Landscape and Urban Planning 94 (2010) 149–157

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Landscape and Urban Planning

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Effect of tree shades in urban planning in hot-arid climatic regions

V.M. Gómez-Munoza, M.A. Porta-Gándarab,∗, J.L. Fernándezc

a CICIMAR-IPN, PO Box 592, La Paz, BCS 23000, Mexicob Engineering Group-CIBNOR, PO Box 128, La Paz, BCS 23000, Mexicoc UNAM, Ciudad Universitaria, 04510, Mexico

a r t i c l e i n f o

Article history:Received 19 March 2009Received in revised form 6 July 2009Accepted 15 September 2009Available online 6 October 2009

a b s t r a c t

The present study is carried out for dry hot climate places, where excessive solar heating is felt throughoutthe year. The effect of tree shadowing buildings is found to reduce heating loads; hence trees have abeneficial effect in energy economics. The emerging economic value of tree shadows in hot climate citiesgrants the development of an appropriate simulation numerical method to establish relative advantageson energy savings related to dwelling envelopes. The results demonstrate that large trees can provide up

Keywords:TCS

to 70% shade during spring and autumn, thus saving a very large amount of energy along the whole year.Hence, economic value of larger trees is greater than that of younger species.

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. Introduction

The emerging energy costs, as well as sustainability consider-tions, have produced an increased value of tree shadows ontowelling envelopes. This new trend runs against the prevailingrchitectural fashion for low rise office buildings that leads to aransparent appearance of the buildings due to large glazed areasf the facade (Kuhn, 2006). Nevertheless, the buildings should reli-bly provide good thermal and visual comfort and have low energyonsumption. This tendency, quite popular in Europe, Asia and thenited States, has also become fashionable in warmer countries likeexico, where solar energy thermal radiation levels are markedly

igher. Shading the building envelopes is therefore most important,oth to reduce energy consumption in cooling loads, improvingnergy economics, and comfort.

Previous research has analyzed the impact of different facadesn the energy consumption of the building (Stec and Van Paassen,005). The best option for solar control appeared to be the tra-itional facade with external shading. Optimization processes,owever, do very seldom include external shading when reduc-

ng HVAC (heating, ventilating and air conditioning) energy loadsor specific comfort conditions. This goal can only be reached if theacade is carefully designed to provide effective protection against

xcessive solar gains. A good option in residence buildings andouses could be that of vegetal shading.

Some authors have provided experimental measurements tovaluate the influence of trees on the heat transfer between a

∗ Corresponding author. Tel.: +52 612 1238484x3614.E-mail address: maporta@cibnor.mx (M.A. Porta-Gándara).

169-2046/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.landurbplan.2009.09.002

© 2009 Elsevier B.V. All rights reserved.

building facade and the environment (Papadakis et al., 2001). Theimpact of the tree covers on reducing electric loads and hence costscan be properly measured. Comparisons were made experimen-tally between shaded and non-shaded areas regarding the air andwall temperatures, the heat transfer toward the wall, wind speedand air humidity. Consequently, the application of plants in build-ings provides a great potential for energy savings, because apartfrom their aesthetic value, they can contribute greatly to reduceelectrical demand peaks.

Some urban programs aim at reducing air pollutant emissions bylowering external ambient temperatures through tree planting andlight-colored surfacing. In some cities where summer temperaturesare high, tree planting is one of the most cost-effective means ofmitigating urban heat islands and associated expenditures for airconditioning. Trees are considered essential to moderating the heatgained by asphalt parking lots, for example. Cooler air temperaturesreduce ozone concentrations by reducing hydrocarbon emissions(as results from gasoline evaporation) that are involved in ozoneformation (McPherson, 2001).

The application of plants and trees to shading buildings pro-vides an efficient passive method of solar control (Parker, 1983).The radiative and thermal loads in the shaded area have proved tobe significantly lower relative to the non-shaded one. In addition,the evaporative cooling effects of the plants have resulted in lowerair temperature around the shaded wall. Apart from the energy sav-ings that can be achieved from the use of trees as shading devices,

there are also significant improvements on the environment whichresult from the reduced emissions due to the energy savings andthe aesthetic influence of trees on urban landscape.

The factors and parameters involved in passive cooling ofcanyon-type streets with trees have been studied by other authors

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50 V.M. Gómez-Munoz et al. / Landscap

Shashua-Bar and Hoffman, 2003). Using a cluster thermal timeonstant model that considers construction materials and its build-ng geometry, they demonstrate that cooling effect depends on themount and extent of the partial shaded area.

