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Advanced Structured Materials
J. M. P. Q. Delgado
A. G. Barbosa de Lima Editors
Transport
Processes and
Separation
Technologies
Advanced Structured Materials
Volume 133
Series Editors
Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of
Applied Sciences, Esslingen, Germany
Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of
Engineering, University of Porto, Porto, Portugal
Holm Altenbach , Faculty of Mechanical Engineering,
Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany
Common engineering materials reach in many applications their limits and new
developments are required to fulfil increasing demands on engineering materials.
The performance of materials can be increased by combining different materials to
achieve better properties than a single constituent or by shaping the material or
constituents in a specific structure. The interaction between material and structure
may arise on different length scales, such as micro-, meso- or macroscale, and offers
possible applications in quite diverse fields.
This book series addresses the fundamental relationship between materials and their
structure on the overall properties (e.g. mechanical, thermal, chemical or magnetic
etc.) and applications.
The topics of Advanced Structured Materials include but are not limited to
• classical fibre-reinforced composites (e.g. glass, carbon or Aramid reinforced
plastics)
• metal matrix composites (MMCs)
• micro porous composites
• micro channel materials
• multilayered materials
• cellular materials (e.g., metallic or polymer foams, sponges, hollow sphere
structures)
• porous materials
• truss structures
• nanocomposite materials
• biomaterials
• nanoporous metals
• concrete
• coated materials
• smart materials
Advanced Structured Materials is indexed in Google Scholar and Scopus.
Editors
Transport Processes
and Separation Technologies
123
Editors
J. M. P. Q. Delgado A. G. Barbosa de Lima
CONSTRUCT-LFC, Department of Civil Department of Mechanical Engineering
Engineering Federal University of Campina Grande
University of Porto Campina Grande, Paraíba, Brazil
Porto, Portugal
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Contents
v
vi Contents
Abstract This chapter is devoted to study heat and mass transfer and dimension
variations of arbitrary-shaped porous materials. The focus is on the drying process
of clay ceramic materials. Here, different topics related to history, manufacturing,
drying process, phenomenological lumped modeling, and parameters estimation are
present and discussed. Emphasis is given to industrial clay bricks, with theoretical
and experimental approaches.
© The Editor(s) (if applicable) and The Author(s), under exclusive license 1
to Springer Nature Switzerland AG 2021
J. M. P. Q. Delgado and A. G. Barbosa de Lima (eds.), Transport Processes
and Separation Technologies, Advanced Structured Materials 133,
https://doi.org/10.1007/978-3-030-47856-8_1
2 A. G. B. de Lima et al.
1.1.1 History
The art of pottery is one of the oldest in the world due mainly to the abundance of
clay and the ease of extraction and fabrication. There is evidence of activity of this
art in almost all peoples of antiquity and to improve their quality of life, man has
always been seeking to perfect the various uses of ceramic materials.
Pottery was invented in the Neolithic (polished stone age) in 25000 BC and during
this period prehistoric man-made wicker baskets with clay, that is, the first objects
were intended to store grain and liquids and were just simple objects. Later, the
plasticity of clays was discovered, where it was noted that by adding water the clay
could be molded, dried in the sun, and hardened when exposed to high temperatures.
Following, ceramics were widely used for various purposes, such as pieces with
nozzles and handles made with relief images, or with living paintings that were
considered decorative objects (Cavalcanti 2010).
Each civilization and each culture have developed its own forms and characteris-
tics in the use of clay, so that pottery is one of the greatest auxiliaries in historical
research. One of the greatest ancient peoples who have strong ties to ceramics is the
Greeks, who for a long time produced the finest pieces in the Mediterranean world.
It was common at that time to sell these products at fairs and there was a continuous
export of generally ovoid and handled vessels (Phoenician amphora), which could
often be used to serve water, wine, and olive oil (Silva 2009).
In addition to the Greeks and Romans, other ancient peoples such as the Byzan-
tines and Arabs were responsible for transmitting their practices throughout Europe,
which consequently have varied styles of construction in their territories. It was
precisely with the growth of civil construction that the manufacture of ceramic pieces
evolved from a more artisanal activity to an industrial one. Initially, around 1850,
the first bricks were made on animal-powered molding machines, only later that the
manufacturing would go through a major leap.
Production systems were stagnant until the nineteenth century, i.e., drying was
still done in the sun, burning in trapezoidal ovens and production was still mostly by
hand. Only with the emergence of the first steam-powered machines, it was possible
to increase production as raw material extraction, preparation, and forming opera-
tions became mechanized. Thus, in the modern era countries like Spain, France and
Germany stood out in the market as producers of red ceramics and as equipment
manufacturers. It is important to highlight that Italy was one of the great pioneers in
the production of bricks in series with good quality (Silva 2016).
Later, in the mid-twentieth century, the technological development of the ceramic
industry boosted the manufacture of high strength and low weight cast structural
blocks, a major evolution compared to previously manufactured solid bricks. At this
time, including Brazil, there was a resurgence of structural masonry with ceramic
products, competing economically with conventional reinforced concrete structures
in medium-sized buildings (up to about 8–10 floors) (Silva 2009).
1 Clay Ceramic Materials: From Fundamentals … 3
Ceramic or ceramic material can be defined as any non-metallic and inorganic mate-
rial whose structure, after heat treatment at high temperatures, is wholly or partially
crystallized. They are composed of total or predominantly ionic interatomic bonds,
but having some covalent character. Ceramics are known to have different raw mate-
rials in their composition, but the main one is clay, which can be defined as an earthy,
thin, and natural material that, by adding water, acquires a certain plasticity and can
be easily molded (Callister 2007; Callister and Rethwisch 2008).
Ceramic materials have a wide range of structural arrangement types. The exis-
tence of several ceramic phases makes possible the combinations of metallic and
non-metallic atoms (which form many structural arrangements) making them widely
applicable in various sectors besides construction. It is noteworthy that the structure
of the ceramic material defines its properties (Silva 2009; Callister 2007; Callister
and Rethwisch 2008).
The ceramic industry sector plays a very important role in Brazil’s economy, with a
share of approximately 1% of GDP. Gaining prominence, the evolution of Brazilian
companies has been very fast, mainly due to the abundance of natural raw material,
alternative sources of energy, and the availability of practical technologies. Among
the regions of the country, the ones that stand out and have a large concentration of
industries are the Southeast and the South; this is because they have higher demo-
graphic density, greater industrial and agricultural activity, better infrastructure, and
better income distribution. It is noteworthy that the other regions of the country have
shown a certain degree of development, especially in the northeast due to the large
4 A. G. B. de Lima et al.
Red ceramics can generate a wide variety of products and for this it goes through
a specific production process, which is sometimes still poorly evolved compared to
other segments of the ceramic industry. However, due to the increasing emergence
of technological innovations in some companies, we can find good quality produc-
tion processes with high production rates. Most of these technological advances are
related to equipment automation and, consequently, the reduction of labor costs.
The production process, exemplified in Fig. 1.1, is common to all red ceramic
companies in general, with slight variations depending on the particular charac-
teristics of each raw material or end product. For example, some companies use
rudimentary equipment and others have more modern equipment, or some have a
much higher degree of production, among other differences.
The production process of pieces with red ceramic comprises several steps that
can be divided into four major stages, namely, extraction and preparation of raw
6 A. G. B. de Lima et al.
The manufacturing process begins with the extraction of clay, which is removed from
the deposits with the aid of backhoes and then transported to storage sheds, which
may be owned by companies or third parties. At this stage, the material goes through
a “rest”, thus undergoing chemical changes and being unpacked. Shed storage also
ensures continued production in rainy seasons. After this phase, we have the dosage,
in which the clays are proportionally dosed in a feeder coffin obeying their ceramic
characteristics.
Following the manufacturing process is disintegration, which is the step respon-
sible for bringing the hardest and most compacted clays to a disintegrator that will
crush the larger clumps of clay to facilitate subsequent operations. Then, the raw
material goes to the mixer, where it will be homogenized, thus allowing the addition
of water in the mixture to obtain adequate moisture and plasticity for extrusion.
1 Clay Ceramic Materials: From Fundamentals … 7
The last step of this first major stage is lamination, which is responsible for a
thickening of the mixture, eliminating air bubbles or clumps that may have remained
so far. With the end of this stage, the raw material already prepared can be directed
to the extruders, which may even have a rolling mill attached to them.
The mechanical conformation stage is responsible for transforming the clay plastic
mass into products with different shapes and sizes. Thus, according to the type of
product to be obtained and also depending on the plasticity characteristics of the
available raw material, it will be possible to choose the appropriate forming system.
The main systems of this stage are extrusion and cutting. Firstly, the clay mass
will take the desired shape upon entering the extruder, which contains a steel plate
perforated in a vacuum chamber. Then, through the manual or automatic cutter, the
extruded block is cut to standard sizes, thus obtaining products such as bricks, tiles,
ceramic tubes, and among others (Oliveira and Bernils 2006).
This stage consists of the drying and burning steps of the already formed parts. This
is where the composition and structure transformations will occur, generating the
final properties of the product, such as color, gloss, porosity, flexural strength, high
temperatures, and among others (Silva 2009).
During drying, a large amount of thermal energy is used to slowly and evenly
evaporate the water added during the molding process. This step usually takes place
inside drying chambers and aims to reduce the moisture content of the products from
20–25% to 3–10% after the process.
An important property of any clay is that it has water in the constitution of its
crystal lattice. Thus, during the drying process, the water that has been added is
easily removed, with the temperature starting from room temperature and reaching
approximately 110 °C. However, water that is in the clay crystal lattice will only be
removed at temperatures above 400 °C and may vary to even higher values depending
on the type of clay.
During the drying process, the clay may contract as the spaces that were occupied
by water inside the material become empty after evaporation. This shrinkage is
proportional to the degree of moisture removed. Thus, it is important to control
the process well, as a possible consequence of this shrinkage is that it can cause
deformation or cracking in the material.
Following is the firing step, in which the product is taken to a kiln and, as well
as drying, will receive an even greater amount of thermal energy. Once these steps
are completed, the product will have lower porosity and greater mechanical strength
and will also be ready for commercialization and use.
8 A. G. B. de Lima et al.
1.1.4.4 Expedition
Shipment is the final stage of the production process, where finished product is
inspected to identify excessively cracked, broken, chipped, or burned products. Then,
the parts are stored in a covered area until they leave for delivery to the customer. In
Brazil, transportation of the parts is usually made by trucks on the highways of the
country.
The thermal processing stage must be performed correctly, otherwise the parts
could present a series of defects and thus, the products will not be able to perform
their respective functions. Given this, the most common defects are as follows (Silva
2009; Silva et al. 2011):
(a) Commitments—This defect is a deformation of the part usually caused by
residual shrinkage stresses, which arise when one side of the material dries faster
than the other, i.e., it is important that the drying is done evenly. Commitments
may also arise due to poor positioning of the product on the drying support.
(b) Cracks—It is important that during the drying process the air velocity and
temperature are controlled, because when we have a very fast drying, it is
common the appearance of cracks, which are nothing more than small fissures
that start at the edges and spread until the center of the piece. Cracks may also
appear in the firing step, which may be by heating or cooling. The heating ones
are characterized by being open, little winding, and with jagged edges, while the
cooling ones are characterized by being closed and very thin, usually S-shaped
edges. It is important to point out that all drying starts must be done with the
plastic-covered part, to prevent a very fast outflow of water that is closer to the
surface, causing a localized shrinkage that can cause cracks.
(c) Black heart—This type of defect is black or gray spots that can be seen along the
cross section of the part and appear after the firing process. The existence of the
“black heart” is associated with the presence of carbon-containing compounds,
which are formed due to the small amount of oxygen, preventing the complete
oxidation of carbon compounds and organic matter.
(d) Efflorescence—Efflorescence occurs on the outer surface of the product and
is a salt deposit accumulated in some regions, which may cause undesirable
stains and colors. This defect appears as the water interferes with salts. If the
piece, after burning, absorbs moisture, the salts will be dissolved; however, if
the external environment becomes dry, the opposite process occurs, the surface
water is evaporated and the crystallization of the salts occurs.
(e) Defects related to steps before or after drying—It is common for small cracks
to occur when the clay paste is improperly mixed in the mixing step. This defect
is most pronounced in areas with higher moisture content and is quite common
in manual manufacturing processes. Finally, it is worth mentioning the problem
of moisture absorption. Depending on the type of clay, if the time elapsed from
the clay leaves the dryer to when it is introduced into the kiln is large and the
ambient absolute humidity is very high, a rehydration (reabsorption) process
may occur, which may cause breakage and/or explosion when material enters
the kiln.
1 Clay Ceramic Materials: From Fundamentals … 9
vapor is greater than the partial pressure of water vapor. Therefore, the ability of air
to absorb water vapor increases with temperature, so that the higher the air temper-
ature, the greater its drying capacity, in fixed conditions of the air relative humidity.
In addition, if the air is warmer, the volume of air needed for drying decreases and,
as a result, the powers of the hoods and air circulators are reduced, reducing drying
costs.
The speed with which the product is dried can be affected by many factors, such
as moisture movement mechanism, product shape, external environment conditions,
and green product porosity. Thus, it is of great importance to verify the influence
of the shape and volume of the pores in the part, because it is inside them that
is the moisture, that even under favorable conditions can be retained inside these
pores. This occurs when the surface of the part is dried very quickly, as the pores,
being very narrow, reduce moisture migration for a rate less than the evaporation
rate. Another important point is that with a higher drying air temperature and lower
relative humidity there will be an increase in drying rate.
The drying process is generally divided into four distinct phases: adaptation,
colloidal water outlet, void formation, and interstitial moisture expulsion. In the first
phase occurs the adaptation of the product to environmental conditions (tempera-
ture, relative humidity, and pressure), in which drying will be performed. In the
second phase, there is evaporation of the colloidal water, and sensible variations in
the dimensions of the part occur due to the approximation of the particles of its
microstructure. Even at this stage water continually migrates to the surface of the
part, constantly forming an evaporating saturated wet film. In the third phase occurs
the disappearance of the water film on the surface of the piece, which provokes
changes in color. The last drying phase, which is not always reached in the dryers
and is often performed in the kilns, is the expulsion of the last amounts of moisture
from interstitial origin, in which the moisture removal rate decreases to near zero
(Silva 2009).
Given the importance and complexity of the drying process, a large number of
researchers have been working intensively on its analysis. Some focus on external air
conditions, such as temperature, relative humidity, and velocity, correlated with the
product’s drying rate, while others consider the internal conditions of the product,
with emphasis on the mechanisms of moisture movement and their effects on it. In
this regard, several drying theories have been proposed to describe heat and mass
transport in capillary porous media, namely,
(a) Liquid diffusion theory;
(b) Vaporization–condensation theory;
(c) Cappilary theory;
(d) Kricher’s theory;
(e) Luikov’s theory;
(f) Philip and De Vrie’s theory;
(g) Berger and Pei’s theory;
(h) Fortes and Okos theory.
1 Clay Ceramic Materials: From Fundamentals … 11
The complexity of the drying process depends on the geometric and thermophys-
ical parameters of the material and thickness of the material layer in study. They
can then be classified in thin-layer drying models (particle level models) and thick-
layer models (dryer models). The dryer mathematical models (thick-layer model)
most used by the researchers take into account the thermophysical properties, drying
kinetics, and mass and energy balance in the device. Some researchers have applied
dryer model to predict drying process of clay ceramic materials with particular refer-
ence to industrial clay bricks (Almeida et al. 2013; Tavares et al. 2014; Almeida et al.
2016; Silva 2018). From a practical point of view, thin-layer drying is very limited.
But to have a good understanding of the thick-layer drying process it is necessary to
have thin-layer equations for the drying kinetics of a particular material under certain
predetermined operating conditions (Macedo 2016).
Several thin-layer mathematical models have been proposed to describe the rate
of moisture loss during drying and can be divided into two large groups: lumped
and distributed models. Distributed models express heat and mass transfer rates as
a function of position within the part and drying time, taking into account external
and internal resistances. Lumped models, on the other hand, express the same rates
only as a function of process time and ignoring the existing internal resistance for
heat end mass transfer.
The following general balance equation (distributed model) has been applied to
predict drying process (by diffusion only) of irregularly-shaped porous body:
d(λΦ)
= ∇ · (Γ Φ ∇Φ) + Φ (1.1)
dt
where λ and Γ Φ are transport properties. Φ is the unknow, Φ is the source term,
and t is the time.
Distributed models based on the liquid diffusion theory have been applied to
predict drying of ceramic porous materials. For example, clay plates (Silva et al.
2009), clay pipes (Santos 2018), roof tiles (Farias et al. 2013; Silva et al. 2012;
Farias et al. 2012), and bricks (Araújo et al. 2019a, b, 2017; Brito et al. 2017; Araújo
et al. 2017; Silva et al. 2011; Lima et al. 2015; Santos et al. 2020).
This chapter addresses the use of the lumped model to describe the drying process.
The equations of the lumped model can be classified as empirical, semi-empirical,
and theoretical. It is noteworthy that in this analysis the effects of temperature and
moisture variation inside the material are neglected during the process.
When it comes to empirical equations, they have a direct link between mois-
ture content and drying time, while semi-empirical ones are analogous to Newton’s
law of cooling, assuming that the drying rate is proportional to the difference
between moisture content of the product and its equilibrium moisture content for
the specified drying conditions. Theoretical equations generally use heat and mass
balances between the product and air surrounding it, taking account different phys-
ical phenomena during the process. Some researchers have applied lumped models
to describe drying process of clay porous materials. For example, clay pipes (Silva
et al. 2016), bricks (Silva 2009; Almeida et al. 2013; Tavares et al. 2014; Silva 2018;
1 Clay Ceramic Materials: From Fundamentals … 13
Fig. 1.2 Representative scheme of the drying process of an arbitrarily-shaped solid based on a
lumped analysis
Silva et al. 2011; Lima et al. 2015), and others geometries (Silva et al. 2016; Lima
et al. 2018; Lima 2017).
For a better understanding of the lumped analysis method (theoretical model),
consider the solid with arbitrary geometry, illustrated in Fig. 1.2.
In this scheme, the arbitrary solid will receive on its surface a flux per unit area
of the potential of interest Φ and has uniformly distributed internal generation per
unit volume. According to what has already been mentioned, when applying the
lumped analysis method, the effects of the potential variation within the material are
neglected. Thus, all flux of Φ received and generated will diffuse instantly through the
solid. In order for this condition to be physically possible and well approximated, the
flux resistance within the solid must be much lower than the flux resistance between
the solid and its vicinity.
Thus, the balance of Φ (potential of interest) can be obtained as follows:
d λΦ
V = Φ S + Φ V (1.2)
dt
in which Φ and Φ are flux of Φ per unit area and source term, respectively. Further,
λ includes transport parameters and S and V are the surface area and volume of the
porous material, respectively.
As an application, in this topic will be developing new research methods and tech-
niques, particularly process modeling and simulation involving heat and mass trans-
port in solid–liquid systems, with particular reference to drying of clayey ceramic
materials, via lumped models.
14 A. G. B. de Lima et al.
The materials used for drying in oven were parallelepiped-shaped ceramic bricks with
8 rectangular holes (industrial ceramic bricks). Figure 1.3 illustrates the test body
model used, as well as the positions where the measurements of length (R1 ), width
(R2 ), height (R3 ), and dimensions that characterize the brick holes, a1 , a2 , a3 , and
a4 , were obtained. Initially, dimensions were measured with a digital caliper, mass
with a digital scale, brick temperature (vertex) with infrared thermometer, and room
temperature and relative humidity with thermohygrometer. Then, the samples were
taken inside the forced-air oven where drying was performed. In this process, the
internal temperature of the oven was set as desired with the temperature controller. At
predefined intervals, the brick was taken from the oven and measured its temperature,
mass, and dimensions.
Table 1.2 summarizes, for each experimental condition, the product, and air data.
Table 1.3 presents, for each operating condition, the dimensions, volume, and surface
area of the sample before the drying process begins.
During the process, measurements were taken every 10 min until the mass had
minimal variation. Then, the measurements were changed every 30 min, and the next
measurements were taken every 60 min until it reached constant mass. Soon after, the
Table 1.2 Experimental air and brick parameters for each drying test (Silva 2009)
T (°C) Air Brick Time, t (h)
UR (%) V(m/s) M o (db) M f (db) M e (db) θ o (°C) θ f (°C)
50 80 0.05 0.13969 0.0 0.00011 20.6 41.0 18.5
60 79 0.06 0.14795 0.0 0.00268 20.5 50.2 13.7
70 69 0.07 0.15414 0.0 0.00076 26.0 64.5 17.8
80 66 0.08 0.15248 0.0 0.00039 21.4 69.2 15.0
90 68 0.09 0.15921 0.0 0.00151 21.0 78.5 11.5
100 52 0.10 0.16903 0.0 0.00038 26.1 93.2 12.3
sample was dried for 24 h at the same drying temperature to obtain the equilibrium
mass and then, for another 24 h at 105 °C to obtain the mass of the dried product.
All experiment was performed by Silva (2009). This author also performed an
adjustment of experimental data related to mass transfer (moisture content) during the
process and proposed an exponential equation with two terms and four parameters.
The equation has the form:
Table 1.3 Brick dimensions before the drying process begins (Silva 2009)
T R1 (mm) R2 (mm) R3 (mm) a1 (mm) a2 (mm) a3 (mm) a4 (mm) V o (mm3 ) S o (mm2 )
(°C)
50 93.36 197.00 200.00 9.04 7.10 7.88 6.30 141,5643.80 371,100.44
60 92.75 195.00 200.00 8.34 7.32 7.11 6.45 1,367,269.30 369,020.69
70 93.16 197.00 203.00 8.54 9.87 7.99 6.96 1,621,580.85 162,158.85
80 92.76 197.00 201.00 8.16 7.20 7.84 6.66 1,408,074.95 37,214.46
90 93.10 197.00 201.00 8.88 7.95 6.57 6.78 1428,426.08 37,233.87
100 92.80 198.00 202.00 1.70 9.41 8.74 8.00 1,734,026.10 36,116.49
A. G. B. de Lima et al.
1 Clay Ceramic Materials: From Fundamentals … 17
Table 1.4 Parameters of Eq. 1.3 obtained after fitting to experimental data of average moisture
content
T (°C) Parameter R (–) Explained
A1 (–) k 1 (mm−1 ) A2 (–) k 2 (mm−1 ) variance (%)
Table 1.5 Parameters of Eq. 1.4 obtained after fitting to experimental data of the vertex temperature
T (°C) Parameter R (–) Explained
B1 (°C) B2 (°C/min) k 1 (–) B3 (min) variance (%)
From the various measurements of the brick dimensions, made during the drying
process, mathematical equations were proposed to calculate the volume and surface
area of the brick (Fig. 1.3). The brick volume at any time t was calculated as follows:
The brick surface area at any time t was determined by using the following
equation:
After determination of the volume and surface area at different moments of drying,
it was possible to adjust them to mathematical models that describe the volumetric
variation and surface area of the brick during the drying process. This procedure was
realized by using Statistica® software (Simplex numerical method and convergence
criterion of 0.00001). For this, a third-degree polynomial model was proposed for
both volume and surface area, as follows:
V (t) = C1 t 3 + C2 t 2 + C3 t + C4 (1.10)
S(t) = D1 t 3 + D2 t 2 + D3 t + D4 (1.11)
The complexity of the drying process depends, among other parameters, on the
analysis taken into account. Distributed models express heat and mass transfer rates
as a function of position within the part and drying time, taking into account external
and internal resistances. Already the lumped models express the same rates only as
a function of the process time and ignoring the existing internal resistance for this
transfer. This study makes use of the lumped model analysis to describe the drying
process of ceramic brick. Thus, from Eq. 1.2, we have the following mass balance:
dM
V = −h m S(M − Me ) + V Ṁ (1.12)
dt
where S and V represent the surface area and volume of the solid at any time t, hm
is the convective mass transfer coefficient, M is the average moisture content, M e is
the equilibrium moisture content of the brick, and t is the time.
Considering M = M − M e , it is valid dM = dM. Therefore, it is possible to
write:
1 Clay Ceramic Materials: From Fundamentals … 19
dM
V = −h m S M + V Ṁ (1.13)
dt
Separating the variables and rearranging the terms, Eq. 1.13 results in:
dM hm S
=− dt (1.14)
(M ) − h m S
V Ṁ V
Since that M = M 0 at t = 0, and that there are no reactions that can generate water
inside the product, it was considered Ṁ = 0. So, Eq. 1.14 can be integrated from the
initial condition. Thus, it is possible to write:
M−Me t
dM hm S
=− dt (1.15)
(M ) V
M0 −Me 0
Putting Eqs. 1.10 and 11.11 into Eq. 1.15 and integrating it, we obtain the following
equation, which defines the mass transfer, considering dimensional variations during
the process:
⎧ ⎡ ⎧ ⎫⎤⎫
⎨ ⎨ a 1 t + a 2 arctan a 3 a 4 + 2t + a 5 log a 6 − t ⎬ ⎬
M = (M0 − Me ) exp⎣−h m ⎦ + Me (1.16)
⎩ ⎩ + a 7 log a 8 + a 9 t + t 2 − a 10 ⎭ ⎭
Similarly, to mass transfer, for heat transfer analysis, considering constant the heat
flux per area unit, the following energy balance is given:
dθ
ρV Cp = [h c S(θ∞ − θ )] + q̇ V (1.17)
dt
where ρ and C p represent the density and specific heat of the brick, respectively,
hc is the convective heat transfer coefficient, θ and θ ∞ represent, respectively, the
average product temperature at any time t and the equilibrium temperature (which
is equal to the drying air temperature).
Considering T = θ∞ − θ , it turns out that dT = −dθ. Then, putting this result
into Eq. 1.17, separating the variables, this equation can be rewritten as follows:
dT hc S
=− dt (1.18)
T + q̇ V ρV Cp
(h c S)
20 A. G. B. de Lima et al.
Since that θ = θ 0 at t = 0, and that there are no chemical reactions that can
generate heat inside the product, it is possible to consider q̇ = 0. So, Eq. 1.18 can be
integrated from the initial condition. Thus, we have that:
θ∞ −θ t
dT hc S
=− dt (1.19)
[T ] ρV Cp
θ∞ −θo 0
Now, putting Eqs. 1.10 and 1.11 into Eq. 1.19 and integrating it, we obtain as results
the following equation, which defines the heat transfer, considering dimensional
variations during the process:
⎧ ⎡ ⎧ ⎫⎤⎫
⎪ ⎪
⎨ b1 t + b2 arc tan b3 b4 + 2t + b5 log b6 − t ⎪
⎬ ⎪
⎨ ⎬
⎢ hc ⎥
θ = θ∞ − (θ∞ − θo ) exp⎣− ⎦
⎪ ρCp ⎪ ⎪
⎭ ⎪
⎩ ⎩ + b7 log b8 + b9 t + t 2 − b10 ⎭
(1.20)
Tables 1.6 and 1.7 summarize the parameters obtained for Eqs. 1.10 and 1.11,
respectively.
Table 1.6 Parameters of Eq. 1.10 that describe the volumetric behavior of the brick during drying
process
T (°C) Parameter R (–) Explained
C1 (m3 /min3 ) C2 (m3 /min2 ) C3 (m3 /min) C4 (m3 ) variance (%)
Table 1.7 Parameters of Eq. 1.11 that describe the surface area behavior of the brick during drying
process
T (°C) Parameter R (–) Explained
D1 (m2 /min3 ) D2 (m2 /min2 ) D3 (m2 /min) D4 (m2 ) variance (%)
Table 1.8 summarizes the coefficients of Eqs. 1.16 and 1.20. With this, it was possible
to adjust these equations to the experimental data of moisture content (Eq. 1.3) and
surface temperature (Eq. 1.4) and to estimate the convective heat transfer and mass
transfer coefficients.
Figures 1.6 and 1.7 illustrate a comparison between the predicted and experimental
brick average moisture content as a function of time for drying at 50 and 100 °C,
respectively.
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Itse baretti näytti vain täydentävän tuon kauniin, puhtaan pään
muotoa, ja värit hänen yksinkertaisessa puserossaan, joka ulottui
korkealle kaulaan, pieni hieno pellavareunus koristeenaan, ja hänen
kapeissa housuissaan olivat vain tukan oman värin vivahduksia.
"Miksi niin?"
VIININKORJUU.
Joka kerta kun hän tuli kotiin, alkoi hänen sydämensä sykkiä
näiden korkeitten, harmaitten muurien edustalla, joissa oli vain yksi
ainoa pitkä, kapea oviaukko, minkä kautta sekä ihmiset että ratsut
kulkivat sisään, ja harvoja, yksinäisiä ikkunoita, jotka vaanivat kuin
sisäänpainuneet silmät.
*****
*****
Sellaisena näkivät nuo kolme nuorta miestä hänet, kun hän pari
päivää myöhemmin ratsasti isänsä kanssa Poggibonsin huvilaan
aloittamaan viininkorjuuta. He piiloutuivat Gentilen huvilamuurin
taakse. Veri läikehti heidän poskillaan, kun he kurottuivat katsomaan
ohikulkevan joukon jälkeen. Iso keltainen olkihattu oli tytöllä
niskassa, ja se loisti kuin pyhimyskehä.
"Ne ovat sanoja, joista teidän itse pitäisi punastua", sanoi Gentile.
"Naiskauneutta ei saa häväistä, vaan sitä on palvottava."
*****
He puhelivat rakkaudesta.
Rinaldo oli hiljattain tullut kotiin Veneziasta, missä hän oli oppinut
tuntemaan rakkauden hurjimmat mysteeriot. Hän kertoi
gondolamatkoista kapeissa kanavissa, silmistä, jotka salamoivat
felzin varjosta, naisista, jotka istuivat espanjanruokokaarten alle kuin
loistavat paratiisilinnut avoimissa häkeissä, ja kahden gondolan
liukuvasta kosketuksesta, ikäänkuin kaksi kättä hiljaa sivelisi toisiaan
sivumennen. Tulisia katseita mustista naamareista. Vilahduksia
parvekkeilta, joilta sataa ruusuja soutajien päälle. Aallonloisketta,
pehmyttä kuin suutelo. Solakoita, kamelikurjen sulilla koristettuja
maureja, jotka ohjaavat gondolaa purppuranvärisellä airolla.
Rakkautta etsivien kuuma virta Rialtosillalla. Öitten seikkailurikkaita
serenaadeja tummien palatsien varjossa. Kuu, ikäänkuin mykkä
vakooja, joka oikealla hetkellä sulkee silmänsä hopeanvalkeilla
pilvillä. Nuoraportaat loiskahtavat veteen. Keinuva nousu
parvekkeelle. Rinaldo ei ollut jättänyt ainoatakaan ruusua
makaamaan, ei yhtään katsetta vastaamatta.
GIOVANNAN VAPAUS.
"Jos haluatte, että pidän teitä jaloina herroina, jotka tietävät, mikä
sopii ja mikä ei, niin ei teidän pidä puhua valheita suuresti
rakastetusta isästäni. Hän on yhtä vähän ovenvartijani kuin tyrannini.
Omasta vapaasta tahdostani minä kieltäydyn seurustelemasta
sukuni vihollisten kanssa."
*****
NOLI ME TANGERE.