes and even the whole thermocell device, allowing ‘total device efficiency’ to be quantified. The deviation between the routinely applied estimation methodology and actual measurement will support the rational development of novel thermal energy harvesting chemistries, materials and devices.


Introduction
Thermoelectrochemistryor electrochemistry where temperature is applied as an active variablehas many implications and applications, given the signicant impact of temperature upon both thermodynamic and kinetic parameters. 1 A key thermodynamic parameter is the temperature effect upon the standard electrode potential. 2,3 A growing application of this is thermogalvanic cells; these are devices that typically comprise of two electrodes at different temperatures in contact with a solution or gel that contains both oxidation states of the same redox couple (i.e. both [Fe(CN) 6 ] 3À and [Fe(CN) 6 ] 4À or Fe 2+ and Fe 3+ ). 4 The presence of a temperature difference (DT) across the two electrodes results in a thermodynamically-driven induced potential difference (DV) between the electrodes, which drives redox processes that results in a ow of electrical current. This, coupled with diffusion of the redox couple between the two electrodesresults in an elegant, purely chemical route for the conversion of a temperature gradient into electricity. 5 Given that ca. two-thirds of the energy from human industry (even down to the human metabolism) is dissipated as low grade waste heat, 6 widespread application could result in major efficiency gains, e.g. such as on industrial piping 7 or skin. [8][9][10] Thermogalvanic electricity production is an entropicallydriven process, and the magnitude of the driving force is normally expressed as the 'thermogalvanic Seebeck coefficient', or S e ; where DS rc is the difference in entropy between the two redox states, n is the number of electrons transferred and F is the Faraday constant. 11 The S e therefore represents the possible potential difference; Soret effects 12 and thermocapacitive effects 13,14 can also be signicant contributors to the potential difference, but are only signicant in fairly unique (typically gelled) systems. The S e is sensitive to the absolute concentration of the redox couple, the ratio of the concentration of the two oxidation states, and the ionic strength. 4 The current generated by this potential difference is also sensitive to the concentration and ratio of the redox species, 4 and especially sensitive to the S e and DT; 11 it's also sensitive to the mass transport of the two redox states (diffusion coefficient, convection, inter-electrode transport distance, etc.), the kinetics of the redox process, and the electrode surface area. 11 Fig. 1(a) shows a general layout for measuring a thermogalvanic cell, where a temperature difference can be easily applied across the two electrodes. The electrical power produced by the two electrodes can then be easily quantied via a variety of routes. 15 Once the cell comes to equilibrium and steady state power output is achieved, the power is oen limited by one dominant resistance, which is typically either kinetics (e.g. rate of electron transfer) or the rate of mass-transport between the two electrodes. 5,15 This one major resistance factor, R, means the thermogalvanic cell displays a linear but inverse relationship between the voltage and the current produced, in line with Ohm's Law, so V ¼ IR. The maximum voltage from the cell is characterised as the open circuit potential (V ocp , in V), and the maximum current as the shortcircuit current density (j sc , in A m À2 ). Power generated follows Watt's Law, so P ¼ IV. The inverse relationship between V ocp and j sc means that thermogalvanic cells typically generate a parabolic power curve, 15 where the maximum power density generated by the cell (P max , W m À2 ) occurs at half the maximum current and half the maximum voltage, such that; Thermogalvanic conversion of thermal energy to electrical energy is not a perfect process and is limited by both Carnot efficiency and non-ideal processes. Fig. 1(b) highlights the scenario if the hot electrode temperature is far above ambient temperature while the cold electrode is close to ambient; conduction of heat will occur through the cell and electrolyte, temperature-difference induced convection in the electrolyte will exacerbate this, and radiative heat loss into the surroundings will also occur. The thermoelectric-inspired 'dimensionless ZT gure of merit' is sometimes referred to in thermogalvanic literature in an effort to consider the competing thermogalvanic current and heat ux processes; where the bigger the ZT the more 'merit' the device has for power production, with s the ionic conductivity of the electrolyte and k the thermal conductivity (steady state, conduction-only, and in the absence of a temperature gradient). This approximate relationship is largely valid for thermoelectrics, since the entire device is typically solid, and electrical conductivity correlates well with charge carrier mobility. However, this is a awed metric for thermogalvanic systems, because the s typically signicantly overestimates the quantity and mobility of the actual charge carriers in the cell because it measures all ions including non- Fig. 1 Diagrams highlighting (a) the general layout of the thermogalvanic cell employed here, whereby electrical power can be generated (and measured) when a temperature difference is applied via the heat source/sinks; (b) the three heat flux mechanisms expected (if the temperature of the heat source, T h , is significantly above T c , and T c is close to the ambient temperature). This figure excludes air-convection and also the metallic wiring, which can be an additional source of conduction; (c) how the heat flux is typically estimated, using an electrolyte-only, conduction-only approximation; and (d) the experimental design employed in this study whereby all heat flux through the heat source into the cell is quantified simultaneous to electrical power quantification.
redox-active supporting electrolyte, 16 while the k signicantly underestimates heat transfer through a mobile convection-prone liquid exposed to a temperature difference. 7 The absolute efficiency, h, is another key (and in theory unambiguous) means of benchmarking thermogalvanic device performance. The absolute efficiency is quantied by the proportion of thermogalvanic electrical power generated by the cell (p max in W) from the corresponding heat ux passing into the cell (q, also in W), as shown in eqn (4). This can then be expressed as the fraction of the theoretically limiting Carnot cycle efficiency, h r , based upon the applied hot and cold electrode temperatures (T h and T c , respectively), as shown in eqn (5); Arguably the magnitude of the efficiency doesn't necessarily matter; if thermal energy that would otherwise be wasted is instead valorised then that can be a net benet. It's been stated that thermogalvanic devices with Carnot-relative efficiencies in the range of 2 to 5% could be commercially competitive, 11 although a full technoeconomic comparison is still required to conrm if these values should be higher or lower. Nevertheless, the more efficient this process is at valorising waste, the more 'green' it is (alongside a series of other considerations). 17,18 Additionally the more green and cost-effective it is, the more 'sustainable' the process is. 19 In an effort to benchmark efficiencies, an 'electrolyte-only' heat ux estimation is commonly employed for thermogalvanic cells. This is exemplied in Fig. 1(c). Typically, the heat ux through the cell housing itself is disregarded, since most early cell constructions were not representative of applicationappropriate cells. Instead, only the body of the electrolyte is considered, and is treated as a 1D heat ux through a 2D solid material using Fourier's Law; where k is the (steady state, conduction-only) thermal conductivity of the electrolyte again, A the cross-sectional area of the electrolyte, and d the distance between the two electrodes. Early adoption of this estimation was mathematically justi-ed, since it was suggested convective heat ux was negligible under their particular circumstances. 11 However, cell design has since diversied, and several studies have highlighted the very signicant role convection can play in certain thermogalvanic cells. [20][21][22] The use of IR cameras in electrolyte measurements explicitly relies upon radiation out of the cell, 20,23 which is also not considered by eqn (6). Numerous application-appropriate cell designs have also been reported, such as integrated into clothing, 8 wearable on skin, 9,10 installed onto hot water pipes, 7 etc., yet heat ux quantication has not advanced in step with this progress. While awed, this estimated heat ux has nevertheless been the most valid approach available for benchmarking the efficiencies of new systems, with several solution-based, nano-structured electrode systems having Carnot-relative efficiencies predicted in excess of 0.5%, 7,23-33 and even >2%. 23,[34][35][36] Two recent studies have gone beyond directly measuring thermogalvanic power and estimating the heat ux. Wu et al. investigated gelled vs. non-gelled electrolytes, and calculated the efficiency via the estimated heat ux; no signicant difference was observed between the two types of electrolytes. 22 However, the cold electrode temperature was maintained using a peltier device, and the power required to keep that electrode isothermal was two-fold higher in the ungelled electrolyte than the gelled electrolyte, demonstrating there was a signicant difference in the genuine bulk heat ux through the cell that was not detected by the thermogalvanic measurements; likely because the thermal apparatus actively compensated for this difference by providing more cooling power. 22 Yu et al. reported the efficiency of a thermogalvanic cell using the estimated heat ux equation, but rather than apply the conduction-only thermal conductivity, k, they used IR imaging to estimate the effective thermal conductivity, k eff . 23 This indirect measurement (via heat radiated from a cell wall) suggested that at an applied temperature difference of 50 K, the k eff reached 1.64 W m À1 K À1 ; 23 far above the k ¼ 0.55 W m À1 K À1 measured in other studies. 34,37 Interestingly, the addition of guanidinium resulted in crystallisation of the [Fe(CN) 6 ] 4À , and the presence of this solid material in the cell reduced k eff to ca. 0.4 W m À1 K À1 . This value of k eff , combined with a signicantly boosted S e value due to the crystallisation process, resulted in a very signicant electrolyte-only Carnot efficiency value of 11.1% being stated (at DT ¼ 40 K). 23 The purpose of this study was to prepare a thermoelectrochemical cell that could unambiguously measure both the heat ux and the thermogalvanic power; the ratio of these two quantied values will then yield the absolute efficiency for an entire thermogalvanic device, for the rst time. Fig. 1(d) describes this concept, whereby a heat ux sensor is introduced as a thermal bottleneck through which all thermal energy from the hot electrode must pass, thus allowing total quantication of the heat ux (q total , which subsequently passes through and out of the cell, via conduction, convection and/or radiation). If the heat ux through the empty cell, q empty , is subtracted it can yield a directly measured and quantied electrolyte-only heat ux, q m . That was performed in this study, and the trends in efficiency compared between those obtained using direct measurements vs. those predicted using the estimated heat ux given via eqn (6). This study was used to achieve unambiguous quantication, then explore the effect of the parameters DT, d and gelled vs. un-gelled electrolytes upon efficiency.

