Method of calculating solar heat transmitted through shaded windows for OTTV in consideration of diffuse radiation diminished

ABSTRACT The energy used in the buildings in subtropical and tropical countries is mainly for the cooling load requirements for air conditioning in summer, which is evaluated by OTTV value. To ensure a satisfactory OTTV value, the top priority of building envelope designing is reasonable WWR with appropriate sun shading devices to reduce the heat penetrated through window glass into the buildings. Therefore, the accurate calculation of solar radiation through window glass with shading devices is very important. The research aims to calibrate the reduced solar radiation coefficient of the shading device (β) from the original method, thereby completing the OTTV evaluation method with high accuracy. The calculation is based on the sunlight projection to determine the window’s shaded area created by shading devices of any shapes with consideration of diffuse radiation diminished. The results of β calculated showed that in tropical climate, the total solar radiation on the window glass is much lower than the previous method. A computer software built from the proposed theory helps the designer easily choose the optimal sun shading solutions. The research conducts the calculation as an example for horizontal-inclined and the combined of horizontal and left/right vertical shading devices.


Introduction
Using the right building envelope will reduce both heat entering the building and use of electrical energy (Berger and Mendes 2017). Therefore, overall thermal transfer value (OTTV), which measures average heat gain transmitted into completely enclosed building through its envelope, is one of the most popular performance-based methods. Compared to thermal insulation standards in cold climates, OTTV is more suitable for buildings in hot climates because it accounts for the solar heat gain at the building envelope (Yik and Chan 1995). For this reason, since the 1980s and early 1990s, OTTV is used as a control measure for building envelope design in building energy regulations of some sub-tropical and tropical countries, such as the Association of Southeast Asian Nations (ASEAN), including Singapore, Indonesia, Malaysia, Philippines and Thailand (Vijayalaxmi 2010;Hui 1997).
The American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) was the first to propose the OTTV method (Refrigerating andAir-Conditioning Engineers 1975, 1980). The OTTV comprises two values: envelope thermal transfer value (ETTV -measure of heat transferred through the walls) and roof transfer value (RTTV -measure of heat transferred through the roof). Calculations for ETTV consist of three major components: (a) conduction through opaque wall (b) conduction through window glass, and (c) solar radiation through window glass. Windows are the most significant components of the building envelope in terms of comfort and energy use per unit surface area (Carmody et al. 2004). According to the research of Ha, the percentage of heat transferred into the middle floors of the building through window glass ranges from 86.96% to 91.94% of the overall heat transferred into the building through the building envelope, depending on building directions (Ha 2016). Thus, to enhance the building's energy performance, it is necessary to protect the windows from solar radiation and regulate the sunlight by fenestration. If installed on the exterior of the building, shading devices can block solar radiation effectively before it passes through the window glass. Kim et al. through their experiment study concluded that "it is clear that the external shading device is much more effective than any other form of internal devices since the internal device absorbs solar heat and radiates to the interior" (Kim et al. 2012). Through testing the effects of various configurations of external shading devices towards the energy consumption based on simulation tools, Shahdan et al. recommend that it is very important to take into consideration the design criteria of each shading device prior to its application on the façade of the building at the early phase of the design (Shahdan, Ahmad, and Hussin 2018). Thus, the use of shading devices to prevent direct radiation helps minimize heat penetrating through windows into the room, thereby reducing OTTV, re-visual comfort (glare) and thermal comfort of the near perimeter windows. Therefore, improving the energy efficiency of the building is the top priority of designing in hot climate region. And accurate calculation of solar radiation through window glass with shading devices is very important to properly determine the OTTV value as well as the cooling load of the air conditioning system in the summer and the energy efficiency of the architectural facade solutions in tropical countries, especially those near the equator with hot and humid tropical climates like Vietnam.

Basic theory and calculation solar heat transmitted shaded window glass methods
In all external shading devices, it is necessary to consider the structures' geometry relating to changing sun positions to determine the times and quantities of sun direct radiation penetrated through the shaded windows. Ewing and Yellott (1976) gave the details of the solar profile angle and tabular data presented for the effectiveness of external shading in four major U.S. latitudes and for all times of day, year, and window orientations (Ewing and Yellott 1976). Rudoy and Duran (1974) developed the cooling load temperature difference/ cooling load factor method (CLTD/CLF), which were derived from the TFM method and used tabulated data to simplify the calculation process. The most current cooling load calculation method is the radiant time series method (RTS) that Spitler et al. developed in 1997 (Spitler, Fisher, andPedersen 1997), which is suitable for peak design cooling load calculations. RTS method is an improvement over all previous methods: to determine solar heat gains through fenestration, it is needed to calculate transmitted and diffuse solar heat gain for each hour, each window.
By ASHRAE 2017 (American Society of Heating Refrigeration and Air-Conditioning Engineers 2017), the glazing solar heat flux q s can be split into that from incident beam radiation q b and incident diffuse radiation q d : The glazing solar heat flux caused by incident beam radiation is calculated from: The glazing solar heat flux caused by incident diffuse radiation is calculated from: of which: E DN -direct normal irradiance, W/m 2 ; E ddiffuse sky irradiance, W/m 2 ; E r -diffuse ground reflected irradiance, W/m 2 ; θ -incident angle; SHGCsolar heat gain coefficient. With shading windows, the general effect of shading is to attenuate solar radiation. Some of the beam radiation may reach the fenestration unaffected by the shade, and this is accounted for by the unshaded fraction Fu. If the shade does not transmit or diffuse solar radiation, the solar heat gain of the fenestration can be approximated by modifying Equation (4) (American Society of Heating Refrigeration and Air-Conditioning Engineers 2017): In Russia, Dang and Bogoslopski (1973) published a method of calculating solar heat transferred through windows which takes into account of solar heat diminished by external shading devices. This method also considers the total heat transferred into the room through the windows, taking into account the indices: direct radiation q s , diffuse radiation q D , heat radiated from the ground q зем ., radiant heat from surrounding buildings q эл . heat from long-wave diffuse radiation from the environment q дл . The method simultaneously takes into account the heat absorbed by the window glass q пог. and the heat transferred into the building is formed by the difference in indoor and outdoor air temperatures q Δt , as well as by the influence of shading devices such as blinds and shielding (Dang and Bogoslovskiy 1973). Then, this calculation method of solar radiation through windows with shading devices was completed in 1978 (Dang 1978). The most remarkable of their calculation method is the definition of reduced diffuse radiation coefficient, which is the hidden sky radiation coefficient from differential equation performing radiation heat exchange between the surface of shading devices and central point of windows. The method of calculating solar heat radiation through windows with shading devices developed by Dang and Bogoslopski was worked out and serves as a database for calculation coefficients K 1 and K 2 in Equation of solar cooling load in the standard (L'vovskiy and Barkalov 1993). In which component K 2 is the coefficient taking into account the reduced diffuse radiation by the shading devices: of which: q p ,q r -intensity of incident direct radiation and incident diffuse radiation, respectively, W/m 2 ; K 1coefficient of direct radiation decreased by both horizontal and vertical shading device; K 2 -coefficient of diffuse radiation decreased by both horizontal and vertical shading device; K 3 -coefficient of heat radiated through curtains and blinds; K 4 -solar heat gain coefficient of windows; A oc -windows area, m 2 . In the research on energy efficiency facade design in high-rise apartments, Ha (2018) developed a method for calculating the solar heat radiation through a window with infinitely sized shading devices by Dang and Bogoslopski to propose for finite horizontal shading devices (Ha 2018). However, this method only can provide the calculation for simple horizontal or vertical shading devices.

