Analysis of transient heat conduction in pyinkado (xylia xylocarpa) using finite element solutions

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.

T BT T  y     y   T   T   T   +   =   x   y   x Eq (2.11) [6-8] 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 : 2   T    T   − 2QT tdA  =  k   + k  A   x    y   Eq (2.12) [3-8] So that T = T0 on ST To change equation (2.11) in two first term in first integral of , we will be had :  T    T   tdA  =  k   + k  A   x   y  = e T T kT e BT BT T e tdA  2e Eq (2.13) [3-8] T T T T =  T e ktAe BT BT T e = T e kT T e = T e K T T e e e Where conductive matrix of element was defined by: kT = ktAe BT BT T Eq (2.14) [6] And conductive matrix of system: K T =  kT Eq (2.15) [3] e When Q =Qe was contans in element; thick of element t = const −  QTtdA = −  e A Vì    Qe NtdA T e = −    e   r Q T T e e  N dA = A , so that calory vector : i e e 95 Eq (2.16) [3] AGU International Journal of Sciences – 2019, Vol (4), 91 – 99 rQ = Qe tAe 1 1T Eq (2.17) [6] Final, we have this function (2.12) in the form:  = T T KT − R T T Eq (2.18) [6,7] Where : K =  kT Eq (2.19) [6-8] R =  (rQ ) Eq (2.20) [6-8] e To minimum functions  must perform so that satisfy with conditions T = T0 on all the note on ST From there, to build problem apply the finite element method to define heating elements on wood following the footsteps such as: RESULTS AND DISCUSSION The objective of this paper is finding the temperature distribution in wood as shown in figure 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 elements with 10 dots is illustrated as Figure • 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 system KTe = R 96 AGU International Journal of Sciences – 2019, Vol (4), 91 – 99 T=1000 10 T=1000 7 5 0,04m 4 3 k=0,17W/m K 0,02m 1 T=300 Figure Conductive problem model Following the diagram with element divided, we conducted to join many elements Table 1: Diagram with element divided Freedom level Elements 1 3 4 5 7 8 10 Following the equation we have: BT = det J  y 23 x  32 y31 x13 y12  x21  We calculated: 50  − 50 50   − 50 50  − 50 B1 =  B = B =    − 100 100    100 − 100  100 − 100  50  − 50 50  − 50 50  − 50 B4 =  B = B =    − 100 100  − 100 100     100 − 100   B7 =  100 − 50 50  − 100  50  − 50 B8 =  − 100 100   97 AGU International Journal of Sciences – 2019, Vol (4), 91 – 99 Apply the equation kT = kAe BT BT (t=1 unit), we calculated heating matrix of element T  0,0425 kT = − 0,0425  kT kT kT − 0,0425 0,2125 − 0,17 − 0,17  0,17  = − 0,17 0,2125  − 0,0425  0,0425 = − 0,0425  − 0,0425 0,2125 − 0,17 − 0,17  0,17  = − 0,17 0,2125  − 0,0425   0,2125  − 0,17  kT =  − 0,17 − 0,0425 0,17  − 0,17 − 0,0425  0,17  0,0425    0,0425  − 0,0425  kT =  − 0,0425 0,0425  − 0,0425  0,17 − 0,17  − 0,17 0,2125    0,2125  − 0,17  kT =  − 0,17 − 0,0425 0,17  − 0,17 − 0,0425  0,17  0,0425    0,0425  − 0,0425  kT =  − 0,0425 0,0425  − 0,0425  0,17 − 0,17  − 0,17 0,2125  Conductive matrix of system (dimensions 10x10) will be constructed based on join table of this element Boundary condition T = 1000C at dot and 10, T = 300C at dot and we have matrix Table 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 98 AGU International Journal of Sciences – 2019, Vol (4), 91 – 99 Figure Conductive problem model CONCLUSIONS REFERENCES 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 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 Element Method using MATLAB (2nded.)Florida: CRC Press LLC ACKNOWLEDGEMENTS Rannacher, R (1999) Finite Element Method for the Incompressible Navier-Stokes Equations Heidelberg: Institute of Applied Mathematics University of Heidelberg I would like to express my endless thanks and gratefulness to my supervisors and my students 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 99 ... 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 Nguyen Luong Dung, (1993) Finite element method in. .. 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. .. 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