The motion of the lymph has a very important role in the immune system, and it is influenced by the porosity of the lymph nodes: more than 90% takes the peripheral path without entering the lymphoid compartment. In this paper, we construct a mathematical model of a lymph node assumed to have a spherical geometry, where the subcapsular sinus is a thin spherical shell near the external wall of the lymph node and the core is a porous material describing the lymphoid compartment. For the mathematical formulation, we assume incompressibility and we use Stokes together with Darcy–Brinkman equation for the flow of the lymph. Thanks to the hypothesis of axisymmetric flow with respect to the azimuthal angle and the use of the stream function approach, we find an explicit solution for the fully developed pulsatile flow in terms of Gegenbauer polynomials. A selected set of plots is provided to show the trend of motion in the case of physiological parameters. Then, a finite element simulation is performed and it is compared with the explicit solution. More than 90% of the lymph remains in the SCS, while the remaining part enters into the LC through a conduit system network   as a spherical region: the subcapsular sinus (SCS) is a thin spherical shell with radii p is the pressure, effective viscosity, k the permeability. The second equation in . Using , we obtaRegarding the pressure, we use  to obtaiFor We use an idealized spherical geometry based on the data obtained from a murine  lymph node: the radius is L is the maximum lymph mean flow of the inlet lymphatic vessel. Here we assume f(t) is a periodic function. The function H is given byThe inlet and outlet conditions are imposed in the upper and lower lymphatic vessel . Fo. Fon the}H(ζ) in  in termsTo impose the boundary condition, we need to expand in Fourier series the time dependence of , as we d1(ζ), by \documentBy the no-slip boundary condition on ecalling \documentContinuity of Continuity of Continuity of normal stress: slip constant which has to be estimated experimentally. Since the expression of the shear stress is The stress jump condition: From .From –34),  and and25) action of , that isgiven in  for \docThis section is devoted to show some plots related to the explicit solution and to make some considerations about the proposed model.Following  with the Brezzi-Pitkäranta stabilization, which consists in adding the term The boundary conditions in the external wall (inlet condition and no-slip boundary condition) are imposed by the penalty method. Moreover, we add the Grad-div stabilization terms the Eq. , with \dIn this section we want to qualitatively compare the results obtained with the numerical simulation with the explicit results exposed in Sect.  the Eq.  with \doIn Figs. In this section we want to show a more complete numerical simulation using the method given in Sect. We use a spherical idealized 2D geometry with the same parameters given in Sect. v et al. , Shanti v et al. , and Jafv et al. ) and theAs we mention in Sect. y direction with the Eq.  is the one given by the Eq. , the bed et al. .Fig. 9ShIn Fig. entclass1pt{minimaWe can see the norm and the velocity behavior in more details in Fig. We proposed a model that describes the pulsatile lymph flow inside a simplified spherical lymph node (LN), using the Darcy–Brinkman equation to describe the lymph flow in the lymphoid compartment  and the Stokes equation to describe the flow inside the subcapsular sinus . We found the explicit solution in terms of Gegenbauer polynomials and we showed the trend of the velocity, the pressure and the shear stress inside the LN; after that, we used this explicit solution to validate the numerical simulations of the model. Finally, we performed a more general numerical simulation with finite elements.This model allows to better understand the fluid behavior inside the LN and how it changes with respect to time. The results obtained by our model are in agreement with the literature (Birmingham et al. Particular attention was paid to the shear stress, because a lot of biological phenomena in the LN depend on it. Among them is the cell adhesion to the exterior of the LC, which is proportional to the shear stress: this is important because inside the LC there is a connection between the lymphatic system and the blood system, and some cells can get access from here to the blood circulation [for instance, tumor cells (Birmingham et al. Let us now make some considerations that can be interesting to improve the model in future. Here we have proposed to use a spherical geometry, but in general the LNs have a spheroidal shape (Giantesio et al. For simplicity, we did not take into account the fluid exchange inside the LN between the lymph in the fibroblastic reticular cells FRC and the blood in the capillaries, although it is important for the fluid regulation of the LN (Tobbia et al. Moreover, in order to close our model and to find the unknown constants in the explicit solution, we used the Ochoa–Tapia boundary conditions, although other boundary conditions can be taken into consideration. For instance, a common choice is to impose the continuity also of the shear stress at the interface, which is tantamount to choose 0 in Eq. . Using t0 in Eq.  and Ango0 in Eq. , where tAnother interesting and important extension of this model would be to couple the flow in the lymph node with the flow in the lymphangion, in order to have more realistic time pulsation and see how the lymph node affects and regulates the global lymph circulation (Bertram et al.


In this work, a latent heat storage system was designed, installed, and tested when it was integrated with a single basin solar still. The latent heat storage system is a shell and spiral finned tubes, where 20 kg of a paraffin-CuO nanocomposite with a weight fraction concentration of 1% was poured into the shell while hot water from the solar collector was being pumped through the spiral finned tubes. In order to identify the effect of implementing the storage system on the performance of the solar still, two experiments were conducted, with and without storage, under approximatelysimilar weather conditions. The proposed design of the storage system succeeded to overcome all challenges associated with using paraffin wax as storage material with the solar still. The obtained results revealed that an improvement in fresh water daily productivity of 4.54% was achieved when the storage system was integrated with the solar still. The economic analysis showed that the cost per 1 L of fresh water was 0.25 $/L when the storage unit was used. This high cost will be significantly reduced when a large-scale system is installed. The single basin solar still (SBSS) is considered the simplest system used for low-capacity water destination Shalaby . Many deA mathematical model of the SBSS with PW as PCM was prepared by Ansari et al. . They us2.day, respectively. Yousef and Hassan  was used with the SBSS for the first time, as far as the authors know, to overcome all difficulties associated with a solar still with built-in PCM. The separate LHSS introduces a perfect solution for the PCM leakage problem and provides high energy storage capacity. In addition,it provides unlimited design options to improve the heat transfer, such as using spiral fins that were implemented in this work. Moreover, nanoparticles of CuO are safely used to improve the thermal conductivity of paraffin wax without any scare from its leakage. The LHSS was integrated with the SBSS and tested under the prevailing weather of Tanta city.The LHSS and SBSS designs and constructions are discussed in detail in this section. The experimental setup and testing procedure are also presented.The proposed system mainly consists of three units: first, the single basin solar still, the LHSS, and a solar water heater (SWH). The LHSS was connected to the SWH in a closed loop when high solar radiation is available, or to the SBSS at night or during periods of low solar radiation (sun fade or cloudy periods).To avoid any corrosions, the solar still basin is made with stainless steel with a 0.7 m × 0.7 m bottom length × width and 0.55 m north and 0.2 m south heights, as shown in Fig. 1In order to improve the LHSS, the thermal storage prototype (TSP) shown in Fig. 2Based on the results obtained from testing the TSP, an enhanced LHSS was designed and constructed, where the latent heat storage system is a shell and spiral finned tubes. Twenty kilograms of a paraffin-CuO nanocomposite with a concentration of 1%  was poured in the shell where it was heated during the charging process by the hot water that flowed through the finned tubes. The used tubes are made of copper, while the attached spiral fins were made of iron. The detailed design of the LHSS is shown in Fig. 32 was designed and constructed to provide the heat needed to completely melt the PCM nanocomposite where a flat sheet made of galvanized iron attached with rectangle copper tubes was used as the absorber, and all the sides and bottom