Hindawi Publishing CorporationJournal of TermodynamicsVolume 2013, Article ID 909162, 10 pageshttp://dx.doi.org/10.1155/2013/909162Research ArticleSecond Law Analysis of a Gas-Liquid Absorption FilmNejib Hidouri, Imen Chermiti, and Ammar Ben BrahimUniversity of Gabes, Engineers National School, Chemical and Process Engineering Department, Applied Termodynamics Unit,Omar Ibn El Khattab Street, 6029 Gab` es, TunisiaCorrespondence should be addressed to Nejib Hidouri; n hidouri@yahoo.comReceived 17 November 2012; Accepted 28 January 2013Academic Editor: Felix SharipovCopyright 2013 Nejib Hidouri et al. Tis is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Tis paper reports an analytical study of the second law in the case of gas absorption into a laminar falling viscous incompressibleliquid flm. Velocity, temperature, and concentration profles are determined and used for the entropy generation calculation.Irreversibilities due to heat transfer, fuid friction, and coupling efects between heat and mass transfer are derived. Te obtainedresults show that entropy generation is mainly due to coupling efects between heat and mass transfer near the gas-liquid interface.Total irreversibility is minimum at the difusion flm thickness. On approaching the liquid flm thickness, entropy generation ismainly due to viscous irreversibility.1. IntroductionGas-liquid (and vapor-liquid) fows occur in a wide varietyof situations such as: power generation, chemical processing,and energy production facilities. Some practical applicationsinclude condensers and reboilers, gas-liquid reactors (used inthe production of pharmaceuticals and specialty chemicals),wetted wall absorbers, power systems, core cooling of nuclearpower plant, and falling flm reactors. In addition to normalgravity applications, gas-liquid (and vapor-liquid) fows alsooccurinmanyplannedspaceoperationssuchas: powercycles, active thermal control systems [1], propulsion devices,and storage and transfer of cryogenic fuids.Gas-liquidfows canbe groupedinto a number ofdiferent fowpatterns. Tesefowpatternsarebasedonthe spatial and temporal distribution of the gas and liquidphases. Gravityis the major reasonfor suchtwo-phasefows to exhibit a wide range of fow patterns with diferentcharacteristics.Gas-liquidabsorptionis one of the most importantseparationprocesses, the gas-liquidcontactors inwhichabsorption takes place are typically spray, trayed, or packedsystems involving falling flms. Falling flm fow has foundwide applications in modern equipments, because it is widelyusedinmodernprocessindustriessuchasvertical con-densers, flm evaporators, distillation columns, and packedand absorption towers.Various works are elaboratedinorder tostudy thecoupled heat and mass transfer processes in gas or vaporabsorption into a liquid flm. Te theoretical analysis of com-bined heat and mass transfer processes in the case of gas orvapor absorption into a laminar and/or turbulent liquid flmas well as the temperature evolution and the concentrationprofles in the flm are developed [25]. Conlisk and Mao[6] considered the condensation of a two-component vapormixture into a two-component liquid flm on a falling flmabsorberinanabsorptionheat pump. Analytical expres-sions for temperature and mass fraction in the falling flmhave been determined. It is important in this situation torecall the experimental works of Miller and Keyhani [7], inwhichtheypresentedexperimental dataofthelocal heatandmass transfer processes inaqueous LiBr absorptioncycle. Teystudiedheatandmasstransferprocessesandpresented measurements of the temperature profle and thebulk concentration along the absorber length of the used testtube. Islam et al. [8] developed a linearized coupled modelfor heat and mass transfer in falling flm absorbers. In orderto test its accuracy, the developed model was compared with anonlinear model and a numerical simulation. Te used modelprovidesanalytical expressionsof heat andmasstransfercoefcients from the experimental data concerning verticalandhorizontal tubeabsorbers. KillionandGarimella[9]reported an interesting reviewabout the main used models to2 Journal of Termodynamicsillustrate coupled heat and mass transfer phenomena duringfalling flmabsorption. Detailed expressions of the governingequations, the boundary conditions, and the methods of res-olution used for modeling combined heat and mass transferproblems have been developed. Other extensive researcheshave been performed on gas absorption in liquid flms withdiferent fow regimes, geometries, and boundary conditions[1012]. In many real cases, the heat efect is small and theabsorption process may be considered isothermal [11, 13].Teimprovementofindustrial systemsaswell astheenergy utilization in any fuid is one of the fundamental prob-lems of the engineering processes, since improved systemswill provide better material processing, energy conservation,andenvironmental efects. Oneof theusedmethodsforpredicting the performance of the engineering processes isthe second law analysis. Te second law of thermodynamicsis applied to determine the irreversibilities in terms of theentropy generation. Entropy generation is the measure of thedestruction of the available work of the system. In the presentwork, entropy generation analysis concerns gas absorptionthrough a laminar falling liquid flm. In the following para-graph, someresearchesaboutentropygenerationanalysisthrough a falling flm will be presented.Te study of laminar falling viscous incompressible liq-uidflmalonganinclinedporousheatedplatehasmanysignifcant applications in thermal engineering and indus-tries, starting from petroleum drilling equipments to variousindustrial exchanger systems [16]. Adeyinka and Naterer [17]investigated the physical signifcance of entropy generation,and a resulting optimization correlation for a laminar flmcondensationonafat platewasdeveloped. Galovi c[18]studiedentropygenerationinevaporator/condensertypesof heat exchangers. Amodel was developedtocalculatethe overall entropy generation of heat exchange for givenvalues of heat transfer efectiveness and ratios of fow rates.Boulama et al. [19] derived ananalytical expressionof entropygenerationrateinafullydeveloped, laminar, andtime-independent simultaneous thermal and solute mixed convec-tion for the case of a nonreacting binary gas mixture fowingupwards in a vertical parallel plate. Carrington and Sun [20]developed diferent expressions of entropy generation dueto the coupling efects between heat and mass transfer ina multicomponent fuid. Tey identifed restrictions on theapplicabilityofexpressionsforentropygenerationduetomass transfer and heat transfer. Tey used an ideal binaryfuid to illustrate the analysis in the case of small pressuregradient inthe difusionfuxdirection. Magherbi et al.[21] reported a numerical study of entropy generation dueto heat transfer, mass transfer, and fuid friction in steadystate for laminar double difusive convection in an inclinedenclosure with heat and mass difusive walls. Tey studiedthe infuences of the inclinationangle of the enclosure,thethermal Grashof number, andthebuoyancyratioontotal entropy generation due to heat transfer, mass transfer,and fuid friction irreversibilities. Saouli and Aboud-Saouli[22]investigatedsecondlawanalysisof alaminarfallingliquid flm along an inclined heated plate. Tey consideredthe upper surface of the liquidflmfree andadiabaticandthelowerwall fxedandimpermeable. Resultsshowthat entropy generation number transversely decreases andincreases for all values of group parameters. Tey found thatthe irreversibility ratio decreases in the transverse directionand increases as the group parameter increases. Makinde andOsalusi [23] analytically investigated entropy generation ofa laminar falling viscous incompressible liquid flm alongan inclined porous heated plate. Velocity and temperatureprofles are obtained and used to calculate entropy gener-ationnumber. Teyfoundthat, neartheinclinedporousheatedplate, heattransferirreversibilitydominates, whilenear the liquid-free surface, viscous dissipation irreversibilitydominates. Gorla and Pratt [24] analytically studied secondlaw of a non-Newtonian laminar falling liquid flm alonganinclinedheatedplate, wheretheuppersurfaceof theliquidflmisconsideredfreeandadiabatic. Velocityandtemperature profles are calculated and used to compute theentropy generation number (

), the irreversibility ratio (),and the Bejan number (Be), for diferent values of the viscousdissipation parameter (Br1), the viscosity index (), andthe dimensionless axial distance (). Results show that heattransfer dominates the viscous irreversibility when 0 < 1,and the fuid friction irreversibility dominates heat transferirreversibility when > 1. For = 1, both heat transferand fuid friction irreversibilities have equal contribution forentropy generation. Recently, Chermiti et al. [25] analyticallyinvestigated entropy generation of an isothermal absorptionof carbondioxideintoalaminar viscous incompressiblefalling liquid flm (H2O). Entropy generation expression isdeveloped, and the velocity profle as well as the absorbedgas concentration is determined and used for the calculationof thedifusiveandtheconvectivemass transfer fuxes,as well as the entropy generation. It was found that, fromthe gas-liquid interface towards the difusion flm thickness

, entropy generation increases towards a maximum valuedue to difusive mass transfer and then decreases towardsaminimumvaluebyconvectivemode. Tisminimumisobtained at the difusion flm thickness. For values of liquidflm thickness greater than the difusion flm thickness (i.e., >

), entropy generation linearly increases due to thefuid friction irreversibility. Near the difusion flmthickness,mass transfer irreversibility dominates. On approaching theliquidflmthickness , viscous irreversibilitydominates.Te increase of the falling flm length induces the decreaseof entropy generation magnitude. Aboud-Saouli et al. [26]presented the application of the second law to gravity-drivenliquid flm along an inclined heated plate in the presence ofa transverse magnetic feld and viscous dissipation efects. Itwas found that entropy generation increases with Hartmannumber, Brinkman number, and the dimensionless group.Irreversibility due to the magnetic feld, the conduction heatin the transverse direction, and the fuid friction increases bythe increase of Hartman number, Brinkman number, and thedimensionless group, respectively.Te main originality of the present work is the secondlaw analysis for the case of gas absorption through a verticallaminarfallingviscousincompressibleliquidflm, bythedetermination of diferent sources of irreversibility. Te studyis based on the analytical fow feld solutions obtained forJournal of Termodynamics 3the velocity, the temperature, and the concentration profles.Tevelocitydistributionwas determinedbysolvingthemomentum equation subject to an appropriate set of bound-ary conditions. Te concentration profle was determined bysolving the species conservation equation with appropriateboundary conditions, and the temperature distribution wasobtainedbyusingthe energyconservationequation. Inmany previous works, authors are interested to determinethe concentration and the temperature profles only. In thecase of vapor absorption [2, 3], a linear relation between thetemperature and the concentration in equilibriumwith vaporat constant pressure is adopted and makes the resolution ofthe systemequations. In this study, velocity, temperature, andconcentration felds are analytically determined and directlyused to evaluate the entropy generation.In the presence of simultaneous heat and mass transfer,four sources of irreversibility have been identifed, which aredue to coupling efects between heat and mass transfer bydifusion, due to heat transfer, due to coupling efects betweenheat and mass transfer by convection and fuid friction.2. Mathematical Formulation2.1. Physical Considerations. Generally, the fuid densitydepends on pressure and temperature.Tus,the presenceof temperature variation or chemical species (thermal andconcentration gradients) may change the fuid density whichcan result in thermal or solute natural convection. Te systemunder consideration is illustrated in Figure 1. It consists of alaminar falling viscous incompressible liquidflm(designatedby the letter B), fowing downona vertical plate under gravity.Tegas, designatedbytheletterA, isabsorbedwithoutchemical reactions. Inthepresentstudy, theassumptionsemployed in the problem formulation are(i) physical absorption (no chemical reactions),(ii) two dimensional fow in steady state regime,(iii) falling laminar fow of a Newtonian incompressibleviscous liquid flm (water),(iv) ammonia as absorbed gas,(v) the difusion flm thickness is too small as comparedto the falling flm thickness (i.e.,

), so thatthe thermophysical properties of the falling flm aresimilar to those of the liquid,(vi) the Boussinesq approximation is valid for assuming alinear variation of the fuid density with temperatureand concentration,(vii) the curvature of the liquid flm is neglected, and thevelocity only depends on the transverse direction (i.e., =

()),(ix) the absorption process takes place at constant pres-sure (i.e., (/) 0),(x) for the falling flm (a thin flm), the axial conductionterms are neglected as compared to the transverseterms (i.e., (2/2) (2/2)). Further, for thethin flm, small values of temperature diference are considered. = 0 = Liquid ow (B)

(, )(, )Gas ux (A)L

()

dFigure 1: Description of gas absorption into a laminar falling liquidflm: typical profles of temperature, velocity, and concentration.2.2. VelocityProfle. UndertheBoussinesqapproximationand for a small density variation, the fuid density linearlyvaries with temperature and species concentration such that = ref [1

( ref)

( ref)] . (1)

ref is the reference density measured at reference tempera-ture (ref) and concentration (ref).Termal and solute expansion coefcients

and

are,respectively, given by

= 1

ref(

),

= 1

ref(

). (2)Te continuity and momentum equations are given by

+

= 0,

ref (

+

) =

+

(

2

2 + 2

2),

ref (

+

) =

+

(

2

2 + 2

2) + .(3)4 Journal of Termodynamics describes the buoyancy forces due to thermal and concen-tration gradients. It is given by =

0 = (

0) . (4)

and 0 are the fuid densities measured at the falling flmthickness ( = ) and at the gas-liquid interface ( = 0),respectively. Since the fow takes place along a vertical plate,thevelocityonlydependsonthedifusionflmthickness(i.e., in transverse direction). Tat is, the transverse velocity 0 and =

() [27]. Further, the absorption processtakes place at constant pressure.Afer rearrangement,thecontinuity and the momentum equations are, respectively,given by

= 0, (5)

ref(

2

2) +

ref= 0. (6)By using (1) and (4), the buoyancy forces are given by = ref [(

(0

) +

(0

))] . (7)

0and

are the gas-liquid interface and the falling flmthickness temperatures, respectively. Tus, (6) can be writtenas follows:

2

2 = ref [

(

) +

(

)] .(8)Tis diferential equation is the liquid motion equation.Te parabolic profle of the velocity enables us to give theappropriate boundary conditions as follows:at = 0, = max,at = , = 0.(9)Te reference density is supposed to be equal to that ofthe liquid (i.e., ref =

). Tus, the solution of (8) is given by () = max(1 (

)2)=

2

2[

(

) +

(

)] (1 (

)2).(10)For small density diferences, thermal and solute expan-sion coefcients are given by [28, 29]

= 1

(

) 1,

= 1

(

) 1, (11)where = 0

, = 0

. (12)Te maximum velocity is, therefore, given by

max =

2.(13)2.3. Concentration Profle. For the gas-liquid absorption sys-temunder consideration, the difusing species is the absorbedgas, which is designated by the letter A (see Figure 1). Tus,the species conservation equation is given by

+

(

2

2 + 2

2 ) = 0. (14)As the Reynolds number of the system is smaller than 25(laminar fow) [29], the above equation is, therefore, given by

max(1 (

)2)

2

2 = 0. (15)For a small penetration distance of the gas into the fallingliquid flmand a short contact time (physical absorption), theflm is supposed to move with a velocity equal to max [30,31]. Finally, the required solution for the gas concentrationthrough the falling flm can be obtained by using the methodof combination of variables (see details in [25]):

= 0(1 erf (

4 (

/max) )). (16)Te transverse and axial molar fuxes

and

of thegas are, respectively, given by

=

,

=

() .(17)2.4. Temperature Profle. Due to the strong dependence of thegas solubility on temperature, it is important to determinethe temperature distribution through the falling liquid flm.Indeed, the gas absorption heat is the most heat infuencingthe temperature profle and leading to a rise in the temper-ature of the liquid phase [32]. Energy conservation equationthroughout the falling liquid flm is given by

+

=

, (

2

2 + 2

2 ). (18)By taking into consideration the above-mentionedassumptions and by introducing the thermal difusivity ofthe fuid given by =

/(

), the energy conservationequation is simply given by

2

2 = max(1 (

)2)

. (19)Journal of Termodynamics 5Te appropriate boundary conditions associated to thisequation are = 0, (, 0) = 0, (20) = 0, =

(0, )

, (21) = ,(, )

= 0. (22)

is the interfacial heat fux given by the heat of absorption.Before proceeding with the solution of the temperatureequation (19), it is convenient to rewrite the equations in adimensionless form. Let us defne the following dimension-less variables: (, ) = (, ) 0 , =

, =

max

2.(23) is the temperature diference given by = ()/

.Equation(19) with the newdimensionless variablesbecomes

2

2 = (1 2)

.(24)Te dimensionless boundary conditions are given by = 0, (, 0) = 0, (25) = 0, (0, )

= 1, = 1, (1, )

= 0.(26)To solve the dimensionless temperature equation(24), thebest approach is to use the separation of variables method[33], so that the dimensionless temperature is given by (, ) = 1() + 2() . (27)Te solution of(27) is the sum of two functions 1()and 2() that depend only on and , respectively. Conse-quently, the separated solutions into the energy equation give

2

1()

2 = (1 2) 2()

.(28)Te two sides of the above equation are equal for anychosen values of and , so both sides are equal to the sameconstant. In other words, the above equation yields a pair ofseparated diferential equations given by

2

1()

2 = 1(1 2) ,

2()

= 1.(29)Te general solution of the dimensionless temperature is,therefore, given by (, ) =

412

1 + 22 1 + 2 + 1 + 3.(30)

1, 2, and 3 are constants of integration.In order to determine the third constant 3, the meanbulk temperature should be introduced. It is given by

() = 10 (, ) . (31)Te frst boundary condition given by (25) gives the meanbulk temperature

(0) = 0. By using this condition andthe two other boundary conditions, the temperature profleis fnally given by(, ) = 0 + [18(

)4 + 34(

)2 (

) + 32 (

max

2) + 1140] .(32)3. Entropy GenerationTe existence of thermal, velocity, and concentration gradi-ents along the absorbing falling flm yields a nonequilibriumstate, which causes entropy generation. Under these condi-tions, the volumetric rate of entropy generation in the fowfeld with both heat and mass transfer and without body forceand chemical reaction efects is given by [34]

= grad (1

) + : grad ()

1

grad (

) .(33)Te frst term of the right hand side of (33) is the entropygeneration due to heat transfer, the second is due to fuidfriction, and the third is due to mass transfer.Te energy fux is the sum of the heat fux

, given bythe Fourier law, and the enthalpy fux due to species difusion.Tat is =

=

grad ()

.(34)Rearrangement of (33) gives

= 1

2 [

grad ()] + 1

2 [(

) grad ()]+ : grad ( )

1

grad (

) .(35)Te frst termof the right-handside of (35) is theirreversibility due to thermal gradients, the second is duetocouplingefects betweenthermal gradients andmass6 Journal of Termodynamicsdifusion, the third is due to fuid friction, and the fourth isdue to mass difusion.Te specifc chemical potential of the gas () is given by

(,

) = ,(

) ,Ln (

)+

(

,

)

(

,

) .(36)

and

are the reference pressure and temperature, respec-tively.

and

are the enthalpy and entropy of the gas, at

and

.Te shear stress is simply given by [28]

=

(

+

). (37)Tus, the local entropy generation rate in a two-dimensional fow with a single difusing species can, there-fore, be written as follows:

=

2 [(

)2 + (

)2] +

2 [

+

]+

(

)2 1

(

+

).(38)Expressions of specifc enthalpy andentropy of theabsorbed gas are, respectively, given by

=

( 0) + 0,

=

Ln (

0) + 0.(39)Byreplacingthefuxes

and

andthespecifcenthalpy and entropy given above by their expressions andby making the derivatives of the specifc chemical potential,the expression of the local entropy generation rate for the caseof gas absorption into a falling liquid flm is, therefore, givenby

=

2 [(

)2 + (

)2] (

2 )(

)(

) [

( 0) + 0 + (

Ln (

0) + 0)]+ (

2 )(

) [

( 0) + 0+(

Ln (

0) + 0)] +

(

)2.(40)Tefrst termof theright-handsideof (40) is theirreversibility due to heat transfer ,th, the second and thethird terms are due to the coupling efects between heat andmass transfer ,-th (by difusion and by convection, resp.),andthelastoneisduetothefuidfriction ,(viscousdissipation irreversibility).Table 1: Gas and liquid physical properties [14, 15].PropertyLiquidInlet temperature,

(K) 298Reference pressure,

(atm) 1(Water)Dynamic viscosity,

(Pa s) 0, 911 103Mass density,

(kg m3) 1000Termal conductivity,

(W ml K1)0,607GasDifusion coefcient,

(m2 s1)1, 9 109Solubility in water at

,

(g L1)540(Ammonia)Reference molar entropy,

(J mol1 K1)192,77Specifc heat,

(J kg1 K1) 2129,454. Results and DiscussionsVelocity and concentration are calculated by using (10) and(16),respectively.Temperature Equation (32) is solved foragivenvalueof temperaturediference . Valuesof are selected in order to ensure a laminar fow of the liquidflm. Te absorption of ammonia into a falling water flm isconsidered. All the following results are calculated using thephysical properties of water and ammonia given in Table 1.Figures 25 show the general behavior of irreversibilitiesdue to thermal gradients, coupling efects between heat andmasstransfer, fuidfriction, andtotal entropygenerationversus the transverse direction(i.e., fromthe gas-liquidinterface to the falling liquid flm thickness ), for knownvalues of the liquid flm length and the temperature difer-ence . Te choice of the temperature diference rangingbetween1Cand5Cisbasedonanexperimental studyelaborated by Mahmoud [35] about heat and mass transferaccompanying ammonia vapor absorption into a solution ofwater-ammonia. Te study shows that for a null fraction ofammonia in the solution and a penetration distance equalto 4 104 m, the solution temperature increases from 22Cto 29C. Using this argument, it is important to consider theabove-mentioned temperature diference values to examinetheevolutionof diferent sourcesof irreversibilityinthesystem.As it can be seen, Figure 2 shows that irreversibility dueto thermal gradients ,th is at its maximum value at the gas-liquid interface, where the gas absorption process begins. Forany fxed value of the temperature diference , entropygeneration due to thermal gradients exponentially decreasesas increases; since the temperature is a fractional polyno-mial function of the transverse coordinate , its derivativegives another polynomial function with lower degree (i.e.,the irreversibility due to thermal gradients). Consequently,thermal irreversibility mainly happens near the gas-liquidinterface, then it decreases and tends towards a minimumconstant value on approaching the falling flm thickness ,where the absorption process is practically absent.Journal of Termodynamics 71040 1 2 300.050.10.150.20.250.30.350.40.450.5 (m)

,th(kJ/Km3s) = 1C = 3C = 5CFigure 2: Entropy generation due to thermal gradients, ,th versus and .0204060801001201401040 1 2 3 (m) = 1C = 3C = 5C

,m-th(kJ/Km3s)Figure 3: Entropy generation due to heat and mass transfer, ,-thversus and .For a fxed value of the transverse distance , it can beobserved that as the temperature gradients increase, entropygeneration due to thermal gradients also increases.Figure 3 illustrates entropy generationdue to the couplingefects between heat and mass transfer, ,-th. It can be seenthat near the gas-liquid interface, entropy generation is at itsmaximum value, since the absorption process mainly takesplace close to the gas-liquid interface, which is accompaniedby the heat of absorption. Tus, entropy generation due tothe coupling efects between heat and mass transfer is similarto the concentration profle, which is maximum at the gas-liquid interface (i.e., at = 0) as found by Chermiti et al.

,

(kJ/Km3s)00.020.040.060.080.10.120.141040 1 2 3 (m) = 1C = 3C = 5CFigure 4: Entropy generation due to fuid friction, , versus and.1040 1 2 3 (m) = 1C = 3C = 5C

(kJ/Km3s)102101100101102103Figure 5: Total entropy generation,

versus and .[25]. Te increase of the transverse distance (i.e., > 0)induces the decrease of ,-th, to become null near the fallingflmthickness(i.e., = ), forall thestudiedvaluesof. Tis is due to the fact that when the transverse dis-tance increases, the absorption process and the fuid motiondecrease. Consequently, irreversibilities due coupling efectsbetween heat and mass transfer by difusion and convectiondecrease. At the flmthickness (= ), the absorption processis insignifcant (no concentration gradients).It could be noticed that, at lower values of temperaturegradient , the gas absorptionproceeds at a faster rate due tothe availability of larger driving potentials for mass and heattransfer.8 Journal of TermodynamicsTroughout the difusion flmthickness region, thedecrease of the temperature diference induces the decrease ofboth irreversibilities due to heat transfer (Figure 2) and cou-pling efects between heat and mass transfer (Figure 3). Tisis attributed to the decrease of temperature and concentrationas well as velocity gradients in this region.Figure 4 shows entropy generation due to fuid friction,

, along the liquid flm. As shown, the analytically obtainedviscous dissipation irreversibility is null near the gas-liquidinterface, sincevelocityis at its maximumvalueat thisregion. As a consequence, the friction efects contribution isinsignifcant near the interface.On approaching the flm thickness, viscous dissipationirreversibility signifcantly increases and it is maximumat thefalling flm thickness ( = ). Te fuid friction irreversibilityexhibits a parabolic profle. Tis behavior is expected, since itsexpression includes the term (2) via the velocity expression(see (40)). Te fuid friction irreversibility magnitude slightlyincreases with the decrease of the temperature diference as it is inversely proportional to the temperature.As a consequence, the viscous dissipation irreversibilitythrough the falling liquid flm increases as the transverse dis-tance increases and, inversely, as the temperature diferencedecreases.Total entropygeneration

behavioris illustratedinFigure 5. As it can be seen, entropy generation is at its max-imum value near the gas-liquid interface for all the studiesvalues of . Tis is attributed, on one hand, to the domi-nant efect of coupled heat and mass transfer irreversibilityinduced by the absorption process and, on the other hand,to the thermal irreversibility. As mentioned, at the interfaceregion, irreversibilityduetofuidfrictionisinsignifcant(Figure 3). Further, the increase of the temperature diferenceinduces the increase of total entropy generation magnitude.On increasing the transverse distance, total entropy gen-eration signifcantly decreases towards a minimum value forall the studied values of the temperature diference. Entropygeneration is minimumat the difusion flmthickness

. Tedecrease of entropy generation from the gas-liquid interfaceto the difusion flm thickness is due to the decrease of bothconvective and difusive irreversibilities, since the gas difu-sion occurs at the beginning of the falling flm thickness andthe mass transfer by convection decreases while approachingthe difusion flm thickness. At this point, the infuence ofboththermal andcoupledefectsbetweenheatandmasstransfer (by difusionandconvection) irreversibilities ontotalentropy generation becomes insignifcant. For >

, totalentropy generation increases towards an asymptotic value,which is mainly due to the viscous dissipation irreversibility.Tis result is consistent with the fndings of Chermiti et al.[25]. For >

, entropy generation slightly increases withthe decrease of the temperature diference as explained above.5. ConclusionsTesecondlawof thermodynamicsisanalyticallyinves-tigatedforthecaseof gasabsorptionthroughalaminarfalling flm. Under some physical assumptions, velocity,temperature, andgas concentrationequations are derivedandused for the entropy generation calculation concerning theabsorption of ammonia in a water falling flm. Te analysisshows that entropy generationis due to heat transfer, couplingefects between heat and mass transfer (by convection anddifusion), and fuid friction. Entropy generation due to fuidfrictionis insignifcant at thegas-liquidinterface, wherethermal and coupled efects between heat and mass transferirreversibilities dominate. At the interface ( = 0), total irre-versibility magnitude is maximum, whereas it is minimum atthe difusion flm thickness ( =

). On approaching theliquid flm thickness ( =

), entropy generation is mainlydue to fuid friction efects. At the difusion flm region (0 <

), entropy generation magnitude increases with theincreaseofthetemperaturediferenceofthefallingflm;this behavior is reversed for values of the transverse distancegreater than the difusion flm thickness (

< ).Nomenclature

: Molar concentration of gas (molm3)

: Isobaric specifc heat of gas (Jkg1K1)

: Molecular difusivity coefcient (m2s1)erf: Error function: Energy fux (Js1m2)

: Mass transfer fux of gas (molm2s1)

: Specifc enthalpy of gas (J/mol)

: Liquid thermal conductivity (Wm1C1): Falling flm length (m): Pressure (Pa)

: Specifc entropy of gas (Jkg1K1): Temperature (K), : Falling flm velocity components (ms1), : Rectangular coordinates (m), :Dimensionless coordinates ().Greek Symbols: Termal difusivity of liquid (m2s1)

: Solute expansion coefcient (m3mol1)

: Termal expansion coefcient (K1)

: Dynamic viscosity (Pas): Falling flm thickness (m)

: Difusion flm thickness (m): Temperature diference through theliquid flm (C)

: Specifc chemical potential of gas(Jmol1): Fluid mass density (kgm3): Stress tensor (Nm2): Dimensionless temperature (): Integration constant ()

: Entropy generation rate per unitvolume (Jm3s1K1).Journal of Termodynamics 9References[1]A. A. M. Delil, Researchissues ontwo-phase loops forspace applications, in Proceedings of the Symposium on SpaceFlight Mechanics, Institute of Space and Astronautical Science,Sagamihara, Japan, 2000.[2]G. Grossman, Analysis of interdifusion in flm absorption,International Journal of Heat and Mass Transfer, vol. 30, no. 1,pp. 205208, 1987.[3]G. Grossman and M. T. Heath, Simultaneous heat and masstransfer inabsorptionof gases inturbulent liquidflms,International Journal of Heat and Mass Transfer, vol. 27, no. 12,pp. 23652376, 1984.[4]L. P. KholpanovandE. Y. Kenig, Coupledmassandheattransferinamulticomponent turbulent fallingliquidflm,International Journal of Heat and Mass Transfer, vol. 36, no. 14,pp. 36473657, 1993.[5]B. J. C. van der Wekken and R. H. 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