Thermal performance of building roof elements

Building and Environment 37 (2002) 665 – 675

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Thermal performance of building roof elements Sami A. Al-Sanea Department of Mechanical Engineering, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia Received 1 April 1999; received in revised form 10 May 2001; accepted 31 May 2001

Abstract The study concerns the evaluation and comparison of the thermal performance of building roof elements subject to periodic changes in ambient temperature, solar radiation and nonlinear radiation exchange. A numerical model, based on the /nite-volume method and using the implicit formulation, is developed and applied for six variants of a typical roof structure used in the construction of buildings in Saudi Arabia. The climatic conditions of the city of Riyadh are employed for representative days for July and January. The study gives the detailed temperature and heat 1ux variations with time and the relative importance of the various heat-transfer components as well as the daily averaged roof heat-transfer load, dynamic R-values and the radiative heat-transfer coe2cient. The results show that the inclusion of a 5-cm thick molded polystyrene layer reduces the roof heat-transfer load to one-third of its value in an identical roof section without insulation. Using a polyurethane layer instead, reduces the load to less than one-quarter. A slightly better thermal performance is achieved by locating the insulation layer closer to the inside surface of the roof structure but this exposes the water proo/ng membrane layer to c 2002 Elsevier Science Ltd. All rights reserved. larger temperature 1uctuations. Keywords: Thermal insulation; Building roofs; Heat transmission; R-value; Finite-volume method

1. Introduction The use of thermal insulation and special types of building materials has increased signi/cantly in recent years in both hot and cold climates. This was due to the increasing demand on the thermal comfort of people inside residential, commercial and governmental buildings besides the ever increasing cost of energy. The thermal design of buildings depends on the indoor conditions required, the outdoor prevailing climatic conditions, and the choice of building construction materials and insulation. Accurate methods of analysis to predict the thermal performance of a whole building envelope or an element are, therefore, sought. A whole building thermal analysis is quite involved since all mechanisms of heat transfer are present and the building components are composite of many layers of di9erent materials. The analysis is often time dependent since the outside ambient temperature, wind speed and solar radiation vary with time. Also, the heat gains due to occupants, equipment, lighting, and solar radiation transmission through fenestration, besides the ventilation and in/ltration of the outside air will have to be accounted for. Therefore, various methods with di9erent levels of simpli/cation exist for building energy calculations such as the transfer function, the degree day and bin methods [1]. Computer codes are also available to perform complicated building load analysis. Mathews

et al. [2] compared the thermal load predictions of six such programs for a number of cases and found signi/cant di9erences in the results of the various methods. Jensen [3] reported procedures for validating complex simulation codes. Utilizing a prede/ned computer program, based on the thermal response factor method, Eben Saleh [4,5] investigated the e9ects of using di9erent insulation materials, thicknesses and arrangements on the thermal performance of buildings. The studies that deal with the thermal performance of speci/c building components, and not the building as a whole, have their own merits. Indeed, computer codes for building energy simulation can bene/t from improvement in the modeling of their components. The problem can thus be reduced, in general, to solving the Fourier heat conduction equation through a composite structure subject to time-dependent boundary conditions. Ozisik [6] described various analytical methods for solution of one-dimensional problems with temperature-independent properties; such methods include the separation of variables, orthogonal expansion technique, Green’s function, Laplace transform and integral method. By separating the thermal capacitance into discrete components, Letherman [7] determined the temperature response of a slab wall to a sinusoidal heat 1ux input wave. Han [8] described a numeric-analytical approach using complex algebra of analyzing linear periodic heating-cooling problems in laminates. Chen and

c 2002 Elsevier Science Ltd. All rights reserved. 0360-1323/02/$ - see front matter PII: S 0 3 6 0 - 1 3 2 3 ( 0 1 ) 0 0 0 7 7 - 4

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S.A. Al-Sanea / Building and Environment 37 (2002) 665–675

Nomenclature Ao Bi c Fo h Is k L N q Q Re Rn t T Tf; i Tf; o Tf; o; mean Tsky v x

amplitude of periodic temperature ◦ variation ( C) Biot number = hHx=k speci/c heat (J=kg K) Fourier number = Ht=(Hx)2 heat-transfer coe2cient (W=m2 K) solar radiation 1ux (W=m2 ) thermal conductivity (W=m K) layer thickness (m) number of layers heat 1ux (W=m2 ) daily total heat 1ux (MJ=m2 day) e9ective resistance (m2 K=W ) nominal resistance (m2 K=W ) time (s) ◦ temperature ( C or K) ◦ indoor air temperature ( C) ◦ outdoor air temperature ( C) ◦ mean outdoor air temperature ( C) sky temperature (K) wind speed (m=s) coordinate direction (m)

