Numerical Simulation of Heat and Mass Transfer in the Evaporation of Glycols Liquid Film along an Insulated Vertical Channel

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1 Colmbia nternational Pblishing American Jornal of Heat and Mass Transfer doi:1.7726/ajhmt Research Article Nmerical Simlation of Heat and Mass Transfer in the Evaporation of lcols iqid Film along an nslated Vertical Channel Abderrahman Nait Alla 1, M barek Feddaoi 1*, and Hicham Meftah 1 Received 4 December 214; Pblished online 28 March 215 The athor(s) 215. Pblished with open access at Abstract A nmerical std is reported to investigate the evaporation in mied convection of a pre glcol liqid film sch as ethlene and proplene glcol. The model solves the copled governing eqations in both phases together with the bondar and interfacial conditions. The sstems of eqations obtained b sing an implicit finite difference method are solved b TDMA method. The inflence of inlet temperatre of liqid film and inlet pressre in the gas flow are eamined. The reslts indicate that the proplene glcol evaporates in more intense wa in comparison to ethlene glcol, de to the volatilit difference. Additionall, n the operating sstem, the heat transfer b sensible mode is more important than latent mode. Kewords: Mied convection; Heat and mass transfer; Evaporation, Ethlene and proplene glcol; Vertical channel 1. ntrodction Heat and mass transfers are important in man processes sch as evaporative cooling for waste heat disposal, cooling of a high temperatre srface b coating it with a phase-change material, liqid film evaporators, trbine blade cooling, combstion premiing, and distillation of a volatile component from a mitre with in volatiles. The evaporation of liqid film in air is an essential phsical phenomenon that ma be encontered in the most diverse fields. Becase of its practical importance, a large nmber of theoretical analses have been made concerning the evaporation of liqid film. A vast amont of works relating to water have been condcted b man athors. Chang et al. (1986) solved the Navier-Stokes eqations sing the thin shear flow assmption. The wall of the *Corresponding m.feddaoi@iz.ac.ma 1 aboratoire énie Energie, Matéria et Sstèmes (EMS), bn Zohr Universit ENSA B.P. 1136, Agadir, Morocco. 59

2 tbe was considered to be covered b a stationar water film at a specified niform temperatre. This std was etended later b Chiang and Kleinstreer (1991) b allowing the liqid film to fall nder the action of gravit and removing the thin shear flow assmptions. Both stdies involve Bossinesq approimation and the flow was assmed to be laminar. The evaporative cooling of liqid film in mied convection channel flows was eplored b Yan et al. (1991); Yan and in (1991). Their reslts show that the liqid film cooling is mainl cased b latent heat transfer associated with its evaporation. The evaporation of the pre liqid film b mied convection of hmid air in a vertical channel has been stdied b Yan (1992). Their std shows that the assmption of etremel thin film thickness is valid onl for low mass flow rate. He et al. (1998) was interested in cooling of the wall in a vertical tbe niforml heated b a water film. Trblent flow is considered in the gas flow and laminar in the liqid film. Two different modes of heat transfer are identified. When water is spplied at relativel high temperatre, the sstem operates in evaporating mode. When it is low the sstem operates in the direct film cooling mode. Debbissi et al. (21) investigated nmericall the copled heat and mass transfers b natral convection dring water evaporation into air in a vertical heated channel b inclding radiative heat transfer between plates. The observed that the evaporative cooling distrbs considerabl the velocit and temperatre profiles in particlar near the eit section of the channel. Feddaoi et al. (21, 23b) stdied the thermal transfer taking place in the event of film evaporation in the presence of a gas flow inside a vertical heated tbe. The reported that the evaporation is improved for a sstem with higher heat fl and smaller inlet liqid film. The also treated the case of inslated wall for both clindrical configration and vertical channel (Feddaoi et al. (23a, 26)) for eamining the effectiveness of evaporative cooling process. As far as the evaporation of other pre liqids films was concerned, Tsa et al. (199) ndertook a nmerical and eperimental std of cooling wall b sing an ethanol film on a vertical plate in the presence of a co-crrent gas flow. t was observed that the interfacial heat fl is predominantl determined b latent heat transfer connected with film evaporation, and significant reslts are obtained for the sstem with a high inlet liqid temperatre or a low inlet liqid film. Senhaji et al. (29) condcted a nmerical std of evaporating liqid film of pre alcohol b mied convection. The considered trblent liqid film falling on the inner face of a vertical tbe with a laminar flow of dr air entering the tbe with a constant temperatre. Recentl, Obella et al. (214) presents a nmerical simlation of the evaporation in laminar mied convection along a vertical channel formed b two parallel plates; one is isothermal and wetted b a thin liqid film of water or acetone, the second plate is inslated. The showed that the dimensionless mass evaporating rate increases noticeabl with the decrease of inlet temperatre and the se of more volatile liqid film. Comptational stdies of liqid film evaporation for glcols have been carried ot b a nmber of investigators. Palen et al. (1994) condcted an eperimental std in the case of mitres of water - ethlene glcol and water - proplene glcol in a vertical tbe. The observed that for some eperimental conditions, the local heat transfer coefficient between the partition and the liqid mitre can fall b 8% in relation to the relative vale of the pre water, a vale that is as low as the one got with the pre ethlene glcol in the same conditions. Agnaon et al. (1998) presented a nmerical analsis of the heat and mass transfer in a binar liqid film flowing on an inclined plate. The most interesting reslts are obtained in mied convection, particlarl in the case of ethlene glcol water mitre. Cherif and Daif (1999) considered the evaporation of a thin binar liqid film b mied convection in a vertical channel. The showed the importance of the film

3 thickness and composition in the mass and heat transfers. Armozi et al. (25) investigated nmericall the evaporation b mied convection of a binar liqid film flowing down of two coaial clinders. The showed that the volatilities of the mitre inflence the heat transferred throgh the latent mode, which is more prononced for a mitre composed of volatile components. Nasr et al. (211) performed a nmerical analsis of the evaporation of binar liqid film. The film is falling down on one plate of a vertical channel nder mied convection. The first plate of a vertical channel is eternall sbmitted to a niform heated fl while the second one is dr and isothermal. The liqid mitre consists of water and ethlene glcol while the gas mitre has three components: dr air, water vapor and ethlene-glcol vapor. Recentl, Debbissi et al. (213) investigated nmericall the evaporation of a binar liqid film flowing on one of two parallel vertical plates nder mied convection channel. This std shows that in some operating conditions, the water evaporates better when mied with ethlene glcol than water alone. t is important to note that the evaporation of the pre sbstances sch as ethlene or proplene glcols have not received mch attention, despite their importance in indstr as the most common antifreeze flids. n addition most of the stdies condcted are not treating the effect of pressre of the srronding gas on the evaporation phenomena. However, when the gas enters with a high pressre, this will decrease the difference between the gas pressre and its satration pressre and conseqentl a disadvantage for the evaporation phenomena. All these considerations motivate the present work b analsing the copled heat and mass transfers dring the evaporation of ethlene glcol and proplene glcol. n the following, the phsical and mathematical model will first be presented, and then the nmerical approach will be described. 2. Analsis 2.1 Phsical model and assmptions We consider a glcol liqid film falling along the internal face of a vertical channel (Fig 1). The liqid flows down over the left wall with an inlet liqid temperatre T and inlet liqid mass flow rate. The flow of dr air enters the channel at temperatre T and constant velocit. For mathematical formlation of the problem, the following simplifing assmptions are taken into consideration: Vapor mitre is ideal gas. The gas flow is laminar. The vapor and liqid phases are in thermodnamic eqilibrim at the interface. Radiation heat transfer, viscos dissipation and other secondar effects are negligible. 61

4 62 Fig. 1. Phsical model 2.2 overning eqations iqid phase - Continit eqation v (1) - Momentm eqation g d dp v (2) - Energ eqation T T v C T C p p (2) as stream - Continit eqation v (4) - Momentm eqation g d dp v (5) - Energ eqation

5 C T C v T p p T - Concentration eqation of vapor w vw D( C D Pv C w pa w ) T (6) (7) n the above eqations,, v, T, w,, and D, respectivel, stand for the densit, aial velocit, transverse velocit, temperatre, mass fraction of vapor, dnamic viscosit, condctivit and mass diffsivit; C p, C pv and C pa are the specific heat of gas, glcol vapor and air, respectivel, and g is the gravitational acceleration. The sbscripts and denote the liqid and gas phases, and the condition at inlet. The pressre gradients in eqations (2) and (5) can be written as: p pd g Ths the term: p pd g ( g ) where the term g represents the boanc forces de to the variations in temperatre and concentration Bondar and interfacial conditions The bondar conditions are: - : ; T T ; w w ; T T T - : v ; T - d : v ; w ; The interfacial matching conditions specified at are described as follows: - Continities of velocit and temperatre: - Continit of shear stress, ; T T, T,,,, (8) Transverse velocit of the air-vapor mitre is dedced b assming that the interface is semipermeable (Eckert and Drake (1972)) that is, the solbilit of air in the liqid film is negligibl small and the -directed velocit of air is zero at the interface: (9) 63

6 D w v (1) 1 w B assming the interface is in thermodnamic eqilibrim and the air-vapor mitre is an ideal gas mitre, the mass fraction of water or glcols vapor at the interface can be evalated b the relation: M v pv, i w (11) M p p M p where a v, i v v, i p v, i, is the partial pressre of vapor at the interface, and the molar masses of vapor and air; - Heat balance at the interface impling T where, h fg is the enthalp of evaporation and M v, and T m hfg, m, the vapor generation rate ) M a, are, respectivel, ( v The local heat echange between the air stream and liqid film depends on two related factors: the interfacial temperatre gradient on the air side reslts in sensible convective heat transfer, and the evaporative mass transfer rate on the liqid film side reslts in latent heat transfer. The total convective heat transfer from the liqid film to the air stream can be epressed as follows: T q qs q m h fg (13) The local Nsselt nmber at the interface defined as: ht 2d q 2d N T T Can be written as: S l N S l, b (12) (14) N N (15) where N, and N, are, respectivel, the local Nsselt nmbers for sensible and latent heat transfer, and are defined as: And N N qs 2 S T q 2 l T d T d T b The local Sherwood nmber is defined as: hm 2d m 1 w 2d Sh D Dw w where the sbscript b denote the blk qantities, the local blk temperatre fraction w b in the channel are respectivel defined as follows: b b (16) (17) (18) T b and the blk mass 64

7 And T d b d w b C d d C p p Td wd d At ever aial location, the overall mass balance in the gas flow and liqid film shold be satisfied: d d d v d d d v d To improve the nderstanding of heat and mass transfer process, a cmlative evaporation rate is introdced: M r m d (19) (2) (21) (22) v d (23) Note that in the above formlation, the variations of the thermophsical proprieties with temperatre and air-vapor composition are inclded. The are calclated from the pre component data b means of miing rles applicable to an mlticomponent mitre. For frther details, the thermophsical proprieties are available in (Reid and Sherwood (1984)). 3. Nmerical Method The conjgated problem is defined b the parabolic sstems eqations, (Eqs. 1 7) with the appropriate bondar conditions are solved b a finite difference nmerical scheme. Since the impossibilit of obtaining an analtic soltion for the non-linear copling differential eqations, the aial convection terms are approimated b the backward difference and the transversal convection and diffsion terms are approimated b the central difference. Each sstem of the finite-difference eqations forms a tridiagonal matri eqation which can be solved b the Thomas algorithm ( Patankar (198)). 3.1 Marching procedre After specifing the flow and thermal conditions, the nmerical soltion is advanced forward and step b step as follows: dp 1. For an aial location, gesses d and. d 65

8 2. Solve the finite-difference forms of eqations (1)-(7) simltaneosl for, T and w. Nmericall integrate (1) and (4) to findv. 3. Check the mass conservation of both liqid film and gas flow b eamining the satisfaction of the eqations (21) and (22). f not, adjst dp d and and repeat procedres 2 to 4. d 4. Check the satisfaction of the convergence of velocit, temperatre and mass fraction. f the relative error between two consective iterations is small enogh, i.e.: n n1 ma i, j i, j 4 1 (24) n ma where represents the variables, T and w. f not, repeat procedres 1 to Stabilit mesh i, j To obtain enhanced accrac in the nmerical comptations, grids are chosen to be non-niform in both aial and transverse directions. Accordingl the grids are compressed towards the interface gas-liqid and towards the entrance of channel. rid independence tests were carried ot b comparing the total heat transfer of ethlene and proplene glcol for different vales of, J and K (Table 1). t is noted that the differences in the cmlative evaporation rate from comptations sing grids ranging from to were alwas less than three percent. n light of those reslts all frther calclations were performed with the grid. n order to fi the position of the liqid-gas interface, the cartesian coordinate transformed into coordinates: - for - for d d Table1 Comparisons of cmlative evaporation rate Mr 1 kgm s for varios grid 1 1 arrangements and T 3K, T 4K,.2kgm s, Re 2, p 1atm, d. 2m d E. gl P. gl E. gl P. gl E. gl P. gl E. gl P. gl E. gl P. gl

9 N : total grid points in the aial direction; J: total grid points in the transverse direction at the gas side; K: total grid points in the transverse direction at the liqid side 3.3 Comparison with previos stdies To frther check the adeqac of the nmerical scheme for the present std, the reslts of the present model were compared with std of Yan (1992). The soltions agree well between the present predictions and those of Yan (1992) as shown in figre 2 which presents the aial evoltion of local Nsselt nmber. n view of these validations, the present nmerical algorithm and emploed grid laot are adeqac to obtain accrate reslts for practical prposes Present prediction Simlation of Yan [Yan, 1992] q W =5W.m -2 q W =1W.m -2 q W =3W.m /(b.Re) Fig. 2. Comparison of or present work with those of Yan (1992). validation of calclated ocal Nsselt nmber along the channel for Re 2, T 2C 4. Reslts and Discssions n this std, the reslts were obtained for impte parameters. The range of each parameter for the analsis is listed in Table 2. The chosen common parameters are the length of the channel 2m, the dr air at T 3K and the width of the channel d. 2 m. Table 2 The ranges of the phsical parameters. T K 3, 4 nlet liqid temperatre, nlet pressre, p atm 1, 2 kgm.1,. 2 nlet Renolds nmber, Re 1, nlet liqid mass flow rate, s For comparing the evaporation of some indstrial flids sch as ethlene glcol and proplene glcol, it is of interest to std the evoltion of some phsical properties which control the evaporation phenomena. Figre 3a presents the effect of the liqid temperatre on the evoltion of satrated pressre for two glcols liqids ethlene and proplene. t is clear that the satrated pressre of proplene glcol is more important than ethlene glcol, conseqentl proplene glcol 67

10 P Sat X1 5 (Pa) CP (J/kg/K) is more volatile than ethlene glcol. The effect of the liqid temperatre on the specific heat capacit of the two liqids is illstrated in figre 3b. t is apparent that the specific heat which represents the amont of heat per nit mass reqired to raise the temperatre b one degree Celsis is higher for proplene glcol. This implies that the specific heat for proplene is greater than ethlene. Figre 4 shows the development of the predicted velocit profiles along the channel of glcols film. The velocit profiles develop gradall from the niform distribtions at the inlet towards the parabolic ones when the flow goes downstream, with the maimm velocit shifting towards the inslated wall. The reslts indicate that in the gas flow, velocit profiles of ethlene and proplene glcol are similar ecept a small difference in the maimm velocit. ndeed,.65 speed ethlene glcol is higher than a proplene glcol. This is de to the lower densit of ethlene glcol than proplene glcol. Figre 5 presents the distribtion of aial temperatre profiles along the channel for varios sections. n the gas flow the temperatre increases as the flow goes downstream which is de to the energ spplied b the liqid film to the gas stream. The later, eplains the liqid temperatre decreases from the inlet to the otlet channel. t is interesting to observe that distribtion of aial temperatre profiles along the channel of glcols film develop in ver similar fashion. This is apparentl de to the small difference between the specific heats of the two glcols stdied. n the liqid phase, the temperatre seems to be constant. This can be eplained b the fact that the liqid film is etremel thin, so the distribtion of temperatre inside the liqid phase in the transverse direction is approimatel constant, ecept at high inlet liqid film (liqid flowing trblentl) when a slight decrease will be noticed from the temperatre at the wall to that at the gas-liqid interface. Figre 6 shows the evoltion of the mass fraction of glcols in the gas for different sections of the channel. Also it is noted that the mass fraction of proplene glcol is higher than the mass fraction of the ethlene glcol close to the interface liqid-gas. This increase is mch more noticeable for more volatile component, i.e. proplene glcol than ethlene glcol (a) Ethlene glcol Proplene glcol 35 (b) Ethlene glcol Proplene glcol T(K) T(K) Fig. 3. Evoltion of thermophsicals propertises of proplene glcol and ethlene glcol with thetemperatre of: (a) Satrated pressre, (b) Specific heat 68

11 T(K) (ms -1 )..2.4 iqid as Ethlene glcol Proplene glcol Fig. 4. Distribtions of aial velocit profile for T 4, Re 2 and K. 2kgm s iqid as 38 b (n=1) Ethlene glcol Proplene glcol Fig. 5. Distribtions of aial temperatre profile for T 4, Re 2. 2kgm s 1 1 K and 69

12 w.25.2 as.15 Ethlene glcol Proplene glcol Fig. 6. Distribtions of aial mass fraction profiles for T 4, Re 2 and. 2kgm s 1 1 T (K) K T (K) P =1atm - T =4K P =2atm - T =4K P =1atm - T =3K 2 4 P =1atm - T =4K P =2atm - T =4K P =1atm - T =3K /d /d (a) Ethlene glcol (b) Proplene glcol Fig. 7. Distribtions of temperatre at the gas-liqid interface along the channel for (a) 1 1 Ethleneglcol (b) Proplene glcol, for Re 2 and. 2kgm s 7

13 (b) Proplene glcol W W (a) Ethlene glcol 2 4 /d 8 P =1atm - T =4K P =2atm - T =4K P =1atm - T =3K 4 /d 8 P =1atm - T =3K P =2atm - T =4K P =1atm - T =4K 1 Fig. 8. Distribtions of mass fraction at the gas-liqid interface along the channel for (a) Ethlene 1 1 glcol (b) Proplene glcol, for Re 2 and. 2kgm s For more detailed analsis of heat and mass transfer characteristics, or attention is focsed on the effect of the inlet pressre and inlet liqid temperatre. Figre 7 shows the effect of the inlet pressre of the gas and inlet liqid temperatre on the interfacial temperatre along the channel. t is noticed that the interfacial temperatre decreases as the flow goes downstream. This decrease is de to the heat transfer from the liqid film to the gas flow. Also the interfacial temperatre profiles for two glcols are develop in ver similar fashion ecept some differences in the otpt of the channel. t is clear that the interfacial temperatre decreases from the inlet to the eit channel for the lower pressre, which is evident for the perfect gas assmed in this std. ndeed when the gas enters with a high pressre, this redces the intermoleclar distance and redces the evaporation. The distribtions of the interfacial mass fraction of glcols vapor along the gas-liqid interface are given in figre 8 for varios inlet pressre and inlet liqid temperatre. t is shown that the interfacial mass fraction decreases with small inlet liqid temperatre or a high inlet pressre. ndeed, for a fied vale of inlet pressre, we observed that the interfacial mass fraction is high for high inlet liqid temperatre. The interfacial mass fraction increases with the decrease of inlet pressre whereas the reverse trend is noticed for the effect of the inlet liqid temperatre along the channel. This is obviosl de to the enhancement of the evaporation in the operating conditions. 1 71

14 q S (Wm -2 ) q S (Wm -2 ) /d P =1atm - T =4K P =2atm - T =4K P =1atm - T =3K 4 /d P =1atm - T =4K P =2atm - T =4K P =1atm - T =3K 8 (a) Ethlene glcol 8 (b) Proplene glcol 1 1 Fig. 9. Aial evoltion of sensible heat fl for (a) Ethlene glcol (b) Proplene glcol, for 1 1 Re 2 and. 2kgm s q (Wm -2 ) q (Wm -2 ) P =1atm - T =4K 4 /d P =2atm - T =4K P =1atm - T =3K (a) Ethlene glcol 4 /d P =1atm - T =4K P =2atm - T =4K P =1atm - T =3K 8 8 (b) Proplene glcol 1 1 Fig. 1. Aial evoltion of latent heat fl for (a) Ethlene glcol (b) Proplene glcol, for 1 1 Re 2 and. 2kgm s 72

15 /d Mr (kgm -1 s -1 ) (a) Ethlene glcol P =1atm - T =4K P =2atm - T =4K P =1atm - T =3K /d Mr (kgm -1 s -1 ) P =1atm - T =4K P =2atm - T =4K P =1atm - T =3K 8 8 (b) Proplene glcol 1 Fig. 11. Aial evoltion of cmlative evaporation rate for (a) Ethlene glcol (b) Proplene glcol, 1 1 for Re 2 and. 2kgm s n figre 9 we present the aial distribtion of the sensible heat fl at the interface. t is remarkable that the heat transfer b sensible mode decreases monotonicall for both glcols for ( d 2) and tends towards an asmptotic vale at the channel eit. The later, can be eplained b the decrease of the gradient of temperatre between the hot liqid film and the cold gas as the flow goes downstream. t is interesting to observe that the distribtion of sensible heat fl at the interface along the channel develop in ver similar fashion for the two glcols. This is apparentl de to the small difference between the specific heat of the two stdied glcols (Fig 3b). t is also clear that the sensible heat q increases whit the increase of the inlet liqid temperatre. Whereas, S the effect of the inlet gas pressre on the sensible heat transfer is not ver prononced. The effects of T and P on the evoltion of latent heat transfer are illstrated in figre 1. The first observations to be drawn from these crves indicate that the glcols are varied in similar fashion, with a lower increase for proplene than ethlene. This is de to the higher satrated pressre of proplene compared to ethlene (Fig 3a). t is also noticed that latent heat transfer is higher for a high inlet liqid temperatre and lower inlet pressre, indicating the evaporation enhancement in theses operating conditions. t remains to notice that the heat transfer b evaporation is lower than the heat transfer b sensible mode de to temperatre gradient. This ascertainment is confirmed b figre 11 which represents the evoltion of the cmlative evaporation rate along the channel. t is clear that the evaporation rate increases when we increase the inlet liqid temperatre of the liqid film. ndeed, when the liqid film enters at 4 K, it is close to its boiling point and hence the evaporation is optimal. However, despite a higher inlet liqid temperatre, the effect of the pressre is remarkable. For T 4K and P 1 atm we can reach more than 2.7% for proplene glcol and 1.7% for ethlene glcol, whereas for other inlet liqid temperatre and inlet pressre, the cmlated evaporation rate does not eceed.8%. n 1 73

16 addition, there is a significant difference between the evaporation of the proplene glcol and ethlene glcol, which apparentl de to the higher volatilit of proplene glcol compared to ethlene glcol as clearl seen from (Fig 3a). 5. Conclsion The analsis presented in this paper has been developed b comparing the evaporation of two glcols liqid film: ethlene glcol and proplene glcol along a vertical channel. The effect of inlet liqid film temperatre and inlet pressre on the momentm, heat and mass transfer in the two phases are eamined in detail. Based on the nmerical reslts obtained, the following conclsion can be drawn: (1). n the operating sstem, the heat transfer b sensible mode is more important than latent mode. (2). Heat and mass transfers b latent mode are more enhanced at lower pressre. (3). Evaporation of glcols is improved for a sstem with higher inlet liqid film temperatre. (4). Proplene glcol evaporates in more intense wa in comparison to ethlene glcol, de to the volatilit difference. References Armozi, M. E., Chesnea, X., and Zeghmati., B. (25). Nmerical std of evaporation b mied convection of a binar liqid film flowing down the wall of two coaial clinders. nt. J. Heat Mass Transf., 41: Chang, C. J., in, T. F., and Yan, W. M. (1986). Natral convection flows in vertical open tbe reslting from combined boanc effects of thermal and mass diffsion. nt. J. Heat Mass Transf., 29: Chiang, H. and Kleinstreer, C. (1991). Analsis of passive cooling in a vertical finite channel sing a falling liqid film and boanc indced gas-vapor flow. nt. J. Heat Mass Transf., 34: Debbissi, C., Orfi, J., and Nasrallah., S. B. (21). Evaporation of water b free convection in a vertical channel inclding effects of wall radiative properties. nt. J. Heat Mass Transf., 44: Debbissi, C. H., Nasr, A., and Nasrallah, S. B. (213). Evaporation of a binar liqid film flowing down the wall of two vertical plates. nt. J. Therm. Sci., page Eckert, E. R.. and Drake, R. M. (1972). Analsis of Heal and Mass Transfer. Chaps 2 and 22. Mcraw- Hill,New York. Feddaoi, M., Belahmidi, E., and Mir., A. (23a). Nmerical simlation of mied convection heat and mass transfer with liqid film cooling along an inslated vertical channel. Heat Mass Transf., 39: Feddaoi, M., Belahmidi, E., Mir, A., and Bendo, A. (21). Nmerical std of the evaporative cooling of liqid film in laminar mied convection tbe flows. nt. J. Therm. Sci., 4: Feddaoi, M., Meftah, H., and Mir., A. (26). The nmerical comptation of the evaporative cooling of falling water film in trblent mied convection inside a vertical tbe. nt. Com. Heat Mass Transf., 33: Feddaoi, M., Mir, A., and Belahmidi, E. (23b). Co-crrent trblent mied convection heat and mass transfer in falling film of water inside a vertical heated tbe. nt. J. Heat Mass Transf., 46:

17 He, S., An, P., i, J., and Jackson, J. D. (1998). Combined heat and mass transfer in niforml heated vertical tbe with water film cooling. nt J Heat Flid Flow, 19: Nasr, A., Debbissi, C. H., and Nasrallah, S. B. (211). Nmerical std of evaporation b mied convection of a binar liqid film. Energ, 6: Obella, M., Feddaoi, M., and Mir, R. (214). Nmerical simlation of mied convection heat and mass transfers with film evaporation of water or acetone in a vertical channel. American Jornal of Heat and Mass Transfer, 1:65-8. Palen, J. W., Wang, Q., and Chen, J. C. (1994). Mass transfer in evaporating falling liqid film mitres. AChE J., 4: Patankar, S. V. (198). Nmerical Heat Transfer and Flid Flow. Hemisphere/Mc raw Hill. New York, Chap.6. Reid, R. and Sherwood, T. (1984). The properties of gases and liqids. Mc raw Hill, New York. Senhaji, S., Feddaoi, M., Medioni, T., and Mir., A. (29). Simltaneos heat and mass transfer inside a vertical tbe in evaporating a heated falling alcohols liqid film into a stream of dr air. Heat Mass Transf., 45: Tsa, Y.., in, T. F., and Yan, W. M. (199). Cooling of a falling liqid film throgh interfacial heat and mass transfer. nt J Mltiphase, 16: Yan, W. M. (1992). Effect of film evaporation on laminar mied convection heat and mass transfer in vertical channel. nt. J. Heat Mass Transf., 35: Yan, W. M. and in, T. F. (1991). Evaporative cooling of liqid film in throgh interfacial heat and mass transfer in a vertical channel : - nmerical std. nt. J. Heat Mass Transf., 34: Yan, W. M., in, T. F., and Tsa, Y.. (1991). Evaporative cooling of liqid film in throgh interfacial heat and mass transfer in a vertical channel : - eperimental std. nt. J. Heat Mass Transf., 34: