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Energy and Buildings 41 (2009) 1276–1287

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Review

Thermal design optimization of lightweight concrete blocks for internal one-way spanning slabs floors by FEM J.J. del Coz Dıáz a,*, P.J. Garcıá Nieto b, J. Domıńguez Hernańdez c, A. Sua´rez Sańchez d a

Department of Construction, University of Oviedo, 33204 Gijoń, Spain Department of Mathematics, University of Oviedo, 33007 Oviedo, Spain c Department of Mechanical Engineering, University of Zaragoza 50018, Spain d Department of Management, University of Oviedo, 33004 Oviedo, Spain b

A R T I C L E I N F O

A B S T R A C T

Article history: Received 15 May 2009 Accepted 10 August 2009

In the present work, numerical thermal analysis is used in order to optimize the lightweight hollow block design for internal floors with respect to the energy saving, by the finite element method (FEM). From an initial block design with 0.57 m 0.25 m 0.20 m constant external dimensions, other five different configurations were built varying the number the vertical and horizontal intermediate bulkheads. Besides, five different compositions of lightweight concrete have been taken into account, giving place to sixty different configurations of the floors, thirty per each heat flow direction: upward and downward heat flows. Based on the non-linear thermal analysis of the different configurations, it is possible to choose the best candidate block from the CTE rule requirements. Mathematically, the non-linearity is due to the radiation boundary condition inside the inner recesses of the blocks. Also, the temperature distribution and thermal characteristic values of the floors, both for downward and upward heat flows, are provided. From the numerical results, we can conclude that the main variables in the thermal performance are: the number of the horizontal intermediate bulkheads and the material conductivities. Therefore, increasing the number of horizontal intermediate bulkheads and decreasing the material conductivities, the best thermal efficiency is obtained. Optimization of the floors is carried out from the finite element analysis by means of the average mass overall thermal efficiency and the equivalent thermal conductivity. In order to select the appropriate floor satisfying the CTE rule requirements, detailed instructions are given. Finally, conclusions of this work are exposed. ß 2009 Elsevier B.V. All rights reserved.

Keywords: Internal hollow block Lightweight concrete Finite element modelling Non-linear complex heat transfer Energy savings Thermal optimization

Contents 1. 2. 3. 4. 5.

6.

7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of a multilayer floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials of an in situ cast floor with one-way spanning slabs . . . . . . . . . . . . . . . . . . . . . . Geometrical models of the multi-holed blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical model of heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Conduction and convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Radiation implementation: radiation matrix method . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Radiation implementation: calculation of view factors by the hidden method . . . . . Numerical simulation method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Validation of the two-dimensional finite element models: mesh independence and 6.3. Two-dimensional finite element models and results . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* Corresponding author. Tel.: +34 985 182042; fax: +34 985 182433. E-mail address: [email protected] (J.J. del Coz Dıáz). 0378-7788/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2009.08.005

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J.J. del Coz Dıáz et al. / Energy and Buildings 41 (2009) 1276–1287

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1. Introduction Buildings today account for about 40% of energy consumption in developed countries according to the Organization for Economic Co-operation and Development (OECD). The effort for transforming the way buildings are designed, constructed, operated and dismantled has ambitious targets: by 2050 new buildings will consume zero net energy from external power supplies and produce zero net carbon dioxide emissions while being economically viable to construct and operate. Constructing buildings that use no net energy from power grids will require a combination of onsite power generation and ultraefficient building materials and equipment. ‘Green’ buildings already are erected in various parts of the world but current cost structure prevents widespread adoption by builders. Existing standards for energy efficiency in buildings will be the starting point for the industry-led alliance. During the last five years, several experimental results and numerical calculations have been developed for describing two and three dimensional heat transfer of walls made up of hollow bricks [1,2]. Most interests are involved in the optimization process applied to obtain the best design from the thermal point of view in case of clay and lightweight concrete bricks [3–6]. In this new and innovative paper, the thermal analysis and optimization of reinforced concrete one-way spanning slabs for internal floors is carried out, taking into account both upward and downward heat flows (see Fig. 1), in which all heat transfer processes for every constituent materials are considered: conduction, convection and radiation inside enclosures. The thermal conditions for the Spanish buildings are defined at present by the new Building Standard Code (named CTE project [7]). The CTE rule, in section ‘energy saving’ DB-HE, introduces modifications in order to improve the requirements of thermal insulation of the building’s enclosure. The energy demand of

