Jamova cesta 2 SI 1000 Ljubljana, Slovenia Jamova cesta Ljubljana, Slovenija

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1 Univerza v Ljuljani Fakulteta za gradeništvo in geodezijo University of Ljuljana Faculty of Civil and eodetic Engineering Jamova cesta Ljuljana, Slovenija Jamova cesta 2 SI 1000 Ljuljana, Slovenia DRU Digitalni repozitorij UL F DRU The Digital Repository zirki je izvirna različica izdajatelja. Prosimo, da se pri navajanju sklicujete na iliografske podatke, kot je navedeno: This is a pulisher s version PDF file. When citing, please refer to the pulisher's iliographic information as follows: Pečenko, R., Hozjan, T., Svensson, S Implementation of fully coupled heat and mass transport model to determine tempertature and moisture state at elevated temperatures. 11th World Congress on Computational Mechanics (WCCM XI) and 5th European Conference on Computational Mechanics (ECCM ) and 6th European Conference on Computational Fluid Dynamics (ECFD I), July 2014, Barcelona, Spain. Datum arhiviranja / Archiving Date:

2 11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM ) 6th European Conference on Computational Fluid Dynamics (ECFD I) IMPLEMENTATION OF FULLY COUPLED HEAT AND MASS TRANSPORT MODEL TO DETERMINE TEMPERATURE AND MOISTURE STATE AT ELEATED TEMPERATURES. R. PECENKO *, T. HOZJAN * AND S. SENSSON * University of Ljuljana Faculty of Civil and eodetic Engineering Department of Mechanics Jamova 2, 1000 Ljuljana, Slovenia roert.pecenko@fgg.uni-lj.si Technical University of Denmark Dept. Civil Engineering Brovej 118, 2800 kgs. Lyngy, Denmark nss@yg.dtu.dk Key Words: Coupled physics, heat transfer, moisture transfer, FEM formulation, phase change. Astract. The aim of this study is to present precise numerical formulation to determine temperature and moisture state of timer in the situation prior pyrolysis. The strong formulations needed for an accurate description of the physics are presented and discussed as well as their coupling terms. From these the weak formulation is deducted. Based on the weak formulation two case studies are conducted. The results of this case are presented. 1 INTRODUCTION In an open porous hygroscopic material such as wood, heat and moisture transport is a complex system of coupled processes. Inside timer, different phases of water can e oserved, i.e., free water, ound water and gas phase of water (water vapour and air). Conservation equations for each phase with the exchange of mass etween the different phases have to e considered. Different transfer phenomenon s can e applied for each phase. The ound water transfer model is assumed to follow Fick's law [1]. More complex is the transfer of gaseous mixture. Transfer of water vapour and air has to e comined y a convective and diffusive model of transport. For the convective part Darcy's law is usually applied and for the diffusive part Fick's law is used [2]. The free water constitutive relation is usually assumed to follow the generalized Darcy's law []. All three processes are connected with each other through the exchange of mass etween the different phases, i.e., the conversion of vapour to free water and vice versa (condensation and evaporation), the conversion of vapour to ound water and vice versa (sorption) and the conversion of free

3 water to ound water and vice versa. Thermal interaction on mass transfer is seen as temperature dependent diffusion coefficients where Fick s law is applied and temperature dependent mass velocity where Darcy s law is used as well as the Soret effect. The equations descriing the conservation of mass must also e supplemented with an equation descriing the conservation of enthalpy [2]. This equation takes three phenomena into account. Firstly, the usual conduction of heat thru solid, descried y Fourier's law. Secondly, the changes in enthalpy resulting from change of phase, i.e. sorption, evaporation and condensation. And finally, the convective transfer of heat, i.e. the effect that heat is carried in the mass flux. These equations with corresponding initial and oundary conditions are generally non-linear and can rarely e if at all solved analytically. Therefore, numerical methods have to e employed. 2 CONSERATION EQUATIONS The coupled system of differential equations for heat and mass transport is derived from the following asic laws for mass and energy conservation. Free water conservation Bound water conservation Water vapour conservation Dry air conservation t t c t J J c E (1) J E c () t A In eqs. (1) (4), J i and i denotes the mass flux and the volume fraction of the phase i. Indices,,, and A stands for free water, ound water, gaseous mixture, water vapour and air. ρ i is phase density defined per unit volume of gaseous mixture, c the concentration of ound water, Ė is the amount of vaporized free water and ċ is the sorption rate which within time interval dt represent the amount of water vapour asored to ound water or vice versa. J A (2) (4) Energy conservation equation T C k T hsorpc EE Cv T. (5) t 2

4 In the eq. (5) conductivity of timer, represents the energy transport y fluid convection. CONSTITUTIE RELATIONS C denotes specific heat of timer, k is the coefficient of thermal hsorp is heat of sorption, E is latent heat of evaporation and Cv.1 Bound water The model for temperature dependent ound water diffusion was proposed y [1]. The asic set of equations is: JB Dc DT T (6) D 0 E D exp RT (7) D T ce (8) D RT 2 E m1 10 (9) In eqs. (6) (10), D and diffusion coefficient, R is universal gas constant, diffusion, m moisture content and 0 dry density of wood. c m (10) 0 D T are diffusion coefficient and thermal coupling ound water E the activation energy for ound water.2 Free water The free water flux is descried y Darcy s law: v J v (11) KK P (12) elocity of free water v is defined y specific permeaility of dry wood K, relative permeaility of water K, dynamic viscosity of water and partial pressure gradient of water. P. Water vapour and air Water vapour and air fluxes are defined y contriution of diffusion (Fick s law) and convection (Darcy s law).

5 J v D A A A A (1) J v D A In the eq. (1) (14), DA represents the diffusion of air or vapour in the gaseous mixture. The expression is proposed y [4]: D A T P 5 (14), (15) where represents the estimated reduction factor due to the hindrance of the diffusion in the cellular structure. is material direction dependent. elocity of gaseous mixture is defined y Darcy s law. KK v P, (16) where the partial pressure of gaseous mixture is according to Dalton s law the sum of partial pressure of air and partial pressure of water vapour: P P P A. The pressures are assumed to follow the ideal gas law: P R T. (17) A A A P R T. (18).4 Sorption Sorption occurs when the driving potentials in the two phases, i.e. water vapour and ound water, are not in equilirium. Here the Hailwood-Horroin isotherm is applied: m h f 2 fh, fh (19) 1 2 where h is the humidity, it is defined as ratio etween partial pressure of water vapour P and saturated vapour pressure P SAT. P h. (20) P The ound water concentration c can e compared to the ound water concentration in equilirium with the vapour pressure P, which is denoted as c l : SAT 4

6 c m. (21) l Model for sorption rate was proposed y [1] and here modified for temperatures aove 100 C: Hccl c T 100C c (22) Hc0 c T 100C where: and: H 0 c C l C1exp C2 c C4 c cl c c C l C1exp C22 c C4 c cl (2) C C exp C h (24) 4 SYSTEM OF DIFERENTIAL EQUATIONS By summing eq. (1) and () the system of equations is reduced to four. The chosen primary unknown variales are T, P, and c. After some algeraic manipulation, the system of differential equations can e written in a form suitale for a finite element solution. T P c C TT CTP CT CTB KTT T KTP P KT KTB c t t t t K T T P c C C C C K T K P K K c t t t t PT PP P PB PT PP P PB T P c C C C C K T K P K K c t t t t T P B T P B T P c C C C C K T K P K K c t t t t BT BP B BB BT BP B BB Comparing to the other equations, the eq. (25) is additionally accounted y this term energy transport y fluid convection is considered. 4.1 Boundary conditions T. (25). (26). (27). (28) KT T. In It is assumed that the water vapor concentration at the (imaginary) oundary etween surrounding air and pores (lumens) at a macroscopic surface is identical to the partial vapor pressure of amient air. Dirichlet oundary condition is applied: 5

7 ,. (29) Also the pressure on the oundary surface is equal to the atmospheric pressure: P P. (0) The ound water is restricted to the wood cell wall and is only exchanged y sorption, therefore the Neumann oundary condition is applied, accordingly to: Boundary condition for heat transfer:, n J 0. (1) T k hqr T T n. (2) 5 FINITE ELEMENT FORMULATION In the matrix form the system of eqs. (25) (28) can e written: with oundary conditions: u C K u K u t 0 u R Rku n, (), (4) In the aove eq., u is vector of asic unknowns TP,,, c, matrices C and K contains coefficients C ij and K ij (i,j = T, P,, ), matrix K contain only coefficient K TT, other components of matrix are equal to zero. The finite element solution is ased on approximation of the unknown vector function: u n node i Ny, (5) i1 where N denotes a matrix of polynomial shape functions, y is vector of discrete node unknowns. Using alerkin method and integration y parts, the system () can e rewritten on the equivalent system of differential equation of first order: where ˆ y C Ky ˆ Fˆ t, (6) nel nel nel nel e e e e e e ; R; ; e1 e1 e1 e1, (7) Cˆ Cˆ Kˆ Kˆ Kˆ Kˆ Rˆ Rˆ u u 6

