Advances in Automatic Control, Modelling & Simulation
Determining the time constant of buildings
STAN IVAN FELICIA ELENA, MIRCEA ION
Department of Electrical Engineering and Aerospace
University of Craiova
Bdul. Decebal, No. 107, Craiova 200440, tel/fax: +40 251 436 447
Romania
ely_felicia@yahoo.com, imircea@elth.ucv.ro
Abstract: This paper is presenting a study on determining the thermal time constant of buildings. Were
considered two types of building an energy efficient building and another building less energy efficient, which
were determined thermal time constants.
Key-Words: buildings, thermal inertia, heat capacity, thermal time constant, coupling thermal mass, internal
coupling coefficient.
χ'j is the internal heat capacity corrected for a
specific building element j, [J/(m2·K)];
Aj is the area of element j, [m2].
For simple hourly method, internal thermal capacity
of the building or site construction, Cm, is calculated
by summing the heat capacities of all structural
elements in direct thermal contact with the air inside
the analyzed area:
1 Method for determining the thermal
time constant of buildings
Thermal inertia of a building can be quantified by
thermal time constant which describes how long it
takes, where the heating/cooling of a building is
discontinuous for building temperature to change.
Thermal time constant is defined as the ratio of a
building's thermal mass and overall heat loss.
Time constant of the building area, τH, is
characterized by internal thermal inertia conditioned
area during the heating period, i.e. cooling. [1]
It is calculated by the formula:
τ=
C ' m / 3600
[h]
Hm
Cm =
(1)
ISBN: 978-1-61804-189-0
⋅ A j [J/K]
(3)
Maximum
thickness [m]
Application
Factor determining the gain or
loss of use
Interrupt
0,10
0,03
Alternatively, it may be decided at national level for
specific applications and types of buildings, use of
default values depending on the type of
construction. In the absence of national standardized
values can be used following default values for
dynamic parameters. [1]
For monthly and seasonal method, internal thermal
capacity of the building for a specific area of
buildings, C'm is calculated by summing the
corrected heat capacities of all structural elements in
direct thermal contact with the air inside the
analyzed area:
'
j
⋅ A j [J/K]
Table 1.Thickness to be considered for internal heat
capacity
1.1. Corrected internal thermal capacity
building C'm [1]
∑χ
j
Cm is the heat capacity [J/K];
χj is the internal heat capacity of the building
surface j element analysis [J/(m2·K)];
τ is the time constant of the building or construction
zone for both heating and / or cooling [h];
C'm is corrected internal thermal capacity building
[J/K].
Coupling coefficient hm is internal thermal mass
[W/K].
C m' =
∑χ
Table 2.The default values for the dynamic
Class of
inertia
(2)
Very
light
202
Monthly and seasonal
method
Hm
τ
C'm (J/K)
(W/K)
(h)
9,2 · Afl
60 000 ·
Afl
1,8
Simple hourly
method
Am
Cm
(m2)
(J/K)
80
2,5 ·
000 ·
Afl
Afl
Advances in Automatic Control, Modelling & Simulation
Class of
inertia
Monthly and seasonal
method
Hm
τ
C'm (J/K)
(W/K)
(h)
Light
9,2 · Afl
83 000 ·
Afl
2,5
Average
9,2 · Afl
124 000 ·
Afl
3,7
Heavy
9,9 · Afl
195 000 ·
Afl
5,5
Very
heavy
10,4 ·
Afl
278 000 ·
Afl
7,4
annual values of heat transfer coefficients of
transmission and ventilation.
Simple hourly
method
Am
Cm
(m2)
(J/K)
110
2,5 ·
000 ·
Afl
Afl
165
2,5 ·
000 ·
Afl
Afl
260
3,0 ·
000 ·
Afl
Afl
370
3,5 ·
000 ·
Afl
Afl
2.2. Simplified hourly method [2]
The model is a simplification of dynamic
simulation, taking into account the following
considerations:
- Same level of transparency, reproducibility and
robustness as monthly method;
- Be clearly specified a limited set of equations,
allowing calculation traceability process;
- Reduce as much as possible of the input data;
- Unambiguous calculation procedure;
- The main advantage it has is monthly intervals
method that allows direct input of hourly models.
Afl is useful surface room [m2];
2 Simplified
method
Coupling thermal mass [1]
schedule.
2.1. Simplified method and data on the dynamic
Internal coupling coefficient of thermal mass of the
building area is obtained using the following
equation for calculating:
H m = (H tr + H ve ) [W/K]
(4)
Htr - heat transfer coefficient by transmission
[W/K];
The values for the coefficient of heat transmission,
Htr,k of k element is calculated in accordance with
ISO/ DIS 13789:2005, taking into account standards
for specific, such as windows (ISO/DIS 100771:2004) walls and roofs (ISO/DIS 6946:2005)
curtain wall (prEN 13947), etc..
Heat transfer coefficient of heat transmission Htr,k
takes into account the temperature θe,k is the
temperature θe for outdoor use. For simple hourly
method requires a distinction between structural
elements glazing (windows, doors, curtain walls,
glazed elements provided) and compact elements.
H ve ,k = ρ a ⋅ c a ⋅ q ve ,k [W/K]
Fig.1. Electrical model of a wall with five resistance
and capacity [2]
Internal coupling coefficient of thermal mass of the
building area is obtained using the following
equation for calculating:
Hm =
(5)
1
1
+
H is H ms
[W/K]
(6)
His and Hms coupling coefficients, calculated for
simple hourly method:
Hve,k is the coefficient of heat transfer by ventilation
air circulation element k in the temperature θs,k
[W/K];
qve,k is the flow of air in the conditioned space,
determined in accordance with standards, [m3/s];
ρa·ca that is heat capacity of air volume=1200[J/
(m3·K)], or ρa·ca = 0, 34 Wh/(m3K);
Htr and Hve values can be calculated as the average
ISBN: 978-1-61804-189-0
1
His = his ⋅ At [W/K]
(7)
At = Rat ⋅ A fl
(8)
With:
203
Advances in Automatic Control, Modelling & Simulation
opaque building elements, Hop, and divided into
Hem and Hms;
- Hv ventilation and process characteristics θsup air;
- His coupling conductance [W/K];
- Cm internal heat capacity [J/K].
His is the coupling conductance between nodes i and
s [W/K];
At is the total area of the room surfaces [m2];
Afl is useful surface room [m2];
His is the heat transfer coefficient between nodes i
and s, with fixed value hsi = 3, 45 [W/ (m2K)];
Rat is the dimensionless ratio of surface areas and
internal floor area, Rat can be assumed to be equal to
4, 5.
Sizing system used to calculate Afl (internal
dimensions, external dimensions or size) may be set
nationally, but must be specified. [3]
If necessary, useful surface area of a building is
determined in a similar manner for each building
area calculation. Sum of all the useful surface area
equals the usable area of the building.
Coupling coefficient table, monthly and seasonal Hm
methods:
Table 3. Heat storage materials and their physical
properties [4]
clay
brick
sand
wood
concrete
glass
aluminum
iron
steel
earth
magnet
water
His = 3, 45·At, At = 4,5·Afl, Hms = 9,1·Am;
Internal heat capacity corrected for monthly and
seasonal method:
Split transmission heat transfer coefficient, Htr, for
opaque elements Hop in Hem and Hms is obtained as
follows:
1
[W/K]
1
1
−
H op H ms
C m2
∑A
j
⋅ χ 2j
[m2]
Material
d [cm]
m [kg/m³]
cs [kJ/(kgK)]
λ [W/(mK)]
Afl [m2]
(9)
Coating
2
Concrete
20
2400
1,08
2,1
1800
1,08
0,87
10,04
U-value: 2, 6439 W/ (m²K)
Heat capacity: 557, 28 kJ/ (m²K)
Exterior wall
Material
d
[cm]
m [kg/m³]
cs[kJ/(kgK)]
λ[W/(mK)]
Afl [m2]
(10)
Exterior
plaster
Polystyrene
Concrete
Interior
plaster
1
10
20
2,5
1800
1,08
0,87
30
1,8
0,04
2400
1,08
2,1
1800
1,08
0,87
10,04
U-value: 0, 35645 W/ (m²K)
Heat capacity: 591, 84 kJ/ (m²K)
2.3. Determine the main variables of the
method:
Foundation plate
The main variables of the model are:
-heat transfer coefficients at the doors, windows,
curtain walls and walls made of glass Hw and
ISBN: 978-1-61804-189-0
Volumetric
heat capacity
[106J/m3K]
1,28
1,51
1,57
1,67
1,76
2,27
2,43
3,57
3,68
3,77
3,89
4,17
Less energy efficient building [5]
Interior wall
And Hms = hms · Am;
Hms is the coupling conductance between nodes m
and s [W/K];
hms is the heat transfer coefficient between nodes m
and with a fixed value hms = 9,1 [W/(m2·K)];
Am is the effective mass area [m2].
