Measuring mean radiant temperature for indoor comfort assessment using low-resolution optical sensors

Abstract

Measuring and controlling human thermal perception-related parameters within the built environment is crucial for ensuring occupant comfort, productivity, well-being, and reduced energy consumption. The human body is sensitive to both convective and radiative thermal effects. Mean radiant temperature represents the comprehensive radiant thermal impact individuals perceive in their surroundings. However, no feasible, robust, and ergonomic methods exist for real-time mean radiant temperature measurements in the built environment. In this paper, we introduce a method for measuring longwave mean radiant temperature utilizing low-resolution infrared temperature sensors. The approach utilizes projective transformations to derive surface temperature distributions from raw infrared thermal data. Our technique is tested in four diverse real-world environments, encompassing different heating methods and room configurations, resulting in a maximum error of ±0.5 °C. The results demonstrate the method’s repeatability and robustness across diverse room sizes, layouts, and scenarios, suggesting its potential integration into room thermostats to improve human comfort while optimizing building energy utilization. We anticipate that this method will revolutionize sensing in the built environment by eliminating the requirement for costly hardware.

This study introduces a method for real-time mean radiant temperature measurement, a key parameter representing the thermal comfort perceived by individuals. This method achieves ±0.5 °C accuracy in diverse environments, showing potential for integration into energy-efficient building systems.

Introduction

Buildings consume a significant amount of energy, accounting for 29% of total end-use energy consumption in the United States, and more than 40% of the residential and commercial building energy consumption is dedicated to providing thermally comfortable environments for occupants through heating and cooling applications^1–3. Thermal comfort depends on several factors, including the level of radiative and convective thermal interaction between the human body and its surroundings. The convective part of this thermal interaction is mainly related to the air temperature and humidity, while the radiative part is primarily driven by the surrounding surface temperatures^4–6. Measuring thermal comfort levels and controlling heating and cooling systems are vital in balancing substantial energy consumption with the occupants’ needs. Room thermostats have been employed to gauge thermal comfort levels and regulate the heating and cooling systems since the late 19th century^7,8. However, a fundamental limitation persists despite the technological advancements from traditional to modern smart thermostats since their inception: they measure only air temperature, neglecting the intricate effects of surrounding surface temperatures on human comfort. This oversight often results in comfort-related issues and inefficiencies in energy usage^9,10.

Mean radiant temperature (t̄_r) represents the holistic measure of effective surrounding temperature, which accounts for the combined effects of radiant heat exchange between surfaces and occupants¹¹. To accurately evaluate the comfort level of a space, it is essential to measure t̄_r in real-time. In addition to enhancing comfort levels, t̄_r -based control offers the added benefit of substantially reducing energy consumption, particularly in radiant systems such as floor heating and ceiling cooling. Radiant heating and cooling systems have gained attention in recent years for high-performance and net zero energy building design since they offer better energy efficiency or reduced operational costs compared to air-based systems^12–14. Radiant systems are designed to directly achieve t̄_r in a room by increasing or decreasing the surface temperature. Subsequently, heated or cooled surfaces activate convection. Thus, these systems require precise control based on real-time t̄_r measurement to achieve optimal efficiency⁹. However, the lack of a feasible and practical t̄_r measurement method for such automation purposes has been a critical barrier, restricting control of radiant systems to air temperature measurement and preventing them from fully optimizing operational efficiency.

Currently, measuring t̄_r requires expensive laboratory-type devices such as net radiometers or black globe sensors which provide an affordable option⁵. Historically, mean radiant temperature measurements relied on the black globe sensor, which has been used since the early 1800s to measure radiant heat exchange^15–17. The contemporary version of the hollow black globe thermometer and associated convective correction correlations have been used since the 1930s¹⁷. Yet, black globe sensors possess several limitations: they exhibit slow response times, have large physical dimensions, require supplementary air velocity measurements, and, as recent research has demonstrated^18,19, are susceptible to convective errors. On the other hand, net radiometers are expensive and directionally dependent, so their use for t̄_r measurement is limited to research studies only.

The black globe method is the most commonly used approach for estimating mean radiant temperature and the standard-sized black globes are reliable in controlled environments²⁰. However, in several cases, certain key parameters must be considered when estimating t̄_r using black globe measurements to ensure lower error margins. First, the geometry of the globe introduces a bias, as horizontal surface temperatures have a greater ratio of influence compared to vertical surfaces, impacting the sensor more than the human body⁵. This can lead to overestimating t̄_r, particularly in environments with radiant floor heating or ceiling cooling systems. Moreover, ISO Standard 7726 recommends using different colored globes for shortwave radiation, for example, medium grey or clothing absorptivity must be taken into account if a black globe is used. For the long wave radiation case, the standard requires a smooth black surface with circa 0.95 emissivity⁵.

