40

DARNIOJI ARCHITEKTŪRA IR STATYBA
2013. No. 1(2) 

JOURNAL OF SUSTAINABLE ARCHITECTURE AND CIVIL ENGINEERING
ISSN 2029–9990

Investigation into Thermal Capacitance of the Building Envelope

Patrikas Bruzgevičius1, Arūnas Burlingis1, Vytautas Stankevičius1, Darius Pupeikis2, 
Rosita Norvaišienė1, Karolis Banionis1

1Institute of Architecture and Construction of Kaunas University of Technology, Tunelio st. 60, LT-44405 Kaunas, Lithuania
2Kaunas University of Technology, Faculty of Civil Engineering and Architecture, Studentu st. 48, LT-51367 Kaunas, Lithuania

* Corresponding author: patrikas.bruzgevicius@yahoo.com

http://dx.doi.org/10.5755/j01.sace.1.2.3543

The purpose of this research is to determine the actual effective thermal capacitance of the building envelope in respect 
of the place of thermal layer in the enclosure. The actual effective thermal capacitance of the building envelope is estimated 
by calculating the thermal capacitance of building enclosure layers in order to find out the active thermal capacitance of 
the enclosure. This research analyses unsteady heat transfer cases. The calculation principle of such transfers is based on 
the finite elements method. The specific value of the Fourier number, which gives calculations the optimal accuracy, is 
determined. This paper also discusses three types of envelope constructions. In all three cases, thermal resistance of the 
envelopes coincides; only the position of thermal insulation layer in the envelopes is different. In all three cases, time delay of 
the indoor air temperature in respect of the outdoor air temperature, as well as decrement factors of the oscillation amplitude 
are calculated. These factors of inertia were determined at the continuous outdoor air temperature oscillation. The research 
revealed that continuous outdoor air temperature oscillations through multi-layered building enclosures are suppressed by 
high thermal capacitance enclosure layers, independently from their position. High thermal capacitance enclosure layers 
oriented to the interior of the premises suppress the fluctuations more efficiently than those oriented to the exterior.

Keywords: building envelope, enclosure, time constant, Fourier number, thermal capacitance, time delay, decrement 
factor.

1. Introduction

When analysing unsteady thermal processes in 
buildings and classifying the buildings according to their 
thermal inertia factors, it is important to properly evaluate the 
influence of building enclosures on the internal temperature 
fluctuation due to the outdoor temperature changes. The 
effective thermal capacitance Ceff of the building envelope, 
thermal time constant τ and actual thermal delay are the 
indicators of the thermal inertia of the building. Therefore, 
it is important to evaluate them appropriately. 

The methodology of the Lithuanian Standards Board 
(LST EN ISO 13790:2008) classifies buildings into inertia 
classes according to the value of time constant. Thermal 
time constant of the building τ is expressed as a proportion 
between the internal active thermal capacitance of the 
building Cm and heat transfer coefficient of the building H 
(LST EN ISO 13790:2008):

)(,
3600/

h
HH

C

VT

m

+
=τ  (1)

where: Cm  – internal active thermal capacitance, J/K; 
HT – heat transfer coefficient of building enclosures, W/K; 
HV – ventilation heat loss coefficient, W/K.

The average active thermal capacitance Cm is defined 
as the heat accumulated inside the building, when the 
internal air temperature oscillates according to the sinusoid 
with the period t-24h and the amplitude 1K (LST EN ISO 
13790:2008). 

The value of the internal thermal capacitance of the 
building depends on the amplitude of temperature oscillation 
in the premises; the allowed fluctuation limits depend on the 
purpose of the premises (Šeduikytė et al. 2008, LST EN ISO 
7730, STR 2.09.02:2005).

The standard (LST EN ISO 13790:2008) evaluates 
active thermal capacitance Cm only inside the building, 
i.e. it evaluates energy costs per year when temperature 
fluctuations occur inside the building, which is not a very 
accurate method for evaluating the complex influence of the 
building envelope on the thermal regime of the premises 
and selecting the capacity of the heating system. Continuous 
fluctuations of the outdoor air temperature affect the building 
enclosures first, and only then these fluctuations make an 
impact on the thermal regime of the premises.

Swedish scientists Adamsson, Dafgard, Rydberg and 
Peterson (Berg-Hallberg 1985) were among the first to 
classify buildings according to the value of time constant. 



41

When classifying buildings, the scientists evaluated the 
thermal capacitance of the building envelope taking into 
account all the layers of the enclosure, or half of the thermal 
capacitance of the enclosure, or the internal layers of the 
enclosure up to thermal insulation layer. 

According to (Kalema et al. 2006), the thermal 
capacitance of the construction taken into account is limited 
to 0.10 m from the surface, or to the first insulation layer. 

When classifying buildings according to the time 
constant factor, Antonopoulos et al. (1995) calculate thermal 
capacitance using the lump capacitance method: 

)/(, KJcC nnnρν∑=   (2)
where: νn, ρn and cn are the volume, density and specific heat, 
respectively, of element n.

Thermal regime inside the building is affected in two 
ways: due to infiltration and through the external building 
envelope. In reality, most of the buildings are made of 
enclosures of different construction types, depending 
on the purpose and technological regime of the building 
and dominant construction traditions in the country. The 
massive layer of multi-layered enclosures can be oriented 
to the exterior or the interior of the building, or it can be 
in the middle of the enclosure. Thermal capacitance is an 
important quality that enables walls to absorb, store and 
later release thermal energy into the building space. 

The aim of this work is to determine the effect of 
massive layer position in the enclosure on the thermal 
inertia of the enclosure.

2. Methods

The accuracy of calculations of unsteady heat transfer 
using the finite elements method depends on the selected 
values of Fourier and Biot numbers. 

The Fourier number is designated by the symbol Fo and 
the equation for every layer of this number is (Stankevičius 
et al. 2000, Incropera et al. 2007, Hagentoft 2001):

2d
za

Fo
D⋅

=  (3)

The Fourier number can also be expressed as:

 

where: ν
n
, ρ

n
 and c

n 
are the volume, density and specific 

heat, respectively, of element n. 

Thermal regime inside the building is affected in two 

ways: due to infiltration and through the external building 

envelope. In reality, most of the buildings are made of 

enclosures of different construction types, depending on the 

purpose and technological regime of the building and 

dominant construction traditions in the country. The massive 

layer of multi-layered enclosures can be oriented to the 

exterior or the interior of the building, or it can be in the 

middle of the enclosure. Thermal capacitance is an 

important quality that enables walls to absorb, store and 

later release thermal energy into the building space.  

The aim of this work is to determine the effect of 

massive layer position in the enclosure on the thermal 

inertia of the enclosure. 

2. Methods 

The accuracy of calculations of unsteady heat transfer 

using the finite elements method depends on the selected 

values of Fourier and Biot numbers.  

The Fourier number is designated by the symbol F
o 
and 

the equation for every layer of this number is (Stankevičius 

et al. 2000, Incropera et al. 2007, Hagentoft 2001): 

 

2

d

za

F
o

Δ⋅

=                   

 (3) 

 

The Fourier number can also be expressed as: 

 

;
2

zA

C

z

RC

z

dc

z

F

sl

sl

slslp

o
Δ⋅=

Δ⋅Λ

=

⋅

Δ

=

⋅⋅

Δ⋅

=

ρ

λ

    (4) 

 

where: a – the thermal diffusivity, (m
2

/s); Δz – the time step, 

(s); d – thickness of conditional layer, (m); λ – thermal 

conductivity, (W/(m⋅K)); ρ – material density, (kg/m
3

); c – 

specific thermal capacity , (J/(kg⋅K)). 

 

The thermal capacitance of the layer:  

 

)/(,
2

.
KmJdcC

psl
⋅⋅⋅= ρ            (5) 

 

where: ρ – material density, (kg/m
3

); c – specific thermal 

capacity , (J/(kg⋅K)); d - thickness of layer, (m).  

 

The thermal resistance of the layer: 

 

WKmdR
sl

/)(,/
2

.
⋅= λ              (6) 

 

where: d – thickness of material, (m); λ – thermal 

conductivity, (W/(m⋅K). 

 

The thermal diffusion of the layer (Pupeikis et al. 

2010): 

 

)/(,/
2

.
KmWd

sl
⋅=Λ λ             (7) 

where: Λ  – thermal conductance, (W/m
2

⋅K); λ – thermal 

conductivity, (W/(m⋅K)); d – thickness of layer, (m). 

 

The thermal diffusivity of the material:  

sm

c

a

p

/;
2

⋅

=

ρ

λ                (8) 

 

where: λ – thermal conductivity, (W/(m⋅K)); ρ – material 

density, (kg/m
3

); c – specific thermal capacity , (J/(kg⋅K)). 

 

The thermal diffusion of the separate layer:  

 

)/1(,
2

s

d

a

A
sl
=                 (9) 

 

where: A
sl
 – thermal diffusion of the material layer [1/s]; a – 

thermal diffusivity of material, [m
2

/s]; d – thickness of 

layer, [m]. 

 

The dependence of layer thickness on the Fourier 

number and time step:  

)(, m

F

za

d

o

Δ⋅

=                (10) 

 

where: a – thermal diffusivity of material, (m
2

/s); d – 

thickness of layer, (m); Δz - the time step, (s); F
o
 - Fourier 

number. 

 

Thermal diffusivity of the material a (m
2

/s; m
2

/h) 

shows the rate of temperature “diffusion/equalization” in the 

object, i.e. the part of 1 m distance per time unit for the 

temperature to equalize in the homogeneous object from the 

initial moment, when the temperature drop between two 

planes (points) at 1 m distance is 1K.  

The inverse proportion 1/a (s/m
2

; h/m
2

) shows the time 

in which the temperature equalizes between two planes 

(points) of the homogeneous object at 1 m distance, when 

the temperature drop at the initial moment is 1K. Suppose, 

there is an EPS with the following characteristics: λ=0,07 

W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 J/(kg⋅K), then, the 

calculations are performed according to (8) a=0,00000209 

m
2

/s≈0,0075m
2

/h and 1/a= 478571 s/m
2 

≈133 h/m
2

.  

Thermal diffusion of the layer A (1/s; 1/h) shows the 

rate of thermal “diffusion/equalization”, i.e., the part of 

layer thickness Y per time unit for the temperature to 

equalize in the homogeneous object from the initial 

moment, when the temperature drop on both sides of the 

layer is 1K. 

The inverse proportion 1/A (s; h) shows the time in 

which the temperature equalizes in the homogeneous layer, 

when layer thickness is Y and temperature drop in the layer 

at the initial moment is 1K. 

Suppose, there is an EPS layer with the following 

characteristics: λ=0,07 W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 

J/(kg⋅K), d= 50mm, then, the calculations are performed 

according to (9): A≈3,01,(1/h) and 1/A≈0,332, (h).  

Fourier number F
o
 - shows the thickness (depth) of 

“equalization” in the layer during the selected time step Δz, 

i.e., the part of layer thickness Y for the temperature to 

equalize in the homogeneous layer from the initial moment 

during the time period Δz, when the temperature drop on 

both sides of the layer, whose thickness is Y, is 1K. 

 (4)

where: a – the thermal diffusivity, (m2/s); Dz – the time 
step, (s); d – thickness of conditional layer, (m); λ – thermal 
conductivity, (W/(m × K)); ρ – material density, (kg/m3); 
c – specific thermal capacity , (J/(kg×K)).

The thermal capacitance of the layer: 

 

where: ν
n
, ρ

n
 and c

n 
are the volume, density and specific 

heat, respectively, of element n. 

Thermal regime inside the building is affected in two 

ways: due to infiltration and through the external building 

envelope. In reality, most of the buildings are made of 

enclosures of different construction types, depending on the 

purpose and technological regime of the building and 

dominant construction traditions in the country. The massive 

layer of multi-layered enclosures can be oriented to the 

exterior or the interior of the building, or it can be in the 

middle of the enclosure. Thermal capacitance is an 

important quality that enables walls to absorb, store and 

later release thermal energy into the building space.  

