Ind. Eng. Chem. Res. 2000, 39, 2091-2100
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Transport Phenomena in Turbulent Falling Films Abdulmalik A. Alhusseini* and John C. Chen Department of Chemical Engineering, Lehigh University, 111 Research Drive, Bethlehem, Pennsylvania 18015
This study was concerned with transport phenomena in turbulent falling films, with negligible interfacial shear. Emphasis was on the development of general expressions for eddy diffusivities, which could then be utilized to predict transport coefficients for various momentum and heatand mass transfer processes. Experiments were pursued to obtain new data for heat transfer during film evaporation, a phenomenon that involves significant boundary layer resistance at both the wall and the free interface. These new data extended the existing database by an order of magnitude in the Prandtl number. When predictions of the new model are assessed against both new and existing experimental information, good agreement was found for film thickness, sensible heat transfer at the wall, mass transfer at the free surface, and film evaporation. Introduction Liquid films draining down vertical or inclined surfaces are commonly termed “falling films”. Process equipment that utilize falling films are used in various industrial applications, as evaporators, condensers, fluid heaters or chillers, absorbers, strippers, and reactors. In many applications, a high film flow rate is desired in order to achieve good transport efficiencies and to avoid film breakdown, resulting in operations in the turbulent film regime. Extensive researches on transport phenomena in falling films have been pursued since the early 1900s. This body of published literature was reviewed by Seban1 in 1978 and more extensively by Yih2 in 1986. Nusselt’s3 classical theory for laminar falling films provided the starting point for all subsequent analyses. Its main shortcoming is the assumption of a smooth liquid film, whereas actual falling films have a wavy interface at practically all flow rates of interest. The wavy laminar regime, which prevails at low flow rates, is characterized by large waves that travel along a thin laminar substrate. These waves carry a significant portion of the mass flow and induce secondary circulation in the laminar bulk, resulting in enhancement of heat and mass transfer above that predicted by the classic Nusselt theory. This enhancement has been confirmed both by experimental measurements (e.g., Chun and Seban,4 Kutateladze and Gogonin,5 and Palen et al.6) and by numerical simulation (Wasden and Dukler7,8). As the flow rate is increased, a gradual transition from wavy laminar to turbulent films takes place. The nature of this transition is not well understood but is known to depend on both the viscosity and surface tension of the fluid, as suggested by Chun and Seban,4 Blangetti et al.,9 and Mudawwar and Elmasri.10 At sufficiently high flow rates, the falling film becomes fully turbulent. Turbulence in falling films is not well understood because of a lack of data on the structure of turbulence in such films. Experimental investigations have been hampered by the difficulties associated with the * Corresponding author. Present address: Department of Chemical Engineering, King Saud University, Riyadh 11421, Saudi Arabia. E-mail:
[email protected].
probing of thin films (typically of thickness less than 1 mm) without interference by, or disruption to, the wavy interface. Numerous models have been proposed to predict transport processes in turbulent films. The vast majority of these models justifiably use the smooth film physical model as the starting point to simplify subsequent analysis. As pointed by Mudawwar and Elmasri,10 the smooth film physical model is a reasonable representation of turbulent films because, unlike the case of laminar films, turbulent wave activity is characterized by very long waves (wavelengths were 2 orders of magnitude greater than the average film thickness in the experimental studies by Kirkpatrick11 and Takahama and Kato12). Turbulent eddy transport is often captured by the classical eddy diffusivity models. A noteworthy exception is the advection-diffusion model developed by McCready and Hanratty13 and McCready et al.14 The advection-diffusion approach to modeling the transient turbulent transport near interfaces is quite interesting and does, in fact, offer a more detailed picture of the transport mechanisms than that offered by the timeaveraged eddy diffusion approach. However, the advection-diffusion approach requires experimental data in the form of a time-varying signal to represent the normal velocity gradient near the free interface. To be useful, such data should be extensive enough to cover a wide range of flow rates and fluid properties. Obviously, this is a major hurdle to overcome especially if the main objective is to develop reliable time-averaged design correlations. A promising alternative to such hard-to-get experimental data is to conduct direct numerical simulation (DNS) studies of the transient Navier-Stokes and continuity equations. However, the DNS approach demands extensive computational resources because the flow should be resolved down to the eddy level. Hopefully, the needed computational resources will become more affordable soon. Until then, it seems reasonable to continue to use the eddy diffusivity models, which serve well at predicting timeaveraged transport rates. It is generally agreed that turbulent films consist of three regions: a viscous boundary layer near the wall, a turbulent core, and a viscous boundary layer near the free interface. The wall boundary layer forms as a result
10.1021/ie9906013 CCC: $19.00 © 2000 American Chemical Society Published on Web 04/21/2000
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of turbulence dampening by the wall, as in turbulent flow past a flat plate. The surface boundary layer is believed to exist because of the suppression of turbulence by surface tension forces at the free interface. The two boundary layers account for most of the resistance to heat and mass transport across turbulent falling films and consequently are the primary targets of interest for modeling. The relative importance of each boundary layer to the overall process depends on the process itself. For example, absorption of a solute from gas into the liquid is governed by the free interface boundary layer because the mass flux vanishes near the solid wall. On the other hand, the resistive contribution of the wall boundary layer dominates in processes such as wall dissolution or sensible-heating of nonvolatile films. For cases of falling films without interfacial shear, the wall boundary layer governs momentum transfer and thus determines the average film thickness. In evaporation or condensation processes, the resistances of both boundary layers are important because the heat flux is nearly invariant across the entire film thickness. Consequently, experimental data with evaporating or condensing films are especially useful for assessment of theoretical models. Unfortunately, condensation experiments tend to have a significant increase of film flow rate down the wall, because of the mass addition; hence, it is difficult to obtain condensation data for fully developed condition. In addition, condensation experiments are often associated with significant vapor shear, which further limits their usefulness when developing a model for the baseline case of zero interfacial shear. Evaporation data at essentially fully developed condition with negligible vapor shear are available in published literature but only for limited Prandtl numbers (generally less than 7); see Alhusseini et al.15 for a recent review of such data. The objectives of the present work were to obtain additional experimental data for transport governed by both boundary layers (i.e., evaporation) over an extended range of Prandtl numbers, to develop a general model for turbulent transport in falling films, and to assess its applicability for the evaporation phenomenon and for various other phenomena that are governed by a single boundary layer (e.g., surface absorption, wall heating). As part of this work, an effort also was made to review existing transport models for turbulent falling films and to compare their predictions against experimental data. Experiment In this section, we briefly describe the experimental setup and procedures used to collect the new evaporation data. A more detailed description can be found in work by Alhusseini et al.15 Evaporative heat transfer coefficients were measured for pure water and propylene glycol films under atmospheric and vacuum conditions to cover a wide range of physical properties. The test section was an electrically heated stainless steel tube 3.81 cm in outside diameter and about 3.05 m in length. The test liquid was distributed evenly on the external surface at the top of the test tube and flowed down in the form of a falling film. Electrical heating was supplied by a dc rectifier through two electrodes spaced to provide a heated length of 2.90 m. The evaporative heat transfer coefficients were calculated using the equation hE ) qw/(Tw - Ts), where qw
Figure 1. Schematic of a wall temperature measurement station.
is the heat flux, Tw is the external wall surface temperature, and Ts is the fluid saturation temperature. The heat flux was calculated from electrical power measurement. The saturation temperature was measured in the vapor phase at the exit of the test section. The external wall temperatures were calculated from the measured inner wall temperatures after correcting for temperature drop through the wall. The inner wall temperatures were measured at nine axial locations, with approximately 30 cm of spacing between locations. To ensure good contact, the bare thermocouples were pressed against the inner wall of the test tube by small sections of high-temperature rubber tubing whose outside diameter was slightly larger than the inside diameter of the test tube, as shown in Figure 1. From these measurements, it was possible to calculate the local heat transfer coefficients along the test section. It was found that thermally developed conditions were established within the upper half of the test section for all runs. In this paper, we report only fully developed heat transfer coefficients measured at the location nearest to the test section exit. All temperature measurements were taken by precalibrated type E thermocouples wired to read differential temperatures. Uncertainty analysis showed that the maximum error in the measured heat transfer coefficients was less than 10%. The experimental results are presented in Table 2, grouped by ranges of Prandtl numbers. The dimensionless heat transfer coefficient, defined as h/E ) hE(ν2/ g)1/3/k, is tabulated as a function of film Reynolds number (Re) and fluid Prandtl and Kapitza numbers (Pr and Ka). To ensure turbulent status, only results for Re > 3000 for water films and Re > 1000 for propylene glycol films are given. It is seen that these data reach a maximum Prandtl number of 48, approximately an order of magnitude greater than previously available data. These new data were included in the collection used for model assessment, as described below.
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 2093 Table 1. Details of the Turbulence Models Evaluated in This Study authors and year Mills and Chung20 (1973)
turbulence model used
0 e y+ < y+ i + + y+ i e y < δ
+ + 1 1 ) - + x1 + 4k2y+2(1 - e-y /Aw)2 ν 2 2 Re1.678 ) 6.47 × 10-4Ka1/3 +2/3 (δ+ - y+)2 ν δ
Prt ) 0.9, k ) 0.4, A+ w ) 26 Hubbard et al.21 (1976)
0 e y+ < y+ i + + y+ i e y < δ
+ + 1 1 ) - + x1 + 4k2y+2(1 - y+/δ+)(1 - e-y /Aw)2 ν 2 2 8.13 × 10-17 Re2m + (δ - y+)2 ) ν Ka δ+2/3
1/2 (ν in m2/s) Prt ) 0.9, k ) 0.4, A+ w ) 26, m ) 695ν
Blangetti et al.9 (1982)
0 e y+ < y+ i + + y+ i e y < δ
+ + 1 1 ) - + x1 + 4k2y+2(1 - y+/δ+)(1 - e-y /Aw)2 ν 2 2 Re1.678 + ) 0.00661Ka1/3 (δ - y+)2 ν δ*
2/3 Prt ) 0.9, k ) 0.4, A+ w ) 26, δ* ) 0.169Re
Mudawwar and Elmasri10 (1986)
1 1 )- + ν 2 2
x1 + 4k y
2 +2
(1 - y+/δ+)2(1 - e-((y
+/A+) w
x1 - y+/δ+(1 - 0.8656Recrit1/2/δ+))
)2
Recrit ) 97/Ka0.1 for heating and Recrit ) 0.04/Ka0.37 for evaporation Prt ) 0.66 + 1.4e-(15y+/δ+), k ) 0.4, A+ w ) 26 proposed model (present study)
0 e y+ < y+ i + + y+ i e y < δ
Prt )
+ + + + 1 1 ) - + x1 + 4k2y+2(1 - e-y /Aw)2(e-cy /δ )2 ν 2 2
(
)
1.199 × 10-16 Re2m δ+ - y+ ) ν Ka δ+2/3 A+ i
2
1 1 + C′Pet 2Pri
x
1 - (C′Pet)2[1 - e(-1/C′Pet xPri)] Pri
0.0675, A+ ) 0.01665Ka-0.17324, k ) 0.4, A+ w ) 26, c ) 2.5, m ) 3.49Ka i C′) 0.2, Pri ) 0.86, Pet ) (/ν)Pr
Assessment of Prior Models Review of Prior Models. The performance of a turbulence model depends on its expression for eddy viscosity and, to a lesser extent, its expressions for Prandtl and Schmidt numbers of turbulence. The earlier models only considered the boundary layer at the solid wall, using eddy viscosity expressions similar to those for single-phase internal flow in channels. Examples of such eddy viscosity expressions are those based on the Diesseler function as used in Dukler’s16 model for film condensation, the Spalding function as used in Ishigai et al.17 model for film heating, and the Van Driest damping function as used in Limberg’s18 model for film heating. Because these earlier models ignore dampening of turbulence at the free interface, they tend to overpredict transport rates for any process that involve finite fluxes at the free surface, e.g., absorption, condensation, or evaporation. Evidence for this conclusion is found in the data generated by Chun and Seban,4 which helped show that all previous models tended to overpredict the measured heat transfer coefficient of evaporation. For example, Seban and Faghri19 noted that Limberg’s model overpredicts Chun and Seban’s data by as much as 30%. Turbulence models that account for both the wall and free interface boundary layers were proposed by a number of researchers as improvements over the singlelayer models. Examples are the models of Mills and Chung20 for film evaporation, Hubbard et al.21 for film evaporation and condensation, Blangetti et al.9 for film condensation, and Mudawwar and Elmasri10 for film heating and evaporation. Details of these models are shown in Table 1, along with expressions developed in
the present study (discussed below). The models use different versions of the Van Driest function to account for the wall boundary layer. As Table 1 indicates, the models of Mills and Chung, Hubbard et al., and Blangetti et al. employ eddy viscosity expressions that consist of two different functions accounting for the wall and free interface boundary layers separately. Mudawwar and Elmasri used a single eddy viscosity function to account for both boundary layers with the eddy viscosity smoothly approaching zero at the free interface with no damping function. While these existing models were successful when compared with experimental data for each specific application, none were assessed against data for a range of different transport processes. It is reasonable to require that a turbulence model, with general expressions for eddy diffusivities of momentum, heat, and mass, should be applicable to all of the different transport processes. Method Used in the Assessment of Prior Models. The following paragraphs describe an attempt to obtain a broader assessment of the prior models, by comparing their predictions for film thickness, heat transfer coefficient of sensible heating, mass transfer coefficient of absorption, and heat transfer coefficient of evaporation against quantitative experimental data or well-accepted empirical correlations. To have a consistent basis of comparison, only models and data without interfacial shear and at fully developed conditions were utilized in this assessment. This is considered to be the baseline case, which all models should predict successfully. This requirement precluded the use of condensation data of Carpenter22 and Blangetti23 (vapor shear effect) and
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wall mass transfer data of Iribarne et al.24 (developing mass transfer). For film thicknesses, model predictions were compared with the widely accepted correlation proposed by Brauer25 for turbulent films
δ+ ) 0.0946Re0.8
(1)
where δ+ is the dimensionless film thickness defined as δ+ ) δu*/ν, Re is the Reynolds number defined as Re ) 4Γ/µ, and Γ is the mass flow rate per unit width. The film thickness was calculated by integrating the momentum equation with the no-slip boundary condition at the wall and zero vapor shear at the interface and solving simultaneously with the continuity equation as given by
∫0y
(
+
u+ )
)( )
σ / 1 + dσ+ ν δ+
1-
and
∫0δ u+ dy+
Γ Re ≡ 4 ) 4 µ
+
(2)
where y+ is the dimensionless distance from the wall, u+ is the dimensionless velocity, and is the eddy diffusivity of momentum. The heat transfer coefficient of sensible heating was calculated by integrating the energy equation with constant heat flux at the wall and zero heat flux at the free interface as given by
h/H )
4 Re
∫0
[
δ+
Re 4
δ+1/3Pr
∫0
y+
][
u+(σ+) dσ+ /
]
1 1 + dy+ Pr Prt ν (3)
where Prt is the Prandtl number of turbulence. The predicted heat transfer coefficients for sensible heating of the liquid were compared with the correlation of Wilke,26 representing experimental data for nonevaporating aqueous ethylene glycol films over a Prandtl number range of 5.4-210 and a Reynolds number range of 10-10 000. Fully turbulent data were, however, limited to runs with Prandtl numbers of 5.4 and 9.4. Wilke’s correlation for the fully turbulent data, which originally was given in terms of Nu ) hHδ/k, can be rewritten in the following form with the aid of Brauer’s film thickness correlation
h/H ≡
hH(ν2/g)1/3 ) 0.0087Re0.4Pr0.344 k
(4)
where hH is the heat transfer coefficient of heating defined as hH ) qw/(Tw - Tm) and Tm is the mixing cup temperature. The mass transfer coefficient of absorption was calculated by integrating the solute’s continuity equation with constant molar flux at the free interface and zero molar flux at the wall as given by K/a ) 4 Re
∫ {∫ δ+
0
δ+1/3Sc
[1 - Re4 ∫
Y+
0
σ+
0
] [Sc1 + Sc1 ν] dσ }u
u+ dξ+ /
Schmidt number of turbulence. Although the true boundary condition at the free interface for the absorption problem is constant concentration rather than constant molar flux, the error introduced by solving for the simpler case of constant flux is very small (less than about 2%) because of the large value of the Schmidt number. The predicted mass transfer coefficients of absorption were compared with the correlation proposed by Won and Mills.27 This correlation was based on the extensive absorption data obtained by Chung and Mills28 and Won and Mills27 using various fluids to cover wide ranges of liquid viscosity and surface tension. The Schmidt number ranged from 83 to 2700 and the Kapitza number (defined as Ka ) µ4g/Fσ3) ranged from 5.48 × 10-12 to 2.68 × 10-8. Most of the data points were in the turbulent region for all Schmidt numbers. Won and Mills’ correlation, in which the mass transfer coefficient was originally nondimensionlized as Ka/ (gν2)1/3, can be rewritten as
+
+
dY+
t
(5)
where Y+ is the dimensionless distance from the free surface, defined as Y+ ) (δ - y)u*/ν and Sct is the
K/a
Ka(ν2/g)1/3 ≡ ) 6.97 × 10-9RemScβKa-1/2 D
(6)
where Ka is the mass transfer coefficient of absorption defined as Ka ) N1/(Ci - Cb), N1 is the molar flux of the solute across the free interface, Ci is the solute’s molar concentration at the free interface, Cb is the solute’s mixing cub molar concentration, D and Sc are the liquid mass diffusivity and the Schmidt number, respectively, m ) 3.49Ka0.0675, and β ) 1 - 0.137Ka-0.055. Finally, heat transfer coefficients of evaporation predicted by various models were compared with our own experimental data, obtained as a part of this work. These data, measured with pure water and propylene glycol films, covered a wide Prandtl number range of 1.73-46.6 and a Reynolds number range of about 3015 000. Data points in the turbulent flow region are listed in Table 2. The heat transfer coefficient of evaporation was calculated from the various models by integrating the energy equation with constant heat flux across the wall and the free interface as given by
h/E )
δ+1/3Pr 1 1 δ+ + 1/ dy+ 0 Pr Pr+ ν
∫
[
]
(7)
where hE is the heat transfer coefficient of evaporation defined as hE ) qw/(Tw - Ts). Predictions Obtained by Prior Models. When the above approach was utilized, four of the published models (listed in Table 1) were evaluated. Models which only take into account the wall boundary layer would be unable to predict absorption or evaporation processes and were not considered. All four selected models account for both the wall and surface boundary layers and thus should be able to deal with the various types of transport in turbulent films. There are other models that account for both boundary layers but suffer from physical inconsistencies, e.g., the use of a discontinuous eddy viscosity expression in the model proposed by Seban and Faghri.19 Such models were not included in the comparison. Results of this evaluation for heat- and mass transfer coefficients are presented in parity plots and discussed below. Predictions for film thicknesses are not plotted but are described in the discussion.
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 2095 Table 2. Data in the Turbulent Region Re
h/E
Pr
Ka
µ (kg/m/s)
F (kg/m3)
15594 15583 15538 15186 15180 13257 13254 13251 13249 11372 11325 11325 11321 11320 11317 11317 11312 11289 11281 9410 9404 9379 7454 7454 7453 5514 5511 5510
0.226 0.230 0.227 0.234 0.236 0.227 0.226 0.232 0.226 0.230 0.219 0.222 0.219 0.216 0.223 0.229 0.224 0.215 0.226 0.222 0.213 0.218 0.204 0.205 0.204 0.188 0.188 0.186
1.68 1.69 1.69 1.74 1.74 1.73 1.73 1.73 1.74 1.72 1.73 1.73 1.73 1.73 1.73 1.73 1.74 1.74 1.74 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73
1.20 × 10-13 1.21 × 10-13 1.22 × 10-13 1.35 × 10-13 1.35 × 10-13 1.34 × 10-13 1.35 × 10-13 1.35 × 10-13 1.35 × 10-13 1.32 × 10-13 1.34 × 10-13 1.34 × 10-13 1.34 × 10-13 1.34 × 10-13 1.34 × 10-13 1.34 × 10-13 1.35 × 10-13 1.36 × 10-13 1.36 × 10-13 1.32 × 10-13 1.33 × 10-13 1.34 × 10-13 1.33 × 10-13 1.33 × 10-13 1.33 × 10-13 1.33 × 10-13 1.33 × 10-13 1.33 × 10-13
2.68 × 10-4 2.68 × 10-4 2.69 × 10-4 2.75 × 10-4 2.75 × 10-4 2.75 × 10-4 2.75 × 10-4 2.75 × 10-4 2.75 × 10-4 2.74 × 10-4 2.75 × 10-4 2.75 × 10-4 2.75 × 10-4 2.75 × 10-4 2.75 × 10-4 2.75 × 10-4 2.75 × 10-4 2.76 × 10-4 2.76 × 10-4 2.74 × 10-4 2.74 × 10-4 2.75 × 10-4 2.74 × 10-4 2.74 × 10-4 2.74 × 10-4 2.74 × 10-4 2.74 × 10-4 2.74 × 10-4
952 952 952 954 954 954 954 954 954 953 954 954 954 954 954 954 954 954 954 953 953 954 953 953 953 953 953 953
8660 8659 8651 7554 7553 7551 6448 6443 6441 5434 5427 5413 4324 4311 4305
0.277 0.273 0.273 0.263 0.262 0.261 0.247 0.247 0.244 0.247 0.246 0.246 0.230 0.225 0.230
3.22 3.22 3.23 3.23 3.23 3.23 3.23 3.23 3.23 3.17 3.17 3.18 3.16 3.17 3.17
1.38 × 10-12 1.38 × 10-12 1.39 × 10-12 1.38 × 10-12 1.38 × 10-12 1.39 × 10-12 1.39 × 10-12 1.39 × 10-12 1.39 × 10-12 1.29 × 10-12 1.30 × 10-12 1.31 × 10-12 1.28 × 10-12 1.29 × 10-12 1.30 × 10-12
4.96 × 10-4 4.96 × 10-4 4.96 × 10-4 4.96 × 10-4 4.96 × 10-4 4.96 × 10-4 4.96 × 10-4 4.97 × 10-4 4.97 × 10-4 4.87 × 10-4 4.88 × 10-4 4.89 × 10-4 4.86 × 10-4 4.87 × 10-4 4.88 × 10-4
980 980 980 980 980 980 980 980 980 980 980 980 979 979 979
6528 6498 6492 6313 5751 5740 5728 5719 5707 4890 4885 4879 4879 4054 4051 4046 3227 3217 3215
0.278 0.284 0.286 0.288 0.276 0.262 0.270 0.262 0.266 0.247 0.253 0.251 0.248 0.242 0.242 0.239 0.244 0.226 0.223
4.40 4.42 4.42 4.56 4.35 4.36 4.37 4.38 4.39 4.37 4.37 4.38 4.38 4.37 4.37 4.38 4.35 4.37 4.37
4.38 × 10-12 4.46 × 10-12 4.48 × 10-12 5.03 × 10-12 4.21 × 10-12 4.24 × 10-12 4.28 × 10-12 4.31 × 10-12 4.34 × 10-12 4.28 × 10-12 4.30 × 10-12 4.32 × 10-12 4.32 × 10-12 4.28 × 10-12 4.29 × 10-12 4.31 × 10-12 4.21 × 10-12 4.27 × 10-12 4.28 × 10-12
6.62 × 10-4 6.66 × 10-4 6.67 × 10-4 6.86 × 10-4 6.56 × 10-4 6.57 × 10-4 6.59 × 10-4 6.60 × 10-4 6.61 × 10-4 6.58 × 10-4 6.59 × 10-4 6.60 × 10-4 6.61 × 10-4 6.59 × 10-4 6.60 × 10-4 6.60 × 10-4 6.56 × 10-4 6.59 × 10-4 6.59 × 10-4
987 988 988 988 987 987 987 987 987 987 987 987 987 987 987 987 987 987 987
1854 1852 1629 1627 1360 1104
0.566 0.573 0.542 0.531 0.477 0.432
39.57 39.60 39.33 39.37 40.01 40.62
1.04 × 10-8 1.04 × 10-8 1.01 × 10-8 1.02 × 10-8 1.10 × 10-8 1.18 × 10-8
2.30 × 10-3 2.31 × 10-3 2.29 × 10-3 2.29 × 10-3 2.34 × 10-3 2.39 × 10-3
963 963 962 962 962 963
1643 1470 1284 1282 1116
0.562 0.475 0.454 0.478 0.437
47.87 47.53 47.47 47.51 46.73
2.58 × 10-8 2.49 × 10-8 2.48 × 10-8 2.49 × 10-8 2.30 × 10-8
2.95 × 10-3 2.92 × 10-3 2.92 × 10-3 2.92 × 10-3 2.86 × 10-3
972 970 970 970 969
Figure 2. Assessment of Mills and Chung’s20 model against experimental information for various transport phenomena in turbulent falling films.
(a) Model of Mills and Chung.20 The predicted heat- and mass transfer coefficients are compared with the experimental data and correlations in Figure 2, in the form of a parity plot. Predictions for the film thickness, which are not shown in Figure 2, were higher than Brauer’s correlation by about 15%. The model consistently overpredicted Wilke’s film heating correlation by up to 100%. The overprediction of the film thickness and the heat transfer coefficient of heating implies that this model overestimates the eddy viscosity near the wall. Predictions for the heat transfer coefficient of evaporation were found to be in good agreement with data at low Prandtl numbers but tended to be higher than data by up to 30% for Prandtl numbers greater than 4.0. Although the free interface eddy viscosity function for this model was deduced from Chung and Mills’ absorption data, predictions for absorption were found to be only in fair agreement with Won and Mills’ correlation. (b) Model of Hubbard et al.21 Predictions for the film thickness were found to be higher than Brauer’s correlation by about 15%. The predicted heat- and masstransfer coefficients are compared with the selected data and correlations in Figure 3. It is seen that this model overpredicted Wilke’s film heating correlation by up to 70%, which, as in the case of Mills and Chung’s model, indicates that the eddy viscosity near the wall is overestimated. For absorption mass transfer, this comparison indicated poor agreement between the model and Won and Mills’ data correlation. While no distinct direction of error was seen, the parity plot shows wide scatter, especially for fluids of higher Sc numbers. In contrast, predictions for evaporation heat transfer were found to be in good agreement with data. This is somewhat surprising in view of the inadequate predictions for film heating and absorption, implying difficulties with diffusivity expressions for both wall and surface boundary layers. Because evaporation involves both boundary layers, it is possible that errors associated with the wall and free interface eddy viscosity functions compensate in this model. (c) Model of Blangetti et al.9 Predictions for the film thickness were found to be higher than Brauer’s correlation by about 15%. The predicted heat- and mass-
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Figure 3. Assessment of Hubbard et al’s.21 model against experimental information for various transport phenomena in turbulent falling films.
Figure 4. Assessment of Blangetti et al.’s9 model against experimental information for various transport phenomena in turbulent falling films.
transfer coefficients are compared with the selected data and correlations in Figure 4. In general, predictions by this model are somewhat similar to that by the model of Mills and Chung. Wilke’s film heating correlation is consistently overpredicted by up to 50%, and the evaporation data are overpredicted by up to 30% for Prandtl numbers greater than 4.0. As in the previous two models, it seems that this model overestimates the eddy viscosity near the wall. Predictions for the absorption case are very similar to that generated by the model of Mills and Chung. (d) Model of Mudawwar and Elmasri.10 Predictions for the film thickness were found to be in very good agreement with Brauer’s correlation. The predicted heat- and mass transfer coefficients are compared with data and correlations in Figure 5. Unlike the above three models, this model produces predictions that are in good agreement with the heating and evaporation cases. The good agreement with Brauer’s film thickness and Wilke’s film heating correlations suggests that the
Figure 5. Assessment of Mudawwar and Elmasri’s10 model against experimental information for various transport phenomena in turbulent falling films.
Mudawwar and Elmasri model uses an accurate expression for the eddy viscosity near the wall. However, predictions for absorption are not in good agreement with Won and Mills’ correlation at Schmidt numbers larger than 800. We must point out that predictions for the absorption case were made with Recrit ) 0 because it gave better agreement with the correlation than with Recrit calculated from the suggested expressions for film evaporation or heating as given in Table 1. We also tried Recrit ) 1000, which seemed to be the actual critical Reynolds number after inspecting Won and Mills’ data, but the results were worse than those for the case Recrit ) 0. For the cases used in this assessment, this model yielded generally better predictions than the other three models, but one still would desire better agreement with mass transfer data for absorption. The overall observation from the above assessment was that each of the models could predict some transport phenomena, but none were in uniform agreement with the available data and correlations for general transport phenomena of different types in falling films. Proposed Model Model Development. A new model for turbulent transport in a falling film is presented here. The underlying objective was to formulate a model which can be applied to predict film thickness (momentum transfer), sensible heating or cooling (heat transfer), absorption or desorption (mass transfer), and evaporation of single-component films (integral momentum and heat transfer). Where possible, this development took advantage of proven expressions for eddy viscosity and turbulent Pr and Sc numbers, which could be applied for appropriate regimes. The analysis is based on the smooth film physical model. First, the Prandtl number of turbulence was taken from an expression proposed by Kays and Crawford29
Prt )
[
x
1 + C′Pet 2Pri
]
1 - (C′Pet)2(1 - e-1/C′PetxPri) Pri
-1
(8)
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 2097
where C′ ) 0.2, Pri ) 0.86, and Pet is the Peclet number of turbulence and is defined as Pet ) (/ν)Pr. This expression allows Prt to increase near the wall and free interface, which is the expected trend in boundary layers. The corresponding Schmidt number of turbulence was also calculated from the same expression, replacing Pr with Sc. The behavior of the eddy viscosity near the wall and in the core region is expected to be similar to that in internal conduit turbulent flows. Hence, the mixing length in the film’s core is expected to scale with the distance from the wall as +
l ) ky
+
It remains to develop an expression for the eddy viscosity near the free interface. As an empirical guide for this development, we elect to emphasize phenomena which are sensitive to transport activity near the free interface, such as mass transfer in absorption or desorption. The mass transfer coefficient of absorption can be calculated by integrating the continuity equation of the solute as given by eq 5. Because Schmidt numbers for liquids are often much greater than unity, eq 5 can be approximated by
K/a ≈
(9)
where l+ is the dimensionless mixing length defined as l+ ) lu*/ν and k is the Von Karman constant which is assigned a value of 0.40. In accordance with the treatment by Limberg,18 deviation from the linear relationship between l+ and y+ in the core region is accounted for by multiplying the right-hand side of eq 9 by a function which has the form, F ) e-cy+/δ+, where c is a constant whose value is to be determined. As in turbulent pipe flows, turbulence dampening in the wall boundary layer is accounted for by multiplying the mixing length in the core region by the well-known Van Driest damping function which is given by Dw ) + 1 - e-y+/Aw, where A+ w is assigned a value of 26. Hence, the mixing length in the wall boundary layer and core region is given by
l+ ) ky+DwF
(10)
Defining the eddy diffusivity of momentum by
| |
du+ ≡ l+2 + ν dy
(11)
| |
]
dY
+
(12)
(13)
It is seen that the limiting behavior of /ν as y+ approaches zero is y+4, as has been found in several prior studies. To determine the constant c in Limberg’s function, we elect to emphasize transport phenomena which are governed by the wall boundary layer and are sensitive to the profile of the eddy viscosity, e.g., sensible heat transfer between fluid and the wall. Using Wilke’s correlation for film heating as the standard, it was determined that a constant of 2.5 is a good value for the Limberg parameter, c.
(14)
The eddy diffusivity is still unknown, but it can be represented by a Taylor series expansion in the distance from the free interface as /ν ) bnY+n + bn+1Y+n+1 + ..., where n is a positive integer whose value will be determined. If the leading term of the expansion is substituted in eq 14, the integral can be evaluated in closed form for the case of constant Sct, giving the following approximate expression for the absorption mass transfer coefficient:
K/a ≈
()
( )
π Sc n sin bn1/nδ+1/3 π n Sct
1/n
(15)
The values of bn and n can now be determined by equating eq 15 to Won and Mills’ absorption correlation. The exponent of the Schmidt number in Won and Mills’ correlation is between 0.4 and 0.6 for all Kapitza numbers covered in their experiments, which suggests that n ) 2 is a reasonable choice. For Sct ) 1, b2 ) (1.199 × 10-16/Ka)(Re2m/δ+2/3), where δ+ can be calculated from Brauer’s correlation. Thus, the leading behavior of /ν near the free interface can be expressed by
()
The velocity gradient in eq 12 can be eliminated with the aid of the bulk flow momentum equation for fully developed conditions and zero vapor shear which, after integration, is given by, 1 - y+/δ+ ) (1 + /ν) (du+/dy+). Near the wall, this relationship can be approximated by, 1 ≈ (1 + /ν) (du+/dy+). Combining this expression with eq 12, we obtain a quadratic equation in /ν with one positive root. The resulting expression for /ν in the wall boundary layer and the core regions is then
-1 1 ) + x1 + 4k2y+2Dw2F2 ν 2 2
[
-1
1.199 × 10-16 Re2m Y+ ) ν Ka δ+2/3 A+ i
and using eq 10, we get
du+ ) k2y+2Dw2F2 + ν dy
∫0
∞
δ+1/3Sc 1 1 + Sc Sct ν
2
(16)
where A+ i can be interpreted as a function to correct for the inadequacies of representing the wavy interface as a smooth surface. Using Won and Mills’ correlation for absorption as the standard of comparison, it was determined that this function can be empirically represented by the equation -0.17324 A+ i ) 0.01665Ka
(17)
It is worth noting that the mass transfer expression given by eq 16 is for absorption and may not apply for mass transfer associated with multicomponent evaporation. A global eddy viscosity expression is constructed by taking the lower value of the wall or free interface eddy viscosity expressions
/ν ) MINIMUM [(/ν)w + (/ν)i]
(18)
where (/ν)w and (/ν)i are given by eqs 13 and 16, respectively. The general characteristics of this turbulence model are illustrated in Figure 6, where /ν is plotted versus y+ for various conditions. It is seen that the slope of /ν is steeper near the free interface, suggesting that the wall boundary layer is thicker and contributes more to
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Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000
Figure 6. Representative behavior of the proposed dimensionless eddy diffusivity of momentum for various conditions.
the total resistance of the film than the free interface boundary layer. When the Reynolds number is reduced from 6000 to 2500, the film becomes thinner and turbulence activity decays as expected. The effect of the Kapitza number is only seen near the free interface. The thickness of the free interface boundary layer is reduced by about 75% when the Kapitza number is increased from 2.98 × 10-13 (water films at 100 °C) to 2.27 × 10-8 (propylene glycol films at 99 °C). This is consistent with the expectation that decreasing surface tension (increasing Ka) leads to an increase in the wave-induced mixing and thinning of the boundary layer at the interface. This new expression for the turbulent eddy viscosity, given by eqs 13, 16, and 18, can be utilized to determine the film thickness by eq 2, the heat transfer coefficient of sensible heating by eq 3, the heat transfer coefficient of evaporation by eq 7, and the mass transfer coefficient of absorption by eq 5 if Pr and Prt are replaced by Sc and Sct, respectively. Assessment of the Proposed Model. This proposed turbulence model was assessed by comparison with the same data and correlations used to validate the existing models. It should be noted that the assessments for film thickness and evaporation are particularly meaningful because these data were not used for the determination of empirical constants in the derivation of this model. Figure 7 shows the dimensionless film thicknesses predicted by the proposed model as a function of film Reynolds number. For comparison, Brauer’s correlation for turbulent films and Nusselt’s theory for laminar films are also plotted. The model is seen to be in very good agreement with Brauer’s correlation at large Reynolds numbers and smoothly merges with Nusselt’s solution at low Reynolds numbers. Because film thickness is governed by momentum transfer through the wall/core boundary layer, this agreement lends credence to the eddy viscosity expression proposed for the wall/ core region. Comparison with evaporation data is an integral test for a turbulence model, because the thermal resistances of the wall and free interface boundary layers are both important for this phenomenon. Because of the multi-
Figure 7. Comparison of film thickness predicted by the proposed model with Brauer’s correlation for turbulent films and Nusselt’s theory for laminar films.
parameter dependence of the dimensionless heattransfer coefficient, it is difficult to show such a comparison on a single graph. One approach is to use asymptotic analysis to obtain the following closed-form approximate expression for h/E:
h/E )
Prδ+1/3 (A1Pr3/4 + A2Pr1/2 + A3Pr1/4 + Ct) + (BKa1/2Pr1/2) (19)
where A1 ) 9.17, A2 ) 0.328π(130 + δ+)/δ+, A3 ) 0.0289(152100 + 2340δ+ + 7δ+2)/δ+2, B ) 2.51 × 106δ+1/3Ka-0.173/Re3.49Ka0.0675, Ct ) 8.82 + 0.0003Re, and δ+ ) 0.0946Re0.8. The method used to obtain the approximate expression is somewhat similar to that used by Sandall et al.30 The derivation procedure is based on the asymptotic expansion of the integral in eq 7 as Pr approaches infinity. The details of this procedure can be found in the work by Alhusseini31 and are omitted here for brevity. Equation 19 can be rearranged to the more compact form
h/E/f (Re,Pr,Ka) ) δ+1/3
(20)
where, f (Re,Pr,Ka) ) Pr/[(A1Pr3/4 + A2Pr1/2 + A3Pr1/4 + Ct) + BKa1/2Pr1/2]. In Figure 8, the turbulent data in the form h/E/f (Re,Pr,Ka) are plotted versus Re. It is seen that the data for all Pr and Ka values collapse on the curve representing δ+1/3. The good agreement between predictions and data for evaporation from turbulent films, without readjustment of the model’s parameters, is encouraging support for the eddy viscosity expressions proposed for both the wall and the free interface boundary layers. Figure 9 presents a combined parity plot to demonstrate the comparison of this model with experimental data/correlations for three different transport phenomena. In addition to the phenomenon of film evaporation already discussed above, assessments for sensible heat transfer at the wall and mass transfer of absorption at
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 2099
Nomenclature
Figure 8. Comparison of the measured evaporative heat transfer coefficients in turbulent falling films with that predicted by the proposed model.
Figure 9. Assessment of the proposed model against experimental information for various transport phenomena in turbulent falling films.
the free surface are all shown in this figure. Uniformly good agreement is indicated for all three transport phenomena. This is particularly pleasing in view of the fairly extended ranges of the Prandtl and Schmidt numbers represented by these data (approximately 1-50 for Pr and 80-30 000 for Sc). Summary A review of existing models for transport processes in turbulent falling films found that each of the models could predict some transport phenomena, but none were in uniform agreement with the data and empirical correlations for all of the transport phenomena. A different model was developed, with new expressions for the eddy diffusivities in the wall-core and the interface boundary layers. Assessing the new model against established correlations representative of experimental data for film thickness, sensible heat transfer at the wall, and absorption at the free interface showed good agreement for all phenomena. To further assess the proposed new model, experiments were performed to collect data on heat transfer in film evaporation, extending the prior database for this process by an order of magnitude in the Prandtl number. Comparison of the new model with these data indicated very good agreement.
A+ i ) parameter in the free interface eddy viscosity function A+ w ) parameter in the Van Driest damping function (26) b2 ) second coefficient in a Taylor expansion of /ν near the free interface c ) constant in the wall eddy viscosity function (2.5) C′ ) constant in the Prt function (0.2) D ) mass diffusivity Dw ) Van Driest damping function F ) function to correct the mixing length in the core region g ) gravitational acceleration h ) heat transfer coefficient h* ) dimensionless heat transfer coefficient, h(ν2/g)1/3/k h/H ) dimensionless heat transfer coefficient of sensible heating, hH(ν2/g)1/3/k / hE ) dimensionless heat transfer coefficient of evaporation, hE(ν2/g)1/3/k k ) liquid thermal conductivity k ) Von Karman constant (0.40) K/a ) dimensionless mass transfer coefficient of absorption, Ka(ν2/g)1/3/D Ka ) Kapitza number, µ4g/Fσ3 l+ ) dimensionless mixing length, lu*/ν m ) exponent of Re in Won and Mills’ correlation n ) exponent of the leading term in a Taylor expansion of /ν near the free interface N1 ) total molar flux of solute across the free interface Pet ) Peclet number of turbulence, /νPr Pr ) Prandtl number, ν/R Pri ) constant in the Prt function (0.86) Prt ) Prandtl number of turbulence qw ) wall heat flux Re ) film’s Reynolds number, 4Γ/µ Sc ) Schmidt number, ν/D Sct ) Schmidt number of turbulence Tm ) mixing cup temperature Ts ) saturation temperature Tw ) wall temperature u ) film’s velocity profile u+ ) film’s dimensionless velocity profile, u/u* u* ) friction velocity, (δg)1/2 y ) distance from the wall y+ ) dimensionless distance from the wall, yu*/ν Y+ ) dimensionless distance from the free interface, (y δ)u*/ν Greek Letters β ) exponent of Sc in Won and Mills’ correlation δ ) film thickness δ+ ) dimensionless film thickness, δ u*/ν ) eddy diffusivity of momentum Γ ) liquid mass flow rate per unit flow width ν ) liquid kinematic viscosity µ ) liquid dynamic viscosity π ) 22/7 F ) liquid density σ ) surface tension
Literature Cited (1) Seban, R. A. Transport to Falling Films. Proceedings of the 6th International Heat Transfer Conference, Toronto, Canada, 1978; Hemisphere: Washington, DC, 1978; p 417. (2) Yih, S. Modeling Heat and Mass Transport in Falling Liquid Films. In Handbook of Heat and Mass Transfer; Cheremisinoff, N. P., Ed.; Gulf Publishing Co.: Houston, TX, 1986; Vol. 2, p 111. (3) Nusselt, N. Die Oberflachenkondenstion der Wasserdapfes. VDI Z 1916, 60, 541. (4) Chun, K. R.; Seban R. A. Heat Transfer to Evaporating Liquid Films. J. Heat Transfer 1971, 93, 391.
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(5) Kutateladze, S. S.; Gogonin, I. I. Heat Transfer in Falling Film Condensation of Slowly Moving Vapor. Int. J. Heat Mass Transfer 1979, 22, 1593. (6) Palen, J. W.; Wang, Q.; Chen, J. C. Falling Film Evaporation of Binary Mixtures. AIChE J. 1994, 40, 207. (7) Wasden, F. K.; Dukler, A. E. Insight into the Hydrodynamics of Free Falling Wavy Films. AIChE J. 1989, 35, 187. (8) Wasden, F. K.; Dukler, A. E. A Numerical Study of Mass Transfer in Free Falling Wavy Films. AIChE J. 1990, 36, 1379. (9) Blangetti, F.; Kerbs, R.; Schlunder, E. U. Condensation in Vertical TubessExperimental Results and Modeling. Chem. Eng. Fundam. 1982, 1, 20. (10) Mudawwar, I. A.; Elmasri, M. A. Momentum and Heat Transfer Across Freely-Falling Turbulent Liquid Films. Int. J. Multiphase Flow 1986, 12, 771. (11) Kirpatrick, A. T. Wave Mechanics of Inclined and Rotating Liquid Films. Ph.D. Dissertation, MIT, Cambridge, MA, 1980. (12) Takahama, H.; Kato, S. Longitudinal Flow Characteristics of Vertically Falling Liquid Films Without Concurrent Gas Flow. Int. J. Heat Mass Transfer 1980, 6, 203. (13) McCready, M. J.; Hanratty, T. J. Concentration Fluctuations Close to a Sheared Gas-Liquid Interface. AIChE J. 1984, 30, 816. (14) McCready, M. J.; Vassiliadou, E.; Hanratty, T. J. Computer Simulation of Turbulent Mass Transfer at a Mobile Interface. AIChE J. 1986, 32, 1108. (15) Alhusseini, A. A.; Tuzla, K.; Chen, J. C. Evaporation from Falling Films of Single Component Liquids. Int. J. Heat Mass Transfer 1998, 41, 1623. (16) Dukler, A. E. Fluid Mechanics and Heat Transfer in Vertical Falling Film Systems. Chem. Eng. Prog. 1960, 56, 1. (17) Ishigai, S.; Nakanisi, S.; Takehara, M.; Oyabu, Z. Hydrodynamics and Heat Transfer of Vertical Falling Liquid Films (Part 2: Analysis by Using Heat Transfer Data). Bull. JSME 1974, 17, 106. (18) Limberg, H. Warmeubergang an Turbulente und Laminare Rieselfilme. Int. J. Heat Mass Transfer 1973, 16, 1691. (19) Seban, R. A.; Faghri, A. Evaporation and Heating with Turbulent Falling Liquid Films. J. Heat Transfer 1976, 98, 315. (20) Mills, A. F.; Chung, D. K. Heat Transfer across Turbulent Falling Films. Int. J. Heat Mass Transfer 1973, 16, 694.
(21) Hubbard, G. L.; Mills, A. F.; Chung, D. K. Heat Transfer across a Turbulent Falling Film with Concurrent Vapor Flow. J. Heat Transfer 1976, 98, 319. (22) Carpenter, F. G. Heat Transfer and Pressure Drop for Condensing Pure Vapors Inside Vertical Tubes at High Vapor Velocities. Ph.D. Dissertation, University of Delaware, Newark, DE, 1948. (23) Blangetti, F. Lokaler Warmeubergang bei der Kondensation mit uberlagerter Zwangskonvection im vertikalen Rohr. Ph.D. Dissertation, University of Karlsruhe, Karlsruhe, Germany, 1979. (24) Iribarne, A.; Grossman, A. D.; Spalding, D. B. A Theoretical and Experimental Investigation of Diffusion-Controlled Electrolytic Mass Transfer Between a Falling Liquid Film and a Wall. Int. J. Heat Mass Transfer 1967, 10, 1661. (25) Brauer, H. Stromung und Warmenbergang bei Rieselfilmen. VDI-Forschungsh. 1956, 457. (26) Wilke, W. Warmeugergang an Rieselfilme. VDI-Forshungsh. 1962, 490. (27) Won, Y. S.; Mills, A. F. Correlation of the Effects of Viscosity and Surface Tension on Gas Absorption Rates into Freely Falling Turbulent Liquid Films. Int. J. Heat Mass Transfer 1982, 25, 223. (28) Chung, D. K.; Mills, A. F. Experimental Study of Gas Absorption into Turbulent Falling Films of Water and Ethylene Glycol Mixtures. Int. J. Heat Mass Transfer 1976, 19, 51. (29) Kays, W. M.; Crawford, M. E. Convective Heat and Mass Transfer, 2nd ed.; McGraw-Hill: New York, 1980. (30) Sandall, O. C.; Hanna, O. T.; Ibanez, G. R. Heating and Evaporation of Turbulent Falling Liquid Films. AIChE J. 1988, 34, 502. (31) Alhusseini, A. A. Heat and Mass Transfer in Falling Film Evaporation of Viscous Liquids. Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 1995.
Received for review August 9, 1999 Revised manuscript received January 19, 2000 Accepted January 26, 2000 IE9906013