The microclimate in urban cities can be modified by means ofree plantation. The consideration of green spaces is growing as anmportant constituent of city planning, since plants improve aes-hetically the spaces that surround the buildings and contribute toontrol the ambient temperature. According to (Ca et al., 1998), aearby park can reduce the surrounding air temperature by up to◦C.

Other work has been performed in desert regions to modifyhe microclimate by means of appropriate species. For instance,ix indigenous trees were investigated in the Negev Desert, where

he relation between shade and the height/width of the tree canopyas shown with simple rectangular shaped trees. It was found that

he width is more important than the height of the trees to pro-ide shading below them and within the vicinity of the tree canopyKotzen, 2003).

ig. 1. Pictures of the three studied tree species, and their different schematic represenndica and (c) Tabachin Delonix regia.

Urban Planning 94 (2010) 149–157

In order to simulate tree shades by means of a mathematicalapproximation, other authors have used geometric shapes for sim-ulation purposes without introducing substantial errors. Severalgeometric solids were used for simulation of tree crowns and theirshadows (McPherson and Rowntree, 1988) and the mean percent-age difference between formula and photo-estimated tree crownprofile was only 1.3%.

The analysis of the benefit that trees can provide in energy sav-ings, as well as other multiple advantages they provide, can behighlighted by determining the community and home owner sav-ings from planting trees, as has been already reported (McPhersonet al., 2000). Accordingly, every large tree produces savings ofapproximately $2600 over a 40-year period in southern California.

Mathematical models are widely used to evaluate tree shad-

ows. The precision and completeness of mathematical modelinghave been pursued by several researchers. For example, a simpli-fied method to study interactions between tree–sun–building toestimate the tree effects on reduction of energy use has been stud-ied (Simpson, 2002). The method took into account the energy use

tations with parallelepipeds. (a) Benjamina Ficus benjamina, (b) Neem Azadirachta

V.M. Gómez-Munoz et al. / Landscape and Urban Planning 94 (2010) 149–157 151

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Table 1Estimation of parameter K for the Von Bertalanffy growth curves, from field data tr,Ltr and L∞ .

Common name Species tr (years) Ltr (m) L∞ (m) K

ig. 2. Parallelepiped parameters representing the main characteristics of the tree:enter of foliage C, and foliage’s extent in the three cartesian directions fx , fy and fz ,espectively.

ithout trees to provide a reference to calculate changes in energyse due to shade. It also provides a useful discussion on the com-lexity of processes that account for tree configuration (species, agend location), building characteristics (window size and locations,uilding orientation, level of insulation and neighboring buildings,mong others) and weather conditions, among the many factorshat can affect the economics of tree shading, and analyzes several

athematical models that can be employed to that effect.Experimental measurements of the energy savings in two one-

torey houses that include the shade of 16 trees of different speciesere performed (Akbari et al., 1997). Based on the experimental

esults, and using DOE-2.1E (Department of Energy Software), itas shown that the energy savings can reach 29% of the total loads

n both sites.A technique to reasonably calculate the effectiveness of a certain

hadowing device or plant was used (White, 1982). A horizontalrojection technique on facade surfaces, using “pseudo shadows”

nd gnomonic diagrams, allows the shading effects on any surfacen any orientation to be seen directly and calculated. According tohe rules of solar geometry and the superposition of real shadowsn a common horizontal plane, the procedure allows the analysis

ig. 3. Projected shadows on three plane elements of a typical construction at 24◦

orth latitude: (a) roof; (b) facade and (c) courtyard by a simulated tree during anquinox at 8:30 h.

Tabachin Delonix regia 10 6.25 8 0.152Benjamina Ficus benjamina 6 5 7 0.208Neem Azadirachta indica 3 4 10 0.170

of solar rights, shading, and direct gains on walls and windows tobe performed, which is very useful in many cases, as, for example,in tall building clusters.

A model to predict the effect of trees as passive cooling optionson buildings has been studied using spherical, conical and cylindri-cal shaped tree approximations (Raeissi and Taheri, 1999). Resultsshow that cooling loads may be reduced between 10% and 40% in atypical house in Iran, with proper tree planting.

Others, by means of numerical methods, have determined thepassive cooling effects of a courtyard by the presence of the wallsof a one-storey building and two large trees planted in front of thesouth wall, using the same tree shape simulation (Safarzadeh andBahadori, 2005).

In any case, design tools are required to assess site and buildinglayouts for passive solar design, whether for external shading orotherwise. The resolution of such issues can be achieved from con-sideration of the angular relationship between the sun, the buildingand any shading devices and obstructing bodies.

A simulation model enables the study of various possible set-tings of a building concerning energy consumption, capacity of thesystem, and indoor comfort quality.

In the present work, a set of Matlab programs were developedto evaluate the tree shading and the blocked solar radiation on one-storey constructions, over facades and roofs, for an arid hot and dryclimate region. The study considers the growth of the trees over 15years and different tree planting patterns in front of the facade, rep-resentative days of the year, and several facade orientations. Threecommon tree species in the city of La Paz, Mexico, were employedin the numerical simulation. This case studied is deemed to be quiterelevant to many geographical regions in the world.

The better understanding of the dynamic properties of treeshadows on housing facades is explored. It is then put in the largerperspective of thermal loads reduction, and hence, energy sav-ings. As stated at first, this approach reveals that trees would have,

apart from their aesthetic and exterior freshening contribution, anincreasing economic value along the years. It must be stressed thatthe wide spread use of larger trees to provide shadow and hencethermal load reduction, will significantly contribute to the larger

Fig. 4. Growth curves of neem, benjamina and tabachin trees.

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owadays worry of energy savings, carbon foot print and thus cli-ate change.

. Methodology

For the tree shadowing evaluation, a set of parameterizationsnd algorithms needs to be developed. It includes the geometricalree representation, tree growth modeling, shading projection andlocked solar radiation simulations.

.1. Tree simplified representation

Since this work deals with trees and the calculation of their shad-ws as projected onto any vertical wall, or facade, and roof, it wille most desirable to put forward an easy although realistic modelf any tree. First it should be considered that trees can adopt manyeometrical shapes.

Fig. 5. Shadow projections of an arbitrary point (x,y,z), calculated from

Urban Planning 94 (2010) 149–157

Three tree varieties were chosen among the most common toprovide shadows on walls in the studied region: tabachin (Delonixregia), benjamina (Ficus benjamina), and neem (Azadirachta indica).The first two species have become very well known in the regionfor many years. On the other hand, the neem was introduced fromIndia toward the later part of the twentieth century, and has provedto be a very dynamic tree, which is very much appreciated becauseof its quick growing rate and lush shadow. Other important reasonfor the selected species was their commercial availability, relativelow cost, abundant foliage and rapid growth.

In order to get an easy way to calculate areas of the tree shadowson the principal elements of dwelling (facades, roof and courtyard),

each one of the tree species considered in this work was simulatedby a geometric simplification of the shape of the tree, representedby the aspect ratio between its height and foliage extent.

The different aspect ratios of the studied trees are illustrated inFig. 1, with a rectangle approximation drawn on top of the picture.

solar angles. On the courtyard: (x′ , y′), and over facade: (y′′ , z′′).

V.M. Gómez-Munoz et al. / Landscape and

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To simulate the change of the shadows originated by the treegrowth along the years, the height of each tree species was calcu-lated from the Von Bertalanffy growth model (Eq. (5)). This model isphysiologically well founded, and requires only two parameter esti-mates, the maximum tree height and a growth parameter (Rammig

ig. 6. Shadow projection (x′′′ , y′′′) on the roof of an arbitrary point (x,y,z), calculatedy means of the solar angles.

he rectangle has been chosen to include the thick of the foliage,nd sparse branches are left out of it. Clearly, other rectangles cane chosen as well, and other geometrical figures can also be useds other authors have done (Raeissi and Taheri, 1999). The simpleethod proposed is quite attractive due to its simplicity.The parameterization of this approximation is described based

n the coordinates of the center of foliage C = (c1,c2,c3) and the threeimensional foliage extents: fx, fy and fz. The foliage was repre-ented by a parallelepiped, whose vertexes are formed by the eightollowing combinations: (c1 ± fx, c2 ± fy, c3 ± fz), which generate theix projecting faces (Fig. 2).

In order to show the instantaneous tree shading, for a partic-lar day, latitude and facade orientation, the six faces of the treeere projected on the building elements of a typical dwelling: roof,

acade, and courtyard (Fig. 3). Each shadow is composed by theix overlapping parallelograms filled in black, corresponding to therojected faces of the parallelepiped of an arbitrary tree.

