Energy and Power En gi neering, 2011, 3, 150-157
doi:10.4236/epe.2011.32019 Published Online May 2011 (http://www.SciRP.org/journal/epe)
Behaviour of Thermodynamic Models with Phase Change
Materials under Periodic Conditions
Amelia Carolina Sparavigna, Salvatore Giurdanella, Matteo Patrucco
Dipartimento di Fisica, Politecnico di Torino Corso Duca degli Abruzzi, Torino, Italy
Received February 18, 2011; revised April 1, 2011; accepted April 7, 2011
Abstract
We study the thermal behaviour of some models in a steady periodic regime. The aim is to simulate the be-
haviour of small environments at the outermost part of our planet, subjected to the periodic solar radiation.
Our approach is based on a method using lumped elements or volumes that simplifies the description of spa-
tially distributed physical systems, through a topology consisting of discrete entities. Our models include
some parts acting as energy storage systems, made with Phase Change Materials (PCMs). The storage is
based on latent heats: the energy is stored during the melting and recovered during the solidification of the
PCM substance. The simulation with lumped elements shows some interesting behaviours of temperatures.
Keywords: Thermodynamics, Heat Exchange, Lumped Volume Models
1. Introduction
The study of the thermal behaviour of macroscopic sys-
tems in a steady periodic regime is quite important be-
cause of its usefulness in investigating the effects pro-
duced by solar radiation on the structures located on the
outermost part of our planet. The aim of these studies is
the simulation of temperature behaviours and heat ex-
changes in local environments. Due to the current condi-
tions created by an increasing average temperature com-
ing from the global warming, these simulations could
help in offering new solutions to reduce the energy con-
sumption or prevent some side effects. Let us consider
for instance the role of permafrost soil in the behaviour
of climate of arctic regions. Permafrost is that soil al-
ways below the freezing point of water, which is located
at high latitudes close to the poles. The extent of perma-
frost can vary as the climate changes: at the same time, it
is thought that permafrost thawing could exacerbate the
global warming by releasing methane and other hydro-
carbons that are powerful greenhouse gases [1]. A model
of the thermodynamic behaviour of permafrost could
then be interesting for a forecast of future environmental
conditions in arctic regions. Here we will propose and
discuss the thermal behaviour of some preliminarily
simple models, useful to simulate those structures in-
cluding solid parts, for instance ice or permafrost or
other materials, which can melt at specific temperatures.
These models are also interesting for simulating the
thermal behaviour of a building with systems for energy
storage. Let us note that a storage system is a fundamen-
tal counterpart of all those systems based on renewable
energies working under periodic conditions.
Among the various methods to store energy, the latent
thermal energy storage using Phase Change Materials
(PCMs) is widely considered as a highly effective one. It
has the advantage of a high density of energy, stored in
an isothermal operation, during the solid/liquid phase
change. In the latent heat storage, the energy is stored
during the melting and released during the solidification
of a PCM substance [2-10]. For what concerns the use of
PCMs, it is necessary to note that some practical difficul-
ties can arise because these materials usually have a low
thermal conductivity and a poor stability of properties
under extended cycling. Sometimes phase segregation
and subcooling can happen. Over the past years, several
studies have been performed to examine the perform-
ances of various latent heat storage systems [11-18] and
on micro-encapsulations, which is the best packaging of
PCM, to have good performances [19,20]. Micro-en-
capsulated PCMs constitutes a portable heat storage sys-
tem, easy to insert in the construction materials for
buildings [21-28].
As a possible approach to simulate the thermal behav-
iour of volumes including an energy storage system with
PCMs, we propose the use of models composed of sev-

A. C. SPARAVIGNA ET AL.
151
eral parts, each obeying the laws of thermodynamics.
These components are interacting with heat exchanges,
some of them being in connection with the external en-
vironment. The behaviour of the models will be dis-
cussed by means of an approach based on lumped ele-
ments, where we simplify the description of spatially
distributed physical systems through a topology consist-
ing of discrete entities, which approximate the behaviour
of the distributed system under certain assumptions. This
method, initially developed for electrical systems, is well
known and useful to solve the problem of heat transport
[29]. The partial differential equations of a continuous
time and space model of the physical system are changed
in ordinary differential equations with a finite number of
parameters. Before discussing the lumped volume
method, let us shortly revise the problem of heat ex-
change.
2. Heat Exchange with the Environment
We will consider in our models some structures with
PCM material surrounded by an environment idealized
as a collection of thermal baths, such as heat sources and
reservoirs. We assume a time-dependent state of the sur-
rounding in the terrestrial conditions, that is, with tem-
peratures of the order of a few hundred Kelvin, oscillat-
ing with a defined period. Let us note that in thermody-
namics, the system can have free inputs, when fuels or
electric powers are used. In the case of the natural envi-
ronment, we have a deterministic input, because it is the
Nature to dictate the conditions. Therefore, we can con-
sider for our models a steady periodic regime for solar
radiation, pressure and temperature, as an ideal case of
the conditions of the outermost part of the planet. Let us
remember that the input black-body solar temperature is
approximately of 400 K, the amplitude of oscillation of
30 K [30,31].
Let us shortly discuss the boundary conditions. Let us
imagine a finite body with a temperature, which is a
function of time and position, and heat inputs and out-
puts. The temperature distribution in the bulk is sub-
jected to the Fourier field equation:
2
T
c
t


T
, (1)
where

is density, c specific heat and thermal
conductivity of the body. T is the temperature field. This
equation can have several boundary conditions. Consider
a finite volume V and its surface divided in two continu-
ous surfaces 1 and 2. The model is shown in Figure
1. The temperature field is defined in the volume of the
body and the two surfaces are in contact with two ther-
mal baths having temperatures

S S


1
Tt and


2
Tt
. We
can fix the temperature at the surface of the body coinci-
dent to the temperatures the local two thermal baths as:



 
  
1
2
1
2
,
,
S
S
Tt Tt
Tt Tt


r
r (2)
We can consider different boundary conditions in the
following way:
    
    
11
22
11
22
,,
,,
SS
SS
TtTt Tt
TtTt Tt



 


 

rn r
rnr (3)
Note that the boundary conditions (2) are of the
Dirichlet type, while the boundary conditions (3) are
known as the Neumann conditions [31].
To have an idea of the magnitude of parameters in-
volved in the problem, we can imagine the oscillating
temperatures of the baths as the following functions:




101
cosTt TAt

 ,
 
202
cosTt TBt

 , with
01 285 KT

, 15AK

, 02 , 275 KTB5K

,
1
2π86400 s 

, for a daily oscillation, for instance.
The physical constants are:11
K1.512 Wm 



1
,
63
mK2.0 10c  J




21
K
, corresponding to concrete and
10 Wm 



 which corresponds to zero-wind
concrete-air interface coupling [31,32].
Let we consider a one-dimensional case: it could be a
wall with a certain thickness h (see Figure 1, lower part,
for the frame of reference). In this case, we have an
equation the boundary conditions of which are:
Figure 1. Upper part, a finite volume V with its surface
divided in two continuous sur faces S1 and S2. The tempera-
ture field is defined in the volume of the body and the two
surfaces are in contact with two thermal baths having tem-
peratures T1(t) and T2(t). Lower part, the frame of reference
for a one-dimensional case, where the body is interacting
with the environment through two surfaces again.

A. C. SPARAVIGNA ET AL.
152



 
1
2
0,
,
TtTt
ThtT t



(4)
or
 
 
1
0
2
0,
,
l
z
r
zh
TzTtTt
TzTtTht




  

  


(5)
The Fourier one-dimensional equation is simply:
2
2
TT
ctdz 


, (6)
which is giving the temperature profile in the wall ac-
cording to the chosen boundary conditions.
3. Lumped Volumes and PCM
It is interesting to consider a volume balance of Equation
(6), as proposed in [31]:
 
2
2
0 00
2
dd
d
,0,
hh
zh z
rl
TTT T
cz z
tz
z
Tt ThtTt Tt
 


 
 
 
 

 
  


1
z



(7)
In performing the volume balance, let us assume that
the thermal conductivity is so high to be considered as
practically infinite. The temperature is no more depend-
ing of the z-coordinate, , and then Equa-
tion (7) turns out to be:
 
,TztTt
 


2
,0,
rl
chTTtThtTtTt     


1


(8)
Equation (8) is a function of time only, and it is de-
pending on the specific heat and mass of the body and its
ability to exchange heat with the environment. The dot is
representing the derivative with respect to time. Equation
(8) describes the behaviour of a lumped volume.
Let us consider a body with a certain volume and sur-
faces as in Figure 1 . The volume balance equation gov-
erning this lumped system is:


 
222
11 1
,
0,
cVTCTSTtTht
STt Tt


  

 


, (9)
where V is the volume and 1, 2 are the two surfaces,
through which the body exchanges the heat with the sur-
rounding [31]. Equation (9) is the basic equation we use
in our models.
S S
Let us consider the simple case shown in Figure 2.
We have a body 1 containing inside a smaller body 2.
Only the body 1 is exchanging heat with the environment.
It is quite simple to write the equations of the volume
balance for the two bodies:
 


 
1111112 221
2212212
EE
CTSTtT tSTtTt
CTST tTt


 

 





, (10)
where 11112 222 ;cVCcVC


.
The system of equations (10) can be solved by nu-
merical methods. The numerical solutions have been
found using a two-step Runge-Kutta method. The time
interval t

, that is the step of time in the numerical so-
lution, was chosen small enough, that its further reduc-
tion was not able to change the final result in an appre-
ciable manner. We considered a temperature difference
to be appreciable, only in the case it is greater than 0.05
K, which is the sensitivity of common instruments.
If the external temperature is oscillating with a certain
period, after a transitory time, the temperatures of the
two bodies are oscillating with the same period, as
shown in Figure 3.
The body 2 could contain a PCM. In this case, when 2
reaches its transition temperature, the heat gained or lost
by body 2 with body 1, that is



12 212
QSTtTt   

,
could be used for melting or solidification of PCM. Dur-
ing the phase change, the temperature of body 2 remains
constant.
A control on the mass percentage of solid/liquid
phases of PCM (body 2) is inserted in the numerical
procedure to solve Equation (10). With an upgrade of
their values according to the heat exchange with body 1,
the quantities of liquid and solid PCM can be evaluated,
according to the specific latent heat H. Let us remember
that the latent heat is the amount of energy in form of
heat required to have a complete phase change of a unit
of mass. The phase change is described by the following
equation:


12 212,melt
QSTtT m   

H (11)
Q is the amount of energy released or absorbed during
the change of phase of the substance, m is the mass of the
substance, and H is the specific latent heat (J/kg−1).
2,melt is the phase transition temperature. In the nu-
merical procedure, we considered, instead of a precise
value of the melting temperature, an interval one degree
wide about it, as in Ref.33. The upper panel of Figure 3
shows the behaviours of the temperatures that we ob-
tained when body 2 is a PCM.
T
Figure 2. A certain finite body 1 is surrounded by surface
S1. It contains a body 2 having surface S2. Only body 1 is in
ontact with the environment. c

A. C. SPARAVIGNA ET AL.
Figure 3. Behaviour of the temperatures T1, T2 and Te, for the model in Figure 2 as a function of time. In the upper part of the
figure, it is supposed that body 2 is a PCM. The period of oscillation is one year. In the specific simulation, the PCM is imag-
ined as a certain quantity of water. Note that, when the water reaches its liquid-solid transition, the temperature remains
constant. If body 2 is made of a material having no phase change, the behaviour is different, as shown in the lower part of the
image. The presence of PCM changes the minimum temperature of body 1.
Figure 3 shows the behaviour of the temperatures T1, T2
and Te, for the model in Figure 2, with suitable parame-
ters of thermal exchanges among bodies. The upper part
of the image corresponds to the case where body 2 is a
PCM. When PCM reaches its phase transition, its tem-
perature remains constant. The quantity of heat gained or
lost in the exchange with body 1 does not change the
temperature of 2, but changes the relative amount of liq-
uid and solid PCM. In the lower part of the figure, it is
shown the case with a body 2 made of a material having
no phase transitions. Note that the presence of a PCM
influences the minimum temperature of body 1, which is
different of a few degrees, as it is possible to see com-
paring the two panels of the figure. The parameters used
in the model of Figure 2 are: C1 = 2.0 × 106 J·K1, C2 =
1.0 × 105 J·K1, α1eS1 = 1.0 W·K1, α12S2 = 0.5 W·K1,
Tmelt = 0˚C; mPCM = 100 kg and H = 333.0 kJ/kg. Varying
the amount of PCM and the exchange parameters αS
with the environment and between the bodies, we can
control, for instance, the amplitude of oscillation of the
temperature in body 1.
4. A Container with a Heat Storage System
The model we want to propose in this section is more
complex. It is supposed to be a container 1, in which we
have inside two bodies: 2 is made of PCM and 3 is a
cavity. The container is supposed to be in thermal con-
tact with a substrate S having a constant temperature and
the environment E, having an oscillating temperature.
Figure 4. Container 1 has inside two bodies: 2 is made of
PCM and 3 is a cavity. The container is supposed to be in
thermal contact with a substrate S at constant temperature
and the environment E, having an oscillating temperature.
C
opyright © 2011 SciRes. EPE

154 A. C. SPARAVIGNA ET AL.
The equations of the volume balance for the three bod-
ies are:
 


  
 
 
111111 11
12 122113 1331
2 212 1212
3 313 1313
EE ESSS
CTSTtT tSTT t
STtTt STtTt
CTSTtTt
CTST tTt






 

 










(12)
The parameters we use for calculations are
C1 = 2.0 × 106 J·K1, C2 = 4.18 × 106 J·K1, C3 = 1.0 ×
105 J·K1, α1ES1E= 2.0 × 106 J·K1, α13S13 = 20 W·K1,
α12S12 = 100 W·K1, α13S13 = 20 W·K1, mPCM = 104 kg
and H = 333.0 kJ/kg. Note that we suppose a high value
of the exchange parameter 12 12 S

between container
and PCM, imaging a high value of the surface between
the bodies. The substrate S is at a fixed temperature,
270.0K
S
T

, that is the temperature of a permafrost soil
for instance.
The temperature of the environment E is oscillating
assuming that the solar radiation changes during the day
and over the year, as shown by the red curve (E) in Fig-
ure 5. We see a red band then, composed by the daily
oscillation and the seasonal oscillation during the year.
The green band (1) represents the temperature of the con-
tainer walls, which is oscillating too with smaller ampli-
tude during the day. The blue curve (2) is the tempera-
ture of the PCM, assumed water. Note that the tempera-
ture (2) is constant for a quite long period during which
the material freezes or melts. Then we can see the tem-
perature of the cavity 3 (purple line).
Figure 5. The temperature of the environment is oscillating with a daily oscillation and with a seasonal oscillation during the
year, as shown by the red curve (E). The green band (1) represents the temperature of the container walls. The blue curve (2)
is the temperature of the PCM, assumed to be water. Note that the temperature (2) is constant when the material is freezing
and melting. The temperature of the cavity 3 is the purple line. In the lower panel, 2 contains a material without the phase
ransition, for instance water always in the liquid phase. t

A. C. SPARAVIGNA ET AL.
155
From the upper panel of Figure 5, we can see that
when the PCM is freezing or melting, according to the
thermal exchange, its temperature is constant. We can
also imagine the same system that, instead of PCM in 2,
has a material that does not possess a phase change, for
instance water that always remains in its liquid state. In
this case, the temperature 2 is oscillating as reported in
the lower panel of Figure 5. It is interesting to compare
the two cases, to evaluate the influence of the presence of
PCM on the local average temperature of cavity 3. Fig-
ure 6 shows this comparison. The minimum tempera-
tures observed in 3 are slightly different, but in the case
of the presence of PCM, during its freezing, cavity 3 has
a temperature (grey curve) which is 5 degrees greater
than that observed without PCM (black curve).
We can then try to increase the amount of PCM: for an
amount that is twice, we have the behaviour shown in
Figure 7, for the same temperatures of environment and
substrate. We see that the increase of PCM is strongly
influencing all the system. It is interesting to note that the
minimum temperature in the cavity 3 is always 5 degrees
above the minimum temperature reached by the system
without PCM (Figure 8 shows the comparison for the
two cases). This simulation shows that with a suitable
Figure 6. The temperature in the cavity 3 has a different behaviour in the case that body 2 is a PCM or not.
Figure 7. The temperature of the environment is oscillating, as shown by the red curve. The green band represents the tem-
perature of the container walls. The blue curve is the temperature of the PCM material 2, assumed water. The temperature
of the cavity 3 is the purple line. Note that the presence of PCM maintains the minimum temperature of the cavity five de-
grees above the case without PCM (see the next figure for comparison).

156 A. C. SPARAVIGNA ET AL.
Figure 8. Temperature in the cavity 3 has a different behaviour, in the case that body 2 is a PCM or not. Increasing the
amount of PCM, the minimum temperature of the cavity 3 is always at higher values (a difference of approximately 5 de-
grees).
choice of the PCM quantities, it is possible to manage the
excursion of the temperature inside the cavity (and in the
walls of container).
5. Conclusions
We proposed the study of the thermal behaviour of
lumped volume models in a steady periodic regime cre-
ated by the solar radiation. The method with lumped
elements is based on the volume balance of the thermal
conductivity equation. The description of spatially dis-
tributed physical systems is approximated with a topol-
ogy consisting of discrete entities.
According to the authors’ knowledge, our proposed
approach to the study of lumped systems under periodic
conditions is new because it is including some elements
acting as energy storage systems. The storage is based on
the latent thermal energy storage with Phase Change
Materials (PCMs), where the energy is stored during the
melting and recovered during the solidification of PCM.
We did simulations with simple models, to show the
method. These models are indicating that a temperature
control can be obtained with a suitable choice of the
PCM. Of course, realistic models simulating buildings or
larger environments can be created, including more ele-
ments in the calculation, to study the possibility to have a
passive control of the temperature inside the specific
environments.
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