Experimental
Electrode material comparison using conventional thermogalvanic assembly Initial experiments used a setup that has been previously reported in detail elsewhere; 4 namely a 6.7 mm diameter cylinder was machined from a block of PMMA (8.4 mm deep). At either end larger, shallow cylinders (10 mm diameter, 0.5 mm deep) were machined to form lips around the main cylinder, into which were inserted solid platinum electrodes (10 mm diameter, 1 mm thick disc, from SurePure Chemetals, USA). This gave a geometric electrode surface area of 35 mm 2 and an interelectrode spacing of 7.4 mm. Different types of carbon electrode materials were then cut into ca. 10 mm circles by hand, and inserted between the platinum electrode and the cylinder, into which the electrolyte was injected. The carbon electrodes were Pyrolytic Graphite Thermal Interface Material (0.017 mm thick, 1750 W m À1 thermal conductivity, RS Components, UK) or two thicknesses of exible graphite 'graphoil' gasket sheets (0.3 mm thick or 1 mm thick, both from Xiaochengshop, China).
The platinum electrodes were temperature controlled by contacting with RS-TX150 thermostatic circulator baths (Grant Instruments Ltd, UK) via copper heat exchangers, as previously described. 4 Thermogalvanic characterisation methodology All thermogalvanic measurements were allowed to reach steady state (unless otherwise specied in the manuscript), and characterisation was performed using the 'sequence of constant voltages' method previously reported, using either 2-point or 5point power curve measurements. 15 Specically, these measurements yielded the voltage and current density (V ocp and j sc ) as well as maximum power density (P max , in W m À2 ). This was converted into the absolute maximum power generated by the cell (p max , in W) by dividing by the exposed electrode surface area (0.000094 m 2 for the lled electrolyte cells; surface area for the three-sided gel cells are specied later). A Keysight B2901A Source Measure Unit and Quick IV soware (Keysight, UK) were used throughout. For ungelled systems, current and voltage were measured for 300 s, and the average of the data from 151-300 s reported; for the gel systems, this expanded to 600 s with the average taken from 301-600 s.
Thermogalvanic assemblies incorporating the heat ux sensor A dedicated thermogalvanic cell was developed to interface reproducibly with a commercial heat ux sensor (gSKIN-XP26 9C calibrated, greenTEC, Switzerland), which was a 10 mm by 10 mm square. The resulting assembly is described visually in Fig. 2 (a photograph is also shown later, in Fig. 4). The thermostatic water baths, copper heat exchangers and measurement techniques were unchanged from above, but the main cell was a hollow rectangular tube (extruded, clear acrylic hollow square tubing, 13 mm diameter and 9.7 mm bore, eBay Co. UK), which was cut to shape to vary the electrode separation distance, and had two injection holes drilled in the top. Squares of the 0.3 mm thick graphite were cut out and held in place at both ends of the cell by compressing it between the cell and the heat exchangers; the heat ux sensor was also compressed between the graphite electrode and hot copper heat exchanger. The electrodes and heat exchanger were both larger than the heat ux sensor, ensuring that all conductive thermal energy passing from the hot side into the thermogalvanic cell had to pass through the heat ux sensor (an irradiation effect could occur through the air gaps, but this has been modelled elsewhere 22 and is expected to be negligible). Repeated use of the graphite resulted in increasing deformation and therefore worsening thermal contact, hence the electrodes could not be glued and each electrode was only used for one experiment before being disposed of. This arrangement gave an electrolyteexposed square electrode surface area, A, of 94 mm 2 .
Photographs and IR images were obtained using an iPhone 11 Pro Max (Apple Inc., USA) and a Seek Thermal CompactXR with iPhone connector (Seek Thermal Inc., USA), respectively.

Gelled electrolyte preparation and cell
Electrolytes were partially or fully gelled by the addition of 1.5 wt% or 3 wt% equivalents of sodium polyacrylate powder (SnoWonder Instant Snow Mix, SnoWonder, USA), which achieved the majority of its swelling within ca. 1 minute. Physically moving the gelled electrolyte into the thermogalvanic cell was found to result in signicant reproducibility issues, due to different packing, trapped air, etc. Reproducible measurement was only achieved by physically removing one side of the rectangular cell housing, changing it from an enclosed square into a U-shape. All measurements reported here were recorded in three cells with electrode separation of 13.6, 20.9 or 29.2 mm; these cells were set up and the heat ux through the empty three-sided cell was measured. Then 0.4 M K 3 /K 4 [Fe(CN) 6 ] electrolyte was added to ll the cell by ca. 80%; for 13.6, 20.9 or 29.2 mm this was 1, 1.57 and 2.15 mL, respectively, affording electrode-electrolyte surface areas of ca. 71, 71 and 73 mm 2 , respectively. The electrolyte was allowed to thermally equilibrate, measured, and then either 1.5 wt% or 3 wt% equivalents of sodium polyacrylate powder was added through the open side of the cell. This resulted in a homogenous dispersion of the powder and reproducibly formed a fully packed, air bubble-free gel. The system was measured using the thermogalvanic characterisation methodology described above, when the output from the heat ux sensor reached steady state (ca. 5 min).