Fig. 1. Scheme of the building’s thermal envelope.

buildings in the CTE rule is limited depending on the city climate and the internal load in their rooms. Besides, in order to avoid decompensations between the thermal qualities of different internal rooms, each one of them should have a transmittance lower than 1.2 W/(m2 K) [7]. The actual possible locations of the internal floors are shown in Fig. 1 for a typical building. This figure also shows the possible thermal flows between houses. With respect to the thermal performance of the concrete slabs, they usually have a high thermal mass. In older buildings, concrete slabs cast directly on the ground can drain heat from a room. In modern construction techniques, concrete slabs are usually cast on top of thicker layers of insulation, for example expanded polystyrene, and may contain underfloor heating. Even so their thermal mass can lead to a delay warming the room when the heating is switched on. This can be an advantage in climates with

Fig. 2. Geometry of different multilayer floors: (a) one-way slab; (b) two-way slab; (c) precast hollow core slab and (d) reinforced concrete slab floor.

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large daily temperature swings, where the slab keeps the building cool by day and warm by night. 2. Geometry of a multilayer floor In Spanish buildings, both for industrial and housing, four different types of multilayer floors are used (see Fig. 2):

In situ cast floor with one-way spanning slabs. In situ cast floor with two-way spanning slabs. Precast concrete floors with hollow core slabs. Reinforced concrete slab and girder floor.

Other authors have previously studied the Life Cycle Analysis (LCA) of two types of slab systems for internal floor structures: in situ cast floors and with precast concrete floors. The most common solution in residential buildings in Spain is the in situ cast floor with one-way spanning slabs [8]. Therefore, this work is applied to the thermal study of this type of floors. 3. Materials of an in situ cast floor with one-way spanning slabs The constituent materials of this kind of floor are the following ones (see Fig. 3): Plant produced concrete. Floor slab and integrative casting have to be made with plant produced concrete (i.e. C-250 concrete). Before casting, the bricks and the beams must be placed, cleaned and thoroughly wetted; vibrate accurately the casting and make sure that the ribbing is well stuffed. The floor slab is completed by a finishing concrete cast in situ which has on top thickness of 5 cm. Weldmesh reinforcement. The supplementary reinforcement (or tensile stress, for negative moments) must be placed at the time of casting in correspondence to the beams so that it is covering will be about 2 cm from the top edge of the finished slab. Prestressed concrete joists. The beams are industrially massproduced for prompt delivery in spans multiple of 20 cm starting from 1.20 up to 7.60 m. The joists are made by 11.5 cm 11 cm beams in prestressed reinforced concrete with five threads of 5 mm diameter. Multi-holed block. In order to study the thermal performance of the floor, we have varied the number of holes of the block, both in horizontal and vertical directions, from a minimum number of three up to a maximum number of twelve recesses. The material used in the multi-holed block is light concrete, due to its good characteristics of thermal insulation. Gypsum plaster. This layer has usually a thickness of 10 mm.

Table 1 Physical properties of the constituent materials. Item

Density (kg/m3)

Conductivity, l (W/(m K))

Plant-produced concrete Weldmesh reinforcement Prestressed concrete joist Gypsum plaster

2200 7850 2200 1100

0.70 60.0 0.70 0.28

Table 2 Physical properties of the lightweight concrete. Composition

Density (kg/m3)

Conductivity, l (W/(m K))

A B C D E

600 800 1000 1200 1400

0.213 0.272 0.347 0.440 0.549

Table 1 shows the physical properties of the constituent materials of the in situ cast floor described above with the exception of lightweight concrete. As it is shown in Table 2, the light concrete has variable density and conductivity. In this work, five typical values of conductivity have taken into account in thermal analysis, corresponding to the different categories of lightweight concrete in the manufacturing process [1,3,4,6]. 4. Geometrical models of the multi-holed blocks In the first place, we have modeled six different types of lightweight concrete hollow blocks (see Fig. 4 below) varying only the number of recessed between them and keeping the same overall dimensions: F1 (three recesses), F2 (six recesses), F3 (nine recesses), F4 (four recesses), F5 (eight recesses) and F6 (twelve recesses). In the second place, we have built an entire floor with each one of the six different types of multi-holed blocks described above. The one-way spanning slab is made of five multi-holed blocks with four joists, including the weldmesh reinforcement, the plantproduced concrete and the gypsum plaster (see Fig. 4). Just as it is shown in Fig. 4, the six types of blocks (F1–F6) have the same external dimensions. These dimensions will be kept constant in the calculation and the subsequent optimization process due to ergonomic reasons. The human body’s response to physical loads in the building of the floors is strongly related to the total mass of the block. This last requirement limits the external maximum dimensions of the block’s design to the values previously indicated above. It is possible to classify the blocks F1–F6 from the number of intermediate bulkheads. In this sense, there are two main groups. The first set has two intermediate bulkheads including blocks F1– F3. The second group has three intermediate bulkheads and it includes blocks F4–F6. The minimum thickness of the intermediate bulkheads is 15 mm in order to keep the mechanical resistance of the entire block in the manufacturing process. 5. Mathematical model of heat transfer The basic equations of heat transfer, namely the energy balance and rate equations are summarized below: 5.1. Conduction and convection The first law of thermodynamics states that thermal energy is conserved. Specializing this to a differential control volume [9– 12]:

rc Fig. 3. Materials of an in situ cast floor with one-way spanning slabs.

@T þ fvgT fLgT þ fLgT fqg ¼ q_ € @t

(1)


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Fig. 4. Geometrical model and dimensions (long width high) of the light concrete multi-holed blocks F1, F2, F3, F4, F5 and F6: 0.57 m 0.20 m 0.25 m (upper) and their corresponding in situ cast floors (lower).

€ heat generation rate per unit volume. q_ ¼

where

r = density.

c = specific heat. T = temperature (=T(x,y,z,t)). t = time. 8 9 @> > > > > > > > > > > @x > > < > @= ¼ vector operator. fLg ¼ > > > > > @y > > > > > > > > ; : @ > @z 8 9 < vx = fvg ¼ vy ¼ velocity vector for mass transport of heat. : ;

vz

{q} = heat flux vector.

It should be realized that terms {L}T and {L}T{q} may also be interpreted as 5T and 5{q}, respectively, where 5 represents the gradient operator and 5 represents the divergence operator. Next, Fourier’s law is used to relate the heat flux vector to thermal gradients [9,13]: fqg ¼ ½DfLgT where

2

(2)

3 0 K xx 0 K yy 0 5 ¼ conductivity matrix: ½D ¼ 4 0 0 0 K zz Kxx, Kyy, Kzz = conductivity in the element x, y, and z directions, respectively.


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Pre-multiplying Eq. (3) by a virtual change in temperature, integrating over the volume of the element, and combining with Eqs. (8) and (9) with some manipulating yields: Z

rc dT

¼

Z

@T þ fvgT fLgT þ fLgT ðdTÞð½DfLgTÞ dV @t

dTq dS2 þ

Z

S2

S3

dTh f ðT B TÞ dS3 þ

Z

€ dTq_dV

(10)

where vol = volume of the element. dT = an allowable virtual temperature (=dT(x,y,z,t)).

5.2. Radiation

Fig. 5. Three-dimensional model with boundary conditions.

Combining Eqs. (1) and (2): @T þ fvgT fLgT ¼ fLgT f½DfLgT g þ q_ € rc @t Expanding Eq. (3) to its more familiar form [11]: @T @T @T @T þ vx þ vy þ vz rc @t @x @y @z @ @T @ @T @ @T Kx þ Ky þ Kz þ q_ € ¼ @x @y @z @x @y @z

(3)

N X d ji

(4)

Specified temperatures acting over surface S1: (5)

where T* is the specified temperature. Specified heat flows acting over surface S2: fqgT fhg ¼ q

(6)

where {h} = unit outward normal vector. q* = specified heat flow. Specified convection surfaces acting over surface S3 (Newton’s law of cooling): fqgT fhg ¼ h f ðT S T B Þ

(7)

where

Note that positive specified heat flow is into the boundary (i.e., in the direction opposite of {h}), which accounts for the negative signs in Eqs. (6) and (7). Combining Eq. (2) with Eqs. (6) and (7) [9–12]:

T

fhg ½DfLgT ¼ h f ðT B T Þ

1 ei

ei

N X 1 Qi ¼ ðd ji F ji Þs Ti4 Ai i¼1

(11)

N = number of radiating surfaces. dji = Kronecker delta. ei = effective emissivity of surface i. A value of ei = 0.88 is adopted for the lightweight concrete [16]. Fji = radiation view factors. Ai = area of surface i. Qi = energy loss of surface i. s = Stefan–Boltzmann constant whose value is 5.67 108 W/ (m2 K4) Ti = absolute temperature of surface i. For a system of two surfaces radiating to each other, Eq. (11) can be simplified to give the heat transfer rate between surfaces i and j as [14,15]: Qi ¼

1 s ðTi4 T 4j Þ ðð1 ei Þ=Ai ei Þ þ ð1=ðAi F i j ÞÞ þ ðð1 e j Þ=A j e j Þ

(12)

where

hf = film coefficient evaluated at (TB + TS)/2 unless otherwise specified for the element. TB = bulk temperature of the adjacent fluid. TS = temperature at the surface of the model.

fhgT ½DfLgT ¼ q

ei

i¼1

F ji

where

It will be assumed that all effects are in the global Cartesian system. Three types of boundary conditions are considered (see Fig. 5). It is presumed that these cover the entire element [12]:

T ¼ T

Radiant energy exchange between neighbouring surfaces of a region, such as internal recesses in the blocks, can produce large effects in the overall heat transfer problem. Though the radiation effects generally enter the heat transfer problem only through the boundary conditions, the coupling is especially strong due to nonlinear dependence of radiation on surface temperature. Extending the Stefan–Boltzmann law for a system of N enclosures or recesses, the energy balance for each surface in the enclosure for a gray diffuse body [14,15], which relates the energy losses to the surface temperatures:

(8) (9)

Ti, Tj = absolute temperatures at surface i and j, respectively. If Aj is much greater than Aj, Eq. (12) reduces to: Q i ¼ sei F 0i j Ai ðTi4 T 4j Þ

(13)

where F 0i j ¼

Fi j F i j ð1 ei Þ þ ei

The view factor, Fij, is defined as the fraction of total radiant energy that leaves surface i which arrives directly on surface j, as it is shown in Fig. 6. It can be expressed by the following


Eqs. (15) and (16): fQ g ¼ K 0 fT g

1281

(21)

3

[K0 ] now includes T terms and is calculated in the same manner as in Eq. (16). To be able to include radiation effects in the recesses of our numerical model, two different stages are necessary. In the first stage, the boundary of all recesses is meshed with uniaxial finite elements. In the second stage, a group of previously assembled elements are treated as a single element termed superelement. This superelement is used to bring in the radiation matrix through the effective conductivity matrix, [Kts], Eq. (19), as well as the view factors required for finding [Kts]. In order to calculate the view factors in the radiation matrix method, the hidden method is used [14]. Fig. 6. View factor calculation terms.

5.4. Radiation implementation: calculation of view factors by the hidden method equation [14,15]: Fi j ¼

1 Ai

Z Z Aj

Ai

cos u i cos u j dA j dAi pr 2

(14)

where:

Fi j ¼

Ai, Aj = areas of surface i and surface j, respectively. r = distance between differential surfaces i and j. ui = angle between Ni and the radius line to surface dAj. uj = angle between Nj and the radius line to surface dAi. Ni, Nj = surface normal of dA1 and dAj.

5.3. Radiation implementation: radiation matrix method In this paper, we have used the radiation matrix method for analysis of the radiation term [15]. In this method, for a system of two radiating surfaces, Eq. (13) can be expanded as: Q i ¼ sei F i j Ai ðTi2 þ T 2j ÞðT i þ T j ÞðT i T j Þ

(15)

or Q i ¼ K 0 ðT i T j Þ

(16)

where K 0 ¼ sei F i j Ai ðTi2 þ T 2j ÞðT i þ T j Þ

The hidden procedure is a method which uses Eq. (14) and assumes that all the variables are constant over the recesses, so that the equation becomes:

(17)

K0 cannot be calculated directly since it is a function of the unknowns Ti and Tj. The temperatures from previous iterations are used to calculate K0 and the solution is computed iteratively. For a more general case, Eq. (11) can be used to construct a single row in the following matrix equation: n o (18) ½C fQ g ¼ ½D T 4

Aj

pr 2

cos ui cos u j

The hidden procedure numerically calculates the view factor in the following conceptual manner. The hidden-line algorithm is first used to determine which surfaces are visible to every other surface. Then, each radiating, or ‘‘viewing’’, surface (i) is enclosed with a hemisphere of unit radius. This hemisphere is oriented in a local coordinate system (X0 Y0 Z0 ), whose center is at the centroid of the surface with the Z axis normal to the surface, the X axis is from node I to node J, and the Y axis orthogonal to the other axes. The receiving, or ‘‘viewed’’, surface (j) is projected onto the hemisphere exactly as it would appear to an observer on surface (i). As it is shown in Fig. 7, the projected area is defined by first extending a line from the center of the hemisphere to each node defining the surface or element. That node is then projected to the point where the line intersects the hemisphere and transformed into the local system (X0 Y0 Z0 ). The view factor, Fij, is determined by counting the number of rays striking the projected surface j and dividing by the total number of rays (Nr) emitted by surface i. This method may violate the radiation reciprocity rule, that is, AiFij 6¼ AjFji. The hidden method requires significantly computer time and it is used in this work because the geometry of the recesses in the lightweight concrete multi-holed block is complex. Some radiating surfaces are isolated completely from the other groups in terms of radiation heat transfer. In this method, the smaller the meshing size of the radiating surface elements, the more accurate is the view factors. Moreover, the summation of the view factors from any radiating surfaces to all other remaining radiating surfaces

Such that: Each row j in ½C ¼ ððd ji =ei Þ F ji ðð1 ei =ei ÞÞÞð1=Ai Þ; i = 1, 2, . . ., N. Each row j in ½D ¼ ðd ji F ji Þs ; i = 1, 2, . . ., N. Solving for {Q}: n o fQ g ¼ K ts T 4

(19)

and therefore: ts K ¼ ½C 1 ½D

(20)

Eq. (19) is analogous to Eq. (11) and can be set up for standard matrix equation solution by the process similar to the steps in

(22)

Fig. 7. Receiving surface projection.


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should be 1.0 for a closed system, such as the internal recesses of the blocks. Both checks are made in this work in order to validate the numerical model proposed. 6. Numerical simulation method

than 2%. The reference model in order to carry out the checking has the shape shown in Fig. 9 (left). Once the area is obtained, the meshing of the same one is carried out with a two-dimensional mesh size of 2 mm (see Fig. 9 (right)). Then we proceed to the analysis of it through two different numerical models and solution procedure:

6.1. General remarks The above governing equations are discretized by the finite element method (FEM) [17–20] and then the thermal behavior of reinforced concrete one-way spanning slabs for internal floors is optimized. The procedure of optimization is based on previous works [3–6]. In order to study the thermal behavior of the floor, the relationship between the light concrete’s density and its thermal conductivity has been previously determined in laboratory tests [4,6]. The curve is fitted by the following expression:

l e ¼ C 1 r2 þ C 2 r þ C 3

(23)

Being C1 = 2.0995 107, C2 = 9.2039 105, and C3 = 1.3787 101 and r is the density of the lightweight concrete. 6.2. Validation of the two-dimensional finite element models: mesh independence and numerical procedure In our preliminary computation, mesh-independence of the solution has been examined for the most complex F6 hollow block in which all processes of heat transfer have been considered [21– 24]. Five sets of the mesh sizes have been checked ranging from 1 to 8 mm. The results of the equivalent thermal conductivity are shown in Fig. 8. Comparing a 4 mm size mesh with a 1 mm size mesh (see Fig. 8), we can conclude that increasing the mesh size there is only no more than 0.1% difference in the equivalent thermal conductivity. In order to reduce the truncation and round-off errors as well as the computational effort, we have chosen the optimum mesh size of 4 mm in the subsequent simulations. The calculation methods pointed out previously should be checked to adapt to what is stated in Annex D of the UNE-EN-1745 Rule [25]. Specifically and according to what is described in section D.3 of the Standard rule [25], the accuracy of the calculation method must be checked by applying it to reference’s cases exposed in section D.4 of the Annex, verifying that the deviation of the conductance obtained with the calculation methods used with respect to the reference’s cases in question must not be greater

Method 1. The problem is solved doing a meshing of the recesses with elements of convection plus radiation [21–24] (see Fig. 9). Method 2. The problem is solved by means of the matrix radiation method [14,15]. The input data adopted to carry out the calculations referred to in section D.4 of the Annex [25] are the following ones: Material conductivity: kmat = 0.35 W/(m K) Boundary conditions: Internal surface resistance of the wall: Rsi = 0.13 m2 K/W and External surface resistance of the wall: Rse = 0.04 m2 K/W. ˙ Thermal heat flux in the lower side: Q=A ¼ 10 W/m2. Film coefficient in recesses: h = 5.933 W/(m2 K) in Method 1 h = 1.7605 W/(m2 K) in Method 2 From the temperature distribution data (see Fig. 10) the equivalent thermal conductivity, leq, for the Method 1 is T1 = 288.903 K T2 = 273.344 K leq = 0.1888 W/(m K) and for the Method 2: T1 = 289.317 K T2 = 273.344 K leq = 0.1844 W/(m K) The value leq obtained according to the reference case is of 0.188 W/(m K). Therefore, the Method 2 has a difference in percentage, that is to say, it is only 0.5%. This difference is less than 2% of deviation required by the Standard rule [25]. Therefore, according to the obtained results, the Method 2 agrees well with the specifications of the Standard and it will be used in subsequent calculations. 6.3. Two-dimensional finite element models and results In order to check the thermal performance of the different types of lightweight concrete multi-holed blocks (F1–F6), six floors (one per each type of block) have been considered. Each one of them is composed of five blocks as it is shown in Fig. 11. Then, we have built the two-dimensional finite element model. For the modelling of this problem, we have used a twodimensional 8-node quadrilateral finite element for the solid area of blocks in order to simulate the thermal conduction phenomenon, a one-dimensional 3-node (plus an extra node) finite element for the recesses of blocks in order to calculate the thermal convection phenomenon and, finally, a one-dimensional 3-node finite element in order to solve the thermal radiation phenomenon inside the recesses of blocks [26,27] (see Figs. 11 and 12). On the one hand, in the model the following external thermal boundary conditions are considered [16]:

Fig. 8. Validation of mesh independence.

Downward heat flow: a q/A = 10 W/m2 heat flow in the upper floor side, a hl = 1/Rsl = 5.88 W/(m2 K) film coefficient in the lower floor


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Fig. 9. Geometrical model of the reference model (left) and FEM model (right).

Fig. 10. Temperature distribution in the reference model for the Method 1 (left) and Method 2 (right).

side, a Rsu = 0.17 m2 K/W surface resistance in the upper floor side and a 273 K ambient temperature. Upward heat flow: a q/A = 10 W/m2 heat flow in the lower floor side, a hu = 1/Rsu = 10 W/(m2 K) film coefficient in the upper floor side, a Rsl = 0.10 m2 K/W surface resistance in the lower floor side and a 273 K ambient temperature. On the other hand, the following internal boundary conditions in the recesses are taking into account [16]: Downward heat flow. The film convection coefficient inside the recesses in this case is 0:44 0:025 (24) ha ¼ max 0:12 d ; d Upward heat flow. The film convection coefficient inside the recesses in this case is 0:025 (25) ha ¼ max 1:95; d Fig. 11. Two-dimensional FE floor: overall view (upper) and a detail (lower).

where d is the thickness of the recesses in the vertical direction.


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Table 3 Numerical results for the mass overall thermal efficiency, ethermal_p (m2 K/W/kg), in percentage for downward heat flow.

Fig. 12. Finite elements: (a) 8-node quadrilateral and (b) one-dimensional 3-node (plus an extra node) elements.

The determination of the thermal efficiency of a floor is based on the well-known expressions [3–6]: U¼

q=A DT

(26)

1 þ Rsu þ Rsl U Rtot ethermal p ¼ M Rtot ¼

lequi ¼

e ð1=UÞ þ Rsu þ Rsl

(27) (28) (29)

On the one hand, Figs. 13 and 14 (left) show the mass overall thermal efficiencies in all analyzed cases, both for downward and upward heat flows, and it reveals that the differences between them are significant, being the block F4 the worst of them. From

Case/block

F1

F2

F3

F4

F5

F6

Case Case Case Case Case

0.40% 0.37% 0.34% 0.32% 0.30%

0.48% 0.43% 0.40% 0.36% 0.34%

0.53% 0.48% 0.43% 0.40% 0.36%

0.40% 0.37% 0.34% 0.31% 0.29%

0.48% 0.43% 0.39% 0.36% 0.33%

0.53% 0.47% 0.43% 0.39% 0.35%

0.34%

0.40%

0.44%

0.34%

0.40%

0.43%

A B C D E

Average

Table 4 Numerical results for the mass overall thermal efficiency, ethermal_p (m2 K/W/kg), in percentage for upward heat flow. Case/block

F1

F2

F3

F4

F5

F6

Case Case Case Case Case

0.29% 0.26% 0.24% 0.22% 0.21%

0.36% 0.32% 0.29% 0.27% 0.25%

0.41% 0.37% 0.33% 0.30% 0.28%

0.29% 0.26% 0.24% 0.22% 0.21%

0.36% 0.32% 0.29% 0.26% 0.24%

0.41% 0.37% 0.33% 0.29% 0.27%

0.25%

0.30%

0.34%

0.24%

0.29%

0.33%

A B C D E

Average

numerical results in Tables 3 and 4, we can see that the best block from the thermal point of view is the block F3, since its average value (0.34% and 0.44% for upward and downward heat flows, respectively) is the biggest one. On the other hand, we show the numerical results for the equivalent thermal conductivity in Figs. 13 and 14 (right). From the

Fig. 13. Mass overall thermal efficiency (left) and equivalent thermal conductivity (right) for downward heat flow in all analyzed cases.

Fig. 14. Mass overall thermal efficiency (left) and equivalent thermal conductivity (right) for upward heat flow in all analyzed cases.


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Fig. 15. Temperature distribution in floors made up of block type F1–F6 in the intermediate Case C (density 1000 kg/m3) for an upward heat flow.

point of view of this parameter the best blocks are F3 and F6 with small differences between them. This behavior corresponds to massive floors since one of the main constituent materials is the concrete, both for upper side of the floor and the prestressed concrete joists. This fact causes a lower mass overall thermal efficiency compared with other building products [3,4].

The results obtained by FEM (see Fig. 15) are processed in order to obtain the thermal characteristic values of the floors, both for downward and upward heat flows, are shown in Table 5. Comparing to the numerical results (Table 5 and Fig. 16) with the CTE rule requirements (cut-off line) [7], we see that only some lightweight concrete hollow blocks analyzed are in agreement

Table 5 Thermal transmittance, overall thermal resistance and equivalent thermal conductivity for all analyzed cases and types of blocks. Case-block

A-F1 A-F2 A-F3 A-F4 A-F5 A-F6 B-F1 B-F2 B-F3 B-F4 B-F5 B-F6 C-F1 C-F2

Downward heat flow

Upward heat flow

U (W/m2 K)

Rtot (m2 K/W)

lequiv (W/(m K))

U (W/m2 K)

Rtot (m2 K/W)

lequiv (W/(m K))

1.577 1.161 0.960 1.503 1.130 0.946 1.678 1.232 1.018 1.600 1.203 1.008 1.772 1.301

0.936 1.150 1.320 0.965 1.172 1.334 0.900 1.103 1.264 0.928 1.122 1.272 0.870 1.062

0.331 0.270 0.235 0.321 0.265 0.232 0.344 0.281 0.245 0.334 0.276 0.244 0.356 0.292

1.976 1.424 1.138 1.942 1.389 1.113 2.109 1.528 1.216 2.073 1.496 1.199 2.230 1.624

0.676 0.860 1.026 0.684 0.877 1.045 0.646 0.815 0.973 0.653 0.828 0.984 0.622 0.779

0.459 0.360 0.302 0.453 0.353 0.297 0.480 0.380 0.319 0.474 0.374 0.315 0.499 0.398


1286 Table 5 (Continued ) Case-block

Downward heat flow 2

C-F3 C-F4 C-F5 C-F6 D-F1 D-F2 D-F3 D-F4 D-F5 D-F6 E-F1 E-F2 E-F3 E-F4 E-F5 E-F6

Upward heat flow 2

U (W/m K)

Rtot (m K/W)

lequiv (W/(m K))

U (W/m2 K)

Rtot (m2 K/W)

lequiv (W/(m K))

1.076 1.693 1.275 1.073 1.859 1.368 1.135 1.782 1.349 1.141 1.937 1.431 1.194 1.865 1.422 1.212

1.214 0.895 1.077 1.216 0.846 1.027 1.168 0.868 1.037 1.164 0.825 0.997 1.127 0.844 1.001 1.116

0.255 0.346 0.288 0.255 0.367 0.302 0.265 0.357 0.299 0.266 0.376 0.311 0.275 0.367 0.310 0.278

1.291 2.195 1.597 1.282 2.335 1.711 1.364 2.304 1.692 1.364 2.405 1.790 1.432 2.378 1.779 1.443

0.928 0.628 0.788 0.933 0.603 0.749 0.889 0.608 0.756 0.889 0.591 0.725 0.856 0.595 0.728 0.851

0.334 0.493 0.393 0.332 0.515 0.414 0.349 0.510 0.410 0.349 0.525 0.427 0.362 0.521 0.426 0.364

Fig. 16. Thermal transmittance versus type of blocks for the thirteen analyzed cases: downward heat flow (upper) and upward heat flow (lower).


with the objective value established in the CTE rule with respect to the thermal transmittance. On the one hand, in case of the downward heat flow, the caseblocks A-F6, A-F3, B-F6, B-F3, C-F6, C-F3, A-F5, D-F3, D-F6, A-F2 and E-F3 satisfy the CTE rule requirements [7]. On the other hand, in case of the upward heat flow, only the case-blocks A-F6, A-F3 and B-F6 fulfil the CTE requirements [7]. Therefore, the case of the upward heat flow is more restrictive than the other case. After examining the results obtained numerically, it can be assumed that the optimization procedure constitutes a reasonable approach to choose the appropriate type of block that satisfies the rule requirements [7]. The finite element model reproduces quite accurately the heat transfer in floors made of different materials (such as gypsum, concrete, steel, lightweight concrete, etc.) and recesses with complex shapes.

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also fulfil all strength and serviceability requirements for housing and industrial structures. From this point of view, the designer can use the results shown in Table 5 and Fig. 16 in order to obtain the best candidate floor configuration according to the CTE rule requirements [7]. Acknowledgements The authors wish to acknowledge the financial support provided by Spanish Ministry of Science and Innovation through the Research Project BIA2008-00058. We also thank to Swanson Analysis Inc. for the use of ANSYS Academic program, the Department of Construction at University of Oviedo and the MAXIT Group. References

7. Conclusions In this work, the finite element method is used for finding approximate solution of the heat transfer equation in twodimensional complex models [21,22]. The finite element model reproduces quite accurately the heat transfer mechanism in floors made up of lightweight aggregate concrete with complex shapes of recesses. In this sense, the key step in engineering analysis is therefore choosing appropriate mathematical models. These models will clearly be selected depending on what phenomena are to be predicted, and it is most important to select mathematical models that are reliable and effective in predicting the quantities sought [23,24]. In the first place, the numerical thermal analysis technique (FEM) has been carried out to study six different kinds of floors in two dimensions, made up of five different compositions of lightweight concrete hollow blocks, according to the previous experimental results. With the variation of the length and width of the holes, it is possible to modify the thermal performance of the blocks and consequently of the entire floor. On the basis of the mass overall thermal efficiency and the equivalent thermal conductivity, we have selected the best candidate block from the thermal point of view. To define the complex geometry of a multilayer floor, it is necessary to use a three-dimensional parametric design program. In this work we have used a fine FEM mesh, with a meshing parameter ranging from 0.001 m in the weldmesh reinforcement to 0.004 m in the concrete and gypsum. Secondly, the equivalent thermal conductivity depends on three heat transfer processes: the heat conduction through the floor, taking into account the different conductivities of the constituent materials, the radiation between surfaces of the recesses and the natural convection in holes [17–20]. According to the numerical results, we see that the thermal transmittance depends on the number of the vertical and horizontal intermediate bulkheads. Specifically, if the number of horizontal intermediate bulkheads is increased, the thermal transmittance grows more than if the number of vertical intermediate ones does. Furthermore, the case of the upward heat flow is more restrictive than the downward heat flow. Thirdly, in order to select the appropriate floor satisfying the CTE requirements [7], Fig. 16 shows the appropriate case and type of block. On the one hand, the overall heat transfer coefficient increases if material conductivity increases and the number of recesses decreases. On the other hand, the bigger mass overall thermal efficiency, the better thermal insulation and the lower floor’s weight. Therefore, the support structure of these floors will be subjected to smaller dead loads and the best block from the average mass overall thermal efficiency point of view was the block F6. Finally, there is an increasing interest to use materials with good physical properties with respect to an energy savings, which

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