DILUTE GAS-LIQUID FLOWS WITH LIQUID FILMS ON WALLS

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Bethanie McKenzie
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1 Forth International Conference on CFD in the Oil and Gas, Metallrgical & Process Indstries SINTEF / NTNU Trondheim, Noray 6-8 Jne 005 DILUTE GAS-LIQUID FLOWS WITH LIQUID FILMS ON WALLS John MORUD 1 1 SINTEF Materials and Chemistry, 7465 Trondheim, NORWAY ABSTRACT A model for dilte -id flos ith films at the alls has been implemented in the CFD code FLUENT. The model consists of transport eqations for droplets in the and for id transport in the film. Exchange of id beteen the to occrs by atomization of the film and by droplet deposition. Predictions are in reasonable agreement ith knon cases, sch as horizontal and vertical pipe flos. Hoever, independent validation of the code is yet to be performed. NOMENCLATURE a C C f,pipe d drop d * D g h k k' A R A R D Sc t U Flid acceleration Droplet concentration Friction factor for pipe Droplet diameter Dimensionless droplet diameter Droplet diffsion coefficient Gravity Film thickness Trblent kinetic energy Constant in atomization model Atomization rate Deposition rate Trblent Schmidt nmber Velocity Gas velocity in pipe * Shear velocity * V drop V Dimensionless settling velocity Droplet impaction velocity Deposition velocity Greek: γ Parameter in Csanady model Γ Film mass flx Γ crit Critical film mass flx Viscosity or statistical mean t Trblent viscosity ρ Density σ Srface tension or standard deviation all Wall shear stress Dimensionless relaxation time Sbscripts: D drop t all To-dimensional operator Droplet phase Gas phase Liqid phase tangential or terminal At the all INTRODUCTION In many to-phase /id flos, id films at alls are important. Examples are condensation in heat exchangers; annlar flo in pipes; and inlet manifolds for chemical reactors ith to-phase feed. Hoever, the leading CFD codes are not presently able to calclate id films. The objective of the present ork as to develop and implement a model for id films at alls; a model for droplet transport in flos; together ith relations for the interaction beteen the to. The interactions are the deposition of droplets at alls and atomization/reentrainment of id from the id film into the de to Kelvin-Helmholtz instabilities. There has been a significant amont of papers on all films, most of hich are on annlar flos in pipes. This paper is heavily inflenced by the ork of Pan and Hanratty (00), ho stdied entrainment in horizontal annlar pipe flo. For droplet diffsion, the papers by Loth (001), Mols and Oliemans (1998) and by Mols et. al. (000) have been particlarly sefl. For droplet deposition at alls, the reslts of McCoy and Hanratty (1997) has been sed. A brief otline of the paper: First, the transport models for id films and droplet transport are described, together ith models for droplet deposition and film atomization. Then, an example calclation for air/ater flo in a horizontal pipe segment is given. Finally, possible extensions are otlined. 1

2 MODEL DESCRIPTION Model overvie The present model consists of the folloing elements: A to-dimensional transport eqation for films at alls. The film is assmed thin compared to the dimensions of the flo geometry. A transport eqation for droplets in a blk phase. A relation for the deposition and entrainment of id beteen the film and the blk phase. In the folloing, these elements are described in detail. Note that the sb models for the film and the droplet phase are independent in the sense that the film model cold be sed together ith other models for the droplet phase. Film model The folloing to-dimensional transport eqation is sed for the film on a srface 1 : ρ h D ( ρh film ) = RD R (1) A t Here, ρ [kg/m 3 ] is the id density, h [m] is the film thickness, R D [kg/m. s] is the rate of droplet deposition, and R A [kg/m. s] is the rate of atomization of the film. The film velocity, film [m/s] is calclated from: h h g all t film = () 3 Here, all [N/m ] is the all shear stress taken as a vector and g t [m/s ] is the component of gravity tangential to the srface. Droplet transport In the present FLUENT implementation, a simple driftflx type of eqation for the droplet phase is implemented C ( C drop D C) = 0 (3) t here C [kg/m 3 ] is the droplet concentration in the, drop is the droplet velocity and D [m /s] is the droplet diffsion coefficient described belo. This model is intended as a stepping stone for later models for droplet size distribtions. Presently, there is only one droplet size. Droplet velocity The droplet velocity, drop, is calclated as follos. It is assmed that the droplet relaxation time is mch loer than a characteristic (lagrangean) time scale of the mean flo. The calclations are based on a correlation for settling velocity by Trton and Clark (1987). The correlation is explicit, and gives the terminal velocity of falling spheres to ithin ~8% of the one calclated from the standard drag crve. A dimensionless particle diameter is given by d = d * drop ( ) 1/3 g a ρ ρ ρ here g [m/s ] is the gravity and a=d/dt is the acceleration of the phase. The Trton-Clark correlation gives a settling velocity on dimensionless form: * = d* d * (5) Finally, the settling velocity, t, is calclated from * ρ = t g-a ( ρ ρ ) The drift beteen the droplets and the is then calclated from = drop t Droplet diffsion ( g-a) g-a Rigoros models for droplet diffsion shold take into accont crossing trajectory, inertial-limit and continity effects. In the present model, e se the transverse diffsion coeffisient of the Csanady (1963) model. With "standard" parameters, this eqates to : D = t / Sct 1 0.7γ γ = drop - k /3 Atomization of the film The model for atomization of the id film follos the ork by Pan and Hanratty (00). The sitation is depicted in Figre 1. 1/3 (4) (6) (7) (8) 1 To emphasize the to-dimensional natre of the eqation, a "D" sbscript has been added to the divergence operator. The all is ths considered to be a to-dimensional manifold.

3 Liqid film Floing Figre 1: Atomization of id film. De to the passing flo, the id film is nstable (Kelvin-Helmholz mechanism), and droplets are torn off the forming aves. For pipes, a correlation for the total rate of id entrainment, R A [kg/m. s], into the is given by: R A ' ku A ρρ = ( Γ Γ crit ) (9) σ here Γ [kg/m. s] is the id film mass flo rate per meter circmference and Γ crit is a critical flo rate. Belo Γ crit there is no atomization. The constant k A is approximately (Pan/Hanratty 00). For CFD calclations in general geometries, the form of the Pan/Hanratty (00) correlation needs some modification. If e define at friction coefficient for the pipe as C = (10) all f, pipe 1/ρU e can eliminate the presence of the velocity, and obtain: R A ' = kaall ρρ ( crit) ρ C σ Γ Γ (11) 1 f, pipe hich is the one implemented in FLUENT. For pipe flos, C f,pipe may be looked p in a standard Moody chart. The critical film flo, Γ crit, is given by : 1 Γ crit = Re LFC, here: 4 3 ReLFC = 7.3log( ω) 44. log( ω) 63log( ω) 439, ρ ω, validity (1.8 < ω < 8) ρ Droplet deposition onto the film (1) The deposition rate of droplets, R D [kg/m. s], onto the film is taken as R =V C (13) D here C is the droplet concentration at the all and V is the mean impaction velocity of droplets hitting the all. To model V, e do the folloing reasoning, similar to Pan/Hanratty (00): The velocity, V drop, of individal droplets in the direction normal to the all is assmed to be normally distribted ith mean, hich is taken as the settling velocity in the direction normal to the all, and a standard deviation σ hich is yet to be determined. V is then the expected velocity toards the all: V = V f( V ) dv drop drop drop 0 (14) Here f(.) is the Gassian velocity distribtion. This integral has an analytic soltion: σ σ e 1erf( ) V = σ (15) π hich is the form implemented. Note that eqation (15) is a general soltion for a Gassian distribtion ith mean velocity and standard deviation σ, here in the present case = 0. It remains to find the standard deviation, σ. For vertical pipes, e have the folloing correlation (McCoy and Hanratty 1997) V 5 / * = for <0. 4 = ( ) V / * for 0.< <.9 (16) V / * = 0.17 for.9 < < V / * = 0.7 / for > here 1 ddropρ* ρ drop = (17) 18 ρ is a dimensionless droplet relaxation time and * [m/s] is the shear velocity. For this particlar case, e have V = σ. Assming π that the standard deviation, σ, is the same for other all inclinations, e get the folloing correlation for the general case: σ / * = 5 π for <0. σ / * = 4 π for 0.< <.9 ( ) σ π σ π hich is the one implemented. / * = 0.17 for.9 < < / * = 0.7 / for > EXAMPLE CALCULATION (18) The model has been implemented in FLUENT sing User- Defined-Fnctions (UDF). As an example, flo in a horizontal pipe segment is calclated. The inpt data is for small ater droplets in air, as given in Table 1. Note that the entrainment rate is zero if Γ<Γcrit. In that case, the film is stable. 3

Description Vale Pipe diameter ( ft) 0.67 m Pipe length 10.0 m Roghness height 6.7. 10-4 m Inlet velocity 10.0 m/s Trblent intensity 10.0 % Inlet droplet concentration 0.01 kg/m 3 Gas density 1.

4 Description Vale Pipe diameter ( ft) 0.67 m Pipe length 10.0 m Roghness height m Inlet velocity 10.0 m/s Trblent intensity 10.0 % Inlet droplet concentration 0.01 kg/m 3 Gas density 1. kg/m 3 Gas viscosity Pa. s Liqid density 1000 kg/m 3 Liqid viscosity Pa. s Srface tension N/m Droplet diameter 50 m Table 1: Simlated conditions. Calclated reslts In the folloing, reslts are presented for: Droplet concentration field Film thickness at alls Film velocity near the inlet Film mass flx Droplet concentration field Figre shos contors of droplet concentration at pipe cross sections. The inlet is to the left. Figre 3: Droplet concentration, [kg/m 3 ], in vertical slice throgh the pipe axis. Pipe inlet to the left. Film thickness Figre 4 shos contors of film thickness at the pipe all as seen from the pipe inlet into the pipe. As can be seen, id tends to accmlate at the bottom of the pipe. Also, minor ringing artifacts may be seen de to the nondissipative natre of the model, bt the ringing is tolerable. Figre 4: Film thickness, [m], at all Figre : Droplet concentration, [kg/m 3 ], in pipe cross sections The effect of the gravity is apparent; the droplets are dran donards, leading to an accmlation of droplets at the loer part of the pipe. In order to see this better, the concentration field is dran in Figre 3 in a vertical slice throgh the pipe axis: In order to see the film thickness better, xy-plots are shon in Figre 5. The crves apply to: The top of the pipe, i.e. the line parallel to the axis, bt one pipe radis above it. The bottom of the pipe The sides of the pipe, i.e. at lines parallel to the axis, bt shifted /- one radis sideays. 4

DISCUSSION Experimental validation Presently, the model has only been benchmarked against knon reslts and existing models for special cases, sch as annlar flos in horizontal and vertical pipes.

5 DISCUSSION Experimental validation Presently, the model has only been benchmarked against knon reslts and existing models for special cases, sch as annlar flos in horizontal and vertical pipes. Ths, the model needs independent experimental validation before one may be certain abot the accracy. Hoever, the sbmodels for entrainment and deposition are validated in the literatre (e.g. Pan/Hanratty 00). Figre 5: Film thickness, [m], as fnction of axial position. The crves apply to i) the bottom of the pipe, ii) the sides of the pipe and iii) the top of the pipe. Film velocity Figre 6 shos the film velocity near the pipe inlet, hich is of the order of 1 cm/s. The effects of gravity on the film is apparent. Choice of droplet model The present droplet model is simple, and is intended as a stepping stone toards a model describing a droplet size distribtion. One particlarly interesting option is the Qadratre Method of Moments (QMOM), as described by Marchiso et. al. (003). Pipe roghness In the present model, the ser has to specify a roghness height for the pipe all. In reality, the presence of the film ill inflence the apparent all roghness and ths the all shear stress. This copling may be inclded in later versions of the code. CONCLUSION Figre 6: Film velocity, [m/s] near the pipe inlet Film mass flx Figre 7 shos the xy-plots of the calclated film mass flx at the top/side/bottom of the pipe. Figre 7: Film mass flx, [kg/m.s], as fnction of axial position. Again, the id accmlation at the bottom is apparent. A model for dilte -id flos ith films at the alls has been implemented in the CFD code FLUENT sing User-Defined Fnctions (UDF). The model consists of transport eqations for droplets in the and for id transport in the film. Exchange of id beteen the to occrs by atomization of the film and by droplet deposition. Predictions are in reasonable agreement ith knon cases, sch as horizontal and vertical pipe flos. Hoever, independent validation of the code is yet to be performed. REFERENCES CSANADY, G.T. (1963). "Trblent diffsion of heavy particles in the atmosphere". J. Atmos. Sci., 0, LOTH, E. (001). "An Elerian trblent diffsion model for particles and bbbles". Int. J. of Mltiphase Flo, 7, MARCHISIO, D.L., VIGIL, R.D. and FOX, R.O. (003). "Implementation of the qadratre method of moments in CFD codes for aggregation breakage problems". Chemical Engineering Science, 58, McCOY, D.D. and HANRATTY, T.J. (1997). "Rate of deposition of droplets in annlar to-phase flo". Int. J. of Mltiphase Flo, 3, MOLS, B., MITTENDORFF, I. and OLIEMANS, R.V.A. (000)."Reslts from a to-dimensional trblent diffsion-model for dispersion and deposition of droplets in horizontal annlar dispersed /id flo", Int. J. of Mltiphase Flo, 6, 6, MOLS, B., and OLIEMANS, R.V.A. (1998). "A trblent diffsion model for particle dispersion and deposition in horizontal tbe flo", Int. J. of Mltiphase Flo, 4, 1,

6 PAN, L. and HANRATTY, T.J., (00). "Correlation of entrainment for annlar flo in vertical pipes", Int. J. of Mltiphase Flo, 8, 3, TURTON, R. and CLARK, N.N. (1987), Poder Technol., 53, 17. 6

Two identical, flat, square plates are immersed in the flow with velocity U. Compare the drag forces experienced by the SHADED areas.

Two identical, flat, square plates are immersed in the flow with velocity U. Compare the drag forces experienced by the SHADED areas. Two identical flat sqare plates are immersed in the flow with velocity U. Compare the drag forces experienced by the SHAE areas. F > F A. A B F > F B. B A C. FA = FB. It depends on whether the bondary