Effective mass area Am, is calculated by the
formula:
Am =
Specific
heat
[J/kgK]
879
837
712
2390
880
837
896
452
465
1840
752
4182
We will determine the value of thermal constant for
a less energy efficient building and energy efficient
building having walls composed of the following
structures:
C'm = 0, 75·Cm
H em =
Material
density
[kg/m3]
1458
1800
2200
700
2000
2710
2710
7900
7840
2050
5177
988
Material
204
Material
Concrete
d
20
Mineral
wool
10
Sound
insulation
2
Plast
er
4
Advances in Automatic Control, Modelling & Simulation
Material
Concrete
Mineral
wool
Sound
insulation
Plast
er
2400
45
20
1080
[cm]
m
[kg/m³]
U-value: 0, 11893 W/ (m²K)
Heat capacity: 920, 67 kJ/ (m²K)
Roof
Material
cs
[kJ/(kgK)]
λ
[W/(mK)]
Afl
[m2]
1,08
1,8
1
1,8
2,1
0,04
0,035
0,87
d [cm]
m [kg/m³]
cs[kJ/(kgK)]
λ[W/(mK)]
Afl [m2]
67,60
Mineral wool
10
45
1,8
0,04
80,93
OSB
15
600
1,98
0,13
U-value: 0, 26073 W/ (m²K)
Heat capacity: 205, 74 kJ/ (m²K)
Table 4.Determine the main variables of simplified
hourly method less energy efficient building
Energy efficient building
Interior wall
Material
d [cm]
m [kg/m³]
cs [kJ/(kgK)]
λ [W/(mK)]
Afl [m2]
Plaster
2
1800
2400
0,35
Concrete
20
2400
1,08
2,1
χ [j/m2K]
Aj [m2]
Cm [J/K]
10,4
C'm [J/K]
U-value: 2, 4249 W/ (m²K)
Heat capacity: 557, 28 kJ/ (m²K)
His[W/K]
Hms[W/K]
At[m2]
Am[m2]
Hm [W/K]
Exterior wall
Material
Concrete
d [cm]
m [kg/m³]
c
s[kJ/(kgK)]
λ
[W/(mK)]
Afl [m2]
20
2400
Mineral
wool
15
45
1,08
2,1
20
30
Plast
er
5
1800
0,504
1,8
1,8
1,8
0,35
0,35
Polystyrene
Foundation plate
d [cm]
m [kg/m³]
cs[kJ/(kgK)]
λ[W/(mK)]
Afl [m2]
35
2400
1,08
2,1
Extruded
polystyrene
ISBN: 978-1-61804-189-0
20
30
1,8
0,04
67,60
Mineral
wool
10
45
0,504
0,04
Exterio
r wall
Foundation
plate
591840
604660
10,40
615513
6
461635
2
35,88
9,877
10,4
1,08
45,75
67,60
40875016
30656262
233,22
435,61
67,6
47,87
668,83
Roof
2057
40
80,93
1665
0538
1248
7904
279,2
72,28
80,93
7,94
351,4
Building type
Less energetically
Energetically efficient
efficient building
building
Afl
Structur
Afl
τ1
Structure
τ2 [h]
[m2]
e
[m2]
[h]
Interior
26,4
Interior
28,93
10.4
10.4
wall
6
wall
9
Exterior
28,0 Exterior
10.4
10.4 32,54
wall
2
wall
Foundati
12,7 Foundat 67.6
67.60
16,64
on plate
3
ion plate
0
80.9
Roof
80.93
9,86
Roof
1,72
3
77,0
Total
79,83
7
10,4
Concrete
Interio
r wall
55728
0
10,04
55950
91
41963
18
35,88
8,162
10,4
0,89
44,047
Table 5. Thermal time constant values,τ for the
analyzed buildings
U-value: 0,096079 W/ (m²K)
Heat capacity: 629, 8 kJ/ (m²K)
Material
Sound
insulation
1
20
1
0,035
We will determine building thermal constant for the
two types of buildings analyzed using computer
relationship (1).
To determine the thermal constant of the building,τ,
for both types of buildings analyzed, using equation
(1) calculation, we determine the main variables
used in the simplified hourly method using relations
(2) and (6).
Roof
Plaster
1
1800
1,08
0,87
Mineral Mineral
wool
wool
20
20
45
45
0,504
0,504
0,04
0,04
80,93
U-value: 0,095511 W/ (m²K)
Heat capacity: 28,712 kJ/ (m²K)
U-value: 0, 29563 W/ (m²K)
Heat capacity: 604, 66 kJ/ (m²K)
Material
d [cm]
m [kg/m³]
cs [kJ/(kgK)]
λ [W/(mK)]
Afl [m2]
Gypsum
plaster
1
1800
1,08
0,7
Sound
insulat
ion
2
20
1
0,035
205
Advances in Automatic Control, Modelling & Simulation
Table 6.Determine the main variables of the
simplified hourly method for building energy
efficient
χ
[j/m2K]
Aj
[m2]
Cm
[J/K]
C'm
[J/K]
His
[W/K]
Hms
[W/K]
At
[m2]
Am
[m2]
Hm
[W/K]
35
Interior
wall
Exterio
r wall
Foundatio
n plate
Roof
557280
629800
920670
28712
10,04
10,40
67,60
80,93
5595091
4196318
654992
0
491244
0
3 Conclusions
This paper presents a study regarding a comparation
for the thermal constant of buildings for two
different buildings: an energy efficient building and
another building less energy efficient
From the analysis it can be seen that the time
constant of less energy-efficient building is smaller
than the time constant for energy efficient buildings,
the total value of τ1 = 77,07 compared with τ2 =
79,83, so that the higher the time constant value is
the inside temperature is more stable to external
temperature variations.
Intermittent heating saves energy even higher as the
room (building) has a smaller time constant, the
higher the temperature is higher and the more usage
time is less. So if you subtract heat transfer
coefficient of the building envelope insulation, high
performance windows that are used in terms of
thermal resistance, energy savings through lower
intermittent heating.
23236
62
17427
47
279,20
85
62237292
46677969
35,88
35,88
233,22
4,40
6,04
545,56
0,760
10,4
10,4
67,6
80,93
0,48
0,66
59,95
0,08
40,29
41,92
778,78
279,97
References:
[1]. ISO TC 163/SC 2/N02, Energy performance of
buildings - Calculation of energy use for space
heating and cooling 2006, pg. 58 -60.
[2]. ISO TC 163/SC 2/N02, „Energy performance
of buildings - Calculation of energy use for
space heating and cooling”, 2006, Annex C,
Full set of equations for simple hourly method
pg. 82.
[3]. ISO TC 163/SC 2/N02, „Energy performance
of buildings - Calculation of energy use for
space heating and cooling”, 2006, pg. 23, 24.
[4]. Tuohy et al, „Masa termică, izolaŃie şi
ventilaŃie în locuinŃe durabile” - 2004 .
[5]. Stan Ivan Felicia E., „Studiul eficienŃei
energetice şi economice a clădirilor”, Anexa
2.1. Caracteristici structuri pereŃi pentru cele
două tipuri de clădiri analizate, 2009.
Thermal time constant for the two buildings analyzed
t1 [h]
30
t2 [h]
V a l u e [h ]
25
20
15
10
5
0
1
2
3
4
Fig. 2. Comparisons between time constants of the
two buildings analyzed
ISBN: 978-1-61804-189-0
206