are insulted by 5 cm of foam sandwiched by two sheets of galvanized iron. The upper side of the SWH was covered by a sheet of glass of 4 mm thickness. The water flowed inside the rectangular tubes in a closed cycle between the SWH with a water storage tank and the LHSS when abundant solar radiation was available while it was disconnected during the night or when the sun seemed to be fading. In this case, the LHSS is reconnected to the solar still using manual valves. The schematic diagram and photograph of the proposed system are exhibited in Fig. 4The solar water heater with an area 0.5 mTg), absorber plate (Tp), and basin water (Tb-w) are hourly measured using a K-type thermocouple. The hourly productivity is collected in a graduated cone and weighted using a digital balance (accuracy 0.0001 g). The solar radiation (I) and surrounding air temperature (Ta) are hourly measured using a prynometer and a thermometer. The second experiment was conducted when the LHSS was integrated with the SWH and the SBSS. In this case, the SBSS and the LHSS integrated with the SWH are separately operated from 8:00 am to 5:00 pm. Then, the LHSS is disconnected from the SWH and reconnected to the SBSS where the basin water is circulated between the SBSS and the LHSS from 5:00 pm to 8:00 am of the next day. Keep in mind that the hourly measurements of the system temperatures and productivity were taken until 11:00 pm. The temperature of water at the outlet of SWH (To-h) and that of water in the storage tank (Tsw) are also measured in this case. The temperature at different points of the PCM was regularly distributed with depth, and the redial axes were also measured using 5 K-type thermocouples. Then, the average value of the PCM temperature (TPCM(av)) was calculated. The accumulated productivity during the period from 11:00 pm to 8:00 am was measured at 8:00 am in the early morning on the next day.In order to identify the effect of using the LHSS on the performance of the SBSS, two experiments were conducted under approximately the same weather conditions. The first experiment was conducted without using the storage system where the conventional solar still was tested during the day when 5 kg water was charged in the basin. The temperatures of the still glass cover , which is closes to the results obtained by Tabrizi et al.  were hourly measured and displayed in Fig. Tb-w-Tst-g) explains the high productivity achieved during the night when the storage system was used, as seen in Fig. In the case of integrating the SBSS with the storage system, the temperature of the different elements of the SBSS and storage system such as 2 and the SWH with an area 0.5 m2 is 1982.8 g while the daily productivity per square meter of the solar field is 2478.5 (g/day.m2). So, an improvement in daily productivity of 4.54% was achieved when the LHSS was integrated with the SBSS. Although this improvement is low compared to that obtained when the PCM was used under the still basin absorber plate, which reaches to 12% , which is assumed to be 12%/year, n is the number of life years, which is assumed to be 10 years, P is the capital cost ($), ECPD is the electrical consumption per day (kWh), EC is the household electrical cost ($/kWh), and AP is the annual productivity (l). Table In this section, we analyzed the cost per liter of the fresh water produced from the SBSS in two cases: first without the LHSS and then when the LHSS is integrated with the system. The cost is affected by the several parameters given in Table 2. The high value of the CPL was found when the LHSS was used mainly due the high consumed power where a 0.35-HP pump was used. Actually, a low-power pump may be used for this small capacity system. In other words, the used pump of the lowest power found in Egypt market, perfectly appropriate a higher capacity system up to 10 times area of the studied system. The calculated values of the CPL of the SBSS with and without the LHSS were found to be 0.25 and 0.171 $/L, respectively. These values are close to the results found by Rahbar and Esfahani (0.2696 $/L) (Rahbar and Esfahani The proposed design of the LHSS has succeeded to overcome all challenges associated with using paraffin wax as storage material with the SBSS.An improvement in daily productivity of 4.54% was achieved when the LHSS was integrated with the SBSS.The highest hourly productivity is directly achieved 1 h after circulating the basin water between the basin and the LHSS.The calculated values of the CPL of the SBSS with and without the LHSS were found to be 0.25 and 0.171 $/L, respectively. These high costs will be significantly reduced when a large-scale system was installed.Using several storage tanks is recommended to be integrated with the SBSS one by one during the night period to increase the operating time and fresh water productivity.It is also recommended to test several types of PCM enhanced by high conductive nanomaterials.The effect of using solar reflectors with the SWH and the SBSS is also recommended for future works.The single-basin solar still is constructed and tested when it is connected to the LHSS and free of the LHSS. The outdoor testing results showed that:


Background: Cauda equina syndrome (CES) is an uncommon condition that occurs due to compression of the terminal portion of the spinal cord. Early recognition and intervention in CES are crucial for an improved prognosis. Delayed diagnosis and action may lead to irreversible adverse effects, i.e., permanent disability, and in some circumstances can lead to litigation.Aim: The aim of this quality improvement project (QIP) was to identify areas for improvement and expedite the management of suspected CES patients presenting to the hospital.st October 2020 to 27th November 2020; a re-audit on the second group of patients was done from 1st June 2021 to 16th July 2021; the third group was re-audited from 1st of January 2022 to 31st of March 2022.Material and methods: This was a retrospective study in which patients admitted to the Poole district hospital were analyzed in three groups with more than 50 patients in each subset group. The first group was audited from 1Results: There were a total of 168 patients in all audit groups, of whom 71% were female. The mean time from getting triaged to having an MRI improved from 13hrs 54mins to 10hrs 39mins. The total inpatient length of stay (LOS) of less than 24 hours was 28% in the first cycle and improved to 54.4% by the third cycle of the audit. Eight patients exhibited a diagnosis of cauda equina syndrome (CES) and were sent to the tertiary care center.Conclusions: This quality improvement project identified delays in requesting the MRI for the diagnosis of CES and was addressed by ED booking the scans directly. This, in turn, reduced the length of stay in the hospital for patients who did not have cauda equina syndrome.  Cauda equina syndrome (CES) is the compression of the terminal portion of the spinal cord and roots of the spinal nerves, beginning at the first lumbar nerve root. Maxter "and" Barr described the condition first in English literature in 1934 ,3.The most common causes of CES are lumber disc herniation and spinal stenosis; other uncommon causes can include epidural hematoma, infections, ankylosing spondylitis, trauma, and neoplasms -8. Due tImportant historical elements to evaluate include a detailed history. It is important to inquire about changes in a patient’s bowel and bladder habits; difficulty voiding, incontinence, or loss of bowel control raises suspicion of CES. Physical examination includes examining sacral nerve roots. Pinprick sensation in the S2-S4 dermatomes, i.e., perianal region, posterior thigh, and perineum, must be checked. Decreased rectal tone, bulbocavernosus reflex, and anal squeeze test should be evaluated.Postoperative spinal surgery patients and those on an anticoagulation regimen should raise suspicion of CES. Postoperative hematomas can lead to undiagnosed CES . For patA post-operative spine patient with back pain presents a unique and perplexing scenario to physicians. Post-operative back pain can be attributed to the expected postoperative course, but clinical suspicion must be high for CES if the back pain is persistent and urinary symptoms are present. A patient's anticoagulation regimen must be considered too. The preferred diagnostic modality for CES is magnetic resonance imaging (MRI) .Early recognition and intervention in CES are crucial for an improved prognosis. Delayed diagnosis and action may lead to irreversible adverse effects and litigation consequences. The objective of this study was to evaluate the current practice of CES in our hospital from the patient's time of arrival to their management plan, try to determine factors that lead to delays in patient management, formulate a checklist for CES patients, and develop a hospital protocol.This was a retrospective study in which patients admitted to the Poole district general hospital were analyzed in three audit cycles: the first two groups included 