Crystallisation of a supercooled spherical nodule in a flow

Transcription

1 EUROTHERM 69 Heat and Mass Transfer in Soid-Liquid Phase hange Processes June 25-27, 2003, Bistra caste, Ljubjana, Sovenia Eds.: B. Sarer, D. Gobin rystaisation of a supercooed spherica nodue in a fow Jean Pierre Dumas*, Kama E Omari, Jean Pierre Bédecarrats, Miche Strub Laboratoire de Thermique, Energétique et Procédés Avenue de Université, BP PAU, rance e-mai: jean-pierre.dumas@univ-pau.fr Key words: atent heat storage, encapsuated nodues, supercooing, crystaisation, convection Abstract The genera purpose of this work concerns the atent heat storage ( cod storage ) by encapsuated nodues of 7.7 cm in diameter containing a Phase hange Materia ( PM ) and fiing a cyindrica tank ( about 2500 nodues per m 3 ). During the storage process, a cod heat transfer iquid fows through the tank to crystaise the PM inside the nodues. The stored energy is reeased when a hotter heat transfer iquid fows through the tank to induce the meting of the PM. We have presented a goba study [1][2] of the behaviour of the tank without differentiating the nodues. But, we highighted the difference between the storage (cooing ) and the discharge ( heating ) due to the supercooing phenomenon. As typicay the veocity in the tank is reativey sma ( a few mm/s ), it is possibe to assume the appearance of natura convection of the cooant when the atent heat is reeased at the crystaisation of the PM. So, we present a study of a singe nodue surrounded by a fowing heat transfer iquid at a temperature T ower than the meting temperature T. At the crystaisation, ony a part of the nodue is soid whereas the remaining iquid is at T. The crystaisation is pursued owing to the heat fux invoved by the difference T T. This text describes a mode to expain the mutua infuence of the natura convection and the kinetics of crystaisation inside the nodue.

2 Jean Pierre Dumas, Kama E Omari, Jean Pierre Bédecarrats, Miche Strub 1 Introduction During the fow of the cod heat transfer fuid, the crystaisation of the PM in the nodues does not occur at the meting temperature T ( typicay 0 for the investigated nodues the PM being water ) but severa degrees ower due to the supercooing phenomenon. The consequence is very important because, in industria appications, a suppementary cooing woud be necessary and consequenty the use of a refrigerating machine with a higher power. The use of nuceating agents reduces this supercooing but does not vanish it and the kinetics of storage is dependent of the aws for the crystaisation of the supercooed iquid unike the discharge, at the heating, which occurs at the meting ( thermodynamic equiibrium ). The great difference is that the crystaisation of the supercooed iquids has an erratic character described by probabiistic aws [2]. Using these aws in the case of a vertica drainage, a mode predicting the evoution with time of the temperature in each point inside the tank has been vaidated by experiments [1][2]. The aim of this paper is to study a singe nodue previousy maintained at a temperature beow the meting temperature by a iquid fowing verticay around it and when, at time t = 0, occurs the crystaisation ( metastabiity breakdown ) inducing a sudden energy reease. This temperature jump of the externa surface induces natura convective streams changing competey the forced convection initiay imposed and modifying the heat fuxes between the nodue and the fuid. Because the kinetics of the crystaisation depends on these fuxes there is a mutua infuence between the phase change and the fow around the nodue. The fow of the iquid is studied by the code LUENT and the crystaisation is anaysed by an origina code using an enthapic method. The two different cacuations are inked when the boundary conditions are expressed. 2 ow around a nodue Two different fows are possibe ( figure 1 ): the norma mode when the forced fow is verticay rising in the same direction as the natura convection or the inverse mode when the forced fow is verticay descending in the opposite direction of the convection. igure 1 Genera scheme of the fow 2

3 EUROTHERM 69, June 25-27, 2003, Bistra caste, Ljubjana, Sovenia The tank generay instaed in the storage oop have a voume reativey important ( from 20 to 500 m 3 ) with sections a few m 2 in area. The duration of the storage is severa hours. So, the mean veocity of the iquid is very ow ( to m/s ) and we wi ony consider this veocity range. The Reynods number based on the nodue diameter U D 5 Re = 35 υ f (1) indicates that the forced fow is aminar. Due to T = T P T the difference between the temperature of the surface of the nodue T P and the temperature of the outer fowing iquid T a natura convection wi appear. The Grashof s number Gr cacuated for a supercooing maximum of 7 K and for the therma properties of the fowing fuid which is propyene gyco has a maximum vaue about < 10 9 [5]. The vaue of Gr < 10 9 indicates that the natura convection wi be aminar[3] inducing ony axisymmetric fows. or simuation of the non stationary fow around the nodue, we have chosen to use the commercia code LUENT based on finite voumes on unstructured grid. Unsteady, viscous and incompressibe Navier-Stockes equations incuding buoyancy effects are considered. The modeisation is two dimensiona with axisymmetrica coordinates ( x, r ) as indicated on the grid of the figure 2. We are in the case where we have a sphere whose outer radius is R e, ocated on the axis of a cyinder whose dimension is sufficienty important to consider as quasi-infinite. This chosen grid permits the study of the norma and inverse modes. The spatia discretisation is fuy impicit. or the couping of the pressure and veocities the agorithm SIMPLE[4] has been chosen. Boundary condition at the wa of the nodue is a no sip condition for veocities, and a time depending heat fux. This heat fux is cacuated by the mean of the unsteady mode described in the next section. ig 2 Grid for the fuid ig 3 Grid for the nodue and its scheme 3 rystaisation of the PM inside the spherica nodue The nodue is competey fied of PM and its voume is considered as constant. So, we negect the existence of a necessary expansion air bubbe but, aso, the modification of the density ρ at the crystaisation. These two assumptions are compementary and we can consider that the 3

4 Jean Pierre Dumas, Kama E Omari, Jean Pierre Bédecarrats, Miche Strub goba voume does not change. But to take into account that, at the beginning, the nodue is not competey fied, the atent heat L is reduced. We aso assume the constancy of the specific heat c. Before the crystaisation, at any points in the fowing iquid and in the sphere is at the temperature T < T where the PM is supercooed. At the metastabiity breakdown, the crystaisation is so rapid that we can first consider that it is adiabatic according to: H fina [ m h (T ) + m h (T )] Mh (T ) = Mc ( T T ) m L H = 0 = (2) initia s s s The mass of the ice which has appeared is: ( T T ) Mc ms = (3) L The soid is ocated near the enveope [2], and in these first moments the symmetry of the sphere is preserved. So, the equation (4) gives the radius of the initia soid-iquid interface ( 0 ) by: r 4 3 ρ πri c L ( T T ) ρ π ( Ri rc ( 0 )) = 3 (4) 3 L We finay obtain : 1 / 3 c L r ( 0 ) = Ri 1 ( T T ) L (5) At t = 0, the radius of the iquid-soid interface is not the inner radius of the nodue R i but r ( 0 ) which depends on the supercooing. We can notice that with the supercooing generay observed with such a type of voume ( a few K ), the thickness of this first crystaised ayer is very sma ( 2% R ). i We have aready presented [5][6] a quasi stationary method to determine the kinetics of crystaisation of the PM inside the nodue. This method is reativey easy to operate but it assumes that the crystaisation was concentric. It is not exacty the case because, as it can be easiy predicted, the natura convection is responsibe of different heat fuxes on the nodue. So, in this paper, we present the resuts using an enthapic method. Inside the PM, the equation for the conduction of heat can be written: H t =.( k T ) (6) with for the variation of the enthapy: T H(T ) = h(t ) h(t ) = ρ c dt + ρ f (T ) L (7) T 4

5 EUROTHERM 69, June 25-27, 2003, Bistra caste, Ljubjana, Sovenia where f (T ) is the fraction of iquid with: 1 f = 0 if if T > T T < T 0 < f < 1 if T = (8) T c is either c s the specific heat of the soid PM (ice) or c the specific heat of the iquid PM, k is either k the heat conductivity of the soid PM (ice) or iquid PM. With the spherica coordinates, (6) becomes: k the heat conductivity of the T ρ c t 1 2 T 1 k T f = r k + sinθ ρ L (9) 2 r r r r sinθ θ r θ t On figure 3 we give the grid inside the nodue end its genera scheme. Indeed the two grids for the fowing iquid ( figure 2 ) and inside the nodue are in connection at the interface fuidnodue. The numerica cacuation is impemented as foows: - knowing at time t the temperatures, given by LUENT, on the interface fuid-nodue T p ( t ), first we take into account the therma resistance of the enveope by T k p k s ( t + t ) = [ TN ( t + t ) T p ( t )] r where k p is the therma conductivity of the enve- R R ope. Ri e i - these temperatures T p ( t ) constitute the boundary conditions to sove the equation (9) by a finite-voume method. The obtained inear system is soved by a Successive Over Reaxation (SOR) method. The fraction of iquid f is corrected as suggested by V.R. Voer [7] and a new inear system is resoved unti convergence. So we obtain T and f at t + t. The points of the surface where 0 < f < 1 precise the ocation of the interface soid-iquid. T - at t + t the fux ϕ ( t + t ) is cacuated by ϕ ( t + t ) = k s ( t + t ) r - these new vaues of the fux are used by the code LUENT as a boundary conditions at the interface nodue-fuid - inside the fuid the equations of baance of mass, energy and momentum are soved by the code LUENT at time t + t - the cacuated temperatures ( t + t ) permit to begin the cacuation for a new step. T p R i 5

6 Jean Pierre Dumas, Kama E Omari, Jean Pierre Bédecarrats, Miche Strub 4 Resuts We made cacuations in different fow configurations, by varying U, the veocity of the cod fuid at the infow, as we as its temperature T. The different studied configurations are given on tabe 1. These configurations were investigated in both norma and inverse mode. Tabe 1: Studied fow configurations configuration U [m/s] T [K] Resuts concerning the fow As an exampe we present the resuts concerning a fow in configuration 7-12 ( tabe 1 ). The cacuation starts from an isotherm fow situation at T where ony forced convection was present. On figure 4 we see the streamines, in the norma mode, at different instants after during the crystaisation of the PM. Just after the beginning of the crystaisation, we observe an acceeration of the fuid which distort the streamines. We have noticed [5] that the axia veocity V x is quicky very important since V x / U 11 as soon as t = 40 s, vaue remaining quasiconstant. We confirm that an important natura convection is estabished near the axis on figure 5 where the isotherms at different instants are given. 6

7 EUROTHERM 69, June 25-27, 2003, Bistra caste, Ljubjana, Sovenia igure 5 Isotherms for the norma mode igure 4 Steamines for the norma mode or the inverse mode, in the same configuration 7-12, the isotherms of the figure 6 confirm that, athough the inet fow is from the top to the bottom, a natura convection appears. But the streamines of the figure 7 indicates an opposition between the forced fow and the natura convection inducing currents of recircuation. The natura convection is the predominant phenomenon because we observe that the maximum axia veocity is the same order that for the norma mode with V x / U 10. 7

8 Jean Pierre Dumas, Kama E Omari, Jean Pierre Bédecarrats, Miche Strub igure 6 Steamines for the inverse mode igure 7 Isotherms for the inverse mode 4.2 Resuts concerning the crystaisation As expained above, the enthapic method permits the determination of the kinetics of phase change. We can see on figure 8, for the case of the configuration 7-24 in the norma mode, the heat fux on the nodue versus the ange θ. We observe that the exchange of energy is the more important for the first instants after the crystaisation with an infuence of the ocation on the nodue. The heat fux is more important at the bottom of the nodue than at the top which is ogica considering the sense of the fow in the norma mode. This heat fux decreases when the crystaisation progresses. igure 8 Heat fuxes at different instants for different anges θ igure 9 Mean heat fux versus time for the norma and inverse modes To compare the resuts for different configurations, we prefer cacuate the mean vaue of the heat fux given by: π 1 ϕ( t ) = ϕ π 0 ( t, θ ) dθ (10) 8

9 EUROTHERM 69, June 25-27, 2003, Bistra caste, Ljubjana, Sovenia or exampe on the figure 9 we observe this mean fux versus time for the configuration 7-24 both in the two modes norma or inverse. We observe a curve with a quasi-inear shape as it has been observed experimentay[2]. We aso observe that there is no difference between the norma and the inverse modes indicating the ow infuence of the forced inet fow. On the figure 10 we observe this mean heat fux for a configurations. We deduce that the tota time to the crystaisation, given when the heat fux is zero, depends strongy on the inet temperature T ( the duration of the crystaisation is arger when T is ower ) but not of the veocity of the fuid. igure 10 Mean fuxes versus time for the different configurations igure 11 Mean interface temperature for the different configurations This resut is confirmed on the figure 11 where we have potted the mean temperature of the interface nodue-fuid defined by: π 1 TP ( t ) = T π 0 p ( t, θ ) dθ (11) As indicated above the cacuation of the fraction of iquid f permits to ocaise the interface iquid-soid inside the nodue. We present on the figure 12 the resuts on the ocaisation of this interface at different instants. We observe that these interfaces are not concentric spheres but the difference is not very important. In this case, we understand why the mode of a quasistationnary crystaisation [5] gives simiar resuts. 9

10 Jean Pierre Dumas, Kama E Omari, Jean Pierre Bédecarrats, Miche Strub igure 12 Locaisation of the interface iquid-soid versus time for the configurations 5-24 (eft) and 9-24 oncusion We have presented a numerica study of the mutua infuence on the fow around a nodue containing a Phase hange Materia and its crystaisation after a supercooing. This work concerns the atent heat storage technoogy. The natura convection which is deveoped around the nodue modifies its temperature of surface and thus modifies the kinetics of crystaisation inside. REERENES [1] J.P. Bédecarrats,. Strub, B. acon, J.P. Dumas, Phase-change therma energy storage using spherica capsues. Performance of a test pant, Internationa Journa of Refrigeration, 19,No.3, (1996) [2] J.P. Bédecarrats, J.P. Dumas, Study of crystaisation of nodues containing a phase change materia for coo therma storage, Int. J. Heat Mass Transfer, 40, No. 1, (1997)

11 EUROTHERM 69, June 25-27, 2003, Bistra caste, Ljubjana, Sovenia [3] B. Gebhart, Heat Transfer, Mc Graw-Hi Book ompany, New York, second edition (1970) [4] V. Patankar, Numerica Heat transfer and uid ow, Hemisphere, Washington D (1980) [5] K. E Omari, Etude de a convection autour d'un nodue sphérique contenant un matériau à changement de phase au cours de sa cristaisation, Thesis, University of Pau (2001) [6] K. E Omari, J.P. Dumas, ow around a spherica nodue during its crystaisation, Internationa Journa of Heat and Technoogy (2003) submitted [7] V.R. Voer, ast impicit finite-difference method for the anaysis of phase change probems, Numerica Heat Transfer Part B, (1990), 17,