LATTICE BOLTZMANN SIMULATION OF A STORAGE TANK WITH AN IMMERSED TUBE BUNDLE HEAT EXCHANGER

Transcription

1 HEFAT th International Conerence on Heat Transer, Flui Mechanics an Thermoynamics July 014 Orlano, Floria LATTICE BOLTZMANN SIMULATION OF A STORAGE TANK WITH AN IMMERSED TUBE BUNDLE HEAT EXCHANGER Yan Su Department o Electromechanical Enineerin, Faculty o Science an Technoloy, University o Macau, Taipa, Macau yansu@umac.mo ABSTRACT A rectanular storae tan with an immerse cylinrical tube bunle heat exchaner has been simulate by Lattice Boltzmann Metho (LBM). The etails o transient temperature istributions an low streamlines in the nearby iel o each tube are clearly emonstrate as well as the complexity o the overall low iel an the extent o the mixin urin ischare. The LBM maes the irectly simulation possible an the computational spee is increase comparin to a conventional porous meium moel simulation. The transient averae Nusselt numbers o the tube bunle have been obtaine, which can also be applie in other applications such as numerical simulations o porous meium moels. NOMENCLATURE c p [J/-K] Thermal capacity o water c [m/s] Lattice velocity vector [/m ] Density istribution unction [K/m ] Temperature istribution unction ᶢ [m/s ] Gravity acceleration L x [m] With o the storae tan L y t U [m] [s] [m/s] Heiht o the storae tan Time Velocity scale x [m] Cartesian axis irection y [m] Cartesian axis irection Special characters ν [m/s ] Kinematic viscocity α [m] Thermal ispersivity ρ [/m ] Density l [m] Lattice lenth scale x [m] Mesh size in x irection y [m] Mesh size in y irection Subscripts L Microscopic lenth scale Macroscopic lenth scale INTRODUCTION Flui low an heat transer o tube bunles immerse in solar storae tan have been stuie wiely or their importance on the Enineerin applications [1]. In early time, either experimental stuy or numerical stuy usin a small unit to present the whole tube bunle has been investiate. It woul be time an space consumin to employ the conventional methos such as inite ierence, inite element, an control volume methos to simulate the transient low an heat transer o the tube bunle heat exchaner. This is because the computation spee an accuracy o these methos are hihly epenent on converence spee an accuracy o the Poisson solver []. Then, porous meium moels are applie to stuy the overall eects base on a representative element volume (REV) [1, ]. Recently, one o the state-o-the-art methos, Lattice Boltzmann Metho (LBM), has been propose to increase the simulation spee an also to aapt to complex eometry omains. A lot o stuy o LBM or natural an mixe convection insie a close enclosure riven by top li an buoyancy orce have been carrie out by introucin a boy orce term into the momentum uation or macroscopic Reynols number up to an the macroscopic Rayleih number up to [4]-[7]. Also conjuate heat transer can be solve easily with LBM metho by moiication o the relaxation time in the enery uation [8]. However, or the present stuy our Rayleih numbers woul be substantially larer than those in the above literature. Moreover, the conventional LBM metho with parameters base on real units cannot explicitly show the eects o the imensionless parameters an mesh size on computational accuracy. In orer to overcome the above limitations o the conventional LBM with units, this paper evelops a numerical moel base on LBM or the solar storae tan with a immerse tube bunle heat exchaner. Compare to the conventional LBM, the propose metho can clearly reveal the relations o the macroscopic, microscopic, an mesoscopic lenth scales. 58

2 NUMERICAL MODEL As shown in Fi. 1, the tilte heat storae tan can be moele as a lui enclosure with a ranomly istribute 40 tube bunle heat exchaner near the top. Two ins o istribution base on loosely istribute tubes an compacte tubes will be investiate in the present stuy. Fiure 1(a) shows a loosely istribute tube bunle, while Fi. (b) shows a compacte tube bunle with the same number o tubes. Durin the ischare proceure, the initial temperature o the tan is at a uniorm hih value T 0. The heat will be taen away by the immerse small tubes with time oin on. The ravity orce is o in the 0 irection start rom the x axis. The lui omain is meshe in size o n m, then the mesh size or both the soli an lui part is n m. The mesoscopic lenth scale, i.e. the lattice mesh size an the lenth scale or the present numerical stuy, l x y L / m L n. The y x/ temperature scale is the initial temperature ierence between the storae lui an the tube wall T, an the reerence temperature is the lui temperature in the mi o the storae tan T. The velocity scale or the natural convection is re U β TL. The corresponin macroscopic Rayleih number is base on the macroscopic lenth scale, i.e. with o the tan L, Ra L β TL ν α (1) (a) The microscopic Rayleih number is base on the microscopic lenth scale, i.e. the tube iameter, Ra β T ν α Denote as imensionless ensity istribution unction an as the imensionless temperature istribution unction, we can obtain the ollowin Lattice Boltzmann Equations (LBE) or the momentum uation an heat transer uation respectively: ( x + c, t + ( ) + F () ( (, t ) Q ( x + c, t + 1) x ) + Then, the respective imensionless ensity, velocity, an temperature can be obtaine as ollows: () (4) ρ, (5) u c/, (6) (b) Fiure 1 Setch o the ranomly istribute tubes in mi o the storae tan: (a) loosely istribute 40 tubes (b) compacte 40tubes The LBM is applie to simulate the transient natural convection insie the solar storae tan with the 40 tubebunle heat exchaner. DQ9 scheme is chosen in the present stuy because previous stuies show that two imensional simulations can capture the heat transer coeicient as well as the three imensional stuy []. For the present problem, the macroscopic lenth scale is the ap heiht o the channel, H. T (7) The local uilibrium istribution unctions or lui low an heat transer are iven by wζ ( c, u)ρ (8) an wζ ( c, u) T, (9) respectively, where ζ ( c,u) can be obtaine base on the Bhatnaar-Gross-Kroo (BGK) moel [8] as c 1 1 u c u u u ζ ( c, u) (10) The corresponin imensionless iscrete velocities are 59

3 (0,0) c, 0 c ( ± 1,0) c, 1,,,4, (11) ( ± 1, ± 1) c, 5,6,7,8. an the weihtin actors are 4/9, 0 w 1/9, 1,,,4, (1) 1/6, 5,6,7,8. Base on the ChapmanEnso Expansion on the imensionless Navier-Stoes an enery uations, similar to the analysis with units [8], the relaxation times or the low an heat transer, an, are eine as, Base on the ChapmanEnso Expansion on the imensionless Navier-Stoes an enery uations, similar to the analysis with units [8], the relaxation times or the low an heat transer, an, are eine as, ν s +, (1) an α s +, (14) The relaxation times or low an heat transer uations are expresse in the orm o the lattice viscosity an thermal ispersivity. The imensionless orce term ue to the buoyancy orce is expresse as, F w c βρ ( T T ) (15) re where is the ravity acceleration. The imensionless heat source term is eine as, V porous Cη(1 φ)( ) t Vtan Q w exp ρc p T (16) 1/ L Pr Ra l As where, η is a imensionless eometry actor, eine Vs as the ratio o the prouct o the microscopic lenth scale an the soli lui interace area to the soli volume in are presentative elementary volume (REV),connects the macroscopic an microscopic ra an heat lux between the soli an lui phases in a porous meium []. An in the present stuy, the shape o the heat exchaners is cyliner, thus η 4 []. Base on a curve it o prior measurements o overall heat transer urin ischare, C 0.6± 0.01, / 0 q L T, an U βl T [9, 1]. ( ) ρc p β Bounce bac bounary conitions are applie on the storae walls as no slip zero velocity conitions,, 0 : 8 (17) o where o is the opposite irection o. The temperature bounary conition or the storae tan walls are,,, n, n 1,6,7 (18) The present problem is a transient problem, zero initial velocity an uniorm environment temperature are selecte as the initial conitions or low an heat transer iels. The corresponin uilibrium istribution unctions are selecte or the LBM scheme as, w ρ 0, at t 0; (19) an wt0, at t 0. (0) Reer to the scale analysis o [1], the averae Nusselt number or the heat transer rom the tubes to the storae tan lui in the present stuy can be estimate by, Nu V η(1 φ)( V ( T porous tan porous L ) l T tan NUMERICAL RESULTS Pr Ra ) U 1/ T t tan. (1) A computational coe or the present LBM is evelope in Fortran 90, which is compile an run on the hih perormance computin cluster with GNU/Linux operatin system (CentOS bi. To maintain hih accuracy, the mesh size n m is set to be , an the lattice size x is smaller than the ruire size or the inite ierence base projection metho stuy on the corresponin computational velocity scale []. Two cases shown in Fi. 1 (a) an (b) are compute to a real time lenth o about 100s, which is corresponin to a ischare o 40% o total storae thermal enery. Firstly, we woul lie to chec the tan averae temperature to valiate the enery conservation o the LBM coe. The tan averae imensionless temperature obtaine by the present LBM simulation or eometry o the Fi. 1(a) is compare with the previous experimental stuy o Liu et al. (005) an the porous meium moel simulation base on inite ierence base projection metho o Su et al. (007) in Fi.. From Fi., we can see that the transient tan averae temperature o the LBM simulation arees very well with previous stuies on or the whole ischare proceure rom all o the proceure rom conuction to convection. An the transient tan averae temperature o the present LBM result an the measure results o Liu et al. (005) are slihtly hiher than the porous meium moel results near the turnin perio at time about 00s. Aroun this time, conuction eects ecrease an convection eects bein to ominate the heat transer. 60

4 Fiure Transient averae imensionless temperature in the whole storae tan Fiure Transient averae Nusselt numbers or the tubes Fiure, shows the transient averae Nusselt number Nu rom the present LBM results o the transient tan averae temperature an the averae temperature in the tube bunle zone base on Eq. (1). As shown in Fi., the transient averae Nusselt numbers Nu ecay with time. They approach to a stable value near 10 or the convection ominate perio, which is consistent with the porous meium stuy o [1] an the experimental measurement o [9]. The present irect simulation results by LBM show that the ecay o the Nusselt number is quicer than the results obtaine by the porous meium moel, an the present preicte Nusselt numbers are more stable than those obtaine by porous meium simulation. More etaile experimental stuies on the initial transient heat transer nee to be one to valiate the Nusselt numbers urin the initial ew secons. Also the loosely pace tubes (Fi. 1(a)) has hiher value o Nu than the compacte tubes (Fi. 1(b)) because the averae ra is less o loosely istribute tubes, an the hot low is more easier to enter the heat exchaner zone than the compacte case. The averae imensionless temperature in the heat exchaner zone ( 5H /9 < y < 8H / 9 ), below the heat exchaner zone ( 0 < y < 5H / 9 ), an above the heat exchaner zone ( 8 H /9 < y < H ) are compare with the tan averae imensionless temperature in Fi.4. All o the our averae temperatures rop quicly in the initial 600s, an the chane o the temperature is slow own with time, which is ue to the rop o the temperature ierence between the tubes an the tan lui. From Fi. 4, we can also see that the tan averae temperature is between the averae temperature o the water below the heat exchane an the averae temperature in the heat exchaner zone. While the water temperature above the heat exchaner is always hiher than the rest two zones, because the small stanation zone above the heat exchane has lower velocity low ue to the buoyancy orce will rive the col plumes own alon the bac o the tan wall. The transient imensionless streamlines an isotherms at t 700 s are plotte in Fi. 5 an 6 respectively. The bottom parts o the pictures present the contour plot in ull tan an the top parts o the pictures present or zoom pictures in the zone ille tubes. Fiure 5 shows that both the lare scale lows an the etaile low aroun the tubes can be shown in the present LBM metho results. The velocity in the pure lui zone is hiher than the velocity in the tube bunle zone ue to the ra o the tube bunle heat exchaner. Fiure 6 shows that the temperature istributions in both lare scale insie the tan an the etaile istributions near each tube are clearly shown either. Fiure 4 Transient averae imensionless temperature or water in, above an below the loosely istribute tube bunle 61

5 Isotherms (a) (b) (c) () (e) () () (h) (i) (j) () (l) (m) (n) (o) (p) Streamlines Fiure Contour plot o streamlines at 1s: bottom picture or ull tan an top picture or zoom in the loosely pace tubes (a) (b) (c) () (e) () () (h) (i) (j) () (l) (m) (n) (o) (p) Fiure 7 Transient isotherms an streamlines o loosely istribute tubes or the initial 100s Isotherms (a) (b) (c) () (e) () () (h) (i) (j) () (l) (m) (n) (o) (p) Streamlines Fiure 4 Contour plot o isotherms at 1s: bottom picture or ull tan an top picture or zoom in the loosely pace tubes (a) (b) (c) () (e) () () (h) (i) (j) () (l) (m) (n) (o) (p) Fiure 8 Transient isotherms an streamlines o compacte tubes or the initial 100s 6