50 patients each, and the third cycle had 68 patients. A data collection tool was designed to include the patient's journey through the hospital. The patient's journey started at the emergency department, moved to radiological imaging, and then a discussion with the neurosurgical team to formulate a plan, followed by discharge from the hospital to either another hospital or home with conservative management. The time intervals were recorded as data points Figure  to documThe collection for these time intervals was sourced from the electronic patient record (EPR) used by these hospitals, and the software used by their emergency department (ED) to collect all time intervals/timestamps.These time intervals were used and recorded into an Excel sheet  for further analysis. Formulas were used to calculate the mean time and range of time intervals for each of the data points, i.e., time intervals while the patient was being assessed at different points of their clinical journey through the hospital. The first group was audited from 1st October 2020 to 27th November 2020, the second group of patients from 1st June 2021 to 16th July 2021, and the third group from the 1st of January to 31st of March 2022.Inclusion criteria were patients over the age of 18 who presented with complaints of lower back pain and neurological symptoms, and those who were referred as suspected CES patients to the orthopedic department. Those with vertebral fractures were excluded from the study, including pathological metastatic cord compression.Analysis was completed with basic statistics of gender and age ranges, including the time intervals for each step of the clinical journey of the patient  cycles; 43.3 ±16.94 ), 49.7 (±20.33), and 46.8 (±16.27), respectively  but there were some from the general practitioner (GP) in the community and other specialty departments across the hospital Table .Table The time of arrival to having an action plan was the time interval from the arrival of the patient to the ED to having a definitive management plan according to their radiological imaging findings. These images were mostly discussed with the spinal team at a tertiary care hospital. This time interval showed a 33.1% improvement from the first cycle to the second cycle and a 43.1 % improvement comparing the third cycle with the first cycle, in terms of the reduction in the total-time interval Table . Patients who were diagnosed with CES on MRI were referred by a GP and were sent to tertiary care urgently. The average time of getting the plan from the tertiary care was 4:06:45 (hours:minutes:seconds) and the average time of admission was 7:37:45 (hours:minutes:seconds) and this was in the second cycle. And another four patients showed CES in the third cycle and were sent via blue light emergency transfer to the tertiary care center. The rate of CES in the second cycle was 8% (n=4), and 5.8% (n=4) for the third cycle. The total number of patients requiring urgent surgical intervention at the tertiary care center was eight out of 168 (4.7%), which is higher than the national standard .The purpose of this study was to assess the time it takes to admit a patient with back pain and reach a definitive diagnosis of CES. It is crucial to diagnose CES early to prevent permanent neurological dysfunction. A review of literature from the 1960s demonstrated it was not possible to recover from neurological dysfunction caused by CES . HoweverThe issues noted in our baseline audit were the time frames of reviewing a patient in the ED and the referral to orthopedics, which is the case at most district general hospitals due to the lack of a neurosurgical department, that were delaying the course of booking the MRI early for a definitive diagnosis. Most tertiary care hospitals will have 24-hour coverage for MRIs but at most district hospitals the service stops after six p.m. So, if there was a delay in reviewing the patient and the subsequent referral, the patient would not have the appropriate imaging and would be delayed till the next morning, prompting an overnight admission for observation.After establishing surgical management of CES as the definitive treatment, the time from diagnosis to intervention becomes essential. Several studies have shown that prolonged symptoms and delayed intervention lead to irreversible damage. Some studies recommend surgery be performed within 48 hours, but the exact timing is controversial ,17. HencThe diagnostic investigation of choice for CES is the MRI. A study found that the typical symptoms of CES such as urinary symptoms, back pain, and perineal symptoms, correlate poorly with the confirmed diagnosis of CES. Hence, an MRI is necessary to reach a definitive diagnosis of CES . AnotherSince evidence heavily supports the role of MRI in the definitive diagnosis of CES, a pathway should be established to allow access to MRI 24/7. This will reduce the timing from presentation to intervention, improve outcomes, and result in fewer litigation cases. It will also reduce the number of unnecessary referrals to the neurosurgical unit and hence decrease the burden and limit the wastage of resources ,22.PDSA 1: Sharing preliminary audit findings with Radiology and EDThe involved stakeholders are the ED and the radiology department, and the findings of the audit were presented to them. Specific timings were identified that could be improved by changes in the practice pertinent to the ED and radiology department, respectively. A consensus was reached that specific slots for imaging for patients suspected of having CES would be reserved by the radiology department, and the scans would be booked directly by the ED. And these scans would be prioritized for vetting by the radiologist. Also, defined criteria and checklist were developed with red flags and strict protocol to document pre-void and post-void bladder scans. It was noted that the delay caused by the ED team by not booking the MRI immediately after reviewing a suspected case of CES was causing significant delays that could be improved if the ED would arrange for the scan before a referral to orthopedics. These delays were demonstrated to the relevant stakeholders via presentations and summary charts to the radiology department and ED. A provisional protocol pathway Figure  was propPDSA 2: Posters and findings of improvement shared with stakeholdersThe second PDSA cycle yielded better time intervals for the patient's journey through the ED and for having a definitive scan for diagnosis. The prompt booking of an MRI post-ED assessment of the patient reduced the MRI delay period from 7:48:19 to 5:01:32 (hours:minutes:seconds), which helped hasten the process of further management. A draft of the pathway that was being tested was made and shown to the stakeholders for improved MRI turnaround time and reduced length of stay (LOS) at the hospital.PDSA 3: Finalizing the pathway for CES patientsThe LOS in the hospital increased in the third cycle of the audit. Through the cycles of the audit after the due presentation and formulation of the protocol pathway (as seen above in Figure To establish a set pathway for these patients, a proposed pathway was initiated with distribution to all the ED personnel and was also shared among the orthopedic department doctors for clarity on how to proceed with these patients.RecommendationsEstablishing a pathway that deals with CES immediately can have a significant impact on the financial burden caused by the majority of the cases. An outpatient pathway could reduce the impact on hospital resources such as manpower and diagnostics caused by inpatient stays without intervention due to delays in the booking of scans. This will also lead to better patient satisfaction as reduced hospital stays and early discharge can prevent the frustrations and anxieties of prolonged hospital stays.Service for such a subset of patients could be improved with a spinal surgeon's clinic appointment after their presentation to the hospital with a scan that does not show CES but has some spinal pathology. In some hospitals, spinal service is not present, such as in district hospitals. This lack can overwhelm a region's local spinal service, but this could be addressed by the regional spinal service via telephone or virtual consultation. This approach could settle patients' anxieties and answer their questions regarding their symptoms.This quality improvement project identified delays in diagnosing CES, which can cause irreversible damage; hence, early diagnosis is important. Time intervals and their analysis showed how to reduce the time to have a definitive plan for these patients in less than 24 hours. The majority of the patients were found not to have CES on the gold standard investigation, but the ones that required urgent surgical decompression were higher than the national standard. In conclusion, the ED must book radiological imaging early after review without delays or referrals to orthopedics to reduce the inpatient hospital stay and the time taken to make a definitive plan.


Testing of lithium-ion batteries depends greatly on accurate temperature control in order to generate reliable experimental data. Reliable data is essential to parameterise and validate battery models, which are essential to speed up and reduce the cost of battery pack design for multiple applications. There are many methods to control the temperature of cells during testing, such as forced air convection, liquid cooling or conduction cooling using cooling plates. Depending on the size and number of cells, conduction cooling can be a complex and costly option. Although easier to implement, forced air cooling is not very effective and can introduce significant errors if used for battery model parametrisation. Existing commercially available immersion baths are not cost effective (∼£3320) and are usually too small to hold even one large pouch cell. Here, we describe an affordable but effective cooling method using immersion cooling. This bath is designed to house eight large lithium-ion pouch cells (300 mm × 350 mm), each immersed in a base oil cooling fluid . The total cost of this setup is only £1670. The rig is constructed using a heater, chilling unit, and a series of pumps. This immersion bath can maintain a temperature within 0.5 °C of the desired set point, it is operational within the temperature range 5–55 °C and has been validated at a temperature range of 25–45 °C. The battery industry is expected to increase in size by an order of magnitude or more over the next few decades, as global society shifts towards a low-carbon economy, by electrifying transportation and storing energy from renewables The performance and lifetime of each cell is unique; it depends on the cell chemistry and the manufacturer Temperature and thermal gradients affect every aspect of lithium-ion cell operation and performance. For example, a cell will have a higher impedance at lower temperatures and be generally less capable of passing current in either charge or discharge et al. observed a cell surface temperature rise of 14 °C during a routine characterisation experiment Heat generation occurs inside every lithium-ion cell during testing which complicates temperature control during characterisation experiments et al. reduced temperature variation by >80 %, compared to equivalent tests conducted in a climate chamber Conduction cooling is an attractive alternative Another alternative method is immersion cooling. Immersion cooling is an effective way to control the temperature of a large number of cells. This is an attractive method because it is better than air cooling and does not have the complications associated with conduction cooling. In addition, this setup is far less geometry dependant and a range of different cells can be cooled.In this paper, we introduce the design of a cost efficient, large immersion bath specifically designed for cell testing. Each bath can accommodate at least eight large format pouch cells, however, the apparatus is easily adaptable to accommodate multiple cell sizes and form factors. The design incorporates temperature control between 5 °C and 55 °C. The dielectric immersion fluid can achieve three orders of magnitude higher rate of convective heat transfer than may be achieved with air.•The capability to both chill and heat the immersion fluid;•Precision temperature control of the immersion fluid to within ±0.5 °C;•Simple and robust temperature control system;•Sized to accommodate eight large lithium-ion pouch cells (>50 Ah);•Low cost.We had five requirements for our immersion bath, as listed below:Immersion baths available on the market target single cell testing applications or are borrowed from other research fields. Bath sizes are typically an order of magnitude smaller than our requirement. This immersion bath design meets the five requirements at a fraction of the cost of the smaller immersion baths in the market and is based on components that can easily be sourced.The immersion bath is made of six main components : tank, tThe heater and chiller work alongside each other to achieve the required temperature without a complex control system. The heater is set to the desired temperature, which it can achieve with an accuracy of ±0.1 °C The design files for the immersion bath can be found in The bill of materials for making of one unit of the immersion bath is shown in Cells and other components are arranged in the bath in two different regions . The firThe primary pump was used to achieve good mixing between the two regions of the bath. By carefully laying out the piping arrangement, oil was pumped to each location where cells were mounted to ensure uniform flow across every cell and hence uniform temperature. The secondary pump was used to allow for good mixing in the proximity of the heating and chilling elements in the oil. This was important for two reasons. The first was that local heating or chilling does not directly affect the temperature probe in the heater. The second reason was that a uniform mixing of the oil was achieved locally at the intake of the primary pump.Temperature stability was one of the key requirements in the design of this apparatus and this was achieved by allowing the heater and chiller to work alongside each other. This was because the chiller counterbalances the unwanted temperature rise if an internal or external heat source other than the heater was to inject heat into the bath. Two main internal heat sources are present, the heat generated by the cells during testing and the heat from the pumps during operation. This is in addition to the variability of the external environmental temperature.In use, the heater switches off automatically when the set temperature is reached. However, significant heat is produced from the pump and dissipated into the bath even when the heater is idle. The chiller, set to cool the bath at a constant rate throughout testing, counterbalances this heating effect. During periods where the chiller is removing excessive heat, the heater is automatically triggered into operation. In this manner, the bath temperature is maintained by the heater’s intermittent operation, without needing a combined control system for the heater and chiller together.1.Drill four holes at locations shown in 2.Build a rectangular frame using aluminum extrusions and brackets. This frame will be referred to as the main frame in the later stages of the assembly.3.It is important to add the insulation to all sides of the tank in the early stages of the assembly to ease the process. This is particularly important for the bottom of the tank. The insulation can be cut to the dimensions required, then attached to the sides of the tank by applying adhesive.4.Attach rectangular frame to the top of the tank using mending plates .Fig. 4Co1-Take measurements across the length of the tank from the primary pump for the desired exit locations of the oil at the different cell locations. Then cut pipe to dimension.2-t-connectors, and reducers should then be connected to the desired locations measured in the previous step. Dry-fit (without applying Poly-Vinyl Chloride (PVC) pipe cement) at this stage to allow for changes that may need to be made.Pipe elbows, 3-Assemble piping arrangement and primary pump in the tank to ensure it is in the correct location and make any necessary adjustments .Fig. 5Pr4-Apply PVC pipe cement to secure the pipe connections.5-Place secondary pump in the desired location see .1-The aluminum extrusion bar to which the heater will be attached to, should be attached to the main frame using angle brackets.2-The heater should then be attached to the bar.3-The chiller should be attached to a copper coil using flexible pipes. Care should be taken such that the water in the loop of the chiller does not mix with the oil.4-The copper coil can then be hung from the edge of the tank in its allocated location.1-Each cell is mounted on a jig – provided by the cell supplier .Fig. 6Tw2-Two jigs should be attached to an aluminum extrusion bar prior to the bar being attached to the main rectangular frame.3-The bars can be attached one by one using angle brackets to the main rectangular frame.4-In our implementation, 56 Ah lithium-ion pouch cells with NMC chemistry are used.At this stage, the bath should be filled with oil or other desired fluid depending on the application. In applications where electrical conductivity is not an issue, de-ionised water could be used, along with inhibitor and biocide to prevent mould growth.1-The cover was laser cut into two sections to allow for easier assembly. Profiles were laser cut on the cover to allow for the cables from the cells to pass through. Depending on the application that this immersion bath will be used for, the lid can be further split into smaller sections to allow an easier assembly process.2-A layer of insulation should then be placed on top of the cover to minimize heat exchange between the fluid within the bath and the environment.1-Turn pumps on2-Turn chiller on3-Turn heater on4-Use It was essential that the immersion bath design facilitated effective and uniform cooling across the jigs that hold the cells. For this reason, as well as ensuring thorough mixing of the immersion fluid, it was important to achieve consistent rates of heat transfer from both surfaces of every jig. This would prevent the build-up of a temperature difference from one surface of a jig to the other.Two sets of tests were performed to confirm that the requirements of thorough bulk coolant mixing, and uniform jig cooling were both met. Preliminary tests were performed with water, and actual validation was carried out using oil. The advantage of performing preliminary tests with water first was that it was easier to drain and make any necessary modifications than it would be once the tank is filled with oil.During both the water tests and the oil tests, the temperature of the cells (or jigs that hold the cells) was measured using K-type thermocouples mounted on the surface at the centre of each jig. Pico Technologies TC-08 units were used for temperature logging. The location where the cells were mounted in shown in The symbols used throughout the validation section are defined in A first batch of tests is performed using water, to ensure the rig is working satisfactorily before adding oil because adding oil to the rig too early would complicate any subsequent modifications or adjustments. The purpose of these tests is twofold: first, to test the overall behaviour of the immersion bath including the heater, chiller, pump and thermocouples; and second, to test the effectiveness by monitoring any temperature imbalances between the compression jig surfaces. Five water tests were conducted without the presence of the cells to avoid short circuiting the cells.The first and second test looked at investigating temperature homogeneity between the seven cells, mounted into the seven jigs in the immersion tank . The firThe third and fourth test looked at temperature stability . Both paThe fifth test analysed the response of the system to a thermal perturbation of 250 kJ, added at the site of jig location one over a period of five seconds in the form of boiling water . The equThis apparatus was designed to be used for oil. All tests so far in this section were performed with water as the immersion fluid. However, oil and water have different fluid properties. It is well established that base oil and water have different thermal properties and therefore are expected to perform differently as immersion fluids for lithium-ion cells.The purpose of these experiments was to assess the overall functioning of the immersion bath and to measure the time taken to reach a stable temperature when using oil. The temperature of the bath was both increased and decreased so that the time taken in both scenarios could be measured. Temperature across different locations in the bath and on the surface of two of the cells in resting were recorded. The results are summarised in An experiment was also carried out while the cell was being electrochemically charged and discharged at constant current (CC). The aim of this experiment was to observe any variation in tank temperature as a result of the cells heating up during electrochemical cycling. One of the cells was cycled at a moderate C-rateThis experiment was carried out at a tank temperature of 25 °C  and at 4We have presented a low-cost, robust temperature control system for immersion cooling of large lithium-ion pouch cells, capable of controlling the temperature of the bath to within 0.5 °C. Despite its large size (150 L), the design presents efficient fluid mixing, leading to homogeneous temperatures across the bath. Besides the thermal management of large lithium-ion batteries, this design is also suitable for the thermal management of supercapacitors, lithium-hybrid capacitors and any other type of large batteries, electrical components, or even non-electrical components. The oil used in the tank may be replaced with other fluids depending on the application and/or budget constraints.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Data assimilation (DA) is a powerful tool to optimally combine uncertain model simulations and observations. Among DA techniques, the particle filter (PF) has gained attention for its capacity to deal with nonlinear systems and for its relaxation of the Gaussian assumption. However, the PF may suffer from degeneracy and sample impoverishment. In this study, we propose an innovative approach, based on a tempered particle filter (TPF), aiming at mitigating PFs issues, thus extending over time the assimilation benefits. Probabilistic flood maps derived from synthetic aperture radar data are assimilated into a flood forecasting model through an iterative process including a particle mutation in order to keep diversity within the ensemble. Results show an improvement of the model forecasts accuracy, with respect to the Open Loop: on average the root mean square error (RMSE) of water levels decrease by 80% at the assimilation time and by 60% 2 days after the assimilation. A comparison with the Sequential Importance Sampling (SIS) is carried out showing that although SIS performances are generally comparable to the TPF ones at the assimilation time, they tend to decrease more quickly. For instance, on average TPF‐based RMSE are 20% lower compared to the SIS‐based ones 2 days after the assimilation. The application of the TPF determines higher critical success index values compared to the SIS. On average the increase in performances lasts for almost 3 days after the assimilation. Our study provides evidence that the application of the variant of the TPF enables more persistent benefits compared to the SIS. We assimilate flood extent maps into a flood forecasting system using a tempered particle filter (TPF)The TPF mitigates degeneracy and enables long‐lasting forecast improvementsThe TPF outperforms a standard particle filter in terms of accuracy of model outputs Tellman et al.  show thaParticle filters (PFs) have gained attention within the research community because of their ability to handle nonlinear and non‐Gaussian systems  steps within the PF proposal step  method described in Abbaszadeh et al.  where GAIn this study, we adopt and evaluate an enhanced PF following the results of the previous studies by Di Mauro et al.  and HostIn this study, we also further investigate additional benefits that come from this new approach. According to Dasgupta et al. , degenerWe carry out twin experiments based on a synthetically generated data set with controlled uncertainty. The SAR observations are synthetically generated from the simulated flood extent maps and assimilated into a coupled hydrologic‐hydraulic model. Two different background ensembles, that is, Open Loops (OLs), are drawn and used: in the first case, the ensemble encompasses the synthetic truth most of the time, in the second case the ensemble is most of the time outside the ensemble range.The objectives of this study are therefore (a) to evaluate whether a principled method, in which the only approximation is the finite ensemble size, can mitigate degeneracy, (b) to evaluate whether the proposed framework improves the prediction accuracy and increases the persistence of the assimilation benefits, (c) to evaluate the efficiency of the method in reducing forecast bias. The paper is structured as follows: Section 2The first part of this section presents the structure of the flood forecasting system. The second part describes the proposed assimilation framework based on a TPF. The experimental design, case study, and the performance metrics used within this experiment are introduced in the last part.2.1SUR representing the root zone, a fast reservoir with storage SFR representing the fast responding components , and a slow reservoir with storage SSR representing slow responding components . A lag function is used at the outlet of the unsaturated soil reservoir to enable a delayed hydrological response of the basin under intense rainfall conditions. The hydraulic model is based on LISFLOOD‐FP  is computed by multiplying the prior probability density function p(xk), which is the probability of the model before any observation is taken into account, with the likelihood p(yk∣xk) that is the probability density that the model state xn produces the observation. In PFs, the prior PDF is drawn from an ensemble of model states of size N called particles. Equation δ is the Dirac delta function. Inserting Equation The observation quation pxk≈∑n=1Wn, hereafter called global weights, were computed by the multiplication of the pixel‐based local weights win, according to the formula by Hostache et al.  is given by Equation r* of the InEff is previously defined. Iterations are stopped when InEff(1) < r* and γS=1−∑s=1S−1γs, where S is the total number of iterations.At each iteration s, the particles with high weights are resampled using the SIR algorithm proposed by Gordon et al. , a variable of the hydrological model, 24 hr prior to the assimilation to regain diversity within the particle ensemble and the mutated value is used as initial condition for a subsequent model simulation over the 24 hr preceding the assimilation time. Mutating the hydrological state variable 24 hr prior to the assimilation time and carrying out the related model simulations is done in order to update the hydrological and hydraulic models more consistently since the water depths simulated by the hydraulic model at a certain time are the result not only of the current but also of the past upstream streamflow conditions.Next, a mutation is applied to the fast run‐off reservoir level  for each particle i and find γ1 such that InEff(1) ≥ r*.Calculate ta‐24 hr.Particles are resampled using the tempered weights. The particles after resampling that are duplicates of particles with high weights are perturbed at time FR are carried out during the 24 hr prior to the assimilation.New hydrodynamic simulations with the mutated levels of the Spmu(y∣x) is compared to the likelihood of the resampled particles pre(y∣x).The likelihood of the mutated particles The resampled particles are replaced by the mutated particles if the ratio of the two is larger than a value randomly taken from the interval .The mutation step is repeated twice.The iteration with a new tempering coefficient is realized.The entire process is repeated until the sum of the tempering coefficients is equal to unity.In detail, the method is structured according to the following time steps Figure :Ensemble2.3The study area is the lower river Severn located in the United Kingdom Figure  every 24The flood event has been simulated using the rainfall and temperature (ERA‐5 data set) time series corresponding to the July 2007 event as input data to the flood forecasting system.mmh while for a “limited” bias case the MBE is 0.14 mmh, meaning that the error of the markedly biased case is 6.56 times larger than for the other case.Further details concerning the hydrological and hydraulic model set‐up as well as the study area of the synthetic experiment, are provided in our previous study  and temporal scales (at the assimilation time and for the subsequent time steps). The filter performances are evaluated in terms of predicted flood extent and water depth maps, as well as local discharge and water levels time series. The performance metrics are assessed by comparing the results of the TPF with those of the OL. Moreover, the TPF is compared with the SIS method applied in our previous study Di Mauro et al. . The locConfusion matrices: a matrix providing the number of false negatives (under‐prediction) and false positives (over‐prediction), together with correct positives and negatives;Contingency maps: maps comparing the simulated flood map with the synthetic truth map;Critical success index (CSI): a metric that evaluates the accuracy of the flood map predictions and is defined as the ratio between the number of pixels correctly predicted as flooded over the sum of predicted flooded pixels . It ranges from 0, complete disagreement, to 1, perfect match;Root mean square error (RMSE): it is given by the square root of the mean of the squares of the deviations of the predicted water levels against the synthetic truth over the hydraulic model domain. It evaluates the prediction errors of a state variable, in our case the water levels.95): it measures the reliability of the ensemble prediction quantiles and it is given by the formula: (Nexceedence/T) ⋅ 100, where Nexceedence is the number of times during the total simulation T where observations fall outside the 95% predictive bounds. The ideal ensemble should fall outside the 95% predictive bounds only the 5% of the time /2N and it should be equal to one. NRR > 1 indicates an insufficient spread, while NRR < 1 indicates the opposite : it is a normalized measure of the ensemble dispersion. It is defined as the ratio of the time‐averaged RMSE of the ensemble mean to the time‐averaged RMSE of the single members of the ensemble over the value The following performance metrics are used:33.13.1.1The flood extent maps are evaluated via different performance metrics: the contingency maps, the CSI and the confusion matrix. The contingency map is derived from the comparison between the simulated flood extent map  and the validation map which is derived from the synthetic truth simulation in our case. The contingency maps, corresponding to three different assimilation time steps , are shown in Figure Yellow and red pixels correspond to errors of under‐prediction (when the model wrongly predicts the pixels as not‐flooded) and over‐prediction (the opposite case), respectively. In Figure The decrease of wrongly predicted pixels is quantified in the confusion matrix reported in Table Time series of CSI are also used to evaluate the TPF performances Figure . They alThis figure shows an improvement of