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Theoretical Study of Stress Reversal Phenomena in Drying of Porous Media.

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Dev. Chem. Eng. Mineral Process., 10(3/4), pp. 261-280, 2002.
Theoretical Study of Stress Reversal
Phenomena in Drying of Porous Media
Stefan Jan Kowalski"
Poznari University of Technology, Institute of Technology & Chemical
Engineering, pl. Marii Sklodowskiej-Curie 2, 65-960 Poznari, Poland
Drying of capillary porous materials may generate internal stresses, the magnitude of
which depenh on both the drying techniques and the mechanical properties of dried
materials. Acoustic emission (EA), a method to monitor for damage control during
drying, shows an enhanced emission of acoustic impulses and energy fiom the
material in some stages of drying. A particularly interesting phenomenon is the
occurrence of enhanced emission of the acoustic impulses at thefinal stage of drying,
that is, at the time when the drying induced stresses are expected to disappear. This
phenomenon is explained by the stress reversal, i.e. the change of stress signs fiom
negative (compressive) inside the material at the beginning to the positive (tensional)
at the final stage of drying. A theoretical analysis is presented in this paper using the
viscoelastic model. Stress reversal was predicted and analyzedfiom drying cylinders
of ceramics-like material and wood.
Introduction
Drying of moist capillary porous materials usually results in shrinkage along with
decrease of moisture content or increase of temperature. The non-uniformity in
moisture and/or temperature distributions may generate internal stresses whose
magnitude depends on the gradients of moisture content and/or temperature. If the
small thermal stresses are neglected, then the meaningful stresses due to the moisture
* (email: kowal@rose.man.poznan.pl)
261
S.J.Kowalski
removal are tensional at the surface and compressive in the core of the dried material,
at least during the first stage of drying. Further drying may change the signs of the
stresses both at the surface and in the core.
The tensional stresses are likely to be responsible for the destruction of dried
materials. One can expect damages of material structure at the surface when the
tensional stresses reach their critical values, and inside the dried material after stress
reversal, i.e. when the core is also under tensional stress.
The reverse of the drying-induced stresses can be detected by the acoustic
emission method (EA) used to monitor drying processes [3,4, 81. One can observe the
number of acoustic signals per time increment that indicates the intensity of
destruction, or the magnitude of elastic energy transported by individual signals that
shows the dimension of the destruction. An enhanced emission of the impulses at the
final stage of drying, i.e. at the time when the drying induced stresses are expected to
disappear, suggests the reversal of the stress signs and appearance of tensional
stresses inside the dried material as found in [8, 101.
The aim of this paper is to show theoretically that the sign of the stress may
change at the final stage of drying. The Maxwell viscoelastic model of drying is used
[5,6]. The stress reversal during drying is illustrated for convective drying of ceramic
and wood samples in the form of a cylinder.
Brief description of the drying model
The thermomechanical model of drying, used in this paper in the analysis of the
drying induced stresses, was developed on the basis of mechanics of continua [5, 6,
111. The dried material is assumed to be a capillary porous medium, the pores of
which are filled with liquid containing gas bubbles. The gas bubbles grow during
drying, so at the final stage of drying the pore space is filled with gas and the
funicular liquid on the pore walls. A number of assumptions,justified from the drying
technology point of view, were applied. The most important ones are:
0
neglecting the accelerations and the kinetic energies of the individual
constituents;
assuming the vapor in the pore gas bubbles to be in the saturated state;
262
Stress Reversal Phenomena in Drying of Porous Media
0
taking the temperatures of all constituents to be equal in the infinitesimal volume
of the body;
0
neglecting the amount of air in pore gas bubbles.
If p a denotes the mass of a constituent, and a = s (skeleton), I (liquid) g (gas) per
unit volume of the body as a whole, then the mass fiaction of the individual
constituent referred to the dry body is: Xu = p a / p S. As the specific volume of water
vapor is a thousand times larger than the specific volume of water, it can be assumed
that X' > > ~ g .
The equations of mass continuity for skeleton and constituent a,obtained 6om the
mass balance, take the form:
p s + pSvv,"j= 0 ,
where
p s g . = -qi + p"
p a represents the rate of mass production per unit volume associated with
phase transition of liquid into vapor; V: = q.is the porous solid velocity; and Uf is
the mass flux of a constituent with respect to the porous solid.
The equation of momentum balance, under the above mentioned assumptions,
becomes the equation of internal equilibrium of surface forces represented by the
stress tensor
oijand the body forces:
where p denotes the total mass density of the body, and g, is the gravity acceleration.
Balance of energy (equilibrium conditions) implies Gibbs' identity of the form:
263
S.J. Kowalski
and the entropy equation:
a
a
where e and s denote the internal energy and entropy of dried material per unit mass
of the dry body; T is the absolute temperature;
sa and p a denote the entropy and the
free enthalpy (chemical potential) of constituent a; qiis the heat flux conducted to the
dried body through its surface; r denotes the volumetric heat supply (radiation), then:
is the strain tensor, being the sum of reversible and irreversible strains, and ui is the
displacement vector of the porous solid (skeleton).
The balance of entropy and the second law of thermodynamics give the inequality
of the form
or, after substitutingequations (4) into (6), one gets:
a
264
a
Stress Reversal Phenomena in Drying of Porous Media
The individual terms in the above inequality have different tensor representation. In
order to fulfil the above inequality, we assume it to be a positively defmed quadratic
form and to take into account the Curie’s principle by construction of the constitutive
relations for the respective thermodynamic fluxes.
The Heat and Mass Transport Equations
In order to satisfy the inequality (7), the sufficient (not necessary) conditions are
applied. Thus, we suggest the following relations between the respective fluxes and
the thermodynamic forces:
Relation (8) expresses the generalized Newton-Cauchy-Poisson’s law for plastic
or viscous bodies [12], where
@ is the tensor of material constants for plastic or
viscous deformations. Relation (9), known as the Fourier’s law, contains the total heat
flux that consists of the heat conducted and the heat transported by the mass fluxes
moving with different velocities with respect to the solid. Relation (10) point out that
the mass fluxes of the individual constituents are proportional to the gradients of
constituent potentials pu and to the gradient of gravitational potential p?
= -gi.
In the case of liquid creating a concave meniscus in pore space, the liquid pressure
under the meniscus surface p ‘ differs from the liquid pressure under flat surface
by the capillary pressure pCLp,
that is
p‘
= p:
p:
+ pa’. It means that potential p’ also
265
S.J. Kowalski
contains the capillary potential defined as pap = pap/ p “ , where plr denotes the
true liquid density.
Relation (1 1) indicate that the difference in chemical potentials of liquid and gas
(strictly: vapor in gas) determine the rate of phase transition of liquid into vapor. The
procedure of the determination of the coefficient w was given [ 11.
Physical Relations
We express Gibbs’ identity (3) with the help of the free energy function: f = e - sT,as
the temperature is designated to be the natural thermodynamic parameter of this
function, and thus it is convenient to consider the isothermal processes as, for
example, the constant drying rate period. Besides, it was mentioned earlier that the
conventional drying processes could be considered as being running close to the
thermodynamic equilibrium states. This allows us to assume the chemical potential of
liquid and gas inside the dried body to be equal, that is, p‘
= p g = p [2]. The
Gibbs’ identity may now be rewritten as:
We assume the free energy to be a function of the reversible strains
&hr), the
temperature T,and the moisture mass fiaction X = X‘ + Xg,where X‘ >> Xg.The
irreversible strains did not appear in the Gibbs’ identity as the work of stresses on
these strains is assumed to be dissipated totally and converted into heat. This work
appears in the entropy equation (4) as a source of heat, and manifests itself in Gibbs’
identity (12) through the increase of temperature.
Applying the chain rule expansion for the time derivative of the fiee energy and
comparing the obtained time derivative with those of (12) one obtains the equations of
state, which in developed form will yield the physical relations. The physical relation
between stress, strain, temperature and moisture mass fraction takes the form [6]:
266
Stress Reversal Phenomena in Dtying of Porous Media
where
c# is the tensor of material constants for reversible strains, and:
is the tensor of thermal-humid strains, where
ICE)and
Kf) are the coefficients of
linear thermal and humid expansion, and 8 = T - To and 8 =
x - xodenote the
relative temperature and moisture mass fraction, respectively.
Maxwell’s model is constructed as the sum of elastic and viscous strains (5). The
physical relation (8) and (13), inverse with respect to strains, substituted into ( 5 ) will
yield the Maxell model of the form:
where Dii = [C$p and D# = [CtA!’
are the tensors of elastic and viscous
compliance.
The equation of force equilibrium (3) and the physical relation (15), and in the
case of significant changes in porosity also the equation of mass continuity (2),,
constitute the basic equations for the analysis of deformations of the dried body. The
equations of mass balance for the moisture and balance of energy, together with the
constitutive relations for mass and heat fluxes, constitute the basic equations for
determination of the temperature and moisture content distribution. Thus, the equation
of mass balance ( 2 ) ~,after substitutingthe mass flux (1 l), can be written as:
where potential p depends on the current thermodynamic state, that is:
where p o is the reference potential. The coefficient
4;) express the permeability of
the medium in Darcy’s sense, i.e. it contains both the geometrical permeability and
267
S.J. Kowalski
the fluid physical properties (viscosity). It can be determined experimentally through
the measurement of the fluid discharge under the action of gravitational force.
The other coefficients,
C(*),
yf’
and
c(x)
, determine the
influence of
additional ingredients on the moisture flow inside the dried material, namely, the
temperature, the actual strain, and the moisture content. The first of the three
coefficients is determined from the measurement of the moisture gradient generated
by the temperature gradient. The second one follows from the symmetry in physical
relations, required for the evaluation of the coefficient of humid expansion (swelling
or shrinkage coefficient) and the elastic modules of the porous solid. The third one is
a measure of capillary uplift, and can be estimated from the equilibrium condition
between the capillary and gravitationalpotentials.
The temperature distribution is determined from the entropy equation (4),which
after utilizing the physical relations for heat and mass fluxes and irreversible strains,
will reduce to:
+
where Q = c$..&,
(i) (i)&kl(0 &)(p;
- gi
- gj)denotes the heat source due to
the irreversible work of stresses on the irreversible strains, and due to the irreversible
moisture flow. It is a non-linear term contributing only slightly to the temperature rise
in comparison with the external heat supplied (conduction or radiation).
The entropy of the dried body is a function of state, and has the following form
where so denotes the reference entropy; c, denotes the specific heat at constant volume
per unit mass of the body; and the remaining coefficients follow the symmetry
conditions in physical relations as being the derivatives of the fiee energy equation.
The material coefficients depend, in general, on the parameters of state and in
particular on the moisture content.
268
Stress Reversal Phenomena in Dving of Porow Media
The fact that the common drying processes operate close to the thermodynamic
equilibrium allows us to neglect the phase transitions inside the body and assume that
they take place mainly at the surface. Thus, the latent heat of evaporation appears in
the boundary condition for the convective heat exchange.
Solution for an Isotropic Cylinder
The analysis of stresses in a ceramic cylinder dried convectively is presented below.
Because the ceramic-like materials suffer most shrinkage and experience maximal
stresses during the constant drying rate period, the analysis will be confined to this
period only. During the constant drying rate period, the temperature of the dried body
is constant and equal to the wet bulb temperature.
Maxwell's physical relation of equation (15) for isotropic material is reduced to:
Afier splitting the strain and stress tensors into the deviatoric and spherical parts, the
above relation can be written as:
where s,, is the deviator and CT the spherical part of the stress tensor; e,, is the deviator
of the strain tensor; E is the volumetric strain; K = A + 2M/3 is the elastic modulus and
K = 77
+ 2 x / 3 is the viscous bulk modulus for the volumetric strains; A and Mare
the elastic, and
x
and q the viscous counterparts, of Lame constants for dried
material. In general, one assumes that the porous material may suffer also a
volumetric viscous flow, because of the void space inside.
For simplicity, we assume here that the relaxation times for the stress deviator
7=
Mq and for the spherical stress so = W Kare the same. Making use of the Laplace
269
S.J.Kowalski
transforms, we can express the total viscoelastic stress (v) by the elastic one (e) with
the help of Borel’s convolution:
The numerical solution for an elastic cylinder is carried out under the assumption
that the displacement in radial direction depends on radius r and time
t,
but the
displacements in the other directions are equal to zero. Having in mind that both the
radial displacement in the middle of the cylinder (r = 0) and the radial stress on the
external surface of the cylinder (r = R) equal zero, the solution for the displacement
and for the radial and circumferential stresses are obtained in the following form:
where
=~
&>e
( ~ +~ 8 ,(see equation
14); and
8, = const is the wet
bulb temperature determined by the drying medium parameters, that is:
In the above formula 8, is the temperature of drying medium; a m a n d a, are the
coefficients of convective heat and mass exchange between the dried material and the
2 70
Stress Reversal Phenomena in Drying of Porous Media
drying medium; p,, and pa are the vapor chemical potentials in the drying medium
respectively close to and far from the dried material boundary; and I is the latent heat
of evaporation.
The distribution of moisture content is determined from the differential equation
(1 6), after substituting the physical relation for the moisture potential (equation 17)
then:
The volumetric deformation, being directly proportional to the moisture content, also
influences the above equation but only through the diffusion coefficient K,,,,
which
has the form:
The second term in the square brackets, which represents the influence of the volume
change, is approximately four orders smaller than for d4,and therefore is negligible.
The boundary conditions for moisture flow are:
where
8, = (pa - p,, + dT)8m)/c(x);
A = a m /A('); and a,,,
is the coefficient
of the convective mass exchange.
The initial condition is
8(r,t)l,=,=8, . The solution to this initial boundary
value problem is:
2 71
S.J.Kowalski
In the above formula an is the n-th root of the characteristic equation
c d 1(aR)= u
o (d),
where Jo and 5, are the Bessel functions of order zero and
of the frst and second kind. The Bessel fbnctions satisfy the orthogonality condition
of the form:
Now, having the solution for moisture distribution, one can evaluate the evident
forms for stress distributions in elastic cylinder, equations (24) and (25), and fmd the
stresses for a viscoelastic cylinder using the convolution (22). Figures 1 and 2 present
the distributions of radial and circumferential stresses along the radius for some
instants of time.
-0,4 4
I
Figure 1. Distribution of radial stresses in ceramic cylinder at various instants.
2 72
Stress Reversal Phenomena in Drying of Porous Media
{
0,8
Figure 2. Distribution of circumferential stresses in a ceramic cylinder at
various times.
Stages of drying [min]
Figure 3. The number ofAE impulses in a given range of time.
2 73
S.J.Kowalski
Figures 1 and 2 illustrate the characteristic development of the stresses. The
cylinder is stress free at the beginning of drying. As the process proceeds, the stresses
rise in parallel to the non-uniformity of the moisture distribution, attaining a
maximum at some instant. When the moisture distribution becomes more uniform, the
stresses start to decrease. Based on the elastic model they would approach zero, but
this is not the case for the viscoelastic model. According to the viscoelastic model the
stresses first change their sign attaining again a maximum, or else then approaching
zero. Arising of the tensional stresses inside the material at the final stage of drying
may explain the appearance of the enhanced acoustic emission (EA) in this time
period (see Figure 3).
Solution for a Wood Cylinder
Wood is an anisotropic material and its drying is rather more complex than for
ceramic-like materials. Therefore, its modeling is much more difficult. The stresses in
wood are generated mainly in the falling rate period, that is, below the fiber saturation
point. In this period, the temperature of wood rises fiom the wet bulb temperature to
the temperature of the drying medium.
Physical relations in the elastic range for wood, as a material of orthotropic
structure, were presented in [7, 91. As the strength of wood is the weakest in radial
and tangential directions with respect to the rings of annual growths, we shall analyze
here the radial and circumferential stresses. The Maxwell physical relations for wood
in the form of a cylinder are given by equations (32 a and b). The material coefficients
(Poisson’s v O and Young’s E,) are functions of moisture content in the range fiom
zero to the fiber saturation point, and satisfy the following symmetry conditions
V,,
2 74
/E, = V,, / E t ,withzi denoting the relaxation time in the i-th
direction.
Stress Reversal Phenomena in Drying of Porous Media
Based on the experimental data for birch wood [8], one has established the
following dependence of Young modules and Poisson ratios on the moisture content:
E, = -4841(X - 0,647)MPa,
V, = -0,68
(X- 0,64),
in the range of 0 I
X
Ep
= -2936(X
V, = -0,41
- 0,668)MPa
(X- 0,668)
(33a)
(33b)
I0,3, where moisture content equal to 0,3 denotes the fiber
saturation point. For moisture content greater than 0,3, the material coefficients are
constant and equal to those in the fiber saturation point.
The thermal-humid strains are postulated to be:
for 0 IX 5 0,3. If the moisture content is greater than the fiber saturation point,
one has to apply X
=
0,3 in equation (34), because wood does not change its
dimensions when the saturation is above the fiber saturation point.
The distribution of the moisture content was determined on the basis of the mass
balance equation after substituting the physical relation for moisture potential, that is:
2 75
S.J. Kowalski
where potential p = f ( e , 8 ) is a function of moisture content and temperature in
equation (17), and the coefficient of moisture transfer A t ) = k / q ( 8 ) is
proportional to the permeability k and inversely proportional to the liquid viscosity
(which depends on temperature).
The distribution of temperature was determined fiom the energy balance equation
of the form:
In our case, the boundary conditions for the mass and heat transfer must satisfy the
symmetry and convective conditions of the form:
-Iae
ar
-o=
4
-1a8
ar
Iso=
0
The initial conditions represent the moisture content
00
and the temperature So
uniformly distributed within the whole cylinder at t = 0.
The evident dependence of the coefficients on the moisture content and the
temperature makes the problem non-linear. The analytical solution of such a problem
is generally not possible, and therefore the numerical method of finite difference is
applied. The finite difference for the steps (n) and ( n - 1) are substituted into the
2 76
Stress Reversal Phenomena in Drying of Porous Media
physical relations (32a,b) to approximate the time derivatives of the strains and
stresses according to the Crank-Nickolson scheme, for example:
w -(&1
;
&rr
-&r)
At
One solves the obtained system of difference equations with respect to stresses 0;
and
o&,and thus:
where pii are the shortening denotations.
There is no shrinkage of wood during the first period of drying when the moisture
content of wood is above the fiber saturation point. The stresses that may appear at
the beginning of this period, if any, are the thermal stresses. The stress-induced
shrinkage appears when the wood surface reaches the fiber saturation point. Figures 4
and 5 presents the shrinkage stresses in a wood cylinder during the second period of
drying.
The filly saturated zone and the zone with moisture content below the fiber
saturation point are clearly observable in Figures 4 and 5. The wood cylinder does not
shrink in the fully saturated zone and therefore the stresses are uniform, although their
values vary with time. However, in the zone with the moisture content below the fiber
saturation point, the stresses vary along the radius depending on the gradient of the
moisture concentration. The radial stresses are initially negative (compressible) in the
whole region except the surface, where they are zero. However, the circumferential
stresses are initially negative (compressive) in the wet core and positive (tensional) in
277
S.J.Kowalski
5
0
-5
-10
-15
-20
-25
!
Figure 4. Radial stresses in woody cylinder at various instants.
the layers close to the surface. After some time, when the dry zone is displaced
sufficiently deep towards the interior of the cylinder, the circumferential stresses start
to change their sign. Also the radial stresses in a part of the region suffer the variation
of their sign. A numerical instability is visible on the interface between the saturated
and unsaturated zone. These instabilities follow from the discontinuity of the first
derivative of the displacement function.
It should be noted that the tensional circumferential stresses reach their maximum
twice. First, close to the surface at the beginning of the process. Second, inside the
cylinder at the end of the process. The tensional stresses are presumably responsible
for the destruction of the wood structure, that is manifested by the enhanced emission
of acoustic signals at the beginning as well as at the end stage of the drying process.
The performance of the intensity of acoustic signals in time for wood has a similar
course to that presented in Figure 3.
2 78
Stress Reversal Phenomena in Drying of Porous Media
7
50
40
30
20
10
0
-10
-20
Figure
5. Circumfirentialstresses in woody cylinder at various instants
Final remarks
The considerations presented in this paper allow us to appreciate the meaning of the
mathematical model adequately describing the mechanical phenomena during drying
of capillary-porous bodies. The experimentally observed enhanced emission of
acoustic signals indicate the enhanced destruction of the material at the final stage of
drying, and may pose a surprise at first sight as they are contrary to the predictions of
the elastic model. The viscoelastic model, taking into account the permanent
deformations of the dried material, reveals the phenomenon of stress reversal, and in
particular the appearance of tensional stresses inside the dried material at the final
stage of drying. The tensional stresses are supposed to be the main reason for the
material destruction in its interior, and the generation of hidden defects. The
numerical results of the viscoelastic model agree in principle with the experimental
observations using the acoustic emission method, which offer an online observation
of the development of the destruction of dried materials.
2 79
S.J Kowalski
Acknowledgements
This work is a part of the research project No 7 T09C 035 21 sponsored by the Polish
State Committee for Scientific Research in the years 2001-2004.
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2. Elwell, D., and Pointon, A.J., 1976, Classical Thermodynamics, WNT, Warszawa
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T e c h(submitted for publication).
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10. Kowalski, S.J., Rajewska, K., and Rybicki, A., 2000, Destruction of Wet materials by Drying, Chem.
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I.
Received 5 April 200 1;Accepted afrer revision: 24 August 200 1.
280
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