Despite the extensive use of black globes over many decades and numerous studies addressing their performance^21–23, several key questions regarding globe diameter, material, and convective effects remain areas of ongoing research. Recent studies^20,24–27 have investigated the material and size of black globes, demonstrating that in a controlled black cylindrical enclosure, standard-sized globes (150 mm diameter) showed negligible errors²⁰. In contrast, smaller globes (38 mm or 50 mm dia.), which have faster response times, exhibited significant uncertainties under high longwave radiative loads (above 30 °C) in both natural and forced environments, largely due to potential inaccuracies in heat transfer modeling^24,25. However, these smaller globes showed lower measurement errors in moderate environments^24,26. Researchers have also been working on alternative materials for the globe. In black enclosures without shortwave effects, polymeric standard-sized globes have shown good agreement with metallic black globes²⁰. However, metallic globes outperformed polyethylene globes under shortwave radiation, providing better surface temperature uniformity²⁷.

Additionally, the metrological performance of different measurement techniques compared to the black globe has been subject of recent research. A study²⁸ comparing methods described in ISO Standard 7726 demonstrated that, as mean radiant temperature increases (with mean air velocity maintained at 0.1 m/s), the difference between globe measurements and other techniques—including contact surface sensors, thermal cameras, infrared thermometers, and net radiometers—also increases. Another recent research^18,19 comparing black globe measurement with radiometer measurements and suggests that the observed differences are due to convective effects, which standard correction correlations fail to fully address.

Alternatively, t̄_r in a space can be calculated if surface temperatures and corresponding areas and view factors are known⁶. In recent years, researchers have started to explore the use of optical sensors for t̄_r measurements in research settings^19,29–36. Many of these studies^29–34 use high-resolution thermal cameras or image-stitching methods. One technique employs mock target infrared (IR) thermography^29–31, which requires stereo calibration and auxiliary equipment such as mock targets. Others require multiple cameras or pan-tilt mechanisms to scan the surrounding surfaces^32–34. Two studies^35–37 used similar approaches requiring a single channel or linear thermopile array and a pan-tilt mechanism. One of them³⁷ coupled the temperature sensor with a Light Detection and Ranging (LiDAR) sensor to measure the distance simultaneously. The pan-tilt mechanism rotates the sensor periodically to measure the temperature of the surrounding points with the corresponding distance. Then, the algorithm creates a point cloud or calculates the surface temperature distribution. This technique may require a few minutes to complete one scan and additional hardware, namely a pan-tilt mechanism and distance sensors.

Using optical sensors is a promising approach to measure t̄_r, as a result, researchers^38–44 have been further investigating the capability of the above-mentioned methods, utilizing pan-tilt mechanism, in the indoor setting by coupling it pyranometer measurements or RGB/depth cameras as well as for several outdoor applications⁴⁵. In addition, one method⁴¹ requires a polished metallic hemisphere, a numerical analysis, and a high-resolution thermal camera. Another approach³⁹ employs depth camera and pan/tilt mechanism. Existing low-resolution sensor-based approaches^35–37 have intrinsic problems, namely extra hardware and require long scan time for one measurement in addition to surface emissivity and calibration-related issues. Moreover, researchers have also investigated measuring t̄_r with a spherical radiometric array⁴³, although this method provides information about radiant asymmetry, it is a point measurement. On the other hand, high-resolution thermal camera-based approaches are not feasible and unsuitable for automation purposes^33,34,38 due to the high costs of -research-grade- hardware and ergonomics limitations. Methods that use RGB or depth camera³⁹ require high computational cost and, more importantly, raise privacy concerns. Therefore, further study is required for a computationally efficient, feasible, and robust solution that provides instantaneous measurement capabilities. In particular, an optical t̄_r measurement method that uses low-resolution stationary sensor(s) and is capable of measuring surface temperature distribution with instantaneous measurement capability has not been reported yet.

In this work, we present a method for measuring indoor -longwave- mean radiant temperature using low-resolution and low-cost IR sensors, eliminating the pan-tilt mechanism requirement of the existing approaches. The surface temperature distribution over the walls and horizontal surfaces is derived from the 2D thermal data using geometric (homographic) transformations (Fig. 1). We have developed a simple and computationally efficient technique to obtain homography matrices by marking the corners of the observed walls. Here, we demonstrate the method using 32 × 32 pixel IR thermal array sensors by testing it in four different real environments for several cases across a range of mean radiant temperatures between 18 °C (64 °F) and 26.8 °C (80 °F). We examine configurations involving four optical sensors (one on each wall), two sensors, and a single-sensor setup. Ground truth t̄_r measurements are obtained via the net radiometer measurements using the plane radiant method as given in ISO standard 7726^5,46. The proposed method agrees with ground truth measurement within ±0.5 °C and outperforms traditional black globe sensors.

Fig. 1. Schematic of the overall workflow of the method.

a The optical sensor captures 2D thermal data; this data may contain temperatures of the adjacent walls, floor, and ceiling. b Surfaces are segmented by marking the corners, represented by letters A-H, and homography matrices are obtained. Colored trapezoids are the fields that the sensor sees on the sidewalls. c Homographic transformations are applied to obtain surface temperature distribution, and t̄_r is computed for the point of interest.

Results

Mean radiant temperature

Mean radiant temperature models the radiative heat transfer between the human body and its enclosure. It is “the temperature of a uniform, black enclosure that exchanges the same amount of thermal radiation with the occupant as the actual enclosure”⁴. In this study, mean radiant temperature refers to the longwave indoor mean radiant temperature only unless specified otherwise. Here, we calculated t̄_r using three different methods: (i) based on surface temperature measurements, (ii) net radiometer measurements, and (iii) black globe sensor measurements. We built our method using the first method and conducted the ground truth measurements over the net radiometer method. Black globe measurements are given here solely for comparison, as it is the most common and traditional way of measuring mean radiant temperature.

t̄_r measurement over surface temperatures

If the surface temperatures are known, one can calculate the mean radiant temperature using the following equation^5,11,

$${\overline{t}}_r={\sum }_{i=1}^N{\left(T_i^4{F}_{P-i}\right)}^{\frac{1}{4}}-273.15$$ 1

Here, N is the number of surfaces, T_i is the temperature of the ith surface in K, and F_P-i is the view factor between the person and the ith surface¹¹. Nomenclature is provided in Supplementary Table 2. This equation is obtained using the following assumptions: surfaces are grey, so emittance of the surface is equal to absorptance of the surface, all surfaces emit and reflect the radiation diffusely distributed, and surface emissivities are high. Since most building materials satisfy these conditions, this equation is reliable and has become a standard way of t̄_r calculation^5,11.

Equations (2) to (4) provide the view factors, and the required fitting coefficients are given in Table 1^5,47. Room surfaces are divided into segments based on the human location and the waist level, which is 1 m for a standing and 0.6 m for a seated person. The values of a, b, and c are determined accordingly. For horizontal surfaces, c represents the distance between the waist and the surface; for instance, c is 1 m for a standing person on the ground, while a and b are the dimensions of the divided segment of the horizontal surface aligned with the centerline of the human body. For the vertical surfaces, a and b similarly represent the dimensions of the divided segment of the vertical surface at the waist level. The distance between the centerline of the human body and the vertical surface is represented with c for the vertical surfaces. In this study, the ground truth measurement device is placed at the center of the room to verify the method.

$${\mathrm{F}}_{\mathrm{p}-\mathrm{i}}=F_{\max }\left(1-e^{-\frac{\frac{a}{c}}{\tau }}\right)\left(1-e^{-\frac{\frac{b}{c}}{\gamma }}\right)$$ 2

$$\mathrm{\tau }=\mathrm{A}+\mathrm{B}\left(\frac{\mathrm{a}}{\mathrm{c}}\right)$$ 3

$$\mathrm{\gamma }=\mathrm{C}+\mathrm{D}\left(\frac{\mathrm{b}}{\mathrm{c}}\right)+E\left(\frac{a}{c}\right)$$ 4

Table 1.

Coefficients for the field of view factor calculations (Eqs. (2) to (4))^5,47

	Fmax	A	B	C	D	E
Seated (vertical surfaces)	0.118	1.216	0.169	0.717	0.087	0.052
Seated (horizontal surfaces)	0.116	1.396	0.130	0.951	0.080	0.055
Standing (vertical surfaces)	0.120	1.242	0.167	0.616	0.082	0.051
Standing (horizontal surfaces)	0.116	1.595	0.128	1.226	0.046	0.044

t̄_r measurement over plane radiant temperature – ground truth

As outlined in ISO 7726, mean radiant temperature (t̄_r) can be calculated using plane radiant temperatures derived from radiometer measurements and projected area factors that account for human body geometry. A net radiometer measures both incoming and outgoing longwave and shortwave thermal radiation and gives the net thermal radiation at a point. It consists of longwave and shortwave thermal radiation measurement components, pyrometers, and pyranometers, respectively. Due to the directional dependence of the measurement technique, sensors need to be directed to all six cardinal directions. For a standing person, when the orientation is not known, the projected area factors are F_u\d = 0.06, for horizontal surfaces (up/down) and F_l\r\f\b = 0.22 for vertical surfaces (left/right/front/back). Mean radiant temperature can be calculated using the following equation⁵:

$${\overline{t}}_r={\mathrm{F}}_{u\backslash d}\left(t_{pr-u}+t_{pr-d}\right)+{\mathrm{F}}_{\mathrm{l\backslash r\backslash f\backslash b}}\left(t_{pr-l}+t_{pr-r}+t_{pr-f}+t_{pr-b}\right)$$ 5

In Eq. (5), the plane radiant temperature in the ith direction, t_pr-i (in °C), is given by the following equation⁵:

$$t_{pr-i}={\left[0.95T_n^4+\frac{P_i}{\sigma }\right]}^{\frac{1}{4}}-273.15$$ 6

In Eq. (6), P_i represents the incoming thermal radiation from the ith direction, measured by the radiometer in W m^–2, T_n is the radiometer temperature, and σ is the Stefan–Boltzmann constant (5.6704 × 10^-8W m⁻² K^–4). Form of this equation may vary depending on the radiometer output.

