Numerical investigation into solidification in a horizontal annulus

This paper presents a numerical investigation into the solidification process in an annulus that is placed horizontally. The method used is a front-tracking method combined with a linear interpolation technique. The inner wall of the annulus is held at cold temperature, which is lower than the melting point of the liquid, while its outer wall is insulated during solidification.

30 Vu Van Truong NUMERICAL INVESTIGATION INTO SOLIDIFICATION IN A HORIZONTAL ANNULUS Vu Van Truong Hanoi University of Science and Technology; truong.vuvan1@hust.edu.vn Abstract - This paper presents a numerical investigation into the solidification process in an annulus that is placed horizontally The method used is a front-tracking method combined with a linear interpolation technique The inner wall of the annulus is held at cold temperature, which is lower than the melting point of the liquid, while its outer wall is insulated during solidification Accordingly, the solidification initially takes place around the inner wall, and evolves in the direction of increasing the annulus radius Various parameters such as the Rayleigh number (Ra), the Stefan number (St), and the initial temperature of the liquid (0) are investigated The numerical results show that increasing any of these parameters enhances the eccentricity of the solidified layer during the initial stages of solidification Moreover, an increase in Ra or St reduces the time for completing solidification However, the solidification process takes longer with increasing 0 In addition, the effect of the annulus eccentricity is also investigated respectively The annulus is filled with a phase change material whose melting point is Tm greater than Tc Initially, the phase change material at the liquid state is at temperature T0 (T0  Tm) To save the computations, the half of the physical domain is investigated, as shown in Figure 1a Since the temperature of the inner wall is lower than the fusion temperature of the liquid, the solidification layer forms around the inner wall and moves outward The fluid and thermal properties of each phase are assumed constant The liquid assumed incompressible is driven by buoyancyinduced natural convection, i.e., Bussinesq approximation Key words - Numerical simulation; solidification; annulus; Rayleigh number; Stefan number Introduction Phase change processes in annuli play an important role in the workings of nature and engineering problems such as thermal energy storage systems, food processing and others For application of latent heat storage, solidification in an annulus is a well known applied technique to store energy [1] Therefore, understanding solidification heat transfer in an annulus plays an important role in designing and operating such systems Accordingly, there have been many works concerned with this solidification problem Akugun et al [2] investigated the melting and solidification processes of paraffin in a tube in shell heat exchanger system, i.e., in an annulus system, to design and construct a novel storage unit for latent heat storage Yazici et al [3] experimentally investigated the solidification process in a horizontal tube-in-shell storage unit The working material was paraffin The solidification behavior was analyzed based on the transient temperature field inside the phase change material Concerning numerical simulations, there are a few numerical studies related to solidification in an annulus For instance, Ismail et al [4] used a boundary immobilization method to study solidification external to a long tube The physical domain was transformed into a computational domain to ease the calculations Kim et al [5] investigated the icing process of water in an annulus with the presence of three phases: ice, water and air The method used was that proposed by the authors for axisymmetric problems [6] Numerical problem and method Figure shows the investigated problem, a solidification layer is formed in an annulus whose inner wall is kept at cold temperature Tc with an insulated outer wall The inner and outer radii of the annulus are Ri and Ro Figure Solidification in an annulus: (a) computational domain and (b) front-tracking representation We treat all phases as one fluid with variable properties such as density , viscosity , thermal conductivity k and heat capacity Cp In terms of the one-fluid representation, the momentum and thermal energy equations are   u  t    uu   p   ( u  uT ) +f   g T  Tm  (1)    C pT  t      C pTu      k T    q  x  x f dS   C p h (2) f  u  (3) Here, u is the velocity vector, p is the pressure, g is the gravitational acceleration, and t is time T and the superscript T denote the temperature and the transpose D/Dt is the material derivative f is the momentum forcing term used to impose the no-slip condition on the solid– liquid interface, and h is the energy forcing term used to impose a constant temperature on the cylinder boundary [7] The last term in Eq (1) is the Boussinesq approximation for density changes due to thermal gradients [8], and is the thermal expansion coefficient of the liquid At the interface, denoted by f, nf is the normal vector to the interface The Dirac delta function δ(x − xf) is zero ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(115).2017 everywhere except at the interfaces xf q is the heat source at the solidification interface, given as q  ks T  T    kl    sVn Lh n s n l 31 function I which is zero in the solid and one in the liquid: (6) I     x  x f n f dS f (4) where the subscripts s and l represent solid and liquid, respectively