ANALYTICAL OPTIMIZATION OF SIMPLE ROOF SHADING DEVICES. Khaled Nassar 1 and Mohamed Aly 1. line below should be left

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1 ANALYTICAL OPTIMIZATION OF SIMPLE ROOF SHADING DEVICES Khale Nassar 1 an Mohame Aly 1 1 The American University in Cairo line below shoul be left ABSTRACT Parametric an algorithmic esign tools have evelope significantly in the last few years with the avent of several commercial an open-source applications an simulation software. These tools have been use extensively in the esign an analysis of various builing elements such as glazing, screens, massing an shaing evices. Typically, ether iterative, parametric or optimization techniques are couple with simulation software to reach an optimum esign for the consiere element. Setting up these problems usually requires time an effort, but sometime the esigners nee a simple tool to esign particular elements. This paper presents an analytic approach for esigning simple roof shaing evices. The propose approach is base on a groun-up analysis of the shaing evice geometry to reach global optimum esigns for the elements. The approach is applie to a simple roof shaing element an it is shown how ifferent objectives an situations can be moelle using this approach. The results are compare to ENERGY PLUS simulations to verify the results. It is shown that comparable results can be achieve with the propose approach without the nee for the numerical simulation an in less time. INTRODUCTION In warm climates it is often important to protect of the builing from unwante solar gain as a key part of any cooling strategy which is most reaily achieve by blocking the sun s rays before they reach the builing (R. Mc Cluney, 1990). It has been shown repeately that concrete roof slabs without thermal insulation lea to thermal stress resulting in negative effects on the thermal comfort of the occupants (Gare et al,. 2005). Effective Shaing is an excellent option for controlling solar raiation an reucing the amount of heat gain in builings. By blocking both the irect an iffuse solar raiation they offer great passive potential in hot environments where cooling loas are significant, while movable shaing aresses nocturnal raiation in night hours A number of publications aresse the effect of shaing wall openings, such as winows, on energy consumption (Sherif et al., 2011), while other publications examine the epenence of the thermal loas reuction on the size of solar shaing systems (Franzetti et al., 2004), (Afnor, 2003) an (Kuhn et al., 2000). Gare (2005) provies a comprehensive list of various research efforts on esigning shaing evices. Liangliang et al 2012 investigate the rotation of an integrate solar panel (BIPV) in a shaing evice in Hong Kong. In orer to maximize the energy generate while reucing the solar raiation ifferent tilt angles were explore. The simulation results showe that the shaing evice BIPV i significantly increase the total energy benefits relative to PV moules. Niccolò et al (2012) evelope an algorithm to esign a ynamic solar shaing system for an office builing situate in Milan, Italy. The stuy aime to efine the analytical metho for efining the optimal movement profile for ynamic shaing system base on the horizontal mobile blins for inoor visual comfort. This stuy allowe to efine the annual movement of shaing evices, to customize the behaviour of every single shaing evice an ensure constant control of their movements. In aition a wie range of simulation tools can be use to assess the effect of shaing on the various aspects of the builing. These tools often require a etaile builing input an some will have a significant run time. While these may be of importance in many situations, it is often the case where the esigners nee a simple tool to esign a particular builing element. Therefore, often a fully flege simulation is not warrante an a simple esign tool (in the form of a spreasheet or an aon) may be require. Although the many previous research efforts aresse the issue of shaing evices for winows, esign of roof shaing evices has not receive an equal amount of attention. The work presente here aresses this gap an tries to evelop a tool for the esign of simple roof shaing evices (RSD). The goal is to esign a simple tool with a user frienly interface to globally optimize very simple typologies of the RSD. In the next section a formulation of the RSD problem is presente

2 Figure 1, Samples of simple roof shaes PROBLEM FORMULATION Often problems relate to optimization of shaing evices can be formulate analytically from the groun up by consiering the geometry of the shaing evice an the builing. In general, most shaing evices try to minimize the amount of irect solar raiation falling on the builing surface. Since the solar beam has a known irection, it is often possible to moel such problems by consiering the amount of shaing the evice will rener on the builing surface. This shae area can be formulate as a function in the esign of the shaing evice an the solar angles (azimuth an altitue) given a particular time. Then the total shae area throughout the year can be calculate by integrating that function over the hours of the ay an over the all the ays in the year. Figure 2, The main variables in the problem; s is the spacing between the center lines of the shaing pitch angle. Simple roof shaes are efines as an array of linear rectilinear shaing elements place in a plane parallel to the roof with each element perpenicular to the roof plane. Samples of this simple configuration are shown in figure 1. The main goal is as the ratio of the global solar raiation (irect, iffuse an reflecte) receive on the roof in presence of shaing evices to the global solar raiation which woul be receive on the roof without the RSD. However, since we usually set up optimization as a minimization problem we will take the I + I + I ω(t) = (1) FS (t) I + FS (t) I + I Where, FS (t) is the instantaneous shaing factor for irect raiation; FS is the instantaneous shaing factor for iffuse raiation; I,I,I are the irect, iffuse an reflecte solar irraiance respectively falling on the roof plane winow without the RSD installe. In this case, obviously no groun reflecte component is consiere. Since we assume that I +I +I are constant regarless of roof orientation (in the case of no obstructions), the main goal not become to maximize FS (t) an FS (t). Figure 2, shows the ifferent variables in the problem. Consier first the situation where the RSD is oriente in along the West-East orientation with all the louvers in the RSD being of unit length (i.e. =1) an a sufficiently large roof shae area such that all the shaow falls on the roof. In this case the FS (t)

3 can be simply by (given by triangles ABC in figure 2), FS (t) = ( () ) (()) (2) (t) is the projecte solar altitue angle at horizontal plane. Figure 3, variation of the Instantaneous Direct Shaing Factor with the sun altitue an 5 ifferent louver pitches. It can be seen from the figure that it captures the variations of the tilt angle of the RSD an shows that even small variations in the pitch of the RSD in both irections can result in significant changes in shaing factor. Note that this figure shows the un-normalize values of the RSD (which can be easily normalize by expressing it as a percentage of the roof shae area, or through the maximum value). (t) For s > an /2 > (t) > 0 s s -.cos((t)) (t).cos((t)) s -.cos((t)) s -.cos(- (t)) (t) (t) For s > an /2 > (t) > s.cos(- (t)) Figure 3, variation of the Instantaneous Direct Shaing Factor with the sun altitue an 5 ifferent RSD louvers pitches (0.0, 1.5, 0.75, 2.25 an 0.1 raians) Note that we consier the absolute ifference between the solar altitue an the angle of the shaing evice to account for any irection for both between 0 an Pi. Now if we consier the fact the RSD can take any orientation on the roof we also nee to a the yaw angle then the FS (t) becomes, sin ( β(t) θ sin ( γ(t) δ FS (t) = (3) sin (β(t)) sin (γ(t)) Where γ(t)is the an δ is the yaw angle given a certain sun position. Next we consier the iffuse shaing factor SFs(t). SFs(t) is epen on the ratio of the sky ome that is expose to the roof an as such is a function of the pitch an yaw of the RSD an therefore is constant with respect to θ or β (as seen in figure ) an is equal to, SF (t) = 1 ( cos θ cos δ ) (4) θ an β can reaily be converte to time t (an vice versa) by the traitional equation of time an the solar position equations. Figure 4, the area of the roof expose to the sky ome. Total Instantaneous Total Shaing Factor ω(t) is sum of the SF(t) an SFs(t) an figure 5 shows the variation of the Instantaneous Total Shaing Factor with the sun altitue an louver pitch, for RSD with an orientation facing ue south. The figure shows the asymmetric nature of the value ω(t) with respect to pitch. The esign parameters that coul be capture here are the pitch, the yaw, the spacing an the epth of the RSD along with the placement location (height of installation h, an the sie offsets of the RSD, l1, l2, etc as shown in figure (2). ω(t) θ Figure 5, the variation of the Instantaneous Total Shaing Factor ω(t) with the sun altitue an louver pitch β

4 So what is the best angle to place the louvers? We integrate the function evelope earlier over all approximate position of the sun to come up with the total shae area. The above is the instantaneous shaing factor at any given point in time. Now the total shaing factor (φ) through the year can be given by integrating the sum of the irect an iffuse raiation throughout the year ω(t) = I SF (t) + I SF (t) φ(θ, γ,, s) = ω = I SF (t) β δ + I SF (t)β δ (5) where a, b, c an are limits to be efine from the geometry of the roof in relation to the RSD. For example a an b are efine by the limits of the roof ege in each irection an is given as a function of the installe height of the RSD as shown by figure. Note that these values can also be use as esign variables, i.e. by change the installe height an the offsets l1, l2, etc the value of the Total Shaing Factor can easily by calculate. Optimum values for the ifferent esign variables can be obtaine by simply equating the partial erivatives of (φ) an fining the solution to this system of equations, φ θ = 0 & φ γ = 0 (6) One simplification here is that I an I are the annual raiation an not calculate for each sun position (i.e. the assumption of an isotropic sky), which effectively means that all sun angles are given equal weight. This is one so that a simpler analytical solution can be foun an this not significantly impact the overall optimum esign in terms of the shaing factor, mainly because these values are more or less proportional to the solar altitue an azimuth. Figure 6, shows the variation of the Total Direct Shaing Factor with the louver pitch an yaw over the entire year (assuming a sufficiently large roof so that all of the RSD shaow falls on the roof), which confirms the overall trens. One of the biggest avantages of this analytical approach is the ability to fin global optimum solution to the problem an the ability to incorporate other objectives. ω(t) Figure 6, Variation of the Total Direct Shaing Factor with the louver pitch an yaw over the entire year INCORPORATING OTHER OBJECTIVES The Total Shaing Factor function (φ) can be moifie to a other objectives easily. For example a linear combination of the Total Shaing Factor an the cost can be achieve by aing a term that counts the number of louvers (using s an ) an multiplies this value by the cost. Weights can then be ae to the Total Shaing Factor an the cost. Also, as was mentione above consierable amount of heat gain occurs from the roofs of builings in hot climates an therefore esigners often opt for incorporating roof shaing into their esigns as architectural features. These architectural esign solutions have to provie shaing on one han but nee to provie a minimum amount of openings for efficient ventilation of the HVAC roof top equipment such as the chillers an cooling towers. An efficient esign of roof shaing is one that provies a workable compromise between the amount of the roof shae an the ventilation provie for the roof top equipment. This compromise can be foun by aing as the reciprocal of the FSs again with a ifferent weight. COMPARISON TO SIMULATION RESULTS In orer to test the valiity of the evelope moel, a number of ENERGY PLUS simulations were conucte to assess the actual annual egree of shaing versus the preicte. A typical roof in Cairo, Egypt was selecte an we calculate the total annual irraiation (measure in KWhr) on the surface given ifferent configurations. The actual variation in the solar altitue in the same site were use in the moel (34 egrees to 84 egrees)

5 Figure 7, variation in the Total Shaing Factor with pitch for a specific RSD configuration. Figure 7, shows variation in the Total Shaing Factor with pitch for a specific RSD configuration. It can be seen that in this case there are two complement angles; one which maximizes the φ an the other minimizing it. In orer to test the propose moel we consiere 30 ifferent scenarios of the roof/rsd configurations (i.e. by varying the,s,l1,l2,etc ) an conucte an exhaustive enumeration of the all the yaw an pitch to etermine (similar to the ones shown in figure 8) which is the best values for these variables. The values of the φ where compare to the total annual incient solar raiation for these 30 cases an the ifferences range from 5 to 9 percent with an average value of 7.5%. Figure 8, sample simulation runs for the RSD at ifferent pitches an a yaw = 0. DEVELOPED TOOL We propose an alternative approach to the optimum esign of solar shaing evices through an analytical technique of the geometry of the shaing evices themselves an the position of the sun. An optimum esign can be foun analytically without the nee for simulation or optimization search techniques. By fining the optimum values of the various esign parameters of the shaing evice though mathematical analytical techniques, we save the time an effort require to set up simulations as well as the ae overhea of coupling the simulation techniques with optimization search algorithms. In aition, the analytical esign of shaing evices

6 opens the oor for their wier implementation an use in the larger professional architectural esign community since no extensive knowlege about builing simulation or optimization will be require. In cases with close form solutions, simple plug-in formulae can be use. A simple tool was evelope where the user can enter the ifferent esign variables interactively, information about the project an the basic geometry of the roof. The tool then procees to fin the optimum yaw an pitch, spacing, etc of the roof esign. This is one in almost realtime as well as proviing the user with sensitivity information about the total shaing factors an the esign variables. The interface of the tool is shown in figure 9. The tools is still a work in progress an other features such as importing from a CAD file an breaking own complex geometry is also planne. Figure 9, The interface of the propose tool CONCLUSIONS Parametric an algorithmic esign tools have evelope significantly in the last few years with the avent of several commercial an open-source applications an simulation software. These tools have been use extensively in the esign an analysis of various builing elements such as glazing, screens, massing an shaing evices. Typically, ether iterative, parametric or optimization techniques are couple with simulation software to reach an optimum esign for the consiere element. This paper argues for an alternate analytic approach that coul be use for various elements. The propose approach is base on a groun-up analysis of the builing geometry to reach global optimum esigns for the elements. The approach is applie to a simple roof shaing element an it is shown how ifferent objectives an situations can be moelle using this approach. One of the biggest avantages of this analytical approach is the ability to fin global optimum solution to the problem. This analytical approach can also be applie to other shaing elements. One main avantage of the propose approach is the spee an ease by which the ifferent esign parameters can be etermine as oppose to setting up a fully flege simulation. One limitation of the moel is the ifficulty in hanling irregular shape roofs, even though such roofs can be approximate by consiering a number of ifferent rectangular roofs next to each other. This however is an important issue for future research. Also other configurations of the RSD can be consiere, e.g. RSD with sub-elements going in the irection perpenicular to the main elements. Also here equal weights are given for all sun angles by consiering total annual raiation values. It may be the case that shaing uring certain times may be more avantageous than others, or even if certain times will require shaing will others will not. This can also be incorporate into the moel in the future. It is important that a similar approach can be use for vertical openings REFERENCES ASHRAE, (American Society of Heating, Refrigerating an Air Conitioning), ASHRAE Hanbook Funamentals, chap. 29. Afnor, EN ISO 15099, Thermal performance of winows, oors an shaing evices etaile calculations. Bekooy R.G., Computer Shaow Analysis Technique for Tilte Winows Shae by Overhangs, Vertical Projections, an Siewalls. Corrao V., Filippi M. an Serra V., A Detaile Analysis of the Effects of External Shaing Systems on Winow Solar Gain, Proceeing of Clima 2000, Napoli. Corrao V. an Serra V., Analysis of solar raiation an implementation of a calculation moel for energy characterization of fenestration systems, Proceeings of Clima 2000 Brussels, Corrao V., Serra V. an Vosilla A., Performance Analysis of External Shaing Devices, Plea The 21th Conference on Passive an Low Energy Architecture. Einhoven, The Netherlans. Franzetti C., Fraisse G., Achar G., Influence of the coupling between aylight an artificial lighting on thermal loas in office builings, Energy an Builings, 36 (2), pp Feuerstein G.R., General Case Analysis of Shaow Area Equations for Winow Overhang, Siefins, an Parapets. Gare F, Davi M, Aelar L, Ottenwelter E., Elaboration of thermal stanars for French tropical islans: presentation of the PERENE project. In: Proceeings of Clima 2005, Lausanne, Switzerlan. Khalil M. H., Sheble S.S., Morsy M. S., Fakhry S., Thermal Performance of Expose Compose Roofs in Very Hot Dry Desert Region in Egypt (Toshky), Proceeings of the Tenth International Conference for Enhance Builing Operations, Kuwait

7 Kuhn T.E., Hler C.B., Platze W.J., Evaluation of overheating protection with sun-shaing systems, Solar Energy, 69 (1) pp Liangliang Sun, Lin Lu, Hongxing Yang, Optimum esign of shaing-type builing-integrate photovoltaic claings with ifferent surface azimuth angles, Applie Energy, Volume 90, Issue 1, February 2012, Pages , ISSN , /j.apenergy Mc Cluney R., Awning shaing algorithm upate, ASHRAE Transactions, vol. 96, Part 1. Niccolò Aste, Rajenra Singh Ahikari, Clauio Del Pero, An Algorithm for Designing Dynamic Solar Shaing System, Energy Proceia, Volume 30, 2012, Pages , ISSN , /j.egypro Sherif A., El-Deeb K., El Zafarany A. an Aly M., Effectiveness of External Wall Shaing in Reucing the Energy Consumption of Desert Builings, ICSDC