the analysis compared to the OL not only at the assimilation time but also over subsequent time steps: on average, CSI improvements persist for more than 3 days after the TPF application.3.1.2To further investigate the TPF assimilation performance we evaluate water level and discharge predictions. This evaluation is carried out first at specific points along the river Severn: in Bewdley (the gauge station located at the upstream boundary of the hydraulic model domain), and in Saxons Lode (within the hydraulic domain). In Figure To generalize the evaluation made for the gauging stations, we evaluate the accuracy of water level predictions globally, using time series of RMSE computed over the entire hydraulic model domain. This index has been calculated at the assimilation time and for subsequent time steps, in order to assess if the assimilation benefits persist in time. In Figure 3.2We showed in Section In Figure We have also compared the performances of the SIS and the TPF using time series of RMSE Figure . As expeTable 95 evaluates the ensemble spread by quantifying the percentage of time the observation falls outside the 95% confidence interval derived from the ensemble. ER95 values should be ideally around 5%, meaning that the observation falls outside of the 95% predictive bounds only 5% of the time. NRR also evaluates the spread of the ensemble, ideal values should be around the unity and lower or higher values indicate a too narrow or too wide ensemble, respectively. Table 95 varies with the different assimilation time steps. In particular, we found that on average, over the different assimilations, the value of ER95 for the TPF is around 7% in Bewdley and 9% in Saxons Lode, which are values close to the target values (5%). Moreover, if we compare these values with those of the SIS that are around 25%, it is clear that TPF substantially outperforms SIS. This highlights a marked degeneracy in the SIS, that is substantially reduced by TPF.Model performances have also been statistically evaluated using the ER95 and the normalized root mean square error ratio (NRR). Both metrics have been used to evaluate the water level ensemble at two different gauge stations (Bewdley and Saxons Lode). ER3.3In this last experiment, we use the same set‐up as in the previous experiment but with the exception of a modified OL. We have introduced a perturbation error to the ERA‐5 rainfall time series so that the bias in the ensemble is 6.56 times larger than in the previous case. The ensemble has significant bias and the synthetic truth is most of the time located outside of the ensemble range as can be seen in Figure FR 24 hr prior to the assimilation time. In the TPF, better initial conditions of the model forecast are defined at each assimilation time via the different iteration and mutation steps, whereas the SIS only defines the relative importance of each particle, without carrying out any better definition of the initial conditions of the model. The runoff that is used as upstream boundaries of the hydraulic model is a function of the storage SFR of the hydrological model. Updating the SFR, and consequently the fast run‐off, represents an effective way to increase the long‐lasting effects of DA since runoff has the highest uncertainty deriving from poorly known rainfall as already pointed out by Matgen et al.  with respect to the OL and it is comparable to the SIS performances. An important aspect that emerges from the results is the persistence of the assimilation benefits. They remain significant even 3 days after the TPF assimilation when compared to the SIS performances; nonetheless, performances start degrading with the onset of rainfall over the headwater catchment and rainfall uncertainty prevails in the forecast uncertainty. We argue that the marked improvement in the forecast skill of the TPF, compared to the SIS, is due to the update of the initial conditions of the hydrological model including Sn et al. . This asn et al. , could eAfter the TPF application, particles move toward the synthetic truth also in the case the truth falls outside the predictive bounds of the OL ensemble. Despite the improvements due to the TPF, performances are not as good as in the previous case. As a consequence, results obtained using the TPF are sometimes similar to those obtained using the SIS, or even slightly less satisfying when rainfall uncertainty dominates the system. The improvements resulting from the update of the initial conditions are vanished after a few days because of the bias in the ensemble and the model moves back to the OL state. The update of the state level of the reservoir has a time‐limited benefit. It is a state variable highly influenced by the inputs, and thus by the rainfall. In our experiment, the rainfall ensemble is obtained by perturbing the deterministic ERA‐5 product using a multiplicative noise. Therefore, when there is low‐intensity rainfall simulated in ERA‐5 the uncertainty is very limited. Moreover, as the rainfall ensemble is not updated, the ensemble analysis goes back to the OL trajectory after a while. This return of the analysis back to the OL is even more rapid when higher rainfall intensity is imposed to the model: the influence of the initial conditions is rapidly overruled by the forcing uncertainty. To increase the time window of the assimilation benefits, the update of hydrological model state variable could be completed by a forcing update or by a parameter update, as in Cooper et al.  where ch4At the time of the assimilation, forecasts are very accurate locally: the forecast overlaps the synthetic truth for all the different assimilation cases and for both analyzed locations. Results are very satisfying at a larger scale as well: RMSE and CSI improve systematically as a result of the assimilation. On average, RMSE values decrease by 80% whereas CSI values increase by 30% as a result of the assimilation;Results are also satisfying across time: the CSI and RMSE are improved up to 3 days after the assimilation;Performances are improved compared to the OL and the SIS filter. The benefits of the newly introduced TPF‐based assimilation are longer persisting when compared to the effects obtained with assimilation techniques used in the previous studies;The new assimilation framework significantly outperforms the SIS. SIS performance indices are generally comparable to the TPF ones at the assimilation time, but they tend to drop more rapidly, in general 2 days after the assimilation. For example, TPF‐based RMSE are 20% lower compared to the SIS‐based ones, 2 days after the assimilation;When the ensemble is markedly biased results are significantly improved by the TPF at the assimilation times and for a few days after. Afterward, TPF and SIS based results are similar because the model state update cannot compensate for a too large bias in the precipitation ensemble.In this paper, we have proposed a new approach based on a TPF to assimilate flood extent maps into a flood forecasting system. The objective of this new DA framework is to mitigate degeneracy and sample impoverishment, well‐known issues in particle filtering. In the proposed TPF method, the number of tuning parameters is small with respect to methods such as PF‐MCMC (Andrieu et al., The proposed DA framework based on a TPF holds promise for improving prediction accuracy for longer lead times. In this study, we have shown a synthetic experiment where rainfall and SAR observations are the only sources of uncertainty. In a future study, it will be interesting to apply and evaluate this enhanced approach on a real test case in a weakly controlled environment.


We investigate the model performance, the effect of hyperparameter tuning, and assess different methods for calculating uncertainties. Deviating from the default hyperparameters used in the extraTrees package results in little difference in overall model performance . However, accuracy is greatly affected by how the final value from the distribution of trees in the random forest is selected . Using the mean value leads to higher residuals between experimental and predicted P and T, whereas using median values produces smaller residuals. Additionally, this work provides two scripts for users to apply the methodology to natural data sets. The first script permits modification and filtering of the model calibration data set. The second script contains premade models, where users can rapidly input their data to recover PT estimates . Additionally, the scripts allow the user to estimate the uncertainty for each analysis, which in some cases is significantly smaller than the reported SEE. These scripts are open source and can be accessed at https://github.com/corinjorgenson/RandomForest-cpx-thermobarometer.Thermobarometry is a fundamental tool to quantitatively interrogate magma plumbing systems and broaden our appreciation of volcanic processes. Developments in random forest‐based machine learning lend themselves to a data‐driven approach to clinopyroxene thermobarometry, allowing users to access large experimental data sets that can be tailored to individual applications in Earth Sciences. We present a methodological assessment of random forest thermobarometry using the R freeware package  Random forest machine learning is a novel approach for clinopyroxene thermobarometry with a comparable error to thermodynamic thermobarometersModel optimization was performed, and model performance is insensitive to most hyperparametersA detailed description of how to access and use the model is included so readers can use the thermobarometer However, the chemical element (or oxide) selected at each node greatly influences the predictive outcome of the tree. Hence, the random forest model is composed of hundreds of decision trees. Therefore, from these hundreds of decision trees, the output (predicted P or T) is the mean value from all decision trees in the case of regressive models. To allow the model to construct reasonable decision trees for prediction of natural data, we input a data set of experimentally derived clinopyroxenes . These regions are removed as they are rare in the calibration data set and we find poorer performance from the model for regions of the calibration data set with sparse data due to the limited extrapolative ability of machine learning. This forms the final calibration data set  training data set is selected from the total calibration data set of experiments. This data set is used to calibrate with the chosen regression strategy . The remaining data are placed into a test data set, which is used to assess the performance of the model. This is commonly achieved by running each composition in the test data set through the regressed model and calculating the standard error estimate or distribution of residual values to the known experimental values Putirka, .The pressure‐temperature distribution of the calibration data set is not uniform—experiments are preferentially run at low pressures. Thus, randomly extracting from the calibration data set unevenly weights the test set to have low pressure experiments, resulting in a poor representation of the SEE. To circumvent this issue, our test data set was uniformly extracted from the calibration data set on a gridded basis  determines the number of individual decision trees, which are used for prediction. A sufficiently high number of trees must be used to provide stability of the variable importance. The number of trees should be considered a convergent variable, where beyond a certain threshold performance improvement is marginal and the number of trees should be minimized to save on computational time without sacrificing performance  are considered at each node. The mtry is more influential on the overall performance of the model and default mtry for extraTrees is the total number of variables divided by three .nL and nR are the number of datapoints assigned to each left or right branch, and var is the negative variance of the data on the left (or right) side of the split for the y variables  where greater than two splits is termed nonbinary splitting. As noted in the extraTrees vignette, optimization may occur when using numRandomCuts between 3 and 5. We found a minor improvement in the SEE (<0.02 kbar), but the increase in computational time negated any positive effects of more splits.The Each decision tree generates a single output value and thus a forest with 300 trees generates 300 pressure or temperature estimates. By default, in regression mode, random forest algorithms consider the mean or the mode values to return an estimate in regression or classification mode, respectively. In addition to the mean, we additionally calculate the median and modal estimates to evaluate the model performance. The median is calculated by taking the middle value from a sorted set of values. Thus, to avoid the rare case where there is an even number of trees, and the two center points are drastically different, we have decided to use an odd number of trees to average the two values.2.32 values . Varying the test data set is one of the largest sources of variation in the SEE and so we resampled the calibration data set 200 times to produce 200 separate tests and training data sets. Then, the average SEE is taken from the distribution of errors for all 200 data set splits. Two hundred runs were chosen as this is the minimum number of runs where the SEEs are normally distributed, thus preserving computational cost while maintaining a representative assessment. When the chemistry of the mineral under consideration is close to one of the analyses from the experiments, the distribution of estimates is characterized by low values of the IQR . Therefore, we also use the IQR to calculate a confidence interval of each estimated value. We recommend users remove natural data estimates for which the IQR is double the models SEE as these are considered outside of the model error .33.1mtry and 101–1001 ntrees where each permutation is run 200 times with the respective test and train data sets to determine the average SEE and R2, calculated using the ideal median pressures and temperatures.Hyperparameter tuning aims to structure the best performing model possible Breiman, . To systmtry has a larger control on the performance of the model as expected from results in previous studies  than the smaller mtry  for both the mean SEE and residuals. At mtry greater than 6, any difference is minor (±0.01 kbar), and so to limit computational cost, an mtry of 6 should be used. This is counter to the package default, which is one third the number of total variables. A similar trend is observed in the calculated IQR. However, when considering data with the inclusion of liquid—crystal pairs, the new maximum mtry is 18 and hence a new mtry needs to be considered. We performed further testing on the model with the increased maximum mtry and found that although the computational time increased, the models followed the same pattern as clinopyroxene only models whereby performance is relatively insensitive to ntree value and the mtry is optimized at about two thirds of the total variables .The mean SEE varies with the number of trees Figure  where thextraTrees also provides the option to vary the number of cuts at each node. This is conceptualized in a classification model for grouping people on the basis of hair color: instead of discriminating between black or blonde hair (binary choice with 1 cut), brown hair and red hair can be considered as additional options (3 cuts). While the default is 1 cut , increasing the number of cuts to 3–5 may yield marginal performance improvements  and median (R2 = 0.861) both of which offer significant improvements compared to the model using a modal estimate for prediction (R2 = 0.782).To evaluate the performance of the mean, median, and modal estimates, we create pressure and temperature models using the entire calibration data set for clinopyroxene. Figure 3.32 changes from 0.86 to 0.90. For temperature, the difference is even more striking where the SEE decreases by almost half from 68.7°C to 44.1°C and the R2 improves from 0.85 to 0.94.Major element partitioning within the crystal structure of clinopyroxene is not solely sensitive to pressure and temperature but also dependent on chemical availability of elements in the residual liquid (melt). Thus, in systems for which both clinopyroxene and equilibrium melt chemistry are available, we have also calibrated a clinopyroxene‐liquid thermobarometer in addition to the clinopyroxene only model we have presented thus far. All points in the calibration data set included liquid. Performance testing of the two models Figure  reveals 44.1extraTrees, is appropriate. However, the distribution is often not a perfect Gaussian curve. Additionally, some voting distributions may have very wide uniform distributions also indicating an estimate with a low degree of certainty. Other voting distributions show sharp peaks at a given value followed by small, wide tails to low and/or high pressure/temperature. Such tails from poorly performing trees lead to overestimates of pressure or temperature due to unfair weighting by the mean of the distribution. Poorly behaving trees can result from elements being selected for decision tree nodes, which do not have a strong control on the variation of clinopyroxene unit cell parameters: these features ultimately govern the relationship between pressure, temperature, and mineral chemistry  are lower than that of the median (0.861). Despite the modal model showing a marginally tighter distribution of residuals, it has a fundamental flaw that is shown in Figure Mean, median, and modal models all perform well although the residuals from the modal and median model are preferable to the mean Figure . Conside4.2Throughout the course of this work, we have optimized each model to give the best representation of the true  pressure and temperature. Though we have tested and optimized each model, there remain datapoints with high residuals, giving a poor estimate relative to the true experimental value  or use the IQR as a metric for post‐estimate filtering. Figure 4.3P ≤ 30 kbar) and the “crustal model” (P ≤ 15 kbar) as chosen for the crustal range on the basis of the average crustal thickness  crust  yields poor model performance (Figure Hyperparameters generally make little difference to the performance of the thermobarometer. The largest effect is the value of ues 1 or  yields pe Figure . Thus, wTwo sets of codes have been created, with detailed comments and instructions, for the Earth sciences community to rapidly predict intensive parameters for natural data or create more tailored models Appendix ppendix ASupporting Information S1Click here for additional data file.