Predicting daylight illuminance by computer simulation techniques

Transcription

1 Lighting Res. Technol. 36,2 (2004) pp. 113± 129 Predicting daylight illuminance by computer simulation techniques DHW Li BSc PhD CEng MCIBSE MHKIE MIEAust, CCS Lau BEng AMIMechE and JC Lam BSc PhD CEng FHKIE MCIBSE MIMechE RPE Building Energy Research Group, Department of Building and Construction, Tat Chee Avenue, Kowloon, Hong Kong, China Received 16 November 2002; revised 6 May 2003; accepted 11 August 2003 The rst step in evaluating the visual performance and energy ef ciency provided by daylight requires an accurate estimation of the amount of daylight entering a building. Traditionally, the daylight performance of a building is often evaluated in terms of daylight factors with the calculations being based on the CIE overcast sky. In general, the daylight factor approach is quite simple to use but it cannot predict the dynamic variations in interior illuminance as sky conditions and the sun s position change. The recently introduced concept of the daylight coef cient, which needs sky luminance data, provides an alternative approach. With the advances in computer technology, the computation of daylight illuminance using the two prediction methods can be conducted by a simulation program. This paper presents a study of the daylight factor and daylight coef cient approaches. The interior daylight illuminance data measured in a scale model and a classroom were compared with the simulated results using RADIANCE computer simulation software. It was found that, in general, the daylight coef cient approach performs better than daylight factor approach. Both methods tend to estimate the daylight illuminance less accurately when the measurement points in the classroom are far away from the window facë ade. It seems that the parameters for inter-re ection calculations including the furniture layout and external obstructions cannot be input into the RADIANCE program with suf cient detail for accurate simulation. List of abbreviations and symbols CIE DFA DCA MBE RMSE CLD International Commission on Illumination daylight factor approach daylight coe cient approach mean bias error root mean square error cloud cover(okas) Address for correspondence: DHW Li, Building Energy Research Group, Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR, China. bcdanny@cityu.edu.hk. h / h / D h / h / b a b D daylight coe cient at a point in a room produced by a small sky element at elevation and azimuth (dimensionless) E in horizontal internal daylight illuminance (lux) E ou t outdoor horizontal illuminance underthe CIEovercastsky(lux) E The illuminance ultimately produced at a point in a room by a small sky element at elevation and azimuth (lux) f r a i; i; r; r bidirectional re ectance transmittance distribution function (S=r) # The Chartered Institution of Building Services Engineers = li108oa

2 a 114 Predicting daylight illuminance by computer simulation techniques I mea measured internal daylight illuminance (lux) I pred Predicted internal daylight illuminance (lux) L h / The luminance of the sky element at elevation h and azimuth / (cd=m 2 ) L e a r; b r emitted luminance or radiance (cd=m 2 or W=(sr.m 2 )) L r a r; b r re ected luminance or radiance (cd=m 2 or W=(sr.m 2 )) L i a i; b i incident luminance or radiance (cd=m 2 or W=(sr.m 2 )) N the number of data points (dimensionless) D S h / the angular size of the sky element at elevation h and azimuth / (sr) SH sunshine hours (hours) polar angle measured from the surface normal (degrees) b azimuthal angle measured from the surface normal (degrees) 1. Introduction Daylighting has long been recognized as crucial and useful strategy in terms of visual comfort and energy-e cient building design. The determination of daylight illuminances in an interior space is a key stage in daylighting studies. In general daylighting designs, the diœuse component is widely used. Traditionally, the illuminance from natural sources is often determined in terms of the daylight factor, which is the ratio of the internal illuminance to the outdoor illuminance simultaneously available on a horizontal plane from the whole of an unobstructed overcast sky, expressed as a percentage. 1 This approach is used in Europe, particularly in Britain, where the method and the associated design aids have been developed to determine daylight factors at various stages of the design process. 2 The daylight factor approach has gained favour owing to its simplicity, but it is not exible enough to predict the dynamic variations in daylight illuminance as the sun s position and sky conditions change. 3 It has been reported that the ratio of indoor to outdoor illuminance can vary greatly. 4 An accurate estimation of the daylight illuminance on a surface depends on the way that light is distributed over the sky vault at a particular time. The actual daylight illuminance of a room is mainly in uenced by the luminance levels and patterns of the sky in the direction of view of the window. 5 The daylight coe - cient concept, 6 which considers the changes in the luminance of the sky elements, oœers a more eœective way of computing indoor daylight illuminances. As the sky is treated as an array of point sources, the daylight coe - cient approach can be used to calculate the re ected sunlight, and is particularly appropriate for innovative daylighting systems with complex optical properties. 7 In our previous work, 8 we presented the general evaluation for the daylight factor and daylight coe cient methods using hand calculation. However, daylight illuminance determination, especially when it is based on daylight coe cients, is very complex and time consuming, though advances in computer techniques have reduced the time for illuminance calculations. Recently, a lighting simulation program, RADIANCE has been used by a number of researchers to compute interior illuminances Reinhart and Herkel 9 simulated the indoor illuminance distributions for two o ce geometries with six diœerent RADI- ANCE based daylight simulation methods including the daylight factor and daylight coe cient approaches based on daylight data from the Freiburg test reference year (TRY). Galasiu and Atif 10 veri ed the accuracy of the RADIANCE program of the ADELINE software in predicting speci c lighting parameters of a real atrium building. Mardaljevic 11 examined the daylight coe cient approach using the RADIANCE computer package and

