This paper describes the one-dimensional, transient heat conduction in a rectangular piece of Pyinkado (Xylia xylocarpa) with cross grain and in an orthotropic wooden cylinder. Computerized solutions of a generalized, nonlinear heat equation are derived by discretizing the time domain using finite element techniques.
Trang 1ANALYSIS OF TRANSIENT HEAT CONDUCTION IN PYINKADO (Xylia xylocarpa) USING
FINITE-ELEMENT SOLUTIONS
Bui Thi Thien Kim1, Hoang Thi Thanh Huong1, Ho Xuan Cac2
1 Nong Lam University
2 Associate Technology And Science Sylvicultre
Information:
Received: 10/09/2018
Accepted: 11/02/2019
Published: 11/2019
Keywords:
Finite element method,
distribution of temperature on
wood
ABSTRACT
This paper describes the one-dimensional, transient heat conduction in a rectangular piece of Pyinkado (Xylia xylocarpa) with cross grain and in an orthotropic wooden cylinder Computerized solutions of a generalized, nonlinear heat equation are derived by discretizing the time domain using finite element techniques A simplified example of linear heat conduction
in cylindrical coordinates illustrates how to apply the finite element solutions The accuracy of the solutions for this special case is evaluated via comparing them with a well-known exact solution The results could give significant information for fire-resistant solutions in house architecture and design
1 INTRODUCTION
In order to analyze an engineering system, a
mathematical model is developed to describe the
behavior of the system The mathematical
expression usually consists of differential
equations and given conditions These
differential equations are usually very difficult to
solve if handled analytically The alternative
way to solve the differential equation is using
numerical methods The numerical method has
some advantages, not only can it solve
complicated equations but also can it reduce the
cost needed to make experiments Moreover,
numerical methods are able to predict the
physical phenomena so it can be studied and
implemented to make some devices
Considering the advantage of the numerical
method, this paper analyzes physics phenomena
in a heat transfer problem on Pyinkado materials
The knowledge of transient temperature profiles plays an important role in some wood uses, for instance, for simulating heat conditioning of logs
in veneer and plywood mills and temperature response of lumber in dry kilns Since wood is a hygroscopic and porous medium, heat transfer may occur by means of several modes: conduction, convection and radiation However, all transfers can be accounted for by using only Fourier's law of heat conduction associated with
an effective thermal conductivity
The values of these components can be determined by knowing the values of the principal conductivities and of the rotational angles between the geometric and the orthotropic axes The likelihood of obtaining an exact solution to the problem of heat conduction
in wood is negligible Therefore, approximated solutions using numerical methods have been
Trang 2developed for special cases The purpose of this
work is to develop a more general,
three-dimensional finite difference solution for heat
conduction in wood that is shaped as a
rectangular piece with cross grain or as an
orthotropic cylinder The derivation of a one- or
two-dimensional solution, then, is straight
forward
2 MATERIALS AND METHODS
2.1 Materials
Specimens of green Pyinkado (Xylia Xylocarpa)
(20x40x150mm), which are applied a series of
experimental runs, were taken from Dong Nai,
Vietnam
2.2 Methods
The basic idea in the finite element method is to
find the solution of a complicated problem by
replacing it by a simpler one Since the actual
problem is replaced by a simpler one; the
solution is only an approximation rather than the
exact one In the finite element method the
solution domain is divided into small domain
called elements; these elements connected each
other by nodes, therefore this method is called
finite element This subdivision process is called
discretization In each element, a convenient approximate solution is assumed and the conditions of overall equilibrium of the structure are derived The derivation is using several mathematical analyses, likewise integration by parts, weighted residual Once this analyses done, each element will form matrix equation, which resulting in global matrix equations These global matrix equations solved by MATLAB to get the solution for each node All this processes are the basic procedures for the finite element method Systematically, each of the basic procedures will be discussed separately for conduction and convection problem
We use finite element method to determine the transient distribution of temperature on wood (Pyinkado) We research of a temperature-dependent T coefficient with the thermal conductivity coefficient k as show in Fig 1 , with the t-thickness and z direction (constant), the amount of heat generated in the element is
