ThammasatInt. J. Sc. Tech.,Vol. 10, No. 3, July-September2005
The Numerical and Experimental Investigationof Heat Transport and Water Infiltration in Granular Packedbeds due to SuppliedHot Water (One-andTwo-DimensionalModels) P. Rattanadecho* Facultyof Engineering,ThammasatUniversity(RangsitCampus), PathumTham,l2l2l, Thailand,E-mail:
[email protected] *Corresponding Author S. Wongwises King Mongkut Universityof TechnologyThonburi, 9l Suksawas48, Rat Burana,Bangkok, 10140,Thailand
Abstract The characteristicsof heat transport and water infiltration in granular packed bed due to supplied hot water are investigatedexperimentallyand numerically.The distributionsof water content and temperature are predicted for one- and two-dimensional models assuming the local thermal equilibrium among water and particles at any specific space.The predicted temperaturedistributions are comparedwith the experimentalresults. Most importantly, the effects of particle sizes,initial water content and suppliedwater flux on heat transportand flow kinetics are examined.It is found that using a largerparticlesizeresultsin a fasterinfiltration rate and forms a wider infiltration layer,especiallyin the direction of gravity. However, an extension of the heated layer is not as much as that of the infiltration layer becausethe temperatureof water infiltration gradually drops due to upstream heat ffanspon. Keywords:
porousmedia,water infiltration,unsaturated flow, numerical published for heat transport in porous media with water infiltration, except the problem of permafrost[9- I 1] and drying technologyI l2- l6]. The purposeof this paper is to clarify the characteristicsof heat transport with water granular packed beds infiltration in numerically. Most experimentally and importantly, the effects of particle sizes, suppliedwater flux and initial water contenton the flow kinetics are examined. The result presentedhere provides a basis for fundamental understanding of heat transport and water infiltration in granularmaterials.
1. Introduction Understandingof heat transportin granular packed bed or porous media with water infiltration due to the force of capillary suction is essential in a variety of soil science and chemical engineering applications such as temperature control of soil, recovery of geothermal energy, thermal energy storage,and variousreactorsin the chemicalindustry.Up to the present time, the related problem of water infiltration in porous media has been investigated both experimentally and numerically by many researchers tl-81. However, few research reports have been
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ThammasatInt. J. Sc. Tech.,Vol. 10,No. 3, July-September2005
4.) The contributionof convectionto energy transportis included. 5.) Darcy's law holds for the liquid and gas phases. 6.) Fick's law holds for the vapor diffusion. 7.) Gravity is includedfor the liquid and gas phases. 8.) Permeabilityof liquid and gas can be expressedin termsof relativepermeability. 9.) The evaporation of water and the condensation of water vapor take place flow processes. simultaneouslyin the unsaturated Evaporation and condensation respectively representa heat sink and a heat source in the granular packed bed. These representationsare included in the present model formulation. In a macroscopic sense, the unsaturatedgranular packedbed is assumedto be homogeneousand isotropic, and water does not bind to the granular beads.Therefore,the volume average model for a homogeneousand isotropic material can be used in the theoretical modeling and analysis.
2. Analysis of Unsaturated Flow in Granular Packed Bed Figure I shows the schematic diagram of the Buckley-Leverett problem. This case is the unsaturatedflow in granular packed bed due to supplied hot water. When the hot water is uniformly supplied at the top surface of the sample, which is initially composed of glass particles and gas, a two-phaseregion with the infiltration front is formed within the packedbed. In this region, co existantflow of liquid water and gasphasesoccurs.However,the flow due to gasphaseis a dominantmechanismat the lower part.
0
s
77 I
v,
T
ll I
Mass Consemation The microscopic mass conservation equations for liquid and gas phases are expressedrespectively,as shownbelow: Liquid phase A , r d .
Figure 1. Schematic diagram of BuckleyLeverett problem By conservationof mass and energy in the unsaturatedgranularpackedbed, the governing equationof mass and energy for all phasescan be derived by using a volume averaged l7l. The main transpoft technique [6, mechanismsthat enablewater infiltration during suppliedhot water into the sampleinclude; l Water flow is driven by capillarypressure gradientand gravity 2. The vapor is driven by the gradientof the partialpressureof the evaporatingspecies.i.e.. vapor diffusion. Assumptions The main assumptions involved in the formulations of the transportmodel are: l.) The unsaturatedgranularpackedbed is rigid. No chemicalreactionsoccur in the sample. 2.) Local thermodynamic equilibrium is assumed. 3.) The gas phase is ideal in the thermodynamicsense.
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'fi1e,')+flc,,,)=o
(l)
Vapor phase d . , - -').1+ \ I ^ ',rJ=o " I . (t [n, ilo, f
, . (2)
In order to completethe systemof equations, the following forms of the generalizedDarcy's law which are the expressionsfor the superficial averagevelocity of the liquid and gasphasesare u s e d[ l 6 ] : '.r KK ',l' tdp" ' - o,'P ' - o n l (3) v' , = '"1 Y*=--l
lrr 10;. Az K K , " l A p- P" t ] 1 ^
/t" Loz
|
,t +n, ,
l
For the velocity of water vapor and air phases,the generalizedFick's law in the form of the partial pressuregradient of the evaporating speciesis applied:
ThammasatInt. J. Sc. Tech.,Vol. 10,No. 3, July-September2005
p,v, = p,v e
- PrD^+[.]
(5)
P"v" = P"v s
- P. o D , + l e - l
(6)
oz\P" )
oz\pr )
where D- is the effective molecular mass diffusion. In Eqs. (3) and (4), the capillary pressure,p., is related to the gas and liquid
Saturation s. l-l
phasesand can be written by:
Figure 2. The relationshipbetween,K. ands" (7)
p,=ps-pt
The capillary pressure, P" , is further assumed to be adequately represented by Leverett's well know -/(s" ) functions. The
Using Darcy's generalizedequation (Eqs.3 and 4 ), massconservationequationsfor liquid and gasphases(Eqs.l and 2), are then rewritten, respectively,as:
^
relationshipbetweenthe capillary pressureand the water saturationis defined bv using Leverett f u n c t i o n s",/ ( s " ) :
'r
a I r