The aim of this work is to determine the effect of 

massive layer position in the enclosure on the thermal 

inertia of the enclosure. 

2. Methods 

The accuracy of calculations of unsteady heat transfer 

using the finite elements method depends on the selected 

values of Fourier and Biot numbers.  

The Fourier number is designated by the symbol F
o 
and 

the equation for every layer of this number is (Stankevičius 

et al. 2000, Incropera et al. 2007, Hagentoft 2001): 

 

2

d

za

F
o

Δ⋅

=                   

 (3) 

 

The Fourier number can also be expressed as: 

 

;
2

zA

C

z

RC

z

dc

z

F

sl

sl

slslp

o
Δ⋅=

Δ⋅Λ

=

⋅

Δ

=

⋅⋅

Δ⋅

=

ρ

λ

    (4) 

 

where: a – the thermal diffusivity, (m
2

/s); Δz – the time step, 

(s); d – thickness of conditional layer, (m); λ – thermal 

conductivity, (W/(m⋅K)); ρ – material density, (kg/m
3

); c – 

specific thermal capacity , (J/(kg⋅K)). 

 

The thermal capacitance of the layer:  

 

)/(,
2

.
KmJdcC

psl
⋅⋅⋅= ρ            (5) 

 

where: ρ – material density, (kg/m
3

); c – specific thermal 

capacity , (J/(kg⋅K)); d - thickness of layer, (m).  

 

The thermal resistance of the layer: 

 

WKmdR
sl

/)(,/
2

.
⋅= λ              (6) 

 

where: d – thickness of material, (m); λ – thermal 

conductivity, (W/(m⋅K). 

 

The thermal diffusion of the layer (Pupeikis et al. 

2010): 

 

)/(,/
2

.
KmWd

sl
⋅=Λ λ             (7) 

where: Λ  – thermal conductance, (W/m
2

⋅K); λ – thermal 

conductivity, (W/(m⋅K)); d – thickness of layer, (m). 

 

The thermal diffusivity of the material:  

sm

c

a

p

/;
2

⋅

=

ρ

λ                (8) 

 

where: λ – thermal conductivity, (W/(m⋅K)); ρ – material 

density, (kg/m
3

); c – specific thermal capacity , (J/(kg⋅K)). 

 

The thermal diffusion of the separate layer:  

 

)/1(,
2

s

d

a

A
sl
=                 (9) 

 

where: A
sl
 – thermal diffusion of the material layer [1/s]; a – 

thermal diffusivity of material, [m
2

/s]; d – thickness of 

layer, [m]. 

 

The dependence of layer thickness on the Fourier 

number and time step:  

)(, m

F

za

d

o

Δ⋅

=                (10) 

 

where: a – thermal diffusivity of material, (m
2

/s); d – 

thickness of layer, (m); Δz - the time step, (s); F
o
 - Fourier 

number. 

 

Thermal diffusivity of the material a (m
2

/s; m
2

/h) 

shows the rate of temperature “diffusion/equalization” in the 

object, i.e. the part of 1 m distance per time unit for the 

temperature to equalize in the homogeneous object from the 

initial moment, when the temperature drop between two 

planes (points) at 1 m distance is 1K.  

The inverse proportion 1/a (s/m
2

; h/m
2

) shows the time 

in which the temperature equalizes between two planes 

(points) of the homogeneous object at 1 m distance, when 

the temperature drop at the initial moment is 1K. Suppose, 

there is an EPS with the following characteristics: λ=0,07 

W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 J/(kg⋅K), then, the 

calculations are performed according to (8) a=0,00000209 

m
2

/s≈0,0075m
2

/h and 1/a= 478571 s/m
2 

≈133 h/m
2

.  

Thermal diffusion of the layer A (1/s; 1/h) shows the 

rate of thermal “diffusion/equalization”, i.e., the part of 

layer thickness Y per time unit for the temperature to 

equalize in the homogeneous object from the initial 

moment, when the temperature drop on both sides of the 

layer is 1K. 

The inverse proportion 1/A (s; h) shows the time in 

which the temperature equalizes in the homogeneous layer, 

when layer thickness is Y and temperature drop in the layer 

at the initial moment is 1K. 

Suppose, there is an EPS layer with the following 

characteristics: λ=0,07 W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 

J/(kg⋅K), d= 50mm, then, the calculations are performed 

according to (9): A≈3,01,(1/h) and 1/A≈0,332, (h).  

Fourier number F
o
 - shows the thickness (depth) of 

“equalization” in the layer during the selected time step Δz, 

i.e., the part of layer thickness Y for the temperature to 

equalize in the homogeneous layer from the initial moment 

during the time period Δz, when the temperature drop on 

both sides of the layer, whose thickness is Y, is 1K. 

 (5)

where: ρ – material density, (kg/m3); c – specific thermal 
capacity , (J/(kg×K)); d – thickness of layer, (m). 

The thermal resistance of the layer:

 

where: ν
n
, ρ

n
 and c

n 
are the volume, density and specific 

heat, respectively, of element n. 

Thermal regime inside the building is affected in two 

ways: due to infiltration and through the external building 

envelope. In reality, most of the buildings are made of 

enclosures of different construction types, depending on the 

purpose and technological regime of the building and 

dominant construction traditions in the country. The massive 

layer of multi-layered enclosures can be oriented to the 

exterior or the interior of the building, or it can be in the 

middle of the enclosure. Thermal capacitance is an 

important quality that enables walls to absorb, store and 

later release thermal energy into the building space.  

The aim of this work is to determine the effect of 

massive layer position in the enclosure on the thermal 

inertia of the enclosure. 

2. Methods 

The accuracy of calculations of unsteady heat transfer 

using the finite elements method depends on the selected 

values of Fourier and Biot numbers.  

The Fourier number is designated by the symbol F
o 
and 

the equation for every layer of this number is (Stankevičius 

et al. 2000, Incropera et al. 2007, Hagentoft 2001): 

 

2

d

za

F
o

Δ⋅

=                   

 (3) 

 

The Fourier number can also be expressed as: 

 

;
2

zA

C

z

RC

z

dc

z

F

sl

sl

slslp

o
Δ⋅=

Δ⋅Λ

=

⋅

Δ

=

⋅⋅

Δ⋅

=

ρ

λ

    (4) 

 

where: a – the thermal diffusivity, (m
2

/s); Δz – the time step, 

(s); d – thickness of conditional layer, (m); λ – thermal 

conductivity, (W/(m⋅K)); ρ – material density, (kg/m
3

); c – 

specific thermal capacity , (J/(kg⋅K)). 

 

The thermal capacitance of the layer:  

 

)/(,
2

.
KmJdcC

psl
⋅⋅⋅= ρ            (5) 

 

where: ρ – material density, (kg/m
3

); c – specific thermal 

capacity , (J/(kg⋅K)); d - thickness of layer, (m).  

 

The thermal resistance of the layer: 

 

WKmdR
sl

/)(,/
2

.
⋅= λ              (6) 

 

where: d – thickness of material, (m); λ – thermal 

conductivity, (W/(m⋅K). 

 

The thermal diffusion of the layer (Pupeikis et al. 

2010): 

 

)/(,/
2

.
KmWd

sl
⋅=Λ λ             (7) 

where: Λ  – thermal conductance, (W/m
2

⋅K); λ – thermal 

conductivity, (W/(m⋅K)); d – thickness of layer, (m). 

 

The thermal diffusivity of the material:  

sm

c

a

p

/;
2

⋅

=

ρ

λ                (8) 

 

where: λ – thermal conductivity, (W/(m⋅K)); ρ – material 

density, (kg/m
3

); c – specific thermal capacity , (J/(kg⋅K)). 

 

The thermal diffusion of the separate layer:  

 

)/1(,
2

s

d

a

A
sl
=                 (9) 

 

where: A
sl
 – thermal diffusion of the material layer [1/s]; a – 

thermal diffusivity of material, [m
2

/s]; d – thickness of 

layer, [m]. 

 

The dependence of layer thickness on the Fourier 

number and time step:  

)(, m

F

za

d

o

Δ⋅

=                (10) 

 

where: a – thermal diffusivity of material, (m
2

/s); d – 

thickness of layer, (m); Δz - the time step, (s); F
o
 - Fourier 

number. 

 

Thermal diffusivity of the material a (m
2

/s; m
2

/h) 

shows the rate of temperature “diffusion/equalization” in the 

object, i.e. the part of 1 m distance per time unit for the 

temperature to equalize in the homogeneous object from the 

initial moment, when the temperature drop between two 

planes (points) at 1 m distance is 1K.  

The inverse proportion 1/a (s/m
2

; h/m
2

) shows the time 

in which the temperature equalizes between two planes 

(points) of the homogeneous object at 1 m distance, when 

the temperature drop at the initial moment is 1K. Suppose, 

there is an EPS with the following characteristics: λ=0,07 

W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 J/(kg⋅K), then, the 

calculations are performed according to (8) a=0,00000209 

m
2

/s≈0,0075m
2

/h and 1/a= 478571 s/m
2 

≈133 h/m
2

.  

Thermal diffusion of the layer A (1/s; 1/h) shows the 

rate of thermal “diffusion/equalization”, i.e., the part of 

layer thickness Y per time unit for the temperature to 

equalize in the homogeneous object from the initial 

moment, when the temperature drop on both sides of the 

layer is 1K. 

The inverse proportion 1/A (s; h) shows the time in 

which the temperature equalizes in the homogeneous layer, 

when layer thickness is Y and temperature drop in the layer 

at the initial moment is 1K. 

Suppose, there is an EPS layer with the following 

characteristics: λ=0,07 W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 

J/(kg⋅K), d= 50mm, then, the calculations are performed 

according to (9): A≈3,01,(1/h) and 1/A≈0,332, (h).  

Fourier number F
o
 - shows the thickness (depth) of 

“equalization” in the layer during the selected time step Δz, 

i.e., the part of layer thickness Y for the temperature to 

equalize in the homogeneous layer from the initial moment 

during the time period Δz, when the temperature drop on 

both sides of the layer, whose thickness is Y, is 1K. 

 (6)

where: d – thickness of material, (m); l – thermal 
conductivity, (W/(m × K).

The thermal diffusion of the layer (Pupeikis et al. 
2010):

 

where: ν
n
, ρ

n
 and c

n 
are the volume, density and specific 

heat, respectively, of element n. 

Thermal regime inside the building is affected in two 

ways: due to infiltration and through the external building 

envelope. In reality, most of the buildings are made of 

enclosures of different construction types, depending on the 

purpose and technological regime of the building and 

dominant construction traditions in the country. The massive 

layer of multi-layered enclosures can be oriented to the 

exterior or the interior of the building, or it can be in the 

middle of the enclosure. Thermal capacitance is an 

important quality that enables walls to absorb, store and 

later release thermal energy into the building space.  

The aim of this work is to determine the effect of 

massive layer position in the enclosure on the thermal 

inertia of the enclosure. 

2. Methods 

The accuracy of calculations of unsteady heat transfer 

using the finite elements method depends on the selected 

values of Fourier and Biot numbers.  

The Fourier number is designated by the symbol F
o 
and 

the equation for every layer of this number is (Stankevičius 

et al. 2000, Incropera et al. 2007, Hagentoft 2001): 

 

2

d

za

F
o

Δ⋅

=                   

 (3) 

 

The Fourier number can also be expressed as: 

 

;
2

zA

C

z

RC

z

dc

z

F

sl

sl

slslp

o
Δ⋅=

Δ⋅Λ

=

⋅

Δ

=

⋅⋅

Δ⋅

=

ρ

λ

    (4) 

 

where: a – the thermal diffusivity, (m
2

/s); Δz – the time step, 

(s); d – thickness of conditional layer, (m); λ – thermal 

conductivity, (W/(m⋅K)); ρ – material density, (kg/m
3

); c – 

specific thermal capacity , (J/(kg⋅K)). 

 

The thermal capacitance of the layer:  

 

)/(,
2

.
KmJdcC

psl
⋅⋅⋅= ρ            (5) 

 

where: ρ – material density, (kg/m
3

); c – specific thermal 

capacity , (J/(kg⋅K)); d - thickness of layer, (m).  

 

The thermal resistance of the layer: 

 

WKmdR
sl

/)(,/
2

.
⋅= λ              (6) 

 

where: d – thickness of material, (m); λ – thermal 

conductivity, (W/(m⋅K). 

 

The thermal diffusion of the layer (Pupeikis et al. 

2010): 

 

)/(,/
2

.
KmWd

sl
⋅=Λ λ             (7) 

where: Λ  – thermal conductance, (W/m
2

⋅K); λ – thermal 

conductivity, (W/(m⋅K)); d – thickness of layer, (m). 

 

The thermal diffusivity of the material:  

sm

c

a

p

/;
2

⋅

=

ρ

λ                (8) 

 

where: λ – thermal conductivity, (W/(m⋅K)); ρ – material 

density, (kg/m
3

); c – specific thermal capacity , (J/(kg⋅K)). 

 

The thermal diffusion of the separate layer:  

 

)/1(,
2

s

d

a

A
sl
=                 (9) 

 

where: A
sl
 – thermal diffusion of the material layer [1/s]; a – 

thermal diffusivity of material, [m
2

/s]; d – thickness of 

layer, [m]. 

 

The dependence of layer thickness on the Fourier 

number and time step:  

)(, m

F

za

d

o

Δ⋅

=                (10) 

 

where: a – thermal diffusivity of material, (m
2

/s); d – 

thickness of layer, (m); Δz - the time step, (s); F
o
 - Fourier 

number. 

 

Thermal diffusivity of the material a (m
2

/s; m
2

/h) 

shows the rate of temperature “diffusion/equalization” in the 

object, i.e. the part of 1 m distance per time unit for the 

temperature to equalize in the homogeneous object from the 

initial moment, when the temperature drop between two 

planes (points) at 1 m distance is 1K.  

The inverse proportion 1/a (s/m
2

; h/m
2

) shows the time 

in which the temperature equalizes between two planes 

(points) of the homogeneous object at 1 m distance, when 

the temperature drop at the initial moment is 1K. Suppose, 

there is an EPS with the following characteristics: λ=0,07 

W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 J/(kg⋅K), then, the 

calculations are performed according to (8) a=0,00000209 

m
2

/s≈0,0075m
2

/h and 1/a= 478571 s/m
2 

≈133 h/m
2

.  

Thermal diffusion of the layer A (1/s; 1/h) shows the 

rate of thermal “diffusion/equalization”, i.e., the part of 

layer thickness Y per time unit for the temperature to 

equalize in the homogeneous object from the initial 

moment, when the temperature drop on both sides of the 

layer is 1K. 

The inverse proportion 1/A (s; h) shows the time in 

which the temperature equalizes in the homogeneous layer, 

when layer thickness is Y and temperature drop in the layer 

at the initial moment is 1K. 

Suppose, there is an EPS layer with the following 

characteristics: λ=0,07 W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 

J/(kg⋅K), d= 50mm, then, the calculations are performed 

according to (9): A≈3,01,(1/h) and 1/A≈0,332, (h).  

Fourier number F
o
 - shows the thickness (depth) of 

“equalization” in the layer during the selected time step Δz, 

i.e., the part of layer thickness Y for the temperature to 

equalize in the homogeneous layer from the initial moment 

during the time period Δz, when the temperature drop on 

both sides of the layer, whose thickness is Y, is 1K. 

 (7)

where: L  – thermal conductance, (W/m2×K); λ – thermal 
conductivity, (W/(m×K)); d – thickness of layer, (m).

The thermal diffusivity of the material: 

sm
c

a
p

/; 2
⋅

=
ρ
l  (8)

where: λ – thermal conductivity, (W/(m×K)); ρ – material 
density, (kg/m3); c – specific thermal capacity , (J/(kg×K)).

The thermal diffusion of the separate layer: 

 

where: ν
n
, ρ

n
 and c

n 
are the volume, density and specific 

heat, respectively, of element n. 

Thermal regime inside the building is affected in two 

ways: due to infiltration and through the external building 

envelope. In reality, most of the buildings are made of 

enclosures of different construction types, depending on the 

purpose and technological regime of the building and 

dominant construction traditions in the country. The massive 

layer of multi-layered enclosures can be oriented to the 

exterior or the interior of the building, or it can be in the 

middle of the enclosure. Thermal capacitance is an 

important quality that enables walls to absorb, store and 

later release thermal energy into the building space.  

The aim of this work is to determine the effect of 

massive layer position in the enclosure on the thermal 

inertia of the enclosure. 

2. Methods 

The accuracy of calculations of unsteady heat transfer 

using the finite elements method depends on the selected 

values of Fourier and Biot numbers.  

The Fourier number is designated by the symbol F
o 
and 

the equation for every layer of this number is (Stankevičius 

et al. 2000, Incropera et al. 2007, Hagentoft 2001): 

 

2

d

za

F
o

Δ⋅

=                   

 (3) 

 

The Fourier number can also be expressed as: 

 

;
2

zA

C

z

RC

z

dc

z

F

sl

sl

slslp

o
Δ⋅=

Δ⋅Λ

=

⋅

Δ

=

⋅⋅

Δ⋅

=

ρ

λ

    (4) 

 

where: a – the thermal diffusivity, (m
2

/s); Δz – the time step, 

(s); d – thickness of conditional layer, (m); λ – thermal 

conductivity, (W/(m⋅K)); ρ – material density, (kg/m
3

); c – 

specific thermal capacity , (J/(kg⋅K)). 

 

The thermal capacitance of the layer:  

 

)/(,
2

.
KmJdcC

psl
⋅⋅⋅= ρ            (5) 

 

where: ρ – material density, (kg/m
3

); c – specific thermal 

capacity , (J/(kg⋅K)); d - thickness of layer, (m).  

 

The thermal resistance of the layer: 

 

WKmdR
sl

/)(,/
2

.
⋅= λ              (6) 

 

where: d – thickness of material, (m); λ – thermal 

conductivity, (W/(m⋅K). 

 

The thermal diffusion of the layer (Pupeikis et al. 

2010): 

 

)/(,/
2

.
KmWd

sl
⋅=Λ λ             (7) 

where: Λ  – thermal conductance, (W/m
2

⋅K); λ – thermal 

conductivity, (W/(m⋅K)); d – thickness of layer, (m). 

 

The thermal diffusivity of the material:  

sm

c

a

p

/;
2

⋅

=

ρ

λ                (8) 

 

where: λ – thermal conductivity, (W/(m⋅K)); ρ – material 

density, (kg/m
3

); c – specific thermal capacity , (J/(kg⋅K)). 

 

The thermal diffusion of the separate layer:  

 

)/1(,
2

s

d

a

A
sl
=                 (9) 

 

where: A
sl
 – thermal diffusion of the material layer [1/s]; a – 

thermal diffusivity of material, [m
2

/s]; d – thickness of 

layer, [m]. 

 

The dependence of layer thickness on the Fourier 

number and time step:  

)(, m

F

za

d

o

Δ⋅

=                (10) 

 

where: a – thermal diffusivity of material, (m
2

/s); d – 

thickness of layer, (m); Δz - the time step, (s); F
o
 - Fourier 

number. 

 

Thermal diffusivity of the material a (m
2

/s; m
2

/h) 

shows the rate of temperature “diffusion/equalization” in the 

object, i.e. the part of 1 m distance per time unit for the 

temperature to equalize in the homogeneous object from the 

initial moment, when the temperature drop between two 

planes (points) at 1 m distance is 1K.  

The inverse proportion 1/a (s/m
2

; h/m
2

) shows the time 

in which the temperature equalizes between two planes 

(points) of the homogeneous object at 1 m distance, when 

the temperature drop at the initial moment is 1K. Suppose, 

there is an EPS with the following characteristics: λ=0,07 

W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 J/(kg⋅K), then, the 

calculations are performed according to (8) a=0,00000209 

m
2

/s≈0,0075m
2

/h and 1/a= 478571 s/m
2 

≈133 h/m
2

.  

Thermal diffusion of the layer A (1/s; 1/h) shows the 

rate of thermal “diffusion/equalization”, i.e., the part of 

layer thickness Y per time unit for the temperature to 

equalize in the homogeneous object from the initial 

moment, when the temperature drop on both sides of the 

layer is 1K. 

The inverse proportion 1/A (s; h) shows the time in 

which the temperature equalizes in the homogeneous layer, 

when layer thickness is Y and temperature drop in the layer 

at the initial moment is 1K. 

Suppose, there is an EPS layer with the following 

characteristics: λ=0,07 W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 

J/(kg⋅K), d= 50mm, then, the calculations are performed 

according to (9): A≈3,01,(1/h) and 1/A≈0,332, (h).  

Fourier number F
o
 - shows the thickness (depth) of 

“equalization” in the layer during the selected time step Δz, 

i.e., the part of layer thickness Y for the temperature to 

equalize in the homogeneous layer from the initial moment 

during the time period Δz, when the temperature drop on 

both sides of the layer, whose thickness is Y, is 1K. 

 (9)

where: Asl – thermal diffusion of the material layer [1/s]; 
a – thermal diffusivity of material, [m2/s]; d – thickness of 
layer, [m].

The dependence of layer thickness on the Fourier 
number and time step: 

)(, m
F

za
d

o

D⋅
=  (10)

where: a – thermal diffusivity of material, (m2/s); 
d – thickness of layer, (m); Dz – the time step, (s); 
Fo – Fourier number.

Thermal diffusivity of the material a (m2/s; m2/h) 
shows the rate of temperature “diffusion/equalization” in 
the object, i.e. the part of 1 m distance per time unit for 
the temperature to equalize in the homogeneous object from 
the initial moment, when the temperature drop between two 
planes (points) at 1 m distance is 1K. 

The inverse proportion 1/a (s/m2; h/m2) shows the 
time in which the temperature equalizes between two planes 
(points) of the homogeneous object at 1 m distance, when the 
temperature drop at the initial moment is 1K. Suppose, there is 
an EPS with the following characteristics: λ  =  0,07 W/(m×K); 
ρ  =  25 kg/m3; cp  =  1340 J/(kg×K), then, the calculations are 
performed according to (8) a  =  0,00000209 m2/s  ≈  0,0075m2/h 
and 1/a  =  478571 s/m2 ≈  133 h/m2. 

Thermal diffusion of the layer A (1/s; 1/h) shows the 
rate of thermal “diffusion/equalization”, i.e., the part of layer 
thickness Y per time unit for the temperature to equalize in 
the homogeneous object from the initial moment, when the 
temperature drop on both sides of the layer is 1K.

The inverse proportion 1/A (s; h) shows the time in 
which the temperature equalizes in the homogeneous layer, 
when layer thickness is Y and temperature drop in the layer 
at the initial moment is 1K.

Suppose, there is an EPS layer with the following 
characteristics: l  =  0,07 W/(m×K); ρ   =  25 kg/m3; 
cp  =  1340 J/(kg×K), d  =  50mm, then, the calculations 
are performed according to (9): A ≈  3,01,(1/h) and 
1/A ≈  0,332, (h). 



42

Fourier number Fo – shows the thickness (depth) of 
“equalization” in the layer during the selected time step 
Δz, i.e., the part of layer thickness Y for the temperature to 
equalize in the homogeneous layer from the initial moment 
during the time period Δz, when the temperature drop on 
both sides of the layer, whose thickness is Y, is 1K.

The inverse proportion 1/Fo – shows how much more 
time it would take, in comparison to the time period Δz, for 
temperature to equalize in the whole homogeneous layer, 
when the thickness of the layer is Y, and the temperature 
drop in the layer at the initial moment is 1K. Suppose, 
there is an EPS layer with the following characteristics:  
λ  =  0,07 W/(m×K); ρ   =  25 kg/m3; cp  =  1340 J/(kg×K), 
d  =  50mm, Dz  =  0,1 h, then, the calculations are performed 
according to (4): Fo ≈  0,301 and 1/Fo ≈  3,322. 

Thus, if Dz  =  0,1×3,322 ≈  0,3322 h, then Fo ≈  0,9999 
and 1/Fo ≈  1,000; in other words, during this time period, the 
temperature equalizes in all the thickness of the layer.

2dc
z

zAF
p

o
⋅⋅
D⋅

=D⋅=
ρ

l  (11)

The Biot number is usually expressed as: 

The inverse proportion 1/F
o
- shows how much more 

time it would take, in comparison to the time period Δz, for 

temperature to equalize in the whole homogeneous layer, 

when the thickness of the layer is Y, and the temperature 

drop in the layer at the initial moment is 1K. Suppose, there 

is an EPS layer with the following characteristics: λ=0,07 

W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 J/(kg⋅K), d= 50mm, Δz=0,1 

h, then, the calculations are performed according to (4): 

F
o
≈0,301 and 1/F

o
≈3,322.  

Thus, if Δz=0,1⋅3,322≈0,3322 h, then F
o
≈ 0,9999 and 

1/F
o
≈1,000; in other words, during this time period, the 

temperature equalizes in all the thickness of the layer. 

 

2

dc

z

zAF

p

o

⋅⋅

Δ⋅

=Δ⋅=

ρ

λ

            (11) 

 

The Biot number is usually expressed as:  

 

λ

dh

Bi

⋅

=                   (12) 

 

where: h – surface heat exchange coefficient, d – thickness 

of the surface layer of the enclosure, λ - thermal 

conductivity coefficient of the surface layer of the material. 

 

The Biot number can also be expressed in the 

following way: 

.

..

..

.

...

/1
pav

slpav

slpav

pav

slpavpav

h

Rh

RhBi

Δ

=

Λ

=Δ⋅=      (13) 

 

where: h
pav.

 – surface heat exchange coefficient; ΔR
pav.sl.

 – 

thermal resistance of the first surface layer; Λ
pav.sl. 

– thermal 

diffusion of the first surface layer (the inverse proportion of 

thermal resistance). 

 

According to (Karbauskaite et al. 2008, Hensen 1994), 

the limit value of the Fourier number is equal to 0.5. The 

values of this number are higher than 0.5 and cause 

significant errors in temperature calculations. In some cases, 

the temperature calculation average error increases 

significantly when the value of the Fourier number reaches 

0.47 (Pupeikis et al. 2012). 

In case of transient heat transfer, the temperature in 

any plane of enclosure is determined by the following 

equation (14) in (Fig. 1): 

 

⋅⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ

Θ−Θ

−
Δ

Θ−Θ

+Θ=Θ
+

−

−

n

nn

n

nn

nn RR

1

1

1

'     (14) 

nnnnnn

cdcd

z

ρρ ⋅⋅+⋅⋅

Δ⋅

⋅

−−− 111

2

 

 

It is understood as (15) or (16),  

 

( )

nn

nnnnnn

z

qq

χχ +

Δ⋅

⋅−+Θ=Θ

−

+−

1

/)1()'1/(

2

'     (15) 

or 

( )
=

⋅−+Θ=Θ

−+

+−

)1/()1(

/)1()'1/(

1

'

nn

nnnnnn

s

qq

    (16) 

'/

)1/()1(

)1/()1(

'

'

nnn

nn

nn

n

t

s

q

Δ+Θ=

Δ

+Θ=

−+

−+

 

 

where: Θ’
n
 – temperature of plane n at the present time 

moment; Δz – time period (calculation time step), (s); d
n
 – 

thickness of layer between “n“ and “n+1“ planes, (m); c
n
 – 

specific thermal capacity of layer between “n“ and “n+1“ 

planes, (J/(kg·K)); ρ
n
 – the material of  layer between “n“ 

and “n+1“ planes density, kg/m
3

; q’
n’/(n-1) 

– the potential of 

heat flow density between “n“ and “n-1“ planes at the time 

moment after time period Δz, (W/m
2

); q’
(n+1)/n’

  –  heat flow 

density between “n“ and “n+1“ planes at the initial time 

moment z, i.e. the heat flow rate due to temperature 

difference (Θ
n
 – Θ

n+1
), W/ m

2

; χ
n-1

 ir χ
n
  – coefficient of area 

thermal capacitance of the layer “n-1” and “n“ , (J/(m
2

⋅K)); 

S
(n+1)/(n-1)

 – the average thermal receptivity of the material of 

both layers “(n-1)“ and  “n”, (W/(m
2

⋅K)); Δt
n/n‘

 – 

temperature change in the plane “n” after the time period Δz 

passed from the initial time moment. 

 

According to (Incropera et al. 2007), it is 

recommended that the first surface layer thickness amounted 

to ½ of the thickness of other layers, since it is important for 

obtaining the required Biot number. For the calculation of a 

unidimensional temperature field, when F
o 

is inside the 

enclosure, the stability criteria is the following: (1-2⋅F
o
)≥ 0, 

from here F
o
≤ 0,5; 

When Bi is displayed on the surface of the wall, the 

stability criteria is the following: (1-2⋅F
o
-2⋅Bi⋅F

o
) ≥ 0, from 

here F
o
⋅ (1+Bi) ≤ 0,5.  

The second criterion with Bi is more sensitive than the 

first one; therefore, it has to be used for determining the 

critical (the largest allowed) calculation time step Δz: 

The positive Bi ≤ 0,1 is preferred. 

The maximum allowed F
o 
is determined applying:  

 

Bi

F
o

+

≤

1

5,0

                 (17) 

 

Using the maximum allowed F
o 

, the maximum 

allowed time step is calculated:  

 

a

dF

z
o

2

⋅

≤Δ                 (18) 

 

The finite elements difference explicit method is used 

for calculating new values using the already know ones. 

 (12)

where: h – surface heat exchange coefficient, d – thickness 
of the surface layer of the enclosure, λ – thermal conductivity 
coefficient of the surface layer of the material.

The Biot number can also be expressed in the following 
way:

The inverse proportion 1/F
o
- shows how much more 

time it would take, in comparison to the time period Δz, for 

temperature to equalize in the whole homogeneous layer, 

when the thickness of the layer is Y, and the temperature 

drop in the layer at the initial moment is 1K. Suppose, there 

is an EPS layer with the following characteristics: λ=0,07 

W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 J/(kg⋅K), d= 50mm, Δz=0,1 

h, then, the calculations are performed according to (4): 

F
o
≈0,301 and 1/F

o
≈3,322.  

Thus, if Δz=0,1⋅3,322≈0,3322 h, then F
o
≈ 0,9999 and 

1/F
o
≈1,000; in other words, during this time period, the 

temperature equalizes in all the thickness of the layer. 

 

2

dc

z

zAF

p

o

⋅⋅

Δ⋅

=Δ⋅=

ρ

λ

            (11) 

 

The Biot number is usually expressed as:  

 

λ

dh

Bi

⋅

=                   (12) 

 

where: h – surface heat exchange coefficient, d – thickness 

of the surface layer of the enclosure, λ - thermal 

conductivity coefficient of the surface layer of the material. 

 

The Biot number can also be expressed in the 

following way: 

.

..

..

.

...

/1
pav

slpav

slpav

pav

slpavpav

h

Rh

RhBi

Δ

=

Λ

=Δ⋅=      (13) 

 

where: h
pav.

 – surface heat exchange coefficient; ΔR
pav.sl.

 – 

thermal resistance of the first surface layer; Λ
pav.sl. 

– thermal 

diffusion of the first surface layer (the inverse proportion of 

thermal resistance). 

 

According to (Karbauskaite et al. 2008, Hensen 1994), 

the limit value of the Fourier number is equal to 0.5. The 

values of this number are higher than 0.5 and cause 

significant errors in temperature calculations. In some cases, 

the temperature calculation average error increases 

significantly when the value of the Fourier number reaches 

0.47 (Pupeikis et al. 2012). 

In case of transient heat transfer, the temperature in 

any plane of enclosure is determined by the following 

equation (14) in (Fig. 1): 

 

⋅
⎟
⎟

⎠

⎞

⎜
⎜

⎝

⎛

Δ

Θ−Θ

−

Δ

Θ−Θ

+Θ=Θ
+

−

−

n

nn

n

nn

nn

RR

1

1

1

'     (14) 

nnnnnn

c

d

c

d

z

ρρ ⋅⋅+⋅⋅

Δ⋅

⋅

−−− 111

2

 

 

It is understood as (15) or (16),  

 

( )

nn

nnnnnn

z

qq

χχ +

Δ⋅

⋅−+Θ=Θ

−

+−

1

/)1()'1/(

2

'     (15) 

or 

( )
=

⋅−+Θ=Θ

−+

+−

)1/()1(

/)1()'1/(

1

'

nn

nnnnnn

s

qq

    (16) 

'/

)1/()1(

)1/()1(

'

'

nnn

nn

nn

n

t

s

q

Δ+Θ=

Δ

+Θ=

−+

−+

 

 

where: Θ’
n
 – temperature of plane n at the present time 

moment; Δz – time period (calculation time step), (s); d
n
 – 

thickness of layer between “n“ and “n+1“ planes, (m); c
n
 – 

specific thermal capacity of layer between “n“ and “n+1“ 

planes, (J/(kg·K)); ρ
n
 – the material of  layer between “n“ 

and “n+1“ planes density, kg/m
3

; q’
n’/(n-1) 

– the potential of 

heat flow density between “n“ and “n-1“ planes at the time 

moment after time period Δz, (W/m
2

); q’
(n+1)/n’

  –  heat flow 

density between “n“ and “n+1“ planes at the initial time 

moment z, i.e. the heat flow rate due to temperature 

difference (Θ
n
 – Θ

n+1
), W/ m

2

; χ
n-1

 ir χ
n
  – coefficient of area 

thermal capacitance of the layer “n-1” and “n“ , (J/(m
2

⋅K)); 

S
(n+1)/(n-1)

 – the average thermal receptivity of the material of 

both layers “(n-1)“ and  “n”, (W/(m
2

⋅K)); Δt
n/n‘

 – 

temperature change in the plane “n” after the time period Δz 

passed from the initial time moment. 

 

According to (Incropera et al. 2007), it is 

recommended that the first surface layer thickness amounted 

to ½ of the thickness of other layers, since it is important for 

obtaining the required Biot number. For the calculation of a 

unidimensional temperature field, when F
o 

is inside the 

enclosure, the stability criteria is the following: (1-2⋅F
o
)≥ 0, 

from here F
o
≤ 0,5; 

When Bi is displayed on the surface of the wall, the 

stability criteria is the following: (1-2⋅F
o
-2⋅Bi⋅F

o
) ≥ 0, from 

here F
o
⋅ (1+Bi) ≤ 0,5.  

The second criterion with Bi is more sensitive than the 

first one; therefore, it has to be used for determining the 

critical (the largest allowed) calculation time step Δz: 

The positive Bi ≤ 0,1 is preferred. 

The maximum allowed F
o 
is determined applying:  

 

Bi

F
o

+

≤

1

5,0

                 (17) 

 

Using the maximum allowed F
o 

, the maximum 

allowed time step is calculated:  

 

a

dF

z
o

2

⋅

≤Δ                 (18) 

 

The finite elements difference explicit method is used 

for calculating new values using the already know ones. 

 (13)

where: hpav. – surface heat exchange coefficient; DRpav.sl. – 
thermal resistance of the first surface layer; Lpav.sl. – thermal 
diffusion of the first surface layer (the inverse proportion of 
thermal resistance).

According to (Karbauskaite et al. 2008, Hensen 
1994), the limit value of the Fourier number is equal to 0.5. 
The values of this number are higher than 0.5 and cause 
significant errors in temperature calculations. In some 
cases, the temperature calculation average error increases 
significantly when the value of the Fourier number reaches 
0.47 (Pupeikis et al. 2012).

In case of transient heat transfer, the temperature in any 
plane of enclosure is determined by the following equation 
(14) in (Fig. 1):

⋅










D
Θ−Θ

−
D

Θ−Θ
+Θ=Θ +

−

−

n

nn

n

nn
nn RR

1

1

1'
 

(14)
nnnnnn cdcd

z
ρρ ⋅⋅+⋅⋅

D⋅
⋅

−−− 111

2

It is understood as (15) or (16), 

( )
nn

nnnnnn
z

qq
cc +

D⋅
⋅−+Θ=Θ

−
+−

1
/)1()'1/(

2
'  (15)

or

( ) =⋅−+Θ=Θ
−+

+−
)1/()1(

/)1()'1/(
1

'
nn

nnnnnn s
qq  

(16)
'/

)1/()1(

)1/()1( '
'

nnn
nn

nn
n ts

q
D+Θ=

D
+Θ=

−+

−+

where: Θ'n – temperature of plane n at the present time 
moment; Δz – time period (calculation time step), (s); 
dn – thickness of layer between “n“ and “n + 1“ planes, (m); 
cn – specific thermal capacity of layer between “n“ and 
“n + 1“ planes, (J/(kg·K)); ρn – the material of  layer between 
“n“ and “n  +  1“ planes density, kg/m3; q'n'/(n–1) – the potential 
of heat flow density between “n“ and “n –  1“ planes at the 
time moment after time period Δz, (W/m2); q'(n+1)/n' – heat 
flow density between “n“ and “n + 1“ planes at the initial 
time moment z, i.e. the heat flow rate due to temperature 
difference (Θn – Θn+1), W/ m2; cn –1 ir cn – coefficient of 
area thermal capacitance of the layer “n – 1” and “n“, 
(J/(m2×K)); S(n+1)/(n-1) – the average thermal receptivity of 
the material of both layers “(n – 1)“ and  “n”, (W/(m2×K)); 
Dtn/n' – temperature change in the plane “n” after the time 
period Δz passed from the initial time moment.

 

Fig. 1. Principle scheme for calculating the interlayer 

temperature of the enclosure 

3. Materials and equipment 

Two building models were made for the experiments. 

Heat transfer coefficient of the internal surface of the 

enclosures of models WM-I and WM-II, which describes 

the surface heat exchange of the enclosure with the 

environment, is the following:  

 

)/(,

1

11

2

4

2

21

3

KmW

TT

s

Nuh

airs

s

−+

⎟

⎠

⎞

⎜

⎝

⎛ +

⋅⋅

+⎟

⎠

⎞

⎜

⎝

⎛
⋅=

εε

σ

λ     (19) 

 

where: σ =5.6704·10
-8

 – Stefan–Boltzmann constant, (W⋅m
-

2

⋅K
-4

); s – thickness of air boundary layer, (m); Nu – Nusselt 

number calculated according to LST EN 673:2011; λ - 

thermal conductivity of still air, W/(m·K); T
s
 – temperature 

of the surface, (K); T
air

 – surround air temperature, (K); ε
1
 – 

emissivity of surface; ε
2
 – emissivity of surrounding. 

 

The measured thermal properties of materials used in 

wall models WM-I and WM-II are presented in (table 1). 

The thermal conductivity values of the materials were 

measured according to (LST EN 12667:2002, Stankevičius 

et al. 2005) in lambda apparatus which meets the 

requirements of (ISO 8301:1991). The density of materials 

was determined in accordance with (LST EN 1602: 1998, 

LST EN 1602: 1998/AC:2003). 

The thermal capacitance values of common 

construction materials used in the experiment were selected 

from literature sources in accordance to the type and density 

of the materials. Both models WM-I and WM-II were made 

of materials commonly used in practice (Fig. 2). Five walls 

of the model 600x600x600 mm ( length x width x height ) 

were made of polystyrene (EPS) boards d
1
=50+50mm. 

Heterogeneous enclosure of the models consists of two 

medium density wood fibre boards (MDF), with the 

thickness of d
2
=18+18mm, and one expanded polystyrene 

(EPS) board, d
1
=50mm.  

Internal and external volumes of the models are 

identical, but the position of MDF massive layer in 

heterogeneous enclosures is different: in WM-I, it is 

oriented to the interior, and in WM-II, it is oriented to the 

exterior.  

 

EPS

d1=(50+50)mm

EPS d1=50mm

MDF d2=18+18mm

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

MDFd2=18+18mm

EPS d1=50mm

EPS

d1=(50+50)mm

T7

T8

T9

T25

T11

T1

T2 T5

T3

T4 T10

 

WM-II WM-I 

Fig. 2. Structural scheme of thermocouple arrangement and models. T- thermocouples  

Fig. 1. Principle scheme for calculating the interlayer temperature 
of the enclosure

According to (Incropera et al. 2007), it is 
recommended that the first surface layer thickness 
amounted to ½ of the thickness of other layers, since it is 
important for obtaining the required Biot number. For the 
calculation of a unidimensional temperature field, when Fo 
is inside the enclosure, the stability criteria is the following: 
(1–2·Fo)  ≥ 0, from here Fo   ≤  0,5;

When Bi is displayed on the surface of the wall, the 
stability criteria is the following: (1–2 · Fo  –  2 · Bi · Fo) ≥ 0, 
from here Fo·(1+Bi) ≤ 0,5. 

The second criterion with Bi is more sensitive than 
the first one; therefore, it has to be used for determining the 
critical (the largest allowed) calculation time step Δz:



43

The positive Bi ≤ 0,1 is preferred.
The maximum allowed Fo is determined applying: 

The inverse proportion 1/F
o
- shows how much more 

time it would take, in comparison to the time period Δz, for 

temperature to equalize in the whole homogeneous layer, 

when the thickness of the layer is Y, and the temperature 

drop in the layer at the initial moment is 1K. Suppose, there 

is an EPS layer with the following characteristics: λ=0,07 

W/(m⋅K); ρ=25 kg/m
3

; c
p
=1340 J/(kg⋅K), d= 50mm, Δz=0,1 

h, then, the calculations are performed according to (4): 

F
o
≈0,301 and 1/F

o
≈3,322.  

Thus, if Δz=0,1⋅3,322≈0,3322 h, then F
o
≈ 0,9999 and 

1/F
o
≈1,000; in other words, during this time period, the 

temperature equalizes in all the thickness of the layer. 

 

2

dc

z

zAF

p

o

⋅⋅

Δ⋅

=Δ⋅=

ρ

λ

            (11) 

 

The Biot number is usually expressed as:  

 

λ

dh

Bi

⋅

=                   (12) 

 

where: h – surface heat exchange coefficient, d – thickness 

of the surface layer of the enclosure, λ - thermal 

conductivity coefficient of the surface layer of the material. 

 

The Biot number can also be expressed in the 

following way: 

.

..

..

.

...

/1
pav

slpav

slpav

pav

slpavpav

h

Rh

RhBi

Δ

=

Λ

=Δ⋅=      (13) 

 

where: h
pav.

 – surface heat exchange coefficient; ΔR
pav.sl.

 – 

thermal resistance of the first surface layer; Λ
pav.sl. 

– thermal 

diffusion of the first surface layer (the inverse proportion of 

thermal resistance). 

 

According to (Karbauskaite et al. 2008, Hensen 1994), 

the limit value of the Fourier number is equal to 0.5. The 

values of this number are higher than 0.5 and cause 

significant errors in temperature calculations. In some cases, 

the temperature calculation average error increases 

significantly when the value of the Fourier number reaches 

0.47 (Pupeikis et al. 2012). 

In case of transient heat transfer, the temperature in 

any plane of enclosure is determined by the following 

equation (14) in (Fig. 1): 

 

⋅
⎟
⎟

⎠

⎞

⎜
⎜

⎝

⎛

Δ

Θ−Θ

−

Δ

Θ−Θ

+Θ=Θ
+

−

−

n

nn

n

nn

nn

RR

1

1

1

'     (14) 

nnnnnn

cdcd

z

ρρ ⋅⋅+⋅⋅

Δ⋅

⋅

−−− 111

2

 

 

It is understood as (15) or (16),  

 

( )

nn

nnnnnn

z

qq

χχ +

Δ⋅

⋅−+Θ=Θ

−

+−

1

/)1()'1/(

2

'     (15) 

or 

( ) =⋅−+Θ=Θ

−+

+−

)1/()1(

/)1()'1/(

1

'

nn

nnnnnn

s

qq     (16) 

'/

)1/()1(

)1/()1(

'

'

nnn

nn

nn

n

t

s

q

Δ+Θ=

Δ

+Θ=

−+

−+

 

 

where: Θ’
n
 – temperature of plane n at the present time 

moment; Δz – time period (calculation time step), (s); d
n
 – 

thickness of layer between “n“ and “n+1“ planes, (m); c
n
 – 

specific thermal capacity of layer between “n“ and “n+1“ 

planes, (J/(kg·K)); ρ
n
 – the material of  layer between “n“ 

and “n+1“ planes density, kg/m
3

; q’
n’/(n-1) 

– the potential of 

heat flow density between “n“ and “n-1“ planes at the time 

moment after time period Δz, (W/m
2

); q’
(n+1)/n’

  –  heat flow 

density between “n“ and “n+1“ planes at the initial time 

moment z, i.e. the heat flow rate due to temperature 

difference (Θ
n
 – Θ

n+1
), W/ m

2

; χ
n-1

 ir χ
n
  – coefficient of area 

thermal capacitance of the layer “n-1” and “n“ , (J/(m
2

⋅K)); 

S
(n+1)/(n-1)

 – the average thermal receptivity of the material of 

both layers “(n-1)“ and  “n”, (W/(m
2

⋅K)); Δt
n/n‘

 – 

temperature change in the plane “n” after the time period Δz 

passed from the initial time moment. 

 

According to (Incropera et al. 2007), it is 

recommended that the first surface layer thickness amounted 

to ½ of the thickness of other layers, since it is important for 

obtaining the required Biot number. For the calculation of a 

unidimensional temperature field, when F
o 

is inside the 

enclosure, the stability criteria is the following: (1-2⋅F
o
)≥ 0, 

from here F
o
≤ 0,5; 

When Bi is displayed on the surface of the wall, the 

stability criteria is the following: (1-2⋅F
o
-2⋅Bi⋅F

o
) ≥ 0, from 

here F
o
⋅ (1+Bi) ≤ 0,5.  

The second criterion with Bi is more sensitive than the 

first one; therefore, it has to be used for determining the 

critical (the largest allowed) calculation time step Δz: 

The positive Bi ≤ 0,1 is preferred. 

The maximum allowed F
o 
is determined applying:  

 

Bi

F
o

+

≤

1

5,0

                 (17) 

 

Using the maximum allowed F
o 

, the maximum 

allowed time step is calculated:  

 

a

dF

z
o

2

⋅

≤Δ                 (18) 

 

The finite elements difference explicit method is used 

for calculating new values using the already know ones. 

 (17)

Using the maximum allowed Fo , the maximum 
allowed time step is calculated: 

a
dF

z o
2⋅

≤D  (18)

The finite elements difference explicit method is used 
for calculating new values using the already know ones.

3. Materials and equipment

Two building models were made for the experiments. 
Heat transfer coefficient of the internal surface of the 
enclosures of models WM-I and WM-II, which describes 
the surface heat exchange of the enclosure with the 
environment, is the following: 

 

Fig. 1. Principle scheme for calculating the interlayer 

temperature of the enclosure 

3. Materials and equipment 

Two building models were made for the experiments. 

Heat transfer coefficient of the internal surface of the 

enclosures of models WM-I and WM-II, which describes 

the surface heat exchange of the enclosure with the 

environment, is the following:  

 

)/(,

1

11

2

4

2

21

3

KmW

TT

s

Nuh

airs

s

−+

⎟

⎠

⎞

⎜

⎝

⎛ +

⋅⋅

+⎟

⎠

⎞

⎜

⎝

⎛
⋅=

εε

σ

λ     (19) 

 

where: σ =5.6704·10
-8

 – Stefan–Boltzmann constant, (W⋅m
-

2

⋅K
-4

); s – thickness of air boundary layer, (m); Nu – Nusselt 

number calculated according to LST EN 673:2011; λ - 

thermal conductivity of still air, W/(m·K); T
s
 – temperature 

of the surface, (K); T
air

 – surround air temperature, (K); ε
1
 – 

emissivity of surface; ε
2
 – emissivity of surrounding. 

 

The measured thermal properties of materials used in 

wall models WM-I and WM-II are presented in (table 1). 

The thermal conductivity values of the materials were 

measured according to (LST EN 12667:2002, Stankevičius 

et al. 2005) in lambda apparatus which meets the 

requirements of (ISO 8301:1991). The density of materials 

was determined in accordance with (LST EN 1602: 1998, 

LST EN 1602: 1998/AC:2003). 

The thermal capacitance values of common 

construction materials used in the experiment were selected 

from literature sources in accordance to the type and density 

of the materials. Both models WM-I and WM-II were made 

of materials commonly used in practice (Fig. 2). Five walls 

of the model 600x600x600 mm ( length x width x height ) 

were made of polystyrene (EPS) boards d
1
=50+50mm. 

Heterogeneous enclosure of the models consists of two 

medium density wood fibre boards (MDF), with the 

thickness of d
2
=18+18mm, and one expanded polystyrene 

(EPS) board, d
1
=50mm.  

Internal and external volumes of the models are 

identical, but the position of MDF massive layer in 

heterogeneous enclosures is different: in WM-I, it is 

oriented to the interior, and in WM-II, it is oriented to the 

exterior.  

 

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

MDFd2=18+18mm

EPS d1=50mm

EPS

d1=(50+50)mm

T7

T8

T9

T11

T1

T2 T5

T3

T4 T10

 

WM-II WM-I 

Fig. 2. Structural scheme of thermocouple arrangement and models. T- thermocouples  

 (19)

where: σ  =  5.6704·10-8 – Stefan–Boltzmann constant, 
(W×m-2×K-4); s – thickness of air boundary layer, (m); 

λ – thermal conductivity of still air, W/(m∙K); Ts – temperature 
of the surface, (K); Tair – surround air temperature, (K); 
ε1 – emissivity of surface; ε2 – emissivity of surrounding.

The measured thermal properties of materials used in 
wall models WM-I and WM-II are presented in (table 1). The 
thermal conductivity values of the materials were measured 
according to (LST EN 12667:2002, Stankevičius et al. 
2005) in lambda apparatus which meets the requirements of 
(ISO 8301:1991). The density of materials was determined 
in accordance with (LST EN 1602: 1998, LST EN 1602: 
1998/AC:2003).

 

Fig. 1. Principle scheme for calculating the interlayer 

temperature of the enclosure 

3. Materials and equipment 

Two building models were made for the experiments. 

Heat transfer coefficient of the internal surface of the 

enclosures of models WM-I and WM-II, which describes 

the surface heat exchange of the enclosure with the 

environment, is the following:  

 

)/(,

1

11

2

4

2

21

3

KmW

TT

s

Nuh

airs

s

−+

⎟

⎠

⎞

⎜

⎝

⎛ +

⋅⋅

+⎟

⎠

⎞

⎜

⎝

⎛
⋅=

εε

σ

λ     (19) 

 

where: σ =5.6704·10
-8

 – Stefan–Boltzmann constant, (W⋅m
-

2

⋅K
-4

); s – thickness of air boundary layer, (m); Nu – Nusselt 

number calculated according to LST EN 673:2011; λ - 

thermal conductivity of still air, W/(m·K); T
s
 – temperature 

of the surface, (K); T
air

 – surround air temperature, (K); ε
1
 – 

emissivity of surface; ε
2
 – emissivity of surrounding. 

 

The measured thermal properties of materials used in 

wall models WM-I and WM-II are presented in (table 1). 

The thermal conductivity values of the materials were 

measured according to (LST EN 12667:2002, Stankevičius 

et al. 2005) in lambda apparatus which meets the 

requirements of (ISO 8301:1991). The density of materials 

was determined in accordance with (LST EN 1602: 1998, 

LST EN 1602: 1998/AC:2003). 

The thermal capacitance values of common 

construction materials used in the experiment were selected 

from literature sources in accordance to the type and density 

of the materials. Both models WM-I and WM-II were made 

of materials commonly used in practice (Fig. 2). Five walls 

were made of polystyrene (EPS) boards d
1
=50+50mm. 

Heterogeneous enclosure of the models consists of two 

medium density wood fibre boards (MDF), with the 

thickness of d
2
=18+18mm, and one expanded polystyrene 

(EPS) board, d
1
=50mm.  

Internal and external volumes of the models are 

identical, but the position of MDF massive layer in 

heterogeneous enclosures is different: in WM-I, it is 

oriented to the interior, and in WM-II, it is oriented to the 

exterior.  

 

EPS

d1=(50+50)mm

EPS d1=50mm

MDF d2=18+18mm

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

EPS

d1=(50+50)mm

MDFd2=18+18mm

EPS d1=50mm

EPS

d1=(50+50)mm

T7

T8

T9

T25

T11

T1

T2 T5

T3

T4 T10

 

WM-II WM-I 

Fig. 2. Structural scheme of thermocouple arrangement and models. T- thermocouples  

Fig. 2. Structural scheme of thermocouple arrangement and models. T- thermocouples

Table 1. The walls studied: WM-I and WM-II with the same values of thermal transmittance U, but with different values of internal areal 
thermal capacity

Walls model WM-I WM-II
Thickness, (mm) 50.0 + 36.0 = 86.0 36.0 + 50.0 = 86.0

Insulation (EPS - expanded polystyrene)
Conductivity λ, (W/m·K) 0.039 0,039
Density ρ, (kg/m³) 15.807 15.807
Thermal capacity c, (J/kg·K) 1450 1450

Mass (MDF – medium density wood fibreboard)
Conductivity λ, (W/m·K) 0.12 0.12
Density ρ, (kg/m³) 787.22 787.22
Thermal capacity c, (J/kg·K) 1430 1430
Thermal transmittance Uwall, (W/(m2∙K) 0.57 0.57

Nu – Nusselt number calculated according to LST EN 673:2011; 



44

The thermal capacitance values of common 
construction materials used in the experiment were selected 
from literature sources in accordance to the type and density 
of the materials. Both models WM-I and WM-II were made 
of materials commonly used in practice (Fig. 2). Five walls 
of the model 600 x 600 x 600 mm (length x width x height) 
were made of polystyrene (EPS) boards d1  =  50  +  50  mm. 
Heterogeneous enclosure of the models consists of two 
medium density wood fibre boards (MDF), with the 
thickness of d2  =  18 + 18 mm, and one expanded polystyrene 
(EPS) board, d1  =  50mm. 

Internal and external volumes of the models are 
identical, but the position of MDF massive layer in 
heterogeneous enclosures is different: in WM-I, it is oriented 
to the interior, and in WM-II, it is oriented to the exterior. 

4. Experimental and results

In order to determine the accuracy of the calculation 
method, the experiment was carried out. On the basis of 
the experiment, the calculation program was created for 
the evaluation of unsteady heat exchange in multi-layered 
enclosures (using MS Excel). In conditional layers of the 
enclosure, temperature fluctuations are calculated using 
finite elements method, and the values of internal enclosure 
layers are estimated according to the equations given in 
Chapter 2. 

The experiment was carried out by imitating climatic 
conditions in the climatic test chamber, in the Laboratory of 
Building Thermal Physics at Institute of Architecture and 
Construction of Kaunas University of Technology, (KTU 
ASI). Two spatial models, differing in the position of their 
high thermal resistance layer, i.e. inside and outside of the 
model, were constructed in the chamber. The aim of the 
experiment was to change the temperature of the surroundings 
by cooling and heating it and observe temperature changes 
in the walls and the interior of the models. In order to 
ensure the accuracy of the experiment, the values of thermal 
parameters of the materials were determined and the 
thermocouples measuring the temperature were calibrated. 

During the experiment, the temperature was measured 
every one second and the average of one minute was recorded 
(one minute time step is also acceptable in calculations). 
Data accumulator DL-3 scanned the values of automatically 
measured temperatures, recorded them and presented in MS 
Excel format. 

During the experiment, the dynamic temperature 
oscillation consisting of seven cooling and heating cycles 
was created in the climatic chamber using cooling and 
heating devices. Before the experiment, steady temperature 
of Θi   =  +18.56 ºC was settled in the climatic chamber and 
inside the models WM-I and WM-II. Dispersion limits of the 
thermocouples (T1...T25) were the following: Ae   = ± 0.25 ºC.

The experiment begins with the cooling cycle, which 
lasts for three hours. The temperature inside the climatic 
chamber is lowered down to Θe  =  (–20) ºC. When this cycle 
is over, a two-hour heating cycle begins and the temperature 
raises up to Θe  =  (+25) ºC. The temperature oscillation 
amplitude is equal to Ae =  (+25 ºC)÷(–20 ºC).

Using the outdoor air temperature oscillation data 
obtained during the experiment, calculations were performed 
in order to determine the temperature of the indoor air and 
wall interlayers. The temperatures were compared with the 
values measured during the experiment. 

The accuracy of the calculations in the non-
homogeneous (multi-layered) enclosure depends on the 
Fourier number, which defines the conditions of gradual 
change and the selection of the rational number of conditional 
layers. When there is a significant difference among the 
Fourier number values of different layers in multi-layered 
enclosure FoMDF =  0.710 and FoEPS =  0.398, the temperature 
distribution dotted curve obtained using the calculation 
program does not correlate with the experimental data 
bolded solid curve (Fig. 3).

-25

-20

-15

-10

-5

0

5

10

15

20

25

1 501 1001 1501 2001 2501

Time, [min]

Te
m

pe
ra

tu
re

, [
°C

]

Calculated, Θis
Measured, Θis

Fig.  3. Correlation of experimental temperature value of the 
internal surface of multi-layered enclosures of WM-I model with 
temperature values obtained using the calculation program, when 
FoMDF =  0.710; FoEPS =  0.398

Thus, the experiment clearly shows that the least 
distorted calculation results were obtained when the 
layers of multi-layered enclosure made of different 
materials were divided into rational layers so the values 
of the Fourier number were closely equal. In case of the 
experimental calculation, the values were FoMDF =  0.177 
and FoEPS =  0.1633. The dotted curve, representing the 
distribution of the temperature calculated using the 
calculation program, completely correlates with the bolded 
solid curve representing the experimental data in (Fig. 4).

-25

-20

-15

-10

-5

0

5

10

15

20

25

1 501 1001 1501 2001 2501

Time, [min]

Te
m

pe
ra

tu
re

, [
°C

]

Calculated, Θis
Measured, Θis

Fig.  4. Correlation of experimental temperature value of the 
internal surface of multi-layered enclosures of WM-I model with 
temperature values obtained using the calculation program, when 
FoMDF =  0.177; FoEPS =  0.1633



45

The calculation program is sufficiently accurate for 
carrying out further research. 

5. The effect of wall‘s massive layer position on time 
delay, decrement factor and inside temperature 

Time delay and decrement factor are very important 
factors for the evaluation of thermal resistance of the 
material. The outdoor air temperature changes chaotically 
during a 24-h period (Bruzgevičius et al. 2012) and 
influences the microclimate of the premises. Building 
enclosures work as a passive microclimate system and 
reduces the effect of the outdoor air temperature on the 
microclimate of the premises. The thermal inertia properties 
of the enclosures help to reduce energy consumption for 
ensuring microclimate comfort in the premises during the 
hot or cold season.

Many construction types of building enclosures are 
applied in practice. The outdoor air temperature wave 
diffuses through building envelope constructions with 
different time delay due to the inertia properties of the 
building envelope. The time delay of temperature fluctuation 
the premises caused by outdoor air temperature fluctuations 
depends on the thermal inertia of enclosures.

The temperature oscillation is assumed to show 
sinusoidal variations during a 24-h period. Time delay Φ is 
defined as (20):

min,min,min ei ΘΘ Θ−Θ=Φ  (20)

max,max,max ei ΘΘ Θ−Θ=Φ  (21)

The proportion of the amplitudes of the indoor and 
outdoor air temperature is called the wave decrement factor, 
which is expressed by the following equation (Cibse guide 
1988, Kontoleon et. al. 2005) as (22):

min,max,

min,max,

ee

ii

e

i

A
A

f
Θ−Θ

Θ−Θ
==  (22)

where: Ae – amplitude of outdoor air temperature oscillation, 
(oC); Ai – amplitude of indoor air temperature oscillation, 

(oC); Θe,max – maximum outdoor air temperature,  (oC); 
Θe,min – minimum outdoor air temperature,  (oC); Θi,max – 
maximum indoor air temperature,  (oC); Θi,min – minimum 
indoor air temperature,  (oC)  

These thermal inertia parameters are ilustrated in 
figure 5.

Under natural conditions, the outdoor air temperature 
is influenced by a number of factors: solar radiation, 
cloudiness, heat absorption and reflection of the ground 
surface, wind speed, atmospheric pressure and many 
other (Rimkus et. al. 2007). All these factors, their values 
and combinations influence the fluctuation of the outdoor 
air temperature. In this research, only the influence of the 
outdoor air temperature is analysed. In simple cases of 
the outdoor air temperature forecast, it is assumed that 
temperature oscillates regularly around its average value 
according to a cosinusoid. According to this method, the 
air temperature Θd after the time period t is expressed as 
(Phokin 2006):

), (
2

cos Ct
T

A oead 







⋅⋅+Θ=Θ
π  (23)

where: Θd – average outdoor temperature, (K); 
Ae – amplitude of outdoor temperature oscillation, (K); 
T – period of temperature oscillation, (h); t – time from the 
beginning of oscillation, (h).

6. Discussion and analysis

Three structural types of wall insulation have been 
taken for investigation of time delay and decrement factor 
(table 2). In wall WM-I the insulation is positioned outdoors; 
in WM-II the insulation is positioned indoors; the WM-III 
is a single-layer EPS insulation. In all three types wall the 
thermal transmittance is the same (Uwall  = 0.57 W/(m2∙K)). 
The temperature values inside these walls are calculated 
according to the equations given in Chapter 2. 

The outdoor temperature oscillation curves, given in 
Fig. 6 are used for simulation of walls WM-I, WM-II and 
WM-III. Periods of outdoor temperature oscillation varies 
from 0.5 to 10 hours from (a) to (f) (Fig. 6).

-25

-20

-15

-10

-5

0

5

10

15

20

25

1 501 1001 1501 2001 2501

Time, [min]

T
e
m

p
e
r
a
t
u

r
e
,
 
[
°
C

]

Calculated, Θis

Measured, Θis

Fig. 4. Correlation of experimental temperature value of the 

internal surface of multi-layered enclosures of WM-I model with 

temperature values obtained using the calculation program, when 

F
oMDF

=0.177;F
oEPS 

=0.1633 

The calculation program is sufficiently accurate for 

carrying out further research.  

4. The effect of wall‘s massive layer position on time 

delay, decrement factor and inside temperature  

Time delay and decrement factor are very important 

factors for the evaluation of thermal resistance of the 

material. The outdoor air temperature changes chaotically 

during a 24-h period (Bruzgevičius et al. 2012) and 

influences the microclimate of the premises. Building 

enclosures work as a passive microclimate system and 

reduces the effect of the outdoor air temperature on the 

microclimate of the premises. The thermal inertia properties 

of the enclosures help to reduce energy consumption for 

ensuring microclimate comfort in the premises during the 

hot or cold season. 

Many construction types of building enclosures are 

applied in practice. The outdoor air temperature wave 

diffuses through building envelope constructions with 

different time delay due to the inertia properties of the 

building envelope. The time delay of temperature 

fluctuation the premises caused by outdoor air temperature 

fluctuations depends on the thermal inertia of enclosures. 

The temperature oscillation is assumed to show 

sinusoidal variations during a 24-h period. Time delay Φ is 

defined as (20): 

 

 

min,min,min ei ΘΘ
Θ−Θ=Φ            (20) 

 

max,max,max ei ΘΘ
Θ−Θ=Φ            (21) 

 

The proportion of the amplitudes of the indoor and 

outdoor air temperature is called the wave decrement factor, 

which is expressed by the following equation (Cibse guide 

1988, Kontoleon et. al. 2005) as (22): 

 

min,max,

min,max,

ee

ii

e

i

A

A

f

Θ−Θ

Θ−Θ

==            (22) 

 

where: A
e
 – amplitude of outdoor air temperature oscillation,

 

(
o

C); A
i
 – amplitude of indoor air temperature oscillation,

 

(
o

C); Θ
e,max

 – maximum outdoor air temperature, 
 

(
o

C); Θ
e,min

 

– minimum outdoor air temperature, 
 

(
o

C); Θ
i,max

 – 

maximum indoor air temperature, 
 

(
o

C); Θ
i,min

 – minimum 

indoor air temperature, 
 

(
o

C);  

 

These thermal inertia parameters are ilustrated in 

figure 5. 

Under natural conditions, the outdoor air temperature 

is influenced by a number of factors: solar radiation, 

cloudiness, heat absorption and reflection of the ground 

surface, wind speed, atmospheric pressure and many other 

(Rimkus et. al. 2007). All these factors, their values and 

combinations influence the fluctuation of the outdoor air 

temperature. In this research, only the influence of the 

outdoor air temperature is analysed. In simple cases of the 

outdoor air temperature forecast, it is assumed that 

temperature oscillates regularly around its average value 

according to a cosinusoid. According to this method, the air 

temperature Θ
d 

after the time period t is expressed as 

(Phokin 2006): 

 

),(

2

cos Ct

T

A

o

ead
⎟

⎠

⎞

⎜

⎝

⎛
⋅⋅+Θ=Θ

π

         (23) 

 

where: Θd – average outdoor temperature, (K); Ae – 

amplitude of outdoor temperature oscillation, (K); T – 

period of temperature oscillation, (h); t – time from the 

 

Fig. 5. Scheme of time delay and decrement factor Fig. 5. Scheme of time delay and decrement factor

.



46

Table 2. The walls studied: WM-I, WM-II, WM-III with the same values of thermal transmittance U

Wall models

WM-I WM-II WM-III

5. Discussion and analysis 

Three structural types of wall insulation have been 

taken for investigation of time delay and decrement factor 

(table 2). In wall WM-I the insulation is positioned 

outdoors; in WM-II the insulation is positioned indoors; the 

WM-III is a single-layer EPS insulation. In all three types 

wall the thermal transmittance is the same (U
wall 

 = 0.57 

W/(m
2

·K)). The temperature values inside these walls are 

calculated according to the equations given in Chapter 2.  

The outdoor temperature oscillation curves, given in 

Fig. 6 are used for simulation of walls WM-I, WM-II and 

WM-III. Periods of outdoor temperature oscillation varies 

from 0.5 to 10 hours from (a) to (f) (Fig. 6). 

 

Table 2. The walls studied: WM-I, WM-II, WM-III with the same values of thermal transmittance U 

 

 

 

 

Wall models 

   

Thickness, (mm) 50.0+36.0=86.0  36.0+50.0=86.0  61.7 

Insulation (EPS - expanded polystyrene)  

Conductivity λ, (W/m·K) 0.039 0.039 0.039 

Density ρ, (kg/m³) 15.807 15.807 15.807 

Thermal capacity c, (J/kg·K) 1450 1450 1450 

Mass (MDF – medium density wood fibreboard) 

Conductivity λ, (W/m·K) 0.12 0.12 - 

Density ρ, (kg/m³) 787.22 787.22 - 

Thermal capacity c, (J/kg·K) 1430 1430 - 

Thermal transmittance U
wall

, W/(m
2

·K) 0.57 0.57 0.57 

Period of temperature oscillation Time delay, φ (h) 

0,5 14 14 11 

1 22 21 20 

2 33 31 29 

4 48 47 42 

5 51 51 46 

10 68 68 53 

Period of temperature oscillation Decrement factor, f (–) 

0,5 0.037 0.037 0.053 

1 0.120 0.120 0.160 

2 0.273 0.264 0.356 

4 0.483 0.483 0.624 

5 0.552 0.552 0.709 

10 0.739 0.739 0.894 

 

a) 

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 0.5 [h] 

T
e

m
p

e
r
a

t
u

r
e

,
 
[
°
C

]

Θe

Θi

 

b)

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 1 [h]

T
e
m
p
e
r
a
t
u
r
e
,
[
°
C
]

Θe

Θi

 

O
n

l
y

 
i
n

s
u

l
a

t
i
o

n
 

E
x

t
e
r
n

a
l
e
 
s
i
d

e
 

 
 
 
M

a
s
s
 

 
I
n

s
u

l
a

t
i
o

n
 

E
x

t
e
r
n

a
l
e
 
s
i
d

e
 

M
a

s
s
 

 In
s
u

l
a

t
i
o

n
 

E
x

t
e
r
n

a
l
e
 
s
i
d

e
 

5. Discussion and analysis 

Three structural types of wall insulation have been 

taken for investigation of time delay and decrement factor 

(table 2). In wall WM-I the insulation is positioned 

outdoors; in WM-II the insulation is positioned indoors; the 

WM-III is a single-layer EPS insulation. In all three types 

wall the thermal transmittance is the same (U
wall 

 = 0.57 

W/(m
2

·K)). The temperature values inside these walls are 

calculated according to the equations given in Chapter 2.  

The outdoor temperature oscillation curves, given in 

Fig. 6 are used for simulation of walls WM-I, WM-II and 

WM-III. Periods of outdoor temperature oscillation varies 

from 0.5 to 10 hours from (a) to (f) (Fig. 6). 

 

Table 2. The walls studied: WM-I, WM-II, WM-III with the same values of thermal transmittance U 

 

 

 

 

Wall models 

WM-I WM-II WM-III 

   

Thickness, (mm) 50.0+36.0=86.0  36.0+50.0=86.0  61.7 

Insulation (EPS - expanded polystyrene)  

Conductivity λ, (W/m·K) 0.039 0.039 0.039 

Density ρ, (kg/m³) 15.807 15.807 15.807 

Thermal capacity c, (J/kg·K) 1450 1450 1450 

Mass (MDF – medium density wood fibreboard) 

Conductivity λ, (W/m·K) 0.12 0.12 - 

Density ρ, (kg/m³) 787.22 787.22 - 

Thermal capacity c, (J/kg·K) 1430 1430 - 

Thermal transmittance U
wall

, W/(m
2

·K) 0.57 0.57 0.57 

Period of temperature oscillation Time delay, φ (h) 

0,5 14 14 11 

1 22 21 20 

2 33 31 29 

4 48 47 42 

5 51 51 46 

10 68 68 53 

Period of temperature oscillation Decrement factor, f (–) 

0,5 0.037 0.037 0.053 

1 0.120 0.120 0.160 

2 0.273 0.264 0.356 

4 0.483 0.483 0.624 

5 0.552 0.552 0.709 

10 0.739 0.739 0.894 

 

a) 

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 0.5 [h] 

T
e

m
p

e
r
a

t
u

r
e

,
 
[
°
C

]

Θe

Θi

 

b)

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 1 [h]

T
e
m
p
e
r
a
t
u
r
e
,
[
°
C
]

Θe

Θi

 

O
n

l
y

 
i
n

s
u

l
a

t
i
o

n
 

E
x

t
e
r
n

a
l
e
 
s
i
d

e
 

 
 
 
M

a
s
s
 

 
I
n

s
u

l
a

t
i
o

n
 

E
x

t
e
r
n

a
l
e
 
s
i
d

e
 

M
a

s
s
 

 In
s
u

l
a

t
i
o

n
 

E
x

t
e
r
n

a
l
e
 
s
i
d

e
 

5. Discussion and analysis 

Three structural types of wall insulation have been 

taken for investigation of time delay and decrement factor 

(table 2). In wall WM-I the insulation is positioned 

outdoors; in WM-II the insulation is positioned indoors; the 

WM-III is a single-layer EPS insulation. In all three types 

wall the thermal transmittance is the same (U
wall 

 = 0.57 

W/(m
2

·K)). The temperature values inside these walls are 

calculated according to the equations given in Chapter 2.  

The outdoor temperature oscillation curves, given in 

Fig. 6 are used for simulation of walls WM-I, WM-II and 

WM-III. Periods of outdoor temperature oscillation varies 

from 0.5 to 10 hours from (a) to (f) (Fig. 6). 

 

Table 2. The walls studied: WM-I, WM-II, WM-III with the same values of thermal transmittance U 

 

 

 

 

Wall models 

WM-I WM-II WM-III 

   

Thickness, (mm) 50.0+36.0=86.0  36.0+50.0=86.0  61.7 

Insulation (EPS - expanded polystyrene)  

Conductivity λ, (W/m·K) 0.039 0.039 0.039 

Density ρ, (kg/m³) 15.807 15.807 15.807 

Thermal capacity c, (J/kg·K) 1450 1450 1450 

Mass (MDF – medium density wood fibreboard) 

Conductivity λ, (W/m·K) 0.12 0.12 - 

Density ρ, (kg/m³) 787.22 787.22 - 

Thermal capacity c, (J/kg·K) 1430 1430 - 

Thermal transmittance U
wall

, W/(m
2

·K) 0.57 0.57 0.57 

Period of temperature oscillation Time delay, φ (h) 

0,5 14 14 11 

1 22 21 20 

2 33 31 29 

4 48 47 42 

5 51 51 46 

10 68 68 53 

Period of temperature oscillation Decrement factor, f (–) 

0,5 0.037 0.037 0.053 

1 0.120 0.120 0.160 

2 0.273 0.264 0.356 

4 0.483 0.483 0.624 

5 0.552 0.552 0.709 

10 0.739 0.739 0.894 

 

a) 

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 0.5 [h] 

T
e

m
p

e
r
a

t
u

r
e

,
 
[
°
C

]

Θe

Θi

 

b)

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 1 [h]

T
e
m
p
e
r
a
t
u
r
e
,
[
°
C
]

Θe

Θi

 

O
n

l
y

 
i
n

s
u

l
a

t
i
o

n
 

E
x

t
e
r
n

a
l
e
 
s
i
d

e
 

 
 
 
M

a
s
s
 

 
I
n

s
u

l
a

t
i
o

n
 

E
x

t
e
r
n

a
l
e
 
s
i
d

e
 

M
a

s
s
 

 In
s
u

l
a

t
i
o

n
 

E
x

t
e
r
n

a
l
e
 
s
i
d

e
 

Thickness, (mm) 50.0 + 36.0 = 86.0 36.0 + 50.0 = 86.0 61.7
Insulation (EPS - expanded polystyrene) 
Conductivity λ, (W/m·K) 0.039 0.039 0.039
Density ρ, (kg/m³) 15.807 15.807 15.807
Thermal capacity c, (J/kg·K) 1450 1450 1450
Mass (MDF – medium density wood fibreboard)
Conductivity λ, (W/m·K) 0.12 0.12 –
Density ρ, (kg/m³) 787.22 787.22 –
Thermal capacity c, (J/kg·K) 1430 1430 –
Thermal transmittance Uwall, W/(m2∙K) 0.57 0.57 0.57
Period of temperature oscillation Time delay, φ (h)

0,5 14 14 11
1 22 21 20
2 33 31 29
4 48 47 42
5 51 51 46
10 68 68 53

Period of temperature oscillation Decrement factor, f (–)
0,5 0.037 0.037 0.053
1 0.120 0.120 0.160
2 0.273 0.264 0.356
4 0.483 0.483 0.624
5 0.552 0.552 0.709
10 0.739 0.739 0.894

The massive (MDF – medium density wood 
fibreboard) materials are characterized by their capability to 
store energy in their thermal mass and cause a shift delay of 
temperature waves from outside to inside. This is caused by 
high thermal conductivity and volumetric thermal capacity 
of massive (MDF) materials. The due to their low thermal 
conductivity and volumetric thermal capacity, insulation 
materials respond like thermal barriers, significantly 
decreasing temperature fluctuations in the direction of the 
heat flow path.

Figure 7 shows the dependence of time delay and 
figure 8 shows the dependence of wave decrement factor 
of the enclosures WM-I, WM-II and WM-III on different 
oscillation periods of the outdoor air temperature (Fig. 6).

When the outdoor air temperature oscillation period 
is 0.5 h, the inertia of the enclosures WM-I and WM-II 
fully suppresses the outdoor air temperature oscillations 
on the inner surface, in comparison to the homogeneous 

EPS enclosure WM-III. When the oscillation period of the 
outdoor air temperature is 1 h, 2 h and 4 h, a 1 min difference 
in time delay values can be noticed due to different positions 
of the inertial massive layer in the enclosures. Time delay 
of fluctuations is more significant when the massive layer 
is oriented to the interior of the model WM-I. When the 
oscillation period of the outdoor air temperature is 5 h and 
10 h, time delay values of the enclosures WM-I and WM-
II become equal. During oscillation periods, the position of 
the inertial massive layer in the enclosure does not affect 
time delay. 

Figure 8 shows decrement factor values of the 
enclosures WM-I, WM-II and WM-III. The closer are the 
decrement factor values to zero, the more inertial is the 
material. Comparing the values of wave decrement factor 
through each of three multi-layered enclosures, the inside 
surface temperature changes faster in model WM-III than 
in other models. 



47

a)

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 0.5 [h] 

T
em

pe
ra

tu
re

, [
°C

]
Θe
Θi

b)

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 1 [h]

T
em

pe
ra

tu
re

,[°
C

]

Θe
Θi

c)

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 2 [h]

T
em

pe
ra

tu
re

, [
°C

]

Θe
Θi

d)

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 4 [h]

T
em

pe
ra

tu
re

, [
°C

]

Θe
Θi

]

e)

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 501 1001 1501

Period of temperature oscillation, 5 [h]

T
em

pe
ra

tu
re

, [
°C

]

Θe
Θi

f)

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

1 1001

Period of temperature oscillation, 10 [h]

T
em

pe
ra

tu
re

, [
°C

]

Θe
Θi

Fig.  6. The impact of outdoor temperature oscillation to indoor temperature. Solid curve shows the exterior air Θe and the dotted curve 
shows the prognosticated internal air temperature Θi. Outdoor temperature curves are used for walls WM-I, WM-II, WM-III simulation. 
Periods of outdoor temperature oscillation varies from 0.5 to 10 hours. (Figures from (a) to (f)) 

51

68

51
48

33

22

14

68

47

31

21

14

53
46

42

29

20

11

0

10

20

30

40

50

60

70

80

0,5 1 2 4 5 10
Period of temperature oscillations, [h]

T
im

e 
de

la
y,

 [m
in

]

I EPS/MDF
II MDF/EPS
III EPS

Fig.  7. The time delay dependant on period of outdoor temperature 
oscillation

0.12

0.273

0.037

0.483

0.739

0.552

0.739

0.552

0.483

0.264

0.12

0.037

0.053
0.16

0.356

0.624

0.709

0.894

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0,5 1 2 4 5 10
Period of temperature oscillations, [h]

A
i/A

e

I EPS/MDF
II MDF/EPS
III EPS

Fig.  8. The decrement factor dependant on period of outdoor 
temperature oscillation



48

7. Conclusions

1. Continuous outdoor air temperature fluctuations 
through multi-layered building enclosures were suppressed 
by high thermal capacitance layers of the enclosure, 
irrespective of their position. High thermal capacitance 
layers of enclosures oriented to the interior of the premises 
suppressed the fluctuations more effectively than the layers 
oriented to the exterior.

2. When classifying buildings according to their 
thermal inertia, thermal time constant should be calculated 
taking into account thermal capacitance of all the layers of 
building enclosures.

3. The least distorted calculation results were 
obtained when the multi-layered enclosure was divided into 
conditional layers so that their number values of the Fourier 
number were close to equal.

References

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Berg-Hallberg E. 1985. Realistic design outdoor temperatures. 
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310–317.

Bruzgevičius P., Burlingis A., Stankevičius V. 2012. The 
construction of Lithuanian hourly meteorological data 
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Issues, Kaunas, 2012.

Cibse guide. Volume A. Design data. 1988. London: The chartered 
institution of building services engineers.

Hagentoft C. E. 2001. Introduction to Building Physics. Lund, 
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Hensen L. M., Nakhi E. A. 1994. Fourier and Biot Numbers and 
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Received 2013 02 20
Accepted after revision 2013 03 05



49

Patrikas BRUZGEVIČIUS. PhD student of Civil Engineering, Researcher at the Laboratory of Thermal Building Physics 
of the Institute of Architecture and Construction, KTU. 
Research interests: unsteady heat transfer, thermal processes in buildings.
Adress: Tunelio st. 60, LT-44405, Kaunas, Lithuania. 
Tel.: +370 37 350779 
E-mail: patrikas.bruzgevicius@yahoo.com., patrikas.bruzgevicius@stud.ktu.lt

Arūnas BURLINGIS. Doctor, Senior Researcher at the Laboratory of Thermal Building Physics at the Institute of 
Architecture and Construction, KTU. 
Research interests: thermal processes in building, thermal and hydro properties of building materials and elements.
Adress: Tunelio st. 60, LT-44405, Kaunas, Lithuania.
Tel.: +370 37 350779
E-mail: arunas.burlingis@asi.lt

Vytautas STANKEVI IUS. Doctor Habil., Full Professor, Chief Researcher at the Laboratory of Thermal Building Physics 
of the Institute of Architecture and Construction, KTU.
Research interests: heat transfer, technical properties of thermal insulations products.
Adress: Tunelio st. 60, LT-44405, Kaunas, Lithuania.
Tel.: +370 37 350779
E-mail: v.stankevicius@ktu.lt

Darius PUPEIKIS. Doctor, Lecturer at the Department of Building Materials of the Faculty of Civil Engineering and 
Architecture, KTU. 
Research interests: unsteady heat transfer, thermal energy balance of buildings.
Adress:  Studentų st. 48, LT-51367 Kaunas, Lithuania.
Tel.:  +370 689 77371 
E-mail: darius.pupeikis@ktu.lt 

Karolis BANIONIS. Doctor, Researcher at the Laboratory of Thermal Building Physics of the institute of Architecture and 
Construction, KTU. 
Research interests: energy efficiency and air permeability of buildings, heat transfer and thermal insulation, thermal impacts 

of solar radiation.
Adress: Tunelio st. 60, LT-44405, Kaunas, Lithuania.
Tel.:  +370 37 350779
E-mail:  karolis.banionis@ktu.lt

Rosita NORVAIŠIENĖ. Doctor, Researcher at the Laboratory of Thermal Building Physics of the institute of Architecture 
and Construction, KTU. 
Research interests: Sustainable building materials;  durability testing of building materials; damage due to moisture 

penetration.
Adress: Tunelio st. 60, LT-44405, Kaunas, Lithuania.
Tel.:  +370 37 350779
E-mail:  rosita.norvaisiene@ktu.lt

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