Heat transfer phenomena in fibrous insulating materials
ANASTASIOS KARAMANOS, AGIS PAPADOPOULOS, DIMITRIOS ANASTASELLOS
Laboratory of Heat Transfer and Environmental Engineering, Department of Mechanical Engineering
Aristotle University Thessaloniki, GR-54124 Thessaloniki, GREECE
Abstract: - Insulation has been and still remains on of the fundamental tools for achieving energy conservation both in
the buildings’ and in the industrial sector. Whilst specific insulating materials are used for each application, according
to the physical and operational requirements, there is a group of materials that is used in both cases, namely inorganic
fibrous materials. The study of industrial insulating materials is a rather hard task, due to the complex phenomena
which take place in the materials’ structure, because of the extreme conditions, in which the material is applied. In
this paper are discussed some of these changes. The study presented focuses on stone wool, which is the most widely
used, and in that sense the most representative inorganic fibrous material. Its performance is based on the air
embodied between its fibers, which produce a low thermal conductivity factor. Under some operational conditions,
however, like the increased presence of moisture, the thermal conductivity factor may change. The objective of this
paper is to describe the theoretical approach and the resulting data on the assessment of the changes in stone wool’s
thermal conductivity, under various operational conditions. In order to achieve that, a description of stone wool’s
chemical composition and structure is given, followed by the physical and mathematical background necessary for
determining the thermal conductivity factor. The results yielded in this way are compared to experimental and
theoretical references available, in order to verify the validity of the adopted approach.
Key-Words: - Fibrous insulating materials; Thermal conductivity; High temperature applications
1 Description of fibrous materials’
structure
The common feature in all insulation materials is their
low thermal conductivity factor λ, usually lower than
0.1 W/mK. This owes to the fact that a quantity of gas,
usually air, is embodied in the material’s mass. Dry
and firm air, when not moving and in small quantities,
has the lowest thermal conductivity factor
(approximately λ = 0,024 W/mK) over a wide range of
temperatures.
The most widely used categories of insulating
materials are inorganic fibrous and organic foamy
ones. In the former group, atmospheric air is “trapped”
between the fibers, whilst in the latter the air is
embodied in bubble form in the material’s mass.
Considering materials applicable under high
temperature conditions, like in industrial applications
or when strict fire protection regulations apply, fibrous
materials arise as the most interesting solutions.
Organic foamy materials cannot be used in such cases,
due to their very poor resistance to high temperatures
and the increased hazard in case of fire X2. applied to,
Therefore, a fibrous insulation material is studied in
this paper.
Stone wool is the most widely used, and hence most
representative, fibrous inorganic materials, accounting
for more than 35% of the European market (ibid). As
some complex phenomena take place in the material’s
structure when applied under varying temperature
conditions, it is necessary to understand its physical
and chemical structure and composition. It is therefore
useful to describe the production process of stone
wool.
The main component of stone wool is an ore called
amphibolite. Initially a mixture is prepared, consisting
mainly of amphibolite and some additives, such as
limestone (to the percentage of 6% per weight
approximately) and various calcium oxides (to the
percentage of 9% approximately). The contents of this
primary mixture can vary, depending on the
manufacturer and on the specific type of stone wool
produced. Still, the foretold mixture can be considered
as the one most widely used and hence a typical one.
The primary mixture is led into a blast furnace, where
it is heated to 1450 – 1520 oC. At this temperature, the
ore is melting and obtains a lava form. Then it sheds in
a cylindrical rotating tank, with a rotation speed of 700
rpm approximately. Therefore, the melted mixture is
centrifuged and escapes from the tank’s surface
through microscopic holes. As it escapes to the
ambient temperature it cools down and becomes solid.
During this process are created fibers, which compose
formless quantities of stone wool.
The next step is to make stone wool to take a compact
form. Special resins are injected to the formless stone
wool which is then compressed in order to obtain its
usual shape of rectangular plates. Then, it is led in
furnaces where at the temperature of 170 – 180 oC
resins are combined with stone wool fibers and get
solid. At the temperature of 250 oC resins harden
giving to stone wool a form of relatively hard plate. It
is important, that vaporization of the embodied water
takes place, as this is produced from the resins’
injection. Finally, the plates of stone wool are
available for use either as they are, or combined with
other materials which form a coating, like aluminum
leafs, particleboards etc.
-
Atmospheric air between fibers
Additional substances used to increase the
product’s resistance to biological impact.
The participating percentage of each component varies
Fig. 1: Stone wool production line
A typical production process of stone wool is depicted
in Fig. 1.
The final features of stone wool can be synopsized as
follows:
- Fibers with circular crossection and very small
diameter (usually d<15μm). The fibers can be
vertically or parallel oriented to the plate’s surface.
In the former case they feature improved
mechanical strength and sound absorption, in the
latter they feature better thermal insulation
properties. In both cases some of the fibers are
affined with each other and some are free.
- Resins used for fibers’ cohesion
depending on the application of the final product. So
does the material’s density. This last parameter, which
is determined by the ratio between the mass of fibers
and the one of the embodied air, is the one that affects
the thermal conductivity factor and therefore the heat
transfer through the material.
2
Creating heat transfer model in a
fibrous material
The heat transfer model that fits better to the insulating
materials’ case is the one of the homogeneous cylinder
with infinite length. The fibers’ length can reach some
millimeters contrary to fiber’s diameter, which reaches
some micrometers. Therefore, the assumption that the
fiber is a cylinder of infinite length can be used safely.
There are four modes of heat transfer in a fibrous
material: (1) Conduction through the solid medium,
i.e. the fibers, (2) conduction through the gas medium,
i.e. the air trapped between the fibers, (3) convection
due to the air in the space between the fibers and (4)
radiation interchange between fibers and air.
Convection is caused by the air molecule’s movement.
Taking into account the fact that, due to the very small
dimensions of the cavities the air’s movement between
the fibers is practically negligible, heat transfer of
convection is minimal and can therefore be omitted
from the calculations, without losing in accuracy. The
participation of each mode of heat transfer in the total
heat transfer for a wide range of temperatures is
presented in Fig. 2. Such a wide range can easily be
met in industrial applications.
β:
the extinction coefficient,
T:
the temperature,
σ=
5.67∙10-8 W∙m-2∙ K-4 (Stefan – Boltzmann
constant)
The extinction coefficient is given by the equation: β =
e∙ρ, where,
ρ:
the insulation material’s density
e:
the specific extinction coefficient
2.2 Heat transfer by means of gas conduction
Having made the assumption that the air in the
insulation material is stationary, the following
equation can be used, which calculates the thermal
conductivity factor by means of gas conduction:
k*g
kg 
2  α 2γ 1
Φ  2Ψ

  Kn
α
γ Pr
(2)
where,
Fig. 2: Participation of each heat transfer mechanism
to the total [11].
The mathematical model which is presented was
selected with two criteria: its appropriateness for the
case of stone wool case and its simplicity which makes
it usable with a reasonable computational effort.
2.1
kg*:
the air’s thermal conductivity factor at
atmospheric pressure,
α:
energy exchange between air’s and fibers’
molecules factor (Its value is between 0 and
1),
γ:
specific heat ratio,
Pr:
Prandtl number
Kn:
Knudsen number, which is calculated using
the equation: Kn = λ/LC
where,

λ
where,
Kb=
T:
dg:
P:
Heat transfer by means of radiation
In order to study the radiative heat flux a system of
simultaneous differential and integral equations should
be used. Several analytical methods were considered
and the one presented in this paper is a fairly simple
one, which still leads to satisfactiry results,
approximating those produced by experiments. The
equation that calculates the thermal conductivity factor
by means of radiation is therefore:
kR 
16  σ 3
Τ
3β
(1)
where,
kR :
thermal conductivity factor through radiation,
The middle free path of air’s molecules

ΚΒ  Τ
2  π  d g2  P
[m] (3)
1.3806∙10-23 [J/K],
temperature [K],
air’s collision diameter [m],
pressure [Pa]
The characteristic length
LC 
π Df

4 f
(4)
where,
Df: fibers’ diameter [m],
f:
the ratio of insulation material density to the
fibers’ material density
Φ, Ψ: parameters values are:
Φ = 1, Ψ=0, when Kn < 0.01
Φ = 1, Ψ=1, when 0.01 < Kn < 10
Φ = 0, Ψ=1, when Kn > 10
2.3
2.5
Heat transfer by means of solid
conduction (fibers)
The most exact equation that calculates the thermal
conductivity factor through fibers is:
kS  f 2  λ**Σ
(5)
where,
λS**: fibers’ material thermal conductivity factor
f:
the ratio of insulation material density to the
fiber’s material density
2.4
Equivalent thermal conductivity factor
(Fibers and air thermal conductivity)
The first step is to combine the air’s thermal
conductivity factor with the fibers’ one. There are
many methods that can be used in order to combine
the thermal conductivity factors of a gas and a solid
medium. In this paper the following equation is used
to calculate the total thermal conductivity factor by
means of conduction:
k cond  k s 
kg  ks
k  ks 
f 
1
 1  z  g

1  f 
k g  k s 
(6)
where,
kg, ks: the thermal conductivity factors through gas
and solid conduction respectively that
calculated using Equations 2 and 3,
z:
a coefficient that takes the values:
1,
when fibers are oriented vertical to the
heat flux direction
2/3,
when fibers are randomly oriented
5/6,
when the 50% of fibers are oriented
vertical to the heat flux direction and
the other 50% is randomly oriented
This method is preferred, because it takes into
consideration the fibers’ orientation, using the z
coefficient. Generally, the fibers’ orientation in an
insulation material depends on the fibers’ material, the
production method and the aimed final result. Using
the production method of stone wool described in
paragraph 1, the fibers take random orientation.
Therefore, the value z=2/3 is being used.
Total equivalent thermal conductivity
factor
Finally, thermal conductivity factors by means of
radiation and conduction are combined, in order to
calculate the total thermal conductivity factor of the
insulating material by using the equation:
k total  k cond  k r  k conv (7)
where,
kcond, kr: The
conductivity
factors
through
conduction and radiation respectively, as
they calculated in Equations 4 and 2.
kconv: Thermal conductivity factor for heat transfer
with convection, which is practically null and
it’s not been taken in notice in calculations.
3 Stone wool and its behavior under
varying temperatures
The advantage of the inorganic fibrous materials in
generally, but more than so of stone wool, lies in the
fact that it can be used in very high temperatures, of up
to 1000 oC. Furthermore, fibrous materials have high
fire resistance and they can be therefore used in any
other sort of applications, from particular buildings
constructions to shipbuilding. Still, their insulating
behavior does change with varying temperature
conditions, a fact that should not be neglected when
designing and dimensioning insulation solutions. The
determination of the change of the thermal
conductivity factor, depending on the change of some
application conditions is discussed in the following
paragraphs.
3.1
Temperature increase
When temperature increases, changes in the thermal
conductivity take place for each heat transfer
mechanism discussed earlier.
Within the frame of this research a calculation
spreadsheet was created using the heat transfer model
described by equations (1) to (7). Temperature is given
as input and the thermal conductivity factor for each
heat transfer mechanism as well as the total one are
extracted. The temperatures of 25, 50, 75, 100, 250,
500 and 1000oC were used as input. The results
depicted in Fig. 3 indicate the variations arising for
each different heat transfer mode. In Fig. 4 is depicted
a similar diagram produced by another research, in
order to allow comparisons with respect to the
described model’s accuracy.
It has to be noticed that fibers split was not taken into
consideration. This effect begins to take place for
temperatures over 200oC, where the binder resins
decompose. In practice however, this phenomenon
becomes important for temperatures over 500oC,
k(T) =
= 3.6-0.49·10-2 T+0.61·10-5 T2-2.58·10-9 T3 (8)
where T is given in K.
b) As density of fibers the mean value of common
ceramic fibers was used and it is ρF = 2700 kg/m3
and for the fibers’ diameter, an average value of
DF = 15·10-6 m was used.
c) The value of extinction coefficient is e= 60 m2/kg,
as it used in a relative study for fibrous material
[13]
d) The density of stone wool varies according to its
type and the intended application, as it can be seen
in Table 1. For this study a heavy type was
considered, like the ones used in industrial
applications. The value used is ρ= 250 kg/m3
EN
12667
EN
826
DIN
4102
Vapor penetrability factor
Maximum operation
temperature [oC]
Tensile strenght at 10%
compression [kPa]
Thermal conductivity
[W/(mK}]
Application
-
Density
[kg/m3]
-
EN
52615
Buildings:
Floors
Industrial
applications
175
200
0.0315
9,18
750
1
0.036
17.1
750
1
0.04
19.74
1000
1
e) The energy exchange between the air’s and the
fibers’ molecules factor α takes values between 0
and 1. In this paper unity was used.
f) Values for specific heat ratio γ where taken from
[15] and linear interposition method used where
the value was not available.
The results of the calculations are summarized in
Table 2 and presented in Fig.3 and Fig. 4.
Table 2: Results of the calculations of the model
described.
Thermal conductivity (W/mK)
Temp
(oC)
kR
kG
kS
kTOT
25
0,0011
0,0262
0,0224
0,0269
50
0,0014
0,0276
0,0220
0,0284
75
0,0017
0,0290
0,0216
0,0300
100
0,0021
0,0304
0,0213
0,0316
250
0,0058
0,0389
0,0200
0,0426
500
0,0186
0,0530
0,0194
0,0677
1000
0,0832
0,0813
0,0165
0,1561
0,18
0,16
0,14
0,12
0,10
0,08
0,06
0,04
0,02
0,00
0
50
Buildings:
Walls
Test
method
Feature
Table 1: Main features and application for stone
wool of various densities.
100
Thermal conductivity [W/(mK)]
because only then the material’s density begins to
change dramatically.
The following assumptions where made for the
calculations:
a) The fibers’ thermo-physical properties are the
same with the ones of amphibolite, as the latter is
the fibers’ raw material. Therefore, the thermal
conductivity factor of the fibers can be calculated
using the Cemark and Raybach approximation,
1982, for rocks: [16]
0.0327
2,34
750
1
250
500
750
Temperature (oC)
Radiation
Solid Conduction
1000
Gas Conduction
TOTAL
Fig. 3: Results of the calculations of the model
described .
Participation of each heat transfer mechanism
to total
3.2
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
25
50
75
100
250
500
1000
Temperature oC
Radiation
Gas conduction
Solid conduction
Fig. 4: Participation of each heat transfer mechanism
to the total.
3.3
Fig. 5: Thermal conductivity versus temperature for
fibrous insulation material. Experimental (1,2
and 3) and calculation (4) results by related
study [9].
As it can be observed, for low temperature, the
thermal conductivity has fairly low values. That
happens because heat transfer through convection has
not been considered in the calculations. The real
thermal conduction should therefore be 5 – 10%
higher that the calculated one. Still, the calculated data
are quite similar to the experimental ones, which are
presented in Fig 5. Finally, the participation of each
heat transfer mechanism to the total is similar to the
data presented in Fig. 3. As it was expected, the
contribution of radiation increases with the
temperature, while the one of conduction through
fibers decreases.
Other phenomena that affect in the thermal
conductivity factor are discussed in the following
paragraphs.
Water absorption
Stone wool is a porous material, with open air spaces
in its maze. That makes it a quite hydrophilic material.
When vapor is diffused in the materials’ maze, it
replaces the air quantities. But, vapor and atmospheric
air have similar thermal conductivity factors (i.e. at 20
o
C, kair = 0.024 W/(m2K) and kvapor = 0.020 W/(m2K))
as well as some other thermo physical properties.
Therefore, the replacement of air, with vapor causes
minimal change in the total thermal conductivity
factor. On the other hand, water’s thermal conductivity
is significantly greater than air’s (at 20oC, kwater =
0.604 W/(m2K)). Therefore, the diffusion of water in
the material’s maze causes a significant increase of the
stone wool’s thermal conductivity factor. This
highlights the danger of vapor diffusing into the
insulating material, as in case of condensation it will
lead to a significant failure of the insulation. Relative
researches concluded that 1% moisture content by
volume can increase k by up to 107%. [17] In that
sense, the use of vapor barriers has to be considered,
with respect to the specific application and the
probability of vapor condensation occurring. This
problem is also associated with the decomposition of
fibers.
Fibers decomposition
Decomposition of fibers can be caused under
conditions of high temperature and high moisture. In
high temperatures, binder resins are polymerized and
stop keeping fibers connected. For increased moisture
conditions, two processes of degradation where noted:
Chemical degradation, where binder resins are
separated from the fibers and chemical hydration,
where the fibers’ surface is affected and fiber lose its
stress resistance. Finally, the previous phenomena lead
to the creation of segmented fibers, on a micro-scale,
and to the decrease of the material’s density on a
macro-scale, because fibers get detached from the
material’s main body. As a matter of fact, fibers’
degradation starts to take place at the point, where the
increase of k becomes non-linear in terms of a k-T or
k-RH(%) function. This effect, which is not rare in the
case of industrial insulation (e.g. when insulated steam
pipes are lead through closed and fairly humid areas)
leads to a significant increase in the material’s thermal
conductivity factor.
4 Conclusions
Stone wool is a powerful insulation tool for the
designer and the constructor. It is one of the few
insulating materials that can be used in a very wide
range of temperatures. Also, it can be used in many
different applications, due to the flexibility and
adaptability of form and shape. However, it is
hydrophilic and water absorption increases its thermal
conductivity factor dramatically. That would cause
damage to the insulated application. Using vapor
barriers, like protective membranes, can solve this
problem. High temperatures cause an increase of the
thermal conductivity factor. This is not straightly
tortuous, but it should be taken into consideration in
primary calculations of each individual application.
Furthermore, when temperatures exceed a certain
level, the fibers are separated; the insulating material
is damaged and should therefore be replaced. In the
case of such high temperature applications, insulation
should be replaced periodically. The methodology
presented in this paper is a simple, but still reliable,
tool which enables the assessment of the materials
performance under realistic conditions of operation,
and therefore also the prediction of possible failures.
[8]
References:
Papers
[1] Kamran Daryabeigi, AIAA 99-1044 / Analysis
and Testing of High Temperature Fibrous
Insulation for Reusable Launch Vechicles, 37th
AIAA Aerospace Sciences Meeting and Exhibit,
Reno, NV, January 11-13, 1999
[2] Kamran Daryabeigi, AIAA 2002-3332 / Heat
Transfer in High – Temperature Fibrous
Insulation, 8th AIAA/ASME Joint Thermophysics
and Heat Transfer Conferencet, St. Louis, MO,
June 24-26, 2002
[3] Walter
W.
Yuen,
COMBINED
CONDUCTIVE/RADIATIVE
HEAT
TRANSFER IN HIGH POROSITY FIBROUS
INSULATION MATERIALS: THEORY AND
EXPERIMENT, The 6th ASME-JSME Thermal
Engineering Joint Conference, March 16-20,
2003
[4] A. Milandri, F. Asllanj, G. Jeandel, J.R. Roche,
Heat transfer by radiation and conduction in
fibrous media without axial symmetry, Journal
of Quantitative Spectroscopy & Radiative
Transfer,74 (2002), p. 585-603
[5] H.J. Rice, P. Göransson, A dynamical model of
light fibrous materials, International Journal of
Mechanical Sciences, 41 (1999), p. 561-579
[6] T. Ohmura, M. Tsuboi and T. Tomimura,
Estimation of the Mean Thermal Conductivity of
Anisotropic Materials, International Journal of
Thermophsysics, Vol. 23, No. 3, May 2002
[7] L. E. Evseeva and S. A Tanaeva, INFLUENCE
OF IMPREGNATION ON THE THERMAL
PROPERTIES OF HEAT- INSULATING
FIBROUS
MATERIALS,
Journal
of
Engineering Physics and Thermophysics, Vol.
77, No. 2, 2004
[11]
[9]
[10]
[12]
[13]
Jintu Fan, Xiaoyin Cheng, Xinhuo Wen, Weiwei
Sun, An improved model of heat and moisture
transfer with phase change and mobile
condensates in fibrous insulation and
comparison
with
experimental
results,
International Journal of Heat and Mass
Transfer, 47 (2004), p. 2343-2352
V. G. Petrov-Denisov, V. S. Zholudov and V. V.
Gur’ev, A METHOD FOR ESTIMATING
HEAT-SHIELDING
PROPERTIES
OF
MINERAL WOOL INSULATION BASED ON
SILICATE FIBROUS MATERIALS, Glass and
Ceramics, Vol. 57, Nos. 9-10, 2000 (Translated
from Seklo i Keramika, No. 9, pp. 24-31,
September, 2000
Derek D. Hass, B. Durga Prasad, David E. Glass
and Karl E. Wiedemann, Reflective Coating on
Fibrous Insulation for Reduced Heat Transfer,
NASA CR-201733, pp. 24, August 1997
Markus Spinnler, Edgar R. F. Winter, Raymond
Viskanta, Thomas Sattelmayer, Theoretical
Studies of High Temperature Multilayer
Thermal Insulations Using Radiation Scaling,
Proceedings of Eurotherm 73 on Computational
Thermal Radiation in Participating Media,
Mons, Belgium, 15-17 April 2003
Papadopoulos A.M., State of the art in thermal
insulation materials and aims for future
developments, Energy and Buildings, in press
J. Hetfleish, U, Heinermanm, J. Fricke,
Experience with Vacuum Insulation Panels,
Proceedings of Nordic District Heating
Seminar, Aarhus, DK, 2 May, 2001
Books
[14] S. I. Karekos, Heat transfer – Insulation (In
Greek), Publications of Technical Chamber of
Greece, 2001
World Wide Web’s sites
[15] http://www.engineeringtoolbox.com/
22_602.html
(Air’s Specific Heat Ratio Values in S. I. Units
and atmospheric pressure)
[16] http://www.unileoben.ac.at/
~geophwww /neu/data/chapter8thermal.pdf
(Part of study published, with title :“Chapter 8:
Thermal Properties of Rocks”)
[17] www.insulation.kingspan.com/
newdiv/issues01.htm
(Kingspan insulation – issues to consider)