Lin [9] extended the hybrid application of the Laplace transform technique and the /nite-element method to include nonlinear radiation boundary conditions in composite layers. Al-Turki and Zaki [10] predicted the thermal performance of building walls in which the general solution was obtained by superposition of the steady-state solution and harmonic terms. They studied the e9ect of insulation and energy storing layers upon the cooling load; the analysis showed that dispersion of the insulation material within the building material was less e9ective than using a continuous equivalent insulation layer placed on the outdoor facade. Zedan and Mujahid [11] developed applications of the Fourier series technique to inverse Laplace transforms to solve the problem of heat conduction in composite plane walls. The temperature distribution was obtained in closed form in the Laplace s-domain and was transformed back to the time domain using a series formula. Using the concept of the sol–air temperature, Threlkeld [12] studied the periodic heat transfer through walls and roofs based on the periodic solution of the heat conduction equation. This latter analysis was used by Kaushika et al. [13] to investigate the solar thermal gain of a honeycomb roof-cover as a means for passive solar space heating and energy conservation. The same method of analysis was used by Bansal et al. [14] to study the effect of external surface color on the thermal behavior of a building. They, also, carried out measurements on a scaled model.

Greek letters Hx Ht ' ( ) * + !

thermal di9usivity (m2 =s) internodal distance (m) time step (s) surface emissivity solar absorptivity density (kg=m3 ) Stefan–Boltzmann constant (W=m2 K 4 ) phase shift angle (rad) frequency (rad=s)

Subscripts c convi convo i o r s st 1; 2

convection inside convection outside convection inside surface or nodal point outside surface radiation exchange solar radiation storage layer number or nodal point.

Yarbrough and Anderson [15] compared the solar heat gains for a 1at concrete roof deck with and without radiation control coating. Reductions in cooling loads ranging from 60% to 85% were calculated. Ozkan [16] investigated the performance of 1at roofs, water proo/ng, and insulation under hot and dry climatic conditions. Insulation materials were tested and properties were compared under di9erent aging conditions. Under the climatic conditions of Greece, Eumorfopoulou and Aravantinos [17] studied the thermal behavior of planted roofs and compared it with a bare roof. It was concluded that planted roofs contributed to the thermal resistance of buildings, but that was not enough to replace the thermal insulation layer. The /nite-di9erence method was utilized by Kuehn and Maldonado [18] to calculate the time-dependent thermal response of a composite wall. The explicit formulation was used and the radiation boundary conditions were included. Using the /nite-di9erence method, Kosny and Christian [19] modeled various metal stud walls and calculated their R-values. For the climatic conditions of Riyadh, Al-Sanea [20] evaluated the thermal performance of di9erent wall structures used in construction of buildings in Saudi Arabia. The study was conducted under one-dimensional periodic conditions using a control-volume /nite-di9erence method. The e9ect of wall orientation was studied and the R-values under dynamic conditions were determined. The objective of the present study is to evaluate and compare the thermal performance of building roof elements


subject to steady periodic changes in ambient temperature, solar radiation and nonlinear radiation exchange. An implicit, control volume /nite-di9erence method is developed and applied for six variants of a typical roof structure used in the construction of buildings in Saudi Arabia. The dynamic R-values of the roofs are determined under the climatic conditions of Riyadh for representative days for July and January.

2. Mathematical formulation The geometrical con/guration is depicted in Fig. 1. The roof section consists of a number of layers with di9erent thicknesses and physical properties. The outside surface is exposed to solar radiation (Is ), convection heat transfer (qc; o ) and radiation exchange with the sky (qr; o ). The inside surface is subject to combined convection and radiation heat transfer (qi ) which relates directly to the air-conditioning load required to maintain the inside design temperature (Tf ; i ). The mathematical model is formulated using the following assumptions.