Q (W/m3) As the amount of heat transferred to the differential volume plus the amount of heat generated must be equal to the amount of heat transferred, we have (Nguyễn Lương Dũng (1993) and Phan Anh Vũ (1994))
1 +
+
+ +
+
= + +
y x
x sti y
y
q q t dy dx x
q q Q t Qdxdy t
dx q
t
dy
From equation (2.1), to find out: − = 0
+
Q y
q x
qx y
Eq.(2.2)
To change qx = − k T x ; qy = − k T y , Qsti=Qsti+1 because steady state so we have heat conduction equation:
0
= +
+
Q y
T k y x
T
k
For the two-dimensional case as shown in Figure 1, heat move along the x y axis Under constant pressure condition, Luikov (1966) equations
Trang 3Differential equation described two dimensional conductive process (2.3) was special case of general equation for Helmholtz’s conductive process (O.C Zienkiewicz (2000);Kwon, Y.W and Bang, H (2000);Zienkiewicz, O C and Taylor, R.L.(1989)
where T is temperature, t time, K thermal conductivity coefficients, respectively
Boundary conditions
Equation (2.3) must be solved with certain boundary conditions We have T = T0 on wallside S T; We will be used triangular elements to solve conductive problem (Figure 1)
We expressed heating field in element through:
T = T1 N1 + T2 N2 + T3 N3 Eq (2.4) [3,4,5]
or: T = NT e
Figure 1 first clas triangular elements in this problem
where:
− −
= 1
T T T
T = 1 2 3 was temperature at dots which was finding unknown with triangular elements, we have
x
1
2
3
1
2
3
= 1
= 1
y
T(x,y)
T1
T3
T2
Reference element
A
(b) (a)
A
A-A
y
x
Q
q y
q x
d x
d y
Trang 4x = N1 x1 + N2 x2+ N3 x3
To find out
+
=
y
T x x
T
T
+
=
y
T x x
T T
Or
=
y T x
T J T
T
J was Jacobian matrix which was defined :
=
31 31
21 21
y x
y x
J
Where: x ij = x i - x j ; y ij = y i - y j và det J = 2 Ae , here A e is the area of the triangle element inverse (1.7), to get:
e
T x
x
y y
J T
T J y
T
x
T
−
−
−
−
=
=
−
1 0 1
0 1 1 det
1
21 31
21 31
1
Eq (2.8)[5,8]
Or :
e
TT B y T x
T
=
Where :
=
−
−
−
−
=
21 13 32
12 31 23
21 31 21 31
21 31 31 21
det
1 det
1
x x x
y y y J x
x x x
y y y y
J
Trang 5Results:
e T T T
T e
T B B T y T x T
y
T x
T y
T
x
T
=
=
+
Set up functions
As reported, we need to solve equation (2.3) with boundary conditions (i) T = T0 on ST We solve this equation similar minimum this function :
tdA QT y
T k x
T
k
A
−
+
=
2
So that T = T0 on S T
To change equation (2.11) in two first term in first integral of , we will be had :
e T T e e
e T T e e
e T T T e T
e
e T T T T e A
T K T T
k T T
B B ktA
T
tdA T B B kT
tdA y
T k x
T k
2
1 2
1 2
1
2
1
2
=
=
=
=
+
=
Where conductive matrix of element was defined by:
T T T e
And conductive matrix of system:
=
e T
When Q =Q e was contans in element; thick of element t = const
e
e
T Q e
e A
T r T
NtdA Q
−
=
Vì NidA Ae
3
1
=
, so that calory vector :
Trang 6 T e
e Q
tA Q
3
Final, we have this function (2.12) in the form:
T R KT
TT − T
=
2
1
Where :
( )
=
e Q
r
To minimum functions must perform so that satisfy with conditions T = T0 on all the note on
S T From there, to build problem apply the finite element method to define heating elements on wood following the footsteps such as:
• Elemental meshing
• Numbering each elements
• Establish the coordinate system and determine the coordinates of the nodes in each element
• Calculate the thermal conductivity of each element
+ Compute the Jacobi matrix of the element + Calculates the matrix B of the element + Calculate thermal conductance ke
• Calculate the thermal conductivity of the coefficient K
• Solve the equation for determining the temperature vector at the Te node in the systemKTe= R
3 RESULTS AND DISCUSSION
The objective of this paper is finding the temperature distribution in wood as shown in figure 4 The wood considered natural material, inside the wood consist a solid substance The initial temperature inside the wood is considered
to 30 the wood heated on the top sidewall while the bottom side is keeps at 30 temperature, meanwhile the left and right of the wood is remain insulated Wood board with a thermal conductivity k = 0.17W / m.K have the boundary conditions The two opposite sides kept at a temperature of 1000C = 373 K and below kept at
300C = 303K; Left side must be insulated Determine the temperature distribution on the wood section The face of wood which was surveyed to divide 8 elements with 10 dots is illustrated as Figure 4
vector at Te dots for system
Trang 7Following the diagram with element divided, we conducted to join many elements
Table 1: Diagram with element divided
Freedom level Elements
Following the equation we have:
=
21 13 32
12 31 23 det
1
x x x
y y y J
BT
We calculated:
−
−
=
100 100 0
0 50 50 1
−
−
=
0 100 100
50 0 50 2
−
−
=
0 100 100
50 50 0
3
B
−
−
=
100 100 0
50 0
50 4
−
−
=
100 100 0
0 50 50 5
−
−
=
0 100 100
50 0 50 6
B
−
−
=
0 100 100
50 50 0
7
−
−
=
100 100 0
50 0
50 8
B
k=0,17W/m.
