The numerical computation of the evaporative cooling of falling water film in turbulent mixed convection inside a vertical tube

International Communications in Heat and Mass Transfer 33 (2006) 917 – 927 www.elsevier.com/locate/ichmt

The numerical computation of the evaporative cooling of falling water film in turbulent mixed convection inside a vertical tube ☆ M. Feddaoui ⁎, H. Meftah, A. Mir Laboratoire de Génie des Procédés et de l'Energie (LGPE) ENSA, B.P. 1136, Agadir — Morocco Available online 17 May 2006

Abstract An analysis has been developed for studying the evaporative cooling of liquid film falling inside a vertical insulated tube in turbulent gas stream is presented. Heat and mass transfer characteristics in air–water system are mainly considered. A low Reynolds number turbulence model of Launder and Sharma is used to simulate the turbulent gas stream and a modified Van Driest model suggested by Yih and Liu is adopted to simulate the turbulent liquid film. The model predictions are first compared with available experimental data for the purpose of validating the model. Parametric computations were performed to investigate the effects of Reynolds number, inlet liquid temperature and inlet liquid mass flow rate on the liquid film cooling mechanism. Results show that significant liquid cooling results for the system with a higher gas flow Reynolds number Re, a lower liquid flow rate Γ0 or a higher inlet liquid temperature TL0. © 2006 Elsevier Ltd. All rights reserved. Keywords: Evaporative cooling; Heat and mass transfer; Bouyancy forces; Turbulent mixed convection

1. Introduction Heat transfer augmentation through cooling liquid film associated with liquid evaporation in turbulent mixed convection is of importance in many engineering applications such as; liquid film evaporation, drying and the evaporative cooling devices in today's use (evaporative fluid coolers, water-cooling towers, protection of system components from a hightemperature gas stream, etc.). Because of its practical importance, evaporating liquid film flow and related problems have received considerable attention. The present paper aims to contribute in studies of the evaporative cooling by analysing combined heat and mass transfer processes to a turbulent gas stream from a liquid film falling along an insulated vertical tube. A vast amount of work, both theoretical and experimental exists in the literature to study the turbulent mixed convection heat transfer and flow in vertical ducts. Cotton and Jackson and Tanaka et al. [1,2] applied a low Reynolds number k ∼ ε turbulent model of Launder and Sharma [3] to predict the turbulent mixed convection heat transfer and flow in vertical pipes. They were fairly successful in predicting heat transfer distribution with the experimental results. Studies including transport processes in the flowing gas, were performed by Lin et al. [4] and Tsay and Yan [5]. They analysed the influence of wetted wall on laminar mixed convection heat and mass transfer in vertical ducts. In their analyses, the liquid film on the wetted wall was assumed to be extremely thin so that it was regarded as a boundary ☆

Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (M. Feddaoui).

0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.04.004

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condition for heat and mass transfer only. Yan [6] published a study to investigate the turbulent mixed convection flow in a vertical channel under the simultaneous influence of the combined buoyancy forces of thermal and mass diffusion. A similar study was conducted by Fedorov et al. [7] along a vertical channel. Analysis of combined heat and mass transfers including transport process in both gas flow and liquid film has been reported in asymmetrically heated vertical channel by numerous investigators; Yan [8] and Cherif and Daïf [9]. The evaporative cooling of liquid film in natural convection channel flows was explored by Yan and Lin [10]. This study was extended later by Yan [11] by analysing the evaporative cooling process of liquid film in turbulent mixed convection channel flows. The numerical calculations indicated that latent heat transfer associated with film evaporation mainly causes the cooling of the liquid film. Recently, He et al. [12] considered a vertical tube with liquid water film cooling. The gas flow was considered to be turbulent and the liquid film to be laminar. Feddaoui et al. [13] investigate numerically the co-current turbulent mixed convection heat and mass transfer in falling film of water inside a vertical heated tube. They applied a low Reynolds number k ∼ ε turbulence model in the gas stream. This literature review shows that the problem of the evaporative cooling of turbulent liquid film in a turbulent gas stream has not received sufficient attention. The purpose of this paper is to enhance our understanding of the liquid film cooling along a vertical tube in a turbulent gas stream, by performing a detailed analysis for interfacial heat and mass transfer in air over a falling water film. 2. Analysis 2.1. Physical model and assumption The geometry of the problem under consideration is a vertical tube with radius R (Fig. 1). The tube wall is thermally insulated. The liquid flow down over the inner wall is fed with an inlet liquid temperature TL0, and inlet liquid mass flow rate Γ0. The flow of moist air enters the tube at temperature T0, and constant velocity u0. Characteristics of evaporative cooling of liquid film in turbulent mixed convection tube flows are studied numerically. In this work, a higher temperature liquid film falling along the tube was cooled down by a lower temperature gas stream. During the evaporation of liquid film, the heat and mass transfer occurs at this problem. An attempt has been made here to model the process with the following simplifying assumptions: - The flow is considered to be incompressible and axisymmetric. - Radiation heat transfer, viscous heat dissipation and other secondary effects are negligible.

Fig. 1. Schematic diagram of the physical system.

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- The thermodynamic equilibrium is assumed at the water–gas interface. - The inertial terms are neglected in the momentum equation of the liquid film as compared with the diffusional term. Moreover, for the thin liquid film the axial transfers of momentum and energy are smaller than those in the radial direction. Under the above assumptions, heat and mass transfer process in both the turbulent liquid film and turbulent gas stream can be described in detail by the appropriate governing equations and interfacial conditions. 2.1.1. Basic equations for the liquid film With the above assumptions, the steady turbulent flow in the liquid film is expressed by the following equations: - Momentum ð1=rÞ∂½rðlL þ lLt Þ∂uL =∂r=∂r þ qL g ¼ 0

ð1Þ

- Energy ∂ðqL uL TL Þ=∂x ¼ ð1=rÞ∂½rðlL =Pr þ lLt =Prt Þ∂TL =∂r=∂r

ð2Þ

2.1.2. Basic equations for gas flow The steady two-dimensional turbulent boundary-layer flow in the gas side is governed by the following conservation equations: - Continuity ∂ðqG uG Þ=∂x þ ð1=rÞ∂ðqG rvG Þ=∂r ¼ 0

ð3Þ

- Momentum ∂ðqG u2G Þ=∂x þ ð1=rÞ∂ðqG rvG uG Þ=∂r ¼ −dpd =dx þ ð1=rÞ∂½rðlG þ lGt Þ∂uG =∂r=∂r þ ðqG −q0 Þg

ð4Þ

- Energy ∂ðqG uG TG Þ=∂x þ ð1=rÞ∂ðqG rvG TG Þ=∂r ¼ ð1=rÞ∂½rðlG =Pr þ lGt =rt Þ∂TG =∂r=∂r þ ðlG =Sc þ lGt =rw Þðcpv −cpa Þ=cpG ∂TG =∂r∂w=∂r - Species concentration

ð5Þ

∂ðqG uG wÞ=∂x þ ð1=rÞ∂ðqG rvG wÞ=∂r ¼ ð1=rÞ∂½rðlG =Sc þ lGt =rw Þ∂w=∂r=∂r

ð6Þ

2.2. Turbulence model 2.2.1. Liquid film turbulence model In this work, consideration is given to a system having a sufficiently high liquid mass flow rate for which the liquid film may flow turbulently. The turbulence model adopted in the liquid film is a unified approach of a modified Van Driest eddy viscosity model suggested by Yih and Liu [14]. The details of the model are presented below: for ðR−rÞ=dx b0:6 :

lLt =lL ¼ −0:5 þ 0:5f1 þ 0:64yþ2 ðs=sw Þ  ½1−expð−yþ ðs=sw Þ1=2 =Aþ Þ2 :f 2 g

for 0:6bðR−rÞ=dx b1:0 : Where : Aþ ¼ 25:1;

lLt =lL jðR−rÞ=dx ¼0:6 ¼ constant f ¼ exp½−1:66ð1−ðs=sw ÞÞ;

yþ ¼ ðR−rÞu⁎ =v

1=2

ð7Þ ð8Þ ð9Þ

2.2.2. Gas flow turbulence model For simulation of turbulence in the gas flow, a low Reynolds number k ∼ ε turbulence model proposed by Launder and Sharma [3] is adopted. The transport equations of k and ε are as follows:

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k-Transport ∂ðqG uG kÞ=∂x þ ð1=rÞ∂ðqG rvG kÞ=∂r ¼ ð1=rÞ∂½rðlG þ lGt =rk Þ∂k=∂r=∂r þ lGt ð∂lG =∂rÞ2 −qG ðe e þ De Þ

ð10Þ

ε˜-Transport ∂ðqG uG eeÞ=∂x þ ð1=rÞ∂ðqG rvG eeÞ=∂r ¼ ð1=rÞ∂½rðlG þ lGt =re Þ∂e e=∂r=∂r e=kÞlGt ð∂uG =∂rÞ2 −qG Ce2 f2 ðe e 2 =kÞ þ Ce1 ðe þ 2lG lGt =qG ð∂2 uG =∂r2 Þ2

ð11Þ

where : lGt ¼ Cl fl qG k 2 =e˜

ð12Þ

The various constants and functions employed in the turbulence model are exactly the same as those originally proposed by Launder and Sharma [3]. 2.3. Boundary and interfacial matching conditions The boundary conditions for this marching type problem are: 3=2

x ¼ 0 : uG ¼ u0 ; TG ¼ T0 ; w ¼ w0 ; k0 ¼ 3=2ðI0 u0 Þ2 ; e˜0 ¼ Cl k0 =0:03R

ð13Þ

Where I0 is the turbulence intensity at the inlet of the tube. r ¼ 0 : ∂uG =∂r ¼ 0; r ¼ R : uL ¼ 0;

∂TG =∂r ¼ 0;

∂w=∂r ¼ 0;

∂TL =∂r ¼ 0

∂k=∂r ¼ 0;

∂e˜=∂r ¼ 0

ð14Þ ð15Þ

The solution from the liquid side and gas side satisfy the following interfacial matching conditions (r = R − δx): (a) Continuities of velocity and temperature uI ðxÞ ¼ uG;I ¼ uL;I ; (b) Continuity of shear stress

TI ðxÞ ¼ TG;I ¼ TL;I

sI ¼ ½leff ∂u=∂rL;I ¼ ½leff ∂u=∂rG;I (c) Vaporising flux at the liquid–gas interface : m I ¼ qDeff =ð1−wI Þ∂w=∂r

ð16Þ ð17Þ

ð18Þ

(d) Heat balance at the interface implying : ½keff ∂T =∂rL;I ¼ ½keff ∂T =∂rG;I þ m I :g

ð19Þ

where μeff = μ + μt, λeff = λ + λt and Dveff = Dv + Dvt respectively, are the effective parameters of viscosity, conductivity and mass diffusivity. (e) Turbulence parameters at the interface The boundary conditions for k and ε˜ at the water–gas interface, consistent with the equations for the low Reynolds number k − ε turbulence model, are taken by considering the interface as a solid wall with transpiration (He et al. [12]); that is, we set the conditions: kI ¼ 0;

e eI ¼ 0ðeI ¼ De Þ

ð20Þ

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(f) Heat and mass transfer parameters The total convective heat transfer rate from the film interface to the air stream can be expressed as follows: qI ¼ qsI þ qS I ¼ tcp ðl=Pr þ l=rt Þ∂T =∂rbG;I þ ½ðl=Sc þ lt =rw Þ=ð1−wI Þ∂w=∂r :g G;I

ð21Þ

For the purpose of generalising the heat transfer results, the local Nusselt number along the interface water–gas is defined as: Nux ¼ hT ð2RÞ=kG ¼ qI ð2RÞ=kG ðTI −Tb Þ ¼ Nus þ NuS

ð22Þ

Where Nus and Nuℓ are the local Nusselt numbers for sensible and latent heat transfer, respectively, and are expressed as follows: Nus ¼ qsI ð2RÞ=kG ðTI −Tb Þ

NuS ¼ qS I ð2RÞ=kG ðTI −Tb Þ

ð23Þ

Basing the local mass-transfer coefficient on the diffusive mass flux, the local Sherwood number is defined as: Shx ¼ hM ð2RÞ=D ¼ mI ð1−wI Þð2RÞ=qG DðwI −wb Þ

ð24Þ

At every axial location, the overall mass balance in the gas flow and liquid film should be satisfied: R Z−dx 2

½ðR−d0 Þ =2qG u0 ¼

Zx rqG uG dr þ

C0 ¼

ð25aÞ

0

0

ZR

rqG vI dx

Zx ðrqudrÞL −

R−dx

rqG vI dx

ð25bÞ

0

Note that in the above formulation, the variations of the thermophysical properties with temperature and air–vapour composition are included. For further details, the thermophysical properties are available in Fujii et al. [15], and Bird et al. [16]. 3. Numerical method The set of coupled non-linear differential equations defined by the parabolic systems, Eqs. (1)–(6), (10) and (11 are solved by a finite difference numerical method. The axial convection terms are approximated by the backward difference and the transversal convection and diffusion terms are approximated by the central difference. In the centreline (r = 0) of the tube, the diffusional terms are singular. A correct representation can be found from an application of L'Hospital's rule. Each system of the finite-difference equations forms a tridiagonal matrix equation, which can be solved by the TDMA Method (Patankar, [17]). The correction of the pressure gradient and axial velocity profile at each axial station in order to satisfy the global mass flow constraint is achieved using a method proposed by Raithby and Schneider [18]. To obtain enhanced accuracy in the numerical computations, grids are chosen to be non-uniform in both axial and radial directions. Accordingly the grids are compressed towards the interface water–gas (the first node in the gas side should be located in the viscous sublayer at most less than one from the interface) and towards the entrance of the tube. During the program tests, solutions for typical case were obtained using different grid sizes to ensure that the solution is grid-independent. It is noted in the separate computations that the differences in the Nux from computations using grids ranging from 51 × 51 × 11 to 201 × 201 × 41 were always within 3%. In light of those results all further calculations were performed with the 101 × 101 × 21 grid. In view of the unavailability of experimental data in the case of vaporising a turbulent liquid film into a turbulent gas flow. A comparison between predicted distributions of wall temperature along the tube and the experimental data of An et al. [19]

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Fig. 2. Variations of wall temperature along the tube.

is given in Fig. 2. It is clear that generally the agreement between our prediction and the experimental study of An et al. [19] is quite well. In view of these validations, the present numerical algorithm is considered to be suitable for the practical purpose. 4. Results and discussion In order to examine the effects of flow conditions on the liquid film cooling mechanism in turbulent mixed convection heat and mass transfer in a vertical tube, results are particularly presented for water film evaporation. The following set conditions are selected in the computation: the relative humidity of the ambient air is assigned as 50% for the air–water vapour mixture at T0 = 20 °C and 1 atm along a vertical tube with diameter d = 0.06 m. The values of these parameters are 0.01, 0.02 and 0.05 kg/s for Γ0; 40 or 60 °C for TL0 and 5000, 1 × 104, 2 × 104 for Re.

Fig. 3. Variations of interfacial and wall temperatures along the tube.

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Fig. 4. Variations of interfacial Nusselt numbers along the tube for Re = 2.104. (a) Sensible heat Nusselt number; (b) latent heat Nusselt number; (c) overall Nusselt number.

Shown in Fig. 3 are the effects of various parameters on the axial distributions of the interface and wall temperatures. The results indicate that TI and Tw decrease monotonically in the flow direction. This feature indicate that energy required for the evaporation must largely come from the internal energy stored in the liquid film (Γ0 cpLΔTL). This is perhaps, due to a reduction in liquid film temperature. Moreover, the curves of TI and Tw almost coincide with each other, except for the results at the inlet. This implies that the temperature drop across the liquid film is negligible. The results indicate that the best liquid film cooling is experienced for a smaller liquid film flow rate Γ0. This is readily understood by realising that the total internal energy stored in the liquid film is larger for a higher Γ0 and that a simple energy balance for the liquid film gives (Γ0cpLΔTL = qL). In addition, a larger temperature decrease results for a higher TL0 as shown in Fig. 3(a) and (b). This is a direct consequence of the difference in water vapour concentration between the interface and the gas stream for a higher TL0, which in turn causes larger film vaporisation and temperature drop. By comparing Fig. 3(a) and (c), it is noted that a larger temperature drop is observed for a system with a higher Reynolds number Re. This is in line with the general concept that for turbulent convection flow, the heat and mass transfer is larger for a higher Re. In such case of higher Reynolds number the buoyancy effect is not significant. As shown in Fig. 3 the interfacial temperature TI decrease

Fig. 5. Variations of local Sherwood number Shx along the tube for Re = 2.104.

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Fig. 6. Effects of liquid flow rate and inlet liquid temperature TL0 on temperature drop of liquid film.

along the tube. Therefore the corresponding mass fraction of water vapour decreases along the tube since the thermodynamic equilibrium is assumed at the interface. Now we turn our attention to improve the understanding of the interfacial heat transfer. The longitudinal variations of the local Nusselt numbers for the sensible and latent heat transfer are presented in Fig. 4 for various cases. The effect of the inlet water temperature is known. A smaller sensible Nusselt number Nus are found for a higher TL0 except for results at the downstream. These smaller values of Nus are the direct consequence of the larger blowing effect (evaporating effect) for systems having a higher TL0. Also it is noted that a smaller Nus is experienced for a larger liquid mass flow rate Γ0. This can be made plausible by noting that a smaller liquid film cooling (i.e. smaller temperature drop) is experienced for a larger liquid mass flow rate. Which in turn causes a greater blowing effect created by the falling liquid film for a larger Γ0. These results indicate that the heat removal that can be directly attributed to sensible heat transfer into the air–vapour mixture by diffusion of heat at the interface is small. However, the turbulent diffusion process in the air–vapour mixture is of importance, which is due to the evaporation of water film. Fig. 4(b) shows the latent Nusselt number along the tube. The larger latent Nusselt number results for systems with a higher inlet water temperature TL0. This is brought about by the larger latent heat transport in connection with the greater liquid film evaporation for higher Γ0. It is of interest to note that the magnitude of NuL is much larger than that of Nus, implying that the heat transfer resulting from latent heat exchange is much more effective. Note also that a larger NuL results for a larger Γ0, this is again due to the larger latent heat transfer for a system with a larger Γ0, which implies a larger interfacial liquid film temperature. Shown in Fig. 4(c) are the distributions of the total Nusselt number Nux. The effectiveness of mass transfer can be represented by Sherwood number. The latter can be directly related to sensible Nusselt number using heat and mass transfer analogy. The variations of local Sherwood number are presented in Fig. 5. A larger Sherwood number results for systems with lower TL0 or smaller liquid flow rate Γ0. This is due to the smaller opposing buoyancy effect for a lower TL0 or a smaller blowing effect for a smaller Γ0. In order to quantify the evaporative cooling process. The total temperature drop of the liquid film as it flows from the inlet to the outlet ΔTL0 = TL0 − TLe is one of the important quantities. Fig. 6 shows the effect of different conditions on the total temperature drop of the liquid film. It is apparent that a larger liquid temperature drop is experienced for a system with higher TL0 or a smaller Γ0. This is apparently due to the effective evaporative cooling for the case with a higher TL0. In addition a larger temperature depression is also found for the case with a higher Reynolds number Re. This confirms the general concept that for turbulent convection flow, the heat transfer is large for a higher Re. In the case of (Re= 2 × 104 and Γ0 = 0.04 kg/s) for TL0 = 80 °C, the temperature drop of liquid film is above 56%.

Fig. 7. Dimensionless dynamic pressure drop along the tube.

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The variations of dimensionless dynamic pressure along the tube for different conditions are shown in Fig. 7. The dynamic pressure is negative and decreases along the tube. This situation is due to the difference in temperature and mass fraction of water vapour in the mixture, which generate the thermal and solutal buoyancy forces. The pressure is slightly influenced by the inlet water film for a higher Reynolds number. In contrast, for a lower Reynolds number, the dynamic pressure decreases when decreasing the liquid mass flowrate. Also as show in Fig. 7(b), the higher-pressure drop is observed for a higher inlet water film temperature. In some cases the mean flow and heat and mass transfers are affected by the buoyancy forces, which arise due to the non-uniformity of fluid temperature and mass fraction of water vapour. The magnitude of the effect can be characterised by a buoyancy parameter, modified by He et al. [12], given by Bo = 8.104 Gr / Re2.625Pr0.8, where Gr = ((ρb − ρI) / ρb)(gd3 / v2) For the low Reynolds number and high inlet water film temperature Fig. 7(b), the buoyancy effect is very pronounced (the buoyancy parameter vary from Bo = 12.85 at the inlet to Bo = 0.5 at the outlet of the tube). In such situation the flow can be laminarised, and heat transfer is then considerably impaired. The strong decrease of temperature and mass fraction at the interface along the tube is shown for a lower inlet water mass flow rate and a higher inlet water film temperature. This can explain the pressure drop in Fig. 7. In contrast the opposite observation is noticed for a lower Reynolds number due to the buoyancy forces. The variations of Reynolds number and the buoyancy parameter along the tube are in opposite sense to that occurring in single-phase flow cases. Reynolds number increases along the tube due to the evaporation of water from the film and the buoyancy parameter decreases.

5. Conclusion A computer code have been developed for modelling turbulent mixed convection heat and mass transfer with the evaporative cooling of liquid film falling inside a vertical insulated tube. Parametric computations were performed to investigate the effects of Reynolds number, inlet liquid temperature and inlet liquid mass flow rate on the liquid film cooling mechanism. The main conclusions from the study are summarised below. (1) When the water film temperature at inlet is relatively high, the energy stored in the water film is mainly required to sustain its evaporation. In contrast, when the inlet water temperature is relatively low, the saturated vapour pressure is low and the convection of heat by the flowing water film serves as the main mechanism for taking away the heat from the wall. Only the evaporation of a small portion of the water film provides a reduction of its temperature. (2) Larger temperature drop is experienced for a rise in inlet liquid temperature TL0 or a reduction of the liquid film flow rate Γ0. (3) The buoyancy effect is very pronounced in the case of high liquid film temperature TL0 and at low Reynolds number Re. In this case, the flow is predicted to be partially laminarised. Heat transfer is then considerably impaired. Nomenclature Bo Buoyancy parameter Cε1 Constant in turbulent k ∼ ε model Cε2 Constants in turbulent k ∼ ε model Cμ Constant in turbulent k ∼ ε model cp Specific heat [J kg− 1 K− 1] D Mass diffusivity [m2 s− 1] d Diameter of the tube [m] f 2, f μ Functions in turbulent k ∼ ε model Gr Grashof number g Gravitational acceleration [m s− 2] hM Mass transfer coefficient [m s− 1] hT Heat transfer coefficient [m s− 1] k Turbulent kinetic energy [m2 s− 2] m˙I Evaporating mass flux [kg m− 2 s− 1] Nuℓ Local Nusselt number (latent heat) Nus Local Nusselt number (sensible heat) Nux Overall Nusselt number p Mixture pressure [Pa] P Dimensionless motion pressure qw Wall heat flux [W m− 2]

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qI qℓI qsI r R Re Ret Shx T TL0 T0 u u⁎ v w y y+ x

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Total heat flux [W m− 2] Latent heat flux [W m− 2] Sensible heat flux [W m− 2] Co-ordinate in r-direction radius of the tube [m] Gas stream Reynolds number, u0d / v0 Turbulent Reynolds number, k2 / vε˜ Interfacial Sherwood number Temperature [K] Inlet liquid film temperature [K] Inlet temperature [K] Axial velocity [m s− 1] Shear velocity, (τw / ρ)1 / 2 [m s− 1] Radial velocity [m s− 1] Mass fraction of vapour Normal distance from the wall, R − r [m] Dimensionless wall co-ordinate, (R − r)u⁎ / v Dimensional axial co-ordinate [m]

Greek symbols Γ0 Inlet liquid mass flow rate [kg s− 1] δx Local liquid film thickness [m] ε Rate dissipation of k [m2 s− 3] ε˜ Modified dissipation variable [m2 s− 3] γ Latent heat of vaporisation [J kg− 1] λ Thermal conductivity [W m− 1 K− 1] τ Shear stress [Pa] μ Dynamic viscosity [kg m− 1 s− 1] ρ Density [kg m− 3] σt Turbulent Prandtl number σw Turbulent Schmidt number σk Turbulent Prandtl number for k σε Turbulent Prandtl number for ε Subscripts b Bulk quantity I Condition at the liquid–gas interface G Mixture (gas + vapour) L Liquid film 0 Condition at inlet t Turbulent w Condition at wall References [1] M.A. Cotton, J.D. Jackson, Vertical tube air flows in the turbulent mixed convection regime calculated using a low-Reynolds-number k ∼ ε model, Int. J. Heat Mass Transfer 33 (1990) 275–286. [2] H. Tanaka, S. Maruyama, S. Hatano, Combined forced and natural convection heat transfer for upward flow in a uniformly heated vertical pipe, Int. J. Heat Mass Transfer 30 (1987) 165–174. [3] B.E. Launder, B.I. Sharma, Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc, Lett. Heat Mass Transf. 1 (1974) 131–138.

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[4] T.F. Lin, C.J. Chang, W.M. Yan, Analysis of combined buoyancy effects of thermal and mass diffusion on laminar forced convection heat transfer in a vertical tube, J. Heat Transfer 110 (1988) 337–344. [5] H.C. Tsay, W.M. Yan, Binary diffusion and heat transfer in laminar mixed convection channel flows with uniform wall heat flux: extremely thin film thickness, Wärme- Stoffübertrag. 26 (1990) 23–31. [6] W.M. Yan, Turbulent mixed convection heat and mass transfer in a wetted channel, J. Heat Transfer 117 (1995) 229–233. [7] A.G. Fedorov, R. Viskanta, A.A. Mohamad, Turbulent heat and mass transfer in an asymmetrically heated, vertical parallel-plate channel, Int. J. Heat Fluid Flow 18 (1997) 307–315. [8] W.M. Yan, Effect of film vaporization on turbulent mixed convection heat and mass transfer in a vertical channel, Int. J. Heat Mass Transfer 38 (1995) 713–722. [9] A.A. Cherif, A. Daïf, Etude numérique du transfert de chaleur et de masse entre deux plaques planes verticales en présence d'un film de liquide binaire ruisselant sur l'une des plaques chauffée, Int. J. Heat Mass Transfer 42 (1999) 2399–2418. [10] W.M. Yan, T.F. Lin, Evaporative cooling of liquid film in through interfacial heat and mass transfer in a vertical channel: II — numerical study, Int. J. Heat Mass Transfer 34 (1991) 1113–1124. [11] W.M. Yan, Evaporative cooling of liquid film in turbulent mixed convection channel flows, Int. J. Heat Mass Transfer 41 (1998) 3719–3729. [12] S. He, P. An, J. Li, J.D. Jackson, Combined heat and mass transfer in a uniformly heated vertical tube with water film cooling, Int. J. Heat Fluid Flow 19 (1998) 401–417. [13] M. Feddaoui, A. Mir, E. Belahmidi, Co-current turbulent mixed convection heat and mass transfer in falling film of water inside a vertical heated tube, Int. J. Heat Mass Transfer 46 (2003) 3497–3509. [14] S. Yih, J. Liu, Prediction of heat transfer in turbulent falling liquid films with or without interfacial shear, AIChE J. 29 (1983) 903–909. [15] T. Fujii, Y. Kato, K. Mihara, Expressions of transport and thermodynamic properties of air, stream and water, Sei San Ga Ken Kyu Jo, Report N. 66, Kyu Shu Dai Gaku, Kyu Shu, Japan. 1977, pp. 81–95. [16] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. [17] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere/Mc Graw Hill, New York, 1980 Chap. 6. [18] G.D. Raithby, G.E. Schneider, Numerical solution of problems in incompressible fluid flow: treatment of the velocity–pressure coupling, Numer. Heat Transf. 2 (1979) 417–440. [19] P. An, J. Li, J.D. Jackson, Study of the cooling of a uniformly heated vertical tube by an ascending flow of air and a falling water film, Int. J. Heat Fluid Flow 20 (1999) 268–279.