Statistical Evaluation and Development of General Thermal Comfort Equations for Naturally Ventilated Buildings in Humid and Dry Hot Climates

Abstract

Thermal comfort has become an important element in the design, construction, and remodeling of buildings, as well as in the understanding of human behavior that considers inhabitants’ mental conditions. The objective of this study is to propose general thermal comfort equations via a rigorous statistical evaluation for regions with hot-humid (HH) and hot-dry (HD) climates. From the information on thermal comfort provided in the specialized literature, 17 equations were gathered for hot-humid climates and nine equations for hot-dry climates. These equations were developed for hot climate conditions in zones where buildings operate with natural ventilation (NV). The statistical analysis includes the normality test of the data distribution, the recognition of outliers, and the applications of significance tests for the comparison of the equation parameters. The equations proposed in this paper can be used to initially evaluate the thermal comfort of buildings in locations where no equations have been developed locally, as well as in the energy evaluations of buildings. The equation proposed in this paper for the hot-humid climate is the following: T_c = 16.9 (±2.19) + 0.37 (±0.08) T_out; that for the hot-dry climate is: T_c = 12.9 (±5.36) + 0.56 (±0.20) T_out. These equations can be used to initially assess the thermal comfort of buildings in locations without locally developed equations, as well as in building energy assessments.

1. Introduction

Thermal comfort has been defined as a mental condition of satisfaction with the thermal environment [1]. This phenomenon has been the object of study from different perspectives, methodological currents, and individual needs. Occasionally, these evaluations have considered subjective aspects (for example, psychological, habitational, aesthetic, etc.). Altogether, all factors are important in users’ perceived satisfaction with built spaces and are directly related to energy consumption [2], due to the negative effects of rapid urbanization and global warming that have also influenced the deterioration of air quality [3]. Specifications to achieve thermal comfort in buildings with several features have been proposed [4,5].

These studies have allowed the development of useful points to consider in new construction projects with a low energy consumption approach [6,7], while considering thermal comfort as a decisive social need [8].

Several general models of thermal comfort have been documented and have gradually participated in establishing the norms issued for standardization systems [9]. These approaches have been developed for many climates and types of buildings, with the most relevant models being the following [6,10,11]: (a) Fanger’s static model, (b) the European adaptive comfort model, (c) the American adaptive comfort model, and (d) bioclimatic Givoni charts. These models have been used to evaluate indoor hydrothermal conditions and minimize thermal discomfort.

In addition, particular models have been developed to stimulate thermal comfort considering the climate and its variations in a specific location, for example [2,12,13,14,15,16,17,18,19,20,21].

Parkinson et al. (2020) indicate the need for specific models for each region as there is no universal model. The authors state the necessity to develop models for regions with hot-dry (located at latitudes from 15° to 55° in both hemispheres) and humid (distributed from the equator up to the tropics) climates [22]. To achieve thermal comfort in tropical regions is even more significant due to extreme weather conditions and elevated internal and external heat gains [23].

Similarly, there have been several case studies where the physical characteristics of the built environment (hospitals, schools, offices, houses, hotels, and theaters) and the type of ventilation (A/C, NV, and mixture) have been as relevant as the climatic conditions [24,25,26,27,28,29,30,31,32]. To date, the adoption of thermal comfort norms has been voluntary. Moreover, national legislation does not impose a specific model to establish set points for buildings’ energy systems [32]. However, these charts have undergone an important development in the last 15 years [9].

Usually, the charts have been used to provide a reference for the indoor temperature in summer and, occasionally, in winter. In this way, the thermal condition in buildings is an alternative for the mitigation of the energy emergency associated with climate problems [9].

In fact, the development of thermal comfort equations is a fundamental element in the design or remodeling of energy saving buildings [33]. De Dear and Brager [34] concluded that temperature is the dominant variable in the definition of thermal comfort conditions. There are several linear equations that link the outdoor temperature (T_out) and the comfort temperature (T_c) in buildings (Equation (1)):

T_c = a + bT_out

(1)

where a represents the intercept of the line and b is the slope of the line. However, there are no universally accepted parameters for coefficients a and b [6,16]. If in Equation (1) we assume that T_c = T_out, the equilibrium temperature (T_eq) (Equation (2)) can be calculated. This temperature is important in order to determine the limits between two zones. In the first zone, when T_out > T_eq, an artificial air conditioning system is required. In the second zone, when T_out < T_eq, comfort can be achieved with strategies such as adequate clothing or thermal insulation.

$${\mathrm{T}}_{\mathrm{eq}}=\frac{\mathrm{a}}{1-\mathrm{b}}$$

(2)

In this context, this study posed the following objectives: (a) document the existing thermal comfort equations in the specialized literature; (b) statistically evaluate parameters a and b associated with thermal comfort equations in the literature; (c) determine if the proposed equations for locations with hot-humid and hot-dry climatic conditions can be integrated in a unified model; and (d) analyze the possibility of established general thermal comfort equations that govern hot climates. These equations could be used as a primary approach in areas where there is no thermal information and for naturally ventilated buildings.

2. Brief Literature Review

Research related to thermal comfort has developed progressively in the past 50 years [23]. Thermal comfort takes a position as a link between a wide spectrum of disciplines [35]. However, from this wide spectrum, three axes can be deduced on which thermal comfort has developed substantially (Figure 1).

Thermal comfort and health have been linked since the developments and postulates of P.O. Fanger. The World Health Organization (WHO) has established that the feeling of satisfaction with temperature is directly linked to the quality of health [36]. Accordingly, thermal comfort must be studied to guarantee satisfactory sensations. The comfort indicators commonly used to evaluate the perception of thermal comfort include the predicted mean vote index (PMV), which was created by Fanger and has been in force for more than 50 years. PMV is defined as the index that predicts the thermal sensation averaged on a standard scale for a group of people and for any given combination of environmental variables of temperature, activity, and types of clothing. This prediction is made by means of votes. Another indicator used and directly related to PMV is the predicted percentage of dissatisfied (PPD). This index is used to predict the number of people dissatisfied with their thermal environment and was proposed with the aim of estimating the number of people who are likely to become uncomfortable with heat or cold; this approach proposes a quantitative prediction of the percentage of these people [37,38]. Some researchers have proposed alternative indicators, such as the need for heart rate [39] and the actual mean vote (AMV), to characterize thermal comfort in tropical areas based on the behavior and psychology of the occupants [40].

2.1. Health and Thermal Comfort

Mendes et al. (2015) [41] studied indoor air quality and thermal comfort in care centers for elderly people. Among their main findings, they found that air temperature, relative humidity, type of air ventilation source, and density of the occupants can influence in the abundance and transmission of pathogen microbes. They made it clear that old or poorly maintained air conditioning, as well as low ventilation rates, can increase these risks.

In recent years, thermal comfort and health in hot climates have been linked in an exceptional way, especially in health workers. These workers are at constant risk of contagion from coronavirus disease (COVID-19). However, an additional danger is related to variable hot conditions and humidity due to working with personal protective equipment [42]. In these cases, it is imperative to develop ventilation strategies; new models of thermal comfort are required to reduce the negative impact of hygrothermal changes, especially in locations with hot, humid, and dry weather, as these are the regions where resources to battle thermal stress are the scarcest. Ding et al. (2020) [43] highlighted that the number of collective coronavirus (COVID-19) infections intensifies in closed spaces. Similarly, the outbreaks of pneumonia caused by this virus have created a significant concern due to the high rate of infection and minimal risk controls, especially in emerging countries.

Another important topic is mental health. Cleary et al. (2019) [44] investigated the impact of exterior temperature on mental health and life quality of elderly people living in care centers. Among the most significant results, the following stand out: (a) the importance of individual autonomy, (b) mobility inside and outside the built space, and (c) freedom of choice. These are due to the fact that geriatric care personnel are the ones who have control over the ventilation and air conditioning systems, directly undermining the residents.

2.2. Energy and Thermal Comfort

Approximately 40% of CO₂ emissions originate from building construction and operation, and approximately 50% of residential and office building operation consumption is due to excessive demand on HVAC systems [45].

Thermal comfort and efficient use, saving, and disposal of energy are linked directly; various systems have been built that combine thermal comfort and energy to reduce excessive consumption and maintain the thermal conditions that users require. Ku et al. (2015) [46] designed a method based on wireless sensors that quantifies temperature, air speed, and relative humidity around the occupants. The data are evaluated based on a predicted mean vote (PMV) model to provide self-adjusting control of the thermal comfort temperature and keep energy consumption at the necessary levels according to the demand of the occupants.

On the other hand, Ref. [47] evaluated the thermal conditions of 69 apartments in Japan. The results of this investigation indicate that people are well adapted to the thermal conditions of their homes and that the comfort temperature has seasonal variations. Nevertheless, the design of buildings adaptable to changes in temperature and the people’s adaptative thermal comfort are important approaches for the design of energy-saving buildings. The latter criterion was irrelevant for the users who participated in the analysis.