Conversion of the heat ux sensor output into heat ux
The potential difference generated across the commercial heat ux sensor (gSKIN-XP26 9C calibrated, greenTEC, Switzerland) was recorded every 0.25 s using chronopotentiometric measurements via a potentiostat (PGSTAT204 potentiostat with NOVA 2 soware, Metrohm, UK). It was found that disabling the auto-ranging on the current and selecting the smallest current option (i.e. highest impedance) improved the reliability of the measurements as it suppressed any parasitic thermoelectric processes. Raw output data is presented later in Fig. 3 and 6.
The heat ux sensor is a thermoelectric module designed for ultra-high resolution of conductive heat ux, and the potential difference generated across the sensor (DV s ) was converted into heat ux using the manufacturer-supplied sensitivity factor of 15.09 mV per W m À2 of heat ux (at 295.65 K). 38 The temperature-corrected sensitivity factor for this particular sensor was determined using the supplied 15.09 + 0.0189 (T s À 295.65) relationship, where T s is the temperature of the sensor in K. This value for T s can be approximated using the temperature of the hot heat exchanger, T h , except the heat ux sensor acted as an additional thermal resistance in-series between the heat exchanger and graphite electrode meaning it was not as hot as the copper. Therefore, T s was instead calculated for each experiment by rst calculating the experienced temperature difference across the cell using: where DT exp is the experienced temperature difference across the two graphite electrodes, V ocp is the measured potential difference (in mV), and S e is the thermogalvanic Seebeck coef-cient of the electrolyte (À1.4 mV K À1 ). This will be equal to or smaller than the applied temperature difference (DT app ), given by T h À T c , where T h represents the temperature of the hot heat exchanger and T c the temperature of the cold heat exchanger. Assuming the majority of the difference between DT app and DT exp is lost over the heat ux sensor, then T s is given by Most experiments used T h ¼ 313.15 K. While the T s varied as a function of cell and experiment, it fell within the narrow range of 308.45 to 309.85 K; the values for DT exp are included in Table  S2. † Arguably an even more accurate value would have been the value halfway between T h and T s (i.e. 0.5(T h + T s )) given that these values represent the temperature gradient across the sensor, but this additional correction factor was found to have only a very minor effect upon the heat ux values.
As such, the measured total heat ux (q total , in W) through the cell is given by: where A s is the known surface area of the heat ux sensor (0.0001 m 2 ), S e , T h and DT app are known, while DV s and V ocp were the simultaneously measured outputs from the heat ux sensor and the thermogalvanic cell, respectively. This value in W could be converted into the total heat ux density (Q total , in W m À2 ) by omitting A s from this calculation. This procedure was performed rst using the cell assembly without electrolyte, and then the cell was lled in situ and the measurement repeated. The heat ux for the empty cell, q empty , was subtracted from the lled cell value, q total , to afford the electrolyte-only, measured heat ux value, q m .

Calculation of the estimated heat ux
The estimated heat ux was calculated using the oen-reported adaption of Fourier's Law (eqn (6)) except the estimated absolute heat ux (q e , in W) was explicitly calculated from experimental data using: where DT exp was experimentally determined as described above, k was the reported 34,37 thermal conductivity of similar concentrations of aqueous potassium ferri/ferrocyanide of 0.55 W m À1 K À1 , A was the cross-sectional area of the electrodeelectrolyte interface at the hot electrode (0.000094 m 2 ), and d was the distance between the two electrodes (ranged between 0.0044 and 0.0399 m). The estimated heat ux density (Q e , in W m À2 ) could be calculated by omitting A from eqn (10).

Calculation of the efficiency values
The absolute maximum power of the thermogalvanic cell (p max ) was measured; the absolute efficiency of the electrolyte's thermogalvanic performance, h, was then determined using both the experimentally measured heat ux (q m ) and the heat ux estimated using Fourier's Law (q e ). The absolute efficiency of conversion was given by the ratio of p max to q (cf. eqn (4)). Efficiency relative to the theoretical Carnot cycle maximum, h r , was determined using the fractional relationship in the Introduction (cf. eqn (5)) except instead of using the applied T h value, the temperature of the heat ux sensor, T s , was used. The T s was determined based upon the measured V ocp , as described by eqn (7) and (8).
Empty cell heat ux and the total device efficiency As noted above, the measured total heat ux, q total , was corrected to measure the electrolyte-only value, q m , by subtracting the heat ux of the empty cell, q empty ; these measured values of q empty are tabulated in the ESI. † It is also possible to predict the heat ux through the acrylic cell using the eqn (10), using the k value for this plastic (ca. 0.20 W m À1 K À1 ) and the surface area of the plastic (0.000075 m 2 ) that made up the face of the hollow square. Typically this predicted value was ca. 40% of the total measured empty cell heat ux, as shown later in this paper. As for the remaining measured heat ux for the empty cell, it is estimated that the majority of this was lost via conduction through the graphite electrode and into the metallic crocodile clip and wiring at the hot electrode side. The remainder would have been lost to the surroundings and air via radiation and air convection, respectively (from the plastic cell body, graphite electrode, metallic clip and wiring).
As these factors were neither systematically investigated nor optimised, the majority of this paper focuses upon electrolyteonly efficiency parameters. However, the measured total device efficiency parameters can be derived by eqn (4) and (5), but using q total instead of q m ; all measured values are tabulated in Tables S2 and S3. †

Results and discussion
Initial selection of the electrode material As summarised in the Introduction, to the best of our knowledge the direct heat ux through thermogalvanic cells has not been quantied. This was achieved here by putting a heat ux sensor thermally in-series with the cell at the 'hot' electrode, as shown in Fig. 1(d). This, in conjunction to the quantication of the absolute thermogalvanic power, allowed unambiguous comparison of the ratio of the two.
A detailed explanation of this set-up is included in the Experimental section. However, a exible electrode material was required to work reproducibly in this set-up, and therefore a range of electrode materials were screened using a conventional thermogalvanic cell setup. This was achieved using a previously-reported 4 thermocell setup, using solid platinum electrodes, lled with 0.  6 ]), exposed to a temperature gradient of 20 K, and measured using the formalised sequence of potentials. 15 Three different types of graphite materials were then evaluated by placing them between the platinum and the electrolyte.
The resulting thermogalvanic steady-state outputs are summarised in Table 1; clearly platinum possessed the optimum performance due to the high output current density expected of such a highly electrocatalytic electrode towards [Fe(CN) 6 ] 3À/4À electron transfer kinetics, 4 but this electrode material was too rm to reproducibly interface with the heat ux sensor and could not be utilised further. Pyrolytic crystalline graphite was bought as a thermal interface material, and it displayed excellent thermal properties (actually increasing the V ocp when placed on top of the Pt), but also demonstrated very poor electrocatalytic properties with the current decreasing ca. 60-fold. Two thicknesses of amorphous gasket (graphoil) graphite were also evaluated; as the thickness increased the thermal resistance also increased, resulting in a drop in V ocp . However, the current was far higher than that recorded at the crystalline graphite and increased as the thickness increased; this is likely due to partial porosity of the graphoil material increasing the electrochemically active surface area, and exposed electrocatalytic edge sites 39 that comes with its expansion and compression during manufacturing. 4 The 0.3 mm thick graphite was chosen as it was suitably thin and deformable while maintaining reasonable electrocatalytic ability, although it's important to note it generates only ca. 35% of the power of pure platinum (40 mW m À2 vs. 114 mW m À2 ).