Research problem statement
The lack of these existing methods of solar heat radiation through window glass calculation is incomplete combination. For the Russian calculation, the theory can not be applied for the complex shading devices. For the calculation of ASHREA and ISO, the calculation misses the diffuse incident radiation value which is reduced from the effect of shading devices. This is acceptable for the cool climate regions where the diffuse solar heat is low. However, in hot climate regions, with a midlevel cloudy such as Vietnam, Thailand, Malaysia, Indonesia and Singapore, the diffuse radiation characteristics account for a very large proportion of the total solar radiation (Nguyen et al. 2020). Therefore, considering the reduced diffuse radiation coefficient due to shading device makes an important contribution to calculating the energy performance of buildings in the summer time. If we apply the methods of calculating solar heat penetrated through windows with shading devices of ASHRAE, ISO to calculate OTTV in hot and humid countries, this will lead to a large error by neglecting the reduced diffuse radiation effect of the shading devices. Furthermore, changing the OTTV index to respond to climate change when the earth is getting warmer (Hwang et al. 2018) is necessary and the OTTV formula can be continuously evolved with the ever-changing climate (Kusumawati, Setyowati, and Purnomo 2021).
The proposed method is based on the theory of shading projection of the shading device on the window surface according to ASHREA as well as other standards. At the same time, it supplements the reduced diffuse radiation coefficient due to the shading device. This combination creates a calculation which can be applied for all types of shading devices with more accurate calculation. In the research, the authors propose the coefficient of radiation decreased by solar shading devices -β. This coefficient helps to easily evaluate the shading effect of each shading device at each time of day. Using software simulation proposed by the authors shows visual analysis of the shading efficiency of shading devices according to the sun apparent with examples for EKE and horizontal shading devices with inclined an angle α. Other common types of shading devices such as continuous horizontal, vertical shading devices with different angles, box-shaped shading devices etc. have been applied by proposed theories to calculate and program in the research project RD 39-18/2018 (Chan et al. 2017) and international publications by the authors (Chan et al. 2021(Chan et al. , 2020Chan, Luong, and Phuong 2019;Ha 2016;Chan 2015). The targets of the research are as follows: • Theoretical basis for calculating the reduction of solar radiation through the window glass with shading devices with the reduced diffuse radiation coefficient K bt and the reduced direct radiation coefficient G. • Programing and applying the proposed theory to calculate reduced solar radiation coefficient β and visual analyze the effectiveness of the shading devices. • Evaluating and analyzing the effectiveness of different shading devices, select the optimum shading devices.

Calculation formula for overall thermal transfer value (OTTV) to evaluate the effectiveness of the building envelope
Required value of the OTTV of the walls in QCVN 09:2013/BXD and the revised version QCVN 09:2017/ BXD (Vietnam) (Ministry of Construction of Vietnam 2017) is developed from the calculating formula of OTTV of the wall (ETTV) in the research project (Chan et al. 2017). The formula to calculate ETTV for the certain "i" wall of the building will be as follow: of which: WWR i -window-to-wall ratio; U o,wall -U-value of opaque wall, W/m 2 .K; U o,win -U-value of glazed windows, W/m 2 .K; α -solar heat gain coefficient of material surface of the dimensionless, solid wall; ΔTtemperature difference between exterior and interior, K; TD eq -equivalent temperature difference, K; I ointensity of the global solar radiation on the surface of walls and glazed windows, W/m 2 ; SHGC -solar heat gain coefficient of glazed windows; β -reduced solar radiation coefficient of the glazed windows due to the effect of the shading devices (also called as External Shading Multiplier -ESM), shortly known as dimensionless "reduced radiation coefficient". When the windows have no shading devices, β = 1. The third term in formula (6) is the heat transferred by solar radiation into the building through the window, named as Q o . When we calculate the whole window with an area of A win and have no shading devices, meaning β = 1, Q o is expressed by: of which: I S ,I D -intensity of the direct solar radiation and sky diffuse radiation on the surface of walls and glazed windows, respectively.

