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1 ISSN Turbulent fil condensation in a vertical tube in presence of non condensable gas Y. Belkassi, K. Gueraoui, N. Hassanain, A. Elbouzidi Tea of vapour gas in fluid echanics and environent, LPT, URAC 13 Faculty of sciences, Mohaed V University, Rabat-Agdal B.P. 114, Rabat, Morocco y.belkassi@gail.co Tea of vapour gas in fluid echanics and environent, LPT, URAC 13 Faculty of sciences, Mohaed V University, Rabat-Agdal B.P. 114, Rabat, Morocco Departent of echanical engineering, Ottawa University, Canada kgueraoui@yahoo.fr Tea of agnetic aterials and applications, LPM, 4 Avenue Ibn Battouta Faculty of sciences, Mohaed V University, Rabat-Agdal B.P. 114 RP, Rabat, Morocco naje.hassanain@gail.co Tea therodynaic energy, Faculty of sciences, Mohaed V University, Rabat-Agdal B.P. 114, Rabat, Maroc Elbouzidi.abdellah@gail.co ABSTRACT This paper presents the siulation of the condensation of ethanol vapour in the presence of non-condensable gas in turbulent flows in a vertical tube. The liquid and gas strea are approached by two coupled turbulent boundary layer. For solving the coupled governing equations for liquid fil and gas flow together with the interfacial atching conditions an iplicit finite difference ethod is eployed. The effect of the influencing paraeters are studied so the effect of inlet Reynolds nuber, the effect of teperature gradient, ass fraction are illustrated. The nuerical results deonstrate that an iportant concentration of no-condensable gas reduces the heat transfer coefficient and fil thickness considerably. The local heat flux and fil thickness increase as tube surface teperature decreases at any bulk concentration of noncondensable gas. Moreover, inlet velocity increases as fil thickness decreases and heat flux increases. Keywords Condensation; Heat and ass transfers; ixed convection; turbulent flow; Vertical tube; Methanol vapour. Council for Innovative Research Peer Review Research Publishing Syste Journal: JOURNAL OF ADVANCES IN PHYSICS Vol. 6, No. 3 japeditor@gail.co 18 P a g e D e c e b r e 1 5, 1 4

2 1. INTRODUCTION ISSN Condensation of frigorific vapour in the presence of non-condensable gases has any applications such as air conditioning, electricity generation, refrigeration, reactor safety, aerospace, desalination and soe heat exchangers. Fro the literature we report a several studies in the lainar case [1-7] of condensation in a vertical tube and other studies in the turbulent case [8-13]. Belkassi et al [] presented a nuerical study of fil condensation by forced convection in a vertical tube; they studied the effect of soe paraeter influence on the phenoenon of condensation in the lainar case. Revankar and Oh [3] was study the coplete condensation of saturated vapor in lainar liquid fil in a vertical tube isotheral wall, the study is theoretical and the odel elaborate have considered the evolution of the shear stresses at the interface and profiles of liquid and vapor velocity. A nuerical study of lainar ixed convection condensation of stea in the presence of non-condensable gases was perfored by Chin et al. [4] the odel adopted is based on the equations of two-diensional two-phase boundary layer by considering the inertia ters, Convection and constraints shear at the interface and variable physical properties. The thickness of the liquid fil is calculated fro the theral balance at the interface. In the sae context Oh [5] has ade a study, the equations of the two-phase boundary layer systes are solved by the finite difference ethod for lainar liquid fil and finite volue for the turbulent stea. Louahlia and Panday [8] have studied nuerically the fil condensation between two horizontal plates by forced convection of refrigerants R134 and R13 and their ixture. The coupled equations of asse conservation, the oentu equation, species and energy are applied in two phases using the boundary layer odels. Panday [9] conducted a nuerical study of turbulent fil condensation inside a vertical tube the walls are isotheral, the saturated vapour of R13 then the ixture of R134a-R13 are studied. Wang and Tu [1] have developed a odel for analyzing the effect of a non-condensable gas in fil condensation of gas-vapor ixture in turbulent flow in a vertical tube. They noticed that the reduction of heat transfer due to the non-condensable gas was ore significant at low pressure and at low Reynolds nuber of the ixture. The proble of condensation of ethanol vapour in the presence of non-condensable gas (air) in turbulent flow along of a vertical tube is tackled theoretically. The walls are isother, the condensate fil on the wall of the pipe is assued to be a thin fil in turbulent flow, the heat and ass transfers in the liquid and vapor phase are governed respectively by classical strea, and forced convection equations, transfers equations are linked at the liquid vapor interface by the continuity of the shear stress and the heat flux densities and by the heat flux density through the wall of the tube. The equations are discredited by the finite difference ethod and solved by Toas algoriths.. ANALYSIS Fig.1 shows a scheatic of the physical odel considered for the condensation process with defined coordinate systes. We consider a forced convection vapor ixture flow downward inside a vertical tube of (L) height and (d) with. The teperature of the wall is aintained at T wall, and the ixture enters the tube with a unifor velocity u, unifor teperaturet, unifor pressure P and unifor gas ass fraction W. The following assuptions were ade in the developent of governing equation: Fig.1 Physical odel for vertical tube condenser - The liquid and gas flows are turbulent and the flow is tow-diensional. - The condensate fil is ipereable to incondensable gas. - Huid air is an ideal ixture of ethanol vapor it is considered a perfect gas. 183 P a g e D e c e b r e 1 5, 1 4

3 ISSN The Soret and Duffour effects are ignored, and the effect of the superficial tension is neglected. - The gas-liquid interface is in the therodynaic equilibriu. The following equations are written for the boundary layers..1 Liquid fil equations Continuity equation Moentu equation ρ l u l / x + (1/r) rρ l v l / r = (1) (ρ l u l u l )/ x + (u l ρ l v l )/ r = dp/dx + (1/r) / r μ l + μ tl u l / r + ρ l. g () Energy equation u l ρ l C Pl T l / x + (v l ρ l C Pl T l )/ r = (1/r) / r r λ l + λ tl T l / r (3). Gas flow equations Continuity equation Moentu equation ρ G u G / x + 1/r rρ G v G / r = ρ G u G u G / x + (ρ G u G v G )/ r = dp/dx + (1/r) r(μ G + μ tg ) u G / r / r + ρ G. g Energy equation (ρ G C PG u G T G )/ x + (ρ G C PG v G T G )/ r = (1/r) / r r λ G + λ tg T G / r Species equation (ρ G u G w G )/ x + (ρ G v G w G )/ r = (1/r) rρ G D G + D tg w/ r / r.3 Turbulence odeling Liquid fil: (4) (5) (6) (7) As outlined in Yih and Liu [1] a odified Van Driest eddy viscosity odel proposed by Yih and Liu was used to siulate the turbulence in the liquid fil. Siilar turbulence odeless was also used to siulate the heat transfer across a turbulent falling fil. The turbulent eddy viscosity is given by: μ Lt μ L =,5 +,5 1 +,64y + τ τ w 1 exp y + τ τ w 1/ A + f 1/ For:,6 < (R r δ) < 1, where f = exp 1,66 1 τ τ w, y + = R r u /v, A + = 5,1 For : R r /δ x <.6 the turbulence eddy viscosity for the liquid fil was taken as constant an equal to its value at R r /δ x =.6. The turbulent conductivity λ Lt can then be obtained by introducing the turbulent Prandtl nuber Pr Lt so λ Lt = μ Lt. C pl /Pr t, where the turbulent Prandtl nuber is evaluated fro Cebeci and Sith [14]. Gas flow : The realizable k ε odel was used to odel turbulence in the present siulation in the gas flow, a low Reynolds nuber odel in evaporation case is adopted [15, 16]. The governing equation for the turbulent kinetic energy and the dissipation rate are, k transport ρ G u k x + ρ Gv k r = 1 r r r μ G + μ tg σ k k r + μ u G Gt r ρ G ε + D v 184 P a g e D e c e b r e 1 5, 1 4

4 ε transport The odel constants and function are, ρ G u ε x + ρ Gv ε r = 1 r r r μ G + μ tg σ ε ρ G C ε f ε f 1 = 1, f = 1.3 exp Re t k + μ Gμ Gt ρ G ε r + C ε ε1f 1 k μ u G Gt r u G r ε = ε D ε, D ε = ν k 1 ISSN , f μ = exp 3.4/ 1 + Re t /5 c ε c k c ε c ε1 c μ Boundary Conditions The boundary conditions of these arching type probles are: x =, u g = u, T g = T, w = w P = P, k = 3(I u g ), ε = c μ k 3/ /κr, w = w r = R, u l = v l =, T = T w r =, v g =, u g / r =, T g / r = r w/ r = k/ r = ε / r = The solution fro the liquid side and gas side satisfy the following interfacial (r = (R δ x ) atching conditions: continuities of velocity and teperature: u I (x) = u g,i = u l,i, T I (x) = T g,i = T l,i (11) Continuity of shear stress: τ I = (μ G + μ tg ) u/ r G,I = (μ l + μ lt ) u/ r l,i (1) Velocity of air vapour ixture at the interface. The transverse velocity coponent of the air vapour ixture at the interface is deduced by assuing the interface to be sei-pereable, that is the solubility of air into the liquid is negligibly, the velocity at interface can be calculated by : v I = (D G D Gt )/(1 w I ) w/ r (14) Heat balance at the interface : λ l ( T l / r) I = λ g ( T G / r) I fg I (15) in which fg is the latent heat of condensation, and I the condensate generation rate. The ass fraction at the interface can be calculated using: M w I = v P v,i (16) M a P G P v,i +M v P v,i Where P and P v,i are the total pressure and the vapour pressure at the interface, respectively. M a and M v are the olecular heights of air and ethanol vapor respectively. The therodynaic proprieties of the liquid fil and gas are considered variable..5 Heat and ass transfer paraeters The total interfacial heat flux: The local Nusselt nuber along the interface gas-liquid is defined as: Nu x ht D tg h TG tg Ih qi D h r I tg( TI Tb ) tg( TI Tb ) fg D h (17) So the total convective heat transfer rate fro the fil interface to the gas strea can be expressed as follows: q T = q s,i + q L,I = λ G + λ tg T G / r fg I (18) 185 P a g e D e c e b r e 1 5, 1 4

5 ISSN The local Sherwood nuber: We defended the local Sherwood nuber relief at the ass transfer coefficient and the diffusive ass flux as: The cuulate rate of condensation: M r = S x = M,xd λ G D v (19) x.dx () M Where the is the condensate ass rate and M is the ass debit of gas at the inlet of tub, and at every axial location, the overall ass balance in the gas flow and liquid fil should be satisfied.. NUMERICAL METHODE The conjugate proble defined by the parabolic systes, equations (1)-(7) and those of turbulence, with the appropriate boundary conditions are solved by a finite difference nuerical schee. Each syste of the atrix equation which can be solved by the Thoas algorith [17]. The correction of the pressure gradient and axial velocity profile at each axial station in order to satisfy the global ass flow constraint is achieved using a ethod proposed by Raithby and Schneider1979 [18]. The discrete equations are resolved line by line fro the inlet to the outlet of the tube since flows under consideration are a boundary layer type. Several grid sizes have been tested to ensure that the results are grid independent Grid independence tests were carried out by coparing the total heat transfer for different values of I, J and K. In light of those results all further calculations were perfored with the 131 (81 31)grids this is an optius esh of this study. So the grid distribution adopted in this study consists of 31, 81, and 131 nodes, respectively, in the transverse direction of the liquid region, transverse direction of the gas region and in the axial direction. In other way, we transfored the Cartesian z coordinates into coordinate syste in this study, such that the centre line is at =, the liquid-ixture interface is at 1, and the wall is at =. The equations that relate the coordinate syste to the r coordinate syste are: η L = R r /δ for R δ r R (1) And η G = r/(r δ) for r R δ () A sensitivity analysis of the nuerical odel for the diension of esh was conducted to deterine the optial esh. Soe diensions of grids were used to calculate the Nusselt nuber defined by the equation (17) as it has found that is the ost severe test. The calculate of the error for grid shows that the error does not exceed 5% in the case (NI = 141, NJ = 81, NL = 31), consequently the grid esh was used for following our study. The table below shows the coparing nuber of local Nusselt at the interface for different esh. P = 1at, T = 4C, T = C, Re = 5, W =.8 x/d NI=131,NJ=51,NL=1 NI=131,NJ=81,NL=31 NI=131,NJ=81,NL= x/d NI=141,NJ=51,NL=1 NI=141,NJ=81,NL=31 NI=141,NJ=81,NL= P a g e D e c e b r e 1 5, 1 4

6 Sh u * (/s) 4. RESULTS AND DISSCUSSION ISSN The figure, shows the profiles of radial velocities for different teperature gradient in the two phases, the radial variations of these values in the liquid phase are lower than in the gas ixture. In the case of a vertical tube, the velocity of the gas ixture is axiu on the axis of the tube and iniu at the interface 1. The condensate velocity profiles in the liquid fil gradually increase along of tube because of his entrainent by the gas 1, strea and his gravity; these values are lower than those of the gas ixture, it due 1,1 to his high viscosity. In other the values of C velocity are iportant for high 1, C C teperature gradient T = 35 C relatively 1,19 in gas ixture. The figures 3, 5 illustrate,4 Re =35 the effect of Reynolds nuber 1,18 P =1 at,3 respectively on the Sherwood nuber sh w =.5, then on the fil thickness of the 1,17 at x=.369 condensate. It shows that the thickness is,1 1,16 greater with increasing Reynolds nuber, 1, 1,1 1, 1,3 1,4 1,5 1,6 1,7 1,8 1,9, thing which we see in figure.3 this eans 1,15 that the transfer of heat and ass are,,1,,3,4,5,6,7,8 ore effective in forced convection. The figure 4 shows the affect of the Reynolds turbulent nuber on the kinetic energy; we chose different values which includes the two regies lainar and turbulent. We aintaining constant the following paraeters, P = 1at, Re = 35, W =.5 and we study the effect of the teperature difference T between the wall and the gas ixture air-stea of ethanol saturated. A inlet teperature T fixed the values adopted are T = 5 C, 3 C, 35 C the results shown in Figures,6,7,8 show that, for the sae values of input debit, P pressure and w vapor concentration, the values of fil thickness, liquid flow and heat flux do not converge to the sae liit at the end of condensation, the increase of T causes a decrease in teperature parietal, the rate of vapor in the ixture, consequently the increase of the difference between the vapor ass fractions (w w b ). This which leads to increased teperature gradients, vapour concentration and the rate of condensation. It follows a significant increase of the liquid ass flow rate (Fig.6) and the heat flux to the wall (Fig.7) Re=1 Re=15 Re=3 Re=3 Re=35, 1,8 1,6 1,4 1, 1,,8 Re=1 Re=15 Re=3 Re=3 Re=35 8,6 6,4 4, x/d Figure.3 Effect of Reynolds nuber on the evolution of the Sherwood nuber.,,,1,,3,4,5,6,7,8,9 1, Figure.5 Effect of Reynolds nuber on the evolution of the thickness of liquid fil. x * 187 P a g e D e c e b r e 1 5, 1 4

7 dp/dx interfac teperature en (Kelvin) Mr ISSN ,5.3 k 1/ /u b,45,4,35,3,5, x=.369 P =1at Re=1 Re=15 Re=3 Re=3 Re= C 3 C 5 C Re=35 P =1 at w =.5,15.1,1.8,5,,,3,4,5,6,7,8,9 1, r/(r- x Figure.4 Effect of Reynolds nuber on the evolution of the turbulent kinetic energy. Figure.6 Effect of Reynolds nuber on the evolution of the rate. The fig.9 show the thickness of condensate it show that it increase with the tube long it increase also with the iportant values of ass fraction. This can be explained by the fact that the non condensable gas accuulating near to interface which liits the heat transfer and consequently the deprivation of ethanol vapour condensation. On the other hand we see that for the fraction values of 4% the thickness is relatively pronounced because the theral resistance near to interface is relatively in regression coparing with 8% case. x * 5 313,5 Nu x Re =35 P =1 at w =.5 C 3 C 5 C 313, 31,5 31, 311,5 311, 31,5 Re =35 P =1 at w = , 39,5 39, C C C x / (d*l) 38,5,1,,3,4,5,6,7,8,9 1, X * Figure.7 Effect of teperature difference on the evolution of the Nusselt nuber. Figure.8 Effect of teperature difference on the evolution of the interfacial teperature. 1,95 1,8 1,65 1,5 1,35 1, 1,5,9,75,6,45,3,15 W=.4 W=.6 W=.8,,,1,,3,4,5,6,7,8,9 1, Figure.9 Effect of ass fraction on the evolution of the thickness of liquid fil. x *,,1,,3,4,5,6,7,8,9 1, ,,1,,3,4,5,6,7,8,9 1, x * w=.4 W=.6 W=.8 Figure.1 Axial pressure gradient with effect of ass fraction 188 P a g e D e c e b r e 1 5, 1 4

8 ISSN In Figure 1 the variations of axial pressure gradients are shown for these three cases of ass fraction. Most of the pressure gradient is predicted fro inlet of the condensation tube to x =.3. Fro Equation (), it can be seen that the vapor pressure gradient is affected by changes in the oentu flux as well as by the interfacial shear (the gravitational pressure gradient is considered to be negligible). The oentu change of the vapor flow tends to increase the pressure in the flow as the condensation occurs at the wall, while the interfacial stress tends to cause a decrease in pressure on other the pressure gradient is affected by the local condensation heat flux, then the inlet large pressure gradient indicates large condensation. 5. Conclusion The contribution to the heat and ass transfer in vertical tube tow diensional for the air ethanol vapour ixture in case of turbulent is the ean objective of our study. A theoretical odel is developed in the turbulent case in the presence of non-condensable gas using a finite difference. The conservation equation for ass, oentu, energy, and species concentration are developed. The resultants presented clearly that the condensation heat transfer coefficients and the rate of condensation decreases considerably in the presence of non-condensable gas in high percentage. The nuerical results of the present theoretical study agree satisfactorily with the data available in literature. REFERENCES [1] Siow, E.C., 1, Nuerical solution of two-phase odel for lainar fil condensation of vapour-gas ixtures in channels. M.Sc. thesis, University of Manitoba, Winnipeg, Manitoba. [] Belkassi, Y., Gueraoui, K., Hassanin, N., 1, Nuerical study in condensing of ethanol vapor in a vertical tube by ixed convection in the presence of non-condensable Gas. Adv Studies Theor Phys. 6, [3] S. Oh, S. T. Revankar Analysis of the coplete condensation in a vertical tube passive condenser, International Counications in Heat and Mass Transfer 3, pp , 5. [4] Y.S. Chin, S.J. Oriston, H.M. Solian Two-phase boundary-layer odel for lainar ixed-convection condensation with a non condensable gas on inclined plates, Heat and Mass Transfer 34, pp , [5] S. Oh Turbulent Heat and Mass Transfer in a Vertical Condenser Tube, Fall Seester ME65 Convective Heat and Mass Transfer, Project Report,3. [6] Y. El haai M. Feddaoui T. Mediouni R. Mir A. Mir. Étude nuérique de la condensation en fil par convection ixte a l interieur d un canal a paroi poreuse. Revue International d Héliotechnique, 4 :18 4, 1. [7] Y. El Haai M. Feddaoui T. Mediouni A. Mir. Nuerical study in condensing a liquid refrigerant (r134a) fil along a vertical channel. International syposiu on convective heat and ass transfer in sustainable energy Haaet Tunisia., :46 49, 9. [8] H. Louahlia, P.K. Panday Transfert therique pour la condensation du R13, du R134a et de leurs élanges en écouleent forcé entre deux plaques planes horizontales: Etude nuérique, Rev. Gén. Ther., Vol. 35, pp , [9] P.K. Panday Two-diensional turbulent fil condensation of vapours flowing inside a vertical tube and between parallel plates: a nuerical approach, Int. Journal of Refrigeration 6, pp , 3. [1] C.Y. Wang, C.J. Tu Effects of non-condensable gas on lainar fil condensation in a vertical tube, International Journal of Heat and Mass Transfer 31, No 11, , [11] Oriston, S.J., Groff, M.K., Solian, H.M., 7, Nuerical solution of fil condensation fro turbulent flow of vapor gas ixtures in vertical tubes. International Journal of Heat and Mass Transfer 5, [1] S. Yih, J. Liu(1983), "Prediction of heat transfer in turbulent falling liquid fils with or without interfacial shear", AIChE Journal, vol. 9, pp [13] A. Wurster, Discussion, Trans. A. Inst. Che. Eng. 38 (194) [14] T. Cebeci, A. M. O. Sith(197), "A finite difference ethod for calculating copressible and turbulent boundary layers", J. Basic Engineering. Trans ASME. vol. 9, pp [15] B. E. Launder, B. I. Shara (1974), "Application of the energy-dissipation of turbulence to calculation of low Reynolds nuber flow near a spinning disc" Lett. Heat Mass transfer, vol. 1, pp [16] M. Feddaoui, H. Meftah, A. Mir, The nuerical coputation of the evaporative cooling of falling water fil in turbulent ixed convection inside a vertical tube, Int. J. Heat Mass Transfer 33 (6) [17] Patankar S.V. [198] "Nuerical Heat Transfer and Fluid Flow". Heisphere/Mc Graw Hill. New York, Chap.6. [18] Raithby, G. D.; Schneider, G. E. [1979] Nuerical solutions of probles in incopressible fluid flow: treatent of the velocity-pressure coupling. Nuer. Heat. Tran., Vol., pp P a g e D e c e b r e 1 5, 1 4

9 ISSN C p D d g h fg h T q t q LI q si T T u W x r k ε Specific heat Mass diffusivity Tube half width Gravitational acceleration Latent heat of condensation Local heat transfer coefficient Mass flow rate Total heat flux Latent heat flux Sensible heat flux Inlet teperature Diensionless teperature T T ) /( T T ) ( wall wall Diensionless velocity in the y-direction u u Diensionless vapor ass fraction W W )/( W ) ( I W Coordinate along the tube Coordinate noral to the tube Kinetic energy Dissipation rate j kg 1 s. s j kg W. kg. W. W. W. K 1 1 K 1 1 K 1 1. s I L G W Thickness of condensate layer Diensionless fil thickness Theral conductivity Transfored coordinate defined by eqs (, 1) Dynaic viscosity Mass density liquid-gas interface indicate Zero, condition at inlet Liquid index / Length Gas index Wall index W. 1 K kg s kg P a g e D e c e b r e 1 5, 1 4