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where is the thermal di9usivity (k=)c) and the subscript j refers to the layer, i.e. j = 1; 2; : : : ; N , k is the thermal conductivity, ) is the density and c is the speci/c heat. The problem is principally concerned with the solution of Eq. (1), applied to all layers, to obtain the variations of temperature and heat-transfer rates subject to prescribed initial and boundary conditions. The initial temperature is taken as uniform and equal to the daily mean of the outside ambient temperature, Tf ; o; mean . Of course, any other value can be used in the model since the steady periodic solution is independent of the initial temperature distribution. The boundary conditions are given as follows: (i) Boundary conditions at the inside surface (x = 0): @T −k1 = hi (Tf ; i − Tx=0 ); (2) @x x=0

where hi is the inside-surface combined heat-transfer coef/cient; from the ASHRAE handbook of fundamentals [1]: hi = 9:26 W=m2 K for upward direction of heat 1ow; and

(i) There is no heat generation. (ii) The layers are in good contact, hence the interface resistance is negligible. (iii) The variation of thermal properties is negligible. (iv) The thickness of the composite roof is small compared to the other dimensions. Hence, a one-dimensional temperature variation is assumed. (v) The convection coe2cient is constant and is based on the direction of heat 1ow and daily average wind speed.

hi = 6:13 W=m2 K for downward direction of heat 1ow:

Based on the above assumptions, the conduction equation governing the heat transfer through the composite roof is given by

The coe2cient (hc; o ) is a function of wind speed (v). Empirical values are taken from Ito et al. [21] as

2

@ Tj 1 @Tj ; = @x2 j @t

(1)

(ii) Boundary conditions at the outside surface (x = L): @T −kN = hc; o (Tx=L − Tf ; o ) − (Is − qr; o ; (3) @x x=L

where hc; o is the outside-surface convection coe2cient, Tf ; o is the outside ambient temperature and ( is the solar absorptivity of the outside surface.

hc; o = 18:63 V 0:605 and, V=

in W=m2 K

0:25 v

if

v ¿ 2 m=s;

0:50 v

if

v ¡ 2 m=s:

(4)

The temperature (Tf ; o ) is /tted by a sinusoidal function over a 24-h period, with t = 0 corresponds to midnight, as Tf ; o = Tf ; o; mean + Ao sin(!t − +);

(5)

where Ao is the amplitude, ! is the frequency, and + is the phase shift angle; these parameters are given in Section 4. The solar radiation (Is ) is calculated for horizontal roofs in Riyadh by using the ASHRAE clear-sky model [1]. The nonlinear radiation exchange (qr; o ) is given by 4 4 qr; o = '*(Tsky − Tx=L );

Fig. 1. Schematic showing a typical composite roof structure.

(6)

where ' is the surface emissivity, * is the Stefan–Boltzmann constant, and Tsky is the sky temperature and is taken equal to (Tf ; o − 12); see for example, Garg [22].

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(iii) Interface node (i) between layers (j) and (j + 1): Tit+Ht =

t+Ht t+Ht (kj =Hxj )Ti−1 + (kj+1 =Hxj+1 )Ti+1 + BTit ; kj =Hxj + kj+1 =Hxj+1 + B

(9)

where B = ()j cj Hxj + )j+1 cj+1 Hxj+1 )=2Ht: (iv) The boundary node on the outside surface: Tnt+Ht =

t+Ht +Bio Tft+Ht + (HxN =kN )((Ist+Ht +qr; o )]+Tnt 2Foo [Tn−1 ;o ; 2Foo (Bio +1)+1 (10)

where Foo = N Ht=(HxN )2 ;

Bio = hc; o HxN =kN ;

and

t+tH 4 qr; o = '*[(Tsky ) − (Tnt+Ht )4 ]:

Fig. 2. Composite roof of N layers showing node arrangements.

3. Numerical solution procedure The solution of the present nonlinear problem is obtained by the /nite-volume method. The composite roof of N layers is discretized into a number of nodes. Next, the /nite-volume equations are derived by applying the energy balance. It is seen from Fig. 2 that there are four types of nodes which are: (i) (ii) (iii) (iv)

boundary node on the inside surface, node 1, interior nodes inside the layers, interface nodes between the layers, and boundary node on the outside surface, node n.

The resulting /nite-volume equations, using the implicit formulation, are summarized as follows: (i) The boundary node on the inside surface: T1t+Ht =

2Foi (T2t+Ht + Bii Tf ; i ) + T1t ; 2Foi (Bii + 1) + 1

(7)

The set of the /nite-volume equations is solved iteratively by using the Gauss–Seidel method. The iterative process within each time step continues until the di9erences between the new and old nodal temperatures are within a predetermined small tolerance value. The solution is carried through a number of cycles until a steady periodic state is fully obtained. It is noted that the nonlinear radiation e9ects are handled within the process of the iterative solution; the temperature at the outside roof surface, which appears on both the left- and right-hand sides of Eq. (10), is calculated using qr; o evaluated from the temperature value at the previous iteration. 4. Roof structures, thermal properties and climatic data The details of six roof structures; namely, R1–R6 are shown schematically in Figs. 3–5. Structures R1 and R2 represent uninsulated roofs with di9erent foam concrete types (I and II, respectively). Structures R3, R4 and R5 represent insulated roofs with di9erent insulation materials as follows: molded polystyrene, extruded polystyrene and polyurethane, respectively. Structure R6 di9ers from R3 only with respect to the location of the insulation layer. The thicknesses of the various layers are: paving tiles = 25 mm,

where Foi = 1 Ht=(Hx1 )2

and

Bii = hi Hx1 =k1 :

(ii) Interior node (i) in layer (j): Tit+Ht =

t+Ht t+Ht Foj (Ti−1 + Ti+1 ) + Tit ; 2Foj + 1

where Foj = j Ht=(Hxj )2 :

(8) Fig. 3. Schematic of roof structures R1 and R2; (1) tiles, (2) mortar bed, (3) sand /ll, (4) membrane, (5) foam concrete, (6) reinforced concrete, and (7) cement plaster.


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following parameters are calculated and used in the study: ◦

Tf ; o; mean = 34:7 C; ◦

Tf ; o; mean = 12:3 C;

◦

Ao = 7:8 C ◦

Ao = 4:5 C

for July; for January;

and

+ = 0:8333:

Fig. 4. Schematic of roof structures R3, R4 and R5; (1) tiles, (2) mortar bed, (3) sand /ll, (4) thermal insulation, (5) membrane, (6) foam concrete, (7) reinforced concrete, and (8) cement plaster.

The daily average wind speed is 4:1 m=s for July and 2:6 m=s for January. The solar radiation 1ux on horizontal roofs Is is calculated from the direct and di9use components based on the ASHRAE clear-sky model [1]. The latitude, longitude and the standard meridian of the local time zone pertinent to the city of Riyadh are speci/c input to the model. 5. Results and discussion 5.1. Introduction

Fig. 5. Schematic of roof structure R6; (1) tiles, (2) mortar bed, (3) sand /ll, (4) membrane, (5) foam concrete, (6) reinforced concrete, (7) thermal insulation, and (8) cement plaster.

Table 1 Thermal properties Material

k (W=m K)

) (kg=m3 )

c (J=kg K)

Paving tile Mortar bed Sand /ll Molded polystyrene Extruded polystyrene Polyurethane Water proo/ng membrane Leveling foam concrete (I) Leveling foam concrete (II) Reinforced concrete Cement plaster

1.73 0.72 0.33 0.036 0.029 0.022 0.19 0.52 0.08 1.73 0.72

2243 1858 1515 24 35 32 1121 1600 300 2243 1858

920 837 800 1213 1213 1590 1675 837 837 920 837

mortar bed = 20 mm, sand /ll = 50 mm, water proo/ng membrane = 4 mm, foam concrete = 75 mm, reinforced concrete slab = 150 mm, cement plaster = 15 mm and insulation layer = 50 mm. The thermal properties of the materials used are summarized in Table 1; see for example, Croy and Dougherty [23]. With regard to the climatic data, the ambient temperature and wind speed values for Riyadh for the year 1993 are used; these data were obtained from the Meteorological and Environmental Protection Agency [24] for the whole year. The temperatures are averaged for each month on an hourly basis using the daily variations for that month. Then, the averaged representative daily variations are /tted by a sinusoidal function as given by Eq. (5). The values of the

The thermal performance of the six roof structures is determined through the time of day; the day represents an average of all days in a month. The months of July and January are chosen to represent typical summer and winter condi◦ tions. The indoor temperature (Tf ; i ) is controlled at 25 C for ◦ July and 23 C for January. The radiative properties are set to ' = 0:9 and ( = 0:4 as appropriate to light-colored surfaces. Steady periodic results are achieved after about three periods (cycles) of computation; the initial transient e9ects, which diminish with time, have no major interest in the present study. All the results are checked for the e9ect of numerical parameters and shown to be both grid and time-step independent. The accuracy of the numerical model is validated by comparisons with exact analytical solutions for a simpli/ed problem. Due to space limitation, this section concentrates on detailed selected results that highlight and compare the thermal performance of the roofs. 5.2. Roof structure R1 This represents a typical uninsulated roof using a foam concrete layer of type I. Fig. 6 displays the temperature distributions across the roof thickness at di9erent times with an interval of 3 h for the representative day in July. The inside surface of the roof corresponds to the left of the /gure and the di9erent layers are shown by the dotted lines. At 3:00, the /gure shows that the temperature is relatively high near the middle of the roof and decreases gradually towards both the inside and outside surfaces. Three hours later, at 6:00, the temperature level across the whole thickness undergoes a further decrease as the stored heat continues to dissipate from both surfaces of the roof. Then a fast increase in outside surface temperature takes place due to sunrise and continues until about 15:00 during which the temperature gradient is reversed as heat is being gained. Thereafter, the surface temperature drops continuously after sunset and the whole process is repeated in a steady periodic fashion.

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Fig. 6. Temperature variations across the roof thickness at di9erent times for structure R1 in July.

The variation of temperature is smooth across each layer with discontinuities in the gradient at interfaces because of di9erent conductivities on both sides of each interface. Due to the thermal storage e9ect, it is noted that while the temperature at the outside surface is decreasing, the temperature deep inside is still increasing. Also, the temperature 1uctuations are reduced signi/cantly with distance from the outside surface. The minimum 1uctuations are calculated at the ◦ inside surface; however, these which amount to about 1:5 C may still be considered large. The inside surface tempera◦ tures are higher than the design room value (25 C) by about ◦ 3 C. The roof heat-transfer load would be proportional to such di9erences in temperature and is expected to be large as will be seen later. The corresponding results for the month of January give rise to similar remarks but with opposite direction of heat 1ow. Fig. 7 shows the temperature variations with time of day at six distinct locations in the roof. In general, the temperature variation is approximately sinusoidal since it is driven by changes in ambient temperature and solar radiation. The inside roof surface temperature (curve 2) shows a maximum di9erence with the space (room) temperature at about 24:00 and a minimum di9erence at about 13:00; these correspond to the maximum and minimum rates of heat gain into the space during the day. The di9erence in temperatures between curves 3 and 4 is relatively small as a result of the small thermal resistance across the membrane layer. The time lag between the maximum temperatures attained by the outside and inside surfaces (curves 5 and 2) is calculated to be about 10 h. Fig. 8 depicts the variations with time of day of the various heat-transfer rate components for roof structure R1 in July. These are: the absorbed solar 1ux (solar abs ), outside-surface convective 1ux (convo ), outside-surface radiation exchange 1ux (rad), and the net of these (net). The inside-surface heat 1ux (convi ) is also shown which relates to the cooling load

Fig. 7. Temperature variations with time at di9erent locations for structure R1 in July; (1) inside room, (2) inside roof surface, (3) inside interface of membrane, (4) outside interface of membrane, (5) outside roof surface, and (6) outside ambient.

Fig. 8. Components and net heat-transfer rate variations with time for structure R1 in July.

required to maintain the controlled room temperature. While the absorbed solar radiation is either positive or zero, which is virtually the dominant source of heat gain, the radiation exchange (rad) is calculated to be negative throughout. Accordingly, radiation exchange contributes to heat loss only which is favorable in summer. The outside-surface convective 1ux (convo ) shows a more complicated behavior since there are losses and gains at di9erent times of day. This depends on the roof outside-surface temperature relative to the ambient air temperature. The results indicate that the losses by radiation exchange are greater than the losses by convection. The daily averaged values of these components are given in Table 2 in which quantitative comparison can be made.


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Table 2 Summary of total and daily-averaged thermal values Code number

Rn (m2 K=W)

Re (m2 K=W)

R1JUL R2JUL R3JUL R4JUL R5JUL R6JUL

0.670 1.463 2.059 2.394 2.943 2.059

0.529 1.156 1.626 1.891 2.323 1.626

R1JAN R2JAN R3JAN R4JAN R5JAN R6JAN

0.628 1.421 2.017 2.352 2.900 2.017

0.735 1.664 2.362 2.755 3.398 2.359

Qc; o (MJ=m2 day)

Qs (MJ=m2 day)

Qr (MJ=m2 day)

hQr (W=m2 K)

1.584 0.725 0.515 0.443 0.361 0.515

−3:031 −3:677 −3:831 −3:886 −3:948 −3:841

11.480 11.480 11.480 11.480 11.480 11.480

−6:866 −7:079 −7:135 −7:153 −7:173 −7:125

5.67 5.68 5.68 5.68 5.69 5.68

−1:261 −0:557 −0:392 −0:336 −0:273 −0:393

−2:884 −2:355 −2:230 −2:188 −2:140 −2:233

7.260 7.260 7.260 7.260 7.260 7.260

−5:638 −5:461 −5:423 −5:408 −5:392 −5:417

4.51 4.50 4.50 4.50 4.49 4.50

Qi (MJ=m2 day)

Fig. 9. Layer energy-storage variations with time for structure R1 in July; (1) cement plaster, (2) reinforced concrete, (3) foam concrete, (4) membrane, (5) sand /ll, (6) mortar bed, and (7) tiles.

The sum of the above three components is the net heat transfer rate at the outside surface (qnet ). Fig. 8 shows a net heat loss starting in the late afternoon and continues until sunrise; at other times, there is a net heat gain which is attributed to solar radiation. The /gure also reveals that much of the heat gained by the roof is stored inside the various layers of the structure and then dissipated to the outside. This has a great advantage in reducing the rate of heat transmission into the space as indicated by relatively small inside-surface heat 1ux (convi ). Further insight into the thermal behavior of the roof section can be obtained by studying the daily variation of the rate of energy storage in each layer as shown in Fig. 9 for July. The rate of energy storage depends upon both the thermal capacitance and temperature 1uctuation. Considering the reinforced concrete slab (curve 2), for example, it is noted that there is a net storage of heat between 11:30 and


22:30 and a net heat dissipation outside this period. The total amount of heat storage must equal the total amount of heat dissipation over a complete cycle (one day). Similar behavior is obtained for other layers but with di9erent rates and time lag. 5.3. Roof structure R3 This represents a typical insulated roof with a 5-cm layer of molded polystyrene placed above the water proo/ng membrane; except for the insulation layer, this structure is identical to R1. Fig. 10 shows the variations of temperature across the roof thickness for July and are to be compared with the results in Fig. 6. It is seen that the presence of the insulation has a marked e9ect in which sharp changes in the temperature slopes are calculated across the interfaces with the insulation layer. Also, the bulk of the temperature drop through the roof takes place across the insulation leaving

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the layers on the inside having a relatively very small temperature drop and also much smaller variations with time. This has two advantages; /rstly, the transmission load is reduced in magnitude and, secondly, the amplitude of load 1uctuation is reduced in size. It is noted that the variation of ◦ temperature on the inside surface amounts to less than 0:3 C ◦ during the whole day and is only 1 C higher than the inside ◦ room temperature of 25 C. These characteristics are superior when compared to those obtained for roof structure R1. With regard to the large temperature variations across the outer layers, a strong resemblance is seen with those for R1. 5.4. Roof structure R6 This represents a typical insulated roof with the thermal insulation placed close to the inside surface; except for the location of the insulation layer, this structure is identical to R3. Fig. 11 displays the variations of temperature across the roof thickness for di9erent times of day. It is seen that the results are similar to those in Fig. 10 for the three layers that lie close to the outside surface. Starting from the membrane layer and moving inwards, the results in Fig. 11 are distinguished by higher temperature levels and larger 1uctuations. The membrane layer is now exposed to a higher temperature and a greater temperature variation with time which are unfavorable since they can cause cracks and property deterioration after a certain period of use. The insulation layer gives rise to a substantial temperature drop across its thickness and acts to reduce the daily temperature 1uctuation signi/cantly. It is seen that the e9ect on the inside surface is quite similar to that for R3, i.e. an inside surface temperature ◦ ◦ of about 26 C (1 C higher than the inside room temperature) and with a very small variation throughout the day. It may be concluded, therefore, that the two structures, R3 and R6, behave similarly with respect to their e9ect on the mean cooling load, further details are given in the next sec-

Fig. 12. Inside-surface heat 1ux variations with time for all roof structures in July.