Introduction of the heat ux sensor thermally in-series
Next, the effect of introducing the heat ux sensor was explored, using the novel rectangular cell design described in detail in the Experimental section. Firstly, the thermogalvanic power was measured. Fig. 3(a)  Power curves were recorded for the same cell with (grey circles) and without (purple squares) the heat ux sensor thermally in-series. The heat ux sensor clearly added additional thermal resistance, reducing the V ocp and therefore the current and the power; despite this, an ideal power curve was still obtained for the thermogalvanic cell power output. Using a thermogalvanic S e value of À1.4 mV K À1 , 4,5,11,40,41 for the electrolyte, it implies an applied DT app ¼ 20.0 K translated into the graphite electrodes experiencing DT exp ¼ 16.4 K with the heat ux sensor in-series.
Next, the output from the heat ux sensor was quantied. Fig. 3(b) displays the voltage output from the heat ux sensor when connected thermally in-series with an empty cell (same cell and conditions as above). A constant output of ca. 7 mV was recorded, corresponding to a heat ux of ca. 45 mW (or heat ux density of 450 W m À2 ) through the sensor. Aer 10 min the 0.4 M K 3 /K 4 [FeCN 6 ] electrolyte was injected to ll the cell, and a sharp spike in voltage (indicating increased heat ux) was observed; this dropped over ca. 5 min as the solution came to temperature, with further spikes and dips in this period due to injecting and extracting liquid from the cell in order to remove all the trapped bubbles. This equilibrated to give a constant value of ca. 44 mV, or a total measured heat ux of q total ¼ 295 mW; this was how the heat ux for a lled cell and the corresponding empty cell were measured, and unless specied otherwise the empty cell-corrected electrolyte heat ux values were used, e.g. q m ¼ 295 mW À 45 mW ¼ 250 mW. Power was characterised for all subsequent cells using 2point measurements, e.g. measuring just V ocp and j sc , and determining the power density using P max ¼ 0.25V ocp j sc . 15 Fig. 3(c) displays a 10 min characterisation measurement aer the cell has come to thermal equilibrium; the V ocp for the rst 300 s is shown and was very stable, followed by I sc measurement (then converted from I sc to j sc by dividing by the electrode surface area). An initial drop in current is observed as concentration gradients are established at the two electrodes, but it rapidly comes to equilibrium as the rate of consumption and mass transport equilibrate, resulting in genuine steady state power generation. 15 The average from 151-300 s was used to quantify both V ocp and j sc . The simultaneous measurement of heat ux through the sensor was also measured (as overlaid in Fig. 3(c)), and this didn't display any signicant changes, even when current was allowed to ow through the thermogalvanic cell.
The cell was found to generate ca. 31 AE 3 mW thermogalvanic power at steady state (from triplicate measurements), whereas the cell-corrected electrolyte heat ux value was ca. 236 AE 17 mW; this equates to an absolute efficiency of 0.013 AE 0.002%, or 0.024 AE 0.003% vs. Carnot efficiency. Conversely, the estimated heat ux using the typical model employed for thermogalvanic cells of 1D transport through a solid (eqn (6)) predicted nearly an order of magnitude less heat ux at 30 AE 1 mW. This results in an estimated Carnot efficiency of 0.19 AE 0.02%, i.e. the estimated efficiency was nearly 8-fold higher than the directly measured efficiency of conversion. This value increases to 9-fold if the total heat ux (q total ) of the entire device is used, rather than the empty cell-corrected electrolyte heat ux value (q m ).
In order to identify the source of this additional heat ux, IR imaging was employed. Fig. 4 compares photos vs. IR images of the setup, and the latter clearly indicates how the top of the cell was signicantly hotter than the bottom, indicating a signicant amount of convection is occurring; this in turn results in signicant radiation heat loss from the top of the cell. Neither convective nor radiation heat-transfer mechanisms are considered by the 1D Fourier's Law prediction (eqn (6)), thus accounting for the very signicant difference observed here between actual measurement of the heat ux vs. the routinely employed estimated heat ux.
The implications of this order-of-magnitude difference, and a comparison against other published values are both discussed in detail at the end of this paper. Given the observed key role of the cells external surface, different electrode separations were evaluated next.

Effect of the inter-electrode separation distance
It has been previously reported that the power density of a thermogalvanic cell drops signicantly with increasing interelectrode separation, but power conversion efficiency will increase. 27,40,42 Given this expected relationship, we set out to measure efficiency over 5 different electrode separations, all at an applied DT ¼ 20 K. The results are summarised in Table 2 (full experimental results for cells without the heat ux sensor are in Table S1, † and with the heat ux sensor in Table S2 †); key results are visualised in Fig. 5.
Measuring the heat ux through the empty and lled cells ( Fig. 5(a)), the heat ux decreased in a linear manner as the electrode separation of the empty cells increases, in line with expectations for frustrated conduction through the longer plastic. A similar trend was observed in the electrolyte-lled cells, meaning that the corrected heat ux was essentially constant for all the cells, regardless of the electrode separation. This trend of largely constant heat ux vs. electrode separation deviates from the calculated heat ux, which considers conduction only and predicts the overall heat ux should decrease with increasing separation. It is likely that as the cell becomes longer, conduction decreases whereas convection and radiation increases, with these two effects thus cancelling each other out. As the electrode separation increases power also decreases exponentially (plotted in Fig. 5(b)), in line with expectations for a mass-transport limited thermogalvanic cell Table 2 Summary of the thermogalvanic power and heat flux measurements as a function of electrode separation distance (d, top) and applied temperature difference (DT, bottom) to afford measured efficiencies (absolute, h m , and relative to the Carnot cycle, h r,m ). These are compared against the estimated heat flux (q e ) and associated estimated efficiencies. Error values indicated by (AE) are the standard deviation of between 3 to 5 repeat measurements. See Table S1 for equivalent studies without more heat flux in-series, and and prior studies. 27,40 This trend was observed both with and without the heat ux sensor in series. Fig. 5(c) compares the Carnot efficiency for the cells as a function of electrode separation, using either the genuinely measured heat ux or estimated heat ux; the absolute efficiency values followed the same trend ( Fig. S1(a) †). Interestingly, the smallest cell shows fair correlation between measured and estimated efficiencies, consistent with the smallest cell being a conduction-dominated system. Our estimated heat ux model predicts increasing efficiency with increasing electrode separation, but with increasingly diminishing returns; this is in excellent agreement with earlier fundamental work. 27 However, the genuinely measured heat ux results indicate decreasing efficiency, due to frustrated ion transport yet convectionboosted heat loss. This complete divergence of prior predictions vs. genuine measurements is of signicance for future cell design and optimisation.

Effect of the applied temperature difference
Following the investigation into inter-electrode separation, the effect of temperature difference was explored, using a xed, 9.5 mm electrode separation. Here the temperature difference was increased from DT app ¼ 20 K to DT app ¼ 40 K, by increasing the temperature of the hot electrode. As shown in Fig. 6(a), the power increased as the DT app increased, in line with expectations. 5,15 While the trends in P max with and without the heat ux sensor in-series deviated vs. DT app , the two trends were comparable if plotted vs. DT exp (as shown in Fig. S1(b)), † as this corrects for the additional thermal resistance introduced by the sensor. Fig. 6(b) displays the raw data output by the heat ux sensor in-series with both the empty cell and the electrolyte-lled cells, whereas Fig. 6(c) plots the actual heat ux as a function of DT app (the same trend exists vs. DT exp ). The heat ux measured for the empty cell increased in a linear manner in line with Fourier's Law, whereas the electrolyte-lled cell increased in a non-linear manner, consistent with enhanced convection at greater values of DT. Also overlaid in Fig. 6(c) is the predicted heat ux calculated solely for conduction through the perspex cell using Fourier's Law (hollow circles), which is only ca. 40% of the heat ux measured for the whole device. Therefore some additional thermal energy is presumably lost to radiation and airconvection, with the majority lost to conduction through the electrode and into the wiring; this parasitic thermal lost is well recognised, 43 and device-design requires two dissimilar thermogalvanic cell chemistries to be employed to help combat this thermal short-circuit. 18,43 This effect is also clearly seen in Fig. 4(b), with the image achieved via IR radiation heat loss, and the clip at the hot electrode also being visually warmer than the background.
Both P max and heat ux increased with increasing DT, but P max increased by a greater magnitude and thus the overall absolute (electrolyte-only) efficiency increased with increasing DT (shown in Fig. S1(c) †). However, this overall efficiency gain was equivalent to expected gains from a Carnot engine at the increased DT, meaning the overall Carnot relative efficiency was independent of DT ( Fig. 6(d)). Once again, the estimated heat ux lacked this nuance, with both the estimated absolute efficiency and estimated Carnot efficiency values increasing with DT. The divergence in these trends, combined with the smaller estimated heat ux, resulted in the estimated vs. measured efficiency values differing by a factor of 4.5 by DT app ¼ 40 K, in this 9.5 mm separation cell.

Effect of gelling the electrolyte
Gelled or 'quasi-solid' electrolytes have appeared numerous times, e.g. 9,10,12-14,44-48 in thermogalvanic cells as a method of producing an electrolyte which is not susceptible to leaking and so supported the development of wearable devices; 9,10 theoretically it also reduces the thermal conductivity through the thermocell. We therefore investigated the effect of gelling the electrolyte, especially since gelation was expected to 'switch off' the convection found to be so inuential earlier in this study. This was performed by adding either 1.5 wt% or 3.0 wt%equivalent of sodium polyacrylate powder; this is a textured, superabsorbent material that can rapidly swell and even gel highly concentrated electrolytes, within seconds. 22 Fig. 7(a) shows photographs of these systems being exposed to the inversion test, which demonstrates that 0.4 M K 3 / K 4 [FeCN 6 ] electrolyte containing 1.5 wt% eq. sodium polyacrylate powder forms a heterogeneous, free-owing slurry, whereas 3.0 wt% eq. results in a genuinely gelled electrolyte. Table S3 † summarises all relevant values, while Fig. 7(b) plots the measurement of the j sc versus time; addition of 1.5 wt% equivalent gelling agent resulted in a slightly slower equilibration time before steady state current was achieved, but otherwise didn't change the j sc , whereas the gelled 3.0 wt% equivalent system failed to reach equilibrium. The latter observation is common with fully-gelled electrolytes, which frustrate the transport of ions to such an extent that concentration imbalances accumulate and persist at the electrode surfaces. 14 The comparison of gelled vs. ungelled electrolytes was also explored as a function of electrode separation. Fig. S2 † plots the measured thermogalvanic powers, both with and without the heat ux sensor in-series, while Fig. 7(c) and (d) summarise the electrolyte-only Carnot efficiency values. The ungelled system (0 wt%, Fig. 7(c)) displayed the expected deviation between estimated and measured values (cf. Fig. 5(c)), but signicant differences were observed in the gelled system (3 wt%, Fig. 7(d)). In the gelled system, measured efficiency still decreased with electrode separation while estimated efficiency increased. However, the estimated efficiency values were signicantly lower, because the power generated by the gelled systems was lower but the predicted heat ux remained unchanged (as k ¼ 0.55 W m À1 K À1 was assumed throughout). Conversely, the measured actual heat ux was signicantly reduced upon gelation; because heat ux was reduced even more than the current was reduced, the measured efficiency actually increased signicantly. This difference means at relatively small electrode separations (<20 mm), the measured gelled electrolyte efficiency actually exceed estimated efficiency.
These results highlight how the assumption of k ¼ 0.55 W m À1 K À1 is awed for both convective (e.g. liquid) and convection-supressed (e.g. gelled) systems. This mirrors the Fig. 6 (a) Plot of the maximum thermogalvanic power produced as a function of applied temperature difference, with (purple square) and without (grey circles) the heat flux sensor thermally in-series; (b) the raw heat flux sensor output as a function of applied temperature difference, with the thermogalvanic cell empty and filled with electrolyte, and (c) the corresponding heat flux derived from the raw data for the electrolytefilled (square) and empty (filled circle) cell; also shown is the estimated heat flux expected by conduction through the perspex cell (empty circles); (d) the Carnot relative efficiency of thermogalvanic conversion using the estimated (green circles) and measured (purple squares) heat flux values, using electrolyte-only values (i.e. filled cell minus empty cell, for the measured heat flux). All measured using an electrode separation of 9.5 mm. All triplicate measurements for DT app ¼ 20 K are shown in (a) and (c), with the error bars in (d) corresponding to the combined errors as the 95% confidence interval (2 standard deviations).  48 Different k eff values were applied to our model such that the estimated electrolyte-only heat ux matched the measured heat ux; this gave k eff ¼ 1.28 W m À1 K À1 for the 0 wt% system in the 13.6 mm cell, and k eff ¼ 0.18 W m À1 K À1 for the 3 wt% system. However, k eff is also a cell-dependant value, and increased in the 20.9 mm cell to 2.14 W m À1 K À1 and 0.39 W m À1 K À1 for the gelled and ungelled systems, respectively, likely due to increased convective and radiative losses as the cell gets larger.
An additional concept that has apparently never been examined before is total device efficiency. Throughout, the electrolyte-only heat ux has been employed, but total heat ux was quantied. The measured Carnot efficiency was recalculated using the total heat ux, and these values are plotted in Fig. 7(e). It demonstrates that device-efficiency was essentially indistinguishable for gelled and ungelled electrolytes. This was because even though heat ux was suppressed more than current ow upon gelation, the additional heat ux through the cell assembly itself became a much more signicant fraction of total heat ux, and reduced the overall total device efficiency. More thermally resistive electrolytes therefore require more thermally resistive cell housings in order to achieve boosts in efficiency in genuine 'real world' whole device applications.
Finally, the 1.5 wt% eq. gelling agent yielded a heterogeneous suspension; the ionic, solid particles appear to frustrate bulk thermal transfer (such as convection, and even to a degree conduction) but remains a highly ionically conductive system, boosting the (genuine) efficiency relative to the electrolyte alone. The measured and estimated Carnot relative efficiency is plotted in Fig. 7(f) for 0 wt%, 1.5 wt% and 3 wt% eq. sodium polyacrylate in the 13.6 mm cell. These values demonstrate the heterogeneous suspension caused by 1.5 wt% eq. possesses optimal conditions and displays the highest efficiency; this was reected in an additionally boosted total device efficiency.
Broadly, these precise measurements prove an oen stated but only previously tentatively proven 22 concept; that gelled and pseudo-gelled electrolytes can result in genuinely more efficient conversion in thermogalvanic cells, provided they selectively frustrate heat ux more than current generation. Table 3 summarises a number of published papers that quote estimated Carnot efficiency values (using directly measured thermogalvanic power and estimated heat ux for the electrolyte only). The table is not an exhaustive summary of the literature, and is restricted to studies using aqueous [Fe(CN) 6 ] 3À/4À ; it primarily summarises fundamental studies employing planar  6 ] solutions containing 0, 1.5 or 3 wt% eq. of sodium acrylate powder (left) before and (right) 60 seconds after being inverted, demonstrating only the 3 wt% eq. system was sufficiently gelled to pass the inversion test. Also (b) the measured j sc vs. time for these systems, showing how both 0 wt% and 1.5 wt% eq. resulted in steady-state current (half the 0 wt% data excluded for clarity) while 3 wt% failed to reach steady state over 10 min (electrode separation of 13.6 mm). Also shown is an electrode separation study, with the estimated and measured electrolyte Carnot efficiencies for (c) 0 wt% and (d) 3 wt% eq. sodium acrylate powder systems. Shown in (e) is the total device Carnot efficiency, showing largely equivalent values for the ungelled (0 wt%) and gelled (3 wt%) systems. Finally (f) plots the various Carnot efficiency values for different wt% values of sodium acrylate powder in the 13.6 mm cell. All calculations in (c-f) used the average from 2 to 4 repeat measurements, with the j sc values averaged from 301 to 600 s; all measured at DT app ¼ 20 K in a partially filled 3-sided cell (see Experimental for full details). platinum electrodes (by Quickenden et al. 24,25,27 or Lee et al. 37 ) who performed limited temperature, orientation and concentration studies, as well as three high surface area carbon electrode studies as exemplars, 23,34,40 and stainless steel as a comparison. 22 A number of other studies that use nanostructured electrodes have been excluded, because that is an additional factor not studied here; others have reported seemingly promising systems, e.g. [Fe(CN) 6 ] 3À/4À /KCl gelatine hydrogels with a S e up to 17 mV K À1 , 12 but didn't report sufficient thermogalvanic characterisation parameters such that their performance can be compared with others.

A comparison of the factors and values vs. previous reports
Also listed in Table 3 are some estimated and genuinely measured efficiencies from this work. Recalling that all measurements were made at amorphous graphite that generates ca. one-third of the power of platinum electrodes, the efficiency values can in theory be converted to approximate planar platinum values by multiplying by 3 (which assumes thermal conduction routes would be unaffected by this substitution, but current roughly tripled). Doing this, the range of estimated Carnot efficiency values recorded in this study at graphite (0.066-0.213%) converted to platinum-equivalent values (ca. 0.2-0.6%) then sit well within the range of values reported by Lee et al. 37 and Quickenden et al. 24,25,27 for similar electrolytes at planar platinum electrodes (0.08-0.6%).
What stands out from the tabulated results is the general assertion of previous reports that (estimated) efficiency will increase with increasing electrode separation, 27,40 which this study mirrors. However, the directly measured efficiency displays the completely opposite trend, which is signicant for future device design.  6 ] 3À/4À electrolyte was used, and a range of experimental conditions (such as electrode material, electrode separation, temperature difference and cell orientation were studied). All estimated efficiency using the estimated heat flux, except ref. 23 6 ] 4À ¼ concentration. c A forest of CNT was drawn onto a 0.3 mm tungsten wire, and wrapped around to form a ca. 3 to 3.5 mm diameter scroll. Results are reported for (i) CNT scroll as prepared, (ii) scroll thermally oxidised, (iii) scroll platinised, and (iv) scroll platinised and compressed. d The heat ux was estimated using the same equation as the other studies, except an effective thermal conductivity was calculated via IR imaging and used, rather than a conduction-only thermal conductivity.
contributions to heat ux were routinely subtracted throughout this work. This is largely because device contributions are so variable across groups, and were never deliberately adjusted or optimised in this study. For examples like the 9.5 mm cell (cf. Fig. 6), the device-only heat ux was a minor part of the total heat ux; some of which was conduction through the plastic cell, and the rest primarily lost to conduction into the wiring and radiation into the surroundings. Heat loss into the wiring has been recognised before, 43 and is relatively easily addressed; for example connecting 100 electrolyte pairs thermally inparallel but electrically in-series 43 will likely reduce this parasitic effect of the external wiring by 100-fold per cell. Thinner wires will also reduce this. Substituting the acrylic cell used here for thinner, more insulating material and insulating the entire device will also reduce this parasitic heat transfer. While the parasitic device heat ux was a minor factor for electrolytecontaining cells, it is more signicant for cells containing gelled electrolyte, as discussed above and shown in Fig. 7. Signicantly, while genuine efficiency was consistently lower than estimated efficiency through this work, gelled and partially-gelled electrolytes with small electrode separation values deviated from this trend and offer clear promise for more efficient thermogalvanic devices, particularly when coupled with good cell, device and electrode design. These synergies can now be unambiguously quantied, especially via the quantication of total heat ux (as opposed to electrolyte-only heat ux) and this is expected to facilitate total-device optimisation studies.

Conclusions
This study has demonstrated methodology by which the efficiency of thermogalvanic conversion (of a heat ux through a temperature gradient into electricity) can be unambiguously quantied. This was demonstrated for just the electrolyte, and for the entire assembly; the latter allows 'total device efficiency' to be quantied. By comparing the measured efficiency vs. the standard route of estimating efficiency, this study shows that they are only comparable for cells with negligible temperature gradients and very small inter-electrode separations. As the temperature gradient increases, the estimated efficiency will overestimate the actual performance by a signicant degree; this difference is even more signicant with increasing electrode separation, with a complete divergence between estimated and measured efficiency values. Conversely, the measured genuine efficiency of fully-gelled electrolytes can actually exceed the estimated efficiency, but only with small inter-electrode separations, and only if parasitic heat ux through the device apparatus is also discounted. Partially-gelled electrolytes were also identied as the optimum system for efficient thermogalvanic heat-to-electricity conversion, both as an electrolyte and as part of a whole device.
The distinction between absolute efficiency and Carnot efficiency is also important; as the applied temperature difference increases the Carnot efficiency of the thermogalvanic cell decreases slightly, but this is slightly misleading as the absolute overall efficiency actually increases signicantly. These observations, combined with this new methodology, will support the rational design of complete thermogalvanic devices, and thus support increasingly efficient waste heat valorisation.

Data availability
The datasets supporting this article have been uploaded as part of the ESI. †

Conflicts of interest
There are no conicts to declare.