Solidification of Porous Material under Natural Convection by Three Phases Modeling

Transcription

1 Solidification of Porous Material under Natural Convection by Three Phases Modeling Hassan Basirat Tabrizi, Meber, IAENG and F. Sadeghpour Abstract The perforance of natural convective flow over a rectangular enclosure for coupled heat and ass transfer under freezing-thawing process is investigated experientally and nuerically. The enclosure filled with an unsaturated porous aterial. The two-diensional odel based on experiental observation for three phases air, water and ice is sought and solved. Each phase in the porous edia is assued in local therodynaic equilibriu (LTE) with each other. Coparison of the experient and odeling is in good agreeent. Index Ters porous edia, coupled heat and ass, solidification, free convection I. INTRODUCTION HIS Study looks to the freezing processes in porous Tedia which is of great interest in environental, energetic, biological and industrial systes, natural freezing and thawing in the cold regions, food preservation, agriculture, separation processes, theral energy storage and freezing of biological tissues. Sorption of oisture in building aterials can cause etal corrosion, structure deterioration and iproper perforance of building insulations. In addition, expansion and deforation of soil fraework during freezing process is an iportant feature in the foundation and road engineering in the cold regions. In food cheistry, it involves the deterination of the cooling rates the ost suited to the food preservation. In bioechanics, a siilar worry is the cryopreservation of organs in view of their further transplantation. Therefore, heat and oisture transfer in porous edia during freezing process is iportant, but reported literature on this subject is not any. Luikov carried out heat and oisture transfer for the drying process in capillary porous bodies [1], His odel is applicable for both hygroscopic and non-hygroscopic aterials and accounts for all fors of water bonding. The physical and therodynaic properties in Luikov s equations are functions of either teperature or oisture content or both. Therefore, this syste of coupled Manuscript received Feb. 22, 2012; revised March 15, H. Basirat Tabrizi is with the Airkabir University of Technology, Departent of Mechanical Engineering, Hafez Ave,Tehran, Iran (corresponding author: ; fax: ; e-ail: hbasirat@aut.ac.ir). F. Sadeghpour was with Airkabir University of Technology, Departent of Mechanical Engineering, Hafez Ave, Tehran, Iran (e-ail: fateeh.sadeghpoor@gail.co). equations is non-linear. Luikov and Mikhailov suggested that if calculations are carried out by zones, the transport coefficients can be taken as constant in each of these zones and Luikov s equations becoe linear [2]. Basirat Tabrizi and Hadullahpur [3] introduced a source ter due to the surface evaporation and used the energy and ass balances based on Luikov s odel to investigate drying process in capillary porous bodies. Hadai et al. [4]-[5] used schee to siulate heat and ass transfer during freezing of huid porous edia. They eployed Lee s three-level odel of the heat and ass transfer during freezing of par-baked bread. The odel accoodates the effect of teperature dependent variables. They indicated ceent pastes still expand if water replaced by benzene, which unlike water, contracted when solidifying. Few odels were presented for partially frozen soil by using coupled heat and oisture transfer [6]-[9]. Bazant et al. [10] introduced a atheatical odel for freeze thaw durability of concrete as a coupled heat and oisture transfer proble. Nevertheless, due to the coplexity of equilibriu relation of unfrozen water, they did not clearly show the closed for of the equations. Freezing (thawing) of soil around the buried pipes used for conveying various fluids [11], cryosurgery and cryopreservation of biological tissue [12], and food processing [13]. The freezing of porous edia has been studied extensively for low porosity edia (porosity below 50% such as rock, sand, soil) whereas few are available for high porosity edia [14]-[16] Studied on phase front propagation in soils. Chatterji investigated on the frost daage of concrete by freeze thaw cycles [17]. The prediction of the teperature and oisture fields in a product needs to understand the physic of the phenoena and the iportance of specific paraeters. This could be of use for food freezing (i.e. bread) or soil freezing, concrete freezing. In ost practical situations, the flow pattern in porous edia is three-diensional. However, two-diensional flow does exist both under laboratory condition and soe in nature. The understanding fro two-diensional analysis ay facilitate the study on the three-diensional situation. This paper investigates the natural convection in the saturated porous edia by assuing two-diensional pattern in freezing processes. II. EXPERIMENTAL SETUP AND TEST PROCEDURES A test rig is designed to easure the variation of teperature of a porous body during freezing. Figure 1 shows this apparatus and it consists of a loop channel, a copression refrigeration cycle, easuring instruents. The

2 evaporator is ebodied into a flat aluinu plate to provide a cold surface. obtained by the conservation of ass and energy of each phase. Equations are based on the following assuptions: a) each phase in the porous edia is in local therodynaic equilibriu b) two-diensional unsteady flow c) for of lack of filtration otion and pressure gradient in porous atrix, oentu equations are neglected d) all thero physical paraeters are assued constant e) no ass transfer fro boundary of porous solids In order to describe the siultaneous heat, oisture in saturated porous edia with phase change following odel is introduced. List of sybols are shown in Table II. Fig.1. Scheatic diagra of apparatus: (1) Air channel, (2) Digital teperature recorder, (3) Cubic saple of porous edia, (4) Cold surface (evaporator), (5) Insulation, (6) Copressor, (7) Condenser, (8) Expansion valve, Location of therocouples in porous ediu A cubic porous edia is placed on the cold surface while air strea oved naturally over. Hence, the upper surface is exposed to the air. Sand is used as porous ediu due to its extensive applications in building aterials. This aterial is packed in a rectangular cubic perforated old in diensions of c in order to configure the porous edia. Three therocouples type K are used to easure the teperature inside the porous edia. They are inserted at the various coordinates. The teperature easureent is carried out every inute during the freezing process. The estiated uncertainty due to experiental instruents was obtained ±1.5 C for teperature easureent. The accuracy due to experiental instruents is shown in Table I. Process Air teperature Wall teperature Sand weight Sand volue fr TABLE I ACCURACY OF MEASUREMENT Quantity 15±0.7 ºC -7 ±0.7 ºC 500±3g 0.4±0.005 III. MODELING The physical proble involves investigation of heat and ass transfer during freezing of capillary porous aterial. During freezing, oisture in porous edia exists in three phases: solid, liquid and gas. Governing equations are Mass balance of liquid phase: (1) Mass balance of solid phase: (2) Mixture energy equation is: 2 (3) Where (4) Here, the field variables include teperature T, liquid content ε l, solid content ε s, gas content ε g and porous content ε p. The phase change rate of condensation which stands for the source ter due to the oisture freezing on the particle surface and negative sign of in (1) eans that the aount of liquid content decreases during experient and therefore in (2) the positive sign eans the aount of solid content increases. The ter can be expressed as [18]: (5) The evaporation coefficient, σ c, is: (6) It can be assued the relation for sphere particle as [19]: 2.. / (8) 3 / (9) The oisture content of the saturated wetting porous ediu at the surface of the solid particle x cl as a function of the teperature and oisture content of the particle is [18]: 1 2 (10) The above functions can be coputed fro the tension curve of the oisture and the sorption characteristic of the solid oisture in the syste. The approxiations are [18]: (11) / (12) 10^ (13) Where n and l are constants (n = 3; l = 0.01). (7)

3 In order to convert to x, we have: / / (14) Also using Eq. (1) follows: / (15) The initial and boundary conditions are eployed according to the stated experient and follows: ε s =0, ε l =0.3, ε g =0.01 at t=0 T= =T s 0 for lower surface for upper surface (16) (17) (18) Sybol c d D l h H k 0 n Nu P w Ra t T W x, y Φ1 Φ2 x β ε µ ρ σ σ c TABLE II UNITS FOR SOLIDIFICATION PROPERTIES Quantity specific heat diaeter of particle diffusivity of liquid in porous ediu specific enthalpy Height of porous layer conduction heat transfer coefficient ass rate of phase change dry porous ediu weight constant value local Nusselt nuber partial pressure of vapor in gas Rayleigh nuber Tie Teperature width of the porous layer spatial coordinates Palanz Teperature function Palanz Moisture function oisture content theral expansion coefficient difference porosity or voidage viscosity density particle freezing coefficient Unit J/ (kg K) 2 /s J/kg M W/K Kg/( 2 s) Kg Pa s C kg/kg 1/k 3 / 3 kg/ (s) kg/ 3 kg/( 2 ) IV. RESULTS AND DISCUSSION Variation of different porosity, thickness, aterial, initial teperature, lower surface teperature is exained and the source ter effect is investigated. Fig. 2 (a, b, c, and d) shows the siulation results of teperature with experiental results at different locations of porous cube (12, 0.5), (6, 0.5), (12, 13), and (6, 13) c. Because of thin porous layer there is large teperature gradient not only in y direction but also in x direction. Nuerical results are about 3% lower than experient results. Absolute errors except in earlier tie of experient are lower than uncertainty of therocouples. By considering the liit of uncertainty, it can be seen that the odel based on ass rate of phase change could predict the variation of teperature and oisture content in porous edia ore closely. Fig. 3 illustrates the introduced source ter effect with the experiental results. The introduced odel predicts uch better than without source ter (or pure conduction). Fig. 4 (a, b) indicates the effect of different initial and surface teperature. The higher initial teperature needs higher tie for freezing. With lower surface teperature, then final teperature is at lower teperature. (c) (d) Fig. 2 Coparison of siulated teperature with the easured teperature of porous saple at pts (12, 0.5), (6, 0.5), (12, 13), and (6, 13) c Fig. 5 shows the effect of thickness. Since all paraeters are constant and just the thickness of sand is varied here, then the equilibriu tie is shorter and the final equilibriu teperature is higher. In short ties, the variation of

4 thickness had iportant role in ass rate and teperatures, because the cold air doesn t have tie to influence inside the porous saple. Finally Fig. 6 illustrates the effect of different porosity; it shows that the porosity doesn t have ain effect in final equilibriu teperature. between experiental and odel results. The ass rate of phase change predicts the behavior of heat and ass transfer characteristics uch better than without source ter (or pure conduction). Fig. 3 Coparison of siulated with the easured teperature of porous saple and odeling without ass rate of phase change at pt. (12, 0.5) Fig. 5 Siulated teperature data for different thickness vs. tie (sec) Fig. 6 Siulated teperature data for different porosity, dot line ε=0.5, black line ε=0.5 Fig. 4 Siulation of different teperature a. initial and b. surface teperature at pt. (12, 0.5) V. CONCLUSION Nuerical study of two-diensional heat and ass transfer in capillary porous edia by introducing a source ter was proposed. In order to validate the odel, an experiental setup was built to easure the teperature of a cubic porous edia during freezing process. The experiental data was obtained and the uncertainty analysis was perfored. A relatively good agreeent was achieved REFERENCES [1] A. V. Luikov, Heat and Mass Transfer in Capillary-Porous Bodies. Oxford: Pergaon Press, 1966, ch. 10, pp [2] A.V. Luikov, Y.A. Mikhailov, Theory of Energy and Mass Transfer. Oxford: Pergaon Press, [3] H. Basirat Tabrizi and F. Hadullahpur, Matheatical odeling of drying based on a surface evaporation source ter for coupled energy and ass Transfer, Int. J Energy Res, vol. 31, no. 15, pp , [4] N. Hadai, J. Y. Monteau and A. L. Bail, Siulation of coupled heat and ass transfer during freezing of a porous huid atrix, Int. J Refrigeration, vol. 27, pp , [5] N. Hadai, J. Y. Monteau and A. L. Bail, Heat and ass transfer in par-baked bread during freezing, Food Res Int., vol. 37, no. 6, pp , [6] G. Akbari, H. Basirat Tabrizi, and E. Daangir, Nuerical and experiental investigation of variable phase transforation nuber effect in porous edia during freezing process, Heat Mass Transfer. vol. 45, pp , [7] R. L. Harlan, Analysis of coupled heat-fluid transport in partially frozen soil, Water Resource Research, vol. 9, no. 5, pp , [8] G. L. Guyon and J. N. Lutin, A coupled heat and oisture transport odel for arctic soil, Water Resource Research, vol. 10, no. 5, pp , 1974.

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