A new proposal for the reduced radiation coefficient through window "β" when having shading devices
The proposal of the coefficient β is based on the analysis of the solar radiation on the window with shading devices. The solar radiation is divided into two portions: The direct solar radiation on the window is affected by shading devices characterized by coefficient G, based on the ASHREA sun projection's method; the reduced diffuse radiation is calculated according to the sky solid angle obscured by shading devices, characterized by coefficient K bt .
Considering that shading device works at the time when sunlight shines on the wall surface, then the glazed window area will have two parts: the shadowed area A shad and the exposure area A e (A shad + A e = A win ). The heat of solar radiation penetrated through the glazed window will be Q shad , including 2 parts: The exposure area A e affected by direct radiation I S , and the total area of glazed window surface -A win affected by diffuse radiation which is reduced by coefficient. So it can be written as follows: of which: K bt -sky diffuse radiation coefficient of shading devices. The coefficient β is the ratio of the two heat amounts Q K and Q K0 : Therefore we have: This is the formula β for any given moment -also known as the instantaneous coefficient β. It should be noted that the cumulative average coefficient β is used for the calculation of OTTV, while the instantaneous coefficient β is used to calculate the amount of heat transferred by solar radiation to the building when calculating the cooling load for the air conditioner from time to time during the day of the unstable temperature in hot season.
To calculate the average value β, the value β i must be immediately calculated at each moment and then averaged: It should be noted that on a certain wall direction, there are times when there is no sunlight, then I S = 0 and G = 0, which is not due to the effect of shading devices. Therefore, these times should be excluded before taking average. From formular (10) it can be seen that in order to calculate β, it is necessary to determine the exposed area and shaded area on the window, in other words, determine the value G -the quantity representing the ratio of area which have direct radiation and value K btreduced diffuse radiation coefficient caused by shading device partially obscures the sky. Intensive value of direct radiation I S and diffuse radiation I D on the window surface at the time of calculation can be determined based on weather data from meteorological stations or satellites and the classical accumulation methods converted from normal direct radiation to radiation on vertical surfaces.
From the formulas (7), (9) and (10), the formula for calculating the instantaneous unit heat (W/m 2 ) which solar radiation transferred into the building through the window can be drawn as follows:

Determining value G -ratio of the exposed area of the window with shading devices
Value G (the reduced direct radiation coefficient) is the ratio of the exposure area. When the sun shines on the walls and windows with shading devices, part of the window surface is exposed by direct and diffuse radiation, while the rest of the window is shaded (i.е. only exposed to diffuse radiation). The ratio G can be calculated as: The initial theoretical basis for determining the x and y coordinates of the projection ray (Figure 1), in other words, x and y are the destination M coordinates on the window plane, which point is determined by the beam passing through the point O -the outer edge of the shading device.
where h -the solar altitude angle, degrees; γ -the azimuth of the sun to wall, degrees; Ω -angle of the lateral projection of the sunbeam, degrees.
• In case of continuous horizontal shading devices, width b, placed at a distance from the top edge of the window d and inclined at an angle α to the horizontal plane (Figure 2a), the angle α > 0 or α <0.
• In case of finite horizontal shading devices placed at a distance from the top edge of the window d and extended to two sides of the window by e (often called the awning). This is horizontal shading device which is very popular in civil buildings. For this type of shading devices, destination point M of the sunbeam can be at seven different zones: zone 1; zone 2a; zone 2b; zone 2c, zone 3a; zone 3b and zone 3c. For each zone, there is a corresponding formula to determine the area exposed to sunlight and shade on the surface of the window glass (Figure 2b). • A case of the combined of horizontal and left/ right vertical shading devices (accepted as EKE shading devices), consisting of a horizontal shading device above the window, perpendicular to or inclined to the wall surface at an angle α and a vertical shading device perpendicular to the wall surface, placed a distance from the side of the window a distance e from the side edge of the window to the left (tp = 1) or right side (tp = 2) ( Figure 3).
Calculation of value G with different Sun-position will be computed and visualized in the next part of our research.

