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Modeling of multi-phase moisture transfer and induced stress in drying clay bricks
ELSEVIER
Applied Clay Science 12 (1997) 189-207
Modeling of multi-phase moisture transfer and induced stress in drying clay bricks Shun-Lung Su Departr~,.nt of Science and Humanity, National Yun Lin Institute of Technology, Tow Liu 640, Taiwan, ROC
Received 4 July 1996; accepted 15 January 1997
Abstract In this study, transfer of heat and mass of all phases and the induced stresses in structural ceramics during the falling rate period of the drying process are considered, Based on the principle:; of non-equilibrium irreversible thermodynamics and assuming an isotropic porous system, the macroscopic laws of conservation and the liquid-vapor equilibrium relation of the green clay brick, a set of nonlinear equations for the prediction of simultaneous mass and heat transfer (luring the drying process is developed. Based on the obtained results of the histories of pore pressure and temperature from the nonlinear equations, the induced stresses can be calculated from the combination of the Biot theory and the Boley theory. The final result reveals that pore pressure instead of temperature in the porous system plays the dominant role in the determination of stress distribution during the falling rate period of a drying process, which is consistent with previous observations.
1. Introduction Ceramic materials used to construct buildings, roads, drainage systems and other engineering projects, are classified as structural ceramics. Clay products are one of the types of structural ceramics that can be used to construct walls of buildings and the like. Many clay forming processes require the addition of water to the material mixture. The clay body thus formed will be fired at high temperature. Without a pre-drying step, the fire will turn the water to steam in the clay body and can damage the body severely. Therefore, removal of most of the water in the body by a drying process must be done prior to firing. This process includes heat transfer from the surrounding environment to the porous system of the clay body, the simultaneous transfer of water vapor out of the body and the development of complicated internal stresses due to the tendency for the 0169-1317/97/$17.00 Copyright © 1997 Elsevier Science B.V. All fights reserved. PII SO 169-1317(97)00003 -3
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S.-L. Su /Applied Clay Science 12 (1997) 189-207
exterior surface zone to shrink before the interior does. A knowledge of these internal stresses is essential in order to identify causes and eliminate warping and cracking in the drying process of the clay body. The drying process can be divided into three periods. The constant rate period is the period that evaporation occurs only from the surface and the rate of evaporation is the same as that from a free water surface. The first falling rate period still allow the moisture evaporate from the surface, which is no longer behaving as a free water surface. As drying progresses, eventually liquid water can not move to the surface and it has to evaporate within the porous body. In this study, only the two falling rate periods will be focused. Modeling simultaneous heat and mass transfer is a complex task. Lewis (1921) studied the transfer of moisture in a porous media by neglecting the effect of heat transfer and concluded the movement of moisture was by diffusion from the interior of the porous body to the surface where it evaporated. Krischer (1940) was first to include the effect of heat transfer in modeling the drying process. Philip and De Vries (1957) extended previous treatments of drying to include the effects of capillary flow and vapor transport. Later, a set of differential equations for the simultaneous transfer of heat and mass in a porous medium was derived by Harmathy (1969) during the falling rate period of a drying process. The results of these equations were given with particular reference to clay bricks. They yielded the histories of pore pressure and temperature during the drying of clay bricks. In a classic series of papers, Biot (1941) developed a theory for the deformation of three dimensional isotropic porous media, assuming the solid component is homogeneous and elastic, or homogeneous but anisotropic (Biot and Willis, 1957). Cooper (1978) has proposed a model to describe the drying of clay bodies in which the stress that causes cracking or warping arises from the uneven removal of moisture. The stress and strain analysis for drying gels has been carded out by Scherer (1987). The pressure distribution in the liquid during drying is shown to obey the diffusion equation and the maximum stress to depend upon the shape of the porous body. In this investigation, a theoretical analysis of this problem is considered in two parts: (1) temperature and moisture transfer through the drying brick are calculated and (2) the Biot and Boley theories are combined to give a thermoelastic theory for calculating the induced stresses. The heat and mass transfer model used here is based on Harmathy (1969) theory. This theory has been extended in this study to include the effect of volume change of the solid component. The phenomena relevant for describing moisture, pressure and temperature distribution are coupled. A diffusion theory with a linear coefficient of diffusivity (Philip, 1972) is not adequate for the description of the behavior of mass transfer in a clay brick because the volume change in a clay brick generally is significant. By using the technique of scanning neutron radiography, Pel et al. (1993) has determined the isothermal moisture diffusivity as a nonlinear function of moisture content during drying the brick and kaolin clay. Transfer of mass in all phases and the transfer of heat must be considered simultaneously. Therefore, a general mathematical model for multi-phase moisture transfer must be constructed by using the principles of non-equilibrium irreversible flows of heat and mass. With the linear phenomenological equations (De
S.-L. Su /Applied Clay Science 12 (1997) 189-207
191
Groot and Mazur, 1962) and the laws of conservation in a macroscopic sense, a set of differential equations for simultaneous heat and moisture transfer in clay brick during the falling rate period of the drying process was developed. The theory of Biot (Biot, 1941; Biot and Willis, 1957) was originally developed to predict the behavior of soils and rocks under load and it excluded the thermal effect. Boley and Weiner (1960) have established the well known thermal stress theory describing the stress, strain and deformations that are caused by nonuniform temperature distributions in a material. In the present study, a constitutive model was derived from the combination of Biot and Boley theories to involve two key factors, pressure and temperal:ure, which controlling the drying behavior of a clay brick. From the simulated results of the distribution of stresses, we may have better understand when and where fractures might occur and these results can be used to avoid warping and cracking during ldae manufacture of clay brick.
2. Constitutive model
2.1. Analysis of mass and heat transfer In tttis analysis, the macroscopic continuum approach was adopted. We considered transport of liquid and its vapor, mixing with chemically inert air, through a porous clay
l-component £T ; ~ , , ; A ' ~
;~component (Solid)
mponent :ous Mixture)
Fig. 1. Volumetric element in a porous clay medium
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192
Table 1 The physical parameters used for the set of transport equations Components
Volume fraction (cm 3 c m - 3)
Mass concentration ( g c m - 3)
Solid Liquid (water) Gas (vapor) Gas (air) Total
es = 1 - e e I = e - eg ~g ~beg 8g ( 1 - 4~)eg es + e I + gg = 1
8s = Ps es = Ps( 1 - 8) 81 = plel = pl(~ - eg) ~g = pg eg = p~(~b~g)+ Pa(l - ~)~g 8g = pgEg = pv(4~Eg) + p a ( 1 - 4~)eg 8g -[- 81 "~ 8ga "~ 8g v = 8
medium such as that shown in Fig. 1. Also, the following assumptions were made: (1) the multi-phase porous system is isotropic, (2) the multi-phase system is locally in chemical equilibrium and in thermodynamic equilibrium, (3) the liquid component is incompressible, (4) the vapor, air and their mixture are ideal gases, (5) no chemical reactions occur within the system and (6) compressional work and viscous dissipation are negligible. For convenience, the volume fraction and the mass concentration for various components in a clay brick are listed in Table 1. In a conventional drying process, the characteristic time for drying is generally large and is of the order of days and, therefore, the transfers of heat and mass are rather slow. In such cases, the conservation of linear momentum equation can be replaced by the Darcy equation (Scheidegger, 1960),
g V= - - -7/e V.(P+~)
(1)
w h e r e / ( denotes the permeability tensor and the mass average velocity I7 is defined by ¢=
~gvVgv -[- ~gal~a -}- ~ l l ~ -'[- ~sVs / / ~
(2)
Since both the liquid component and solid component possess rather small mobility in comparison with the vapor fluxes, one assumes that 1~= I ~ = 0
(3)
Consequently, the macroscopic law of conservation of mass for each component is --
-- -F
at a~gv
--
cgt
a~g~ - -at +
(4)
+ I7. ngv = r
(5)
v. nga = o
(6)
and r~, = - r 1= r
(7)
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A explicit form of the energy equation for a porous system which consists of solid, liquid, vapor and air can be written as follows (Lykov and Mikhailov, 1965) 0
~t (8~v h~v + 8~ah.a + ~,h, + 8sh~) + V-(.~vh~ + "~ah.a + .~h, + "~hs)
aP = - V'q+
0--7 + V"
(VP) + / z ~ +
~b,
(8)
where t~tp is the viscous stress tensor. Eqs. (1)-(8) constitute the transport equations of moisture transfer in a clay brick under file aforementioned constraints. In ~Le gaseous phase, molecular mass and heat transfer occur simultaneously each from art associated driving force. The molecular mass flux is driven mainly by a mass concentration gradient and the conductive heat flux in the gaseous phase is mainly under the influence by temperature gradient. Using the linear law of irreversible thermodynamics and the Onsage reciprocal relations and neglecting the cross effects (the Dufour and Soret effects), the molecular heat and mass fluxes in gaseous phase can be expressed as (Huang, 1979) q~: = - eg KgVT
(9)
MwM~
J~:~ = - Sg~ = - p g 6 g O - - - ~ -
Vdp,
(10)
where the molecular mass M of the gaseous mixture is M = M w q b + M~(1 - dp).
(11)
With Eq. (3), it can be shown that the mass average velocity of the ~l~orous system l~ is a negligible fraction of the mass average velocity of gaseous mixture Vg. Therefore the terms n~hs, nlh I and IT-(VP) can be omitted. Applying the assumption (6) and substituting the continuity equations into Eq. (8), the energy equation becomes
Ohgv Ohga 60hl Ohs 8gv--~-t + S g a - ~ t + ,-~t + S s - ~ t + F Q + n g , ' V h g v + n g a ' V h g ~ +
V.q
OP O~ - 0-¥ + ha P~-~ = 0
(12)
Wi~L the aid of the Table 1, F_xlS. (1), (9) and (10), the equations of continuity for vapor and air Eqs. (5) and (6) and after some algebraic manipulation of the energy Eq. (12), three second-order nonlinear partial differential equations are derived: o4J oe or A i -~t + B i -~t + C i ~ t = D,~7 2~ + EiV2p + FiV2T + Gi(Vdp) 2 + Hi(VP) 2
+~,(vr)~+J,(V6.Ve) +K,(v6.vr) +L,(ve.vr)
i = l , 2, 3,
(13)
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S.-L. Su /Applied Clay Science 12 (1997) 189-207
where the coefficients A i ..... L i are functions of the dependent variables th, P ~ d T These coefficients are defined in a general f o ~ as follows:
PM w 0---~+~b-~ +6g
31=
=
__ +~--
P M w OT
E1 = - ,
'
BI=
P M w OP + qb OP +
ao,
OT
D 1 = - -
'
F l=0,
M
G1 = ~
~
P
'
,
~(M w-M~)
,
~g
=
--
~g
OP
+
,)
Jl=
OP + - - P
K1 = ~
I1 =0,
'
OT
T
+--
~bG
'
L1 =
~Tg
OWg
-
08g
8g(1 -- $ )
OT
r
kg ( 1
H2=
T,:(1-¢,) (~o,+ ~) ~
~b) ¢
'
eg ( 1 - ~b )
J2 = -g [(1 - th)
08g
,
- M w D [ 0% M 0¢
I 2 = 0,
+
L2
'
M
6 2= - -
- s¢
eg
P
-g.D~g
D2
'
F 2 = 0,
--
- M wD
T
B 2 = (1 - th) 0--P +
g2
"%
OT
08g
A 2 = (1 - ~6) - ~ - - 6g,
C2 = ( 1 - ~ b )
)
+ ~ '
8g ] M (Mw - M a )
S.-L.Su/AppliedClayScience12 (1997)189-207
195
[On (OmOSgl] plQ~+hsps ~+-~]j'
a3=-
Om
(OmO,,)
B 3 = m - e - P l Q - ~ p -hsps 0-7 + Op ' Ca = Pv % ~bCpv + Pa( 1 - 6)%Cpa +
- hs Ps
G3=0,
(0m 0,g) OT + OT
13 = 0%
E3 = 0 ,
J3
F 3 = K,
0,
MwMaDpg %
K3 = OTg O--dp
and
D 3=0,
OK 0% OT '
H 3 = 0,
OK 0%
'
Om plmCpl + Ps( 1 - 6)Cps - PlQ O---f
M2
( C P ' - Cpv)'
Jq OK 0% = 'rig [Pa(1--¢)Cpa-}-Pv~Cpv] "t-OTg O---if'
where m = m(r) is the volumetric moisture content per unit volume of the clay brick. With the definition of r from Kelvin's equation, the liquid-vapor equilibrium relation can be ,expressed as a function of the variables ~b, P and T. Permeability kg for the gaseous mixture in an isotropic porous medium is generally expressed as fractions of the single phase permeability. The ratio kg/k~ is called relative permeability and denoted by ~. Thus Eq. (13) form a complete set of nonlinear differential equations for describing the drying of clay brick during the failing rate period of a drying process.
2.2. Analysis of induced stresses There are several basic assumptions that has been made in this analysis: the stress, strain, pressure and temperature relations are assumed to be linear; the elastic strain is infinitesimal; and all points of the solid component are taken as elastic isotropic with the same bulk modulus. Therefore, based upon these assumptions from the macroscopic point of view, the strains in drying clay brick can be expressed as a function of stresses, pore pressure and temperature.
6ij= eij(orij, P, T).
(14)
Using the isotropic property, we can derive the well known expressions in the Biot theory (Biot, 1941)
1(
EiJ = " ~
where
O'ij
1 "~- 1~O'kkrij
),
"~- 3 H Prij + °tTrij'
tt is a physical constant which must be determined experimentally.
(15)
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Since our present concern is with quasi-static phenomena, the equilibrium conditions (neglecting body force) must be obeyed:
(16)
O'ij,j = O.
By the aid of Eqs. (15) and (16), we can derive six mutually independent compatibility equations in terms of cri3, P and T.
[(1+P)OriJ--l"OrkkSij]'nn÷trkk'iJ÷
k(1 - 2 v ) 7-1
[(P~ij)'nn+P'iJ]
+ 2 G a ( 1 + v)[(rso),n, , + T,i, ] = 0 .
(17)
3. Application This set of governing equations has been applied to determine the moisture migration in a thin clay slab during drying. The specimen used in this study was 19.5 cm 2 slab that was 1 cm thick. The material properties, initial and boundary conditions of present simulated model are adopted from Harmathy (1969) in order to make comparisons with his model. For one-dimensional mass and heat transfer in a slab of thickness 2h, the Eq. (13) has the following form:
at~ OP aT A i -'~t + B i -~t + C i - ~ c92d~
02P
O2T
Odp
2
= D'T~ + ~ ' T ~ + F ' - ~ + °' -ff + I , [ ~ ) +z ox ox +K, ~ . ~
÷ Hi
T~ J +L, o x ~
i=1,2,3.
(18) The boundary conditions on the surface of the clay brick are (i -- 0 for x = 0; i = 1 for x -- 2h)
0-~ = ( - 1 ) % OT
i hT
(4,-4,.),
(19)
O"X= ( - 1) - ~ - ( T - r.)
(20)
P = Patm,
(21)
and
where ~b., T. and P® respectively are the mole fraction of water vapor, temperature and
S..L. Su /Applied Clay Science 12 (1997) 189-207
197
pressure of the surroundings of the clay surface. The initial conditions for the present study are ~b( X, 0) = ~bmi,
P ( X, 0) = Pat~,
T( X, 0) = Tini .
(22)
Therefore Eqs. (18)-(22) form a nonlinear boundary value problem for the simultaneous mass and heat transfer in a clay slab. In the drying process, the surface of clay slab is subjected to a stream of air of varied or consl~rit velocity, temperature and humidity. When the evaporation of water at the surface of clay slab starts, the temperature at the surface falls due to the loss of heat in the form of the latent heat of the water evaporated. The difference in temperature between the slab surface and the surrounding air causes heat to be transferred from the air to the surface. As the difference in temperature increases, so does the rate of transfer of heat. The rate of evaporation of water from the surface at which heat is conducted through the air is proportional to temperature difference between the air and the surface. One cart calculate the rate of evaporation as follows (Ford, 1986): Rate of evaporation = ~hT ( r a -- ~,)
(23)
If the thickness of the slab is 2h, with X - Y axis in the midplane of the slab, the surfaces are at Z = h and - h . By extending the Boley theory (Boley and Weiner, 1960), the model of a clay slab with pressure and temperature variations through the thickness has been formulated. The normal stress in the X - Y plane of the slab is --
k(1 [.n("i---'~21,"'~')
[
X--T-l-Z-- ~
-e+
~.~fhhPdZ3Zfh -~ ]oLEI.~_..~ _
+
h
Zjh 1
hTdZ-t--~--.~
hTZdZ .
ZdZ
+
(24)
ff P and T are symmetrical, those are P ( Z ) = P ( - Z ) and T ( Z ) = T ( - Z ) and then the last terms inside the brackets drop out in F-xt. (24). In this convention, positive values of o- are tensions. 3.1. Numerical analysis
The purpose of this analysis is to develop a numerical method for solving the coupled set of differential Eq. (18). These equations can be expressed in a vectorial form; Uk, k = 1, 2, 3. They represent the mole fraction of water vapor in the gaseous mixture, ~b, pore pressure, P and temperature, T, which are the functions of the space coordinate, r, and time, t. A fully implicit finite difference scheme incorporating quasi-linearization of the nonlinear coefficients and the mixed derivatives is employed. Those nonlinear coefficients ~tre first calculated at the previous time step at which all dependent variables are known lind the system of equations is solved for new values of the dependent variables
198
Su /Applied Clay Science 12 (1997) 189-207
S.-L
at present time step. The coefficients can then be updated by utilizing the solutions just obtained at present time step. The calculations are repeated until the changes of dependent variables are smaller than the required tolerance. The terms of time derivative are rewritten using the first-order forward finite difference scheme. The second derivatives with respect to space coordinate, r, are rewritten using the fully implicit central finite difference scheme. In quasilinearization of the mixed derivatives, a scheme of Newton linearization is applied to the mixed derivatives and is proceeded as follows:
(OUmOUn) ~r j
~-- (OUm)i(OUn)k+l~ ~ 1
- - (2ar)
"~- ~(OUnlk(OUm) ]j~ k+lor
j
Or ij -
-~r ~r(OUmOU~) kj
-lTk*lvn,j_1 + Uo.,+1) k*l [( -U:,j_ 1 -[- U:,j.I)( '
-{- ( -- ukj_ l -.I-ukj+ l ) ( -- uk,+jl 1 ...{-Umk,+jll ) - - ( - - u k j _ , "]- ukj+l)(--gk,j_ 1 "{- Umk,j+l)]
.
(25)
Combining the discrete boundary conditions with the set of implicit finite difference equations, after being quasilinearized, results in a three by three block-tridiagonal system of equations in the form AX = B, where A is a 3 ( N - 2) x 3 ( N - 2) sparse unsymmetric matrix, X and B are 3(N - 2) vectors with N being the number of nodes
L0' 0.9.
0al0.?. E
all.
0.1, 0.0'
i
. . . . . . . .
t
. . . . . . . .
i
. . . . . . . .
i
. . . . . . . .
i
. . . . . . . .
i
I~r,DIU$ [A] Fig. 2. Generalized liquid-vapor equilibrium relation. (Ha.rmathy, 1969; Fig. 2)
199
S.-L Su /Applied Clay Science 12 (1997) 189-207
in the r-direction. Thus the normal stress can be calculated at the current time step after the convergent pore pressure and temperature are sought.
4. R e s u l t s a n d d i s c u s s i o n
The vapor-liquid equilibrium relation for a clay brick which was tested by Harmathy (1969) is given in Fig. 2. The quantitative comparisons of average moisture content in three different cases among Harmathy's theoretical results, Experimental results (Harmathy, 1967) and the results obtained from the present model are shown in Fig. 3. The present model shows a better agreement to the experimental data than predictions from Harmathy's model. In this paper, only the overall results of case 1 as shown in Fig. 3 were plotted and
0.08 t
: PRESENT MODEL . . . .
EXP DATA : HARMATBY MODEL
6
8
0.07
I 0
2
4
10
12
Time | brs ] Fig. 3. Comparisons of average moisture content in drying clay slab at both sides.
200
S.-L. Su /Applied Clay Science 12 (1997) 189-207
o~o [ (a)
....
P~sE~ MODE~
0.0~ ]
....
HARMATHYMODEL
0.00 0.07"
i
0.06. 0.0~"
--
0.04"
0.03. 0.02
0-01' I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t=12 hr 0.00 0.0
0.2
0.4
0.0
0.8
1.0
0.4 0.6 × [cm]
0.8
1.0
X {cml 90O 800 700 ~.
61111
o <
g. 21111
lOO 0 0.0
0.2
Fig. 4. (a) Distribution of moisture in drying 1 cm thick clay slab at both sides. (b) Distribution of pressure in drying 1 cm thick clay slab at both sides.
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201
46 (C)
t--nh,
,111' 44
t ~ 3.0
38
--
t~li
--
t'l-O
--
t--L5
0.0
O&
0.4
0.6
0.8
1.0
Xleml Fig. 4 (continued). (c) Distribution of temperature in drying 1 e m thick clay slab at both sides.
Table 2 Values of the constants used in the equations Symbol
Value
Unit
R
8"3149×107
gcm2 s-2 K-l re°l-1
"qs
1"83X 1 0 - 4
gcm-I
Cl~ Cr~ Cpv Cpw Q e Kg Ks Kw Ma H Mw
5.67×10 -5 1.0063X 107 8.15X 106 1.8646×107 4.1793×107 2 . 4 4 1 8 x 10 l° 0.8 2.613X 103 2.207X 10 s 6.16X 104 28.952 0.3741 X 107 18.016
gs-3K -4 cm 2 s - 2 K -1 cm 2 $-2 K-1 cm2 s - 2 K - 1 c m 2 s - 2 K -1 cm 2 s -2
/~ D p, Pw at v Es Gs ks B
2 . 5 × 10 - x ° 0.256 2.75 0.99707 9.0X 10 - 6 0.25 2.068 x 101° 8.2736X 109 1.379×101° 18.39 2.478×10 -6 0.5 0.28
cm 2 c m 2 s -1 gcm -3 gcm -3 K -1
I.C.
Value
Unit
B.C.
Value
Unit
Min i Patm Ti~i
0.105 1,01325 X 106 306.2
cm 3 c m - 3 g cm s - 2 K
t/~ T~
0.010539 318.9
mol m o l - 1 K
gems -3 gcms -3 gcm s -3 gmol -l cm 3 s gg mol- 1
Symbol
K -l K-l K- t
hoF 1
h ~
Value
Unit s-t
g c m -1 s - 2 gem -] s-2 cm 3 sg -1 m o l c m - 2 s -1 cm cm 3 cm- 3
S.-L. Su//AppliedClayScience12 (1997)189-207
202
0.020
.
I
0.016
Z O
0.014
FIRST FALLING RATE PERIOD
F-
SECOND FALLING RATE PERIOD 0.008 -~
f
O.OIM
0.002 0.000 0
2
4
6
8
I)
12
Time [ hrs ] Fig. 5. Rate of evaporation curve in drying l c m thick clay slab at both sides.
used as the numerical example. The clay slab of case 1 was dried both sides at specified boundary conditions for 12 h. The physical parameters, empirical constants, initial conditions and boundary conditions of case 1 are listed in Table 2. The results of case 1 obtained from present model for the histories of moisture content and pore pressure are compared with those obtained using Harmathy's model and are plotted in Fig. 4a and b. The present model took into consideration the volume change of the solid component inside the clay body, whereas this change was neglected in Harmathy's model. The values for the peaks of moisture content and pore pressure were higher for the present model compared to the Harmathy's model, throughout the drying process. As the solid volume increases due to the thermal effect, the porosity decreases and it changes the elastic properties of clay body. These changes may have contributed to the inaccurate from Harmathy's model. The temperature distribution in the clay slab for different drying times was plotted in Fig. 4c. There was a relative sharp rise in temperature after 2.5 h to 3.0 h, which indicates the transient period from first to second falling rate period as shown in Fig. 5. The rate of evaporation on the surface of clay slab (Fig. 5) showed clearly the distinction between different periods: first falling rate and second falling rate period. During the first falling rate period (from 0 to 3 h), the continued migration of liquid water toward the drying surface, coupled with the decrease in the rate of evaporation, caused the curves for moisture distribution (Fig. 4a) during this period to level out to some extent. The duration of the first falling rate period was relatively short. In this period, moisture movement depended strongly on the effect
S.-L. Su /Applied Clay Science 12 (1997) 189-207
203
1
i
1 --1 --1
-I 0.0
0.2
0.4
0.6
0.8
1.0
X [cm] Fig. 6. Distributionof normal stress in drying 1 cm thickclay slab at both sides.
of surroundings. The moisture distribution curves are all convex (Fig. 4a). For the second :railing rate period, 3 to 12 h, the duration was very long and the curves of moisture distribution were no longer convex (Fig. 4a). The brick system at and close to the surface contained insufficient moisture for liquid migration to continue and moisture that sub~iequently migrates to the clay surface did so as vapor. The result revealed that in this period the evaporation-condensation mechanism played the dominant role. In other words, both the topological parameters of the clay brick system and the temperature gradient were important factors during this period of the drying process. The topological paramet~rs include pore structure and pore shape. All these observations from numerical simulation are consistent with the experimental results given in Harmathy (1969). The normal stress of in the plane of the slab is plotted in Fig. 6. The maximum tensile stress occurred at the surface, which is shown in this plot and had been reported by Cooper ,(1978). The stress distribution curves in Figs. 6 and 7 qualitatively agreed with the analysis of Scherer (1987) during the falling rate period of drying process. From the history of surface tension stress, the maximum tensile stress and the peak pore pressure were observed to occur simultaneously. This phenomenon indicates that pore pressure, instead tff temperature, plays the dominant role in the determination of stress distribution during the falling rate period of drying process. Since the total shrinkage stopped after
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S.-L. Su / A p p l i e d Clay Science 12 (1997) 1 8 9 - 2 0 7
3°° I ~ x - O . O 200.
r.~ u~ ,.~ +<
ms 0
0 Z -:100
-:mo~
- 300, 0
xmO,37S
V x-O.S
~ :i
4
6 8 Time [ brs ]
10
)2
Fig. 7. The history of normal stress in drying 1 cm thick clay slab at both sides.
the end of first falling rate period (Ford, 1986), the tensile stresses decreased rapidly as shown in Fig. 7. This result was also consistent with Scherer (1987).
5. Conclusion
An analysis has been presented to describe the moisture transfer and induced stresses that develop during the falling rate period of the drying process of clay slabs in which the solid component was assumed to be elastic. From this study, several conclusions can be made. They are: (1) In Harmathy's model, the solid matrix is rigid and the porosity is unchangeable during the drying process. In the model developed in this study, the solid matrix is free to expand due to the drying effect that causes the decrease of porosity inside the clay slab. (2) The transient period from the first falling rate period to the second falling rate period caused a relatively sharp rise in temperature inside the clay body. (3) For the second falling rate period of the drying process, the duration was relatively long and the evaporation--condensation mechanism played the dominant role in moisture transfer. (4) The porosity decreased due to thermal expansion of the solid component during drying and in doing so changed the elastic properties. The porosity effects on the
S.-L Su /Applied Clay Science 12 (1997) 189-207
205
distributions of moisture and pore pressure were significant, in turn, on the distribution of stresses. (5) Owing to the use of different material properties, it was only possible to obtain the qualitative agreement in stress distribution with Scherer's analysis. (6) "Ilae maximum tensile stress occurred at the surface and caused surface cracks.
6. Nomenclature
B Cp Cpi D E e G h hD hDF hi
hT k ki
ki Ki
m M n ni P Pi
q qi (qi)j
Q R t T T~ T~ T~ V~ V/
characteristic length of the brick measured from the edge (cm) effective heat capacity at constant pressure (cm 2 s-2 K - 1) specific heat of ith-component (cm 2 s-2 K - I ) diffusivity of the gaseous mixture (cm 2 s- l) apparent Young's modulus (g cm -~ s -2) mass rate of evaporation per unit volume (g cm- 3 s- 1) apparent shear modulus of elasticity (g cm -1 s -2) macroscopic enthalpy (era 2 s- 2) effective mass transfer coefficient (mol cm -2 s- 1) mass transfer coefficient due to forced convection (mol cm -2 s- 1) enthalpy of ith-component (cm 2 s -2) effective heat transfer coefficient (g s -3 K -1) apparent bulk modulus (g cm -1 s -2) bulk modulus of ith-component (g cm-1 s- 2) coefficient of permeability of ith-component (cm 3 s g-1) effective thermal conductivity of ith-component (g cm s -3 K -1) volumetric moisture content (cm 3 cm -3) molecular weight of the gaseous mixture (g mol-1) topological constant of the clay brick absolute mass flux of ith-component (g cm -2 s - I ) pore pressure (g cm -1 s -2) pore partial pressure of ith-component apparent heat flux (g s-3) mass flow rate of ith-component ( g c m -2 s- l) mass flow rate of ith-component in jth-direction latent heat of evaporation of free water (cm 2 s-2) gas constant (g cm 2 s- 2 K - 1 mol- l) time (s) temperature rise of bulk volume (K) temperature of air (K) temperature of the enclosure of the system (K) surface temperature of clay brick (K) volume of ith-component per unit bulk volume (cm 3 cm -3) velocity of ith-component (cm s - I )
S.-L. Su /Applied Clay Science 12 (1997) 189-207
206
Greek: e eg Pi th e~j
porosity of the porous system (cm 3 cm -3) volume fraction of the gaseous mixture (cm 3 cm -3) density of ith-component (g cm -3) mole fraction of water vapor of the gaseous mixture ( m o l / m o l ) elastic strain elastic stress (g cm -1 s -2) Stefan Boltzman constant (g s -3 K -4) Poisson ratio linear thermal expansion coefficient of porous media (cm K -1) mass concentration of ith component per unit volume of the porous media (g
orij trs v a 6i
cm -3)
0",7 F~ 0 7/ ~b
Kronecker delta relative permeability mass rate of evaporation of ith-component per unit bulk volume ( g c m -3 s - 1) relative saturation (0 = re~e) coefficient of fluid viscosity (g c m - 1 s - 1) gravitational potential (g c m - 1 s - 2) rate of heat source (g cm-1 s-3
Subscripts: a atm g ga gv 1 s v w
air atmosphere gaseous mixture air in a gaseous mixture vapor in a gaseous mixture liquid solid vapor water ambient
References Biot, M.A., 1941. General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155-161. Biot, M.A., Willis, D.G., 1957. The elastic coefficients of the theory of consolidation. J. Appl. Mech 24, 594-599. Boley, B.A., Weiner, J.H., 1960. Theory of Thermal Stresses. John Wiley, New York. Cooper, A.R., 1978. Quantitative Theory of Cracking and Warping During the Drying of Clay Bodies. Ceramic Processing before Firing. Wiley Interscience. De Groot, S.R., Mazur, P., 1962. Nonequilibrium Thermodynamics. North Holland Pub. Co., Amsterdam. Ford, R.W., 1986. Ceramic Drying. Pergamon Press, New York. Harmathy, T.Z., 1969. Simultaneous moisture and heat transfer in porous systems with particular reference to drying. I&EC Fundamentals 8, 92-103.
S.-L. Su /Applied Clay Science 12 (1997) 189-207
207
Harmathy, T.Z., 1967. Simultaneous Moisture and Heat Transfer in Porous Systems with Particular Reference to Drying, Ph.D. dissertation, University of Technology of Vienna, Vienna. Huang, C.L., 1979. Multi-phase moisture transfer in porous media subject to temperature gradient. Int. J. Heat Mass Transfer 22, 1295-1307. Krischer, O., 1940. Heat and Mass Transfer in Drying. VDI-Forschmgsh, Berlin. Lewis, W.K., 1921. Ind. Eng. Chem. 13, 427-432. Lykov, A.V., Mikhailov, Y.A., 1965. Theory of Heat and Mass Transfer. Daniel Davery Inc., New York. Pel, L., Ketelaars, A.A.J., Adan, O.C.G., Van Well, A.A., 1993. Determination of moisture diffusivity in porous media using scanning neutron radiography. Int. J. Heat Mass Transfer 36, 1261-1267. Philip, J.R., De Vries, D.A., 1957. Moisture movement in porous materials under temperature gradients. Trans. Am. Geophys. Union 38. Philip, J.K., 1972. Hydrostatics and Hydrodynamics in Swelling Media. Fundamentals of Transport Phenomena in Porous Media. Elsevier, New York. Scheidegger, A.E., 1960. The Physics of Flow through Porous Media, 2nd Ed. University of Toronto Press. Scherer, G.W., 1987. Drying gels. V. Rigid gels. J. Non-Cryst. Solids 92, 122.
Applied Clay Science 12 (1997) 189-207
Modeling of multi-phase moisture transfer and induced stress in drying clay bricks Shun-Lung Su Departr~,.nt of Science and Humanity, National Yun Lin Institute of Technology, Tow Liu 640, Taiwan, ROC
Received 4 July 1996; accepted 15 January 1997
Abstract In this study, transfer of heat and mass of all phases and the induced stresses in structural ceramics during the falling rate period of the drying process are considered, Based on the principle:; of non-equilibrium irreversible thermodynamics and assuming an isotropic porous system, the macroscopic laws of conservation and the liquid-vapor equilibrium relation of the green clay brick, a set of nonlinear equations for the prediction of simultaneous mass and heat transfer (luring the drying process is developed. Based on the obtained results of the histories of pore pressure and temperature from the nonlinear equations, the induced stresses can be calculated from the combination of the Biot theory and the Boley theory. The final result reveals that pore pressure instead of temperature in the porous system plays the dominant role in the determination of stress distribution during the falling rate period of a drying process, which is consistent with previous observations.
1. Introduction Ceramic materials used to construct buildings, roads, drainage systems and other engineering projects, are classified as structural ceramics. Clay products are one of the types of structural ceramics that can be used to construct walls of buildings and the like. Many clay forming processes require the addition of water to the material mixture. The clay body thus formed will be fired at high temperature. Without a pre-drying step, the fire will turn the water to steam in the clay body and can damage the body severely. Therefore, removal of most of the water in the body by a drying process must be done prior to firing. This process includes heat transfer from the surrounding environment to the porous system of the clay body, the simultaneous transfer of water vapor out of the body and the development of complicated internal stresses due to the tendency for the 0169-1317/97/$17.00 Copyright © 1997 Elsevier Science B.V. All fights reserved. PII SO 169-1317(97)00003 -3
190
S.-L. Su /Applied Clay Science 12 (1997) 189-207
exterior surface zone to shrink before the interior does. A knowledge of these internal stresses is essential in order to identify causes and eliminate warping and cracking in the drying process of the clay body. The drying process can be divided into three periods. The constant rate period is the period that evaporation occurs only from the surface and the rate of evaporation is the same as that from a free water surface. The first falling rate period still allow the moisture evaporate from the surface, which is no longer behaving as a free water surface. As drying progresses, eventually liquid water can not move to the surface and it has to evaporate within the porous body. In this study, only the two falling rate periods will be focused. Modeling simultaneous heat and mass transfer is a complex task. Lewis (1921) studied the transfer of moisture in a porous media by neglecting the effect of heat transfer and concluded the movement of moisture was by diffusion from the interior of the porous body to the surface where it evaporated. Krischer (1940) was first to include the effect of heat transfer in modeling the drying process. Philip and De Vries (1957) extended previous treatments of drying to include the effects of capillary flow and vapor transport. Later, a set of differential equations for the simultaneous transfer of heat and mass in a porous medium was derived by Harmathy (1969) during the falling rate period of a drying process. The results of these equations were given with particular reference to clay bricks. They yielded the histories of pore pressure and temperature during the drying of clay bricks. In a classic series of papers, Biot (1941) developed a theory for the deformation of three dimensional isotropic porous media, assuming the solid component is homogeneous and elastic, or homogeneous but anisotropic (Biot and Willis, 1957). Cooper (1978) has proposed a model to describe the drying of clay bodies in which the stress that causes cracking or warping arises from the uneven removal of moisture. The stress and strain analysis for drying gels has been carded out by Scherer (1987). The pressure distribution in the liquid during drying is shown to obey the diffusion equation and the maximum stress to depend upon the shape of the porous body. In this investigation, a theoretical analysis of this problem is considered in two parts: (1) temperature and moisture transfer through the drying brick are calculated and (2) the Biot and Boley theories are combined to give a thermoelastic theory for calculating the induced stresses. The heat and mass transfer model used here is based on Harmathy (1969) theory. This theory has been extended in this study to include the effect of volume change of the solid component. The phenomena relevant for describing moisture, pressure and temperature distribution are coupled. A diffusion theory with a linear coefficient of diffusivity (Philip, 1972) is not adequate for the description of the behavior of mass transfer in a clay brick because the volume change in a clay brick generally is significant. By using the technique of scanning neutron radiography, Pel et al. (1993) has determined the isothermal moisture diffusivity as a nonlinear function of moisture content during drying the brick and kaolin clay. Transfer of mass in all phases and the transfer of heat must be considered simultaneously. Therefore, a general mathematical model for multi-phase moisture transfer must be constructed by using the principles of non-equilibrium irreversible flows of heat and mass. With the linear phenomenological equations (De
S.-L. Su /Applied Clay Science 12 (1997) 189-207
191
Groot and Mazur, 1962) and the laws of conservation in a macroscopic sense, a set of differential equations for simultaneous heat and moisture transfer in clay brick during the falling rate period of the drying process was developed. The theory of Biot (Biot, 1941; Biot and Willis, 1957) was originally developed to predict the behavior of soils and rocks under load and it excluded the thermal effect. Boley and Weiner (1960) have established the well known thermal stress theory describing the stress, strain and deformations that are caused by nonuniform temperature distributions in a material. In the present study, a constitutive model was derived from the combination of Biot and Boley theories to involve two key factors, pressure and temperal:ure, which controlling the drying behavior of a clay brick. From the simulated results of the distribution of stresses, we may have better understand when and where fractures might occur and these results can be used to avoid warping and cracking during ldae manufacture of clay brick.
2. Constitutive model
2.1. Analysis of mass and heat transfer In tttis analysis, the macroscopic continuum approach was adopted. We considered transport of liquid and its vapor, mixing with chemically inert air, through a porous clay
l-component £T ; ~ , , ; A ' ~
;~component (Solid)
mponent :ous Mixture)
Fig. 1. Volumetric element in a porous clay medium
S.-L. Su /Applied Clay Science 12 (1997) 189-207
192
Table 1 The physical parameters used for the set of transport equations Components
Volume fraction (cm 3 c m - 3)
Mass concentration ( g c m - 3)
Solid Liquid (water) Gas (vapor) Gas (air) Total
es = 1 - e e I = e - eg ~g ~beg 8g ( 1 - 4~)eg es + e I + gg = 1
8s = Ps es = Ps( 1 - 8) 81 = plel = pl(~ - eg) ~g = pg eg = p~(~b~g)+ Pa(l - ~)~g 8g = pgEg = pv(4~Eg) + p a ( 1 - 4~)eg 8g -[- 81 "~ 8ga "~ 8g v = 8
medium such as that shown in Fig. 1. Also, the following assumptions were made: (1) the multi-phase porous system is isotropic, (2) the multi-phase system is locally in chemical equilibrium and in thermodynamic equilibrium, (3) the liquid component is incompressible, (4) the vapor, air and their mixture are ideal gases, (5) no chemical reactions occur within the system and (6) compressional work and viscous dissipation are negligible. For convenience, the volume fraction and the mass concentration for various components in a clay brick are listed in Table 1. In a conventional drying process, the characteristic time for drying is generally large and is of the order of days and, therefore, the transfers of heat and mass are rather slow. In such cases, the conservation of linear momentum equation can be replaced by the Darcy equation (Scheidegger, 1960),
g V= - - -7/e V.(P+~)
(1)
w h e r e / ( denotes the permeability tensor and the mass average velocity I7 is defined by ¢=
~gvVgv -[- ~gal~a -}- ~ l l ~ -'[- ~sVs / / ~
(2)
Since both the liquid component and solid component possess rather small mobility in comparison with the vapor fluxes, one assumes that 1~= I ~ = 0
(3)
Consequently, the macroscopic law of conservation of mass for each component is --
-- -F
at a~gv
--
cgt
a~g~ - -at +
(4)
+ I7. ngv = r
(5)
v. nga = o
(6)
and r~, = - r 1= r
(7)
S.-L. Su/Applied ClayScience12 (1997)189-207
193
A explicit form of the energy equation for a porous system which consists of solid, liquid, vapor and air can be written as follows (Lykov and Mikhailov, 1965) 0
~t (8~v h~v + 8~ah.a + ~,h, + 8sh~) + V-(.~vh~ + "~ah.a + .~h, + "~hs)
aP = - V'q+
0--7 + V"
(VP) + / z ~ +
~b,
(8)
where t~tp is the viscous stress tensor. Eqs. (1)-(8) constitute the transport equations of moisture transfer in a clay brick under file aforementioned constraints. In ~Le gaseous phase, molecular mass and heat transfer occur simultaneously each from art associated driving force. The molecular mass flux is driven mainly by a mass concentration gradient and the conductive heat flux in the gaseous phase is mainly under the influence by temperature gradient. Using the linear law of irreversible thermodynamics and the Onsage reciprocal relations and neglecting the cross effects (the Dufour and Soret effects), the molecular heat and mass fluxes in gaseous phase can be expressed as (Huang, 1979) q~: = - eg KgVT
(9)
MwM~
J~:~ = - Sg~ = - p g 6 g O - - - ~ -
Vdp,
(10)
where the molecular mass M of the gaseous mixture is M = M w q b + M~(1 - dp).
(11)
With Eq. (3), it can be shown that the mass average velocity of the ~l~orous system l~ is a negligible fraction of the mass average velocity of gaseous mixture Vg. Therefore the terms n~hs, nlh I and IT-(VP) can be omitted. Applying the assumption (6) and substituting the continuity equations into Eq. (8), the energy equation becomes
Ohgv Ohga 60hl Ohs 8gv--~-t + S g a - ~ t + ,-~t + S s - ~ t + F Q + n g , ' V h g v + n g a ' V h g ~ +
V.q
OP O~ - 0-¥ + ha P~-~ = 0
(12)
Wi~L the aid of the Table 1, F_xlS. (1), (9) and (10), the equations of continuity for vapor and air Eqs. (5) and (6) and after some algebraic manipulation of the energy Eq. (12), three second-order nonlinear partial differential equations are derived: o4J oe or A i -~t + B i -~t + C i ~ t = D,~7 2~ + EiV2p + FiV2T + Gi(Vdp) 2 + Hi(VP) 2
+~,(vr)~+J,(V6.Ve) +K,(v6.vr) +L,(ve.vr)
i = l , 2, 3,
(13)
194
S.-L. Su /Applied Clay Science 12 (1997) 189-207
where the coefficients A i ..... L i are functions of the dependent variables th, P ~ d T These coefficients are defined in a general f o ~ as follows:
PM w 0---~+~b-~ +6g
31=
=
__ +~--
P M w OT
E1 = - ,
'
BI=
P M w OP + qb OP +
ao,
OT
D 1 = - -
'
F l=0,
M
G1 = ~
~
P
'
,
~(M w-M~)
,
~g
=
--
~g
OP
+
,)
Jl=
OP + - - P
K1 = ~
I1 =0,
'
OT
T
+--
~bG
'
L1 =
~Tg
OWg
-
08g
8g(1 -- $ )
OT
r
kg ( 1
H2=
T,:(1-¢,) (~o,+ ~) ~
~b) ¢
'
eg ( 1 - ~b )
J2 = -g [(1 - th)
08g
,
- M w D [ 0% M 0¢
I 2 = 0,
+
L2
'
M
6 2= - -
- s¢
eg
P
-g.D~g
D2
'
F 2 = 0,
--
- M wD
T
B 2 = (1 - th) 0--P +
g2
"%
OT
08g
A 2 = (1 - ~6) - ~ - - 6g,
C2 = ( 1 - ~ b )
)
+ ~ '
8g ] M (Mw - M a )
S.-L.Su/AppliedClayScience12 (1997)189-207
195
[On (OmOSgl] plQ~+hsps ~+-~]j'
a3=-
Om
(OmO,,)
B 3 = m - e - P l Q - ~ p -hsps 0-7 + Op ' Ca = Pv % ~bCpv + Pa( 1 - 6)%Cpa +
- hs Ps
G3=0,
(0m 0,g) OT + OT
13 = 0%
E3 = 0 ,
J3
F 3 = K,
0,
MwMaDpg %
K3 = OTg O--dp
and
D 3=0,
OK 0% OT '
H 3 = 0,
OK 0%
'
Om plmCpl + Ps( 1 - 6)Cps - PlQ O---f
M2
( C P ' - Cpv)'
Jq OK 0% = 'rig [Pa(1--¢)Cpa-}-Pv~Cpv] "t-OTg O---if'
where m = m(r) is the volumetric moisture content per unit volume of the clay brick. With the definition of r from Kelvin's equation, the liquid-vapor equilibrium relation can be ,expressed as a function of the variables ~b, P and T. Permeability kg for the gaseous mixture in an isotropic porous medium is generally expressed as fractions of the single phase permeability. The ratio kg/k~ is called relative permeability and denoted by ~. Thus Eq. (13) form a complete set of nonlinear differential equations for describing the drying of clay brick during the failing rate period of a drying process.
2.2. Analysis of induced stresses There are several basic assumptions that has been made in this analysis: the stress, strain, pressure and temperature relations are assumed to be linear; the elastic strain is infinitesimal; and all points of the solid component are taken as elastic isotropic with the same bulk modulus. Therefore, based upon these assumptions from the macroscopic point of view, the strains in drying clay brick can be expressed as a function of stresses, pore pressure and temperature.
6ij= eij(orij, P, T).
(14)
Using the isotropic property, we can derive the well known expressions in the Biot theory (Biot, 1941)
1(
EiJ = " ~
where
O'ij
1 "~- 1~O'kkrij
),
"~- 3 H Prij + °tTrij'
tt is a physical constant which must be determined experimentally.
(15)
S.-L. Su /Applied Clay Science 12 (1997) 189-207
196
Since our present concern is with quasi-static phenomena, the equilibrium conditions (neglecting body force) must be obeyed:
(16)
O'ij,j = O.
By the aid of Eqs. (15) and (16), we can derive six mutually independent compatibility equations in terms of cri3, P and T.
[(1+P)OriJ--l"OrkkSij]'nn÷trkk'iJ÷
k(1 - 2 v ) 7-1
[(P~ij)'nn+P'iJ]
+ 2 G a ( 1 + v)[(rso),n, , + T,i, ] = 0 .
(17)
3. Application This set of governing equations has been applied to determine the moisture migration in a thin clay slab during drying. The specimen used in this study was 19.5 cm 2 slab that was 1 cm thick. The material properties, initial and boundary conditions of present simulated model are adopted from Harmathy (1969) in order to make comparisons with his model. For one-dimensional mass and heat transfer in a slab of thickness 2h, the Eq. (13) has the following form:
at~ OP aT A i -'~t + B i -~t + C i - ~ c92d~
02P
O2T
Odp
2
= D'T~ + ~ ' T ~ + F ' - ~ + °' -ff + I , [ ~ ) +z ox ox +K, ~ . ~
÷ Hi
T~ J +L, o x ~
i=1,2,3.
(18) The boundary conditions on the surface of the clay brick are (i -- 0 for x = 0; i = 1 for x -- 2h)
0-~ = ( - 1 ) % OT
i hT
(4,-4,.),
(19)
O"X= ( - 1) - ~ - ( T - r.)
(20)
P = Patm,
(21)
and
where ~b., T. and P® respectively are the mole fraction of water vapor, temperature and
S..L. Su /Applied Clay Science 12 (1997) 189-207
197
pressure of the surroundings of the clay surface. The initial conditions for the present study are ~b( X, 0) = ~bmi,
P ( X, 0) = Pat~,
T( X, 0) = Tini .
(22)
Therefore Eqs. (18)-(22) form a nonlinear boundary value problem for the simultaneous mass and heat transfer in a clay slab. In the drying process, the surface of clay slab is subjected to a stream of air of varied or consl~rit velocity, temperature and humidity. When the evaporation of water at the surface of clay slab starts, the temperature at the surface falls due to the loss of heat in the form of the latent heat of the water evaporated. The difference in temperature between the slab surface and the surrounding air causes heat to be transferred from the air to the surface. As the difference in temperature increases, so does the rate of transfer of heat. The rate of evaporation of water from the surface at which heat is conducted through the air is proportional to temperature difference between the air and the surface. One cart calculate the rate of evaporation as follows (Ford, 1986): Rate of evaporation = ~hT ( r a -- ~,)
(23)
If the thickness of the slab is 2h, with X - Y axis in the midplane of the slab, the surfaces are at Z = h and - h . By extending the Boley theory (Boley and Weiner, 1960), the model of a clay slab with pressure and temperature variations through the thickness has been formulated. The normal stress in the X - Y plane of the slab is --
k(1 [.n("i---'~21,"'~')
[
X--T-l-Z-- ~
-e+
~.~fhhPdZ3Zfh -~ ]oLEI.~_..~ _
+
h
Zjh 1
hTdZ-t--~--.~
hTZdZ .
ZdZ
+
(24)
ff P and T are symmetrical, those are P ( Z ) = P ( - Z ) and T ( Z ) = T ( - Z ) and then the last terms inside the brackets drop out in F-xt. (24). In this convention, positive values of o- are tensions. 3.1. Numerical analysis
The purpose of this analysis is to develop a numerical method for solving the coupled set of differential Eq. (18). These equations can be expressed in a vectorial form; Uk, k = 1, 2, 3. They represent the mole fraction of water vapor in the gaseous mixture, ~b, pore pressure, P and temperature, T, which are the functions of the space coordinate, r, and time, t. A fully implicit finite difference scheme incorporating quasi-linearization of the nonlinear coefficients and the mixed derivatives is employed. Those nonlinear coefficients ~tre first calculated at the previous time step at which all dependent variables are known lind the system of equations is solved for new values of the dependent variables
198
Su /Applied Clay Science 12 (1997) 189-207
S.-L
at present time step. The coefficients can then be updated by utilizing the solutions just obtained at present time step. The calculations are repeated until the changes of dependent variables are smaller than the required tolerance. The terms of time derivative are rewritten using the first-order forward finite difference scheme. The second derivatives with respect to space coordinate, r, are rewritten using the fully implicit central finite difference scheme. In quasilinearization of the mixed derivatives, a scheme of Newton linearization is applied to the mixed derivatives and is proceeded as follows:
(OUmOUn) ~r j
~-- (OUm)i(OUn)k+l~ ~ 1
- - (2ar)
"~- ~(OUnlk(OUm) ]j~ k+lor
j
Or ij -
-~r ~r(OUmOU~) kj
-lTk*lvn,j_1 + Uo.,+1) k*l [( -U:,j_ 1 -[- U:,j.I)( '
-{- ( -- ukj_ l -.I-ukj+ l ) ( -- uk,+jl 1 ...{-Umk,+jll ) - - ( - - u k j _ , "]- ukj+l)(--gk,j_ 1 "{- Umk,j+l)]
.
(25)
Combining the discrete boundary conditions with the set of implicit finite difference equations, after being quasilinearized, results in a three by three block-tridiagonal system of equations in the form AX = B, where A is a 3 ( N - 2) x 3 ( N - 2) sparse unsymmetric matrix, X and B are 3(N - 2) vectors with N being the number of nodes
L0' 0.9.
0al0.?. E
all.
0.1, 0.0'
i
. . . . . . . .
t
. . . . . . . .
i
. . . . . . . .
i
. . . . . . . .
i
. . . . . . . .
i
I~r,DIU$ [A] Fig. 2. Generalized liquid-vapor equilibrium relation. (Ha.rmathy, 1969; Fig. 2)
199
S.-L Su /Applied Clay Science 12 (1997) 189-207
in the r-direction. Thus the normal stress can be calculated at the current time step after the convergent pore pressure and temperature are sought.
4. R e s u l t s a n d d i s c u s s i o n
The vapor-liquid equilibrium relation for a clay brick which was tested by Harmathy (1969) is given in Fig. 2. The quantitative comparisons of average moisture content in three different cases among Harmathy's theoretical results, Experimental results (Harmathy, 1967) and the results obtained from the present model are shown in Fig. 3. The present model shows a better agreement to the experimental data than predictions from Harmathy's model. In this paper, only the overall results of case 1 as shown in Fig. 3 were plotted and
0.08 t
: PRESENT MODEL . . . .
EXP DATA : HARMATBY MODEL
6
8
0.07
I 0
2
4
10
12
Time | brs ] Fig. 3. Comparisons of average moisture content in drying clay slab at both sides.
200
S.-L. Su /Applied Clay Science 12 (1997) 189-207
o~o [ (a)
....
P~sE~ MODE~
0.0~ ]
....
HARMATHYMODEL
0.00 0.07"
i
0.06. 0.0~"
--
0.04"
0.03. 0.02
0-01' I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t=12 hr 0.00 0.0
0.2
0.4
0.0
0.8
1.0
0.4 0.6 × [cm]
0.8
1.0
X {cml 90O 800 700 ~.
61111
o <
g. 21111
lOO 0 0.0
0.2
Fig. 4. (a) Distribution of moisture in drying 1 cm thick clay slab at both sides. (b) Distribution of pressure in drying 1 cm thick clay slab at both sides.
S.-L Su /Applied Clay Science 12 (1997) 189-207
201
46 (C)
t--nh,
,111' 44
t ~ 3.0
38
--
t~li
--
t'l-O
--
t--L5
0.0
O&
0.4
0.6
0.8
1.0
Xleml Fig. 4 (continued). (c) Distribution of temperature in drying 1 e m thick clay slab at both sides.
Table 2 Values of the constants used in the equations Symbol
Value
Unit
R
8"3149×107
gcm2 s-2 K-l re°l-1
"qs
1"83X 1 0 - 4
gcm-I
Cl~ Cr~ Cpv Cpw Q e Kg Ks Kw Ma H Mw
5.67×10 -5 1.0063X 107 8.15X 106 1.8646×107 4.1793×107 2 . 4 4 1 8 x 10 l° 0.8 2.613X 103 2.207X 10 s 6.16X 104 28.952 0.3741 X 107 18.016
gs-3K -4 cm 2 s - 2 K -1 cm 2 $-2 K-1 cm2 s - 2 K - 1 c m 2 s - 2 K -1 cm 2 s -2
/~ D p, Pw at v Es Gs ks B
2 . 5 × 10 - x ° 0.256 2.75 0.99707 9.0X 10 - 6 0.25 2.068 x 101° 8.2736X 109 1.379×101° 18.39 2.478×10 -6 0.5 0.28
cm 2 c m 2 s -1 gcm -3 gcm -3 K -1
I.C.
Value
Unit
B.C.
Value
Unit
Min i Patm Ti~i
0.105 1,01325 X 106 306.2
cm 3 c m - 3 g cm s - 2 K
t/~ T~
0.010539 318.9
mol m o l - 1 K
gems -3 gcms -3 gcm s -3 gmol -l cm 3 s gg mol- 1
Symbol
K -l K-l K- t
hoF 1
h ~
Value
Unit s-t
g c m -1 s - 2 gem -] s-2 cm 3 sg -1 m o l c m - 2 s -1 cm cm 3 cm- 3
S.-L. Su//AppliedClayScience12 (1997)189-207
202
0.020
.
I
0.016
Z O
0.014
FIRST FALLING RATE PERIOD
F-
SECOND FALLING RATE PERIOD 0.008 -~
f
O.OIM
0.002 0.000 0
2
4
6
8
I)
12
Time [ hrs ] Fig. 5. Rate of evaporation curve in drying l c m thick clay slab at both sides.
used as the numerical example. The clay slab of case 1 was dried both sides at specified boundary conditions for 12 h. The physical parameters, empirical constants, initial conditions and boundary conditions of case 1 are listed in Table 2. The results of case 1 obtained from present model for the histories of moisture content and pore pressure are compared with those obtained using Harmathy's model and are plotted in Fig. 4a and b. The present model took into consideration the volume change of the solid component inside the clay body, whereas this change was neglected in Harmathy's model. The values for the peaks of moisture content and pore pressure were higher for the present model compared to the Harmathy's model, throughout the drying process. As the solid volume increases due to the thermal effect, the porosity decreases and it changes the elastic properties of clay body. These changes may have contributed to the inaccurate from Harmathy's model. The temperature distribution in the clay slab for different drying times was plotted in Fig. 4c. There was a relative sharp rise in temperature after 2.5 h to 3.0 h, which indicates the transient period from first to second falling rate period as shown in Fig. 5. The rate of evaporation on the surface of clay slab (Fig. 5) showed clearly the distinction between different periods: first falling rate and second falling rate period. During the first falling rate period (from 0 to 3 h), the continued migration of liquid water toward the drying surface, coupled with the decrease in the rate of evaporation, caused the curves for moisture distribution (Fig. 4a) during this period to level out to some extent. The duration of the first falling rate period was relatively short. In this period, moisture movement depended strongly on the effect
S.-L. Su /Applied Clay Science 12 (1997) 189-207
203
1
i
1 --1 --1
-I 0.0
0.2
0.4
0.6
0.8
1.0
X [cm] Fig. 6. Distributionof normal stress in drying 1 cm thickclay slab at both sides.
of surroundings. The moisture distribution curves are all convex (Fig. 4a). For the second :railing rate period, 3 to 12 h, the duration was very long and the curves of moisture distribution were no longer convex (Fig. 4a). The brick system at and close to the surface contained insufficient moisture for liquid migration to continue and moisture that sub~iequently migrates to the clay surface did so as vapor. The result revealed that in this period the evaporation-condensation mechanism played the dominant role. In other words, both the topological parameters of the clay brick system and the temperature gradient were important factors during this period of the drying process. The topological paramet~rs include pore structure and pore shape. All these observations from numerical simulation are consistent with the experimental results given in Harmathy (1969). The normal stress of in the plane of the slab is plotted in Fig. 6. The maximum tensile stress occurred at the surface, which is shown in this plot and had been reported by Cooper ,(1978). The stress distribution curves in Figs. 6 and 7 qualitatively agreed with the analysis of Scherer (1987) during the falling rate period of drying process. From the history of surface tension stress, the maximum tensile stress and the peak pore pressure were observed to occur simultaneously. This phenomenon indicates that pore pressure, instead tff temperature, plays the dominant role in the determination of stress distribution during the falling rate period of drying process. Since the total shrinkage stopped after
204
S.-L. Su / A p p l i e d Clay Science 12 (1997) 1 8 9 - 2 0 7
3°° I ~ x - O . O 200.
r.~ u~ ,.~ +<
ms 0
0 Z -:100
-:mo~
- 300, 0
xmO,37S
V x-O.S
~ :i
4
6 8 Time [ brs ]
10
)2
Fig. 7. The history of normal stress in drying 1 cm thick clay slab at both sides.
the end of first falling rate period (Ford, 1986), the tensile stresses decreased rapidly as shown in Fig. 7. This result was also consistent with Scherer (1987).
5. Conclusion
An analysis has been presented to describe the moisture transfer and induced stresses that develop during the falling rate period of the drying process of clay slabs in which the solid component was assumed to be elastic. From this study, several conclusions can be made. They are: (1) In Harmathy's model, the solid matrix is rigid and the porosity is unchangeable during the drying process. In the model developed in this study, the solid matrix is free to expand due to the drying effect that causes the decrease of porosity inside the clay slab. (2) The transient period from the first falling rate period to the second falling rate period caused a relatively sharp rise in temperature inside the clay body. (3) For the second falling rate period of the drying process, the duration was relatively long and the evaporation--condensation mechanism played the dominant role in moisture transfer. (4) The porosity decreased due to thermal expansion of the solid component during drying and in doing so changed the elastic properties. The porosity effects on the
S.-L Su /Applied Clay Science 12 (1997) 189-207
205
distributions of moisture and pore pressure were significant, in turn, on the distribution of stresses. (5) Owing to the use of different material properties, it was only possible to obtain the qualitative agreement in stress distribution with Scherer's analysis. (6) "Ilae maximum tensile stress occurred at the surface and caused surface cracks.
6. Nomenclature
B Cp Cpi D E e G h hD hDF hi
hT k ki
ki Ki
m M n ni P Pi
q qi (qi)j
Q R t T T~ T~ T~ V~ V/
characteristic length of the brick measured from the edge (cm) effective heat capacity at constant pressure (cm 2 s-2 K - 1) specific heat of ith-component (cm 2 s-2 K - I ) diffusivity of the gaseous mixture (cm 2 s- l) apparent Young's modulus (g cm -~ s -2) mass rate of evaporation per unit volume (g cm- 3 s- 1) apparent shear modulus of elasticity (g cm -1 s -2) macroscopic enthalpy (era 2 s- 2) effective mass transfer coefficient (mol cm -2 s- 1) mass transfer coefficient due to forced convection (mol cm -2 s- 1) enthalpy of ith-component (cm 2 s -2) effective heat transfer coefficient (g s -3 K -1) apparent bulk modulus (g cm -1 s -2) bulk modulus of ith-component (g cm-1 s- 2) coefficient of permeability of ith-component (cm 3 s g-1) effective thermal conductivity of ith-component (g cm s -3 K -1) volumetric moisture content (cm 3 cm -3) molecular weight of the gaseous mixture (g mol-1) topological constant of the clay brick absolute mass flux of ith-component (g cm -2 s - I ) pore pressure (g cm -1 s -2) pore partial pressure of ith-component apparent heat flux (g s-3) mass flow rate of ith-component ( g c m -2 s- l) mass flow rate of ith-component in jth-direction latent heat of evaporation of free water (cm 2 s-2) gas constant (g cm 2 s- 2 K - 1 mol- l) time (s) temperature rise of bulk volume (K) temperature of air (K) temperature of the enclosure of the system (K) surface temperature of clay brick (K) volume of ith-component per unit bulk volume (cm 3 cm -3) velocity of ith-component (cm s - I )
S.-L. Su /Applied Clay Science 12 (1997) 189-207
206
Greek: e eg Pi th e~j
porosity of the porous system (cm 3 cm -3) volume fraction of the gaseous mixture (cm 3 cm -3) density of ith-component (g cm -3) mole fraction of water vapor of the gaseous mixture ( m o l / m o l ) elastic strain elastic stress (g cm -1 s -2) Stefan Boltzman constant (g s -3 K -4) Poisson ratio linear thermal expansion coefficient of porous media (cm K -1) mass concentration of ith component per unit volume of the porous media (g
orij trs v a 6i
cm -3)
0",7 F~ 0 7/ ~b
Kronecker delta relative permeability mass rate of evaporation of ith-component per unit bulk volume ( g c m -3 s - 1) relative saturation (0 = re~e) coefficient of fluid viscosity (g c m - 1 s - 1) gravitational potential (g c m - 1 s - 2) rate of heat source (g cm-1 s-3
Subscripts: a atm g ga gv 1 s v w
air atmosphere gaseous mixture air in a gaseous mixture vapor in a gaseous mixture liquid solid vapor water ambient
References Biot, M.A., 1941. General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155-161. Biot, M.A., Willis, D.G., 1957. The elastic coefficients of the theory of consolidation. J. Appl. Mech 24, 594-599. Boley, B.A., Weiner, J.H., 1960. Theory of Thermal Stresses. John Wiley, New York. Cooper, A.R., 1978. Quantitative Theory of Cracking and Warping During the Drying of Clay Bodies. Ceramic Processing before Firing. Wiley Interscience. De Groot, S.R., Mazur, P., 1962. Nonequilibrium Thermodynamics. North Holland Pub. Co., Amsterdam. Ford, R.W., 1986. Ceramic Drying. Pergamon Press, New York. Harmathy, T.Z., 1969. Simultaneous moisture and heat transfer in porous systems with particular reference to drying. I&EC Fundamentals 8, 92-103.
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Harmathy, T.Z., 1967. Simultaneous Moisture and Heat Transfer in Porous Systems with Particular Reference to Drying, Ph.D. dissertation, University of Technology of Vienna, Vienna. Huang, C.L., 1979. Multi-phase moisture transfer in porous media subject to temperature gradient. Int. J. Heat Mass Transfer 22, 1295-1307. Krischer, O., 1940. Heat and Mass Transfer in Drying. VDI-Forschmgsh, Berlin. Lewis, W.K., 1921. Ind. Eng. Chem. 13, 427-432. Lykov, A.V., Mikhailov, Y.A., 1965. Theory of Heat and Mass Transfer. Daniel Davery Inc., New York. Pel, L., Ketelaars, A.A.J., Adan, O.C.G., Van Well, A.A., 1993. Determination of moisture diffusivity in porous media using scanning neutron radiography. Int. J. Heat Mass Transfer 36, 1261-1267. Philip, J.R., De Vries, D.A., 1957. Moisture movement in porous materials under temperature gradients. Trans. Am. Geophys. Union 38. Philip, J.K., 1972. Hydrostatics and Hydrodynamics in Swelling Media. Fundamentals of Transport Phenomena in Porous Media. Elsevier, New York. Scheidegger, A.E., 1960. The Physics of Flow through Porous Media, 2nd Ed. University of Toronto Press. Scherer, G.W., 1987. Drying gels. V. Rigid gels. J. Non-Cryst. Solids 92, 122.