K
2
3
1
5
2
0,02m 0,04m
T=1000
Figure 4 Conductive problem model
T=300
4
5
T=1000
1
7
6
6
10
8
9
Trang 8Apply the equation k =T kAeBT TBT (t=1 unit), we calculated heating matrix of element
−
−
−
−
=
17 , 0 17 , 0 0
17 , 0 2125 , 0 0425 , 0
0 0425 , 0 0425 , 0 1
T
k
−
−
−
−
=
0425 , 0 0 0425 , 0
0 17
, 0 17 , 0
0425 , 0 17 , 0 2125 , 0 2
T
k
−
−
−
−
=
0425 , 0 0425 , 0 0
0425 , 0 2125 , 0 17 , 0
0 17
, 0 17
, 0 3
T
k
−
−
−
−
=
2125 , 0 17 , 0 0425 , 0
17 , 0 17
, 0 0
0425 , 0 0
0425 , 0 4
T
k
−
−
−
−
=
17 , 0 17 , 0 0
17 , 0 2125 , 0 0425 , 0
0 0425 , 0 0425 , 0 5
T
−
−
−
−
=
0425 , 0 0 0425 , 0
0 17
, 0 17 , 0
0425 , 0 17 , 0 2125 , 0 6
T
k
−
−
−
−
=
0425 , 0 0425 , 0 0
0425 , 0 2125 , 0 17 , 0
0 17
, 0 17
, 0 7
T
−
−
−
−
=
2125 , 0 17 , 0 0425 , 0
17 , 0 17
, 0 0
0425 , 0 0
0425 , 0 8
T
k
Conductive matrix of system (dimensions 10x10) will be constructed based on join table of this
element Boundary condition T = 1000C at dot 9 and 10, T = 300C at dot 1 and 2 we have matrix
Table 2 Matrix KT=R
We use Matlab programs to solve matrix (KT = R) with results:
T1 = 300C = 303K; T2 = 300C = 303K; T3 = 47,450C = 320,45K ; T4= 47,450C = 320,45K; T5= 64,970C
= 337,97K; T6 = 64,970C = 337,97K; T7 = 82,480C = 355,48K; T8 = 82,480C = 355,48K; T9 = 1000C
= 373K; T10 = 1000C = 373K
Trang 9Figure 5 Conductive problem model
4 CONCLUSIONS
This paper presents the study and
implementation of finite element method to find
the temperature distribution inside wood
material (Pyinkado) The temperature
distribution were analyzed by the help of the
contour chart It was shown that the temperature
parameter governs the conduction on the heated
sidewall As the temperature increase, the
temperature in the sidewall becomes increase
too The results of the heated sidewall
conduction yield to the boundary condition for
the convection diffusion inside the wood It was
shown in contour map for each layer inside of
wood The temperature distribution inside the
wood was dominated by the temperature
conditions on the heated sidewall The study will
contribute to predicting the heat transfer process
in the logs, which will be useful for the drying
and thermal processing of wood in the wood
processing industry
5 ACKNOWLEDGEMENTS
I would like to express my endless thanks and
gratefulness to my supervisors and my students
REFERENCES Nguyen Luong Dung, (1993) Finite element
method in mechanics Ho Chi Minh City University of Technology, Vietnam
Phan Anh Vu (1994), Finite element method in
structural calculations, Ho Chi Minh City
University of Technology Ho Chi Minh,
Youth Publishing House
Dehghan, M (2004) On Numerical Solution of
the One-Dimensional Convection-Diffusion Equation Mathematical Problems in Engineering 2005
O.C Zienkiewicz (2000), The finite element
method, McGraw - Hill Kwon, Y.W and Bang, H (2000), The Finite
(2nded.)Florida: CRC Press LLC
Rannacher, R (1999) Finite Element Method for the Incompressible Navier-Stokes Equations Heidelberg: Institute of Applied Mathematics University of Heidelberg Rao, S (2005) The Finite Element Method in Engineering (4thed.).Oxford: Elsevier Inc Zienkiewicz, O C and Taylor, R.L.(1989), The Finite Element Method, Vol1 Basic Formulation and Linier Problems, 4thEd., McGraw Hill, London