t̄_r measurement over black globe sensor

The black globe sensor is the traditional and most common method of t̄_r measurements. Under natural convection conditions, one can calculate t̄_r over a black globe sensor measurement using the following equation⁵:

$${\overline{t}}_r={\left[{\left(T_{gl}\right)}^4+\frac{0.25\times {10}^8}{ϵ}{\left(\frac{∣T_{gl}-T_{air}∣}{D}\right)}^{\frac{1}{4}}\left(T_{gl}-T_a\right)\right]}^{\frac{1}{4}}-273.15$$ 7

If there is an air draught, air velocity measurement is also required for the convective corrections⁵:

$${\overline{t}}_r={\left[{\left(T_{gl}\right)}^4+\frac{1,10\times {10}^8{V}_a^{0,6}}{ϵD^{0,4}}\left(T_{gl}-T_{air}\right)\right]}^{\frac{1}{4}}-273.15$$ 8

In Eqs. (7) and (8), T_gl is the black globe temperature in Kelvin, T_air is the dry bulb air temperature in Kelvin, V_a air velocity in m s^-1, ε is the emissivity of the black globe surface (0.95 for the matte black painted globe), and D is the diameter of the globe in meters (ISO 7726⁵ recommends 0.15 m diameter globe). We used the method with a standard-sized black globe sensor and standard correlations⁵ as a secondary comparison.

Projective transformations

Projective transformations extract surface temperature distribution from the raw thermal measurements. For projective transformation, transformed points in homogenous coordinates, ${\tilde{\mathit{x}}}^{\prime }$, can be calculated using the following equation:

$${\tilde{\mathit{x}}}^{\prime }=\tilde{\mathit{H}}\tilde{\mathit{x}}$$ 9

Here, the tilde, $~$, denotes the homogenous coordinates. $\tilde{\mathit{H}}$, is a 3 × 3 arbitrary matrix called the homography matrix, and $\tilde{\mathit{x}}$ is the coordinates before the transformation⁴⁸. Transformed homogenous coordinates must be normalized to obtain the results in inhomogeneous coordinates, ${\mathit{x}}^{\prime },$ using the following equations^47–49. We implemented the geometric transformations using the standard built-in functions of OpenCV⁵⁰, an open-source image processing library. Details are given in (Supplementary Eq. 1) through (Supplementary Eq. 5).

$$x^{\prime }=\frac{h_{11}x+h_{12}y+h_{13}}{h_{31}x+h_{32}y+h_{33}}$$ 10

$$y^{\prime }=\frac{h_{21}x+h_{22}y+h_{23}}{h_{31}x+h_{32}y+h_{33}}$$ 11

Experiments and data

We tested the method in four different representative real environments: three living spaces with different aspect ratios and an office space (Supplementary Figs. 1 and 2). The first room was an empty bedroom where we tested five different cases (Fig. 2): empty room, high-temperature heat source, human presence, short wave radiation, and a convective heat source. The second room was a furnished living room with two opposite exterior walls, while the third room was an empty living room with two perpendicular exterior walls. Lastly, we tested the system in an office space.

Fig. 2. Cases for the bedroom experiments.

a Empty room, b High-temperature heat source (electric radiator), c Human presence, d Short wave radiation (blinds are on), e Dominant convective heating.

The experimental spaces were selected to include varying sizes, aspect ratios, layouts, orientations, and window-to-wall ratios to ensure the method’s robustness across diverse real-world scenarios. The different aspect ratios, window orientations, and layouts, such as one exterior window, two exterior windows perpendicular to each other, and windows facing opposite directions, allowed us to evaluate the method’s performance under varying spatial geometries. By incorporating both furnished and unfurnished spaces, we aimed to examine how interior elements would impact the measurements. These varying properties -specifically size and aspect ratio- also allowed us to assess and determine the required minimum number of IR sensors needed to conduct accurate measurements with the method.

The selection of the five cases for testing in the bedroom was guided by the need to evaluate the method under a range of typical and challenging indoor conditions. Each case was carefully designed to include key factors that influence thermal dynamics in real-world environments. In Case 1, the room was left empty with closed blinds to serve as a baseline, eliminating external influences like solar radiation and internal heat sources, allowing us to observe the method’s performance in a basic setting. Case 2 introduced a high-temperature source in the form of an electric radiator, simulating a common heating scenario. This allowed us to examine how the method responds to localized heat sources and varying set points.

Case 3 incorporated human presence, allowing us to test the method under occupancy conditions. Case 4 focused on the impact of solar radiation by conducting experiments with open blinds during the day. Finally, Case 5 tested the method under conditions of dominant convective heating, which is a common scenario in offices and modern multifamily residences. Together, this case selection aimed to evaluate the method’s robustness and adaptability to different heating sources, human presence, and solar radiation, ensuring its effectiveness across a broad spectrum of indoor environments.

