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doi:10.1016/j.bu
�Correspondof Engineering,
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Building and Environment 43 (2008) 37–43
www.elsevier.com/locate/buildenv
A new stable finite volume method for predicting thermalperformance of a whole building
C. Luo�, B. Moghtaderi, H. Sugo, A. Page
School of Engineering, Faculty of Engineering & Built Environment, The University of Newcastle, University Drive, Callaghan NSW 2308, Australia
Received 2 August 2006; received in revised form 7 November 2006; accepted 27 November 2006
Abstract
Discretised governing equations involving only temperatures and heat fluxes at both surfaces of a solid wall layer were obtained by
combining a new stable finite volume scheme for the two inner nodes of the wall layer with the surface diffusion equations (discretised by
third order equations). The finite volume scheme for the inner nodes of the layer is proved to be stable with its truncation error being
OðDx4;Dx2Dt2Þ. A special analytical solution for a solid wall was used to evaluate different schemes for the inner nodes, showing that the
new proposed scheme performs better than all other schemes for time steps of 3600 and 600 s. Finally, this scheme was used to simulate a
whole house and the predicted zonal air temperature, and surface temperatures agreed well with measured values.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Finite volume method; Finite difference method; Fourier stability analysis; Thermal zone model; Fourier diffusion equation
1. Introduction
A thermal zonal model assumes that all independentvariables are distributed uniformly across the building wallsand within the zones (such as room air or the air cavity in acavity wall) and combines the heat balance equations forsolid walls with the energy balance equations for the zonalair. To simplify the heat balance equations for solid walls,the response factor method and the conduction transferfunction (CTF) method have been adopted in a number ofcommercial and scientific software programs, such asTRNSYS, Energy Plus (CTF) and AccuRate (ResponseFactor). To avoid solving the governing equations for theinner nodes of a wall layer, SUNCODE used an explicitdifference scheme to solve the wall conduction equations.However, it is well known that the explicit difference schemehas a stability problem. Because of potential stabilityproblems and the storage requirements for the inner nodes,SUNCODE is only suitable for residential or small
e front matter r 2007 Elsevier Ltd. All rights reserved.
ildenv.2006.11.037
ing author. Department of Chemical Engineering, School
The University of Newcastle, University Drive, Callaghan
stralia. Tel.: +61 2 4921 6951; fax: +61 2 4921 8692.
ess: Caimao.Luo@newcastle.edu.au (C. Luo).
commercial buildings. The inner node equations preventresearchers using the finite difference method for modellingthe thermal performance of buildings. Therefore, to utilisethe finite difference method in the thermal performancemodelling of buildings, it is necessary to construct a schemewhich does not involve the inner nodes.Tsilingiris [1,2] used the Laasonen scheme to develop a
finite difference method to study the thermal behaviour ofwalls. He utilized the effective thermal capacitance andthermal conductivity to simplify the parallel layers of thecomposite material with Dx ¼ 1 cm and Dt ¼ 60–300 s (or1–5min). It is not made clear by the author whether or notthe method could be applied to a whole building.The study reported in this paper focuses on a finite
difference/finite volume method which can be easilyapplied to a whole building with reasonable accuracy fortime steps ranging from 1 to 3600 s (or 1 h). By using twoinner nodes in a construction layer as shown in Fig. 1, allvariables from the two inner nodes can be removed bycombining the implicit discretised finite difference/finitevolume equations for the inner nodes with equations forthe two surface nodes. It is shown that this new implicitscheme for the two inner nodes gives better performancethan the existing implicit schemes.
ARTICLE IN PRESS
Ts0 T2 Ts1
q"s0 "s1q
T1
Fig. 1. Illustration of variables for a single layer in a brickwork wall.
C. Luo et al. / Building and Environment 43 (2008) 37–4338
2. Discretised governing equations for inner nodes
The key differences for existing building thermalperformance software come from the treatment of solidwalls. The Fourier diffusion equation for a solid wall is
rCpqT
qt¼ k
q2Tqx2
(1)
in which r denotes density, Cp is the specific capacity,k is the thermal conductivity of the layer, T is thetemperature across the layer, t is the time and x is thedistance across the layer. A new finite volume schemeand a hybrid finite volume–finite difference scheme fordiscretizing the partial differential equation are presentedbelow.
2.1. Finite volume discretised schemes for the solid Fourier
heat transfer equation
Because of the large amount of cpu time consumed insolving the resulting system of equations, for any singlelayer (in this case a masonry wall), only four nodes areconsidered as illustrated in Fig. 1. For the inner nodes, T1
and T2, the spatially discretised equations are obtained byintegrating the domains ½xs0;x2� and ½x1;xs1� with Simp-son’s 1
3integration law, respectively, as follows:
1
3DxrbCpb
qT s0
qtþ 4
qT1
qtþ
qT2
qt
� �
¼ �kqT
qx
����s0
þ kqT
qx
����2
¼2k
DxðT s0 � 2T1 þ T2Þ, ð2Þ
1
3DxrbCpb
qT1
qtþ 4
qT2
qtþ
qT s1
qt
� �
¼ �kqT
qx
����1
þ kqT
qx
����s1
¼2k
DxðT1 � 2T2 þ T s1Þ. ð3Þ
Integrating Eqs. (2) and (3) over the time domain ½t� Dt; t�,it reads
L2t rCpb
27kDt½ðT s0 þ 4T1 þ T2Þ
n� ðT s0 þ 4T1 þ T2Þ
n�1�
¼ ½ðT s0 � 2T1 þ T2Þnþ ðT s0 � 2T1 þ T2Þ
n�1�, ð4Þ
L2t rCpb
27kDt½ðT1 þ 4T2 þ T s1Þ
n� ðT1 þ 4T2 þ T s1Þ
n�1�
¼ ½ðT1 � 2T2 þ T s1Þnþ ðT1 � 2T2 þ T s1Þ
n�1�. ð5Þ
In which n denotes the instant time t; ðn� 1Þ denotes timeðt� DtÞ, rCp denotes the product of the density r and itsspecific capacity Cp, Lt is the thickness of the material andk is the thermal conductivity of the layer. The truncationerror (T.E.) for this finite volume scheme is
T.E. ¼rCpb
k
1
6
q4Tqx4
����n
Dx4 �1
12
q4Tqx2qt2
����n
Dx4Dt
"
þ1
36
q5T
qx4qt
����n
Dx6 �1
72
q6Tqx4qt2
����n
Dx6Dt
#
�1
6
q4T
qx2qt2
����n
Dx2Dt2 þ1
48
q6Tqx4qt2
����n
Dx4Dt2. ð6Þ
This scheme is termed as Scheme I in this paper. It shouldbe noted that a discretised scheme can also be obtained byapplying Simpson 1
3integration over the time domain
½t� 2Dt; t�. However, in this case this scheme was proven tobe unstable.
2.2. A hybrid finite volume—finite difference scheme
Starting from Eqs. (2) and (3) obtained by applyingfinite volume method, a finite difference scheme of secondorder of accuracy for discretizing the q=qt terms of theleft-hand sides of Eqs. (2) and (3), employing qj=qt ¼
ð3jn � 4jn�1 þ jn�2Þ=2Dt, can be obtained as follows:
L2t rCpb
108kDt½3ðT s0 þ 4T1 þ T2Þ
n� 4ðT s0 þ 4T1 þ T2Þ
n�1
þ ðT s0 þ 4T1 þ T2Þn�2� ¼ ðT s0 � 2T1 þ T2Þ
n, ð7Þ
L2t rCpb
108kDt½3ðT1 þ 4T2 þ T s1Þ
n� 4ðT1 þ 4T2 þ T s1Þ
n�1
þ ðT1 þ 4T2 þ T s1Þn�2� ¼ ðT1 � 2T2 þ T s1Þ
n. ð8Þ
This scheme is termed as Scheme II in the paper.
2.3. A fifth order accuracy of hybrid finite volume—finite
difference scheme
Starting from Eqs. (2) and (3) obtained by applying thefinite volume method, a finite difference scheme of secondorder of accuracy for discretizing the q=qt terms of the left-hand sides of Eqs. (2) and (3), employing qj=qt ¼ ð50jn�
96jn�1 þ 72jn�2 � 32jn�3 þ 6jn�4Þ=24Dt, can be obtained
ARTICLE IN PRESSC. Luo et al. / Building and Environment 43 (2008) 37–43 39
as follows:
L2t rCpb
1296kDt½50ðT s0 þ 4T1 þ T2Þ
n� 96ðT s0 þ 4T1 þ T2Þ
n�1
þ 72ðT s0 þ 4T1 þ T2Þn�2� 32ðT s0 þ 4T1 þ T2Þ
n�3
þ 6ðT s0 þ 4T1 þ T2Þn�4� ¼ ðT s0 � 2T1 þ T2Þ
n, ð9Þ
L2t rCpb
1296kDt½50ðT1 þ 4T2 þ T s1Þ
n� 96ðT1 þ 4T2 þ T s1Þ
n�1
þ 72ðT1 þ 4T2 þ T s1Þn�2� 32ðT1 þ 4T2 þ T s1Þ
n�3
þ 6ðT1 þ 4T2 þ T s1Þn�4� ¼ ðT1 � 2T2 þ T s1Þ
n. ð10Þ
This scheme is termed as Scheme III in the paper.
3. Fourier stability analyses for the new inner nodes schemes
Assuming Tni ¼ T0rnejbxx for a time t and location of x,
substituting this equation into Eq. (4) for Scheme I,through some algebra operations, we can obtain
L2trCpb
27kDtðr� 1ÞðcosðbxDxÞ þ 2Þ
¼ �ðrþ 1Þð1� cosðbxDxÞÞ. ð11Þ
Recombining (11), and defining b ¼ L2trCpb
=27kDt, itbecomes
r ¼bðcosðbxDxÞ þ 2Þ � ð1� cosðbxDxÞÞ
bðcosðbxDxÞ þ 2Þ þ ð1� cosðbxDxÞÞ. (12)
Because ð1�cosðbxDxÞÞ is non-negative, and bðcosðbxDxÞþ2Þis positive, the denominator in Eq. (12) is larger than or equalto the numerator of Eq. (12), leading to the conclusion thatScheme I is unconditionally stable since �1prp1.
For Scheme II, the amplitude factor equation is
r2
3r2 � 4rþ 1¼ �
L2trCpb
108kDt
ð2þ cosðbxDxÞÞ
ð1� cosðbxDxÞÞ. (13)
It can be shown that jrjp1 (refer to Appendix A),indicating that Scheme II is also unconditionally stable.
Similarly for Scheme III, the amplitude factor equationis
r3 þ 3r2 þ 3rþ 1
ðr3 � 1Þ¼ �
3L2trCpb
216kDt
ð2þ cosðbxDxÞÞ
ð1� cosðbxDxÞÞ. (14)
It is difficult to prove in general that �1prp1, however,for all cases studied, Scheme III was also unconditionallystable.
4. Resulting discretised governing equations for the surface
nodes
The governing equations for surface nodes T s0 and T s1 are
q00s0 � �kqT
qx
����s0
� �¼ 0 or q00s0 þ k
qT
qx
����s0
¼ 0, (15)
�q00s1 þ �kqT
qx
����s1
� �¼ 0 or q00s1 þ k
qT
qx
����s1
¼ 0. (16)
Considering that
qT
qx
����s0
¼1
3Dxð�5:5T s0 þ 9T1 � 4:5T2 þ T s1Þ þOðDx3Þ
(17)
and
qT
qx
����cs1
¼1
3Dxð5:5Tcs1 � 9T2 þ 4:5T1 � T s0Þ þOðDx3Þ.
(18)
Rearranging Eqs. (15)–(18), we can obtain
T1 ¼ �3Dxð2q00s0 � q00s1Þ
13:5kþ
10T s0
13:5þ
3:5T s1
13:5, (19)
T2 ¼ �3Dxðq00s0 � 2q00s1Þ
13:5kþ
3:5T s0
13:5þ
10T s1
13:5. (20)
Substituting Eqs. (19) and (20) back in Eqs. (4) and (5), theresulting discretised equations for Scheme I are
1
162kL2trbCpb
57T s0 þ 24T s1 �9Lt
kq00s0 þ
6Lt
kq00s1
� �n"
� 57T s0 þ 24T s1 �9Lt
kq00s0 þ
6Lt
kq00s1
� �n�1#
¼Dt
2�T s0 þ T s1 þ
Lt
kq00s0
� �n"
þ �T s0 þ T s1 þLt
kq00s0
� �n�1#, ð21Þ
1
162kL2trbCpb
24T s0 þ 57T s1 �6Lt
kq00s0 þ
9Lt
kq00s1
� �n"
� 24T s0 þ 57T s1 �6Lt
kq00s0 þ
9Lt
kq00s1
� �n�1#
¼Dt
2T s0 � T s1 �
Lt
kq00s1
� �n"
þ T s0 � T s1 �Lt
kq00s1
� �n�1#. ð22Þ
Thus, in Eqs. (21) and (22), the variables of the inner nodesare removed.Similarly, the resulting discretised equations for Scheme
II are
3 57T s0 þ 24T s1 �9Lt
kq00s0 þ
6Lt
kq00s1
� �n
� 4 57T s0 þ 24T s1 �9Lt
kq00s0 þ
6Lt
kq00s1
� �n�1
ARTICLE IN PRESS
0.0E+00
1.0E-01
2.0E-01
3.0E-01
4.0E-01
5.0E-01
6.0E-01
7.0E-01
8.0E-01
9.0E-01
0 4 8 12 16 20 24
Time (Hour)
T/T
i (-
)
Finite Volume Scheme
Analytic Solution
Crank-Nicolson Scheme
Bhattacharya Scheme
Fig. 2. Evolvement of temperature at the surface of 0.5m thick brickwork
at ðT i ¼Þ10�C exposed to 0 �C atmosphere with one surface kept at
adiabatic condition using different schemes: Dx ¼ 0:5=3m and
Dt ¼ 3600 s.
-1.0E+01
-8.0E+00
-6.0E+00
-4.0E+00
-2.0E+00
0.0E+00
2.0E+00
4.0E+00
6.0E+00
0 4 8 12 16 20 24
Time (Hour)
Rela
tive E
rror
(%)
Finite Volume Scheme
Crank-Nicolson Scheme
Bhattacharya Scheme
Fig. 3. Relative error of temperature evolvement at the surface of 0.5m
thick brickwork at 10 �C exposed to 0 �C atmosphere with one surface
kept at adiabatic condition using different schemes: Dx ¼ 0:5=3m and
Dt ¼ 3600 s.
C. Luo et al. / Building and Environment 43 (2008) 37–4340
þ 57T s0 þ 24T s1 �9Lt
kq00s0 þ
6Lt
kq00s1
� �n�2
¼324kDt
L2trbCpb
�T s0 þ T s1 þLt
kq00s0
� �n
, ð23Þ
3 24T s0 þ 57T s1 �6Lt
kq00s0 þ
9Lt
kq00s1
� �n
� 4 24T s0 þ 57T s1 �6Lt
kq00s0 þ
9Lt
kq00s1
� �n�1
þ 24T s0 þ 57T s1 �6Lt
kq00s0 þ
9Lt
kq00s1
� �n�2
¼324kDt
L2trbCpb
T s0 � T s1 �Lt
kq00s1
� �n
. ð24Þ
5. Comparison of the schemes against analytical solutions
and full scale measurements
The Crank–Nicolson Scheme and the Bhattacharyascheme [3] were chosen also to discretize the internalnodes. It is not difficult to obtain similar ‘‘resultingdiscretised equations’’ which only involve temperaturesand heat fluxes at both surfaces for a construction layerand hence these are not presented in the current paper. Allof the derived schemes are evaluated against two analyticalsolutions for a construction layer and one set of actual fullscale measurements from a housing test module.
Case I: A constant temperature construction layer with
one surface kept at an adiabatic condition exposed to the
atmosphere. Assuming the initial temperature of theconstruction layer is set as T i, and the atmospheretemperature is T0 ð¼ 0 �CÞ, and the convective heattransfer coefficient between the layer surface and theatmosphere is hi (W/Km2). The initial condition andboundary conditions for this case can be summarized as thefollowing:
Initial condition
Tðx; t ¼ 0Þ ¼ T i. (25)
Boundary condition at the adiabatic surface
�kqT
qx
����x¼�L
¼ 0. (26)
Boundary condition at x ¼ L
�kqT
qx
����x¼L
¼ hiðT � T0Þ ¼ hiT . (27)
And according to [4], the analytical solution for this case is
Tðx; tÞ ¼ T i
X1n¼1
2B cosðunx=LÞ
ðBþ B2 þ u2nÞ cos un
e�ðunktÞ=ðrCpL2Þ. (28)
In which B represents the Biot number ð¼ hiL=kÞ, andu1; u2; . . . ; un are roots of the following equation:
u tan u ¼ B. (29)
To calculate the numerical solution for this case forScheme I, Eqs. (21) and (22) provide two equations forfour variables T s0, qs0, T s1 and qs1. Therefore, two moreequations are needed for a solution of T s0, qs0, T s1 and qs1
qs0 ¼ 0, (30)
qs1 ¼ hiT s1. (31)
For other schemes, the complementary equations (30) and(31) are the same, combined with different layer surfacegoverning equations.
ARTICLE IN PRESS
0.0E+00
2.0E-01
0 4 8 12 16 24 32 40
Rela
tive E
rror
(%)
Finite Volume Scheme
Crank-Nicolson Scheme
Bhattacharya Scheme
1.0E-01
-1.0E-01
-2.0E-01
-3.0E-01
-4.0E-01
-5.0E-01
-6.0E-01
-7.0E-01
Time (Hour)
362820
Fig. 4. Relative error of temperature evolvement at the surface of 0.11
thick brickwork at 30 �C exposed to atmospheric temperature of 20 �C
using different schemes: Dx ¼ 0:11=3m and Dt ¼ 3600 s.
2.0E+01
2.0E+01
2.1E+01
2.1E+01
2.2E+01
2.2E+01
2.3E+01
2.3E+01
0 4 8 12 16 20 24 28 32 36 40
Tem
pera
ture
(C
entigra
de)
Finite Volume Scheme
Analytic Solution
Crank-Nicolson Scheme
Bhattacharya Scheme
Time (Hour)
Fig. 5. Temperature evolvement at the surface of 0:1m thick plaster board
wall at 30 �C exposed as atmospheric temperature of 20 �C using different
schemes: Dx ¼ 0:0333m and Dt ¼ 3600 s.
C. Luo et al. / Building and Environment 43 (2008) 37–43 41
Figs. 2 and 3 show that time evolving history of thesurface temperature T s1 and its relative error for abrickwork layer with a thickness of 0.5m. The time stepis 3600 s as tabulated in Table 1. For a time tp4 h, theresults using the Bhattacharya’s implicit scheme almostrepeat the results using the Crank–Nicolson scheme, whilethe finite volume scheme prediction is not as good as thosetwo schemes. However, for a time in excess of 4 h, the finitevolume scheme predictions agree much better with theanalytical solution than the predictions using Bhattachar-ya’s implicit scheme and the Crank–Nicolson implicitscheme.
Case II: A constant temperature construction layer
exposed to the atmosphere. In this case the initialtemperature of the construction layer is set as T i, theatmospheric temperature as T0, and the convective heattransfer coefficient between the layer surface and theatmosphere as hi (W/Km2). The initial and boundaryconditions for this case can be summarized as the following:
Initial condition
Tðx; t ¼ 0Þ ¼ T i. (32)
Boundary condition at
x ¼ �L �kqT
qx
����x¼�L
¼ hiðT0 � TÞ. (33)
Boundary condition at
x ¼ L �kqT
qx
����x¼L
¼ hiðT � T0Þ. (34)
And according to [5], the analytical solution for this caseis
Tðx; tÞ ¼ T0 þ ðT i � T0ÞX1n¼1
4 sin un cosunx
L
� �2un þ sinð2unÞ
e�u2nkt=rCpL2.
(35)
In which B is also defined as the Biot number ð¼ hiL=kÞ,and u1; u2; . . . ; un are roots of Eq. (29).
The thermal properties and the discretised parametersused are presented in Table 1. Fig. 4 shows the relativeerrors of the temperature at the surface of 0.11mthick brickwork with a temperature of 30 1C which isexposed to atmospheric conditions at 20 1C, using differentschemes. The space step Dx is 0:11
3m and the time step
Dt is 3600 s. The relative error is calculated by the
Table 1
Properties of case studies I–III
Cases Materials rCp ðJ=m3 KÞ k (W/mK)
I Brickwork 1:290� 106 0.743
II (1) Brickwork 1:290� 106 0.743
II (2) Plaster board 7:88� 105 0.0096
II (3) Plaster board 7:88� 105 0.0096
following expression:
er ¼T c � Ta
Ta� 100%, (36)
in which er represents the relative error, T c is the calculatedtemperature and Ta is the analytical temperature. It can be
Dt (s) Dx (m) kDt=ðrCpDx2Þ (–) Bi ¼ hiL=k (–)
3600 0.167 0.075 1.984
3600 0.037 1.542 0.437
3600 0.033 0.0395 26.79
3600 0.0083 0.632 26.79
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2.4E+01
0 4 8 12 16 20 24 28 32 36 40
Time (Hour)
T (
Centigra
de)
dx=L/3
Analytic Solution
dx = L/6
dx=L/12
2.3E+01
2.3E+01
2.2E+01
2.2E+01
2.1E+01
2.1E+01
2.0E+01
2.0E+01
Fig. 6. Effect of different Dx on the predicted temperature profile using
Scheme I ðL ¼ 0:1mÞ.
0.0
5.0
10.0
15.0
20.0
25.0
30.0
0 100
Room
air tem
pera
ture
(C
entigra
de)
Time (hour)
200 300 400 500 600 700 800
Crank-Nicolson schemeFinite volume schemeMeasurements
Fig. 7. Comparisons of the room air temperature obtained by numerical
predictions with the Crank–Nicolson scheme and the finite volume scheme
and by measurements for the cavity brick test module at the University of
Newcastle in December, 2003. The time step is 1 h.
0.0
10.0
20.0
30.0
40.0
50.0
60.0
0
Roof air tem
pera
ture
(C
entigra
de)
Time (Hours from the first day of the month)
100 200 300 400 500 600 700 800
Crank-Nicolson schemeFinite volume method
Measurements
Fig. 8. Comparisons of the roof air temperature obtained by numerical
predictions with the Crank–Nicolson scheme and the finite volume scheme
with the measurements for the cavity brick test module at the University of
Newcastle in December, 2003. The time step is 1 h.
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
0
Tem
pera
ture
of th
e n
ort
h w
all
cavity a
ir (
Centigra
de)
Time (Hours from the first day of the month)
100 200 300 400 500 600 700 800
Crank-Nicolson scheme
Finite volume schemeMeasurements
Fig. 9. Comparisons of the north wall cavity air temperature obtained by
numerical predictions with the Crank–Nicolson scheme and the finite
volume scheme with the measurements for the cavity brick test module at
the University of Newcastle in December, 2003. The time step is 1 h.
C. Luo et al. / Building and Environment 43 (2008) 37–4342
observed from Fig. 4 that the finite volume scheme predictsbetter results than the Crank–Nicolson Scheme and theBhattacharya Scheme for periods t of less than 12h; for t
greater than 12h, the finite volume scheme’s accuracy issimilar to that of the Bhattacharya Scheme, with bothschemes performing better than the Crank–Nicolson Scheme.Fig. 5 shows the temperature comparisons for the threeschemes and the analytical results for a plaster board/timberstud wall with a thickness of 0.1m. To study the spatial stepeffect on the accuracy of the prediction, Fig. 6 shows theevolution of temperature with time for different spatial stepsranging from L=3 to L=12, in which L is the thickness of theplaster board. It can be observed that refining the spatialsteps will improve the prediction accuracy.
Case III: Full scale house measurements. As part of alarger research program, the Thermal Research Group hasconstructed several full scale housing test modules, on thecampus of the University of Newcastle. The modules have
been constructed using typical Australian practice, withwalling systems which include cavity brickwork (two skinsof 110mm thick brickwork with a 50mm cavity) and brickveneer (an external skin of 110mm brickwork, a 50mmcavity and an internal timber stud wall with plaster boardlining). Extensive observations of the thermal performance ofthese modules have been carried out over a period of severalyears. Details of the testing program have been reportedelsewhere [6].In this example, the temporal evolvements of the room air
temperature, the roof space air temperature, the wall cavityair temperature and the room surface temperature arestudied. The results using the different schemes are shown inFigs. 7–10. Although the converged solution for the aboveanalytical cases can be obtained using the Bhattacharya
ARTICLE IN PRESS
0.00 700
Tem
pera
ture
of th
e e
ast ro
mm
surf
ace (
Centigra
de)
35.0
30.0
25.0
20.0
15.0
10.0
5.0
Time (Hours from the first day of the month)
100 200 300 400 500 600 800
MeasurementsFinite volume scheme
Crank-Nicolson scheme
Fig. 10. Comparisons of the eastern wall internal surface temperature
obtained by numerical predictions with the Crank–Nicolson scheme and
the finite volume scheme with the measurements for the cavity brick test
module at the University of Newcastle in December, 2003. The time
step is 1 h.
C. Luo et al. / Building and Environment 43 (2008) 37–43 43
Scheme, this scheme fails to predict a converged solution forthe practical house cases. Therefore, only predictions fromtwo schemes, the finite volume scheme and the Crank–Ni-colson scheme, are compared with the measurements for aone month period. To avoid errors from the conversion ofthe solar radiation on the surface to the total heat flux on theouter surface, the hourly measured temperature profiles ofthe outer surfaces and roof surface of the monitored housewere chosen as the input data. In this comparison, six zones(the room air, the roof air and the four cavity air gaps) werechosen to simulate the thermal performance of the cavitybrick module.
The predicted temperature difference using the finitevolume and the Crank–Nicolson scheme is as small as0.05 1C as shown in Figs. 7–10. The comparison betweenthe numerical predictions and the observed measurementsare excellent except for the initial time. This is becausethe initial solution is assumed to be the steady statesolution of all the inner nodes. If the monitored heatfluxes of the outer surface were used as the input data,this ‘‘delayed’’ period would be much longer. Thisgood comparison between the numerical predictions andmeasurements indicates that the proposed numericalmethod can be used for predicting thermal performanceof a building for long periods.
6. Conclusion
A new numerical method to avoid the discretisedequations for inner nodes is proposed and validated byanalytical solutions and practical building measurements.For the discretised numerical schemes for the computa-tional nodes in the building wall layers, a new finite volume
scheme based on Simpson’s 13rule is proposed and proved
to be stable, and with a higher accuracy than otherschemes. The software based on this new model cantherefore predict satisfactorily the temperature profiles ofthe zonal air and the wall surfaces.
Acknowledgements
This work has been supported by Think Brick Australia(formally the Clay Brick and Paver Institute) and theAustralian Research Council. The support of bothorganisations is gratefully acknowledged.
Appendix A. About the Fourier stability of Scheme II
AssumingL2t rCpb
108kDtð2þcosðbxDxÞÞð1�cosðbxDxÞÞ
¼ C, then Eq. (13) can berewritten as
r2
ð3r2 � 4rþ 1Þ¼ �C or 3þ
1
C
� �r2 � 4rþ 1 ¼ 0.
(37)
The roots of Eq. (37) are
r ¼4�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16� 4ð3þ 1=CÞ
p2ð3þ 1=CÞ
¼2�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1=C
p3þ 1=C
. (38)
If 1� 1=CX0, or, CX1 then r will be a real number, ifC ¼ 1, then r ¼ 0:5; else if C ¼ 1, r ¼ 1 or 1
3, satisfying
that rp1.
If 1� 1=Co0, or, Co1, r ¼2�i
ffiffiffiffiffiffiffiffiffiffiffi1=C�1p
3þ1=C¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þð1=CÞ�1p
3þ1=C
eif ¼ 1ffiffiffiffiffiffiffiffiffiffiffi3þ1=Cp eifo 1ffiffiffiffiffiffi
3þ1p eif ¼ 1
2eif, leading to the conclu-
sion that jrjo 12o1.
Therefore, for all cases, amplitude factor jrjp1 indicatesthat Scheme II is unconditionally stable.
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