In order to calculate the projected shadows, let Rect be the rep-esenting rectangle of the dwelling element in the image, Apix the

otal area of Rect in pixels of any color, and AS,pix (t) the instanta-eous total area of the projected shadow inside Rect, which can bexpressed as:

pix = (b − a) · (d − c) (1)

ig. 7. Shadow fraction on each one of the three dwelling elements on an east facingacade.

Urban Planning 94 (2010) 149–157 153

where a and b are the horizontal pixels; c and d are the verticalpixels, delimiting Rect,

AS,pix(t) =∑

(i,j) ∈ RectSpix(i, j) (2)

where Spix(i,j) = 1 for a black pixel (i,j), or Spix(i,j) = 0 otherwise.The shaded fraction area PS(t) is calculated by:

PS(t) = AS(t)A

= AS,pix(t)Apix

(3)

Thus:

AS(t) = A · PS(t) (4)

where A is the area (m2) of the dwelling element.The area size of each shadow was calculated by image proce-

dures, by means of detecting and counting black pixels on the figureinside the projecting surfaces of each of the three dwelling ele-ments. This novel procedure completely eliminates the problem ofshadows overlapping one onto another. Finally, shading area frac-tions are figured out by expressing the shaded area proportionatelyto each of those elements.

2.2. Tree growth curve parameters

Fig. 8. Shading patterns for a specific 6 years old tabachin tree during particular daynumbers (close to each curve), for southern (upper three figures) and eastern (lowerthree figures) orientation.

1 e and Urban Planning 94 (2010) 149–157

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t al., 2007).

(t) = L∞(1 − exp(−K · t)) (5)

here L(t) is the height of the tree (m) at any age t (years), L∞ theaximum mean height of the tree (m), and K the growth rate.For each tree species, the parameter K of the model was esti-

ated from a known height Ltr at age tr (Eq. (6)) during the earlyears, when the tree is growing rapidly. The former variables, asell as the value of L∞, are usually well known by people with

nowledge about these trees. The expression results in:

= − 1tr

ln[

L∞ − LtrL∞

](6)

he aspect ratio of the tree was assumed constant along the growthf the tree through the years, considered as an inherent character-stic of each species.

From field data obtained in the region of the study (Table 1),he growth curve for each tree was estimated for the first 15 yearsf growth (Fig. 4). Neem grows to a higher size and does it moreapidly than the other two species. Tabachin and benjamina trees,n the other hand, have similar growth curves, although theirspect ratio is different.

.3. Shadow projections

The shadow projections, both horizontal and vertical, of an

rbitrary point in space (x,y,z), were calculated from the parameter-zation shown in Fig. 5, where the center of the facade on the floor ishe origin of the coordinate system, x represents the distance fromhe facade, y the offset of the point respect to the central orthogonaline to the facade, and z the height of the point from the floor.

ig. 9. Shading patterns for southern orientation facade on a specific day (355) forwo trees of several ages (tabachin top three figures and neem bottom three).

Fig. 10. Comparison between the amount of solar radiation that a building receiveswhen no tree is present and when a 10 years old neem is included, on the roof (upperfigure) and southern facade (lower figure), during winter solstice (day 355). The greyarea in the figure represents the solar radiation blocked by the tree.

Horizontal projection (x′, y′) on the floor of courtyard:

x′ = x − z

tan �and y′ = y − h sin � (7)

Vertical projection (y′′, z′′) on the facade:

y′′ = y − x tan � and z′′ = x′zx′ − x

(8)

where

h = z

tan ˇ

and the solar angles were calculated according to Duffie andBeckman (1991): � is the profile solar angle, � the differencebetween solar azimuth and surface azimuth angles, and ˇ the solaraltitude angle.

Similarly, the horizontal projection (x′′′, y′′′) on the roof isdeduced from Fig. 6:

x′′′ = x − z − H

tan �and y′′′ = y − (x − x′′′) tan � (9)

where H is the height of the roof.

2.4. Blocked solar energy

The instantaneous beam radiation Gb(t) in W/m2, on the non-shadowed areas was calculated by the usual method (Duffie andBeckman, 1991), for a specific day of the year, latitude and max-

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mum beam radiation on a region, as well as a given slope andzimuth of the dwelling element (facade or roof).

The subsequent total solar energy incident on each surface RT(t)n W, was calculated multiplying the former instantaneous radia-ion by the total area A, of the dwelling element:

T(t) = Gb(t)A (10)

he corresponding radiation on a shaded dwelling element RS(t) in, was calculated from:

S(t) = Gb(t)(1 − PS(t))A (11)

he instantaneous blocked solar radiation is calculated from Eq.12):

B(t) = RT(t) − RS(t) (12)

ig. 11. Blocked energy (MJ per day) on the roof (dashed line), facade (thin line) and totnd benjamina (right column). Variable day numbers and facade orientations are indicate

Urban Planning 94 (2010) 149–157 155

RB(t) is integrated along the solar day in order to obtain the totalblocked solar energy EB in MJ-day:

EB = 3600

106

∫ sunset

sunrise

RB(t)dt (13)

3. Results

With the previously described method, the relative shadow atree projects onto a dwelling can be plotted along the day. Asa case study, the following results were applied to a dwellingwhich measures 3 m high, 14.7 m long and 12 m wide, located in

La Paz, BCS, Mexico (24◦N, 110◦W). This procedure is illustratedin Fig. 7, which contains an example of the time evolution of theshading fraction along the spring equinox for each dwelling ele-ment. In this case, a 6 years old tabachin tree is employed, whichexplains why the shadow is more intense on the facade than on

al (thick line), and the way it varies as two typical trees grow: neem (left column)d at right.

1 e and Urban Planning 94 (2010) 149–157

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he roof, a behavior that will change as the tree grows. The tree islanted 3 m away from the east oriented facade, and that explainshat the shadow disappears around midday from this wall. Thehadow on the roof is reduced quickly as time progresses, beingmaximum in the early hours of the day. This tree produces a

hadow on the courtyard during the whole day, with a maximum

alue at noon. This shadow reduces the albedo heating on thewelling, a contribution that is not further analyzed in the presentork.

A similar analysis can be performed for any day, for differentacade orientations, for a number of trees and for different ages of

ig. 12. Blocked energy corresponding to the shading project of nine neem trees,hree planted in front of each one of the southern, eastern and western facadesor three representative days of the year. Offsets are −6, 0 and 6 m, and separationistance from the facade is 3 m.

Fig. 13. Fraction of the received energy on the facades and roof for three represen-tative days of the year, corresponding to the shading project.

them. A particular case, where two trees are studied, reveals thechange of shadow patterns with date, as shown in Fig. 8. The effectof the orientation is thus depicted.

On the upper part of the figure, a 6 years old tabachin is studiedwhen planted 3 m in front of the southern facade. The numbers onthe curves represent the days in the year for which each curve wasplotted. The most effective shadow is found on day 80, when it isprojected onto the southern facade all day long. On the other hand,on the summer solstice (day 172), a very brief shadow around noonis observed. A tendency that can be detected on the three dwellingelements is that the shadow fraction is higher away from noon,although the roof gets better shading in winter than in summer. Itcan also be noticed that the shade on the courtyard is pretty muchthe same for any day.

In the lower part of Fig. 8, the dwelling was studied for the sametree and configuration described above, except that the shadedfacade is now facing east. There are now large changes in the shad-ing pattern for facade and roof, but not so much for the courtyard.The facade and the courtyard are shadowed very intensely (around50%) along the year during the shading hours. The roof gets a shadeonly during the very first hours in the morning, although whencompared with the upper part of the figure, it can be seen that theshadow on the roof is more intense. Notice that the shading patterndoes not change significantly over the year.

The changing patterns of the shadows on the dwelling are nowexplored as tree ages. The exploration is performed for day num-ber 355 on a southern facing facade, with a tree separated 3 m fromthe center of the facade. On the upper part of Fig. 9, tabachin shad-ing patterns are shown along the day for a tree of ages 2, 4, 6 and 8years. A similar analysis is performed on the lower part of the figure,which is performed for a neem. A notorious change in tree shadowwith age is found on the roof. Geometric differences between thefoliage of the trees produce a clear difference in the shading pat-tern, particularly on the courtyard, when trees of the same age arecompared. The flattened pattern of the shadow that the tabachinproduces is due to its foliage being extended more along the hori-zontal direction than other trees. A similar shade advantage of neemis seen on the shading of the roof. The effect of shading the facade isnot that different between both trees, which attain very much thesame minimum values at noon.

The previous exercise can be extended to calculate the amountof direct solar radiation a tree can block before it hits a dwelling. In

this case, for simplification, the radiation received by the courtyardis not included. Fig. 10 illustrates both roof (top) and facade (bot-tom) effect of a 10 years old neem, during the winter solstice ontothe southern facade, when the tree is 3 m away from the wall. Each

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V.M. Gómez-Munoz et al. / Landscap

pper curve is the solar radiation that would have been receivednto the roof and facade if no tree was present, and each lower curveorresponds to the radiation received when the tree is planted.

The grey area in the drawings corresponds to the radiationlocked by the tree, for the whole dwelling envelope area, for whichhe dimensions were previously described. The blocked radiationt noon is a maximum on the roof, with a value of around 25 kW,hile the maximum effect on the facade occurs around hours 10

nd 14, with about 18 kW. The addition of these two shading effectss quite sizeable when compared with the total energy that woulde received without a tree: 32.83% and 75.67% of blocked radiationn the roof and southern facade, respectively.

A comparison between two trees, benjamina and neem, is nowerformed in Fig. 11. The blocked energy, expressed in MJ-day, islotted for various representative dates and orientations. The shad-

ng effect on the roof is plotted with interrupted lines, the effect onhe facade is represented by a continuous thin line, and the additionf both is plotted with a thick continuous line. The most outstand-ng difference is that, given its fastest growing rate and its highestnal size, a neem blocks more that three times more energy thanbenjamina, because the roof is the main solar collector in this

articular dwelling, and the neem blocks always more energy onhe roof after 6–8 years of age, whereas the benjamina producesonsistently a larger shadow on the facade than on the roof. Southrientation is only included for day 80, since in the other dates,loser to summer solstice, the sun describes an apparent trajectoryloser to the zenith. In general, a neem provides a better shadowhan a benjamina, principally when it grows to an age enough totart shadowing the roof, whereas the benjamina is more suitableo shadow the facade.

.1. Shading project example

An analysis of a shadowing project with three trees in each of thehree sunbathed facades was then performed, a total of nine neemrees. The northern facade, that receives almost no direct solar radi-tion at 24◦N latitude, has no trees. All trees are separated 3 m fromhe walls; one tree is centered along the wall and the other two areeparated 6 m on each side of the tree on the center. The buildinglan is 14.7 m × 12 m, and height is 3 m. Blocked energy is calcu-

ated along the first 10 years of tree growth. Fig. 12 shows the resultsor three typical days, an equinox and the two solstices. Total solarnergy that would be received if no trees are present is figured andritten on top of each graph, in MJ-day. The evolution of the typical

nergy that is blocked, for each studied day, along the 10 years ofife of the trees, is illustrated. As can be seen, after the fourth year,he capacity of the trees to shadow the facades remains almost con-tant. However, the radiation that is blocked onto the roof increasesith time. Hence, total energy blocked is a growing quantity.

As a result, a graph can be plotted with the incident energyhat must be eliminated from the dwelling for comfort, when norees are present. Such a calculation would allow representinghe real economic value of a tree that provides a useful shadow,hich increases with its age. Fig. 13 corresponds to the energy that

rrives on the analyzed building as trees grow. The energy receivedecreases from 100% to a very low value at great tree age, a shadow-

ng that also depends on the date. At the end of 10 years the allowedncident energy on the building is less than 50% of the total received.

. Conclusions

With the results previously discussed, it is economically morettractive to plant large trees instead than planting small trees and

Urban Planning 94 (2010) 149–157 157

nursing them to grow along the years. However, there are still lit-tle local technological skills in La Paz, as well as many other smallcities, to transplant trees larger than 3 m tall. The growing savingsrelated to shadows of older trees suggest that a new market forlarger trees might be emerging. This might be good news given theadded value of trees in sustainable urban development, in the viewof their proven usefulness as temperature arresters and humidityproviders, as well as the aesthetic worth and well-being contribu-tion they signify.

Working with traditional vector handling of sunbeams and styl-ized computer-simulated trees, a procedure has been depicted toproject the shadow of a tree onto any facade and roof, at any time,in any location. The quantification of the shadow, including theoverlapping of tree parts, is very easily performed by means ofimage processes. The procedure can be extended to any numberof trees and facade and roof situations, so that the simultaneouslyshadowing of several trees can be easily handled.

The growth parameter to account for tree growing along theyears was calculated with an original approach, using the VonBertalanffy model. The generally accepted solar physics models areemployed in the present development.

Based on the present results, shadows can be evaluated asenergy savings inside buildings. In the case of La Paz, where treesare relatively cheap and energy is expensive, the relative economicvalue of the tree’s shadow as energy saver will be probably decisiveof its generalization in the foreseeable future.

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