In each space, four optical sensors were mounted at the center of each wall. For the proposed method, we analyzed three different sensor combinations: a four-sensor (4CAM) solution, a two-sensor (2CAM) solution, and a one-sensor (1CAM) solution (Supplementary Fig. 3). The 4CAM solution utilized four optical IR sensors mounted at the geometric center of each wall, eliminating the need for extrapolations. The 2CAM solution utilized two sensors. In the rooms with a single exterior wall, one IR sensor was mounted on the exterior wall and the other on the opposite wall. In the cases where the rooms had multiple exterior walls, the 2CAM solution employed sensors mounted on the walls opposite the exterior walls. This configuration requires no extrapolation or minimal extrapolations to fill the unseen portions of the room. Finally, the 1CAM solution employed only one optical sensor located on the wall opposite the exterior wall. However, it should be noted that this configuration was utilized only in rooms with a single exterior wall and involved assumptions and extrapolations.

To calculate the ground truth t̄_r, we used Eq. (6) to calculate the plane radiant temperature for each direction, where P_i is the pyrgeometer measurement in the corresponding direction. The same equation was used to integrate the shortwave thermal radiation effects from the sun into the t̄_r; in this case, P_i represented the sum of the pyrgeometer and pyranometer measurements. For all experiments, except for the dominant convective heating case, we used Eq. (7) to calculate the t̄_r measurement over the black globe. Equation (8) was used for the convective heating case. In the Bedroom, the mean radiant temperature was also calculated over contact temperature sensor measurements attached to the walls, and measurements agreed with the ground truth within ±1.1 °C. The vertical partition of the surfaces was set at the height of 1 m to align with the placement of the ground truth measurement sensor.

Experiments were conducted over a range of mean radiant temperatures between 18 °C (64 °F) and 26.8 °C (80 °F). Throughout the experiments, the average difference between air temperature and the mean radiant temperature was 0.97 °C, while the maximum difference reached 3.5 °C. Sample results for each case of the bedroom experiments and samples for the other spaces are given in Fig. 3a–h. As can be seen, besides the simple cases, under cyclic heating (Fig. 3b) and incoming shortwave thermal radiation (Fig. 3d), the proposed method agrees well with the ground truth measurement. According to the results, the proposed method is capable of measuring mean radiant temperature with a maximum 0.96 °C error (±0.5 °C with calibration) with a ± 0.12 °C standard deviation (Fig. 3i). The maximum uncertainty of the method is calculated as ±0.5 °C. Detailed experimental matrices are presented in Supplementary Table 1, and the corresponding measurements for each space are given in Supplementary Fig. 4 through Supplementary Fig. 11.

Fig. 3. Sample experimental data and total errors.

a Bedroom – Case-1, Empty room b Bedroom – Case-2, high-temperature radiator c Bedroom – Case-3, Human presence d Bedroom – Case-4, Shortwave thermal radiation e Bedroom – Case-5, Convective heater f Living Room # 1 g Living Room # 2 h Office space i Errors for all spaces, E_4CAM, E_2CAM and E_{BlackGlobe-SW} are for all spaces, E_1CAM is only for the rooms which have only one exterior wall. The 6-min averaged data of the proposed method is used to calculate errors and match the ground truth method sampling interval. Mean values are shown with filled circles. E_4CAM: sample size (n) = 2905, mean: 0.46, max 0.95, E_2CAM: n = 2905, mean: 0.57, max 0.92, E_1CAM: n = 1098, mean: 0.65, max 0.96, E_{BlackGlobe-SW}: n = 2905, mean: 0.06, max −1.97. Source data are provided as a Source Data file.

Measurement errors for the black globe, E_{BlackGlobe-SW,} is determined by comparing them with the mean radiant temperature measurements that included shortwave measurements, where P_i is the summation of the pyrgeometer and pyranometer measurements. The maximum error for the black globe is observed as –1.97 °C with a 0.2 °C standard deviation when the outliers are counted (Fig. 3i).

Discussion

This study introduced a feasible, and robust indoor mean radiant temperature measurement method. The main advantages of the introduced method over the other recent approaches^35–37 are the ability to conduct instantaneous measurement, computational efficiency, and elimination of requirements for auxiliary hardware, namely distance sensors and pan-tilt mechanisms. The method was tested in four different representative, furnished, and unfurnished real environments over a range of mean radiant temperatures between 18 °C (64 °F) and 26.8 °C (80 °F) for 290 h. Three different sensor configurations were analyzed to assess the minimum number of optical IR sensors required: four, two, and one sensor. The method yielded a maximum error of ±0.96 °C with the one-sensor configuration. Circa 0.5 °C systematic error is observed, and when that offset is calibrated, errors reduce below ±0.5 °C in all three cases. It should be noted that the one-sensor solution was only applied to experimental spaces with one exterior wall. It yielded comparable results to the two- and four-sensor configurations in these spaces. This layout type is quite common for many residential and office spaces. Thus, these reliable results obtained from the 1CAM configuration are significant.

We also compared our method with the traditional and most common t̄_r measurement technique, black globe measurements. Experimental results demonstrate that our method significantly outperforms the black globe technique, which yielded a maximum error of -1.97 °C. The measurement uncertainty for our method is ±0.5 °C while for the black globe method, the propagated uncertainty is ±0.8 °C according to the GUM^51,52. Results show that the difference between the black globe measurement and the ground truth increased as incoming solar radiation increased. Thus, the errors associated with the black globe method are largely due to underestimation or overestimation of solar effects or limitations in the convective correlations, as discussed earlier. Additionally, the slow response time of the black globe method increases the time lag-related errors between the globe temperature and the ground truth method, as the latter had a faster response time.

Furthermore, we analyzed the error margins across different spaces and investigated the potential causes. In each space, the maximum error levels were within a similar range, above 0.69 °C, indicating that the method produces consistent measurement errors across various room layouts. For the bedroom, the minimum and maximum errors were 0.23 °C and 0.89 °C, respectively, while for living room #1, these values were 0.48 °C and 0.95 °C. In living room #2, the minimum and maximum errors were 0.08 °C and 0.69 °C, and for the office space, they were 0.18 °C and 0.81 °C. In the bedroom experiments, all cases yielded similar error ranges between 0.6 °C and 0.9 °C. We observed that the difference between the proposed method and radiometer measurements slightly increased with rising temperatures in all cases, suggesting that the radiometer has a slight temperature dependency, consistent with the calibration experiments. The results also indicate that room dimensions may be a factor to consider, as larger areas are represented by fewer pixels as the distance increases.

Computational and hardware costs are important considerations. The sensors used to demonstrate the method cost approximately one-quarter of the price of commercial black globe sensors (excluding the measurement device and anemometer) and about one-twentieth, or less, of the cost of a typical net radiometer, depending on the radiometer’s configuration. From the computational cost perspective, our method requires 5.6E + 04 floating-point operations (flops), a common metric to represent computational complexity⁵³, for the position-based calibration procedure and 1.2E + 05 flops per measurement to obtain temperature distribution using when used 32 × 32 pixel IR sensors, that results in a total of 6144 pixels for the temperature distribution of the space. On the other hand, the single channel pan/tilt mechanism approach³⁷ requires ~4.4E + 05 flops in the optimized case to obtain temperature distribution using measurements from 4841 points. This calculation even excludes the computations required for the physical control of the pan/tilt mechanism. Compared to the single channel pan/tilt mechanism approach³⁷, our technique requires at least 4.5 times less computation to provide surface temperature distribution in a space for the same resolution.

There are a few potential limitations of the method that we would like to highlight. First, as mentioned earlier, the method only captures longwave mean radiant temperature. Shortwave effects can be incorporated through pyranometer measurements or estimated using the methodology outlined in ASHRAE Standard 55. Second, mean radiant temperature can vary across space due to spatial effects, such as hot and cold surfaces. The method captures the surface temperature distribution of the entire space and accounts for the distance and exposure effects over angle factors. Therefore, it can estimate mean radiant temperature distribution in the space, as it is not a point measurement, unlike the black globe or net radiometer. Provided error margins are validated for the center of the spaces; however, the method is able to incorporate the distance-related effects, as shown in a recent simulation study⁵⁴ that compared the plane radiant method, the angle factor method and the averaging methods at both the center of the room and near the window. The results showed that the two methods align well by capturing the distance-related effects in both locations during the hottest day and time interval, while simple averaging deviates from these two methods. Third, the method inherits the characteristic limitations of IR measurement techniques, meaning distance and surface emissivity should be considered. Nonetheless, as discussed earlier, Eq. (1) is developed for high emissivity surfaces and most building materials satisfy that condition. Lastly, the method relies on a proper location-based calibration process, which must be carried out accurately. For future studies, the measurement interval for the ground truth technique could be reduced from 6 to 1 min if a pan-tilt mechanism is used. Alternatively, six pairs of radiometers could be employed to increase the sampling rate. It is important to note that this is expected to reduce the errors caused by the time lag between the proposed method and the ground truth method. Additionally, higher-resolution sensors could be used to enhance accuracy and quantify uncertainties related to low resolution. Testing different sensor layouts could also help minimize distance-related errors. In this study, we presented the method in its most basic form to demonstrate its lower limits. In the future, machine learning and other advanced algorithms could be integrated to improve the location-based calibration procedure, potentially reducing uncertainties.

Results show that this method is robust and gives repeatable, scalable, and predictable results in various real environments for different heating system scenarios. It can be used in field studies for t̄_r measurements. We conducted tests on rooms measuring up to 7.75 m in dimension. For larger rooms, different sensor layouts might be considered. Since the method can capture temperature distribution in space, other useful information, namely thermal losses from the surfaces, can also be extracted from the captured data for the field studies.

It can be anticipated that this method is promising for room thermostat applications, especially for controlling radiant heating and cooling systems, which are now a popular choice for high-performance buildings.

Methods

Developed method

The developed method captures 2D thermal data using IR thermal array sensors and then processes it to extract the necessary information for Mean Radiant Temperature (t̄_r) calculation. Depending on the position and orientation of the sensor, this 2D thermal image may contain temperature data for multiple walls. Suitable geometrical transformations are applied to establish a mapping between the temperature distribution information (represented as pixels or matrix cells) and the corresponding walls. Such transformations can be applied using homography matrices.

The computation of homography matrices during the installation stage is required for the transformations. For this, at least four points per wall between the room geometry and the thermal image need to be matched. Then, the homography transformation matrices are computed using the OpenCV library. This process is called ‘position-based calibration of the optical sensor’ since it relies on the sensor’s location and position (Supplementary Fig. 13).

To achieve this, optical sensors are first installed on the walls, and then small silicon electric heaters are placed on the corners of the walls to mark the corners in the thermal data, as shown in (Supplementary Fig. 12). The dimensions of the spaces were provided in the form of scaled drawings. Next, the corners of the scaled drawings and the marked thermal data are matched manually to segment the horizontal and vertical surfaces seen by the sensors (Fig. 1b).

Following the position-based calibration step, the method can continuously measure the t̄_r by following these steps: optical sensors capture the 2D raw thermal data, then homography transformations are applied to the raw thermal data to map the temperature distribution to the horizontal and vertical surfaces as schematically shown in Fig. 1c. Finally, t̄_r is calculated for the position of interest in the space using Eq. (1).

Data acquisition setup

We used four OMRON D6T−2L-01A IR thermal array sensors (±0.5 °C (avg temp.), 90° field of view) to capture the raw temperature measurements for the proposed method. Each sensor was connected to an OMRON sensor evaluation board (2JCIE-EV01-AR1), which in turn was connected to an Arduino MKR Wi-Fi 1010 (ABX00023) board. This Arduino board featured an onboard Wi-Fi module for data transmission. The collected data is pushed to the ThinkSpeak® IoT server over Wi-Fi and stored there. The IR thermal array sensors used in this study (Omron D6T-32) have a 90° field of view angle in both horizontal and vertical directions.

The net-radiometer method was used as the ground truth measurement. A pyrgeometer (Apogee Inst. SL-510-SS, ±0.5 °C or ±5%) and a pyranometer (Apogee Inst. SP−420, ±3%) were used to measure longwave and shortwave thermal radiations, respectively. Due to the directional dependency of the measurement technique, these sensors need to be directed to all six cardinal coordinates. To accomplish this, we developed a custom-made pan-tilt mechanism using servo motors. The pyranometer and the pyrgeometer were mounted on the pan-tilt mechanism, which periodically rotated the sensors to all six cardinal coordinates every 30 s. This mechanism was controlled by an Arduino board, which also logged the directions of the sensors to the cloud. In the post-processing, timestamps of sensor directions and the corresponding measurements were matched (Supplementary Fig. 15). The pyranometer had a digital output over USB and was directly connected to the computer, while the Pyrgeometer was connected to a datalogger (Campbell Scientific, CR300).

For the measurement of mean radiant temperature over the black globe sensor, we used a standard-sized metallic 15 cm (6 in) diameter black globe sensor with a DS18B20 temperature sensor (Maxim Integrated DS18B20, ±0.5 °C) and a hot wire anemometer (Testo – 0635 1032, ±0.03 m s⁻¹) with a Testo 440 measurement device (Testo 440 - 0560 4401). Data acquisition devices are given in Table 2.

Table 2.

Measurement device specifications

Sensor
Contact surface temperature measurement	Maxim Integrated DS18B20, ±0.5 °C
Reference temperature measurement device	OMEGA HH42A, ±0.02 °C
Reference temperature sensor	OMEGA ON-403-PP, ±0.1 °C
Pyregeometer	Apogee Inst. SL-510-SS ± 5%
Pyranometer	Apogee Inst. SP-420, ±3%
Black globe	Standard sized globe (Rust-Oleum 7220830) with DS18B20, ±0.5 °C
Hot wire anemometer	Testo – 0635 1032, ±0.03 m/s–1
Thermal array sensor	OMRON D6T-2L-01A, ±0.5 °C (avg temp.), 90° FOV
Omron Sensor Evaluation Board	2JCIE-EV01-AR1
Open source board	Arduino MKR Wi-Fi 1010 (ABX00023)
Datalogger	Campbell Scientific, CR300

Sensor calibrations

Optical sensors were calibrated using three heated plates (black, gray, and white), each of which had contact temperature sensors attached to its surface (Supplementary Fig. 14). For surface temperature measurements, Maxim Integrated DS18B20 (±0.5 °C) digital temperature sensors were used, which had a 0.0625 °C resolution with up to 12-bit integrated analog-to-digital converter (ADC). We calibrated all the contact temperature sensors, air temperature sensors, and the black globe temperature sensor using the five-point calibration method using a NIST-calibrated ultra-precise temperature measurement device (OMEGA HH42A, ±0.02 °C) with a thermistor probe (OMEGA ON-403-PP, ±0.1 °C).

We prepared three aluminum plates, each measuring 20 cm × 20 cm, and painted them black, gray, and white, respectively. On the backside of each plate, we placed eight 4 cm × 10 cm silicon pad heaters to achieve a homogeneous surface temperature. Contact temperature sensors were attached to the surface of the plates using painted aluminum tape with the same paint as the surface. The optical sensors were placed 2.5 cm from the plate to limit their measurement with the 5 cm region at the center of the plate to avoid any possible nonuniform temperature distribution around the edges. During the calibration experiments, the plates were heated up to 45 °C and then allowed to achieve equilibrium with the ambient temperature during the calibration experiments. These experiments were repeated for all three plates and each optical sensor. We observed that the averaged temperature obtained over each calibrated optical sensor agreed with the contact measurement sensor, which had ±0.5 °C accuracy.

Experiments

Experiments were conducted in four representative real environments: three living spaces with different aspect ratios and an office space. Four optical sensors were used in each space, and the sensors were mounted on the center of each wall. Supplementary Fig. 1 shows the layouts of the spaces, and Supplementary Table 1 gives the experimental matrix.

Bedroom

The first room was a typical 3.2 m × 3.2 m × 2.55 m (height) bedroom in a 1960s townhouse building. It had one exterior wall facing southwest and two identical 1 m × 1.2 m windows with regular white vinyl blinds. The room was painted in regular white/beige, and a low-temperature radiator was positioned alongside the exterior wall (Supplementary Figs. 1a and 2a). Five cases were examined in the bedroom to test the method under varying conditions, as shown in Fig. 2.

Case 1: Empty room - The room was empty, and blinds were closed to prevent/reduce incoming solar radiation.

Case 2: High-Temperature Source (electric radiator) - The room was empty, but an electric oil radiator was placed in front of the exterior wall at the horizontal center. The surface of the radiator could reach up to ~70 °C (158 °F), and it had an on-off control with an integrated thermostat, allowing the following set points: 18, 21, 24, 27 °C (65, 70, 75, 80 °F).

Case 3: Human presence - A human subject was sitting in the room in a stationary position.

Case 4: Incoming shortwave radiation - The room was empty, and experiments were conducted during the day with open blinds to test the effects of the solar radiation coming from the windows.

Case 5: Dominant convective heating - A convective electric heater was placed in front of the exterior wall (at the horizontal center) to test the method under conditions with convective heating dominating.

Living Room #1

This was a furnished living room with two exterior walls on opposite sides, facing southwest and northeast. The aspect ratio of the room was 1:2. Supplementary Figs. 1b and 2b show the sensor layout and a view of the room. In this room, blinds were kept off in all experiments, and only the regular and high-temperature radiator cases were tested.

Living Room #2

This space was another living room with two perpendicular exterior walls facing northwest and southwest. The room’s dimensions were 3.5 m × 5.6 m, with an aspect ratio of ~3:5. The room was unfurnished, and only this case was tested. Supplementary Figs. 1c and 2c show the sensor layout and a view of the room.

Office space

This room was a typical office space with 6 m × 7.75 m dimensions. It contained five cubicles, and the window-to-wall ratio was higher compared to the previous spaces. Supplementary Figs. 1d and 2d show the sensor layout and a view of the room.

Data processing

The raw IR thermal array sensor measurements were logged every minute. The pyranometer and the pyrgeometer measurements for the net-radiometer method were logged every second. The pan-tilt mechanism for the net-radiometer method completed one cycle every six minutes. It rotated to each cardinal direction and remained in that direction for 30 s. After completing the rotation, it returned to the front (exterior wall) direction and remained in that position for 3 min. Supplementary Fig. 15 demonstrates the typical thermal radiation measurement cycle over the pan-tilt mechanism after segmentation, where each color represents measurements from a specific direction.

In the post-processing, to avoid the potential mismeasurements due to the rotational movement of the sensors, only the middle 2/5 portion of each step was taken into account. This portion of the measurements was averaged and extrapolated over the corresponding cycle for each direction.

Author contributions

F.E. conceptualized the idea, developed the method, conducted the experiments, analyzed the data and developed the original manuscript. S.B. and R.G. provided supervision. All authors contributed to the scientific discussions and were involved in the writing of the manuscript.

Peer review

Peer review information

Nature Communications thanks Jae-Hun Jo, and the other, anonymous, reviewer for their contribution to the peer review of this work. A peer review file is available.

Data availability

Data that support the plots within this paper are provided in the Source Data and 10.5281/zenodo.14174178. Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-55122-z.