Reduced diffuse radiation coefficient caused by shading device partially obscures the sky -K bt
When there are shading devices, the area of the entire window A win is affected by diffuse radiation I D with the mitigation coefficient K bt . According to documents (Dang and Bogoslovskiy 1973;Bogoslovskiy et al. 1992), for continuous horizontal and vertical shading devices, coefficient K bt depends on "sky solid angle": δ -for the continuous   horizontal shading devices perpendicular to the wall surface and τ -for the continuous vertical shading devices perpendicular to the wall surface (Table 1).
For programming convenience, from the data given in Table 1, regression equations were drawn.
• For continuous horizontal shading device, perpendicular with wall surface: K hor bt ¼ 2:10 À 5 :δ 2 À 0:015δ þ 0:983 with R2 ¼ 0:999 (14) • For continuous vertical shading device, perpendicular with wall surface: K ver bt ¼ 2:10 À 5 :τ 2 i À 0:007τ i þ 0:992 with R2 ¼ 0:999 The angels δ and τ in equations (14) and (15) are given in decimal units. The results in Table 1 are only for the continuous horizontal shading devices, perpendicular to the wall surface. For the finite long and oblique shading devices, the calculation is developed according to the following formulas: (1) Determining the "sky solid angles" δ and the coefficient K bt of the continuous horizontal shading device perpendicular or inclined to the wall surface at an angle α (Figure 4).
It should be noted that when α = 0° Figure 4b becomes Figure 4a, and the general formula will be: of which: (2) Determining the "sky solid angles" δ, φ and coefficient K bt of finite shading device 1 "Sky solid angles" of the finite horizontal shading device include 2 different angles: Angle δ (Figure 4) due to overhang of the shading device and angle φ ( Figure 5) due to the length of the shading device. Finding the formula to determine the coefficient K bt of the window with finite horizontal shading device (K lim:hor bt ) is based on the coefficient K hor bt of the continuous horizontal shading devices. For continuous horizontal shading devices: angle φ goes up to 90° and then K hor bt is determined by following formula (14). In the case of finite horizontal shading device, angle φ gradually decreases to 0, then the coefficient K lim:hor bt will gradually increase and reach the value K bt = 1 (considered as no shading devices). The difference of the coefficient K bt between φ = 90° and φ = 0° is: ∆ = 1 -K hor bt . Then, the formula to calculate K lim:hor bt for the window with the finite horizontal shading devices can be as follows:    in which K hor bt can be obtained from Table 6, 7 or can be calculated from formula (14) when we know the "sky solid angle" δ of this shading device.
(3) Determining the "sky solid angles" δ, τ and coefficient K bt of EKE shading device An EKE shading device is sized by two "sky solid angles" which is angle δ, τ ( Figure 6): K ver bt is the diffuse reduction because the shading device shades the sky by angle τ from two sides. When there are no vertical shading devices, K ver bt =1. When there is only one vertical shading device on the left or on the right of the window, the value K 0 ver bt in this case is only equal to only ½ of the K bt when there are two vertical shading devices on both sides of window, that is: Coefficient K bt of the EKE shading device is calculated by the coefficient K lim:hor bt for the angle δ and K 0 ver bt for the angle τ will be:

Results
In summary, the main purpose of the calculation of the    solar heat penetrated through windows with shading devices is to determine the coefficients β with G and K bt hourly during the daytime, in each month of the year in local constructions in Hanoi (21.02°N) and Moscow (56°N) for the comparison.
Hanoi is a city with a tropical monsoon climate with cold winters. Hanoi has an intertropical solar radiation regime; in the year the sun has two passes through the zenith, resulting in high solar radiation in summer. The daytime of the Sun on June 21st is more than 13 h and on December 21st is more than 11 h. Buildings in Hanoi have typical forms of tropical architecture with outer sunshading devices/elements with a variety of forms: horizontal devices vertical devices, egg-crate devices, EKE devices, canopies, verandas, eaves, canvas, balconies, loggias, etc.
On the other hand, Moscow is a city with a temperate climate, under an extratropical solar regime, and with low beam angle and low solar radiation. The daytime of the Sun on June 21st is more than 17 h and on December 21st is more than 7 h. Buildings in Moscow have high thermal insulation requirements; their sunshading devices are not recommended on façades with north, northeast and northwest direction. However, in facades with south, southeast and southwest direction, sun-shading devices/elements (horizontal devices vertical devices, egg-crate devices, loggias) are often used to reduce solar radiation heat in summer and glare.
For the convenience of the calculation, a software has been set up for calculating and simulating visual analysis. The window parameters with shading devices used in the calculation and simulation of coefficient G and β by programming software are given in Table 2.  The simulation results show the shadow of the finite horizontal and EKE shading devices in two cases of TP1 and TP2 on the window surface presented in Tables 3 and   5 for Hanoi and Moscow. The calculation time used in the simulation is from 6:00 to 18:00. Simulation data to calculate coefficients G and β are shown in Tables 4 and 6.

Discussion
Following the variation of coefficients β for the effect of shading devices shown clearly during the daytime from 9:00 AM to 12:00 PM in Hanoi, from 11:00 AM to 02:00 PM in Moscow (slash area). The average value β shows the effectiveness of shading during the daytime from 6:00 AM to 6:00 PM. For Hanoi and Moscow as shown in the diagrams in Figures 8 and Figures 9, the shading effect is the worst at the highest coefficient avg.β for southeast window when we use horizontal shading devices. EKE shading devices show better effects because they can provide shading in the afternoon after 12:00 for Hanoi and after 14:00 for Moscow.
Considering the EKE shading devices for Moscow and Hanoi: • For Hanoi, using EKE-TP1 on the southeast direction is not appropriate because the left vertical shading device does not shade on the window surface. Thus, the shading effect is poor: Observing the β graph in Figure 7, it can be seen that the effect of EKE-TP2 in the early morning hours from 6:00 to 9:00 o'clock is better than EKE-TP1. Avg.β instantaneous-daily equal 0.600 for EKE-TP1 while avg.β instantaneousdaily equal 0.489 for EKE-TP2 in the same southeast direction. • For Moscow, on the southeast direction, the difference in shading efficiency based on coefficient avg.β of EKE-TP1 and EKE-TP2 is not much. The reason is that in Moscow on the southeast direction, the sun shines from 5:00 (even earlier) to 14:00 in the case of TP1, the shading efficiency is lower in the early morning, but from 11:00 to 14:00, the shading efficiency is higher than TP2. Therefore, considering the daily average shading efficiency, coefficient avg G and avg.β in the case TP1 is roughly equal to TP2. In all cases, when the window direction is the southeast direction, using EKE shading devices, TP2 has better shading efficiency due to the effectiveness of shading in the early hours. It should be emphasized that considering the shading effectiveness over time with the coefficient β have great importance when considering the sun protection requirement for the building. Direct radiation in the early hours for hygiene and epidemiological reasons is sometimes compulsory for some buildings such as kindergartens, hospitals, bedrooms etc., while direct radiation at noon, from after 9:00 AM to 2:00 PM to be minimized due to many dangers and discomfort caused by great intensity. Comparing TP1 and TP2 in Moscow, TP2 is better in terms of effectiveness of avg.β, but in terms of shading efficiency over time, TP1 is more efficient with a low β-value and allows early sunlight to enter the room and limits heat gain at 11:00-14:00 ( Figure 9). So, depending on the specific case, we should consider the overall shading effectiveness and by time to decide the most appropriate shading devices. • On the southeast and northwest direction, it is similar but permutation for each other: shading efficiency of EKE-TP1 in southeast direction is similar to EKE-TP2 in southwest and shading efficiency of EKE-TP2 in southeast is similar to EKE-TP1 in southwest.
From the above observations, our recommendation is that the TP2 form is more effective for the southeast facade in Hanoi. For Moscow, the TP1 form has the best effect for the southeast facade.
The withdrawn comments are completely based on the calculation of the effectiveness of shading through the coefficient avg.β, regardless of the architectural aesthetic factor and the cost of fabrication and installation of the shading devices. It should be noted that the above comments are still valid when the size of windows and shading devices change, provided the ratios R1 = b/H; R2 = b/B; Rd = d/H and Re = e/B remain the same as in the above calculation example. The above comments can be considered as a guide to shading solutions for designers and project owners. The most comprehensive and certain method to compare and choose shading devices with high shading effect on a certain direction is that designer must calculate several types of shading devices with different specific parameters which are considered as the most suitable by the developed software, then choose the shading device with the lowest avg.β coefficient or choose the shading devices with the avg.β that is economically-technically reasonable after considering the shading requirements in necessary space.

Comparing the results calculated by the proposed research with the theory according to ASHRAE and ISO
When we compare the results calculated according to ASHRAE and ISO methods without considering the reduced diffuse radiation coefficient K bt (K bt =1) and proposed method with K bt ≠ 1, it is found that the shading effectiveness and preventing diffuse radiation into the building is significant in the case of EKE shading devices which is demonstrated to be effective for the southeast and southwest facades. The effect is more pronounced in the tropical climate of Vietnam, where there is a high diffuse radiation (Figure 10). Reducing to minimize diffuse radiation in the presence of shading devices, the total solar radiation transmitted through the window is also calculated more accurately according to the proposed method (Tables 4, 5 and 8).

Expanding the application case
The research uses the above-mentioned theoretical method to calculate the effectiveness for the five most popular types of shading devices in Vietnam as well as in many other countries around the world including: (1) A continuous horizontal shading device, placed at distance d from the top edge of the window, perpendicular or inclined to the wall surface an optional α angle; (2) A finite horizontal shading device (also called as awning) placed at an optional distance d from the window top edge, perpendicular or inclined to the wall surface at an optional α angle and extending from the sides of the window a distance of e; (3) An egg-crate shading device, consisting of a horizontal shading device with an inclined angle α, placed at distance d above the window and 2-sides vertical shading devices. The vertical shading devices are placed at a distance e from the side of the window; (4) A continuous vertical shading device, placed at an optional distance e from the side of the window, perpendicular or inclined to the wall surface an optional angle α; (5) An EKE shading device: similar to the egg-crate shading devices, but with only one vertical panel on the left or right side of the window; The five-above mentioned shading devices are divided into 2 groups: The 1 st Group consists of the shading devices which are symmetric across the vertical axis through the center of the window (on the façade) -which is also the north-south axis (on the plan). The 2 nd Group is formed of the shading devices which are asymmetric across the north-south axis. The grouping is to facilitate the calculation and programming for efficient calculation of shading devices in an architectural design.
• 1 st Group (named KCCN-TH1): includes three shading devices symmetric across the vertical axis through the center of the window -which is also the north-south axis including: (a) a continuous horizontal panel; (b) a finite horizontal shading device; and (c) an egg-crate shading device. The meaning of symmetry here is also understood as follows: The windows with these types of shading devices if orient to east and west, then the reduced solar radiation coefficient β during the day will be symmetrical through 12:00 o'clock at noon. Specifically, the value β at 6:00 on the window orients to the east is the same as the value β at 18:00 on the window orients to the west; the value β at 8:00 on the window orients to the east is the same as the value β at 16:00 on the window orients to the west. It is similar to the other pairs of symmetrical directions across the north-south axis such as northeast and northwest; southeast and southwest; east southeast and west southwest etc. In this case, we just calculate the coefficient for nine different directions or groups of directions: north, north-northeast and north-northwest, northeast and northwest, eastnortheast, and west-northwest, east and west, east-southeast and west-southwest, southeast and northwest, south-southeast and southsouthwest. • 2 nd Group (named KCCN-TH2): Shading devices are asymmetric across the vertical axis through the center of the window -which is also the northsouth axis consisting of (a) continuous vertical shading devices, perpendicular or oblique to the wall surface -symbolized as "dg -ngh" (inclined vertical shading device); (b) an EKE shading device. For the two types of asymmetric shading device in the 2nd Group, it is necessary to calculate the coefficient β for all 16 directions.

Conclusions
The research proposes a method to evaluate the effectiveness of using shading devices with the reduced solar radiation coefficient β, the evaluation of the effectiveness of shading devices can be considered according to the average value or by the hour. This helps to choose an appropriate shading device based on energy efficiency requirements and ensures the required sunlight during the day. For hot climates, the consideration of the diffuse radiation reduction coefficient into the buildings through windows with shading devices is considered for the first time with the consideration sky diffuse coefficient K bt . With hot climates, architectural characteristics need to be designed with different types of shading device, the reduced diffuse radiation by shading devices should not be ignored. The new calculation proposal helps quickly, intuitively, and comprehensively improve the accuracy in calculating OTTV value.
The authors have developed a computational application for five typical types of shading devices, divided into two groups corresponding to design symmetry features: the 1 st Group includes three symmetrical shading devices across the north-south axis and the 2 nd Group includes two types of asymmetric shading devices across the north-south axis. Corresponding to these two groups of shading devices, two separate computer software named KCCN-TH1 and KCCN-TH2 have been prepared. Using these software, users and designers can directly check the coefficient avg.β or instantaneous β without approximate interpolation when tabulating for windows with shading devices of relative size of Rd and Rb form. The results calculated by the software include two types of file. The file type of the cumulative mean factor β (avg.β) is used to calculate OTTV and the file type instantaneous coefficient β is used to calculate the cooling load for air-conditioning systems.