Absorption and Evaporation Mass Transfer Simulation of a Droplet by Finite Volume Method and SIMPLEC algorithm

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1 roeedins of the 4th WE International Conferene on Finite Differenes - Finite Elements - Finite Volumes - Boundary Elements bsorption and Evaporation Mass Transfer imulation of a Droplet by Finite Volume Method and IMLEC alorithm DEGH TORFI, EYED MOHMMD HOEINI NEJD, KOLOUMEH OGBI Department of Mehanial Enineerin Islami zad University, usanerd Branh hahid Ghadiri sl t, usanerd IRN st@siau.a.ir, hosseyni@hotmail.om, k.obi@siau.a.ir bstrat: In this paper, a numerial model is developed to simulate sinle droplet heat and mass transfer in a twopiees solution with saturated solvent vapor environment by finite volume method and transient IMLEC alorithm in spherial oordinates system. For simulation of the mass transfer, dimensionless equations of motion, heat transfer and mass transfer (based on mass ratio) are solved simultaneously. ll the thermodynami and transitional solution properties have been onsidered as a funtion of temperature and onentration. The numerial analysis results are presented for mass transfer of lithium bromide solution droplet in 3K and initial onentration of 5%. Verifiation of method is done by ompare these numerial results with analytial and numerial analysis of other studies. Droplet Growth Chart, averae temperature and onentration, variation of dra oeffiient diarams, Nusselt number and flow line, temperature and onentration and temperature distribution ontours, penetration rate of mass and the level of tanential veloity at droplet surfae as the modelin results are presented. The results are shown that.5% inrease of onentration of droplet ause inrease 8C of mean temperature of droplet. Key-Words: bsorption, Evaporation, Heat and Mass Transfer, Finite volume, IMLEC, Droplet. 1. Introdution The effet of droplets and bubbles in many industrial and natural proesses is known for most of Enineers and researhers in different fields. In Industrial proesses suh as fuel injetion, oolin fluids in oolin towers, most of the dryin solution and many other industrial proesses, droplet behavior onsidered one of the important parameters in system desin. Therefore modelin of hydrodynami and thermal behavior of droplets and small fluid partiles has been studied by many researhers in last deades. rakash and irinano studied an analytial solve of mass transfer from sinle one-partial droplet [1]. Haywood et al. presented numerially analysis to solve this problem []. Jazayeri and Hosseini Nejad studied droplet evaporation and ondensation in steam environment with numerial simulations in above studies, equations of motion and enery in droplet and surroundins are resolved [3]. Numerial modelin of mass transfer with multi-piees environment was studied by Chan et al. for first time with solvin equations of motion, enery and onentration [4]. In this model the fluid in droplet was onsidered as sinle-phase surroundin steam as two-piees in this study, the mass transfer from multi-piee droplet in sinle-piees environment has been studied. erform the important assumptions in this model an be assumed laminar flow, the droplets remain spherial and usin the thermodynami equilibrium level member noted. Thermodynami properties onentration has been onsidered as a funtion of temperature.. Governin Equations Considerin the mentioned assumptions, the equations of problem is as follows..1. Mass Conservation ( ) t.. Momentum Conservation D Dt )1( )(.3. Distribution of Chemial Component inside the Droplet ( w t ) ( w ) ( D B w ) )3(.4. Enery Conservation around Droplet In the absene of enery soure and sink, and reardless of the effets of radiation heat transfer and ompressibility and visosity, the enery equation the IBN:

2 roeedins of the 4th WE International Conferene on Finite Differenes - Finite Elements - Finite Volumes - Boundary Elements problem assumin thermal ondutivity from Fourier law, is written as follows: ( T) ( T ) ( t E k ( ) k T ).5. Internal Enery Conservation in Droplet k ( T) ( T ) ( T ) E t E k 1 ( ) C h w. T D E B w )4( )5( )6( )7( 3. Boundary Conditions 3.1. ymmetri Boundary Conditions The boundary is symmetry axis of problem, where the boundary onditions are:,,, T, w r )8( 3.. Free Flow Boundary Condition This boundary onsists of two parts; inlet and outlet of flow at inlet ( nd r r ) Veloity and temperature values an be assumed equal to their values in free flow. Boundary ondition at output flow ( nd r r ) ould be assumed as zero radial radient Liquid-Gas Common urfae For derive ommon surfae boundary ondition, interate the ontrol volume equations that surrounds ommon surfae that limit to zero the details of the boundary onditions derivation for the radial veloity is iven in Ref [5] Radial veloity of liquid: r, m Radial veloity of vapor phase: r, m hear stress onservation at the level of: ~ ~ r, l r, )9( )11( )11( Boundary ondition equations for onentration and temperature in ommon surfae is more omplex. Mass enetration to droplet is assoiated with the enery release due to phase hane. Boundary ondition equations of problem in ommon surfae relative vapor exhane between the environment and droplet. In Common surfae, for ahieve to onsistent thermal boundary ondition, thermodynami equilibrium assumption is used. This means that temperature of droplet surfae is onsidered equal to fluid saturation temperature (at loal pressure and onentration). ( ) R T w, Tl, T Usin enery onservation ondition: T T k k m L r r l )1( )13( )14( )15( Calulations of onentration at droplet surfae are as follows. ordin to Fik law: ( ) D B w )16( Where and B are Chemial omponents in the fluid and veloities nd re alulated as follows. m m w B m l l )17( )18( )19( Thus if veloities and are known, onentration ould be alulated with equation (16) 4. Non-Dimensionin In order to eneralize the problem and presentin results in eneral dimensionless parameters, equations and boundary onditions must be non-dimensioned. IBN:

3 roeedins of the 4th WE International Conferene on Finite Differenes - Finite Elements - Finite Volumes - Boundary Elements 4.1. Equations Non-Dimensionin Conservation dimensionless equations form with dimensionless parameters defined before ould be obtained in eneral form as: r ( VR) V (( r ) rˆ ˆ t V )1( V ( ) R In this equation General variable, and distribution oeffiient and is soure term trin replae nd Conservation equations for eah Table 1 is shown. Replaed terms and for eah onservation equation are shown in table 1. Conservation Equation of this ase, not inherently inlude soure term and it is appeared Beause of seleted dimensionless parameters and inlusion of equations in the overall form appearane is above. Notieable point is that in seond sentene, the time derivative of droplet radius is used in equation and modified veloity vetor are r shown as ( ) rˆ ˆ r. R 4.. Non-Dimensionin Boundary Conditions The dimensionless form of Boundary onditions of symmetri axis, droplet enter and free flow are similar to dimensional form Boundary onditions at ommon surfae Dimensionless form of boundary onditions in ommon surfae has differene to dimensional form Radial veloity (liquid phase): 1 m r, V Radial veloity (as phase): 1 m r, V Continuity of shear stress: )1( )( 1 m )6( V lw Relationship of the averae and loal Nusselt number and dra oeffiient (frition, pressure, and overall thrust) also are written as follows. (ll expressions are written in vapor phase) T k r r1 Nu ( ) (1 T ) Nu C C F d Nu ( ) d 4 Re RV V r1 r1 ( ~ r os d C T os r d sin ~ os ) d r1 r1 rr r1 )7( )8( )9( )31( )31( Thrust dra omponent is due to asymmetri evaporation and ondensation on the surfae droplet. 5. Numerial Model and Validation In order to solve hihly oupled equations, finite volume method and IMLEC alorithm is used With interation of equations on the ontrol volume element in time raned, disrete form of equations are obtained that with usin the alorithm IMLEC and moved Network ould be solved in transient form Computation is done in a network with 15 (5 * 3) elements in droplet and 5 (5 * 5) elements outside of droplets and forwardin speed of 1e-6se that written by Visual C++ prorammin lanuae inludin 67 sub prorams. More details and explanations are available in Ref [5]. ~ ~ r, l r, )3( Conservation of enery: T T Re R k k r m L r r Known onentration at surfae w( ) Re DB w RV )4( )5( Fi 1. Flow Field Networkin. IBN:

4 roeedins of the 4th WE International Conferene on Finite Differenes - Finite Elements - Finite Volumes - Boundary Elements fter investiations and solve the flow around solid sphere and around a liquid droplet flow, IMLEC alorithm was hosen due to its onverene time less than the IMLE alorithm then all Modelin is presented in IMLEC alorithm. For selet suitable Disrete method, isothermal flow around a solid sphere is solved by ode written by this study and Fluent ode separately that in both ode, weakness of upwind different and hybrid methods was evident. On the other hand, the modified entral differene method solve is reatly aordane with results of solve obtained with the preise seond order methods Fluent ode and experimental results were. Thus the analysis performed by the modified entral differene method is used. In the analysis, isothermal flow around a solid sphere solution in performed in different Reynolds numbers and dra oeffiient, and vortex behind the sphere and Vortex separation point in the this analysis and other numerial and experimental results are ompared. For isothermal flow around a spherial drop, analytial relations presented by Clift et al. to alulate the dra fore has been omparin with results of this study [6]. lso a Comparison is done between parameters of isothermal air flow around a droplet of water in the present analysis and results reported by LeClair & Hamiele [7] and Haywood et al. [] that shown reasonably ompliant as shown in Table 1. In Fi. evaporation of water droplet at uniform temperature 3K, Radius.5mm and initial Re 1 at superheated water vapor at 34K and 1ka is shown and in Fi. 3 variation of averae Nu for Evaporation of water droplet is ompared. Condensation of water vapor on a water droplet with uniform temperature 31K, Radius.5mm and initial Re 1 at superheated water vapor at 35K and 4ka is shown in Fi. 4 and in Fi. 5 omparison of variation of CD for ondensation of water vapor on a water droplet is shown. Fi. 6 showed absorption of water vapor by a Li-Br droplet at uniform initial temperature 3K, Radius.5mm and initial Re 1 at superheated water vapor at 3K and in Fi. 7 omparison of variation of CD as funtion of times for absorption of water vapor by a Li-Br droplet is shown. Table 1. Comparison of isothermal parameters of air flow around water droplet. / 55 l Re=1 C resent result.513 Haywood et al.[].511 LeClair & Hamiele [7].49 C F C D l,mx l,mx Fi. Evaporation of water droplet at uniform temperature at superheated water vapor Nu_resent Result Nu_Haywood Nu_Chan Time ( Dimensionless ) Fi 3. Comparison of variation of verae Nusselt number for Evaporation of water droplet. Fi 4. Condensation of water vapor on a water droplet with uniform temperature at superheated water CD_resent Result CD_Haywood CD Chian Fi 5. Comparison of variation of CD ondensation of water vapor on a water droplet. IBN:

5 loal mass fration dimensionless tempreture roeedins of the 4th WE International Conferene on Finite Differenes - Finite Elements - Finite Volumes - Boundary Elements Fi 6. bsorption of water vapor by a Li-Br droplet at uniform temperature at superheated water vapor CD_resent Result CD_Haywood CD_Chian Fi 7. Comparison of variation of CD for absorption of water vapor by a Li-Br droplet. 6. Conlusions The results of numerial analysis are presented for mass transfer of lithium bromide solution droplet in 3K and initial onentration of 5%. In fiure 8, surfae temperature and distribution in droplets surfae at 6 different times is plotted. Temperature boundary onditions on droplet surfae is depends on pressure and is funtion of droplet's absorption rate of vapor and also absorption rate of vapor depends on surfae temperature and its neihbors. On the other hand, the surfae pressure obtained from veloity profiles around droplets Beause of hih sution on the surfae of droplet, veloity profile and thus the surfae pressure distribution has also haned. Thus adsorption is funtion of three important parameters; pressure, temperature and surfae onentration that in this ase the three parameters stronly depend on eah other. In fiures 8 and 9 hih dependeny between these three parameters is shown. The overall dra oeffiient that onsist of three omponents pressure dra frition and thrust dra is dereases in initial moments and then tart to inrease after reahin the minimum point at approximately t=15s, But pressure and thrust dra inrease in initial moments and then after reahin maximum point approximately t=5s are startin to derease. s shown in Fiure 4, the major omponent of dra in absorption proess of droplet is fritional dra omponent Fritional dra omponent in the initial moments is too hih due to hih vauum pressure of droplet and too low thikness of boundary layer on droplet surfae, and then beome steady state with dereasin sution rate. In fiures 1 and 11, onstant temperature and onentration lines at four different times is plotted. s ould be seen in Fiures, onentration penetration power is muh weaker than temperature and onentration boundary layer formed in droplet surfae is very thin nle(de.) Fi. 8: Distribution of Droplet's urfae Temperature Fi. 9: olution Mass Ratio at Droplet urfae IBN:

6 roeedins of the 4th WE International Conferene on Finite Differenes - Finite Elements - Finite Volumes - Boundary Elements Fi. 1: Constant Temperature Lines for Various Times Fi. 11: Constant Conentration Lines for Various Times. Referenes 1- rakash,., irinano, W.., (198), Theory of onvetive droplet vaporization with unsteady heat transfer in the irulatin liquid phase, Int. J. Heat Mass Transfer, Vol. 3, pp R. J. Haywood, N. Nafzier & M. Renksizbulut, (1989), detailed examination of as and liquid phase transient proesses in onvetive droplet evaporation, ME J. Heat Transfer, Vol. 111, pp Jazayeri,.., Hoseini Nejad,. M., (3), olvin heat transfer and movement equations for sinle droplet with onsiderin ondensation and vaporization by finite element method, 11 Th int. onf. of mehanial enineerin vol., pp , Iran. 4- Chan C. H. Raja, M.., irinano, W.., 199, Numerial analysis of onvitin, vaporizin fuel droplet with variable properties, Int. J. Heat Mass Transfer, Vol. 35, pp Hoseini Nejad,. M., (3), Numerial simulation of hydrodynamis, heat and mass transfer of two-piees droplet M.. thesis, K.N. Toosi University of tehnoloy, Tehran, Iran. 6-Clift, R., Grae, J. R., Weber, M. E., (1978), Bubbles, Drops and partiles, ademi press, U. IBN: