Journal ArticleDOI

# 3D Modeling and Visualization of Non- Stationary Temperature Distribution during Heating of Frozen Wood

01 Jan 2013-Drvna Industrija (Šumarski fakultet Sveučilišta u Zagrebu)-Vol. 64, Iss: 4, pp 293-303
TL;DR: In this paper, a 3D mathematical model has been developed, solved, and verified for the transient nonlinear heat conduction in frozen and non-frozen wood with prismatic shape at arbitrary initial and boundary conditions encountered in practice.
Abstract: A 3-dimensional mathematical model has been developed, solved, and verified for the transient non-linear heat conduction in frozen and non-frozen wood with prismatic shape at arbitrary initial and boundary conditions encountered in practice. The model takes into account for the first time the fiber saturation point of each wood species, u fsp , and the impact of the temperature on u fsp of frozen and non-frozen wood, which are then used to compute the current values of the thermal and physical characteristics in each separate volume point of the material subjected to defrosting. This paper presents solutions of the model with the explicit form of the finite-difference method. Results of simu- lation investigation of the impact of frozen bound water, as well as of bound and free water, on 3D temperature distribution in the volume of beech and oak prisms with dimensions 0.4 x 0.4 x 0.8 m during their defrosting at the temperature of the processing medium of 80 °C are presented, analyzed and visualized through color contour plots.

### 3D modeliranje i vizualizacija nestacionarne

• Distribucije temperature tijekom zagrijavanja smrznutog drva Original scientific paper Izvorni znanstveni rad Received – prispjelo: 27.
• This paper presents solutions of the model with the explicit form of the finite-difference method.
• 3D mathematical model, frozen wood, finite difference method, temperature distribution, contour plots Sažetak Kreiran je i riješen 3D matematički model te provjeren za nelinearno provođenje topline u smrznutome i nesmrznutom drvu prizmatičnog oblika pri proizvoljnim početnim i rubnim uvjetima koji se susreću u praksi, also known as Keywords.
• Rad prikazuje rješenja modela s eksplicitnim oblikom metode konačnih razlika.

### 1. UVOD

• For the optimization of the control of the heating process of wood in veneer and plywood mills, it is necessary to know the temperature distribution at every moment of the process (Shubin, 1990; Trebula and Klement, 2003; Pervan, 2009).
• The heat energy, required for melting the ice, formed from bound water in the wood, has not been taken into account in these models.
• The models assume that the fiber saturation point is identical for all wood species (i.e. ufsp = 0.3 kg·kg-1 = const) and that the melting of the ice, formed from free water in the wood, occurs at 0 ºC.
• The complications and deficiencies indicated in these models have been overcome by a 2-dimensional mathematical model of the transient non-linear heat conduction in frozen and non-frozen logs suggested by Deliiski (2004, 2011).
• This paper also presents the results of simulation investigation of the impact of the frozen bound water and free water on 3D temperature distribution in the volume of beech and oak prisms with dimensions 0.4 x 0.4 x 0.8 m during their defrosting at the temperature of the processing medium of 80 °C.

### 2.1. 3D matematički model procesa odmrzavanja prizmatičnoga drvnog materijala

• Other equations quoted by the above authors present mathematical descriptions of wood density, r, and of its thermal conductivity, l, in different anatomical directions.
• This has been done using the method presented by Deliiski (2013) during the update of the mathematical description of l.

### 3.1. Computation of 3D temperature distribution in

• Izračun 3D raspodjele temperature u smrznutome drvnom materijalu tijekom njegova odmrzavanja Besides taking into account the stability condition for solving the 3D model, the value of the step tD is calculated so as to be divisible by the input value of INT, using the software package.
• It must be noted that there are no such almost horizontal sections in the change of wood temperature during defrosting of the ice formed only by bound water in the wood (Fig. 4).

### 3.2 Color visualization of 3D non-stationary

• Temperature distribution in prisms during defrosting 3.2.
• The results obtained by Visual Fortran for 3D temperature distribution in the volume of wooden prisms undergoing defrosting have been subjected to the following visualization with the help of the software Excel 2010.
• For the solution of the model, an explicit form of the finite-difference method is used, with the possibility of excluding any model simplifications.
• This paper was written as a part of the solution of the project “Modelling and Visualization of Wood Defrosting Processes in Technologies for Wood Thermal Treatment”, supported by the Scientific Research Sector of the University of Forestry in Sofia (Project 114/2011).

### 5. LITERATURA

• Modeling and Technologies for Steaming of Wood Materials in Autoclaves, also known as 3. Deliiski, N., 2003a.
• Steinhagen, H. P., 1991: Heat Transfer Computation for a Long, Frozen Log Heated in Agitated Water or Steam - A Practical Recipe. 21. Videlov, H., 2003: Drying and Thermal Treatment of Wood.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

............... Deliiski: 3D Modeling and Visualization of Non-Stationary Temperature ...
DRVNA INDUSTRIJA 64 (4) 293-303 (2013) 293
Nencho Deliiski
1
3D Modeling and
Visualization of Non-
Stationary Temperature
Distribution during Heating
of Frozen Wood
3D modeliranje i vizualizacija nestacionarne
distribucije temperature tijekom zagrijavanja
smrznutog drva

Received – prispjelo: 27. 1. 2013.
Accepted – prihvaćeno: 6. 11. 2013.
UDK: 630*812.141; 630*847
doi:10.5552/drind.2013.1306
AbSTRAcTA 3-dimensional mathematical model has been developed, solved, and veried for the transient
non-linear heat conduction in frozen and non-frozen wood with prismatic shape at arbitrary initial and boundary
conditions encountered in practice. The model takes into account for the rst time the ber saturation point of
each wood species, u
fsp
, and the impact of the temperature on u
fsp
of frozen and non-frozen wood, which are then
used to compute the current values of the thermal and physical characteristics in each separate volume point of
the material subjected to defrosting.
This paper presents solutions of the model with the explicit form of the nite-difference method. Results of simu-
lation investigation of the impact of frozen bound water, as well as of bound and free water, on 3D temperature
distribution in the volume of beech and oak prisms with dimensions 0.4 x 0.4 x 0.8 m during their defrosting at the
temperature of the processing medium of 80 °C are presented, analyzed and visualized through color contour plots.
Keywords: 3D mathematical model, frozen wood, nite difference method, temperature distribution, contour plots
Kreiran je i riješen 3D matematički model te provjeren za nelinearno provođenje topline u smrz-
nutome i nesmrznutom drvu prizmatičnog oblika pri proizvoljnim početnim i rubnim uvjetima koji se susreću u
praksi. Prvi put model uzima u obzir točku zasićenosti vlakanaca za svaku vrstu drva (u
fsp
) i utjecaj temperature na
u
fsp
smrznutoga i nesmrznutog drva, koji se primjenjuju pri izračunavanju trenutačne vrijednosti termo-zikalnih
svojstava u svakoj posebno deniranoj točki volumena materijala koji se odmrzava.
Rad prikazuje rješenja modela s eksplicitnim oblikom metode konačnih razlika. Rezultati simulacijskih istraživanja
o utjecaju zamrznute vezane vode te vezane i slobodne vode na 3D raspodjelu temperature u volumenu bukovih
i hrastovih prizmi dimenzija 0,4 x 0,4 x 0,8 m tijekom odmrzavanja pri temperaturi procesnog medija od 80
о
C
prezentirani su i analizirani te vizualizirani crtežima u boji.
Ključne riječi : 3D matematički model, smrznuto drvo, metoda konačnih razlika, raspodjela temperature , kon-
turni crteži
1
Author is professor at Faculty of Forest Industry, University of Forestry, Soa, Bulgaria.
1
Autor je profesor Fakulteta šumske industrije Šumarskog sveučilišta, Soja, Bugarska.

Deliiski: 3D Modeling and Visualization of Non-Stationary Temperature ... ...............
294 DRVNA INDUSTRIJA 64 (4) 293-303 (2013)
1 INTRODUCTION
1. UVOD
For the optimization of the control of the heating
process of wood in veneer and plywood mills, it is nec-
essary to know the temperature distribution at every
moment of the process (Shubin, 1990; Trebula and
Klement, 2003; Pervan, 2009). Considerable contribu-
tion was made to the calculation of non-stationary dis-
tribution of temperature in frozen and non-frozen logs,
and to the duration of their heating (Steinhagen, 1986,
1991). Later on, 1-dimensional and 2-dimensional
models were developed and solved (Steinhagen et al.,
1987; Steinhagen and Lee, 1988; Khattabi and Steinha-
gen, 1992, 1993, 1995), whose applications are limited
only to wood with moisture content above ber satura-
tion point.
The heat energy, required for melting the ice,
formed from bound water in the wood, has not been
taken into account in these models. The models assume
that the ber saturation point is identical for all wood
species (i.e. u
fsp
= 0.3 kg·kg
-1
= const) and that the melt-
ing of the ice, formed from free water in the wood, oc-
curs at 0 ºC.
However, it is known that there are signicant
differences between the ber saturation point of differ-
ent wood species (Požgai et. al., 1997; Videlov, 2003)
and that, depending on this point, the quantity of the
ice formed from free water in the wood melts at a tem-
perature in the range between -2 ºC and -1 ºC (Chudi-
nov, 1968, 1984). The complications and deciencies
indicated in these models have been overcome by a
2-dimensional mathematical model of the transient
non-linear heat conduction in frozen and non-frozen
logs suggested by Deliiski (2004, 2011).
This paper presents the development, verication
and solutions of an analog 3-dimensional mathematical
model of the transient non-linear heat conduction in
frozen and non-frozen wood with prismatic shape at
arbitrary initial and boundary conditions encountered
in practice. The model takes into account for the rst
time the ber saturation point of each wood species,
u
fsp
, and the impact of the temperature on u
fsp
of frozen
and non-frozen wood, which are then used to compute
the current values of the thermal and physical charac-
teristics in each separate volume point of the material
subjected to defrosting.
This paper also presents the results of simulation
investigation of the impact of the frozen bound water
and free water on 3D temperature distribution in the
volume of beech and oak prisms with dimensions 0.4 x
0.4 x 0.8 m during their defrosting at the temperature
of the processing medium of 80 °C.
2 MATHERIAL AND METHODS
2. MATERIJAL I METODE
2.1. 3D mathematical model of the defrosting
process of prismatic wood materials
2.1. 3D matematički model procesa odmrzavanja
prizmatičnoga drvnog materijala
The defrosting process of prismatic wood materi-
als during their thermal treatment can be described by
a non-linear differential equation of the thermal-con-
ductivity, using the Cartesian coordinates (Deliiski,
2003a):
( )
( )
( )
( )
( )
( )
( )
( )
t
l
+
t
l
+
+
t
l
=
t
t
r
z
zyxT
uT
zy
zyxT
uT
y
x
zyxT
uT
x
zyxT
uTuTc
,,,
,
,,,
,
,,,
,
,,,
),(,
pt
re
(1)
After the differentiation of the right side of equa-
tion (1) on the spatial coordinates x, y, and z, excluding
the arguments in the brackets for shortening of the re-
cord, the following mathematical model is obtained of
the non-stationary defrosting of wood materials with
prismatic shape subjected to heating:
2
p
2
2
p
2
t
2
2
t
2
r
2
2
re
l
+
l+
l
+
l+
+
l
+
l=
t
r
z
T
T
z
T
y
T
T
y
T
x
T
T
x
TT
c
(2)
with an initial condition
( )
0
0,,, TzyxT =
, (3)
and a boundary condition
( ) ( ) ( ) ( )
t=t=t=t
m
,0,,,,0,,,,0 TyxTzxTzyT
. (4)
For the solution of the system of equations (2) to
(4), it is necessary to make a mathematical description
of thermal and physical characteristics of the wood, c
e
,
l
r
, l
t
, l
p
, and of its density, r. Equations in (Deliiski,
2003a, 2011) and (Deliiski and Dzurenda, 2010) pre-
sent a mathematical description of the effective spe-
cic heat capacity coefcient, c
e
, of the frozen wood as
a sum of the capacities of the wood itself, c, and the ice
produced by freezing of the free water, c
fw
, and of the
hygroscopically bound water, c
bw
. Other equations
quoted by the above authors present mathematical de-
scriptions of wood density,
r
, and of its thermal con-
ductivity, l, in different anatomical directions.
The given mathematical descriptions of
e
c
,
r
l
,
t
l
, and
p
l
(Deliiski, 2011), which are part of the
model (2) to (4), have now been updated by taking into
account, for the rst time, the inuence of the ber
saturation point of wood species on the values of ther-
mal and physical characteristics during wood defrost-
ing, and the inuence of the temperature on ber satu-
ration point of frozen and non-frozen wood. This has
been done using the method presented by Deliiski
(2013) during the update of the mathematical descrip-
tion of l.
2.2. Transformation of 3D model to a form suitable
for programming
2.2. Transformacija 3D modela u odgovarajući oblik
za programiranje
The following system of equations (Equation 5)
has been derived by passing to nal increases in equa-
tion (2) with the usage of the same, as well as by the
explicit form of the nite-difference method described
by Deliiski (2003a, 2011) and taking into account the

............... Deliiski: 3D Modeling and Visualization of Non-Stationary Temperature ...
DRVNA INDUSTRIJA 64 (4) 293-303 (2013) 295
mathematical description of the thermal conductivity,
l, in different anatomical directions.
Since in practice prismatic materials subjected to
thermal treatment usually do not have a clear radial or
clear tangential orientation, and are partially radially or
partially tangentially oriented, then in equation (5) in-
0
l
in the observed two ana-
tomical directions, their average arithmetic value can
be used, as it determines the thermal conductivity at 0
°С perpendicular to the wood bers (Equation 6):
(6)
Also, the thermal conductivity at 0 °С in the di-
rection parallel to the bers λ
0p
can be expressed
through
0cr
l
using the equation
(7)
where the coefcient
depends on the
wood species (Deliiski, 2003a).
(5)
For uniformity of the calculations, it is reasona-
ble to use one step of the calculation mesh along the
spatial coordinates
=
=
zD
(see Fig. 1). Taking
into consideration this condition and equations (6) and
(7), the system of equations (5) becomes equation (8).
The initial condition (3) in the model is presented
using the following nite differences equation:
0
0
,,
TT
kji
=
. (9)
(8)
The boundary conditions (4) acquire the follow-
ing form suitatable for programming:
1
m
1
1,,
1
,1,
1
,,1
++++
===
nn
ji
n
ki
n
kj
TTTT . (10)
The presentation of a non-linear differential
equation (2) from the mathematical model through its
discrete analogue (8) corresponds to the setting of the
coordinate system and positioning of the nodes in the
mesh shown in Fig. 1, in which a non-stationary 3D
temperature distribution in prismatic wood materials
during their defrosting is calculated. The calculation
mesh for the solution of the model through the nite-
difference method is built on a 1/8 part of the prism
volume, because of its mirror symmetry with the re-
maining 7/8 parts of the prism volume.
The setting of the coordinate system, shown in
Fig. 1 allows, with the help of only one system of equa-
tions (8), to calculate the change in the temperature in
any mesh node of the volume of the prism subjected to
defrosting at the moment (n + 1)Dt using the already
calculated values of T at the preceding moment nDt.
Wide experimental studies have been performed
for the determination of a 1-, 2- and 3-dimensional
temperature distribution in the volume of frozen and
non-frozen oak, beech, poplar and pine








»
¼
º
«
¬
ª
w
Ww
O
w
w
»
¼
º
«
¬
ª
w
Ww
O
w
w
»
¼
º
«
¬
ª
w
Ww
O
w
w
Ww
Ww
U
z
zyxT
uT
zy
zyxT
uT
y
x
zyxT
uT
x
zyxT
uTuTc
,,,
,
,,,
,
,,,
,
,,,
),(,
pt
re
2
p
2
2
p
2
t
2
2
t
2
r
2
2
re
¸
¹
·
¨
§
w
w
w
Ow
w
w
O
¸
¸
¹
·
¨
¨
§
w
w
w
Ow
w
w
O
¸
¹
·
¨
§
w
w
w
Ow
w
w
O
Ww
w
U
z
T
T
z
T
y
T
T
y
T
x
T
T
x
TT
c

0
0,,, TzyxT
,

W W W W
m
,0,,,,0,,,,0 TyxTzxTzyT
.

>@


>@


>@

°
°
°
°
°
¿
°
°
°
°
°
¾
½
°
°
°
°
°
¯
°
°
°
°
°
®
»
¼
º
«
¬
ª
EE
'
O
»
»
¼
º
«
«
¬
ª
E
E
'
O
»
»
¼
º
«
«
¬
ª
E
E
'
O
U
W'J
2
1,,,,,,1,,1,,,,
2
0p
2
,1,,,,,,1,,1,
,,
2
0t
2
,,1,,,,,,1,,1
,,
2
0r
e
,,
1
,,
215,2731
2
15,2731
2
15,2731
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
TTTTTT
z
TTTTT
T
y
TTTTT
T
x
c
TT
.
2
0t0r
0cr
OO
O
.
0crp/cr0p
O O K ,
0cr
0p
p/cr
O
O
K








»
¼
º
«
¬
ª
w
Ww
O
w
w
»
¼
º
«
¬
ª
w
Ww
O
w
w
»
¼
º
«
¬
ª
w
Ww
O
w
w
Ww
Ww
U
z
zyxT
uT
zy
zyxT
uT
y
x
zyxT
uT
x
zyxT
uTuTc
,,,
,
,,,
,
,,,
,
,,,
),(,
pt
re
2
p
2
2
p
2
t
2
2
t
2
r
2
2
re
¸
¹
·
¨
§
w
w
w
Ow
w
w
O
¸
¸
¹
·
¨
¨
§
w
w
w
Ow
w
w
O
¸
¹
·
¨
§
w
w
w
Ow
w
w
O
Ww
w
U
z
T
T
z
T
y
T
T
y
T
x
T
T
x
TT
c

0
0,,, TzyxT
,

W W W W
m
,0,,,,0,,,,0 TyxTzxTzyT
.

>@


>@


>@

°
°
°
°
°
¿
°
°
°
°
°
¾
½
°
°
°
°
°
¯
°
°
°
°
°
®
»
¼
º
«
¬
ª
EE
'
O
»
»
¼
º
«
«
¬
ª
E
E
'
O
»
»
¼
º
«
«
¬
ª
E
E
'
O
U
W'J
2
1,,,,,,1,,1,,,,
2
0p
2
,1,,,,,,1,,1,
,,
2
0t
2
,,1,
,,,,,1,,1
,,
2
0r
e
,,
1
,,
215,2731
2
15,2731
2
15,2731
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
TTTTTT
z
TTTTT
T
y
TTTTT
T
x
c
TT
.
2
0t0r
0cr
OO
O
.
0crp/cr0p
O O K
,
0cr
0p
p/cr
O
O
K








»
¼
º
«
¬
ª
w
Ww
O
w
w
»
¼
º
«
¬
ª
w
Ww
O
w
w
»
¼
º
«
¬
ª
w
Ww
O
w
w
Ww
Ww
U
z
zyxT
uT
zy
zyxT
uT
y
x
zyxT
uT
x
zyxT
uTuTc
,,,
,
,,,
,
,,,
,
,,,
),(,
pt
re
2
p
2
2
p
2
t
2
2
t
2
r
2
2
re
¸
¹
·
¨
§
w
w
w
Ow
w
w
O
¸
¸
¹
·
¨
¨
§
w
w
w
Ow
w
w
O
¸
¹
·
¨
§
w
w
w
Ow
w
w
O
Ww
w
U
z
T
T
z
T
y
T
T
y
T
x
T
T
x
TT
c

0
0,,, TzyxT
,

W W W W
m
,0,,,,0,,,,0 TyxTzxTzyT
.

>@


>@


>@

°
°
°
°
°
¿
°
°
°
°
°
¾
½
°
°
°
°
°
¯
°
°
°
°
°
®
»
¼
º
«
¬
ª
EE
'
O
»
»
¼
º
«
«
¬
ª
E
E
'
O
»
»
¼
º
«
«
¬
ª
E
E
'
O
U
W'J
2
1,,,,,,1,,1,,,,
2
0p
2
,1,,,,,,1,,1,
,,
2
0t
2
,,1,
,,,,,1,,1
,,
2
0r
e
,,
1
,,
215,2731
2
15,2731
2
15,2731
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
TTTTTT
z
TTTTT
T
y
TTTTT
T
x
c
TT
.
2
0t0r
0cr
OO
O
.
0crp/cr0p
O O K ,
0cr
0p
p/cr
O
O
K
2
>@
°
°
°
°
°
¿
°
°
°
°
°
¾
½
°
°
°
°
°
¯
°
°
°
°
°
®
»
»
¼
º
«
«
¬
ª
E
»
»
¼
º
«
«
¬
ª
E
'U
W'JO
2
1,,,,p/cr
2
,1,,,
2
,,1,,
,,p/cr1,,1,,p/cr
,1,,1,,,1,,1
,,
2
e
0cr
,,
1
,,
)(
)()(
)24()(
)15,273(1
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
TTK
TTTT
TKTTK
TTTT
T
xc
TT
.
0
0
,,
TT
kji
.
1
m
1
1,,
1
,1,
1
,,1
nn
ji
n
ki
n
kj
TTTT .

Deliiski: 3D Modeling and Visualization of Non-Stationary Temperature ... ...............
296 DRVNA INDUSTRIJA 64 (4) 293-303 (2013)
Figure 1 Positioning of nodes in a 3D calculation mesh of a discretized wooden prism
Slika 1. Pozicioniranje čvorova u 3D računskoj mreži u diskretiziranoj drvenoj prizmi
prismatic materials during their thermal treatment. The
values of the coefcient
in equation (8) have
been determined through the solution of the model
with the same initial and boundary conditions in order
to achieve maximum conformity between the calculat-
ed and experimental results.
It has been determined that the coefcient
p/cr
K
has the following values: for oak K
p/cr
=1.76, for beech
K
p/cr
=1.88, for poplar K
p/cr
=2.03, and for pine K
p/cr
=2.26
(Deliiski, 2003a, 2011).
3 RESULTS AND DISCUSSION
3. REZULTATI I RASPRAVA
3.1. Computation of 3D temperature distribution in
frozen wood during its defrosting
3.1. Izračun 3D raspodjele temperature u smrznutome
drvnom materijalu tijekom njegova odmrzavanja
For the numerical solution of the above presented
mathematical model, a software package has been de-
veloped in FORTRAN and integrated in the calculation
environment of Visual Fortran Professional developed
by Microsoft, as a part of the Windows Ofce software
(Deliiski, 2011).
With the help of this software package, 3D tem-
perature changes of beechwood (Fagus Silvatica L.)
and oakwood (Quercus petraea Liebl.) prisms with di-
mensions d = 0.4 m, b = 0.4 m, L = 0.8 m, initial tem-
perature of t
0
= -40 °C and two values of wood mois-
ture content u = 0.3 kg·kg
-1
and u = 0.6 kg·kg
-1
have
been studied during their 20 h heating with the inter-
mediate stage of melting at the heating temperature of
t
m
= 80 °C. The prisms with u = 0.3 kg·kg
-1
contain the
maximum possible quantity of frozen bound water in
beech and oak wood and contain no ice in the cell lu-
mens (i.e. contain no ice from free water). The prisms
with u = 0.6 kg·kg
-1
not only contain frozen bound wa-
ter but also contain a signicant quantity of frozen free
water.
The heating medium temperature, t
m
, increases
exponentially from t
m0
= t
0
to t
m
= 80 °C = const with
the time constant of 1800 s. This increasing of t
m
at the
beginning of the heating process of prisms can be seen
in Fig. 4 and 5. The values of d, b, L, t
m
, and u have
been selected so as to correspond to cases often en-
countered in practice.
The duration of 20 h of the prism heating at t
m
=
80 °C has been proven suitable for complete melting of
the ice in the studied prisms. The calculations have
been done with average values of r
b
= 560 kg·m
-3
and
= 0.31 kg·kg
-1
of the beech wood and of r
b
= 670
kg·m
-3
and
= 0.29 kg·kg
-1
of the oak wood (Vide-
lov, 2003; Deliiski and Dzurenda, 2010).
The computations have been carried out in a step
on the spatial coordinates
= 0.001 m = 10 mm, i.e.
with the nodes M = 1 + [d/(2
xD
)] = 21 and N = 1 + [b/
(2
)] = 21 along the x and y coordinates, respective-
ly, and KD = 1 + [L/(2
xD
)] = 41 along the z coordi-
nate. This means that the calculation meshes in the vol-
ume of the prisms consist of 21 x 21 х 41 = 18 081
nodes in total.
The step on time coordinate,
, which is deter-
mined by the software package that keeps the stability
condition (Deliiski, 2011) of 3D solution of the explic-
it form of the nite-difference method and takes into
account the maximum values of λ and с
е
during wood
defrosting process, is as follows:
▪ for beech wood: Δτ = 30 at u = 0.3 kg·kg
-1
and Δτ =
25 at u = 0.6 kg·kg
-1
;
▪ for oak wood: Δτ = 40 at u = 0.3 kg·kg
-1
and Δτ = 30
at u = 0.6 kg·kg
-1
.
It takes 30 to 45 s to compute the temperature
distribution in the volume of each of the studied prisms
during a 20 h thermal treatment using the above values
2
>@
°
°
°
°
°
¿
°
°
°
°
°
¾
½
°
°
°
°
°
¯
°
°
°
°
°
®
»
»
¼
º
«
«
¬
ª
E
»
»
¼
º
«
«
¬
ª
E
'U
W'JO
2
1,,,,p/cr
2
,1,,,
2
,,1,,
,,p/cr1,,1,,p/cr
,1,,1,,,1,,1
,,
2
e
0cr
,,
1
,,
)(
)()(
)24()(
)15,273(1
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
n
kji
TTK
TTTT
TKTTK
TTTT
T
xc
TT
.
0
0
,,
TT
kji
.
1
m
1
1,,
1
,1,
1
,,1
nn
ji
n
ki
n
kj
TTTT .