Vn is the velocity normal to the solidification front and Lh is the latent heat These above-mentioned equations are solved by the front-tracking method combined with interpolation techniques on a staggered grid with second order accuracy in time and space [7] The momentum, energy and mass conservation equations are discrete using an explicit predictor-corrector time-integration method and a secondorder centered difference approximation for the spatial derivatives The discrete equations are solved on a fixed, staggered grid using the MAC method [9] The interface between two phases is represented by finite discrete points on a stationary grid (Figure 1b) which are updated by xnf 1  xnf  n f Vn t (5) where the superscripts n and n+1 are the current and next time levels Vn is given by Vn   q f  s Lh  Positions of Accordingly, the values of the material property fields at every location are given as   l I s  1  I s s (7) where  stands for ,  Cp, or k A more detailed description of the method used in this study can be found in [7] Numerical parameters We choose the annulus width, i.e., R = Ro - Ri, as a scaling length, and  c  l Cl R2 kl as the characteristic time scale The characteristic velocity scale is taken to be U c  R  c With these above choices, the dynamics of the problem is governed by the following dimensionless parameters: Prandtl number Pr, Stefan number St, Rayleigh number Ra, initial dimensionless temperature of the liquid 0, density ratio sl, thermal conductivity ratio ksl, and heat capacity ratio Cpsl, and the annulus eccentricity L: the interface points are used to construct an indicator Figure Solidification in an annulus with St = 0.1, Ra = 105, 0 = 2, and L = In (a) and (b), the left shows the isotherms (plotted every  = 0.2) and the right shows the velocity field normalized by Uc The dash lines are the annulus walls Pr  C pl l kl ,St  C pl Tm  Tc  Lh ,Ra  g l (Th  Tm )R3C pl  l2 l kl , C T T  k y  yco 0  c ,  sl  s ,ksl  s ,C psl  ps ,L  ci (8) Tm  Tc l kl C pl R Here, yci and yco are respectively the y coordinates of the center of the inner and outer walls The temperature is non  T  Tc  Tm  Tc  dimensionalised as The dimensionless time is = t/c In this study, we use paraffin as a phase change material with assumed identical thermal and fluid properties, and thus Pr = 115, sl = ksl = Cpsl = We here focus on the effects of the inner wall position relative to the outer wall (the annulus eccentricity L), the Stefan and Rayleigh numbers, and the dimensionless temperature 0 To save the computational time, half annulus is considered as shown in Figure 1a The computational domain is WH = 1.6R3.2R The center of the outer wall is at x = 0, y = 0.5H while the center of the inner wall will be varied, depending on L The boundary conditions are shown in Figure 1a The annulus configuration investigated in this study is based on [2] Method validations have been extensively carried out in our previous works [7,10], and thus are not presented in this paper Results and discussion Figure shows the temporal evolution of the solidification front with the temperature and velocity fields At early times of the solidification process (Figure 2a), downward flow arises along the solid–liquid interface, at which the temperature is m = 1.0, because the density of the liquid increases with a decrease in the temperature As a result, a thermally stratified region is gradually formed near the solidification interface After hitting the bottom, the flow goes up along the outer wall, and forms a circulation in the 32 annulus As time progresses, the temperature of the liquid phase decreases to near the fusion temperature m Consequently, the circulation flow is suppressed At  = 0.42 (Figure 2b), the liquid temperature is uniform in the entire annulus, and almost no flows are evident at this time, and the solidification process is merely controlled by conduction Vu Van Truong absolute value of the eccentricity E, as shown in Figure Figure also indicates that increasing Ra enhances the evolution of the solidification region In other words, the solidification evolves faster as Ra increases 4.2 Effect of the Stefan number, St Figure shows the effect of the Stefan number on the solidification process in terms of the eccentricity and the area with Ra = 106, 0 = 2, and L = Increasing St enhances the eccentricity as shown in Figure This is understandable since increasing St corresponds to increasing sensible heat, and thus increasing the bouyancy effect In addition, an increase in St results in an increase in the growth rate of the solidification interface [see Eq (4)], and thus reduces the time for completing solidification, as shown in Figure Figure Temporal variations of the eccentricity E defined in the text , and of the area of the solidified phase normalized by the annulus area, A Figure shows the temporal evolution of the area of the solid phase and the variation with time of the eccentricity of the solidified region defined as E   ya  ycin  R where ya is the y coordinate of the center of mass of the solidified region The circulation flow reduces the solidification growth at the top while it enhances the solidification at the bottom (Figure 2) Thus, the solidification layer forms eccentrically around the inner wall The circulation is strong at early stages of solidification, and then weaker as time progresses Accordingly, |E| is high at the initial stages of solidification and gradually decreases, as shown in Figure Figure also indicates that the slope of A() is high at initial and then gradually decreases, indicating that the solidification rate