tion. However, when interest lies beyond the steady-periodic conditions, the two roof sections are expected to behave differently during the transient periods of operation. This point is left for further research as, indeed, with the di9erent effects the two arrangements would have on possible moisture condensation in wet climates. 5.5. Comparisons between all roofs The inside-surface heat 1ux variations with time for all six roof structures (R1–R6) are presented in Fig. 12 for July. The /gure reveals a relatively large heat 1ux 1uctuation for structure R1 and a much greater mean value as compared to the results of the other structures. Roof R2 shows a more moderate 1uctuation and a mean value that is about one half of that calculated for R1. The thermally insulated structures (R3–R6) show much smaller 1uctuations and mean values that are particularly low. The heat 1ux for structure R1 reaches a minimum value at about 13:00 and a maximum value at about midnight. For structures R3–R6, the heat 1ux reaches its minimum at about 14:30 and its maximum at about 2:30; this time lag is due to the added e9ect of insulation. It is interesting to note that the heat 1ux variations with time for roofs R3 and R6 are close but with a slight di9erence in favor of R6. The calculations reveal that the daily average inside-surface heat 1uxes are the same for roofs R3 and R6; however, the peak load for R6 is about 3% smaller than that for R3. Fig. 13 presents the energy storage variations with time of day for the concrete slab for all roofs. The results show that the concrete slab stores heat at various rates from about 12:00 to 24:00. The maximum storage rate is calculated at about 17:00 for R1, R2 and R6 and at about 18:00 for R3, R4 and R5. The slab dissipates heat from about 24:00 to 12:00 with a maximum rate of dissipation calculated between 7:00 and 8:00. It is also noted that the energy storage behavior


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Fig. 13. Energy storage variations with time of the reinforced concrete slab for all roof structures in July.

of the concrete slabs in R1 and R6 are very similar despite the di9erences between the two roofs with regard to the general performance. This is because the concrete slab in R6 is located on the outside of the insulation layer and, hence, behaves similarly to the slab in R1 which has no insulation. The slab heat storage variations with time for R3, R4 and R5 are relatively very small due to the smaller temperature 1uctuations as a result of placing the insulation on the outside of the slab. The overall thermal performance of the roofs is presented in Fig. 14 in terms of the daily total heat-transfer load per square meter of roof structure (Q in MJ=m2 day). For July, structure R1 shows an extremely high heat-transfer load (1:584 MJ=m2 day) as compared to the other structures. Taking this value as a reference load for comparison, structure R2 gives about 46% of the reference load. This gives the e9ect of using the light weight leveling foam concrete with a higher thermal resistance. With regard to the e9ect of using thermal insulation, structures R3, R4 and R5 give the following percentages of the reference load: 33%, 28% and 23%, respectively. It is also noted that roof R6 reduces the heat-transfer load by an equal amount to that achieved by R3. Similar remarks can be drawn for the heating loads of the roofs in January. Figs. 15 and 16 give the corresponding thermal resistances (R-values) of the various roof structures. A distinction is made between the nominal resistance (Rn ) and the e9ective, or dynamic, resistance (Re ). The Rn values, presented in Fig. 15, are calculated as the sum of the various conductive and convective resistance components in the structure. The Re values, presented in Fig. 16, are calculated from the daily averaged di9erence between the outside ambient and inside room temperatures divided by the daily averaged heat-transfer rate at the inside surface. Therefore, the Re values contain implicitly the e9ect of the outside-surface radiation exchange and the e9ect of the solar 1ux absorbed

Fig. 14. Daily total heat-transfer loads per square meter for all roof structures in July and January.

Fig. 15. Total nominal thermal resistances for all roof structures in July and January.

by the roof within the framework of the total resistance. By all standards, the R-value for roof structure R1 is considered too small, and such a roof would demand a considerably large cooling or heating load as was seen in Fig. 14. It is interesting to note that while the July and January Rn

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Fig. 16. Total e9ective thermal resistances for all roof structures in July and January.