Panraluk and Sreshthaputra (2019) [48], through their research, determined that guidelines can be developed and applied in Thailand to improve the adequate indoor environment for the elderly (who comprised their analysis sample) in the hot season and to analyze the energy usage. According to the data recovered, it was determined that during the period of 8:00 a.m. to 12:00 p.m., natural ventilation should be used along with fans that produce an actual air velocity of 0.64 to 0.73 m/s. From 12:00 p.m. until 4:00 p.m., the air conditioning must be adjusted to 26.00–26.50 °C with an actual air velocity of 0.06–0.22 m/s.

The results also showed that the developed guidelines could improve the level of thermal comfort from “slightly cold” to “neutral” and reduce the use of energy in the hot season by 16.56% due to the reduction in the cooling load and the fan operation of the air conditioning systems. In fact, these findings can be applied as guidelines to improve many buildings in Thailand.

Ramírez et al. (2020), through their review [49], indicate that several results demonstrate the link between heat gain and comfort temperature in built spaces, making it clear that the disproportionate use of mechanical refrigeration systems is a significant concern due to excessive electrical consumption. Therefore, it is relevant to know the alternatives that can be used in hot climates to minimize electricity consumption and provide thermal comfort. To this effect, it is necessary to consider the type of building, the activities the users carry out, and the kind of clothes worn by users, and to orient the built space to the criteria of sustainable development and energy efficiency.

2.3. Buildings and Thermal Comfort

The construction of modern buildings, and even the architectural renovations of existing buildings, maintain a close link with thermal comfort, and under an ideal scheme they also maintain it with the efficient use of energy [33]. As a result, the diverse passive techniques of architecture, as well as the use of organic change phase materials, orientation of built spaces, and the multiple HVAC solutions available on the market for housing, have improved significantly and can directly contribute to adding value to the property and minimize the thermal effects that alter the sensations of the users [50].

Gagnon et al. (2018) [51] established that the process of building design must be modified globally. This concept must be undertaken under a comprehensive design scheme to create sustainable buildings. Thus, multiple performance variables must be considered to achieve optimum energetic consumption and comfort in the spaces that causes satisfaction among users. Similarly, passive architectural techniques must be included from the design phase [49]. In contrast, Gou et al. (2017) proposed a route for multi-objective optimization [52] and passive techniques for the new constructions in Shanghai, China, with a special focus on improving indoor thermal comfort and reducing the energy demand of residential buildings. However, the implementation of passive designs or strategies is sensitive to the local climate and topology, as well as to the building design, the façade, envelope materials, and landscape design. It is due to these factors that different potential combinations of optimum solutions need to be developed for various regions.

In fact, several studies focused on testing the thermal effects of new building materials for different orientation configurations and variable climatic regions have been conducted. Alpuche et al. (2014) [53] investigated the use of colors with low solar absorptive coefficients in the construction of exterior walls. The results indicate a reduction in heat gain in the building can be achieved and, as a direct effect, the need for thermal comfort can be satisfied without making excessive use of mechanical cooling systems in a dry hot climate. Ramírez et al. (2020) [49] identified that those passive techniques that minimize heat gain due to external phenomena, and those that reduce heat gain due to internal phenomena, are two different segments, the latter including the use of energy-consuming equipment, user activity, and basal metabolism. Given this, they suggest that, at least in hot-humid and hot-dry climates, the use of natural ventilation techniques, air–ground heat exchangers, and solar control films should be encouraged in order to reduce the dependency on mechanical cooling systems.

Thus, several techniques have been studied and tested, and some are already commercially available. Still, it is necessary to establish guidelines to orientate existing buildings to low energy criteria, having as an ideal model the near zero energy building (NZEB). Although it is true that to design and build these types of buildings is still a challenge due to their structural complexity in terms of thermal comfort, it is necessary for the well-being of global society to look for techniques that achieve an adequate thermal comfort, and to migrate energy consumption habits to low levels [54].

3. Materials and Methods

To achieve the goals proposed, the following work scheme was carried out:

• A database was elaborated compiling the parameters (slope, intercept, equilibrium temperature, and applicability interval) of thermal comfort equations for hot-humid and hot-dry zones. Two conditions were considered relevant for an equation to be included: the type of climate is specified, and the equations are calculated using data gathered in spaces (no matter the type) with natural ventilation (NV) or free running (FR). This database enabled understanding of the chronological development in terms of thermal comfort, and served as a repository of data regarding construction parameters of adaptive models and case studies.
• The Shapiro–Wilk test for normality was applied to the parameters of the models for hot-humid and hot-dry climates, and the significance level was α = 0.05. The relevance of applying this test is to determine whether the parameters of the comfort equations follow or approximate a normal distribution.
• Outliers were identified and eliminated through discordance tests for univariate samples that tended to be normal. Deviation/extension, Grubbs, Dixon, and high-order moments tests [55] were applied. Considering the rigor of any statistical analysis, it was necessary to use these tests to rule out outliers, as well as to characterize what arises from them.
• The mean and variance of the parameters associated with humid and dry climates were determined and compared using the F-test (Fisher test) and Student’s t-test. These tests were conducted using 5% and 1% significance levels.
• The zones with the highest probability of transit for the comfort equations by type of climate were determined. The authors consider it significant to graphically denote the trend and the points of incidence of the parameters that make up each equation; this allows determination of the differences in the plane between HD and HH climates.
• Average thermal comfort equations were proposed for hot-humid and hot-dry climates, including buildings with adequate natural ventilation or free running.

A graph showing the conditions and steps of the analysis is presented in Figure 2.

For the development of this paper, it was necessary to use three well-known statistical tools:

(a)

Shapiro–Wilk normality test

This test is applied to data that are hypothesized to be normally distributed. To apply this test, the number of data is required to be less than 50. Details on the procedure for applying this test can be found in Razali and Wah (2011) [56].

(b)

Determination of deviated data for normal distribution

There are multiple tests to detect deviated data in a sample that is supposed to be normal. The work of Barnett and Lewis (1994) [57] provides the details for the application of said tests, their statistics, and their tables of critical values. Verma (2020) [58] updated the tables of critical values, determining them for values of n of great magnitude.

(c)

F-test and Student’s t-test, for comparison of samples

These tests were used to compare whether two samples belong to the same universe. Details of the procedure for the application of these tests can be consulted in [58].

4. Results

4.1. Database

The database included 17 hot-humid cases that were examined (Table 1). The first equation was reported by [12] for Pakistan, while the most recent equation was from Lopez et al. (2019) for Mexico [32]. It should be said that, generally, the parameters are reported without including their associated uncertainty or the validation of the regression model. Furthermore, data from nine models of hot-dry climates were gathered (Table 2). The oldest model is the one proposed by [59] for the area of San Francisco, California. The most recent model is the one proposed by [60] for residential buildings in Libya. As in the previous case, the information of the regression models is limited. In Table 1 and Table 2, the equilibrium temperature according to Equation (2) is indicated. In the hot-humid climate, the equilibrium temperature ranges from 21.47 to 32.09 °C; while in the hot-dry climate, the equilibrium temperature ranges from 20.04 to 34.29 °C.

4.2. Evaluation of the Normality in Population Samples

The common procedures to evaluate whether a random sample of independent observations comes from a normally distributed population are graphical methods, numerical methods, and normality tests. The latter includes the Shapiro–Wilk test, which originally was restricted to a sample size of less than 50 [56]. The W statistic of this test has shown a greater sensitivity to non-normality; in comparative studies on statistical tests it has been determined that this test has better power than others, such as the Anderson Darling and Kolmogorov–Smirnov tests [61].

As indicated, the parameters of slope and intercept of the linear thermal comfort equations are grouped according to the type of climate where they are used. The Shapiro–Wilk test with a significance level of α = 0.05 was applied to these parameters. Then, parameters with p-values higher than α are considered to follow a normal distribution (Table 3). In both cases, the evaluation demonstrated the normality of the two population samples.

Table 1. Parameters of the equations of thermal comfort in a hot-humid climate.

Reference		n	a 1	b 1	Temperature of Equilibrium Teq (°C)	Applicability Range for Tout (°C) 1
Nicol and Roaf (1996)	[12]	4927	17.00	0.38	27.42	[5, 35]
Humphreys and Nicol (1998)	[62]	n.a. 2	11.9	0.534	25.54	Undefined
Nicol et al. (1999)	[13]	n.a. 2	18.5	0.36	28.91	[5, 35]
de Dear and Brager (2002)	[14]	n.a. 2	17.8	0.31	25.80	[10, 33.5]
Bouden and Ghrab (2005)	[16]	200	10.35	0.5179	21.47	[8, 35]
Zain et al. (2007)	[63]	n.a. 2	17.6	0.31	25.51	Undefined
Rijal et al. (2009)	[26]	601	15.4	0.516	31.82	>10
Nguyen et al. (2012)	[18]	583	18.83	0.341	28.57	[26, 34]
Toe and Kubota (2013)	[2]	2837	13.80	0.57	32.09	[24.9, 31.2]
Indraganti et al. (2014)	[29]	1352	21.40	0.26	28.92	[10, 36]
Luo et al. (2015)	[64]	834	14.64	0.41	24.81	[17.4, 29.4]
Singh et al. (2015)	[65]	n.a. 2	22.69	0.15	26.69	[10, 35]
Damiati et al. (2016)	[66]	n.a. 2	18.8	0.33	28.05	Undefined
Singh et al. (2017)	[67]	15	16.937	0.3568	26.33	[5, 25]
de Dear et al. (2018)	[31]	76	16.75	0.26	22.64	[8, 27]
Du et al. (2019)	[5]	n.a. 2	16.28	0.39	26.69	[5, 30]
López et al. (2019)	[32]	181	18.45	0.32	27.13	Undefined

¹ The number of decimals for parameters a, b and T_out is stated according to the original work. ² n.a. = not available data.

Table 1. Parameters of the equations of thermal comfort in a hot-humid climate.

Reference		n	a 1	b 1	Temperature of Equilibrium Teq (°C)	Applicability Range for Tout (°C) 1
Nicol and Roaf (1996)	[12]	4927	17.00	0.38	27.42	[5, 35]
Humphreys and Nicol (1998)	[62]	n.a. 2	11.9	0.534	25.54	Undefined
Nicol et al. (1999)	[13]	n.a. 2	18.5	0.36	28.91	[5, 35]
de Dear and Brager (2002)	[14]	n.a. 2	17.8	0.31	25.80	[10, 33.5]
Bouden and Ghrab (2005)	[16]	200	10.35	0.5179	21.47	[8, 35]
Zain et al. (2007)	[63]	n.a. 2	17.6	0.31	25.51	Undefined
Rijal et al. (2009)	[26]	601	15.4	0.516	31.82	>10
Nguyen et al. (2012)	[18]	583	18.83	0.341	28.57	[26, 34]
Toe and Kubota (2013)	[2]	2837	13.80	0.57	32.09	[24.9, 31.2]
Indraganti et al. (2014)	[29]	1352	21.40	0.26	28.92	[10, 36]
Luo et al. (2015)	[64]	834	14.64	0.41	24.81	[17.4, 29.4]
Singh et al. (2015)	[65]	n.a. 2	22.69	0.15	26.69	[10, 35]
Damiati et al. (2016)	[66]	n.a. 2	18.8	0.33	28.05	Undefined
Singh et al. (2017)	[67]	15	16.937	0.3568	26.33	[5, 25]
de Dear et al. (2018)	[31]	76	16.75	0.26	22.64	[8, 27]
Du et al. (2019)	[5]	n.a. 2	16.28	0.39	26.69	[5, 30]
López et al. (2019)	[32]	181	18.45	0.32	27.13	Undefined

¹ The number of decimals for parameters a, b and T_out is stated according to the original work. ² n.a. = not available data.

Table 2. Parameters of the equations of thermal comfort in a hot-dry climate.

Reference		n	a 1	b 1	Temperature of Equilibrium Teq (°C)	Applicability Range for Tout (°C) 1
Schiller et al. (1988)	[59]	221	5.41	0.73	20.04	Undefined
Heidari and Sharples (2002)	[15]	n.a. 2	18.10	0.292	25.56	[0, 40]
Heidari and Sharples (2002)	[15]	n.a. 2	17.3	0.36	27.03	[0, 40]
Farghal and Wagner (2008)	[25]	770	4.59	0.859	32.55	Undefined
Gómez et al. (2009)	[68]	150	15.6	0.545	34.29	[17, 31]
Toe and Kubota (2013)	[2]	2837	13.7	0.58	32.62	[24.8, 33.7]
Toe and Kubota (2013)	[2]	2837	14.3	0.56	32.50	[19.4, 30.5]
Toe and Kubota (2013)	[2]	2102	12.4	0.63	33.51	Undefined
Gabril et al. (2015)	[60]	160	14.26	0.47	26.91	[10, 35]

¹ The number of decimals for parameters a, b and T_out is stated according to the original work. ² n.a. = not available data.

Table 2. Parameters of the equations of thermal comfort in a hot-dry climate.

Reference		n	a 1	b 1	Temperature of Equilibrium Teq (°C)	Applicability Range for Tout (°C) 1
Schiller et al. (1988)	[59]	221	5.41	0.73	20.04	Undefined
Heidari and Sharples (2002)	[15]	n.a. 2	18.10	0.292	25.56	[0, 40]
Heidari and Sharples (2002)	[15]	n.a. 2	17.3	0.36	27.03	[0, 40]
Farghal and Wagner (2008)	[25]	770	4.59	0.859	32.55	Undefined
Gómez et al. (2009)	[68]	150	15.6	0.545	34.29	[17, 31]
Toe and Kubota (2013)	[2]	2837	13.7	0.58	32.62	[24.8, 33.7]
Toe and Kubota (2013)	[2]	2837	14.3	0.56	32.50	[19.4, 30.5]
Toe and Kubota (2013)	[2]	2102	12.4	0.63	33.51	Undefined
Gabril et al. (2015)	[60]	160	14.26	0.47	26.91	[10, 35]

¹ The number of decimals for parameters a, b and T_out is stated according to the original work. ² n.a. = not available data.

Table 3. Results of the Shapiro–Wilk test of normality applied to the parameters of the linear thermal comfort equations.

Climate	Parameters	p-Value of Shapiro-Wilk	(α = 0.05)
			n
	a	0.7863
Hot and humid	b	0.4109	17
	Teq	0.7241
	a	0.0830
Hot and dry	b	0.9933	9
	Teq	0.1055

Table 3. Results of the Shapiro–Wilk test of normality applied to the parameters of the linear thermal comfort equations.

Climate	Parameters	p-Value of Shapiro-Wilk	(α = 0.05)
			n
	a	0.7863
Hot and humid	b	0.4109	17
	Teq	0.7241
	a	0.0830
Hot and dry	b	0.9933	9
	Teq	0.1055

4.3. Identification and Elimination of Outliers in Population Samples

Before estimating the average values of the parameters associated with linear thermal comfort equations (slope, intercept, and equilibrium temperature), the outliers were identified and eliminated by applying statistical discordance of deviation (TN1), Grubbs’ (TN4), Dixon’s (TN9 and TN10), and high-order moments (TN14 and TN15) tests [55]. The evaluation confirmed that the population samples of the parameters associated with the hot-humid climate (

Table 4. Results of the discordance tests of the equations for a hot-humid climate.

Test Code	TN Value Calculated		Critical Values of the Tests (Significance Level)
	Intercept	Slope	95%	99%	99.5%	99.9%
Intercept		n = 17	$\overline{X}$ = 16.8898		s = 3.088872
Slope		n = 17	$\overline{X}$ = 0.3715		s = 0.111008
TN1	2.117	1.995	2.475	2.785		3.103
TN4	0.7658	0.7875	0.5933	0.4848
TN9	0.140	0.286	0.359	0.460	0.495
TN10	0.182	0.298	0.388	0.493	0.529
TN14	0.30	0.18	0.81	1.15
TN15	3.03	2.58	4.16	5.34

) and dry climate (

Table 5. Results of the discordance tests of the equations for a hot-dry climate.

Test Code	TN Value Calculated		Critical Values of the Tests (Significance Level)
	Intercept	Slope	95%	99%	99.5%	99.9%
Intercept		n = 9	$\overline{X}$ = 12.8511		s = 4.51570
Slope		n = 9	$\overline{X}$ = 0.5584		s = 0.16456
TN1	1.829	1.826	2.110	2.323		2.492
TN4	0.8311	0.5830	0.3742	0.2411
TN9	0.064	0.258	0.512	0.635	0.677
TN10	0.074	0.258	0.580	0.701	0.739
TN14	0.863	0.128	0.976	1.431
TN15	2.38	2.37	3.86	4.82

) were free from outliers. Similarly, the samples from the equilibrium temperatures in both climates did not show any evidence of outliers (

Table 6. Results of the discordance tests at equilibrium temperatures for a hot-humid climate.

Test Code	TN Value Calculated	Critical Values of the Tests (Significance Level)
		95%	99%	99.5%	99.9%
Hot humid		n = 17	$\overline{X}$ = 26.96		s = 2.750377
TN1	1.9976	2.475	2.785		3.108
TN4	0.7693	0.593	0.4848
TN9	0.1130	0.359	0.420	0.495
TN10	0.1570	0.388	0.493	0.529
TN14	0.0155	0.81	1.15
TN15	2.9897	4.16	5.34

and

Table 7. Results of the discordance tests at equilibrium temperatures for a hot-dry climate.

Test Code	TN Value Calculated	Critical Values of the Tests (Significance Level)
		95%	99%	99.5%	99.9%
Hot dry		n = 9	$\overline{X}$ = 29.45		s = 4.53250
TN1	2.0761	2.110	2.323		2.492
TN4	0.5212	0.374	0.2411
TN9	0.4097	0.512	0.635	0.677
TN10	0.4387	0.580	0.701	0.739
TN14	0.7763	0.976	1.431
TN15	2.4333	3.86	4.82

).

4.4. Comparison of the Variance and Mean between Hot-Humid and Hot-Dry Climates

After applying the normality test and the detection of outliers, an F-test (Fisher test) (

Table 8. Results of the F-test (Fisher test) for the analyzed comfort equations.

Variable		$\overline{\mathit{X}}$	s	n	F Value	F Critical Value
						95%	99%
Intercept	aHumid	16.8898	3.08887225	17	2.4043
	aDry	12.8511	4.78962797	9		2.59	3.89
Slope	bHumid	0.3715	0.11100754	17	2.4723
	bDry	0.5584	0.17454520	9

) and a Student’s t-test were conducted to evaluate the slope, intercept, and equilibrium temperature data. Their aims were to respectively determine if meaningful differences in the variances and means between the climates under study exist. Finally, based on the results obtained from the F-test (Fisher test) (Table 8), it was determined that there were no significant differences in the variances linked to the intercept and slope parameters for both climates at the 5% and 1% significance levels. These results imply that a comparable statistical level of experimental uncertainty was observed for both hot and dry conditions.

Table 9. Results of Student’s t-test for the analyzed comfort equations.

Variable		$\overline{\mathit{X}}$	s	n	t-Value	t-Critical Value
						95%	99%
Intercept	aHumid	16.8898	3.08887225	17	2.6176
	aDry	12.8511	4.78962797	9		2.064	2.797
Slope	bHumid	0.3715	0.11100754	17	3.3458
	bDry	0.5584	0.17454520	9

shows the estimated t-statistics for humid and dry conditions. These figures are very high compared to the critical values at the 5% and 1% significance levels. A significant difference exists in the slope and intercept between the hot-dry and humid climates. Therefore, a comfort equation is proposed for each climatic condition.

4.5. Influence of Zones with the Highest Probability in Plots of Indoor vs. Outdoor Temperatures

In order to establish the dominant thermal conditions, plots were created to contrast the outdoor (x-axis) and comfort (y-axis) temperatures in humid (Figure 3a,b) and dry (Figure 4a,b) climates. Then, the x–y spaces were divided into regions of probability of events (rectangles where the counted number of lines whose track goes through a specific region takes place). Within this frame, the regression lines of the equations under study were drawn (Figure 3 and Figure 4).

In this way, the frequency density was determined by counting. From these results, the most likely tracks for humid and dry conditions were drawn. Thus, the exercise showed that the zones of highest probability are linearly distributed. This makes it possible to generate an integrated model in both cases.

The colors were used to denote in each box the incidence of the regression lines of the thermal comfort models reported in Table 1 and Table 2; for example, when one to three lines (linear equations) coincide in a region, the color green is used in the plane; when more than 10 straight lines coincide, the color red is used. The coincidences of these models in the plane are also observed in Figure 3b and Figure 4b. These incidence graphs made it possible to detect the regions in the plane in which the thermal comfort models developed under different conditions can coincide.

4.6. Influence of Zones with the Highest Probability in Plots of Indoor vs. Outdoor Temperatures

As previously discussed, the statistically significant differences for the average intercept and slope values between hot-humid and hot-dry climates made it necessary to develop specific linear equations for each climatic condition (Equation (3): humid and Equation (4): dry). Furthermore, coefficients a and b for both equations were reported, including their 1% significance limits:

T_c = 16.9(±2.19) + 0.37(±0.08) T_out

(3)

T_c = 12.9(±5.36) + 0.56(±0.20) T_out

(4)

According to the results of this work, these models will be useful for conducting preliminary evaluations in locations where no experimental data are available. The consistency of these approximations is hence demonstrated in the next plot. Figure 5a shows a plot of the slope (b) vs. the intercept (a) for hot-humid climate equations. In addition, Figure 5b shows the plot for the hot-dry climate equations.

(a)

There is an inverse correlation between the parameters (Pearson correlation coefficients: R_humid = 0.7666 and R_dry = 0.8663); and

(b)

The majority of dots are situated within the rectangle, representing the 1% significance limits.

The red triangle represents the average value of the variables.

Considering the statistical tests carried out on the thermal comfort equations and the proposal of two general models for climates type HH and HD, respectively, it is also necessary to specify the relevance of the thermal comfort requirement, which can be analytically defined as the difference between the room temperature (outside temperature) and the thermal comfort temperature (expected temperature as a function of the room temperature), as well as for the cases of the general thermal comfort equations proposed in this paper. The equations are presented to estimate the comfort requirement. Equation (5) is presented to estimate the thermal comfort requirement in HH climates, while Equation (6) corresponds to the thermal comfort requirement in HD climates:

∆T_(HH) = 0.63 T_out − 16.9

(5)

∆T_(HD) = 0.44 T_out − 12.9

(6)

where ∆T_(HH) represents the thermal comfort requirement for the hot-humid climate, and ∆T_(HD) represents the thermal comfort requirement for the hot-dry climate.

An example of the application of Equations (5) and (6) is presented below, taking as a case study the climatic variation that prevails in Mexico.

Figure 6 shows the estimated thermal comfort requirement in Ciudad Victoria (Tamaulipas) which is located in a HD climate. Temperature data by region were obtained from weather stations of the National Water Commission of Mexico [69].

Using Equation (6) for the hot-dry climate, Figure 6 shows the thermal comfort required in the six months with the highest temperature in Ciudad Victoria, with a thermal gradient from 1.10 to 4.33 °C, which represents the magnitude to be removed by ventilation and air conditioning systems.

Figure 7 presents the application of Equation (5) for the town of Tampico (Tamaulipas) located in a warm humid climate.

In Figure 7, it is observed that the thermal comfort requirement is higher in the hot-humid climate compared to Figure 6; for this case there is a gradient from 2.23 to 4.72 °C. Thus, the requirement of thermal comfort is more evident or necessary in the hot-humid climate, as manifested in the mentioned example.

Figure 8 presents the thermal comfort requirements using Equations (5) and (6), in the same plane, where it is observed that under the same thermal scale the requirement is higher in the hot-humid climate condition.

5. Discussion

The statistical methodology used in this paper consisted mainly of the determination of normality, detection of deviated data, and comparison of means. This allowed us to obtain two equations: Equation (3) for the hot-humid climate and Equation (4) for the hot-dry climate. These equations show similarity with one of the equations used globally [14], which is commonly applied in buildings located in different climatic regions. However, when applying these equations with real data, they provide significantly different results than when applying a particular equation in a given region, such as that of [14].

It should be noted that the generation of a single model to estimate thermal comfort temperatures in hot climates (humid and dry) is not statistically possible, at least with the data used in this research. However, this situation did lead to the generation of specific models for each climatic condition. Equation (3), referring to the hot-humid climate, coincides in terms of both the slope and intercept with 11 of the 17 equations reported in Table 1. These coincidences are given considering the significance limit at 1%. On the other hand, Equation (4), referring to the hot-dry climate, coincides with the slope and intercept of seven equations of the nine reported in Table 2. These slope and intercept concordances can be interpreted as indicators of the validity of the equations developed and proposed in this research, since they have characteristics that agree with those reported in the literature.

The applications of the two generated thermal comfort equations allow the evaluation of new buildings to ensure they satisfy the comfort of users in places where comfort models have not been developed, particularly in regions with hot climates. Thus, the equations obtained in this work can serve as a reference for future applications of comfort studies.

6. Conclusions

During this study, 17 thermal comfort equations were identified for the HH climate, and nine others were identified for the HD climate. A rigorous statistical analysis was applied to these sets of equations. Due to the nature of the equations and the results of the statistical tests used, which resulted in a significant difference in the means, it is not possible to determine a general model, However, the authors propose two independent equations for HD and HH climates.

General linear thermal comfort equations have been developed for hot-humid and hot-dry climates from statistical parameters related to local models distributed worldwide. According to the Shapiro–Wilk test, Gaussian behavior was inferred for the intercept and slope data. Furthermore, outliers were not detected in the samples despite the implementation of various outlier tests. Nevertheless, statistically significant differences were identified in the regression parameters between hot-humid and hot-dry climates. Thus, specific linear equations were inferred for each climate condition. The consistency of the models was demonstrated using slope–intercept plots with the 1% significance limits for these variables.

The equation proposed for the hot-humid climate has a higher intercept value, while the slope is lower, while the proposed equation for the hot-dry climate has a lower intercept and higher slope:

T_c = 16.9(±2.19) + 0.37(±0.08) T_out

(7)

The proposed equation for the hot-dry climate denotes a lower intercept and a higher slope, as shown below:

T_c = 12.9(±5.36) + 0.56(±0.20) T_out

(8)

The proposal of these new models will allow analyses and projections of thermal comfort temperatures to be conducted in regions with similar climates, provided that a built environment operating under NV is considered. This work also highlighted that, in regions where there are no field data for analysis of comfort, these equations can address this lack in a tentative way be providing approximate evaluations of the thermal comfort required by the users.

The results obtained in this paper allowed the proposed objectives to be satisfied. In addition, this paper will also contribute to the understanding of thermal comfort from the perspective of statistical analysis and its relationships with other disciplines. It is also a starting point to developing work that allows linking low-energy air conditioning alternatives with the comfort needs of warm regions.