is high at the beginning due to high thermal gradient at the solidification interface Next, we will investigate the effects of some parameters on the solidification process Figure Effects of the Stefan number St on the eccentricity E and the area A with Ra = 106, 0 = and L = 4.3 Effect of the initial temperature of the liquid,0 4.1 Effect of the Rayleigh number, Ra Figure Effects of the initial temperature of the liquid on the eccentricity E and the area A with St = 0.1, Ra = 10 6, and L = Figure Effects of the Rayleigh number Ra on the eccentricity E and the area A Figure shows us the effect of Ra on the solidification process with St = 0.1, 0 = 2, and L = The Rayleigh number is associated with buoyancy-driven flow, and thus increasing Ra leads to a stronger circulation flow Accordingly, an increase in the Rayleigh number results in an increase in the Figure shows the temporal variations of the eccentricity and the area of the solid phase with St = 0.1, Ra = 10 6, and L = 0, and for three initial temperatures 0 = 1.0, 1.5 and 2.0 At 0 = 1.0, initially the temperature of the liquid is equal to the melting point, and thus the circulation flows in the annulus is weaker than at higher initial temperatures, i.e 0 = 1.5 and 2.0 Accordingly, the effect of the circulations on the eccentricity E decreases as 0 decreases from 2.0 to 1.0, as shown in Figure Looking at the temporal evolution of A, we see that at the same time, the solidified layer is smaller at a higher 0, i.e., 0 = 1.5 or 2.0 However, increasing 0 ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(115).2017 from 1.5 to 2.0 has a minor effect on the form of the solidified layer as shown in Figure 4.4 Effect of the annulus eccentricity, L Figure shows the effect of the eccentricity L of the annulus on the solidification process in the annulus with St = 0.1, Ra = 106, and 0 = for three cases: L = - 0.4 (eccentricity), 0.0 (concentricity), and 0.4 (eccentricity) Figure 7a compares two cases L = - 0.4 (left) and L = 0.4 (right) at  = 0.792 For L = 0.4, there are two circulation flows in the annulus with the clockwise one near the solidified layer As L increases to 0.4, only one weaker 33 circulation occurs (Figure 7a) Accordingly, at this time the case with L = 0.4 produces a wider solidified layer in comparison with L = - 0.4 As a result, the solidification process with L = 0.4 finishes sooner than L = - 0.4 as shown in Figure 7b An interesting point in Figure 7b is that the eccentric cases (L = - 0.4 and 0.4) complete the solidification process more slowly than the concentric case (L = 0.0) This accords with industrial applications that the concentric annulus has been widely used in thermal storage systems since it saves time for storing energy [1] Figure Effects of the annulus eccentricity L on the solidification process with St = 0.1, Ra = 10 6, and 0 = 2: (a) the temperature and the velocity fields at  = 0.792 and (b) temporal evolution of the area of the solidified phase normalized by the annulus area, A In (a), the left frame is for L = -0.4, and the right frame is for L = 0.4 Conclusion We have presented the results of a numerical simulation of solidification in an annulus The method used is the fronttracking that represents the interface by connected elements The numerical simulations are performed by varying various parameters to investigate their effects on the solidification process The results show that during the initial stages of solidification, the eccentricity of the solidified phase increases as any of the Rayleigh number Ra, the Stefan number St and the initial temperature of the liquid 0 increases The solidification process finishes sooner with higher Ra and higher St or with lower 0 In addition, the investigation into the annulus eccentricity indicates that the solidification process finishes sooner with the concentric case Acknowledgments This research is supported by Hanoi University of Science and Technology (HUST) under grant number T2016-PC-028 REFERENCES [1] F Agyenim, N Hewitt, P Eames, M Smyth, A review of materials, heat transfer and phase change problem formulation for latent heat thermal energy storage systems (LHTESS), Renew Sust Energ Rev 14 (2010) 615–628 [2] M Akgün, O Aydın, K Kaygusuz, Experimental study on melting/solidification characteristics of a paraffin as PCM, Energy Conversion and Management 48 (2007) 669–678 [3] M.Y Yazici, M Avci, O Aydin, M Akgun, On the effect of eccentricity of a horizontal tube-in-shell storage unit on solidification of a PCM, Applied Thermal Engineering 64 (2014) 1–9 [4] K.A.R Ismail, F.A.M Lino, R.C.R da Silva, A.B de Jesus, L.C Paixão, Experimentally validated two dimensional numerical model for the solidification of PCM along a horizontal long tube, Int J Therm Sci 75 (2014) 184–193 [5] C.-J Kim, S.T Ro, J.S Lee, M.G Kim, Two-dimensional freezing of water filled between vertical 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Editors received the paper on 04/10/2016, its review was completed on 14/02/2017) ... since increasing St corresponds to increasing sensible heat, and thus increasing the bouyancy effect In addition, an increase in St results in an increase in the growth rate of the solidification. .. progresses Accordingly, |E| is high at the initial stages of solidification and gradually decreases, as shown in Figure Figure also indicates that the slope of A( ) is high at initial and then gradually... decreases, indicating that the solidification rate is high at the beginning due to high thermal gradient at the solidification interface Next, we will investigate the effects of some parameters