6 Fiure 7 an 8 present the transient isotherms an streamlines o the initial 100s or the loosely istribute tubes an the compacte tubes, respectively. From Fi. 7, we can see that the tan temperature ecay with time an there are stratiications o thermal lui alon the ravity irection, which was not observe so clearly in the porous meium moel simulations o Su et al. (007). Fiure 7 (a) an (b) show that the heat transer is initially ominate by conuction when time is small. Fiure 7(c) show that the col thermal plume initially enerate alon the tube walls an the main col plume rops own alon the bac wall, which is consistent with the porous meium simulation o Su et al. (007). Then mushroom structure col plumes enerate as shown in Fi. 7 (c) an (). An the mushroom structure will be estroye base on the thermal ispersion insie water uner the heat exchaners. When reach the bottom o the tan, the col plume will bump bac rom the bottom wall as shown in Fi. 7(e). Shown in Fi. 7() an (), althouh the col plume rops own rom the bac wall, the hot water will be pushe bac into the heat exchaner zone rom mile part o the heat exchaners. This phenomenon was not observe by the previous porous meium moel stuy. Blue colore col plume bein to reach the bottom o the tan an stay there ue to buoyancy orce as shown in Fi. 7 (h) an (i). Then the col plumes bump bac beore they reach the bottom o the tan because hih ensity col water will be stae at riht bottom corner. Ater that the tan is more stratiie an the velocities o lui in the tan ecay with time, as shown in Fi. 7 (m) to (p). This is also ue to the temperature ierence between the soli wall an the tan water ecrease with time. There is lare scale lui motion uner the heat exchaner tubes while the lui is almost stae above the heat exchaner tubes. An insie the heat exchaner, the lui low irections hihly epen on the istribution o the tubes. Fiure 8 shows the similar heat transer proceure as those in Fi.7, exceptin that the time is elaye in the conuction perio. An the stratiication o the storae is more obvious. ACKNOWLEDGEMENT The present stuy was supporte by FDCT/115/01/A, MYRG151(Y1-L)-FST11-SY, an MYRG080(Y1-L)- FST1-SY. REFERENCES [1] Su Y. an Davison, J.H., Multi-zone Porous Enclosure Moel o Thermal/Flui Processes urin Dischare o an Incline Rectanular Storae Vessel via an Immerse Heat Exchaner, ASME J. o Solar Enery Enineerin, 19, , 007. [] Luan H.B., Xu H., Chen L., Sun D.L., He Y.L., Tao W.Q. (011) Evaluation o the couplin scheme o FVM an LBM or lui lows aroun complex eometries. International journal o heat an mass transer, 54: [] Su Y., Davison, J.H., Kulaci, F.A., A Geometry Factor or Natural Convection in Open Cell Metal Foam, International Journal o Heat an Mass Transer, 01; 6(1): [4] Guo Y., Bennacer R., Shen S., Ameziani D.E., Bouzii M. (010) Simulation o mixe convection in slener rectanular cavity with lattice boltzmann metho. International journal o Numerical methos or Heat & lui low, 0 (1), pp [5] Monal B., Mishra S.C. (009) Simulation o natural convection in the presence o volumetric raiation usin the Lattice Boltzmann Metho. Numerical Heat Transer, Part A, 55, [6] Lai F.H., Yan Y.T. Lattice Boltzmann simulation o natural convection heat transer o Al O /water nanoluis in a square enclosure. International journal o thermal sciences. 50: [7] Mehrizi A.A., Farhai, M., Aroozi, H.H., Seihi K., Darz A.A.R. (01) Mixe convection heat traner in a ventilate cavity with hot obstacle: eect o nanolui an outlet port location. 9: [8] Taroh A., Mohama A.A., Jian L. (01) Simulation o conjuate heat transer usin the lattice boltzmann metho. Numerical Heat Transer, part A, 6: [9] Liu, W. Davison, J.H., Kulaci, F.A., an Mantell, S. C., 00. Natural Convection rom a horizontal tube heat exchaner immerse in a tilte enclosure, Journal o Solar Enery Enineerin, 14,1, CONCLUSION LBM simulations or ischare proceure o a ranomly istribute 40 tube bunle heat exchane immerse near the top o a solar storae tan have been one. The transient tan averae temperature an the averae Nussult numbers ecay with time an aree with previous experimental an numerical stuy in inite ierence base projection metho. The macroscopic, microscopic, an mesocopic lenth scales are eine rom lenth scales o the storae tan, the small tube iameter, an the mesh size respectively. The present LBM results clearly show both macroscopic an microscopic transient low an temperature istributions. 6