Oleg Zubelevich
email: ozubel@yandex.ru

On the Meshchersky Equation

In this short note we formulate a version of the momentum theorem for the variable mass systems. We deduce the momentum theorem from the fundamental laws of continuum mechanics.

The Main Theorem

Let D ⊂ R q be an open and bounded domain with C 1 -smooth boundary ∂D. The cases q = 1, 2, 3 are physically reasonable.

The space R q is endowed with an inertial frame of reference Ox 1 , . . . , x q , where x = (x 1 , . . . , x q ) are the standard Cartesian coordinates:

x = x i e i ∈ R q .
A theorem we formulate below has an invariant form but we use the Cartesian coordinates just for convenience.

Let (•, •) denote the standard inner product in R q . Note also that in the Cartesian frame there is no need to distinguish co-and contravariant components of tensors.

Let

w(t, x) = w i (t, x)e i , w i ∈ C 1 (R q+1 ) stand for a vector field in R q and g t t 0 (x) be its flow:

d dt g t t 0 (x) = w(t, g t t 0 (x)), g t 0 t 0 (x) = x.
We assume that g t t 0 (x) is defined for all real t, t 0 and for all x ∈ R q . Introduce a notation D(t) = g t t 0 (D). Loosely speaking w is a velocity of the volume D(t).

Assume that the domain D(t) is filled with a continuous media with a mass density ρ(t, x) and a flow velocity field v(t, x) = v i (t, x)e i ;

v i , ρ ∈ C 1 (Ω), Ω = {(t, x) | x ∈ D(t), t ∈ R} ⊂ R q+1 .
The local conservation of mass equation [START_REF] Sedov | Mechanics of Continuous Media[END_REF] is

ρ t + div(ρv) = 0 or ρ t + ∂(ρv i ) ∂x i = 0. (1.1)
Let F (t, x) = F i (t, x)e i , F i ∈ C(Ω) stand for a force per unit mass. So that the whole matter in D(t) experiences a net force

G(t) = D(t) ρ(t, x)F (t, x)dV,
where dV is the volume element. We denote the stress tensor by p ij (t, x). The boundary ∂D(t) experiences the following external contact net force

P (t) = ∂D(t) p n dS, p n = p ij n j e i ,
where n = n i e i is the outer unit normal to ∂D(t); dS is the area element of the surface;

p ij ∈ C 1 (Ω). The local equation of linear momentum balance is ρ v k t + ∂v k ∂x i v i = ρF k + ∂p kj ∂x j . (1.2) Let Q(t) = D(t) v(t, x)ρ(t, x)dV
stand for the linear momentum of D(t).

Theorem 1. The equation of linear momentum balance for the volume D(t) is as follows

Q = G + P + R, where R = - ∂D(t) v(v -w, n)ρdS.
Proof of the Theorem. By the well-known theorem from the integral calculus and due to (1.2) it follows that

d dt Q k = D(t) v k t ρ + v k ρ t + div (ρv k w) dV = D(t) F k - ∂v k ∂x j v j ρ + ∂p kl ∂x l + v k ρ t dV + D(t) div (ρv k w)dV.
Integration by parts gives

D(t) ∂v k ∂x j v j ρdV = ∂D(t) v k v j n j ρdS - D(t) v k div(ρv)dV, D (t)
∂p kl ∂x l dV = ∂D(t)

p kl n l dS,

D(t) div (ρv k w)dV = ∂D(t)
ρv k w j n j dS.

To finish the proof it remains to employ (1.1).

The Theorem is proved. The angular momentum theorem is derived by the same way:

d dt D(t) x × vρdV = D(t)
x × F ρdV

+ ∂D(t)
x × p n dS + ∂D(t)

x × v(wv, n)ρdS.

Here we use p ij = p ji .


Negin Nazarian

Sijie Liu

Manon Kohler

Jason K W Lee

Clayton Miler

Chow

Sharifah Badriyah Alhadad

Alberto Martilli

Matias Quintana

Leslie Norford

Project Coolbit: Can your Watch Predict Heat Stress 1 and Thermal Comfort Sensation? 2

Global climate is changing as a result of anthropogenic warming, leading to higher daily excursions 16 of temperature in cities. Such elevated temperatures have great implications on human thermal comfort 17 and heat stress, which should be closely monitored. Current methods for heat exposure assessments 18 (surveys, microclimate measurements, and laboratory experiments), however, present several limitations: 19 measurements are scattered in time and space and data gathered on outdoor thermal stress and comfort 20 often does not include physiological and behavioral parameters. To address these shortcomings, Project

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Coolbit aims to introduce a human-centric approach to thermal comfort assessments. In this study, we 22 propose and evaluate the use of wrist-mounted wearable devices to monitor environmental and physiological 23 responses that span a wide range of spatial and temporal distributions. We introduce an integrated 24 wearable weather station that records a) microclimate parameters (such as air temperature and humidity), 25 b) physiological parameters (heart rate, skin temperature and humidity), and c) subjective feedback. The 26 feasibility of this methodology to assess thermal comfort and heat stress is then evaluated using two 27 sets of experiments: controlled-environment physiological data collection, and outdoor environmental data 28 collection. We find that using the data obtained through the wrist-mounted wearables, core temperature 29 can be predicted non-invasively with 95 percent of target attainment (PTA) within ±0.27 • C. Additionally, 30 a direct connection between the air temperature at the wrist (Ta,w) and the perceived activity level (PAV) of 31 individuals was drawn. We observe that with increased Ta,w, the desire for physical activity is significantly 32 reduced, reaching "Transition only" PAV level at 36 • C. These assessments reveal that the wearable 33 methodology provides a comprehensive and accurate representation of human heat exposure, which can be

stress as "human-centric" as opposed to "one-size-fits-all". Additionally, wearables travel with the individu-73 als in realistic exposure scenarios and, therefore, data collected by wearables combined with GPS data can 74 provide a spatiotemporal distribution of environmental parameters and individual's exposure. However, to 75 date, there is no wearable sensing (neither commercial nor in academic use) available nor tested that can 76 combine all the parameters relevant to thermal comfort and heat stress. Accordingly, we propose and test a 77 methodology based on wearable devices here, which can ultimately enable a comprehensive yet unsupervised 78 and non-invasive assessment of thermal comfort and heat stress in the built environment.

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The current generation of wearables monitors such physiological parameters as heart rate, which helps under-80 stand various aspects of human wellbeing and health including sleep quality. However, to assess thermal stress 81 and comfort, the measurements should be extended. Human skin is the mediator between the environment 82 and human body and, therefore, skin temperature and conductance play a major role in thermoregulatory 83 processes involved in thermal comfort and heat stress [27,[START_REF] Soo | Estimation of thermal sensation based on wrist skin temperatures[END_REF]. Sim et al. [START_REF] Soo | Estimation of thermal sensation based on wrist skin temperatures[END_REF] showed that wrist skin tem- non-invasive and continuous method of estimating the human core temperature, the main factor in determin-87 ing heat stress, from sequential heart rate observations. They showed that out of 52,000 observations, 95% 88 of all core temperature estimates fell within ±0.63 • C of measurements. This, in addition to the advancement 89 of ubiquitous sensing in the built environment, has opened new doors to use wearables for thermal exposure 90 assessments. Nonetheless, this is an emerging field and there are only a handful of studies that investigate 91 wearable solutions [START_REF] Lin | Numerical simulation studies of the different 410 vegetation patterns' effects on outdoor pedestrian thermal comfort[END_REF][START_REF] Nakayoshi | Outdoor thermal physiology along 432 human pathways: a study using a wearable measurement system[END_REF][START_REF] Liu | Personal thermal 435 comfort models with wearable sensors[END_REF][START_REF] Abdallah | Sensing occupant comfort using 446 wearable technologies[END_REF]. Among these, Nakayoshi et al. [START_REF] Nakayoshi | Outdoor thermal physiology along 432 human pathways: a study using a wearable measurement system[END_REF] represents a comprehensive measure-92 ment of thermal comfort, monitoring four relevant environmental parameters in the proximity of the human 93 body as well as physiological responses (heart rate and skin temperature) and subjective feedback, and found 94 a correlation between skin temperature and thermal comfort index in a semi-controlled testing environment.

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The wearable system involved five sensing units worn by the participants on multiple locations: hat, belt, 96 hand, and forehead skin, and carried in a small sash. Although comprehensive for research purposes, this 97 methodology cannot be employed in unsupervised settings and is considered impractical for implementation 98 in real-life applications. Wrist-mounted wearables can address such concerns regarding scalability in realistic 99 applications, particularly as smartwatches have recently dominated the wearable tech worldwide [31]. How-100 ever, the challenges of using wearables as sensing methodologies are still numerous and, to date, reliability of 101 wrist-mounted wearable data collection under dynamic use are not fully assessed. There is an urgent need to 102 quantitatively assess the performance of wearables for heat exposure monitoring, which motivates the present 103 study. Here, we propose an Integrated Wearable Weather Station for unsupervised assessment of urban heat 104 impact on individuals (Sec 2.1), and further discuss experiments that rigorously assess the feasibility of this 105 methodology for various heat exposure evaluations in the built environment (Sec. 2.2). In Secs. 3.1 and 106 3.2, we evaluate the prediction of body core temperature (as the main predictor of heat stress) and thermal 107 comfort sensation using collected data and lastly, we discuss the implication of these findings as well as future to thermal exposure assessments [START_REF] Michael | Emerging wearable physiological monitoring technologies and decision 449 aids for health and performance[END_REF][START_REF] Friedl | Real time physiological status monitoring (RT-PSM): accomplishments, requirements, and research 452 roadmap[END_REF][START_REF] Mark | Wearable physiological monitoring for human 455 thermal-work strain optimization[END_REF]. Smart devices further provide interfaces for continuous interaction 123 with users and data collection regarding behavioral patterns of human activities. Tapping on these emerg-124 ing potentials, we propose wrist-mounted smart wearable devices as a novel approach to obtaining dynamic 125 (spatially and temporally variable) data on thermal exposure. Integrated Wearable Weather Stations (Fig. Combined, this methodology aims to provide a comprehensive, integrated, and personalized assessment that 130 improves our understanding of personal thermal comfort and heat stress in cities. As the first example of a wearable weather station for heat exposure assessments, we employed Fitbit smart-132 watches [START_REF]Fitbit official site for activity trackers[END_REF] worn on the wrist that are equipped with the PurePulse (photoplethysmography) technology for heart rate monitoring [START_REF] Bai | Comparative evaluation of heart 459 rate-based monitors: Apple watch vs fitbit charge hr[END_REF][START_REF] Bai | Comparative evaluation of heart 459 rate-based monitors: Apple watch vs fitbit charge hr[END_REF][START_REF] Haghayegh | Accuracy of 461 PurePulse photoplethysmography technology of fitbit charge 2 for assessment of heart rate during sleep. 462[END_REF]. The Fitbit smartwatches are then equipped with two coin-sized iButton environmental sensors (i.e. wireless data logger in the form of a 1.3 cm radius stainless steel button [START_REF]Overview of ibutton R sensors and Temperature/Humidity data loggers -applica-464 tion note -maxim[END_REF]), 135 which are placed at the inner and outer face of the watch strap to collect temperature and humidity of air and 136 skin. The iButton Hygrochron temperature/humidity data loggers (DS1923) are attached to the Fitbit devices 137 with a 3D printed harness (Fig. 2) and measure the air/skin temperature and humidity ranging between -20 138 to 85 • C and 0 to 100% with ±0.5 • C and ±0.6% accuracy, respectively. The use of the smartwatch app for 139 obtaining subjective feedback was also assessed in a separate study (preliminary work presented by Jayathissa 140 et al. [START_REF] Jayathissa | Is your clock-face cozie? a smartwatch 467 methodology for the in-situ collection of occupant comfort data[END_REF]). The feasibility of such integrated sensing for inferring the personalized heat exposure in the built 141 environment is assessed in Sec. 2.2. to approximate the number of steps taken, calories burned, floor climbed, and length of time performing exercises. The PurePulse (photoplethysmography) technology for heart rate monitoring [START_REF] Bai | Comparative evaluation of heart 459 rate-based monitors: Apple watch vs fitbit charge hr[END_REF][START_REF] Bai | Comparative evaluation of heart 459 rate-based monitors: Apple watch vs fitbit charge hr[END_REF][START_REF] Haghayegh | Accuracy of 461 PurePulse photoplethysmography technology of fitbit charge 2 for assessment of heart rate during sleep. 462[END_REF] uses LED lights installed at the back of the instrument to detect blood volume changes that are due to capillary expansion and contractions, and has been shown to tracks heart rate well when compared to three-lead electrocardiography [START_REF] Haghayegh | Accuracy of 461 PurePulse photoplethysmography technology of fitbit charge 2 for assessment of heart rate during sleep. 462[END_REF]. The design and use of the smartwatch app will be further discussed in future studies. heat (such as heart rate and skin temperature and humidity at the wrist) and further evaluate the relationship between wearable data and the body core temperature as the main indicator for heat strain [START_REF] Mark | Estimation of human core temperature from sequential heart rate observations[END_REF]. to 20, resembling "very light" to "extremely hard" and is subjective. In our experiment, the three stages of 170 activity (each 15-minutes) corresponded to RPE of 8-9, 10-11 and 12-13 to induce "fairly light," "moderate," 171 and "somewhat hard" efforts in individuals, respectively. Accordingly, although generally healthy adults were 172 targeted, there was no required threshold level of fitness. The run/walk exercise on the treadmill in this 173 experiment resulted in a change in the metabolic rate and therefore body core temperature, which is needed 174 for assessment of this methodology in a range of daily human activities. Here, we evaluate the physiological data collection using wearable sensors and their correlation with heat 208 strain (indicated by body core temperature). First, we compared the skin temperature obtained at the wrist

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with temperature distribution at different body locations as well as air temperature (Fig. 6). Chest and thigh 210 exhibit the highest skin temperature (∼ 31-37 • C), but also resemble body areas that were mostly covered by 211 participants. Skin temperature at the wrist (T s,w ) shows the lowest median and minimum value compared to 212 other body parts, while being consistently higher than the air temperature at wrist in the studied conditions.

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We observe that although the variability in ambient air temperature (T a ) is very small (∼ 28-30 to the sensor being placed at the proximity of the human body that acts as a heat source. This indicates that 216 air temperature at the wrist, alone, cannot determine the ambient air temperature in the built environment 217 as it exhibits the combined effect of environmental conditions (T a ) and physiological responses (T s ). However,

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it is worth noting that it may be feasible to predict ambient air temperature using air and skin temperature 219 at the wrist considering the heat exchange between human skin surface and thermal environment [START_REF] Nazarian | Project coolbit 476 updates: Personal thermal comfort assessments using wearable devices[END_REF][START_REF] Pharo | Heat exchange between human skin surface and thermal environ-479 ment[END_REF].

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Additionally, we compared wrist skin temperature with mean skin temperature [START_REF] Nl Ramanathan | A new weighting system for mean surface temperature of the human body[END_REF] for each participant (Fig.

Prediction and Impacts of Thermal Sensation Vote

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Comparing participants' thermal sensation with local microclimate parameters (such as WBGT obtained 266 from fixed monitoring stations) yielded similar results to findings of Yang et al. [START_REF] Yang | Thermal comfort in outdoor urban spaces in 504 singapore[END_REF] and Heng and Chow [START_REF] Su | How 'hot'is too hot? evaluating acceptable outdoor thermal comfort 506 ranges in an equatorial urban park[END_REF],

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indicating a linear relationship between thermal comfort indices and aggregated thermal sensation vote. Here,

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we extend the analysis to compare thermal sensation and satisfaction votes, TSV and TCV respectively, with 269 data obtained from wearable devices such as skin and air temperature at the wrist (Fig. 10). We observe that 270 thermal sensation vote (Fig. 10 -right, ranging from "Very Cold" to "Very Hot") exhibits a positive correlation 271 with the air and skin temperature at wrist and a stronger correlation compared to ambient air temperature.

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For thermal comfort vote (Fig. 10 -left, ranging from "Extremely Satisfied" to "Extremely Dissatisfied"),

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the median and distribution of air temperature measured at the wrist exhibit the most significant correlation 274 with thermal satisfaction, which indicates the ability of T a,w for predicting comfort. Additionally, Fig. 10 275

shows that as TSV moves towards hotter sensations or TCV moves to higher dissatisfaction, the difference 276 between temperatures at the wrist and ambient air temperature decreases. This difference dominates the rate 277 of sensible heat transfer from the skin, which is critically important for human comfort and satisfaction [START_REF] Edward | The skin's role in human thermoregulation and comfort[END_REF]. Pearson regression coefficient and errorbars (indicating the standard deviation of TSV for the binned data) are also presented for each variable. The temperature-humidity index is adopted from Steadman [START_REF] Robert G Steadman | The assessment of sultriness. part i: A temperature-humidity index based on human 516 physiology and clothing science[END_REF] to combine air temperature and relative humidity monitored at the wrist.

"Neutral" and "Slightly Warm" can lead to satisfaction of the thermal environment. This is in line with 302 previous studies that demonstrated that a) "Neutral temperature" does not necessarily indicate thermal 303 comfort and satisfaction [START_REF] Middel | Impact of shade on outdoor thermal comfort-a seasonal 406 field study in tempe, arizona[END_REF] and b) Singapore residents tend to have a higher tolerance to colder indoor 304 conditions, especially considering the high humidity level outdoors. More importantly, by comparing the 305 Perceived Activity Vote with TSV (Fig. 12 -right), we find that the desire to do an activity is significantly 306 affected by the thermal environment. Participants may be willing to do an extended activity in cold thermal 307 sensations but a warm condition directly translated to shortened or lack of activity in our experiments.

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Moreover, using the binned data (Fig. 13), we can draw a direct connection between the air temperature 309 at the wrist and perceived activity level: with increased T a,w , the desire for physical activity is significantly go about their daily lives [START_REF] Evan R Kuras | Opportunities and 523 challenges for personal heat exposure research[END_REF]. The personalized and real-time assessment of urban heat exposure, which further provides comprehensive assessments of impacts on human life, is yet to be achieved.

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Using environmental and physiological data obtained from the watch, we were also able to predict the overall 343 sensation of participant groups. However, when regression models are applied to individualized responses, 344 only ∼ 35 -40% of responses are accurately predicted which is similar to previous thermal comfort models 345 [START_REF] María | Suitability of different comfort indices for the prediction 510 of thermal conditions in tree-covered outdoor spaces in arid cities[END_REF]. This is due to the subjective nature of thermal comfort that includes individual preferences based on 346 behavioral, cultural, and climatic backgrounds. To account for these, we aim to extend the data collection 347 period and employ machine learning techniques that incorporate individualized behavioral patterns to train 348 personal comfort models [START_REF] Kim | Personal comfort models: 513 predicting individuals' thermal preference using occupant heating and cooling behavior and machine 514 learning[END_REF]. More importantly, we demonstrated that this methodology can quantify the 349 indirect impact of heat on health through the change in physical activity level and lifestyle. To the best of 350 our knowledge, this is the first study that quantified the impact of urban heat on activity level, which opens 351 new doors for heat-health assessments. We plan to extend this study to quantify the impacts of more realistic 352 thermal environments on perceived and actual activity levels of individuals.

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This study represents the first methodology to monitor personal heat exposure in a non-intrusive yet quanti- The final estimate of T C,t was calculated with the preliminary estimate (T C,t-1 ), the error between the observed variables (HR t , T sw,t , and T aw,t ) and the estimated ones using the T C,t-1 :

T C,t = T T C,t + K t (        HR t T sw,t T aw,t        -(H × T T C,t ) + H 0 ) (7)
Finally, the current core temperature estimate error variance is computed as

P t = (1 -K t H)P t T . (8)
A c c e p t e d M a n u s c r i p t

[12] E Johansson, S Thorsson, R Emmanuel, and E Krüger. Instruments and methods in outdoor thermal

  84peratures can be used to predict whole-body thermal sensation. Additionally, several studies have used heart 85 rate data to indicate thermal stress in the built environment. A study by Buller et al. 2013 introduced a 86

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  research that can extend this methodology for real-time and unsupervised evaluation of urban heat impacts 109 on human life in Sec. 4. 110 A c c e p t e d M a n u s c r i p t 2.1 Integrated Wearable Weather Station: Human-Centric Assessment of Ther-112 mal Comfort and Heat Stress 113 Innovative methods of obtaining data are needed to assess dynamic exposure to thermal environments in 114 cities in a human-centric way. However, the assessment of thermal exposure is notoriously complex as it 115 requires consideration of three critical components: a) environmental factors, b) physiological thermoregulation 116 mechanisms of the human body, and c) subjective psychological perceptions and behavioral patterns as well 117 as cultural and climatic backgrounds of individuals. Accordingly, the integration of all these components into 118 one sensing unit hasn't been accomplished so far. 119 Unprecedented potentials are now emerging through the rise of Internet-of-Things sensing and wearable tech-120 nologies for fitness, performance, and health tracking. Various wearable sensors have enabled continuous 121 and real-time monitoring of physiological parameters over the last few years, with limited attention given 122

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1 )

 1 proposed here aim to record 1) microclimate parameters (such as air temperature and humidity) in the 127 immediate environment of individuals, 2) physiological responses to heat (including heart rate, skin tempera-128 ture and humidity), and 3) human activity and subjective feedback with regards to the thermal environment.

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Figure 1 :

 1 Figure 1: Schematic of an Integrated Wearable Weather Station for personalized assessment of urban heat exposure. In this format, three components of heat exposure are captured: 1) environmental parameters (such as air temperature, humidity, or radiation) are recorded on the outside of the strap (right image), 2) physiological response to heat exposure (including skin temperature and humidity) is captured based on sensors placed on the inner strap (left image), and 3) the smartwatch app is used to monitor activity level, location, and individuals' momentary assessment of heat exposure (center).

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Figure 2 :

 2 Figure 2: Fitbit Ionic and sensor attachments deployed here for monitoring of a) temperature and humidity of air in the proximity of human body, b) heart rate, skin temperature, and humidity at wrist, and c) activity level and momentary feedback. Two iButton sensors [38] are placed on the watch strap using a 3D printed harness. The Fitbit acitivity trackers used have a 3-axis accelerometer to track the wearer's motion patterns (e.g. those that indicate walking, swimming or cycling)

142 2 . 2

 22 Experimental Campaigns: Assessing the Robustness of Data Collection using 143 Wrist-mounted Devices 144 Detailed measurements are carried out and compared with conventional sensing methods to investigate the 145 accuracy of physiological and environmental data collected by wearables. We conducted two sets of experi-146 ments: 1) controlled-environment experiments in a climate chamber, and 2) semi-controlled experiments in a 147 range of indoor-outdoor built environments. In the first experiment, the environmental conditions were kept 148 unchanged while the metabolic rate was varied based on activity level, while the second experiment focused 149 on the changes in the heat exposure and thermal comfort based on microclimate characteristics. The detailed 150 setup and specification of each experiment are discussed here. Ethics approval for conducting human subject 151 research was received from NUS Institutional Review Boards (Reference code N-18-071).

152 2 . 2 . 1 5 A

 2215 Controlled-Environment Test in the Climate Chamber 153 We conducted controlled-environment experiments in a climate chamber (Fig. 3) at the Department of Phys-154 iology of the National University of Singapore. These experiments aim to collect physiological responses to 155 c c e p t e d M a n u s c r i p t

Figure 3 :

 3 Figure 3: Controlled-environment experiment at NUS Department of Physiology. A participant is walking on a treadmill while wearing the wrist-mounted sensor arrangement (Fig. 2), chest-wrapped heart rate monitoring (Polar A300), and iButton sensors for recording skin temperature at various body parts. The environmental parameters in the climate chamber as well as ratings of thermal sensation (ASHRAE's 7point scale) and RPE are continuously monitored throughout the experiment. Data from the ingested telemetric capsules were collected by the wireless data-recording devices and the body core temperature is monitored closely such that the experiment is ceased if the threshold of 40 • C is reached.

175 2 . 2 . 2 6 A

 2226 Semi-Controlled Environment Test in the Built Environment176We further conducted environmental monitoring campaigns to evaluate and compare the wrist-mounted sensor 177 data with microclimate measurements at fixed locations and calibrate the subjective individual thermal sen-178 sation with objective environmental measurements. Semi-controlled environment tests were performed where 179 participants walked through a predefined path (covering different built environment characteristics -Fig.4), 180 while passing through a network of sensors and answering thermal comfort surveys (Fig.5).181Fifteen sessions were organized over six days in October -November (inter-monsoon period), distributed 182 c c e p t e d M a n u s c r i p t

Figure 4 : 7 Page 7 Figure 5 : 1

 47751 Figure 4: Map and photos of sensor network in the semi-controlled experiment. Left: map of the studied area at NUS UTown Campus with location of sensors and survey stations identified. Right: different characteristics of built environment selected for the experiment. Five survey stations were selected among an indoor (air-conditioned) environment, a semi-covered outdoor environment, covered outdoor locations (distinguished by a presence or lack thereof of ceiling fan), and fully exposed locations with different sky view factors.

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6 -

 6 right) and found a linear relationship for all participants. The majority of participants, however, showed 222 lower wrist temperature compared to mean skin temperature, as temperatures at body extremities are usually 223 lower. Nonetheless, we observed that wrist (skin and air) temperatures (T a,w and T s,w ) better describe the 224 thermal comfort sensation of individuals (Sec. 3.2) and therefore these parameters are used for further analyses.225Further monitoring relative humidity at the wrist (RH w ), we observe that for some participants, RH w reaches 226 8 be higher than RH reported in other body parts due to the rubber wristband, we note that the onset of 228 sweating in different individuals (a critical determinant for the physiological strain and acclimatization) is A c c e p t e d M a n u s c r i p t

Figure 7 :

 7 Figure 7: Left: Comparison of heart rate data (1-min average) obtained from Fitbit wearables (Fig. 2) with high-accuracy chest strap sensors (Polar A300). The 1:1 relationship is represented by a dashed line. Right: Distribution of observed core temperature as a function of heart rate. Each ID represents a unique participant in the controlled-environment experiment.

Figure 8 :

 8 Figure 8: Left: schematic of various physiological and environmental parameters used to predict body core temperature, includingheart rate, wrist skin temperature (Ta,w) and wrist air temperature (Ts,w). Skin humidity (RHw) was also considered in the prediction algorithm but did not increase the accuracy of results. Right: prediction of body core temperature (30-sec average) for different genders (red: male, black: female) using a Kalman filter compared with measurements in the climate chamber.

Figure 8 Figure 9 :

 89 Figure 8 shows the schematic of physiological measurements for the core temperature prediction, as well251

278 11 A c c e p t e d M a n u s c r i p tFigure 11 :

 1111 Figure 11: Distribution of thermal sensation vote as a function of wearable data (binned based on wrist air temperature).

13 A

 13 310 reduced, reaching "Transition only" PAV level at 36 • C. This is the first quantification of thermal comfort 311 impact on human activity and demonstrates the ability of this methodology to not only predict the overall 312 thermal comfort, but further contribute to quantification of the indirect impact on health through the loss of 313 activity and change in lifestyle.314 c c e p t e d M a n u s c r i p t

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 1516 tative way, which enables us to better determine the links between climatic variables and human health and 355 wellbeing, design effective mitigation and adaptation strategies, and prepare emergency responses to extreme 356 conditions. Such knowledge can ultimately transform the way we understand and design for optimized ex-357 posure. However, we note the deployment of wearable sensors is done for a limited number of participants 358 so far. To fully realize the impact of this methodology, the sensor array needs to be further developed and 359 of 21 AUTHOR SUBMITTED MANUSCRIPT -ERL-109222.R1 A c c e p t e d M a n u s c r i p t

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  Such knowledge gaps and methodological limitations motivated the present study. Here, we proposed wearable 322 sensing for human-centric heat exposure assessments, such that we connect humans with their immediate We demonstrated that body core temperature (T c ) can be predicted non-invasively with high accuracy: using 331 data from 15 participants, T c was predicted using heart rate, skin temperature, and air temperature at wrist 332 (obtained from wearable devices) and 95% of predicted results fell within 0.27 • C of measurements obtained 333 from telemetric capsules. This is among the best performances seen in the literature and presents the most 334 viable option as smartwatches are easily worn and carried on the wrist at all times. However, it should be noted

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	324	environment. We introduced an integrated personalized methodology, i.e. wearable weather stations, that
	325	in one wrist-mounted sensing unit can record environmental parameters, physiological responses, and human
	326	activities and feedback. We then addressed the feasibility of this methodology using two sets of experiments:
	327	1) controlled-environment experiments in a climate chamber focused on physiological responses and 2) semi-
	328	controlled experiments in the built environment focused on thermal comfort. The objectives were to answer
	329	two questions: 1) can this wearable sensing predict heat strain? and b) what information regarding thermal
	330	sensation can be derived using personalized monitoring?

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that due to the limited number of participants in this study (15 in total), it was not statistically meaningful to 336 train the data using a segment of the sample size for testing and more importantly, a relatively homogeneous 337 participant profile is considered here. Accordingly, it is critical that measurements and testings are extended 338 to increase the number of participants with diverse profiles (such as age, gender, BMI, acclimatization status, 339 fitness, and health conditions). We further plan to extend the measurements to higher T c and HR ranges 340 in collaboration With the NUS Department of Physiology and by studying healthy adults that can complete 341 maximum physical activity tests in experimental settings.

AUTHOR SUBMITTED MANUSCRIPT -ERL-109222.R1

A c c e p t e d M a n u s c r i p t saturation during the experiments (in both controlled and semi-controlled settings). Although this value may captured using the wearables which can be the subject for future research on personalized heat exposure. compared with air temperature measurements (near-wrist and ambient). Right: Distribution of skin temperature at wrist compared to mean body skin temperature for each individuals [START_REF] Nl Ramanathan | A new weighting system for mean surface temperature of the human body[END_REF][START_REF] Liu | Evaluation of calculation methods of mean 481 skin temperature for use in thermal comfort study[END_REF]. Each ID represents a unique participant in the controlledenvironment experiment and the 1:1 relationship is represented by a dashed line.

We further compared the wrist-mounted heart rate data with highly accurate measurements obtained from 231 chest strap sensors (Fig. 7 -left) as well as body core temperature obtained from telemetric capsules (Fig. 7 232 -right). The comparison is in agreement with previous studies that deemed Fitbit satisfactory for heart rate 233 monitoring [START_REF] Bai | Comparative evaluation of heart 459 rate-based monitors: Apple watch vs fitbit charge hr[END_REF]. We observed that 83% of heart rate data falls within the desired ±5 bpm accuracy level, with 234 significantly smaller error observed for HR > 120 that is particularly of interest for heat strain assessments.

235

Additionally, it is found that a significant majority of the error is attributed to two participants. After 236 evaluating temperature measurements at the wrist for these participants (not shown), we find that this error 237 has been introduced due to the way the smartwatches were worn during the experiment. This is particularly 238 important for future deployments and motivates means to ensure that wearables are worn correctly. An 239 example of such interventions can be a smartwatch function that monitors the wearable pressure on the wrist 240 and triggers an alarm on the smartwatch in response.

241

Lastly, we focused on non-invasive prediction of body core temperature (T c ) using physiological and environ-242 mental data by the wrist-mounted sensors. We observed that core temperature is positively correlated with 243 heart rate data (Fig. 7), which is in close agreement with the reported role of metabolic rate on heat strain [START_REF] Lee | Ther-483 moregulation, pacing and fluid balance during mass participation distance running in a warm and humid 484 environment[END_REF].

244

However, given i) potential errors in heart rate monitoring using Fitbit watches (Fig. 7-left) and ii) moderate 245 performance in T c prediction when only heart rate data are used [START_REF] Mark | Esti-487 mation of human core temperature from sequential heart rate observations[END_REF][START_REF] David P Looney | Estimating resting core 491 temperature using heart rate[END_REF], we revisited the core temperature 246 predictions using sequential air and skin temperature at wrist (T a,w and T s,w ). A Kalman filter (also known as 247 linear quadratic estimation [START_REF] Emil | A new approach to linear filtering and prediction problems[END_REF]) was employed to estimate core temperature (T c ) using the variables obtained 248 from wearable sensors. The KF model (further explained in Appendix A) is used extensively for T c estimation 249 using non-invasive measurements, mainly heart rate [START_REF] Mark | Esti-487 mation of human core temperature from sequential heart rate observations[END_REF][START_REF] Mark | Estimation of human 495 internal temperature from wearable physiological sensors[END_REF][START_REF] Seng | Tracking body core temperature in military 498 thermal environments: An extended kalman filter approach[END_REF][START_REF] Alexander P Welles | Estimation of core body temperature from skin temperature, heat flux, and heart 502 rate using a kalman filter[END_REF]. Next, to analyze and predict the respondents' thermal sensation and comfort, we binned the data based on 279 wrist air temperature (T a,w ) into 0.5 • C intervals [START_REF] Middel | Impact of shade on outdoor thermal comfort-a seasonal 406 field study in tempe, arizona[END_REF] and calculated the mean TSV and TCV in each bin 280 (Fig. 11). We observe that the correlation between heart rate and TSV is weak, but skin temperature and assessed in previous studies [START_REF] María | Suitability of different comfort indices for the prediction 510 of thermal conditions in tree-covered outdoor spaces in arid cities[END_REF]. This further motivates the development of personal comfort models [START_REF] Kim | Personal comfort models: 513 predicting individuals' thermal preference using occupant heating and cooling behavior and machine 514 learning[END_REF] 288 based on long-term data collection and consideration of behavioral and subjective factors, which is a focus in 289 future developments of Project Coolbit.

290

To extend our assessment regarding the relationship between thermal sensation and satisfaction and the 291 consequent impact on human life, we show the correlations between Thermal Comfort Vote, Thermal Sensation 292 Vote, and Perceived Activity Vote (PAV) obtained in our experiments (Fig. 12). PAV is introduced here to 293 assess the impact of the thermal environment on human activity and lifestyle, which is a critical factor indirectly 294 contributing to heat-related health outcomes. For example, an uncomfortable thermal environment can result 295 in less desire to perform physical activity, which further contributes to health challenges such as obesity, mental 296 health, and high blood cholesterol and pressure level. Identifying such links between the thermal environment 297 and activity level is, therefore, considered as one of the motivations and advantages of using activity-tracker 298 wearables in this study. Assessing the perceived activity vote enables us to not only analyze and predict 299 thermal comfort but also quantify the implications on health and wellbeing in the built environment.

300

In Fig. 12 (left), we observe that for the climate of Singapore, TSV corresponding to 'cool", "Slightly Cool", and built environment characteristics. For PAV, participants are asked to rank the level of activity that they perceive suitable based on the thermal environment. For example, "Extended/intense" activity vote indicates that participants are comfortable to perform intense activities or stay for an extended period in this thermal environment while "No activity" indicates that participants find this thermal condition extremely uncomfortable or unhealthy for any activity. Heat exposure has a wide range of adverse effects on the human body and is considered a public health hazard 316 [START_REF] Kovats | Heat stress and public health: a critical review[END_REF][START_REF] Bi | The effects of extreme heat on human mortality and morbidity in australia: 519 implications for public health[END_REF]. Additionally, thermal discomfort in urban spaces has been associated with loss in productivity, cognitive 317 performance, and wellbeing of individuals [START_REF] Zhang | Effects of moderate thermal environments on cognitive 376 performance: A multidisciplinary review[END_REF]. However, despite decades of climatological, epidemiological, and 318 physiological research on this topic, little is known about actual thermal conditions people experience as they transferred from the previous time point status T C,t-1 , which can be defined as:

Where A and A 0 are the weights learned by the linear regression of T C,t against T C,t-1 with the 15 seconds 366 time step. w is the transition model noise with a zero mean normal Gaussian distribution with covariance Q.

367

In this case, Q is the standard deviation of minute difference of T c .

368

The observation model was defined as a linear model of observed variables against the hidden variable T C,t .

369

Here we used heart rate (HR t ), skin temperature at wrist (T sw,t ), and air temperature at wrist (T aw,t ) as 370 inputs. The observation models of these two models can be represented as follows:

Where H and H 0 are the weight matrix learned by linear regression of T c against HR , T sw , and T aw . v 372 is the observation model noise with a zero mean normal Gaussian distribution with covariance R. R is the 373 covariance matrix of 15 second difference of HR, T sw , and T aw .

374

In our analysis, at each new 15 seconds time step (t), the KF provided a new estimate of T C,t and its error variance P C,t based on the observed HR t , T sw,t and T aw,t by iteratively calculating Eqs. 1 -6. First, a preliminary estimated T C,t was computed using Eqs. 1 -2. The associated error variance was calculated as

where the initial P t was set as 0 and the superscript T means the transposed matrix. The Kalman gain K t was then estimated by