3 DHW Li et al. 115 measured data from the Building Research Establishment (BRE) unfurnished o ce. He concluded that the daylight coe cient could be used as a basis for e cient computation of the long-term daylighting performance of buildings. Very little work has been done with the RADIANCE program for simulating the indoor daylight illuminance based on measured weather data for a real furnished room with the windows facing moderate external obstructions. This paper analyses the two indoor daylight illuminance prediction techniques, namely the daylight factor approach (DFA) and the daylight coe cient approach (DCA) using the RADIANCE simulation package. A comparative study of the two methods against measured data in a scale model and a classroom located at the City University of Hong Kong is reported. 2. Daylight illuminance prediction approaches and RADIANCE program The DFA was used in conjunction with the horizontal outdoor illuminances to obtain the internal daylight illuminances. The DCA, however, is more sophisticated and relies on explicit modelling of the sky luminance distribution that changes with time. By de nition, the DFA is unable to account for the contribution of direct sunlight, but the method is much easier to use than the DCA. It would be useful to know their relative performances under various situations. Since the two approaches are based on RADIANCE, the same set of geometry, material, weather input les and simulation parameters can be used for all simulations, and their results can be readily compared. The characteristics of the RADIANCE program and the two daylight illuminance prediction approaches are described as follows: 2.1 Daylight factor Mathematically, DF can be expressed as follows: DF ˆ Ein E out 100% 1 where E in is the horizontal internal daylight illuminance (lux); E out is the outdoor horizontal illuminance under the CIE overcast sky (lux). Overcast skies are considered to provide the worst daylighting conditions and the sunlight is completely impeded without any direct component. The empirical Moon and Spencer 12 equation for the luminance distribution of an overcast sky was adopted by the International Commission on Illumination (CIE) in 1955 as the standard for the overcast sky luminance. The sky luminance distribution is symmetrical about the zenith and change with the elevation above the horizon. Under the CIE overcast sky, the internal daylight illuminance for a particular point can be analytically computed. It means that the indoor daylight illuminance results can be precisely determined for this simple sky pattern. Enarun and Littlefair 13 have reported that the CIE overcast sky performed the best among all world-wide models adopted for fully overcast skies in southern England. Our earlier work 14,15 on sky luminance patterns also found that the CIE standard overcast sky showed a good agreement with the measured Hong Kong overcast sky luminance data. The CIE overcast sky is included in the RADIANCE package Daylight coe cient Daylight illuminances inside a room are not in general proportional to the external illuminance, but depend on the exact sky luminance distribution at that time. This is because a point in a room will receive direct light only from certain areas of the sky and the illuminance within a room is not equally sensitive to changes in the luminance of diœerent parts of the sky. The DF cannot account for daylight illuminance from non-overcast sky luminance distributions or direct sunlight.

4 116 Predicting daylight illuminance by computer simulation techniques In 1983, the concept of DC that relates the luminance distribution of the sky to the illuminance at a point in a room was developed. The daylight coe cient, D h /, is expressed as. 6 D h / ˆ D E h / L h / D S h / 2 where D E h / is the illuminance ultimately produced at a point in a room by a small sky element at elevation h and azimuth / (lux); L h / is the luminance of the element (cd=m 2 ); D S h / is the angular size of the sky element (sr). The daylight coe cient, D h /, depends on the geometry of the room and its exterior environment, the re ectances of many surfaces, and the transmittance of the windows. It is, however, independent of sky luminance distribution because D E h / changes in proportion to L h /. Hence, the building characteristics and the surrounding climatic conditions are separated. The total daylight illuminance, E, received at the point in the room is the integral of Equation (2). For all but the simplest cases (e.g., direct light from a uniform or the CIE standard overcast sky via glass with a constant light transmission value on an unobstructed horizontal surface), it is not possible to compute analytically. Instead of integration, numerical techniques can be used: E ˆ X D h / L h / D S h / 3 h / Measured sky luminance data can be used in RADIANCE simulations to compute the DC and hence the internal daylight illuminance. 16 The measured values need to be interpolated to nd values on a regular grid. However, such interpolation can introduce data distortion as the measured data are based on discrete measured results rather than continuous analytical functions. Also, for the measurement of the sky distributions, the celestial hemisphere is split into 145 circular angular patches as recommended by Tregenza. 1 7 The division of the sky can avoid any double counting but it leads to uncovered regions of the sky. If the sky patterns are close to the CIE overcast skies, the internal daylight illuminances predicted using DFA would be comparable or even more accurate than those based on DCA. 2.3 RADIANCE simulation package RADIANCE is a computer simulation package for simulating and visualizing lighting in and around architectural environments. It can examine advanced lighting regimes using the backward ray-tracing technique, in which the light is traced from the observer to the dominant light sources to calculate the luminances required for visualization. This process can be expressed by an integral equation as follows: 16,18 L r a r; b r ˆ L e a r; b r Z 2p Z p 0 0 L i a i; b i f r a i; b i; a r; b r j cos a ij sin a ida idb i 4 where a is the polar angle measured from the surface normal (degrees); b is the azimuthal angle measured from the surface normal (degrees); L e a r; b r is the emitted luminance or radiance (cd=m 2 or W=(sr.m 2 )); L r a r; b r is the re ected luminance or radiance (cd=m 2 or W=(sr.m 2 )); L i a i; b i is the incident luminance or radiance (cd=m 2 or W=(sr.m 2 )); f r a i; b i; a r; b r is the bidirectional re ectance transmittance distribution function (s=r). Equation (4) indicates the essential recursive nature of the ray-tracing process, which continues until the error is small enough to quit. RADIANCE uses the same calculation techniques for both illuminance modelling (e.g., computing daylight factors) and creating synthetic images (i.e., luminance modelling). 18 At present, RADIANCE is available in two versions. The original one developed at Lawrence Berkeley Laboratory (Berkeley, USA) is free of charge for UNIX workstation. The other is a slightly limited MS-DOS version included

5 DHW Li et al. 117 within ADELINE (an integrated lighting design and analysis tool). 19 In the present study, the analysis was based on the ADE- LINE 1.0 RADIANCE Measuring station Predicting the indoor illuminance requires both accurate outdoor illuminance and sky luminance databases. A measuring station was established at the City University of Hong Kong to record the daylight illuminances and sky luminance distributions in Initially, only horizontal surface measurements were made. In 1996, the measuring station was extended to record vertical outdoor illuminance on the four cardinal surfaces, facing the north, east, south and west. The measuring station was upgraded in 1999 with the installation of a sky scanner to record the luminances of the whole sky patches. 21 The measuring station is inside a laboratory space on the top oor of the City University of Hong Kong, which is not located in a high-density urban area. The surrounding buildings are of similar height to the University block. All sensors were installed on the rooftop in a position relatively free from external obstructions, and readily accessible for inspection and general cleaning. Data collection started before sunrise and nished after sunset. All measurements were referred to true solar time (TST). 3.1 Daylight illuminance measurement Illuminance sensors (T-10M) manufactured and calibrated by Minolta of Japan were used for the outdoor illuminance measurements. The silicon photocells with cosine and colour corrections, measure illuminances up to about 300 klux with an accuracy of 2%. A multipoint illuminance measurement system was used. The main body adapter and a receptor head adapter were connected by an extension cable and all remaining receptor head adapters were then connected with other extension cables. Data-management software was used to capture the measured results from the main body adapter, which were fed into a microcomputer for storage. The measured data were displayed in real time for individual receptors and a measurement interval of 1 min was set. 3.2 Sky luminance measurement The sky luminance distribution was measured by means of a sky scanner (EKO MS-300LR), which was manufactured and calibrated by the EKO Company of Japan. It measures the luminance at 145 points (shown in Figure 1) in the sky by scanning the sky dome. The full view angle of the scanner is 11 giving a sky coverage of approximately 68%. 17 The important parts of the sky scanner were housed in a weatherproof casing allowing continuous outdoor operation. Output data from the scanner were recorded on a microcomputer placed inside the laboratory space on the top oor. A Visual Basic computer program was used to capture and transmit the measured data. To safeguard the sensor, light was prevented from falling on it when the luminance was greater than 35 kcd=m 2 by using an automatic shutter. Each scanning period was about 4 min and measurements were taken every 10 min. 4. Field measurements A scale model and a fully air-conditioned classroom at the City University of Hong Kong were selected for this present study. The interior daylight illuminance measurements were carried out by means of a multipoint illuminance measurement system with main body adapter (T-A20) and the required receptor head adapters (T-A21) connected serially. The time for each reading was carefully recorded so that the `synchronous measurement of horizontal outdoor illuminance and sky luminance data could be selected for calculation. The interior daylight illuminance

6 118 Predicting daylight illuminance by computer simulation techniques Figure 1 Measurement points for the sky scanner data were collected every minute and averaged over 10-minute intervals. The measurements were made in July For a fair evaluation of the RADIANCE program, the sky and sun luminances should not vary substantially during the scan. The whole set of measured data was rejected when horizontal outdoor illuminance data showed a diœerence of 10% or more of the mean values. 4.1 Scale model The dimensions of the model were 500 mm (depth) 400 mm (along the aperture) 245 mm (height). The model represented a scale of 1:10 rectangular cellular o ce with an aperture area of 400 mm 140 mm (no glazing). Figure 2 illustrates the four measurement points for the model room. An additional photocell, which was tted with an aligned shadow-ring to block direct sunlight, was used to obtain the horizontal external diffuse illuminance. This reading was compared with the data measured in our measuring station as a double check. A 5-point illuminance measurement system was, therefore, employed for the scale model measurement. The model room was covered with black paper with a re ectance of 0.05 inside so that the direct

7 DHW Li et al. 119 Figure 2 Plan view of the scale model showing the positions of the measurement points sky component of illuminance was mainly recorded. The measurements for the scale model were conducted on the rooftop of the City University of Hong Kong with the aperture facing south in a position relatively free from external obstruction. The purpose of the scale model testing was, therefore, to evaluate the DFA and DCA predicted by RADIANCE program when the room received mainly diœuse component directly from the sky (with very limited contributions from the internally and externally re ected components). 4.2 Classroom measurement The indoor daylight illuminance analyses for the scale model were limited to light of a diœuse nature directly from the sky. The direct sunlight and the light re ected externally and internally can be important sources of interior lighting. The comparative study on the two interior daylight illuminance approaches was further assessed based on classroom measurements. The classroom used for the daylight illuminance measurement faced north-west (320 ) and was located at the fth oor. The dimensions were 6010 mm (depth) 8960 mm (along the window) 2400 mm (height). Since the University is located near residential areas and faces several buildings and hillsides, certain parts of the sky are blocked by these external obstructions. Internally, furniture in the classroom included a number of tables. The working plane was taken at desk height (i.e., 750 mm from the oor). The general view of the classroom is shown in Figure 3. Figure 4 presents the classroom layout plan and the locations for the ve measurement points.

8 120 Predicting daylight illuminance by computer simulation techniques Figure 3 Interior view of the classroom 5. Simulation setting The simulation model consists of two parts: the scale model and classroom including external obstructions, which are comprised of opaque and transparent materials, and the self luminous sources that represent the sky and sun. The built-in `Gensky sub-program and measured luminance data were used to generate the sky conditions for DFA and DCA, respectively, for the indoor illuminance calculations. The same set of geometry, material and simulation parameters was used for the two approaches by the RADIANCE program. The key simulation parameters for daylight illuminance determinations are the number of re ections and resolution of the inter-re ection calculation, which are referred to as the ambient parameters. A convergence test was carried out to obtain the settings for Figure 4 Positions of the measurement points in the classroom

9 DHW Li et al. 121 Table 1 Set of ambient parameters used for simulation Table 2 Parameters for the classroom Ambient calculation Scale model Classroom model Ambient bounces 7 7 Ambient division Ambient sampling Ambient accuracy Ambient resolution 8 64 the ambient parameters so that the accuracy and simulation time of the ray-tracing calculations in RADIANCE were at an acceptable level. Table 1 shows the ambient parameter setting for the simulations. 5.1 Scale and classroom models Geometrically, the models generated for the simulations were close to the actual dimensions of the scale model and classroom. Without window glazing and furniture, the light transmittance of 1 and internal re ectance of 0.05 were set for the scale model. For the classroom, the furniture including tables, notice board, white board, window bars, glazing panes and curtains (pulled up during the data measurements), were modelled as discrete elements. The re ectances used in the classroom were obtained by selecting the sample (with known re ectance value) which most closely resembled the colour and brightness of the surface. 22 The results were also checked against measurements using luminance and ratio of illuminance meters. 22 The glazing transmittance, which is the ratio of illuminance behind and in front of the glazing, was obtained using the illuminance sensors under overcast sky conditions. 23 Externally, two circular ground planes with a common re ectivity of 0.2 and radii of 2 m and 40 m were set for the scale model and classroom, respectively. For the classroom, the nonluminous external objects including hillsides and building blocks were modelled in RADIANCE. The physical dimensions were determined from a map and on-site measurement, and the re ectance values were obtained according to their colours and textures Dimensions Length (mm) 8960 Depth (mm) 6010 Height (mm) 2400 Window area (m 2 ) 10 Working plane height (mm) 750 Window head height (mm) 2085 Sill height (mm) 790 Window reveal thickness (mm) 135 Re ectances Ceiling 0.55 Partitions 0.35 Carpet 0.15 Tables 0.20 Outside ground 0.20 External obstruction Transmittance Window (including light transmittance and dirt) 0.56 (similar to the approach for internal materials). The classroom details are summarized in Table 2. It can be seen that, in general, all room parameters appear to be suitable for daylighting schemes. 5.2 Sky and sun models The luminance of each sky patch may be constant in that it may take its form from a mathematical function (e.g., sky model), or may be based on discrete data values (e.g., measured sky luminance patterns). To use the measured sky luminance data in a RADI- ANCE simulation, the data needed to be used as a pattern modi er (i.e., conversion of sky luminance data to RADIANCE format) to a constant luminance sky. 24 In this way, the measured sky luminance values were converted into regular grid forms using interpolation. For `out of range measurements (points close to the solar position), an estimation of the sky luminance was made from a simple average of the luminance at nearby points. It has been pointed out that simple averaging methods would underestimate the sky luminance around the sun position. However, on bright sunny days, the direct solar radiation would dominate and such an approximation would not cause a substantial loss of

10 122 Predicting daylight illuminance by computer simulation techniques accuracy. 18 The other component of the model sky is the sun. The sun luminance was calculated from the direct normal illuminance (direct component on a surface perpendicular to the sun s rays) and solar disc angle, which was taken to be Sky conditions In interpreting sky conditions, climatic parameters such as sunshine hours (SH) and cloud cover (CLD) were always used. It seems obvious that more cloud results in less sunshine and vice versa. The sun is considered to be shining when an object under its direct light casts a well de ned shadow. Sunny skies mean that SH ˆ 1 25 dominated by a direct component and the CLD tends to be small. A cloudy sky indicates that SH ˆ 0 25 with no direct daylight illuminance and the sun is blocked by cloud. Based on these criteria, the skies were classi ed for illuminance calculations in both approaches. In total, 8 and 12 sets of data for the scale model and classroom measurements, respectively, (including sunny and cloudy days) were selected for the study. 6. Data analysis Graphical and statistical techniques are useful for data analysis. The accuracy of the two approaches was assessed by two widely used statistics, namely mean bias error (MBE) and root mean square error (RMSE). They are given as: MBE ˆ 1 µ X Ipred I mea N RMSE ˆ I mea 5 ( " 1 X #) 2 1=2 Ipred I mea N I mea 6 where I m ea is the measured internal daylight illuminance (lux); I p red is the predicted internal daylight illuminance (lux); N is the number of data points (dimensionless). The MBE provides information on the long-term performance of the models. The RMSE gives information on the short-term performance, and indicates the scattering of data around the models. 6.1 Scale model The data set consisted of four sunny and four cloudy skies and the details are listed in Table 3. As shown in Table 3, the sun was in the south with high elevation around noon and in the west in the afternoon. The interior daylight illuminances for the four grid points according to the DFA and DCA were determined. Figure 5 displays the predicted results from the two approaches for points 1 and 2, which received daylight from a large part of the sky. It can be seen that both models can be used to estimate the indoor daylight illuminances reasonably well under cloudy skies. As Table 3 Measured sky conditions for the scale model Sky conditions Date Time Solar altitude, degrees of arc Solar azimuth, degrees of arc SH CLD, okas Sunny skies 12 July : July : July : July : Cloudy skies 19 July : July : July : July :

11 DHW Li et al. 123 Figure 5 Performance of the daylight factor and daylight coef cient approaches for scale model grid points 1 and 2 the sky was covered by cloud, diœuse daylight was the main component and the sky pattern would be close to overcast. The good performance of DFA is not surprising, given that the prevailing cloudy conditions in Hong Kong are dominated by the CIE standard overcast sky type. 14,15 Under sunny skies, the predictive ability of DCA is far better than that of DFA. With a small amount of cloud, the sky luminance pattern is strongly aœected by the sun position. 2 Generally, peak sky luminance would appear near to the solar position decreasing rapidly with the angular distance from the sun. The results indicate that the luminances seen from points 1 and 2 were smaller than when there is a standard CIE overcast sky. If the outdoor illuminance and sky luminance data were known, the DCA could be used to calculate accurately the direct light from the sky. This may be due mainly to the advantage of jointly utilizing the two data sets. Similarly, the estimated results for points 3 and 4 are plotted in Figure 6. As the points are far away from the aperture, the measured illuminances are less than 1200 lux. Again, DFA performs Figure 6 Performance of the daylight factor and daylight coef cient approaches for scale model grid points 3 and 4

12 124 Predicting daylight illuminance by computer simulation techniques Table 4 Summary of MBE and RMSE for interior daylight illuminance using daylight factor and daylight coefficient approaches (scale model) Sky conditions Grid point Daylight factor Daylight coef cient MBE (%) RMSE(%) MBE (%) RMSE(%) Sunny skies Cloudy skies poorly under sunny skies, substantially overestimating the interior daylight illuminance. The data generated by DCA show good agreement with the measured data. Under cloudy sky conditions, the DFA is on a par with the DCA and there appears to be no repeated trend in overestimation or underestimation by either approach. The MBE and RMSE for each grid point were computed and are presented in Table 4. The statistical evaluations show that for sunny sky conditions the DFA is inferior to DCA. The two approaches tend to overestimate the interior daylight illuminances, causing the MBE and RMSE values to be very close. Grid point 2 has the highest RMSE for both DFA (67%) and DCA (13.6%) for sunny skies. Under cloudy skies, the errors given by the two approaches are approximately of the same order of magnitude. The MBE values of DFA range from an underestimation of 23.7% at grid point 4 to an overestimation of 7.8% at grid point 1 and the maximum RMSE is 27.4%, appearing at grid point 4. Apart from grid 2, the DCA tends to underestimate the internal daylight illuminance. Its performance is slightly better than DFA with the RMSE results ranging from 14.3 to 23.6%. 6.2 Classroom measurement In total, 12 sets of data were used for the analysis and the details are summarized in Table 5. Since the classroom faced north-west, Table 5 Measured sky conditions for the classroom Sky conditions Date Time Solar altitude, degrees of arc Solar azimuth, degrees of arc SH CLD, okas Sun shaded or overcast 10 July : skies 16 July : July : July : July : July : Sun facing under 11 July : non-overcast skies 11 July : July : July : July : July :

13 DHW Li et al. 125 Figure 7 Performance of the daylight factor and daylight coef cient approaches for grid points 1 to 3 of the classroom the window facë ade was diœusely illuminated to a low value in the morning or under overcast sky conditions and direct sunlight giving high illuminances in the afternoon under non-overcast skies. In general, the data set can be classi ed according to whether direct component penetrates into the classroom. Two categorizations, namely sunlit under non-overcast sky conditions (containing a certain amount of direct sunlight) and sun shaded or overcast skies (excluding the direct component) were used. The indoor daylight illuminances for the ve grid points were computed by means of DFA and DCA using RADIANCE software. The predicted values for the rst row (points 1 to 3) when the sun was `seen by the window wall (i.e., including direct sunlight) are plotted in Figure 7. It can be observed that the data simulated by DCA follow the trend of the measured values quite closely. The peak diœerence is less than 380 lux corresponding to 16% of the measured value. The DFA, however, is considered to underperform. When the light directly from the sky consisted of only the direct component, the measured illuminances were far greater than the illuminances from diœuse daylight modelled by DFA. The predicted illuminance based on DFA can be less than one-third of the measured value. Likewise, the estimated illuminance data for the three points containing no direct sunlight (less than 2000 lux) are also shown in the Figure 7. Both the DFA and DCA generated illuminances are reasonably close to the measured data. The DFA is capable of producing data as reliable as DCA under such a condition. As the direct sky component is of a diœuse nature, the sky luminance distribution is close to the CIE standard sky pattern, contributing to the good performance of DFA. The predicted illuminances for points 4 and 5 are displayed in Figure 8. In general, the accuracy of the two approaches for predicting the illuminances at points 4 and 5 is not as good as the accuracy for points 1 to 3. The DFA signi cantly underestimates the daylight illuminances, although the magnitude of the underestimation is less than that shown in Figure 7, when the window facë ade receives direct sunlight. Unlike DFA, DCA only slightly overestimates the indoor illuminance. It appears that both approaches tend to overestimate the internal re ected daylight. Since points 4 and 5 are not near the window wall, the direct sky component becomes less important and the inter-re ected light can be dominant. It has been pointed out that the

14 126 Predicting daylight illuminance by computer simulation techniques Figure 8 Performance of the daylight factor and daylight coef cient approaches for grid points 4 and 5 of the classroom convergence trend becomes more apparent as the number of ambient bounces increases. 11 Although, the maximum ambient bounce of 7 was set, 18 the errors were not insigni cant. This indicates that the daylight factors and daylight coe cients predicted by RADI- ANCE may not fully take into account of the eœect of the complex inter-re ections for the furnished classroom with moderate external obstructions. The MBE and RMSE for the ve grid points were determined and are listed in Table 6. The statistical results indicate that the DCA can estimate interior illuminances to a satisfactory accuracy for the ve grid points under various sky luminance patterns. The DCA performs well in computing the illuminance if direct sunlight is included in the calculations. The illuminances of the rst row of grid points (points 1 to 3) were modelled more accurately than the illuminances of the grid points in the second row (points 4 and 5). The illuminances exhibit relatively low MBE and RMSE values for the points Table 6 Summary of MBE and RMSE for interior daylight illuminance using daylight factor and daylight coefficient approaches (classroom) Sky conditions Grid point Daylight factor Daylight coef cient MBE (%) RMSE(%) MBE (%) RMSE(%) Sun shaded or overcast skies Sun-facing under non-overcast skies

15 DHW Li et al. 127 in the rst row. The smallest RMSE, of 5.2%, occurs at grid point 2, where the sky component consists of the direct component. The maximum RMSEs for the rst and second rows are 20.9 and 35.1%, respectively. The high RMSE for the second row is due mainly to the internal furniture layout and the external obstructions. Apparently, the DFA performs less accurately than DCA particularly when the direct component is included in the sky component. Without direct sunlight, the sky luminance distribution is akin to that from overcast patterns. When the window wall could `see the centre of the sun disk, the predicted daylight illuminances of DFA are always far less than the measured values, resulting in very large MBE and RMSE results. The predictive ability of DFA is, thus, comparable to that of DCA when the window facë ade is shaded from direct sunlight or under overcast sky conditions. It is markedly inferior for the other situations. This supports the argument that DFA cannot model the dynamic variations in sky pattern. The RMSE range from 11.4% for grid point 2 to over 70% for grid point Conclusions A study of two indoor daylight prediction methods, namely the daylight factor and daylight coe cient using the RADIANCE program and measured data in a scale model and a classroom has been carried out. Eight sets of illuminance data for the scale model and 12 sets of illuminance data for the classroom were recorded and analysed. The scale model was covered with black paper of near zero re ectance to test the performance of the two approaches when the direct daylight component was mainly considered. It was found that the daylight coe cient approach (DCA) is more accurate than the daylight factor approach (DFA) under sunny skies. When the sky becomes overcast, both approaches produce similar results, with acceptable accuracy. To evaluate the eœects due to external obstructions and inter-re ections, the study was extended to measure data in a classroom. The DCA gives more accurate results than DFA for the points near the window when the daylight directly received by the facë ade contains direct sunlight. When the daylight component is mainly diœuse, the predictive ability of DCA is similar to that of DFA. When the measurement points are far away from the window wall, neither of the results produced by the two approaches are in very good agreement with the measured values. It seems that the parameters, including details of the furniture and external obstructions, cannot be input to RADIANCE with su cient accuracy for inter-re ected component calculations. The DCA provides an accurate way of predicting the interior daylight illuminance data. The RADIANCE program is an appropriate tool for calculating daylight coe cients. The present analysis using a scale model and a furnished classroom can be considered to be a preliminary study on the DCA and the RADI- ANCE program. Checking the accuracy of the DCA and the RADIANCE software, especially the inter-re ected components, will be carried out in the future when more reliable measured data for modelling various room layouts and external obstructions are available. Acknowledgements The work described in this paper was supported by a grant from the City University of Hong Kong (Project No.: ). CCS Lau is supported by a City University of Hong Kong Studentship. The authors would like to thank J Mardaljevic for sending copies of his publications and CY Leung, TW Huan and TS Tai for their help with data collection. 8. References 1 Hopkinson RG, Petherbridge P, Longmore J. Daylighting. London: Heinemann, 1966.

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