values of a particular roof are very close (the di9erences are due to the surface convective resistance), the Re values for July are noticeably smaller than their counterparts for January. Therefore, the July values give more stringent conditions on the allowed R-values for the roofs. The di9erences, for any one structure, between Rn and Re on the one hand and those due to the seasonal variations on the other, make the Re values of a more general use since they represent the dynamic nature of the heat-transfer process and also re1ect the climatic conditions. 5.6. Outside-surface radiative heat-transfer coe7cient (hr; o ) This is calculated by the model based on the outsidesurface temperature of the roof, which is unknown and is determined iteratively by the calculations, and Tsky which is known from the climatic data as a function of time. A typical variation of hr; o with time of day is shown in Fig. 17 for roof structure R1 in July. This resembles to a good degree the outside roof surface temperature variation shown earlier as curve (5) in Fig. 7. The daily averaged value is calculated to be 5:67 W=m2 K; the daily di9erence between the minimum and maximum values amounts to about 20%. Similar trends of variations are calculated for all roofs with the daily averaged values for July are about 25% higher than those for January. These results are summarized in Table 2. 6. Conclusions A numerical model based on an implicit /nite-volume method was developed for calculating the time-dependent temperature variation in composite layers under nonlinear boundary conditions. The model was applied for the simu-

Fig. 17. Outside-surface radiative heat-transfer coe2cient variation with time for structure R1 in July.

lation and comparison of the thermal characteristics of six variants of a typical roof structure used in building constructions in the Kingdom of Saudi Arabia. The investigation was carried out under steady periodic state using the climatic conditions of Riyadh. The results showed that, as expected, the dominant source of energy gain by the roof was the absorbed solar radiation. While this acted adversely in summer and favorably in winter, the radiation exchange with the sky and the convection heat transfer acted in exactly the opposite manner. In general, the contribution of the radiation exchange was more than twice that of the heat convection. The results, when compared with a reference uninsulated roof section using a heavy weight concrete foam as a leveling layer, produced the following: 45% of the reference daily average heat-transfer load when using a light weight concrete foam; 32%, 27% and 22% of the reference daily average heat-transfer load when using a 5-cm thick layer of insulation made of molded polystyrene, extruded polystyrene and polyurethane, respectively. Placing the thermal insulation layer closer to the inside surface of the roof section showed a little favorable e9ect on the instantaneous heat-transfer load; however, this had exposed the water proo/ng membrane layer to larger temperature 1uctuations. It is recommended that future studies should include, as well, the economic side of the problem, moisture transport and condensation in roof sections under certain climatic conditions, interface and voids resistance between the layers, and the initial thermal transient e9ects. Appendix Tabulated total and daily averaged thermal values Table 2 gives a summary of the important total and daily averaged thermal values calculated for all roofs in July and January. The /rst column in the table gives the identi/ca-


tion code number, the second column gives the roof nominal thermal resistance (Rn ), while the third column gives the effective thermal resistance (Re ). Columns four to seven give the daily total quantities of heat-transfer per square meter of roof: Qi for the inside-surface heat transfer (+ve relates to cooling load and −ve relates to heating load), Qc; o for the outside-surface heat convection (−ve means loss from roof to outside ambient), Qs for the absorbed solar radiation (+ve means gain; the absorptivity is 0.4), and Qr for the radiation exchange between outside-surface and sky (−ve means loss from roof to sky; the emissivity is 0.9). Finally, the last column gives the daily mean outside-surface radiative heat-transfer coe2cient. From an overall energy balance for the roof section over 24 h, Qc; o + Qs + Qr must equal to Qi under steady periodic conditions. It is interesting to note that the largest daily total heat-transfer component at the roof outside surface is the absorbed solar radiation which acts adversely in July and favorably in January. It is also noted that the daily contribution of the radiation exchange at the outside surface can be more than twice that of the convection. This is due to the relative magnitudes of the radiative and convective heat-transfer coe2cients and to partial cancellation of the convective 1ux when integrated over the whole day, as can be seen for example in Fig. 8. References [1] ASHRAE. Handbook of Fundamentals. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., 1993. [2] Mathews EH, Shuttleworth AG, Rousseau PG. Validation and further development of a novel thermal analysis method. Building and Environment 1994;29(2):207–15. [3] Jensen SO. Validation of building energy simulation programs: a methodology. Energy and Buildings 1995;22:133–44. [4] Eben Saleh MA. Thermal insulation of buildings in a newly built environment of a hot dry climate: the Saudi Arabian experience. International Journal of Ambient Energy 1990;11(3):157–68. [5] Eben Saleh MA. Impact of thermal insulation location on buildings in hot dry climates. Solar and Wind Technology 1990;7(4): 393–406. [6] Ozisik MN. Heat Conduction, 2nd ed. New York: Wiley, 1993. [7] Letherman KM. A rational criterion for accuracy of modelling of periodic heat conduction in plane slabs. Building and Environment 1977;12:127–30.

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