8 ˆ e T e ; ˆ e T e ; ˆ e T e C N C Nd K N K Nd K N K Nd; e e e (8) ˆ e T e e ; ˆe T e e K NKRNd R NKRd R k, e A finite difference scheme is used for the time discretisation. The calculation time is k divided on time increments 1 k t, t, within each time interval linear variation of nodal quantities is assumed. The eq. (0) has to e satisfied in each time step, i.e., at the time k k1 t t t. t is the magnitude of time step and is predefined time step: 0 (explicit method), 1/2(Crank-Nicolson method), 2/(alerkin method), 1 (implicit method). Considering time discretisation leads system (6) to: e (9) k k k, (40) C y F k ˆ 1 ˆ k 1 ˆ ˆ k 1 ˆ k ˆ k 1 C K tc and F tc 1 K y 1 F F. (41) The system is solved iteratively in each time step. Implicit method is applied. 6 EXAMPLE The example presents the spatial and time development of temperature and moisture state of timer specimen exposed to varying outside conditions. Size of timer specimen is /L = 1/5 cm, it is modeled with 200 four-node quadrilateral finite elements. Temperature and water vapour oundary condition are shown in Fig.1. On the exposed side the emissivity of timer surface and convection factor are 0.7 and h q 25 W/m 2 K. Thermal conductivity of timer is taken in accordance with [5]. Other parameters are shown in Tale 1. It should e noted that in this case free water within the specimen was not taken into account kg/m K 1 Tale 1: Input data 1000kg/m R J/molK 16 K J/kgK R J/kgK 0.0 (tangential direction) R A D 710 m/s Initial values are: c kg/m, kg/m P,0 0.1MPa T 0 2C Sorption parameters: f 1 = 2.22 f 2 = 15.7 f = C C C C 60 C

9 Figure 1: Boundary conditions The results at different specimen location are show in Fig.2 Figure 2: Temperature, moisture content, water vapour and pore pressures distriution over time in different specimen location The increase of temperatures inside of specimen is expected. Maximum oserved temperatures at points 1, 2 and (x=0 mm, x= mm and x=20 mm) are 200 C, 160 C and 120 C. The increase is faster for point 1. Partial pressure of gaseous mixture is approximated y the ideal gas law, and it is directly correlated to the temperature. This correlation is well oserved in Fig.2(d) where the increase of temperatures results in the increase of pressure 8

10 inside the specimen. As mentioned, sorption occurs where driving potentials in the two phases are not in equilirium. At elevated temperatures, the water vapour migrates much faster than ound water, consequently the driving potential etween the two phases and sorption increases. In eq.(22) it is assumed that for temperatures aove 100 C, only desorption can occur. It is considered that the pressure does not have any impact on sorption. The result of all phenomenons is phase change of ound water into water vapour. Fig.2() and Fig.2(c) present all this processes descried. As the moisture content decreases with time, the concentration of water vapour increases. Fig.() present the variales distriution along the specimen for different times. Figure : Temperature, moisture content, water vapour and pore pressures distriution along the specimen for different times REFERENCES [1] Frandsen, H.L. Selected Constitutive models for simulating the hygromechanical response of wood. Department of Civil Engineering Aalorg University (PhD Thesis), [2] Tenchev, R.T., Li, L.Y. and Purkiss, J.A. Finite element analysis of coupled heat and moisture transfer in concrete sujected to fire. Numerical Heat Transfer, Part A (2001) 9: [] Perre, P., Turner, I.W. A -D version of Transpore: a comprehensive heat and mass 9

11 transfer computational model for simulating the drying of porous media. International Journal of Heat and Mass Transfer (1999) 42: [4] Cengel, Y. A. Heat transfer: A practical approach. WCB/Mcraw-Hill (1998) [5] EN : 2005 Eurocode 5: Design of timer structures - Part 1-2: eneral - Structural fire design 10