Peculiarities of the momentum transport in laminar liquid film with cross curvature

Transcription

1 Peculiarities of the momentum transport in laminar liquid film with cross curvature J. Gylys & S. Sinkunas Department of Thermal and Nuclear Energy, Kaunas University of Technology, Lithuania Abstract The procedure for the prediction of hydrodynamic parameters of laminar liquid film flowing down on the external surface of vertical tube is presented. An analytical study evaluating the influence of cross curvature on film thickness and shear stress distribution using curvature correction factor and correction coefficient is performed. Relations for the determination of correction parameters are also found. It is obtained that under the influence of convex surface curvature the film thickness and shear stress decrease changing local velocity distribution simultaneously. Applied prediction limits of curvature variation are presented. 1 Introduction Gravity fluid-film flows are exploited in a wide scope of industrial apparatus, as well as in cooling open heat-emitting surfaces, and here laminar flows are used in many cases. Thermal treatment of chemical products and food-stuffs in their falling film flows is favoured. The heat transfer is intensive and the short-term contact with the heating surface prevents the products from polymerisation, overheating or loss of their important properties. Most of the film apparatus for the thermal treatment of liquid products consist of vertical tubes with falling films on their external surfaces [1-3]. In the case when a liquid film flows down over a vertical tube the curvature of its surface effects heat transfer characteristics on the surface, transversal shear stress distribution and correspondingly the thickness of the liquid film. A film of large cross curvature exists when its thickness is of the same range as the radius of a tube. This paper is related to the evaluation of cross curvature and hydrodynamic parameters of a laminar liquid film. Publications on the film flow hydrodynamics are numerous. They are collected in references [1-10], but mostly deal with

2 538 Advances in Fluid Mechanics HI vertical or inclined plane film flow. At the same time the operation of real film heat exchangers is based on vertical tubes. In this case the cross curvature of the wetted surface and the film lead to alteration of hydrodynamic parameters of the liquid film. Simultaneously the intensity of the heat exchange between a wetted surface and a liquid film is altered. In order to apply a film flow efficiently, the knowledge of exact hydrodynamic parameters is very important. In such studies, the influence of the cross curvature is of specific interest. The paper presents some results in this field where the studies are not numerous. 2 Problem and assumptions Let us consider a stabilised gravity film flow on the external surface of a vertical tube. We assume a stabilised gravity film flow when the film thickness and average velocity are not changeable in flow direction. In this case the thickness of the hydrodynamic boundary layer is equal to the film thickness. When the ratio of the film thickness and the radius of the tube (s#=j/r) is negligible, the regularities of momentum transport for film flow on the surface of a vertical tube are of small difference from the regularities for vertical plane surface (/?=<*>, #=0). In the case of values c*»0 regularities of momentum transport for film flow on the surface of a vertical tube significantly depend on the cross curvature of the wetted surface. 3 Modelling of cross curvature influence on hydrodynamic parameters of the film For the elementary ring-shape volume of liquid film flow the following force balance equation can be formulated: 2nrpgdrdx - 2%r%b + In (r + dr XT + dr )dx = 0. (1) Reconstructing the force balance equation, we have pgrdr + Tdr + rdr + drdr - 0. (2) The last member in eqn (2) is a small value of the second range and can be ignored. After some transformations, one can get the following differential equation describing the momentum transport in the film on the surface of a vertical tube: r + T + pgr = 0. (3) dr Solving eqn (3) in the limits of variable radius r from R to R+d we can obtain the following expression:

3 Advances in Fluid Mechanics III r (4) For the analysis of momentum transport in the film it is more convenient to express the variable radius r using y or the distance from the wetted surface. Variable radius equals r=r+y. (5) Substituting eqn (5) into eqn (4) we obtain [*g-*+>\ (6) and also the following form T= (7) Note that when surrounding medium density is close to the film flow density or (l-pg/p)«l, we replace g(l-pjp) for g. Calculation of the shear stress on the wetted surface of the tube (j=0) has been based on eqn (7) and it is equal to: R (8) We obtain the variation of non-dimensional shear stress across gravitational liquid film by dividing eqn (7) by eqn (8): (9) R 2R and s- (10)

4 540 Advances in Fluid Mechanics III where the curvature correction factor which corrects shear stress for cross curvature of the film is defined as ^ (ID For plane wetted surface (#=00 andafc=0) the curvature correction factor C#f=l and eqn (10) then becomes (12) Figure 1: Variation of the curvature correction factor in the film with different cross curvature: 1 - %f=0.1; 2-0.2; 3-0.4; 4-0.6; 5-0.8; In such a way the curvature correction factor, CM, corrects shear stress deviation for cross curvature of the wetted surface. Variation of the curvature correction factor for different values of relative cross curvature is presented in Figure 1. The decrease of curvature correction factor is observed as the distance from the wetted surface and relative cross curvature increases. Figure 2 shows the influence of cross curvature of the film on the distribution of relative shear stress across the film. On the external surface of the tube the value of shear stress inside the film decreases as the cross curvature increases. The shear stress for laminar film flow is defined as follows

5 Advances in Fluid Mechanics III 541 T = dw dr (13) Figure 2: Variation of the relative shear stress in the film with different cross curvature: 1 - c*=0; 2-0.2; 3-0.4; 4-0.6; 5-0.8; Placing the above relation into eqn (3) one can formulate the equation of motion for liquid film in cylindrical coordinates: i Idw ^ g r dr v (14) Solving eqn (14) with the following boundary conditions w=0, for r=r\ = 0, for r=r+8 dr (15) we obtain 4v (16)

6 542 Advances in Fluid Mechanics HI Substituting eqn (15) into eqn (16) we can obtain integration constants, 2v 4v 2v (17) and then velocity distribution across the film 2v R (is) or (19) Mean velocity for film flow on the external tube surface is defined in the form {R+8 wrdr "=&3- \rdr JR and the following relation is obtained: -. (21) Mass flow rate of liquid film is defined as G = wpf = 27tRwp8(l + 0.5^ ) (22) and substitution of eqn (21) into eqn (22) leads to: 0.25 }. (23) Defining Reynolds number for film flow as Rc =-^_ (24) 27tRpv and substituting eqn (23) into eqn (24), we can rearranged the above equation as:

7 Advances in Fluid Mechanics III 543 From eqn (25) thickness of the film may be evaluated as 0.25 }. (25). fv * Re fl + ER f [in(l + R)-0.75]+(I + R)^-0.25f^. (26) Thickness of the film on the vertical plane surface can be found easily, e. g. Tananayko and Vorontsov [2]: (27) Solving a set of eqns (26) and (27) we obtain the relation for the determination of the thickness of the film on vertical convex surface: (28) or ^ = ^Q,, (29) where the curvature coefficient which corrects film thickness for cross curvature of the film is defined as Rfi Finally, the thickness of laminar gravitational liquid film on the vertical convex surface is calculated as 8= l^re CRg. (31) Thus curvature coefficient, C^, corrects the deviation of the film thickness for cross curvature of the wetted surface. Eqn (30) shows that the value of correction coefficient depends on the cross curvature of the wetted surface only. Graphically eqn (30) is presented in Figure 3. The curvature correction coefficient, and correspondingly the film thickness, decrease as the cross

8 544 Advances in Fluid Mechanics III curvature of the wetted surface increases. The analysis shows 22% decrease of the laminar film thickness for their relative cross curvature increase from 0 to 1. An expression for the calculation of the curvature correction coefficient by eqn (30) is sufficiently complicated. A relation of a simple form can replace eqn (30). In order to find a more simple relation for the replacement of eqn (30) detailed analysis was provided on the base of which the following equation has been developed: (32) Figure 3: Relationship of curvature correction coefficient with different cross curvature: 1 - eqn (30); 2 - eqn (32). Comparison of eqns (30) and (32) is clear from Figure 3. Calculations showed that replacing exact eqn (30) by more simple eqn (32) makes an error less than 1% for the interval of relative cross curvature from 0 to 1. The case with values of relative cross curvature exceeding 1 practically is not actual. Therefore, with sufficient accuracy we have <5= C, (33)

9 Advances in Fluid Mechanics III 545 In engineering calculations the cross curvature radius of the wetted surface usually are known and the thickness of the film flowing down on a vertical plane surface can be estimated from (27). Then, a relation describing the thickness of the film flowing down on a convex surface results from eqn (33): (34) Substituting eqn (27) into eqn (34) we obtain }%-ll. (35) As it was mentioned above, in engineering calculations relative cross curvature of the filma&<! and eqn (32) is valid for this interval ofa^. For this reason it is betterfirstto estimate relative cross curvature from the relation: and then film thickness n = 1.67 Jl + LQ9(Re/Ga»y* -11 (36) V ) 4 Conclusions 6 = s^r. (37) A simple model of a liquid film on a vertical surface with cross curvature based on force balance equations has been presented in the paper. The model permits to evaluate the influence of cross curvature on hydrodynamic parameters of the film. The model can be useful in calculations of film hydrodynamic parameters in an apparatus of a vertical tube and other installations. The following observations can be made and conclusions drawn from this study: The model allows theoretical evaluation of the influence of cross curvature on shear stress distribution across the film using curvature correction factor. The relationship for the definition of this factor has been presented in the paper. It is determined that for the film flow on a vertical convex surface the value of curvature correction factor is less than 1 and its value decreases as the distance from the wetted surface and relative cross curvature increases. It has been estimated that the cross curvature of the wetted surface decreases shear stress within the film. The cross curvature of the wetted surface changes the distribution of the local velocities in the film. An equation for the calculation of local velocities in the film with respect to the curvature of the wetted surface is given in the paper. In the paper the evaluation of cross curvature influence on film thickness is also presented. Alteration of the film thickness due to the cross curvature

10 546 Advances in Fluid Mechanics III evaluates the curvature correction coefficient. The relationship for an applied engineering case is suggested. Nomenclature g - gravitational acceleration, m/s^; R - cross curvature of the wetted surface (tube external radius), m; r - variable radius, m\ w, w - local and average velocities of the stabilised film respectively, m/s; x - longitudinal coordinate; v - transversal coordinate (distance from the wetted surface), m; G - liquid mass flow rate, kg/s; F - film mass flow rate per unit parameter (wetted density), kg/(ms);/- cross sectional area of film flow, m^; d - thickness of stabilised film on external surface of vertical tube, m; J<, - thickness of stabilised film on vertical plane surface, m; v - kinematic viscosity, nf/s; // - dynamic viscosity, kg/(ms); p - liquid density, kg/nf; pg - surrounding gas (vapour) density, kg/nf; r - shear stress in the film, Pa;?%, - shear stress at the wall, Pa; GCIR - Galileo number, grvv*; Re - Reynolds number of liquid film, 4F/(pv); SR - relative cross curvature of the film, 8/R. References [1] Mikielewicz, J. & Mikielewicz, D. New Approaches to modelling of impinging jets. Progress in Engineering Heat Transfer. Proc. of the Third Baltic Heat Transfer Conference, IFFM Publishers: Gdansk, pp , [2] Tananayko, Y.M. & Vorontsoy, E.G. Calculation and Investigation Methods of Film Flow Processes (Russ.), Technika: Kiev, [3] Gimbutis, G.J. Heat Transfer of Falling Liquid Film (Russ.), Mokslas: Vilnius, [4] Ganchev, E.G. Cooling of Elements in Nuclear Reactors by Film Plow (Russ.), Energoatomizdat: Moscow, [5] Ganic, E.N. & Mastanaiah, K. Hydrodynamics and heat transfer in falling filmflow.low Reynolds Number Flow Heat Exch. Proc. 4 NATO ASI Heat Transfer, Ankara, 1981, Washington e.a. Berlin, pp , [6] Salazar, R.P. & Marshall, E. Statistical Properties of the Thickness of a Falling Liquid Film. Acta Mechanics 29, pp , [7] Ruschak, K. & Scriven, L.E. Developing flow on a vertical wall. Fluid Mechanics, 81(2), pp , [8] Alekseenko, S.V., Nakoryakov, V.E. & Pokusaiev, E.G. Wavy Flow of Liquid Film, Nauka: Novosibirsk, [9] Wang, C.Y. Liquid film flowing slowly down a wavy incline. AIChE Journal, 27(2), pp , [10] Rahman, M M., Hankey, W.L. & Faghri, A. Analysis of thefluidflowand heat transfer in a thin liquid film in the presence and absence of gravity. Int. J. Heat Mass Transfer, 34(1), pp , 1991.