............... Deliiski: 3D Modeling and Visualization of Non-Stationary Temperature ...
DRVNA INDUSTRIJA 64 (4) 293-303 (2013) 297
for
with the help of Intel Pentium (4) CPU 3.0
GHz processor. Using the input data for solving the
model, the value for the interval (INT) is given in sec-
onds. After completing each INT from the beginning of
the process, the calculated temperature distribution in
the prism volume is recorded on computer hard-drive.
The records can be consequently seen on a monitor,
graphically processed, and/or printed. Besides taking
into account the stability condition for solving the 3D
model, the value of the step
is calculated so as to
be divisible by the input value of INT, using the soft-
ware package.
Fig. 2 and 3 show the tables with the computed
temperature distribution in 121 nodes of the calcula-
tion mesh in the central cross-section of the beech
prisms at every 5 h of the defrosting process.
Fig. 4 and 5 shows the temperature change of the
surface of beech and oak prisms subjected to defrost-
ing, which is equal to t
m
, as well as of t in 6 character-
istic points of their volume.
The rst three characteristic points with coordi-
nates (d/4, b/8, L/8), (d/4, b/4, L/4), and (d/4, b/4, L/2)
allow for the tracking of the inuence on the defrosting
process of the gap from the prisms base (see Fig. 1 –
Figure 2 Change in t in the nodes of the calculation mesh, situated in the central cross section of a beech prism with
dimensions 0.4 x 0.4 x 0.8 m and u = 0.3 kg·kg
-1
during every 5 h of defrosting at t
m
= 80°C
Slika 2. Promjene temperature u čvorovima računske mreže smještenima na središnjemu poprečnom presjeku bukove prizme
dimenzija 0,4 x 0,4 x 0,8 m i u = 0,3 kg·kg
-1
tijekom svakih 5 h odmrzavanja pri temperaturi t
m
= 80°C

##### Citations
More filters
01 Jan 2013
TL;DR: In this paper, a mathematical model and an approach for calculating the specific heat energy needed for melting of the ice in the wood above the hydroscopi c diapason, ice q, have been suggested.
Abstract: A mathematical model and an approach for calculation of the specific heat energy needed for melting of the ice in the wood above the hydroscopi c diapason, ice q , have been suggested. The model takes into account to a maximum degree the physics of the processes of melting of the ice, formed by both bound and free water in the wood. It reflects the influence of the temperature, wood moisture content, wood density, and for the first time also the influence of fiber saturation point fsp u of each wood type on ice q during wood defrosting and the influence of temper ature on fsp u of frozen wood. An equation for calculation of the specific heat e nergy needed for melting of the frozen bound water in the wood above the hygroscopic diapason, bwm

3 citations

Journal ArticleDOI
TL;DR: In this article, a 2D non-stationary temperature distribution in logs subjected to freezing and change in the latent heat fluxes of the free and bound water in logs during the freezing process is investigated.
Abstract: This paper suggests a methodology for mathematical modelling and research of two interconnected problems: 2D non-stationary temperature distribution in logs subjected to freezing and change in the latent heat fluxes of the free and bound water in logs during the freezing process. For the purpose of this methodology, a 2-dimensional mathematical model has been created, solved, and verified for the transient non-linear heat conduction in logs during their freezing at convective boundary conditions. The model includes a mathematical description of the specific latent heat fluxes, qLHv-fw and qLHv-bw, formed by the freezing of the free and bound water in the logs, respectively. The paper presents solutions of the model with explicit form of the finite-difference method in the calculation environment of Visual Fortran Professional and its verification in accordance with our own experimental studies. The paper presents the results of simulation analysis of 2D non-stationary temperature distribution in the longitudinal section of pine log with a diameter of 0.24 m, length of 0.48 m, and moisture content above the hygroscopic range during its 30-hour freezing in a freezer at the temperature of the processing air medium of approximately –30 °C. The change in the latent heat fluxes qLHv-fw and qLHv-bw during the log freezing is presented, visualized, and analyzed.

2 citations

##### References
More filters
Journal Article
TL;DR: In this paper, an enthalpy method to compute transient temperatures of logs is presented, which handles phase change at a distinct temperature, which is an advantage over a previous (temperature) method.
Abstract: This paper discusses an enthalpy method to compute transient temperatures of logs. The logs may be initially frozen. It is assumed that the logs are subjected to radial heating in agitated water. The method handles phase change at a distinct temperature, which is an advantage over a previous (temperature) method. Calculations for four test logs were performed by a computerized, explicit finite-difference scheme called LOGHEAT. Model and experiment closely agreed with each other. Simplified "by hand" calculations were also satisfactory.

35 citations

Book ChapterDOI
21 Oct 2011
TL;DR: In this article, Deliiski et al. studied the influence of a few dozen factors on the heating and cooling processes of the capillary tube vessels, small pores or sharp, narrow indentions of bigger pores, and concluded that the correct and effective control of the heating or cooling processes is possible only when its physics and the weight of influence of each of the mentioned above as well as of many other specific factors for the concrete capillary porous body are well understood.
Abstract: Capillarity is a well known phenomenon in physics and engineering. Porous materials such as soil, sand, rocks, mineral building elements (cement stone or concrete, gypsum stone or plasterboards, bricks, mortar, etc.), biological products (wood, grains, fruit, etc.) have microscopic capillaries and pores which cause a mixture of transfer mechanisms to occur simultaneously when subjected to heating or cooling. In the most general case each capillary porous material is a peculiar system characterized by the extremely close contact of three intermixed phases: gas (air), liquid (water) and solid. Water may appear in them as physically bounded water and capillary water (Chudinov, 1968, Twardowski, Richinski & Traple, 2006). Both the bounded water and the capillary water can be found in liquid or hard aggregate condition. Physically bounded water co-operates with the surface of a solid phase of the materials and has different properties than the free water. The maximum amount of bounded water in porous materials corresponds to the maximal hygroscopicity, i.e. moisture absorbed by the material at the 100% relative vapour pressure. The maximum hygroscopicity of the biological capillary porous bodies is known as fibre saturation point. Capillary water fills the capillary tube vessels, small pores or sharp, narrow indentions of bigger pores. It is not bound physically and is called free water. Free water is not in the same thermodynamic state as liquid water: energy is required to overcome the capillary forces, which arise between the free water and the solid phase of the materials. For the optimization of the heating and/or cooling processes in the capillary porous bodies, it is required that the distribution of the temperature and moisture fields in the bodies and the consumed energy for their heating at every moment of the process are known. The intensity of heating or cooling and the consumption of energy depend on the dimensions and the initial temperature and moisture content of the bodies, on the texture and microstructural features of the porous materials, on their anisotropy and on the content and aggregate condition of the water in them, on the law of change and the values of the temperature and humidity of the heating or cooling medium, etc. (Deliiski, 2004, 2009). The correct and effective control of the heating and cooling processes is possible only when its physics and the weight of the influence of each of the mentioned above as well as of many other specific factors for the concrete capillary porous body are well understood. The summary of the influence of a few dozen factors on the heating or cooling processes of the

33 citations

Journal ArticleDOI
A. Khattabi
TL;DR: In this article, the authors present a theoretical solution to the heating of logs, frozen or not, having any length-to-diameter ratio, based on the enthalpy method and makes use of an explicit finite-difference scheme.
Abstract: This paper presents a theoretical solution to the heating of logs, frozen or not, having any length-to-diameter ratio. The solution is based on the enthalpy method and makes use of an explicit finite-difference scheme. The model has been verified for nonfrozen logs, assuming a longitudinal vs. radial thermal diffusivity ratio of two. The diffusivity of frozen logs still needs to be experimentally determined before the model can be computerized for the whole range of interest.

25 citations

Journal Article
TL;DR: In this paper, a temperature method and an enthalpy method for axisymmetric thawing and heating of frozen logs were used to calculate the transient temperature profiles of the logs, and the computerized finite difference program HEAT was used in conjunction with this method.
Abstract: Transient temperature profiles of frozen logs subjected to axisymmetric thawing and heating were calculated by a temperature method and an enthalpy method. The present paper discusses only the temperature method, which uses the conventional (temperature) formulation of the nonlinear heat conduction equation. This approach required the specification of a thawing temperature interval over which the latent heal was incorporated in the specific heat. Thermal properties were varied with position and temperature, and changed discontinuously with the phase. The log surface temperature was specified. The computerized finite-difference program HEAT was used in conjunction with this method. Computed temperature profiles were in overall agreement with experimental data obtained from heating logs in agitated water.

23 citations

Journal ArticleDOI
TL;DR: In this paper, an exemplary computation of transient temperature as a function of heating time for a log that was initially frozen and is heated in agitated water or steam is presented. But this method requires the log length must be at least four log diameters for this method to work correctly.
Abstract: This paper presents, step by step, an exemplary computation of transient temperature as a function of heating time for a log that was initially frozen and is heated in agitated water or steam. An unfrozen log heating situation can be handled likewise and is less complex. Log length must be at least four log diameters for this method to work correctly. The method, based on an enthalpy approach with finite-difference technique, can be computerized and applied for parametric studies and process control in veneer- and plywood mills, waferboard mills, etc.

23 citations