Optimization of the configuration of 290 � 140 � 90 hollow clay brickswith 3-D numerical simulation by finite volume method

L.P. Li a, Z.G. Wu a, Y.L. He a, G. Lauriat b, W.Q. Tao a,*a School of Energy & Power Engineering, Xi’an Jiaotong University, Xi’an, Chinab LETEM, Universite de Marne-la-Valle, France

Energy and Buildings 40 (2008) 1790–1798

A R T I C L E I N F O

Article history:

Received 27 January 2008

Received in revised form 18 March 2008

Accepted 18 March 2008

Keywords:

Hollow clay bricks

Numerical simulation

Equivalent thermal conductivity

Natural convection

Surface radiation

A B S T R A C T

This paper is aimed at finding the optimum configuration of the number of holes and their arrangement

for the 290 � 140 � 90 hollow clay bricks with 3-D numerical simulation by a home-made code with

finite volume method. Seventy-two kinds of configurations with different hole number and arrays are

chosen elaborately and their equivalent thermal conductivities are numerically predicted. In addition,

the effects of the hole surface radiation and the indoor–outdoor temperature difference on the equivalent

thermal conductivity are also investigated. The major findings are as follows. The radiation of the hole

surfaces makes heat transfer enhanced and the equivalent thermal conductivity enlarged in some extent,

ranging from 25.8% to 4.6%. The optimum configuration has eight holes in length, four holes in width and

one holes in height, whose equivalent thermal conductivity is the lowest and of 0.400 W/(m K),which is

only 59% of the highest thermal conductivity of the all cases studied. When the indoor–outdoor

temperature difference varies from 50 8C to 20 8C, the equivalent thermal conductivity of the 72 kinds of

hollow bricks does not vary too much, usually within �5%. Especially, the equivalent thermal conductivity

of the optimum configuration holds no change within this variation range of indoor–outdoor temperature

difference.

� 2008 Elsevier B.V. All rights reserved.

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1. Introduction

Hollow clay bricks used as building materials have obviousadvantages in improving thermal insulation performance ofbuilding walls, separating sound as well as reducing building wallloads. Thermal insulation performance of the hollow clay brickmainly depends on the hole configuration, that is, on the number ofholes, their arrangement and the void fraction, and investigationon the hole configuration effect is of great significance to improveits thermal insulation performance. It is well-known that the heattransfer and fluid flow process within a hollow clay brick is atypical complicated combined mode problem, with conduction,convection and surface radiation being all involved. Theoreticallyspeaking, both experimental and numerical methods can be usedto find an optimum configuration of a hollow brick under certaincondition. However, because of the large number of the possiblevariants of the hole configurations experimental study is very cost-expensive. Meanwhile apart from surface radiation the transportprocess in the hollow brick is conduction and laminar flow innature, and this is a typical case that numerical simulation can play

* Corresponding author.

E-mail address: wqtao@mail.xjtu.edu.cn (W.Q. Tao).

0378-7788/$ – see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.enbuild.2008.03.010

a great role. This is probably the main reason that most relatedliteratures are of numerical study. In the following, a brief reviewof literatures in recent 10 years is presented. In [1] 2-D simulationwas conducted for enclosures similar to the hollow clay bricks withsingle or double holes. Surface radiation was taken into account bytreating the radiation energy as the additional source term of thecontrol volume bounded with the surface. The results emphasizethe effect of the thermal boundary condition on the total heattransfer. Lorente et al. [2] studied the heat flux and thermalresistance of a hollow brick with a single vertical hole by usingsimplified analytical model. They also studied the influence ofenvironment temperatures on thermal resistance of the walls builtwith some shapes of vertical hollow bricks [3]. When time enteredinto 21th century, study on the hollow brick related problemsreceives more interests of researchers because the world-wideenergy shortage. Castro Cadoso et al. [4] numerically studied astructure with 12 rectangular holes for which the horizontal topand bottom surfaces were adiabatic and vertical left and rightsurfaces maintained at constant but different temperatures. Theeffect of the hole surface radiation was examined. Hinojosa et al.[5] predicted the Nusselt number for the natural convection andsurface thermal radiation in a square tilted open cavity. They foundthat the heat transfer via radiation is in the same order of naturalconvection. In [6], indoor thermal environment of office space

Nomenclature

Aw1 area of inner surface of calculation

Aw2 area of outer surface of calculation

cpf specific heat of air (k J/(kg K))

G nominal flow rate of cross section in holes

(G =Rrju jdydz, kg/m3)

h1 convection heat transfer coefficient of inner surface

of wall (W/(m2 K))

h2 convection heat transfer coefficient of outer sur-

face of wall (W/(m2 K))

J radiosity (W/m2)

L1, M1, N1 the grid number in x, y and z direction,

respectively

p Pressure (Pa)

Peff effective pressure (Pa)

Pr Prandtl number

qr net radiant heat flux (W/m2)

Qinwall total heat transfer rate cross inner surface of wall

(W)

Qoutwall total heat transfer rate cross outer surface of wall

(W)

Rmax the largest mass residual in whole computation

field

Sc,ad,air additional source term of control volume at air side

Sc,ad,clay additional source term of control volume at clay

side

T temperature (K)

Tc reference temperature (K)

Tf1 indoor temperature (K)

Tf2 environment temperature (K)

Tw1 average temperature of inner surface of wall (K)

Tw2 average temperature of outer surface of wall (K)

u velocity component in x direction (m/s)

v velocity component in y direction (m/s)

V!

velocity vector in x–y plane

w velocity component in z direction (m/s)

X1, Y1, Z1 the length of calculation unit in x, y and z

direction, respectively (m)

XIJ view factor

x, y, z coordinate (m)

Greek lettersb volumetric thermal expansion coefficient (K�1)

e surface emissivity

h viscosity (kg/(m s))

lf thermal conductivity of air (W/(m K))

lnat equivalent thermal conductivity when radiation

neglected (W/(m K))

lrad equivalent thermal conductivity when radiation

considered (W/(m K))

ls thermal conductivity of clay (W/(m K))

leq equivalent thermal conductivity (W/(m K))

r mass density (kg/m3)

rc reference mass density (kg/m3)

rf mass density of air (kg/m3)

L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798 1791

controlled by cooling panel system was investigated by numericalsimulation and field measurement, and the results showed theimportance of radiation in the thermal environment simulation. In[7], natural convection and radiation heat transfer in a cavityhaving a square body at its center had been simulated. The resultsindicate that the heat transfer by convection and radiation areusually of the same order of magnitude and the radiationhomogenizes the temperature inside the cavity. In [8] couplednatural convection–conduction effects on the heat transferthrough three kinds of hollow clay bricks had been studied usingcommercial software FLUENT, in which the first one was a typicalhollow brick with three identical holes, the second was filled withordinary polystyrene bars and the third filled with hollowpolystyrene bars. The results show that, compared with the first,the second can reduce the heat transfer rate by 36% while the thirdcan reduce only by 6% due to the air motion inside the holes in thepolystyrene bars. All of above studies are of 2-D simulation. In[9,10] 3-D numerical thermal analysis of some light concretehollow brick walls was conducted by the finite element method inwhich the parameter of mass overall thermal efficiency was usedto evaluate the economic efficiency of light concrete hollow brick.Lauriat and Desrayaud [11] simulated the conjugate naturalconvection in partially open enclosures with surface radiationbeing taken into account. In [12] detailed 3-D numericalsimulation about 240 � 115 � 90 hollow clay bricks was con-ducted for 50 kinds of different hole configuration and theoptimum configuration was found which is wildly used toconstruct 240 mm walls in China. In addition, the parameter ofmass overall thermal efficiency was adopted in [12] and found thatit is not suitable to characterize the hollow clay brick performance.

The major purpose of this paper is to find out the optimumconfiguration of the 290� 140 � 90 hollow clay bricks. By optimumconfiguration we mean that the heat transfer rate through the wall ofthe configuration under given conditions is the least. This is animportant way to save building energy either in summer or in winterseasons. The overall heat transfer process from the inner side of aroom to the environment is consisted of three steps: from inner sideto the inner surface of the wall via convection and radiation heattransfer, from the inner surface of the wall to the outside surface ofthe wall through heat conduction and from the outside surface of thewall to the environment via convection and radiation. The thermalresistance of the conduction process can be represented by d/leq,where leq is the equivalent thermal conductivity of the brickstructure [13,14]. As indicated above, within the hollow brick, heat istransferred via conduction, convection and radiation, and differentconfiguration of the hollow brick will affect the relative importanceof conduction, convection and surface radiation in the heat transferprocess from inner surface to outer surface. However, the overallthermal insulation performance of the hollow brick can berepresented by the equivalent thermal conductivity. The equivalentthermal conductivity is the thermal conductivity of an equivalentsolid brick which can conduct the same heat under the same indoor–outdoor conditions and the same brick thickness. Strictly speaking,the heat transfer process through the wall is unsteady in nature,because either the indoor or the outdoor conditions are actuallychanging from time to time. However, from engineering point ofview, thermal design of the building can be conducted for sometypical situations of indoor and outdoor conditions with a steadystate assumption. In this study numerical simulation of the steadyheat transfer process through the hollow brick will be conducted forsome typical combinations of indoor and outdoor conditions. Themajor purpose of the study is to find out the hole configuration withwhich the brick has the lowest equivalent thermal conductivity. Inorder to do that, 72 kinds of different hole configurations aredesigned.

L.P. Li et al. / Energy and Buildings 40 (2008) 1790–17981792

2. Mathematical formulation

The 290 � 140 � 90 hollow clay bricks is one type of brickscommonly used in China to construct building walls with 300 mmthickness. Their exterior sizes are 290 mm in length, 140 mm inwidth and 90 mm in height. For simplicity, the denotation ofLm1Wm2Hm3 is used to represent the brick with m1 holes inlength, m2 holes in width and m3 holes in height such asdenotation L8W4H1 represents the brick having eight holes inlength, four holes in width and one hole in height. The denotationsof some selected hollow bricks from all 72 kinds are listed inTable 1.

From the building wall structure the smallest computationaldomain (unit) consists of four blocks of the bricks that haveidentical configuration and is conglutinated by mortar accordingto the manner of putting bricks. It is considered that itsboundaries are repeated periodically in both up/down and left/right directions. Fig. 1 shows one of such 72 computationaldomains consisting of four blocks of L8W4H1. It should be notedthat all the holes are rectangular in cross section as shown inFig. 1. It is obvious that the heat transfer phenomenon is three-dimension in nature, i.e., both the temperature (of the solid partand fluid part) and the fluid flow velocity in the holes vary withthe three coordinates. Hence, their equivalent thermal con-ductivities are numerically calculated with 3-D numericalsimulation of the temperature and air velocity in Cartesiancoordinates by using a home-made code of finite volumemethod.

As indicated above the equivalent thermal conductivity of thehollow clay bricks depends on the hole number, their arrangement(arrays) and void fraction. The 72 kinds have two things incommon: there is only one hole in the height and in length andwidth directions the holes are uniformly distributed. The holenumber varies from 1 to 14 in length and 1 to 6 in width. All ofthem have the largest void fraction at given hole number and

Table 1Comparison lnat with lrad for 290 � 140 � 90 bricks

Kinds Holes number Void fraction (%)

L01W1H1 1 79.8

L04W1H1 4 71.0

L07W1H1 7 65.0

L10W1H1 10 59.1

L14W1H1 14 50.2

L01W2H1 2 73.2

L04W2H1 8 65.0

L07W2H1 14 59.6

L10W2H1 20 54.2

L14W2H1 28 46.1

L01W3H1 3 66.5

L04W3H1 12 59.1

L07W3H1 21 54.2

L10W3H1 30 49.3

L14W3H1 42 41.9

L01W4H1 4 63.2

L04W4H1 16 56.1

L08W4H1 32 50.3

L10W4H1 40 46.8

L14W4H1 56 39.8

L01W5h1 5 59.9

L04W5H1 20 53.2

L07W5H1 35 48.8

L10W5H1 50 44.3

L14W5H1 70 37.7

L01W6H1 6 53.2

L04W6H1 24 47.3

L07W6H1 42 43.3

L10W6H1 60 39.4

L14W6H1 84 33.5

arrays. Here, the void fraction means the percentage of the volumefilled by air divided by the total volume of the hollow clay brick.The largest void fraction means that for given hole number andarray, the hull of the brick is 10 mm, the ribs parting the holes are7.5–10 mm, which are the essential need for hollow clay bricks notto be split.

Based on the fundamental physical process, the followingapproximations can be adopted in our numerical model: (1) Thehollow clay bricks and the air in the holes are of constant thermalphysical properties; (2) Air flow in the holes of the hollow claybricks is incompressible, steady state and laminar; (3) Whenradiation is considered air in the holes is a non-participatingmedium; (4) The Boussinesq assumption is adopted for the naturalconvection in the holes [15]; (5) the convective boundaryconditions are taken at the indoor and outdoor surfaces of thecomputation domain while the other four surfaces are consideredas adiabatic (Fig. 1(a)); (6) The mortar is assumed to have the sameconductivity as the clay brick; (7) The hole surface is gray anddiffusive.

The numerical prediction of the equivalent thermal con-ductivity will be performed under given third kind of boundaryconditions for both indoor and outdoor sides. That is the fluidtemperatures and the total heat transfer coefficients of the innerand outer surfaces of the wall are prespecified. Then in order topredict the equivalent thermal conductivity of the brickstructure, its two surfaces temperatures have to be known. Inorder to obtain these two surface temperatures the partialdifferential equation of the temperature in the brick structureshould be solved under the given boundary conditions. Sincewithin the brick holes air flows exist and the fluid velocity willaffect the heat transfer process, the momentum equations of thefluid in the holes should be simultaneously solved. Thus, todetermine the equivalent thermal conductivity, the governingequations for the temperature and fluid velocity in the brickshould be simultaneously solved.

lnat (W/(m K)) lrad (W/(m K)) (lrad � lnat)/lrad (%)

0.680 0.905 24.9

0.426 0.554 23.1

0.423 0.522 19.0

0.429 0.511 16.0

0.433 0.498 13.1

0.628 0.807 22.2

0.356 0.450 20.9

0.351 0.422 16.8

0.361 0.419 13.8

0.384 0.429 10.5

0.631 0.781 19.2

0.355 0.431 17.6

0.351 0.408 14.0

0.364 0.410 11.2

0.393 0.428 8.2

0.626 0.758 17.4

0.355 0.421 15.7

0.355 0.400 11.3

0.369 0.408 9.6

0.402 0.431 6.7

0.608 0.726 16.3

0.358 0.417 14.1

0.358 0.401 10.7

0.377 0.411 8.3

0.414 0.439 5.7

0.566 0.670 15.5

0.374 0.424 11.8

0.379 0.415 8.7

0.402 0.430 6.5

0.440 0.461 4.6

Fig. 1. Computational domain.

L.P. Li et al. / Energy and Buildi

The governing equations for the temperature and velocity are asfollows [16]:

@ðru2Þ@x

þ@ðruvÞ@y

þ @ðruwÞ@z

¼ � @ peff

@xþh

@2u

@x2þ @2

u

@y2þ @2

u

@z2

!

@ðruvÞ@x

þ @ðrv2Þ@y

þ @ðrvwÞ@z

¼ � @ peff

@yþh

@2v

@x2þ @2

v

@y2þ @2

v

@z2

!þ rcgbðT � TcÞ

@ðruwÞ@x

þ @ðrvwÞ@y

þ @ðrw2Þ@z

¼ � @ peff

@zþh

@2w

@x2þ @2

w

@y2þ@2

w

@z2

!

@ðruTÞ@x

þ @ðrvTÞ@y

þ @ðrwTÞ@z

¼ lcp

@2T

@x2þ @2

T

@y2þ @2

T

@z2

!(1)

where, peff = p � rcgy.The boundary conditions of the governing equations are

x ¼ 0 u ¼ v ¼ w ¼ 0 h2ðT f2 � Tw2Þ ¼ �ls@T

@x

x ¼ X1 u ¼ v ¼ w ¼ 0 h1ðTw1 � T f1Þ ¼ �ls@T

@x

(2a)

y ¼ 0 u ¼ v ¼ w ¼ 0@T

@y¼ 0

y ¼ Y1 u ¼ v ¼ w ¼ 0@T

@y¼ 0

(2b)

z ¼ 0 u ¼ v ¼ w ¼ 0@T

@z¼ 0

z ¼ Z1 u ¼ v ¼ w ¼ 0@T

@z¼ 0

(2c)

where X1, Y1 and Z1 express the size of the computation domain inx, y and z direction, respectively. Eq. (2a) expresses the no-slipboundary condition of the three velocity components and the thirdkind boundary condition of the temperature at the inner and outersurfaces. Eqs. (2b) and (2c) represent the symmetric boundarycondition, i.e., adiabatic, at the boundaries of y and z coordinatesfor the temperature. It is interesting to note that the abovegoverning equations are valid for both solid part of the brick andthe air in the hole. Here, an idea from [17] is adopted, i.e., the solidis regarded as a special fluid with very large viscosity. This is a verysuccessful numerical technique to treat the complicated fluid flowproblems where both the fluid and solid temperature should besimultaneously predicted, and has been widely adopted in manynumerical simulation [16]. With such a technique, the surfaces ofthe holes are no longer the computational boundaries, rather theyare the interfaces within the computational domain, and no anyspecial treatment is needed for such interfaces when the governingequations are solved. The details of the numerical technique will bepresented below.

According to general building construction engineering prac-tice, following thermal physical properties are used in simulation:

Tf1 = 293 K; Tf2 = 253 K; h1 = 8.72 W/(m2 K); h2 = 23.26 W/(m2 K); Pr = 0.707; cpf = 1005 J/(kg K), lf = 0.0244 W/(m K), h =17.2 10�6 kg/(m s), rf = 1.293 kg/m3, ls = 0.755 W/(m K), e = 0.85.

3. Numerical methods

The governing equations are discretized with the finite volumemethod [16,17]. Discretization of convection term uses the SGSDscheme [18], which is a stability-guaranteed second-orderdifference scheme, and the SIMPLE is adopted as pressure–velocitysolution algorithm. Discretized algebraic equations are solved inthe whole computational domain by TDMA + ADI method [16,17].As indicated above the surfaces parting solid and fluid areasbecome inner parts of the simulation domain. To guarantee thesuccess of such simple and efficient numerical treatment, theequivalent diffusion coefficient at the separating surfaces shouldbe calculated by the harmonic mean [16,17,19]. The thermal

ngs 40 (2008) 1790–1798 1793

conductivity of solid and fluid regions are adopted individuallywhile the specific heat for the solid area should be replaced by thevalue of fluid area in order to guarantee the heat flux continuity atthe separating surfaces [16,19]. By adopting a very large value ofviscosity amounting to1030 in the momentum equation for thesolid part of the hollow brick, the zero velocity of solid area can beactually gained.

From the simulated temperature field the equivalent thermalconductivity can be determined. The relationship of the equivalentthermal conductivity and the predicted temperature field can befound as follows.

L.P. Li et al. / Energy and Buildings 40 (2008) 1790–17981794

3.1. Formulae for calculation of equivalent thermal conductivity

The equivalent thermal conductivity can be calculated accord-ing to the following relationship of heat flux equilibrium

h1ðT f1 � TW1Þ ¼leq

X1ðTW1 � TW2Þ ¼ h2ðTW2 � Tf2Þ (3)

then

leq ¼ X1h1T f1 � Tw1

Tw1 � Tw2(4a)

or

leq ¼ X1h2Tw2 � T f2

Tw1 � Tw2(4b)

where

Tw1 ¼1

Aw1

Xi2Aw1

AðiÞTwðiÞ

24

35 (5a)

Tw2 ¼1

Aw2

Xi2Aw2

AðiÞTwðiÞ

24

35 (5b)

where Tw(i) is the temperature of the grid points at the inneror outer surfaces of the computational domain and h1,h2 arethe specified heat transfer coefficients at the inner and outsurfaces.

3.2. Radiation between hole inner surfaces

The radiant heat transfer between the internal surfaces of theholes in hollow clay bricks should not be neglected [1,4,11,12]. Theradiant heat flux at inner surfaces can be treated as additionalsource terms in the control volumes close to the inner surfaces[1,12], which can be calculated as follows:

For the control volumes at the air side (see Fig. 2)

Sc;ad;air ¼ �qrdx�e =ls

dx�e =ls þ dxþe =lf� 1

Dxþ(6a)

For the control volumes at the clay side

Sc;ad;clay ¼ �qrdxþe =lf

dx�e =ls þ dxþe =lf� 1

Dx�(6b)

where, qr is the net radiant heat flux at the inner surfaces of theholes in the hollow clay bricks. As can be seen from Fig. 1 eachhole is composed of six surfaces: top, bottom, and four lateralsurfaces. These six surfaces are further assumed to be atindividual constant temperatures and taken as the computa-tional radiation surfaces, for each of which the radiosity J(I), andthe heat flux qr(I) can be determined according to radiative heat

XIJ ¼2

pXZln

1þ X2

� �1þ Z

2� �

1þ X2 þ Z

2

24

35

1=2

þ X 1þ Z2

� �1=2tan�1 X

1þ Z2

� �1=2

8><>:

For perpendicular rectangles with a common edge

XIJ ¼1

pX

Xtan�1 1

XþYtan�1 1

Y� Y

2 þ X2

� �1=2tan�1 1

Y2 þ X

2� �1=2

þ1

4ln

1þ X2

� �1þ Y

2� �

1þ X2 þ Y

2

X2

1þ X2 þ Y

2� �

1þ X2

� �X

2 þ Y2

� �264

375

X2

Y2

1þ�

1þ Y2

� �264

8>>><>>>:

0BBBBBBBBB@

transfer theory [13,14]:

qrðIÞ ¼eðIÞ

1� eðIÞ C0TðIÞ100

� �4

� JðIÞ" #

;

JðIÞ ¼ C0eðIÞTðIÞ100

� �4

þ ð1� eðIÞÞX

J

XIJJðJÞ (7)

where, I = 1, 2, 3, 4, 5, 6 and J = 1, 2, 3, 4, 5, 6; C0 = 5.67; T (I) is themean temperature of the surface (I) of the six-surface enclosure.

It is to be noted that for each surface of a hole it may be discretizedby several smaller surfaces depending on the generated grid. Thus,the mean temperature T (I) should be calculated as follows

TðIÞ4 ¼ 1

AðIÞXi2 I

AðiÞTðiÞ4" #

(8)

or

TðIÞ ¼ 1

AðIÞXi2 I

AðiÞTðiÞ" #

(9)

where T(i) is a local temperature of the interface I, which can bereadily obtained during the iterative solution process by inter-polation from the temperatures of the neighboring grid pointsshown below.

In Eqs. (8) and (9) the expression i 2 I stands for all the gridsbelonging to the I surface of the enclosure. Our numerical practicesshow that the final results obtained from Eqs. (8) and (9) are almostthe same while Eq. (9) can get a bit faster convergence because nofourth power calculation is needed in this definition.

The interface temperature T(i) can be interpolated frompredicted neighboring grid points according to the heat fluxbalance. For example, the temperature of e-interface in Fig. 2 canbe calculated with the following equation

TðiÞ ¼ Te ¼TPls=dx�e þ TElf=dxþe � qrðeÞ

ls=dx�e þ lf=dxþe(10)

3.3. View factor calculation

It can be seen that in order to determine the interfacetemperature the surface radiative flux is required (Eq. (10)). Todetermine the surface heat flux from Eq. (7) the view factor data foreach surface of the six-surface enclosure should be supplied. Therelationship between every two surfaces of the enclosure is eithertwo aligned parallel rectangular planes or two perpendicularrectangular planes. For such combinations, the view factor, XIJ, canbe calculated according to the following equation [13,14]:

For aligned parallel rectangles

þ Z 1þ X2

� �1=2tan�1 Z

1þ X2

� �1=2� Xtan�1X � Ztan�1Z

9>=>; (11)

Y2 þ X

2�

Y2 þ X

2� �

375

Y29>>>=>>>;

1CCCCCCCCCA

(12)

Fig. 2. The control volumes at the sides of e-interface.

L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798 1795

Obviously, the solution process is iterative in nature. Both T(I)and qr(I) should be updated during iteration procedure. Besides, inorder to ensure the convergence of iteration, the relaxation forvelocity components and pressure is absolutely necessary for allcases.

3.4. Convergence criteria

Convergence criteria of the iteration procedure for the resultedalgebraic equations discretized from the governing equation,Eq. (1), are taken as follows:

Rmax/G < 10�6; jQinwall � Qoutwallj/min (Qinwall, Qoutwall)<10�2;and

lkþ300 � l

k

lkþ300

���������� � 10�6

for all cases to be simulated.

3.5. Grid-independence examination

In the preliminary computations, the test of grid independenceof the solution has been carried on for the most complicatedL14W6H1 hollow clay bricks. The results are drawn in Fig. 3.Compared case 1 with case 4, when the total grid number increasesfrom 122 � 104 to 949 � 104, lrad has an increment no more than

Fig. 3. The validation of grid independence.

1.3%. So the grid system of case 1, which is L1 = 122, M1 = 82 andN1 = 122, has been chosen for all kinds of 290 � 140 � 90 hollowclay bricks to keep a balance between numerical accuracy andcomputational cost. This set of grid system can ensure that there isat least 720 control volumes in the smallest holes. All grids aredistributed evenly in the computation domain.

4. Simulation results and discussion

In the following presentation, the effects of the hole config-uration (without surface radiation), the surface radiation and theindoor–outdoor temperature difference on the equivalent thermalconductivity will be discussed in order, followed by some typicalvelocity and temperature fields. Simulations are conducted for twoseries of conditions. In the first series the surface radiation is notconsidered, and the heat is transferred only by conduction andnatural convection. The resulting equivalent thermal conductivityis denoted by lnat. In the second series, radiation is taken intoaccount and the resulting equivalent thermal conductivity isexpressed by lrad.

4.1. The influence of the hole number on the equivalent thermal

conductivity

Simulation started on the cases in which surface radiation in theholes was neglected. The results of lnat for 72 kinds of290 � 140 � 90 hollow clay bricks are drawn in Fig. 4 and someselected values are listed in Table 1.

From this graph we can see that when the holes in widthdirection are 1–6, with the increase of the hole number in lengthfrom one to two holes lnat decreases sharply, it further drops offwith the increase of lengthwise holes from 2 to 4,and reaches itsminimum at holes of 4–6. Then it increases gradually. When thelengthwise hole number is greater than 4, widthwise holenumber shows some effect on lnat: the more the widthwiseholes, the larger the value of lnat. This variation tendency can beunderstood as follows. If surface radiation in the holes isneglected, the main influences on the equivalent thermalconductivity are the heat conduction through the clay andnatural convection in the holes. When the hole number is notmuch, the increase of the hole number leads to such a situationthat the deterioration of natural convection is larger than theenhancement in heat conduction through the increased rib,which makes lnat smaller. Otherwise, the increase of the holenumber leads to an opposite results and makes lnat larger.Therefore, whether lnat increases or decreases with holenumber depends on which factor is dominant.

Fig. 4. lnat for the 290 � 140 � 90 hollow clay bricks.

Fig. 5. lrad for the 290 � 140 � 90 hollow clay bricks.

L.P. Li et al. / Energy and Buildings 40 (2008) 1790–17981796

4.2. The influence of the surface radiation on the equivalent thermal

conductivity

The equivalent thermal conductivities of 72 kinds of290� 140 � 90 hollow clay bricks with the effects of conduction,convection and surface radiation all being considered simulta-neously are drawn in Fig. 5 and some selected values are listed inTable 1. Compared withlnat in Fig. 4, it can be observed that bothlnat

and lrad have the same variation tendency: with the increase in holenumber the equivalent thermal conductivity decreases first, reachesthe minimum and then increases gradually in some extent.However, the surface radiation enhances the equivalent thermal

Table 2The temperature influence on lrad for 290 � 140 � 90 bricks

Kinds Tf1 � Tf2 = 50 8C lrad

(W/(m K))

Tf1 � Tf2 = 40 8C lrad

(W/(m K))

Tf1 � Tf2 = 3

(W/(m K))

L01W1H1 0.922 0.905 0.885

L03W1H1 0.594 0.586 0.577

L07W1H1 0.530 0.522 0.513

L10W1H1 0.521 0.511 0.499

L14W1H1 0.510 0.498 0.485

L01W2H1 0.822 0.807 0.787

L02W2H1 0.569 0.561 0.552

L07W2H1 0.424 0.422 0.419

L09W2H1 0.419 0.417 0.414

L14W2H1 0.432 0.429 0.427

L01W3H1 0.799 0.781 0.758

L02W3H1 0.545 0.537 0.527

L07W3H1 0.409 0.408 0.407

L09W3H1 0.406 0.405 0.404

L14W3H1 0.428 0.428 0.429

L01W4H1 0.781 0.758 0.729

L02W4H1 0.533 0.523 0.510

L07W4H1 0.402 0.401 0.401

L09W4H1 0.401 0.401 0.401

L14W4H1 0.431 0.431 0.433

L01W5H1 0.755 0.726 0.691

L02W5H1 0.524 0.512 0.499

L07W5H1 0.402 0.401 0.401

L09W5H1 0.403 0.403 0.404

L14W5H1 0.438 0.439 0.441

L01W6H1 0.703 0.670 0.631

L03W6H1 0.449 0.444 0.438

L08W6H1 0.417 0.417 0.418

L10W6H1 0.429 0.430 0.432

L14W6H1 0.460 0.461 0.463

conductivity in some extent. The L2W1H1 hollow clay brick has thelargest increment of 25.8% while the L14W6H1 hollow clay brick hasthe lowest increment of 4.6%. Generally speaking, the more the holenumber, the less the surface radiation effect. The detailedcomparison between lrad and lnat can be found in Table 1.

From Fig. 5, following features may be noted. First, the brickhaving only one hole in both width and length has significantlyhigher equivalent thermal conductivity than all the other cases.Second, for the case with at least two holes in width, the effect ofhole number in width becomes much weaker, as all the curves with2–6 holes in width direction being more or less compacted. And forthe cases with 3�7 holes in length direction, the equivalentthermal conductivity of 5 holes in width is the least. The characterof the hole number effect is due to the complicated heat transferprocess caused by conduction, convection and surface radiationtaking place in the hollow clay bricks simultaneously. With theincrease of the hole number convection and surface radiation in theholes are deteriorated while conduction through the clay may beenhanced. Therefore, the equivalent thermal conductivity dependson the total effects of conduction, convection and surface radiation.

Accordingtothe simulation results, theoptimumconfiguration ofthe 290 � 140 � 90 hollow clay bricks is the L08W4H1 brick, whichhas the lowestlrad of 0.400 W/(m K). Compared with the lowestlnat

of0.350 W/(m K), there isanincrementof12.5%.Therefore, the effectof surface radiation in the holes cannot be neglected. It is interestingto note that this lowest value is quite close to the one of240 � 115� 90 brick, which is 0.419 W/(m K), and the related voidfraction is not different too much either (from 46.9% to 50.3%)

4.3. The effect of temperature difference between indoor and outdoor

on the equivalent thermal conductivity

By keeping the indoor temperature at 20 8C while changingoutdoor temperatures from �30 8C to 0 8C, the effects of the

0 8C lrad Tf1 � Tf2 = 20 8C lrad

(W/(m K))

Maximum difference

(compared with Tf1 � Tf2 = 40 8C) (%)

0.857 1.9 to �5.3

0.563 1.4 to �3.9

0.499 1.5 to �4.4

0.484 2.0 to �5.3

0.468 2.4 to �6.0

0.760 1.9 to �5.8

0.538 1.4 to �4.1

0.416 0.5 to �1.4

0.410 0.5 to �1.7

0.423 0.7 to �1.4

0.725 2.3 to �7.2

0.512 1.5 to �4.7

0.405 0.2 to �0.7

0.403 0.2 to �0.5

0.429 0.0 to 0.2

0.688 3.0 to �9.2

0.494 1.9 to �5.5

0.400 0.2 to �0.2

0.401 0.0 to 0.0

0.435 0.0 to 0.9

0.644 4.0 to �11.3

0.482 2.3 to �5.9

0.402 0.2 to �0.2

0.405 0.0 to 0.5

0.443 �0.2 to 0.9

0.584 4.9 to �12.8

0.430 1.1 to �3.2

0.420 0.0 to 0.7

0.434 �0.2 to 0.9

0.464 �0.2 to 0.7

Fig. 6. Isotherms for the L1W1H1 hollow clay brick (z = 0.02).

Fig. 8. V vectors for the L1W1H1 hollow clay brick (z = 0.02 and z = 0.21).

L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798 1797

indoor–outdoor temperature difference on lrad were simulated.Some simulation results are listed in Table 2.

We can see from this table that the effect of temperaturedifference on lrad with different hole number and arrays is not thesame. The values of lrad for some kinds of bricks decrease when thetemperature differences decrease while for other kinds lrad

increase in some extent. Except L1W6H1 hollow clay brick forwhich the maximum difference percentage is 12.8% between thecase of Tf1 � Tf2 = 20 8C and Tf1 � Tf2 = 40 8C, most kinds of brickshave the percentage difference no more than 5%. It is wonderful tonote that the hollow clay brick with the optimum configuration(L8W4H1) has the lrad almost constant when outdoor tempera-tures changes. Besides, the nearer to the optimum configurationthe smaller the lrad change with the temperature difference. Thisimplies that the optimum configuration does not change with theindoor–outdoor temperature difference. In [12] the similar resultwas obtained for the 240 � 115 � 90 hollow clay bricks.

4.4. Isothermals and velocity vectors for some representative hollow

clay bricks

Three representative kinds of 290 � 140 � 90 hollow claybricks are selected to present flow and temperature fields. Theseare L01W1H1, L08W4H1 and L14W6H1: L01W1H1 has thesmallest hole number (one hole) and the largest void fraction(79.8%), L08W4H1 is the optimum configuration (void fraction of50.3%) and L14W6H1has the most holes number (84) and the

Fig. 7. Isotherms for the L1W1H1 hollow clay brick (z = 0.21).

smallest void fraction(33.5%). The isothermals and velocity vectorsfor these three kinds of hollow clay bricks at some cross sections(z = 0.02 and z = 0.21) are shown in Figs. 6–11. It should be notedthat these figures are drawn for the entire computational domainwhich is composed of four identical bricks with two orientations.

From these figures we can note the following features. Firstfrom the isotherms with the increase in hole number the naturalconvection in the holes becomes more and more conduction-predominated. The isotherms in Figs. 6 and 7 show very strongconvection, characterized by the horizontal part of the isotherms inthe lower part of the enclosure. While in Fig. 9 isotherms ofL8W4H1 hollow clay brick are more or less parallel to each otherand finally almost all the isotherms of L14W6H1 hollow clay brick,either in the solid clay or in the holes, exhibit quite well parallelcharacter (Fig. 11), characterized by pure heat conduction. Secondfrom the velocity fields, the flow in L01W1H1 is highly 3-D innature. This can be witnessed by the velocity fields in two crosssections: at z = 0.02, the flow field belongs to conduction-predominated type, while the flow in z = 0.21 exhibits strongconvection, and boundary layer type flow can be observed alongthe two vertical surfaces of the hole (Fig. 8). With the increase inthe hole number convection becomes weaker and weaker, and forthe L14W6H1 the magnitude of velocity has a two-order differencecompared with that of L01W1H1. Only the V vectors for theL8W4H1 brick are shown in Fig. 10, and those for L14W6H1 brick

Fig. 9. Isotherms for the L8W4H1 hollow clay brick (z = 0.21).

Fig. 10. V vectors for the L8W4H1 hollow clay brick (z = 0.02 and z = 0.21).

Fig. 11. Isotherms for the L14W6H1 hollow clay brick (z = 0.21).

L.P. Li et al. / Energy and Buildings 40 (2008) 1790–17981798

are so small that they can hardly be spotted with the same printedscale and are not shown by figures.

5. Conclusions

Through an extensively numerical study on the influence of thehole number and arrays, with and without surface radiation as wellas variable indoor–outdoor temperature difference on theequivalent thermal conductivity of 290 � 140 � 90 hollow claybricks, following conclusions can be made.

1. T

he surface radiation in the holes makes the equivalent thermalconductivity augmented in some extent, with the maximumincrement being 25.8% for L2W1H1 brick and the minimumincrement being 4.6% for L14W6H1. With the increase in holenumber the surface radiation influence decreases.

2. T

he hole number and their arrays have crucial influence on theequivalent thermal conductivity. The brick with only one hole inboth length direction and width direction has the largestequivalent thermal conductivity. The L8W4H1 has the smallestequivalent thermal conductivity of 0.400 W/(m K), which is theoptimum configuration of the 290 � 140 � 90 hollow claybricks. The effect of hole number in length direction is largerthan that in width direction. For the hole number of 2–6 in widthdirection their variation curves of the equivalent thermal

conductivity with the length direction hole number are quiteclose to each other. The reason is explained in the context.

3. W

ithin the range of 20–50 8C of the indoor–outdoor tempera-ture difference, its effect on the equivalent thermal conductivityis not very significant, with most percentage difference beingwithin �5%. Interestingly, for the optimum configurationL8W4H1, its equivalent thermal conductivity almost does notchange with the temperature difference. Moreover, the nearer tothe optimum configuration, the smaller the change of theirequivalent thermal conductivity with the temperature difference.

4. C

ompared with the results in [12], it is found that the two kindsof hollow clay bricks (290 � 140 � 90 vs. 240 � 115 � 90) havemuch similar variation trends about the influence of the holenumber and arrays, surface radiation and indoor–outdoortemperature difference. Therefore, it can be anticipated thatabove variation trends may also be suitable to other types ofhollow clay bricks. However, to optimize the configuration ofeach type of hollow clay bricks individual simulation is required.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (50636050) and the National FundamentalProjects of R&D of China (2007CB206902).

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