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The modelling of falling film chemical reactors
Chemical Engineering Science 54 (1999) 1871—1881
The modelling of falling film chemical reactors Federico I. Talens-Alesson* TALENCO Chemical Engineering Consulting, P.O. Box 1035, 08902 Hospitalet de Llobregat, Spain Received 5 January 1998; received in revised form 4 December 1998; accepted 5 December 1998
Abstract Spatially averaged film thickness models have been the preferred approach for simulation of falling film reactors. This paper will discuss both the general and specific limitations of available models. An improved model is proposed, which has been found to perform well in industrially relevant conditions. For its specific application to sulphonation/sulphation reactions yielding anionic surfactants, the improved model provides enhanced predictions about reactor performance, including relative colour intensity at different operational conditions. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Falling; Film; Multiphase; Sulphonation; Modelling; Friction
1. Introduction Falling film reactors are jacketed columns or clusters of columns where liquid and gas flow in annular or annular-mist flow. The gas phase is a mixture of a gaseous or vapourised reagent diluted with an inert carrier. These reactors display outstanding heat removal capability, with the mix between reagents leading to milder reaction conditions than in stirred batch reactors. For this reason the substitution of batch processes by continuous processes in film reactors, in order to improve the reaction progress and safety conditions, has become a recent trend in the chemical industry. Examples of which can be seen in both ethoxylation (Dicoi and Canavas, 1993; Talens et al., 1998) or hydrogenation. This recent move towards film reactors as substitutes to batch reactors, combined with the industrial relevance of conventional applications of falling film reactors (sulphonation, chlorination), justifies the necessity and importance of their accurate mathematical modelling. Sulphonation of linear alkylbenzenes with SO is a classical study case for mathematical modelling, as the physical properties of the liquid phase (Bromstro¨m, 1975) and reaction kinetics and thermodynamics (Mann et al., 1982) are well characterised. Some facts about the reac-
* Tel.: 0034 3 3311 811; fax: 0034 3 33 11 811; e-mail: f — [email protected].
tion and its product will be discussed next to highlight the relevance of the model results. 1.1. Alkylbenzene sulphonation Sulphonation of linear alkylbenzenes (LAB) is a typical reaction for the synthesis of anionic surfactants. Linear alkylbenzene sulphonates (LABS) are mainly produced in falling film reactors using SO . The reaction product contains (by weight) approximately 90% LABS, about 3—4% of unreacted SO and unreacted LAB. The prod uct is allowed to age until LABS content exceeds 95%, with SO and LAB contents about 1—2% each. The operation parameters are; the molar ratio between SO and LAB, the molar fraction of SO in the gas phase, and the temperature of the cooling water. The molar ratio is typically set about 1.1, with molar fractions below 4%. The quality parameters of a sulphonated/sulphated surfactant are its surfactant content (‘active matter’), its unreacted SO content, and its colour. Anionic surfac tants in aqueous solutions have colours ranging from yellow to reddish orange. The intensity of the colour has no relevance from the point of view of actual surfactant performance. However, lightly coloured products are preferred so that the household cleaning formulations they are incorporated into may have dyes added for marketing reasons. Low colour is indicative of good heat dissipation in the reactor, as there has been shown to exist a clear relationship between colour and temperature
0009-2509/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 4 9 7 - 7
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(Hurlbert et al., 1967; Haumer, 1970; Bromstrom, 1974). The latter authors found that for sulphonation runs in a batch reactor, an expression can be derived for colour as a function of a pseudo-kinetic equation. This expression related colour to the inverse of temperature and to sulphonate concentration. As colour intensity is linked to heat accumulation, it is also linked to reaction rate and therefore to mass transfer. This is important in reactions like sulphonation and sulphation, which are highly exothermic. From a design point of view, accurate prediction of the temperature reached within the reactor, or better of the exposure of the reaction mixture to high temperature is important. This is in order to predict/ avoid pyrolisis, surfactant colouring, effects on the solubility of SO , and the occurrence of side reactions like dioxane production in the sulphation of ethoxylated alcohol, etc. Modelling of heat and mass transfer should include at least the prediction of relative colour intensities amongst different operation conditions as a index of undesired side effects due to heat accumulation. There is no published literature about the source of colour although the existence of a light absorption maximum at about 420 nm for aqueous solutions of anionic surfactant suggests conjugated double bonds. The preferential solubility of the colouring substance in water after solvent extraction (a routine procedure for determination of unsulphonated fraction of the sulphonated product) indicates ionic or strongly polar nature. 1.2. Review of sulphonation reactor models The earlier model was proposed by Johnson and Crynes (1974). Their model assumed chemisorption of SO at the gas—liquid interface and perfect mixing within the liquid film. The model therefore assumed that mass transfer was controlled by the gas-phase resistance: higher conversions were predicted for higher SO molar frac tions in the gas phase, against experimental evidence (Hurlbert, 1967). Additionally, assuming chemisorption also meant that the model could not explain the existence of unreacted SO in the liquid phase, which is present in concentration too high to allow its explanation by entrainment of SO -containing gas droplets in the liquid film. Johnson and Crynes included a modification of the Gilliland—Sherwood (1934) equation: Sh"0.023 Re Sc
1.3. Microscopic mass balance: the Davis—Van Ouwerkerk—Venkatesh model The model by Davis et al. (1976) introduced the solution of the microscopic mass balances in the liquid film, which was an enhancement over the previous model. However, the model was based on the hypotheses of molecular diffusion in the liquid phase and chemisorption and provided similar results (i.e., smooth increase of conversion along the reactor and gas-phase-controlled mass transfer) to those predicted by Johnson and Crynes. 1.4. Eddy diffusivitiy models Gutierrez et al. (1988) and Dabir et al. (1996) incorporated the assumptions of wave-induced turbulent diffusivity (even in laminar liquid film flow) and physical adsorption (assuming ideal solubility of SO following Henry’s law) of SO in the liquid phase. Gutie´rrez et al. incorporated the Yih—Liu (1983) turbulent viscosivity model to predict both heat and mass transfer and the McCready—Hanratty (1984) correlation to predict gas—liquid heat and mass transfer coefficients. A set of parameters was introduced and adjusted to minimise the deviations from a data set for sulphonation of dodecylbenzene (DDB). Operational conditions covered a range of SO /DDB molar ratios from 1 to 1.15, and SO /N molar ratios from 0.04 to 0.12. These conditions are representative of the operating conditions used in the industry over the last few decades. The model was able to predict the extremely sharp increase in conversion taking place within the initial 10% length of the reactor (Fig. 1). It also predicted mass transfer to be initially controlled by resistance at the gas phase and later, due to the dramatic increase in the liquid viscosity (from about 0.05 to about 10 kg m\ s\) by resistance at the liquid phase. Finally, the model could also predict SO concentrations in the liquid phase, showing that it would accumulate
(1)
that involves the multiplying of the right-hand term by a factor of 2. The authors justified this factor as an expression of the influence of interfacial waviness on gas—liquid transfer. However, the original equation was derived from evaporation data of water and various organic liquids (including n-butanol, chlorobenzene and aniline) from wavy falling films. This inconsistency will be discussed later.
Fig. 1. An example of longitudinal conversion profile for sulphonation of DDB. Johnson and Crynes, Davis et al. and Dabir et al fail to predict the sharp rise in conversion at the beginning of the reactor.
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near the gas—liquid interface, where the local composition of the liquid was almost 100% dodecylbenzenesulphonic acid (DDBS).There were however two flaws in this model: first, the numerical solution of the differential equations had involved an arbitrarily defined uneven partition of the domain of the liquid phase. This increases the error introduced when approximating the differential equations by finite differences (Ka´lnay de Rivas, 1971). Second, the fitted parameters, which incorporated a compensation for the mathematical error of the model thus hiding it, precluded any analysis about the limitations imposed on the model by assuming spatially averaged film thickness. One of the fitting parameters increased the friction factor (calculated by means of the Blasius equation) and indirectly the heat and mass transfer coefficients in the gas phase. The other parameter reduced the heat and mass transfer coefficients. Dabir et al. (1996) developed another model and tested it against the Gutie´rrez data set, with the aim of solving the model’s apparent lack of transportability. It will be shown later that lack of transportability is unavoidable at the present state of the technique. Their model was based on the modified Gilliland—Sherwood proposed by Johnson and Crynes, the Lamourelle—Sandall (1972) equation for eddy diffusivity, and the Henstock—Hanratty (1976) correlation for the calculation of the friction factor. Yao and Sylvester (1987) had already pointed out that this last correlation was unreliable. In fact, Theofanous and Amarasooriya (1992) had tested several models proposed amongst others by Liles and Mahaffy (1984), Owen and Hewitt (1987) and Govan against an experimental pressure drop database. Predictions were consistently poor. Lack of transportability of existing correlations is an admitted limitation in the prediction of frictional drag in annular flow (Asali et al., 1985). Organic liquid films present two additional difficulties: first, the additional effect of surface tension. Talens et al. (1996) showed than this caused a strong drift between experimental and predicted value, due to the fact that existing correlations do not take into account surface tension except in calculation of liquid droplet entrainment and estimation of its effect on the apparent gas density. Second, most techniques (Alekseenko et al., 1994) are inadequate to provide data on film thickness and local velocity for organic compounds: This, in particular, disables models for interfacial friction based on the availability of experimental film thickness data (e.g. Persen, cited by Theofanous and Amarasooriya). In summary, not only do existing correlations lack transportability, but also most have been developed for water or aqueous solutions with properties very different from those of hydrocarbon compounds. Errors introduced in the calculation of shear stress by the use of correlations from the literature must be taken into account when modelling mass and heat transfer in annular flow, especially in reacting systems with radical
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Fig. 2. An example of the predictions of the Yih and Liu turbulent viscosity model. The liquid is assumed to be DDB at the top of the column, and the figures give the SO molar fraction in the gas phase, assuming 1 : 1 DDB/SO molar ratio and a 2.18;10\ kmol/s\ flow rate for both reagents in a 0.01388 m diameter reactor.
changes of properties: Van Driest-type turbulent viscosity models (like Lamourelle—Sandall and Yih—Liu) are strongly sensitive to shear stress. Fig. 2 shows predictions of turbulent viscosity from the Yih—Liu model. The liquid film is assumed to be unreacted DDB at the top of a reactor (0.01388 m internal diameter). Liquid flow rate of 2.18;10\ kmol s\. Turbulent viscosities are calculated for liquid circulating in annular flow in contact with two different SO /N mixtures, with the restriction that the DDB/SO molar ratio is 1, and only the carrier flow rate varies. The resulting gas flow rates are 5.45; 10\ kmol s\ (for a SO /N mixture containing 4% of SO ) and 1.82;10\ kmol s\. (for a SO /N mixture containing 12% SO ). The difference in turbulent viscosi ties near the interface is dramatic: the liquid film falling cocurrently with the 12% SO mixture is predicted to be almost laminar. The other is strongly turbulent near the gas—liquid interface. The correction proposed by Johnson and Crynes to the Gilliland and Sherwood equation introduces additional distortions. Interpreting their work with the Maxwell—Stefan theory (Taylor and Krishna, 1993) suggests that the factor of 2 is a Stefan drift factor: as DDBS diffuses from the interface towards the reactor wall, it drags along SO from the region near the interface into the liquid film. This inter-species drag accelerates the transfer rate as there is a species motion which did not exist in the experiments by Gilliland and Sherwood, being their films homogeneous in composition. Fig. 3 schematically shows this concept. A sublayer exists near the gas—liquid interface where major species have strong concentration gradients. Drag effects caused by the friction between molecules of different species are important there. Data from the model proposed in this paper (Represented by dots in Fig. 3), show that it is acceptable to assume linear variation of the concentrations in this
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sublayer near the gas—liquid interface. Therefore, a simplified Maxwell—Stefan analysis may be used to semiquantitatively discuss the weight of the Stefan drift factor on mass transfer within the reactor. Eqs. (2) and (3) give the incremental forms of the Maxwell—Stefan force balance for SO and DDBS, which
Let us consider both DDB and DDBS concentration profiles stationary (v "v "0), which is equiva"" 1 "" lent to Johnson and Crynes assumption of homogeneous liquid film from the point of view of the Maxwell—Stefan theory (i.e. no species motion means no species frictional drag). Eq. (4) is then simplified yielding Eq. (5).
v " 1-
Dx /xN #[xN Dx /(xN (D )/d)]/(xN /(D /d)#xN (D /d)) 1- 1- "" 1 1- "" 1 "" \"" 1 1- 1-\"" 1 "" "" \"" 1 , [xN xN /(D /d)]/(xN /(D /d)#xN /(D /d))!(xN /(D /d)#xN /(D /d)) "" 1 1- 1-\"" 1 1- 1-\"" 1 "" "" \"" 1 "" 1-\"" "" 1 1-\"" 1 (4) are acceptable in many engineering problems (Wesselingh and Krishna, 1990). Convective effects are ignored, as these should be incorporated later into the flux calculations. Dx 1-"xN "" xN 1-
v !v v !v "" 1- #xN "" 1 1- , "" 1 (D (D )/d )/d "" \1- "" \1-
(2)
Dx v !v "" 1"xN 1- "" 1 #xN 1- (D "" xN )/d "" 1 "" 1\1-
v !v "" "" 1 . (D )/d "" 1\"" (3)
The term including x can be neglected on the right1- hand side of Eq. (3). The value is small, and therefore the friction between SO and DDBS molecules has little effect on the force balance for DDBS. Eqs. (2) and (3) yield Eq. (4) if v "0 is assumed (no DDB is leaving "" the film). This is an acceptable approximation to the behaviour of the reactor. Eq. (4) gives the Maxwell— Stefan velocity for SO within the liquid film, in a system where DDBS is both diffusing towards the reactor wall and being generated through the film, and DDB is effectively being depleted through the film, with its motion virtually unnoticeable.
Fig. 3. The dots represent data from a simulation run with a SO 0.1 molar fraction in the gas carrier, at 0.425 m of the top of the column, with the same SO and DDB molar ratios.
1 v " . 1- xN /(D )/d#xN /(D )/d "" 1-\"" "" 1 1-\"" 1
(5)
The simulation results are shown in Table 1. The three leftmost columns show the species velocities predicted by Eqs. (4) (a general Maxwell—Stefan prediction) and Eq. (5) (a Johnson—Crynes restricted prediction) as well as their ratio. SO velocity is approximately 5 times larger con sidering inter-species drag than under the assumptions made by Johnson and Crynes. Hence the need for a correction factor in the Gilliland and Sherwood correlation to adjust the predictions to their experimental results. Johnson and Crynes gave an averaged value for all the reactor length and for all their operational conditions, and the estimate of the ratio between Eqs. (4) and (5) (equivalent to the correction factor by Johnson and Crynes) given in the third column of Table 1 has been calculated very crudely, but even so the results of the calculations are in good qualitative agreement. These distortions had a clear effect on the predictions. The database produced by Gutie´rrez included conversion values at a distance of 0.4, 0.975 and 2.0 m, obtained with reactors of different length and assuming negligible exit effects. This provides experimental data against which predicted axial conversion profiles can be compared. The Dabir model under-predicted (deviations between 8% to 18%) conversions at 0.4 m of the top of the reactor, but gave reasonable agreemnent with experimental data at 0.975 m (within 10% and mostly within 5%), and over-predicted conversion at 2.0 m (deviations between 4% and 18%) from the top of the reactor. Their predicted value of 335 K for the interfacial temperature peak was also poor. Industrially run film pilot reactors reach an average 370 K in the liquid film. Gutie´rrez’s model had predicted interfacial liquid temperatures to be about 390 K, which is more realistic. The failure in the prediction of interfacial film temperature is the inevitable consequence of low initial conversion predictions: a slower increase in conversion at the top of the reactor results in reaction heat being released more evenly along the reactor and being dissipated more readily. This effect is beneficial in suppressing undesirable side reactions as mentioned above. It is the basis of the
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F.I. Talens-Alesson/Chemical Engineering Science 54 (1999) 1871—1881 Table 1 M—S species velocities, calculated for the hypothesis of (1) non-homogenenous film composition and (2) homogeneous film composition % SO
d (m)
X 1-
Xd 1-
X ""
Xd ""
d (10— m)
» — +1 10\ m s\
»— (! 10— m s\
» — /» — +1 (!
08 0.08 0.08 0.1 0.1 0.1 0.12 0.12 0.12 0.12
0.17 0.42 1.39 0.17 0.42 1.39 0.06 0.17 0.42 1.39
0 0 0 0 0 0 0 0 0 0
0.004 0.011 0.013 0.009 0.017 0.018 0.001 0.014 0.023 0.023
0.523 0.451 0.316 0.571 0.476 0.338 0.711 0.618 0.504 0.364
0.085 0.050 0.029 0.053 0.032 0.017 0.071 0.039 0.022 0.011
4.9 6.6 7.6 5.7 6.7 8.0 3.5 5.5 6.5 8.5
15.2 11.5 9.3 14.8 12.2 9.6 26.8 16.4 13.2 9.5
3.49 2.44 1.91 3.04 2.39 1.83 5.57 3.21 2.51 1.76
4.36 4.73 4.88 4.86 5.07 5.21 4.81 5.11 5.27 5.38
Table 2 Dynamic viscosity, density and surface tension of typical sulphonation/sulphation organic reagents, as a function of temperature
k (kg m\ s\) o (kg m\) p (N m\)
Dodecylbenzene
Lauryl alcohol
Lauryl alcohol 2OE
2.72;10\ exp(2980/¹) 975—0.504 T 0.026
1.968;10\ exp(2667/¹) 1056—0.66 T 0.028
1.61;10\ exp(2790/¹) 1006—0.329 T 0.038
variable cross section falling film reactor suggested by Talens et al. (1996b).
2. Semiempirical approach to the design of falling film reactors 2.1. Prediction of gas—liquid frictional drag As already stated, accurate prediction of friction factors is a bottleneck in the modelling process. As an alternative to the fitted Blasius equation proposed by Gutie´rrez, Talens (1991) proposed a modification of a explicit Colebrook equation (Zygrang and Sylvester, 1982):
1 e/d 2.51 e/d 13 "!4 log ! log # C 3.7 Re 3.7 Re D
,
Fig. 4. Apparent friction factors for dodecanol, diethoxylated dodecanol and DDB. Even though dodecanol 2OE is more viscid than DDB, it builds up resistance to frictional drag more slowly due to its higher surface tension.
(6)
where the film roughness e, was replaced by Ud. U is a function of the interfacial liquid velocity and d is the film thickness, which sometimes is directly taken as the value of the roughness. U was derived from experimental pressure drop data in annular flow taken in the same reactor originally used by Gutie´rrez for a set of three organic liquids (dodecylbenzene, dodecanol, and dodecanol 2OE, technical grade, supplied by KAO Corporation, Barbera´ del Valle´s, Spain) whose properties (Bromstrom, 1974; Talens et al., 1994) are given in Table 2. Assuming laminar film flow, the film thickness, shear stress, interfacial velocity and apparent friction factor were calculated by means of the
set of equations given in the Appendix. Dodecylbenzene and dodecanol had very similar surface tensions (0.026 and 0.028 N/m, respectively) while the dodecanol 2OE had a surface tension of 0.038 N/m. The estimated friction factors (Fig. 4) for the fluids with similar surface tension have the same general shape, with a maximum about the same Re . Gutierrez¨s fitting factors assume % that the friction factors along the reactor are 14 times higher than those given by Blasius equation for a smooth wall. Fig. 4 shows that this overestimates friction factors for Dodecylbenzene (white circles) by a factor of 2, and there is even more of an overestimation below Re " % 6000. For dodecanol 2OE (with a less deformable
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interface) resistance shows to build up more slowly. The existence of conditions for the onset of large disturbance waves (responsible for strong gas—liquid interactions) and of threshold wave heights required to cause additional roughness are described in the literature (Hewitt and Hall—Taylor, 1970; Andreussi and Zanelli, 1982; Takahama et al., 1983; Okada and Fujita, 1993). These facts can explain why there is a sharp increase in roughness, as the onset of large waves is an abrupt condition, and why the interface with the higher surface tension (less deformable) does not build up additional resistance so quickly. The data for dodecanol 2OE were discarded, and the correlation assumes a liquid surface tension of about 0.027 N/m. The friction factors were substituted in the left-hand side of the Zygrang and Sylvester equation, and the calculated film thickness was substituted in the righthand side of the equation. Then the apparent roughness factor U was calculated by iteration to fit the equation for each set of values. The results can be seen in Fig. 5. It must be stressed that this correlation assumes that pure dodecanol or dodecylbenzene and their mixtures with their sulphonation and sulphation product have approximately the same surface tension: in order for LABS to have surfactant properties in organic phase it must be forming a salt like, for instance, its Calcium or Isopropylamine salts (Porter, 1994). Dispersion is strong and regression coefficients are poor, but the numerical agreement with actual pressure drop data is better than with the Gutierrez ‘black-box’ fit. The following equations are represented as solid lines in Fig. 5: ¸n(U)"3.59!5.1» , (7) '* ¸n(U)"20.55» !0.93. (8) '* Eq. (7) is valid for interfacial liquid velocities above 0.175 m s\, and Eq. (8) is valid below that value. The contradiction in taking into account turbulent viscosity and assuming parabolic velocity profiles is easily justified
Fig. 6. Kinematic viscosity of DDB is about 5;10\ m s\, larger than the values of turbulent viscosity. However, the associated turbulent diffusivity is high when compared with molecular diffusivity.
for the systems and conditions involved in this particular case. While eddy viscosity is important enough to cause non-negligible turbulent diffusivities (Fig. 6), its value is still far smaller than the kinematic viscosity of the liquid, and therefore laminar velocity profiles can be assumed without significant error. 2.2. Heat and mass transfer equations The complete set of Eqs. [(A.1)—(A.14)] is listed in the Appendix together with details on the linearisation procedure. Essentially it includes the Yih and Liu (1983) turbulent viscosity model and its expansion for the calculation of turbulent diffusivities, using the MCCready— Hanratty equation for the gas—phase transfer coefficient. 2.3. Colour intensity It is possible to estimate the relative colour intensity (or the likelihood of by-products affected by higher temperatures) in different runs for the same reaction system under different conditions by solving the following integral (Talens et al., 1993): * B
e 2C1 dr dz.
CI"
Fig. 5. The increased roughness vs the interfacial liquid velocity. Surface tension has a strong effect (Talens et al., 1996b).
(9)
This is done after the model has been solved and composition and temperature profiles over the length and width of the liquid film are available. Eq. (9) assumes that colour is formed in a first-order kinetic equation affecting the sulphonate, as suggested by Bromstro¨m (1974). For a given reaction, with constant frequency factor and activation energy, the integral CI will be proportional to the integration of the differential equation for the first order kinetic equation. Therefore, comparison between
F.I. Talens-Alesson/Chemical Engineering Science 54 (1999) 1871—1881
Fig. 7. Colour Integral as a function of SO fraction in the gas. The relative colour intensity predicted by the model is in agreement with observations by the industrialists: a strong decrease in colour is observed as the carrier gas provides stronger shear stress.
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Fig. 8. The a priori model presented here predicts the sharp increase in conversion, and the shift between control of mass transfer by the gas phase and control by the liquid phase.
the value CI for different runs will show under which conditions heat-induced side reactions are more likely (Fig. 7).
3. Model results The model retains the ability to predict sharp increases in conversion at the top of the reactor (Fig. 8). Higher conversions are initially linked to higher SO molar fraction in the gas (lower overall gas flow rate) while later there is a shift and the higher conversion is finally achieved with the lower SO molar fraction in the gas (higher overall gas flow rate). This means that at the top of the reactor mass transfer is controlled by the gas phase, and that in the lower part of the reactor it is the liquid phase which controls the mass transfer: the higher the turbulence induced by the gas on the liquid, the higher the amount of SO transferred into the film and the higher the conversion. Fig. 9 shows schematically the fast conversion region at the top of the reactor (linked with gas-phase control) and the slow conversion region at the bottom (linked with liquid-phase control). Fig. 10 shows a simulation at a given reactor length (0.425 m). Depletion of DDB near the interface is faster when the gas flow rate is lower. As turbulence is lower, convective effects are less important. Notice that the profiles in the sub-layer near the gas—liquid interface are almost linear. This linearity is also the case in Fig. 11. SO molar fraction in the gas is 4%. The effect between 0.175 and 0.425 m is interesting and indicative of the change in phase control on mass transfer: The viscosity increases with conversion and distance from the top of the reactor, and diffusion of DDB towards the interface becomes relevant as controlling step of the reaction. Figs. 10 and 11 confirm that turbulent diffusivity values are significant even though turbulent viscosity
Fig. 9. Schematic view of the falling film reactor reactor.
values are several orders of magnitude lower than the kinematic viscosity of DDB. The flat portion of the solid curve to the right-hand side gives the molecular diffusivity calculated by the Wilke—Chang equation. The right side gives the turbulent diffusivities obtained from the Yih—Liu and Cebeci equations. Fig. 12 gives some radial temperature profiles at different distances from the top of the reactor. The almost flat profiles indicate that heat removal at the reactor wall is a crucial factor. A strong initial temperature increase is observed. The temperature is initially predicted to be almost 400 K. Later, as differential conversion along the reactor becomes near zero, the temperature in the film almost equals that in the cooling jacket (298 K).
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Fig. 10. The height for this simulation of radial profiles is 0.425 m. Higher overall carrier gas rate leads to stronger renewal of the unreacted DDB near the gas—liquid interface. Note that in all cases there is a sublayer with almost linear variation of DDB concentration.
Fig. 12. Radial temperature profiles. The temperature increases strongly at the beginning, and afterwards heat dissipates faster.
Fig. 13. The model gives better predictions at higher shear stress. Predictions are poorer for lower shear stress, and this seems to be the essential defect of spatially averaged film thickness models. Fig. 11. The effect of conversion and associated increase of viscosity on the radial profiles, calculated at the local composition and temperature for each distance from the interface.
Fig. 13 leads to an interesting reflection on the limits of spatially averaged film thickness models. At higher shear stress (lower SO molar fractions) there is a better agree ment between experimental and predicted conversions. In practice, the Yih and Liu model predicts higher or lower turbulence near the gas—liquid interface as a consequence of gas shear stress, but not a radical change in the depth of the turbulent zone. However, actual films consist of a substrate film and waves, and changes in shear stress modify the thickness of the substrate film (Nencini and Andreussi, 1982). The figure suggests that at lower shear stress the fraction of liquid in the substrate film (and under non-turbulent flow conditions) has increased significantly and the result of this is that conversion is much poorer. On the other hand, for highly sheared films, a spatially averaged film thickness model seems to
behave well. However, the case studied is a A#B gives C reaction, with no significant parallel or subsequent reactions, and it is reasonable to assume that predictions from spatially averaged film thickness models will be less accurate when dealing with more complex reactions, such as polymerisation.
4. Conclusions The main conclusions which can be drawn, from the point of view of modelling strategies, are: (1) The Johnson and Crynes modification to the Gilliland and Sherwood correlation should be avoided. Using it would cause convective effects to be accounted for twice in the model: once in the turbulent diffusivity coefficients, and again implicitly in the Johnson and Crynes correlation.
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(2) Correlations for frictional drag between gas and liquid should be obtained for each specific reactor. Both general transportability problems of existing correlations and technical obstacles for data acquisition in organic films make advisable not to use equations from the literature, which are essentially derived for aqueous films. (3) A reasonable approach could be the one describe here, where interfacial effects are associated with a roughness factor. The user should pay attention to the fact that spatially averaged film thickness is an assumption, which seems to work better at high gas—liquid shear stresses.
Notation C D CI d D D R h % DH 0 K % K * k u
friction factor colour integral, dimensionless column inner diameter, m diffusivity coefficient at infinite dilution, m s\ turbulent diffusivity heat transfer coefficient, J m\ s\ reaction enthalpy, J kmol\ overall mass transfer coefficient, kmol m\ s\ liquid thermal conductivity, J s\ m\ K\ reaction kinetic coefficient, s\ average gas velocity in the liquid-free reactor core, m s\ u characteristic turbulence velocity, m s\, v species diffusion velocity, m s\ v interfacial liquid velocity, m s\ '* x molar fraction x average molar fraction y> non-dimensional distance to the wall: y(q g/o) /l U Pr Prandtl number Re gas Reynolds number: ou(d!2d) k\ Sc, Sc Schmidt number, turbulent Schmidt number R C mass flow rate per unit periphery, kg m\ s\ d film thickness, m e roughness, m k viscosity, kg m\ s\ l, l kinematic viscosity, kinematic turbulent viscos# ity, m s\ o liquid density, kg m\ * q ,q shear stress at the gas—liquid interface, shear % 5 stress at the wall U roughness enhancement factor
Acknowledgements The author wishes to acknowledge the insight gained through his co-operation with Prof. B.J. Azzopardi, Dr. J. Hills, and Mr. W. Clark, from the University of Nottingham, UK, as an advisor to EPSRC. Project GR/K
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77044. The author also wishes to mention the involvement of Drs. S. Esplugas-Vidal, C. Mans-Teixido, and J. Costa-Lo´pez, in his redundancy from the Department of Chemical Engineering of the University of Barcelona, Spain. References Agrawal, A.K., & Peckoever, R.S. (1980). Nonuniform grid generation for boundary-layer problems, Comput. Phys. Commun., 19, 171—178. Alekseenko, S.V., Nakoyakov, V.E., & Pokusave, B.G. (1994). ¼ave flow of liquid films, New York: Begell House. Asali, J.C., Hanratty, T.J., & Andreussi, P. (1985). Interfacial Drag and film height for vertical annular flow. A.I.Ch.E. J., 31, 895—902. Bromstro¨m, A. (1974). Rate of color formation in sulfonation of dodecylbenzene with gaseous surfur trioxide. JAOCS, 51, 507—508. Bromstro¨m, A. (1975). Mathematical model for simulating the sulphonation of dodecylbenzene with gaseous sulphur trioxide in an industrial reactor of Votator type. ¹rans. Instn. Chem. Engrs., 53, 29—33. Dabir, B., Riazi, M.R., & Davoudirad, H.R. (1996). Modelling of Falling Film Reactors. Chem. Engng. Sci., 51, 2553—2558. Davis, E.J., Van Ouwerkerk, M., & Venkatesh, S. (1979). An Analysis of the falling film gas—liquid reactor. Chem. Engng Sci., 34, 539—550. Dicoi, O., & Canavas, C. (1993). German Patent DE 41 28 827 A 1. Gilliland, E.R., & Sherwood, T.K. (1934). Diffusion of vapors into air streams. Ind. Engng Chem., 26, 516—523. Gutie´rrez, J., Mans, C., & Costa, J. (1988). Improved mathematical model for a falling film sulfonation reactor. Ind. Engng Chem. Res., 27, 1701—1707. Haumer, J. (1970). Determination of optimal conditions for sulfonation of alkylbenzenes with SO gas (Czech). »eda »yzk. Potravin. Prum., 21, 115—137. Henstock, W.H., & Hanratty, T.J. (1976). The Interfacial drag and the height of the wall layer in annular flows. A.I.Ch.E. J., 22, 990—1000. Hewitt, G.F., & Hall-Taylor, N.S. (1990). Annular two-phase flow, Oxford: Pergamon Press. Hurlbert, R.C., Knott, R.F., & Cheney, H.A. (1967). Apparatus for small scale sulfonation with SO . Soap Chemi. Specialties, 88—94 and 100. Johnson, G.R., & Crynes, B.L. (1974). Modelling of a thin-film sulfur trioxide sulfonation reactor. Ind. Engng Chem. Process Des. Dev., 13, 6—14. Ka´lnay de Rivas, E. (1972). On the use of nonuniform grids in finitedifference equations. J. Comput. Phys., 10, 202—210. Lamourelle, A.P., & Sandall, O.C. (1972). Gas absorption into a turbulent liquid. Chem. Engng Sci., 27, 1035—1043. Lapidus (1962). Digital computations for chemical engineers. New York: McGraw-Hill. Liles, D.R., & Mahaffy, J.H. (1984). ¹RAC-PF1/MOD1 An advanced best estimate computer program for pressurised water reactor thermal-hydraulic analysis. Los Alamos National Laboratory Report. Mann, R., Knysh, P., & Allan, J.C. (1982). Exothermic gas absorption with complex reaction: sulfonation and discoloration in the absorption of sulfur trioxide in dodecylbenzene. In J. Wei & K.C. Georgakis (Eds.), ACS Simposium series 196, (pp. 441—456). McCready, M.J., & Hanratty, T.J. (1984). Gas transfer at water surfaces, (Vol. 283). D. Reidel Dordrecht, Holland: Publishing Company. Nencini, F., & Andreussi, P. (1982). Studies of the behavior of disturbance waves in annular two-phase flow. Can. J. Chem. Engng, 60, 459—465. Okada, O., & Fujita, H. (1993). Behavior of liquid films and droplets in the non-equilibrium region of a downward annular mist flow (comparison of porous and central nozzle mixing methods). Int. J. Multiphase Flow, 19, 79—89.
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Owen, D.F.G., & Hewitt, G.F. (1987). An improved annular two-phase flow model. 3rd International Conference on Multiphase Flow (BHRA), The Hague. Porter, M.R. (1994). Handbook of surfactants (pp. 138—143). Bury St. Edmunds: Blackie Academic and Professional. Takahama, H., Okada, O., Fujita, H., & Mizuno, A. (1983). Study on annular mist flow in pipe (2nd. report, Behavior of water film in the non-equilibrium region of downward annular mist flow with low water flow rate). Bull. HSME, 26, 2091—2099. Talens, F.I. (1991). Fluid dynamics and transport in wetted wall columns (Spa.) Ph.D. thesis, University of Barcelona, Spain. Talens, F.I., Gutie´rrez, J.M., & Mans, C. (1993). Prediction of colour and other quality parameters in falling film reactors. ¹enside, 30, 331—335. Talens, F.I., Gutie´rrez, J.M., & Mans, C. (1994). Effect of surface tension and viscosity on the efficiency in wetted wall reactors. ¹enside, 31, 6—8. Talens, F.I., Hreczuch, W., Bekierz, G., & Szymanowski, J. (1987). Ethoxylation of alcohols in falling film reactor, J. Disp. Sci. ¹echnol., 18, 423—433. Talens, F.I., Pereira, G., Chenlo, F., & Vazquez, G. (1996a). Device for research into falling film wave structures under shear stress. A.I.Ch.E. J., 42, 3293—3295. Talens, F.I., Gutie´rrez, J., & Mans, C. (1996b). An improved falling-film reactor for viscous liquids. JAOCS, 73, 857—862. Taylor, R., & Krishna, R. (1993). Multicomponent mass transfer. New York: Wiley. Theofanous, T.G., & Amarasooriya, W.H. (1992). Physical benchmarking exercise. In: Hewitt, G.F., Delhaye, J.M., & Zuber, N. (Eds.), Multiphase science and technology (Vol. 6, pp. 2—12). Washington, DC: Hemisphere. Wesselingh, J.A., & Krishna, R. (1990). Mass transfer. Chichester: Ellis Hurwood. Yao, S.C., & Sylvester, N.D. (1987). A Mechanistic model of two-phase annular-mist flow in vertical pipes. A.I.Ch.E. J., 33, 1008—1012. Yih, S., & Liu, J. (1982). Prediction of heat transfer in turbulent falling liquid films with or without interfacial shear. A.I.Ch.E. J., 29, 903—909. Zygrang, D.J., & Sylvester, N.D. (1982). Explicit approximations to the solution of Colebrook’s friction factor equation. A.I.Ch.E. J., 28, 514—515.
where q is calculated from experimental pressure drop % data, it is possible to obtain a value of the correction factor U which can be correlated to an estimated interfacial velocity
Appendix A
B>"Sc\ C (log Sc)G\ (A.10) G G with C "34.96, C "28.97, C "13.95, C "6.33, C "!1.186 Mass transport equations
A.1. Calculation of average film thickness, roughness correction factor average interfacial liquid velocity The initial value for the iterative calculation of the film thickness d can be obtained from
3Ck . g o ! * By combining d"
(A.1)
(A.4)
A.2. Simulation model Mass transfer coefficient in the gas phase is taken as (McCready and Hanratty, 1984) k %"0.8 Sc\ , (A.5) u where the turbulent velocity u is defined as q u " % . (A.6) o % Turbulent viscosity in the liquid phase (Yih and Liu, 1982):
l q #"!0.5#.5 1#.64 (y>) l q 5 !y>(q/q ) 5 ; 1!exp A>
,
(A.2)
and
1 (Ud)/d 2;2.51 (Ud)/d 13 "!4 log ! log # C 3.7 Re 3.7 Re D
,
(A.3)
(A.7)
q q * (y>/d>) (A.8) "1! q q #q 5 % * Turbulent diffusivity in the liquid phase follows the Cebeci equation (see Yih and Liu, 1982) l 1!exp (!y>(q/q ) /A>) 5 Sc " R " R D 1!exp (!y>(q/q ) /B>) R 5 with A>"25.1 and
l 8
(A.9)
*C * *C "" " (D #D ) "" !kC C , (A.11) "" R *y "" 1- *z *y
*C * *C 1-" l (D #D ) 1- !kC C . 8 *z 1R "" 1- *y *y
o g d q d C" * ! ! % 3k 2k
g y q y v " ! dy! # % . '* l 2 k
(A.12)
The equations are solved considering that the curvature of the film is negligible, and that radial symmetry exists. The solution involves linearisation by finite differences using the Laasonen implicit forms for the first- and second-order derivatives in Eqs. (16) and (17) (Lapidus, 1962). This guarantees that the solution is always convergent. The assumption of linear behaviour may be
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F.I. Talens-Alesson/Chemical Engineering Science 54 (1999) 1871—1881
a source of error. To obtain a better result the computational grid was made non-uniform by using a transformation function which converts a coordinate a made non-dimensional by dividing the coordinate x by the magnitude ¸ of the dimension in the domain into a variable q. This guarantees that no additional errors are introduced in the calculations. The transformation equation chosen (Eiseman, cited by Agrawal and Peckover, 1980) was q"0.5(atanh ma/atanhm)#0.5a. The regrouping parameter m, which must be between 0 and 1, was taken 0.001 in the radial direction and 0.9999 in the axial direction after considering the stability of the equations. This particular form of Eiseman transformation equation ensures that the reorganisation of the computational grid does not leave an unacceptably low number of points in the domain region where variations of the
properties are mild. The resulting linear equations are solved by the Thomas method (Lapidus, 1962). A.3. Heat transport equations The heat transport equations follow the Prandtl analogy, and are equivalent to the ones used for mass transfer. v 8
*oC ¹ * *¹ N " !K #(DH) kC C , (A.13) * 0 "" 1- *z *y *y
h %"8 Pr\ . (A.14) u The model is solved for dodecylbenzenesulphonic acid (DDBS), SO , and ¹. Dodecylbenzene (DDB) is cal culated from the overall mass balance.
The modelling of falling film chemical reactors Federico I. Talens-Alesson* TALENCO Chemical Engineering Consulting, P.O. Box 1035, 08902 Hospitalet de Llobregat, Spain Received 5 January 1998; received in revised form 4 December 1998; accepted 5 December 1998
Abstract Spatially averaged film thickness models have been the preferred approach for simulation of falling film reactors. This paper will discuss both the general and specific limitations of available models. An improved model is proposed, which has been found to perform well in industrially relevant conditions. For its specific application to sulphonation/sulphation reactions yielding anionic surfactants, the improved model provides enhanced predictions about reactor performance, including relative colour intensity at different operational conditions. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Falling; Film; Multiphase; Sulphonation; Modelling; Friction
1. Introduction Falling film reactors are jacketed columns or clusters of columns where liquid and gas flow in annular or annular-mist flow. The gas phase is a mixture of a gaseous or vapourised reagent diluted with an inert carrier. These reactors display outstanding heat removal capability, with the mix between reagents leading to milder reaction conditions than in stirred batch reactors. For this reason the substitution of batch processes by continuous processes in film reactors, in order to improve the reaction progress and safety conditions, has become a recent trend in the chemical industry. Examples of which can be seen in both ethoxylation (Dicoi and Canavas, 1993; Talens et al., 1998) or hydrogenation. This recent move towards film reactors as substitutes to batch reactors, combined with the industrial relevance of conventional applications of falling film reactors (sulphonation, chlorination), justifies the necessity and importance of their accurate mathematical modelling. Sulphonation of linear alkylbenzenes with SO is a classical study case for mathematical modelling, as the physical properties of the liquid phase (Bromstro¨m, 1975) and reaction kinetics and thermodynamics (Mann et al., 1982) are well characterised. Some facts about the reac-
* Tel.: 0034 3 3311 811; fax: 0034 3 33 11 811; e-mail: f — [email protected].
tion and its product will be discussed next to highlight the relevance of the model results. 1.1. Alkylbenzene sulphonation Sulphonation of linear alkylbenzenes (LAB) is a typical reaction for the synthesis of anionic surfactants. Linear alkylbenzene sulphonates (LABS) are mainly produced in falling film reactors using SO . The reaction product contains (by weight) approximately 90% LABS, about 3—4% of unreacted SO and unreacted LAB. The prod uct is allowed to age until LABS content exceeds 95%, with SO and LAB contents about 1—2% each. The operation parameters are; the molar ratio between SO and LAB, the molar fraction of SO in the gas phase, and the temperature of the cooling water. The molar ratio is typically set about 1.1, with molar fractions below 4%. The quality parameters of a sulphonated/sulphated surfactant are its surfactant content (‘active matter’), its unreacted SO content, and its colour. Anionic surfac tants in aqueous solutions have colours ranging from yellow to reddish orange. The intensity of the colour has no relevance from the point of view of actual surfactant performance. However, lightly coloured products are preferred so that the household cleaning formulations they are incorporated into may have dyes added for marketing reasons. Low colour is indicative of good heat dissipation in the reactor, as there has been shown to exist a clear relationship between colour and temperature
0009-2509/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 4 9 7 - 7
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(Hurlbert et al., 1967; Haumer, 1970; Bromstrom, 1974). The latter authors found that for sulphonation runs in a batch reactor, an expression can be derived for colour as a function of a pseudo-kinetic equation. This expression related colour to the inverse of temperature and to sulphonate concentration. As colour intensity is linked to heat accumulation, it is also linked to reaction rate and therefore to mass transfer. This is important in reactions like sulphonation and sulphation, which are highly exothermic. From a design point of view, accurate prediction of the temperature reached within the reactor, or better of the exposure of the reaction mixture to high temperature is important. This is in order to predict/ avoid pyrolisis, surfactant colouring, effects on the solubility of SO , and the occurrence of side reactions like dioxane production in the sulphation of ethoxylated alcohol, etc. Modelling of heat and mass transfer should include at least the prediction of relative colour intensities amongst different operation conditions as a index of undesired side effects due to heat accumulation. There is no published literature about the source of colour although the existence of a light absorption maximum at about 420 nm for aqueous solutions of anionic surfactant suggests conjugated double bonds. The preferential solubility of the colouring substance in water after solvent extraction (a routine procedure for determination of unsulphonated fraction of the sulphonated product) indicates ionic or strongly polar nature. 1.2. Review of sulphonation reactor models The earlier model was proposed by Johnson and Crynes (1974). Their model assumed chemisorption of SO at the gas—liquid interface and perfect mixing within the liquid film. The model therefore assumed that mass transfer was controlled by the gas-phase resistance: higher conversions were predicted for higher SO molar frac tions in the gas phase, against experimental evidence (Hurlbert, 1967). Additionally, assuming chemisorption also meant that the model could not explain the existence of unreacted SO in the liquid phase, which is present in concentration too high to allow its explanation by entrainment of SO -containing gas droplets in the liquid film. Johnson and Crynes included a modification of the Gilliland—Sherwood (1934) equation: Sh"0.023 Re Sc
1.3. Microscopic mass balance: the Davis—Van Ouwerkerk—Venkatesh model The model by Davis et al. (1976) introduced the solution of the microscopic mass balances in the liquid film, which was an enhancement over the previous model. However, the model was based on the hypotheses of molecular diffusion in the liquid phase and chemisorption and provided similar results (i.e., smooth increase of conversion along the reactor and gas-phase-controlled mass transfer) to those predicted by Johnson and Crynes. 1.4. Eddy diffusivitiy models Gutierrez et al. (1988) and Dabir et al. (1996) incorporated the assumptions of wave-induced turbulent diffusivity (even in laminar liquid film flow) and physical adsorption (assuming ideal solubility of SO following Henry’s law) of SO in the liquid phase. Gutie´rrez et al. incorporated the Yih—Liu (1983) turbulent viscosivity model to predict both heat and mass transfer and the McCready—Hanratty (1984) correlation to predict gas—liquid heat and mass transfer coefficients. A set of parameters was introduced and adjusted to minimise the deviations from a data set for sulphonation of dodecylbenzene (DDB). Operational conditions covered a range of SO /DDB molar ratios from 1 to 1.15, and SO /N molar ratios from 0.04 to 0.12. These conditions are representative of the operating conditions used in the industry over the last few decades. The model was able to predict the extremely sharp increase in conversion taking place within the initial 10% length of the reactor (Fig. 1). It also predicted mass transfer to be initially controlled by resistance at the gas phase and later, due to the dramatic increase in the liquid viscosity (from about 0.05 to about 10 kg m\ s\) by resistance at the liquid phase. Finally, the model could also predict SO concentrations in the liquid phase, showing that it would accumulate
(1)
that involves the multiplying of the right-hand term by a factor of 2. The authors justified this factor as an expression of the influence of interfacial waviness on gas—liquid transfer. However, the original equation was derived from evaporation data of water and various organic liquids (including n-butanol, chlorobenzene and aniline) from wavy falling films. This inconsistency will be discussed later.
Fig. 1. An example of longitudinal conversion profile for sulphonation of DDB. Johnson and Crynes, Davis et al. and Dabir et al fail to predict the sharp rise in conversion at the beginning of the reactor.
F.I. Talens-Alesson/Chemical Engineering Science 54 (1999) 1871—1881
near the gas—liquid interface, where the local composition of the liquid was almost 100% dodecylbenzenesulphonic acid (DDBS).There were however two flaws in this model: first, the numerical solution of the differential equations had involved an arbitrarily defined uneven partition of the domain of the liquid phase. This increases the error introduced when approximating the differential equations by finite differences (Ka´lnay de Rivas, 1971). Second, the fitted parameters, which incorporated a compensation for the mathematical error of the model thus hiding it, precluded any analysis about the limitations imposed on the model by assuming spatially averaged film thickness. One of the fitting parameters increased the friction factor (calculated by means of the Blasius equation) and indirectly the heat and mass transfer coefficients in the gas phase. The other parameter reduced the heat and mass transfer coefficients. Dabir et al. (1996) developed another model and tested it against the Gutie´rrez data set, with the aim of solving the model’s apparent lack of transportability. It will be shown later that lack of transportability is unavoidable at the present state of the technique. Their model was based on the modified Gilliland—Sherwood proposed by Johnson and Crynes, the Lamourelle—Sandall (1972) equation for eddy diffusivity, and the Henstock—Hanratty (1976) correlation for the calculation of the friction factor. Yao and Sylvester (1987) had already pointed out that this last correlation was unreliable. In fact, Theofanous and Amarasooriya (1992) had tested several models proposed amongst others by Liles and Mahaffy (1984), Owen and Hewitt (1987) and Govan against an experimental pressure drop database. Predictions were consistently poor. Lack of transportability of existing correlations is an admitted limitation in the prediction of frictional drag in annular flow (Asali et al., 1985). Organic liquid films present two additional difficulties: first, the additional effect of surface tension. Talens et al. (1996) showed than this caused a strong drift between experimental and predicted value, due to the fact that existing correlations do not take into account surface tension except in calculation of liquid droplet entrainment and estimation of its effect on the apparent gas density. Second, most techniques (Alekseenko et al., 1994) are inadequate to provide data on film thickness and local velocity for organic compounds: This, in particular, disables models for interfacial friction based on the availability of experimental film thickness data (e.g. Persen, cited by Theofanous and Amarasooriya). In summary, not only do existing correlations lack transportability, but also most have been developed for water or aqueous solutions with properties very different from those of hydrocarbon compounds. Errors introduced in the calculation of shear stress by the use of correlations from the literature must be taken into account when modelling mass and heat transfer in annular flow, especially in reacting systems with radical
1873
Fig. 2. An example of the predictions of the Yih and Liu turbulent viscosity model. The liquid is assumed to be DDB at the top of the column, and the figures give the SO molar fraction in the gas phase, assuming 1 : 1 DDB/SO molar ratio and a 2.18;10\ kmol/s\ flow rate for both reagents in a 0.01388 m diameter reactor.
changes of properties: Van Driest-type turbulent viscosity models (like Lamourelle—Sandall and Yih—Liu) are strongly sensitive to shear stress. Fig. 2 shows predictions of turbulent viscosity from the Yih—Liu model. The liquid film is assumed to be unreacted DDB at the top of a reactor (0.01388 m internal diameter). Liquid flow rate of 2.18;10\ kmol s\. Turbulent viscosities are calculated for liquid circulating in annular flow in contact with two different SO /N mixtures, with the restriction that the DDB/SO molar ratio is 1, and only the carrier flow rate varies. The resulting gas flow rates are 5.45; 10\ kmol s\ (for a SO /N mixture containing 4% of SO ) and 1.82;10\ kmol s\. (for a SO /N mixture containing 12% SO ). The difference in turbulent viscosi ties near the interface is dramatic: the liquid film falling cocurrently with the 12% SO mixture is predicted to be almost laminar. The other is strongly turbulent near the gas—liquid interface. The correction proposed by Johnson and Crynes to the Gilliland and Sherwood equation introduces additional distortions. Interpreting their work with the Maxwell—Stefan theory (Taylor and Krishna, 1993) suggests that the factor of 2 is a Stefan drift factor: as DDBS diffuses from the interface towards the reactor wall, it drags along SO from the region near the interface into the liquid film. This inter-species drag accelerates the transfer rate as there is a species motion which did not exist in the experiments by Gilliland and Sherwood, being their films homogeneous in composition. Fig. 3 schematically shows this concept. A sublayer exists near the gas—liquid interface where major species have strong concentration gradients. Drag effects caused by the friction between molecules of different species are important there. Data from the model proposed in this paper (Represented by dots in Fig. 3), show that it is acceptable to assume linear variation of the concentrations in this
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F.I. Talens-Alesson/Chemical Engineering Science 54 (1999) 1871—1881
sublayer near the gas—liquid interface. Therefore, a simplified Maxwell—Stefan analysis may be used to semiquantitatively discuss the weight of the Stefan drift factor on mass transfer within the reactor. Eqs. (2) and (3) give the incremental forms of the Maxwell—Stefan force balance for SO and DDBS, which
Let us consider both DDB and DDBS concentration profiles stationary (v "v "0), which is equiva"" 1 "" lent to Johnson and Crynes assumption of homogeneous liquid film from the point of view of the Maxwell—Stefan theory (i.e. no species motion means no species frictional drag). Eq. (4) is then simplified yielding Eq. (5).
v " 1-
Dx /xN #[xN Dx /(xN (D )/d)]/(xN /(D /d)#xN (D /d)) 1- 1- "" 1 1- "" 1 "" \"" 1 1- 1-\"" 1 "" "" \"" 1 , [xN xN /(D /d)]/(xN /(D /d)#xN /(D /d))!(xN /(D /d)#xN /(D /d)) "" 1 1- 1-\"" 1 1- 1-\"" 1 "" "" \"" 1 "" 1-\"" "" 1 1-\"" 1 (4) are acceptable in many engineering problems (Wesselingh and Krishna, 1990). Convective effects are ignored, as these should be incorporated later into the flux calculations. Dx 1-"xN "" xN 1-
v !v v !v "" 1- #xN "" 1 1- , "" 1 (D (D )/d )/d "" \1- "" \1-
(2)
Dx v !v "" 1"xN 1- "" 1 #xN 1- (D "" xN )/d "" 1 "" 1\1-
v !v "" "" 1 . (D )/d "" 1\"" (3)
The term including x can be neglected on the right1- hand side of Eq. (3). The value is small, and therefore the friction between SO and DDBS molecules has little effect on the force balance for DDBS. Eqs. (2) and (3) yield Eq. (4) if v "0 is assumed (no DDB is leaving "" the film). This is an acceptable approximation to the behaviour of the reactor. Eq. (4) gives the Maxwell— Stefan velocity for SO within the liquid film, in a system where DDBS is both diffusing towards the reactor wall and being generated through the film, and DDB is effectively being depleted through the film, with its motion virtually unnoticeable.
Fig. 3. The dots represent data from a simulation run with a SO 0.1 molar fraction in the gas carrier, at 0.425 m of the top of the column, with the same SO and DDB molar ratios.
1 v " . 1- xN /(D )/d#xN /(D )/d "" 1-\"" "" 1 1-\"" 1
(5)
The simulation results are shown in Table 1. The three leftmost columns show the species velocities predicted by Eqs. (4) (a general Maxwell—Stefan prediction) and Eq. (5) (a Johnson—Crynes restricted prediction) as well as their ratio. SO velocity is approximately 5 times larger con sidering inter-species drag than under the assumptions made by Johnson and Crynes. Hence the need for a correction factor in the Gilliland and Sherwood correlation to adjust the predictions to their experimental results. Johnson and Crynes gave an averaged value for all the reactor length and for all their operational conditions, and the estimate of the ratio between Eqs. (4) and (5) (equivalent to the correction factor by Johnson and Crynes) given in the third column of Table 1 has been calculated very crudely, but even so the results of the calculations are in good qualitative agreement. These distortions had a clear effect on the predictions. The database produced by Gutie´rrez included conversion values at a distance of 0.4, 0.975 and 2.0 m, obtained with reactors of different length and assuming negligible exit effects. This provides experimental data against which predicted axial conversion profiles can be compared. The Dabir model under-predicted (deviations between 8% to 18%) conversions at 0.4 m of the top of the reactor, but gave reasonable agreemnent with experimental data at 0.975 m (within 10% and mostly within 5%), and over-predicted conversion at 2.0 m (deviations between 4% and 18%) from the top of the reactor. Their predicted value of 335 K for the interfacial temperature peak was also poor. Industrially run film pilot reactors reach an average 370 K in the liquid film. Gutie´rrez’s model had predicted interfacial liquid temperatures to be about 390 K, which is more realistic. The failure in the prediction of interfacial film temperature is the inevitable consequence of low initial conversion predictions: a slower increase in conversion at the top of the reactor results in reaction heat being released more evenly along the reactor and being dissipated more readily. This effect is beneficial in suppressing undesirable side reactions as mentioned above. It is the basis of the
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F.I. Talens-Alesson/Chemical Engineering Science 54 (1999) 1871—1881 Table 1 M—S species velocities, calculated for the hypothesis of (1) non-homogenenous film composition and (2) homogeneous film composition % SO
d (m)
X 1-
Xd 1-
X ""
Xd ""
d (10— m)
» — +1 10\ m s\
»— (! 10— m s\
» — /» — +1 (!
08 0.08 0.08 0.1 0.1 0.1 0.12 0.12 0.12 0.12
0.17 0.42 1.39 0.17 0.42 1.39 0.06 0.17 0.42 1.39
0 0 0 0 0 0 0 0 0 0
0.004 0.011 0.013 0.009 0.017 0.018 0.001 0.014 0.023 0.023
0.523 0.451 0.316 0.571 0.476 0.338 0.711 0.618 0.504 0.364
0.085 0.050 0.029 0.053 0.032 0.017 0.071 0.039 0.022 0.011
4.9 6.6 7.6 5.7 6.7 8.0 3.5 5.5 6.5 8.5
15.2 11.5 9.3 14.8 12.2 9.6 26.8 16.4 13.2 9.5
3.49 2.44 1.91 3.04 2.39 1.83 5.57 3.21 2.51 1.76
4.36 4.73 4.88 4.86 5.07 5.21 4.81 5.11 5.27 5.38
Table 2 Dynamic viscosity, density and surface tension of typical sulphonation/sulphation organic reagents, as a function of temperature
k (kg m\ s\) o (kg m\) p (N m\)
Dodecylbenzene
Lauryl alcohol
Lauryl alcohol 2OE
2.72;10\ exp(2980/¹) 975—0.504 T 0.026
1.968;10\ exp(2667/¹) 1056—0.66 T 0.028
1.61;10\ exp(2790/¹) 1006—0.329 T 0.038
variable cross section falling film reactor suggested by Talens et al. (1996b).
2. Semiempirical approach to the design of falling film reactors 2.1. Prediction of gas—liquid frictional drag As already stated, accurate prediction of friction factors is a bottleneck in the modelling process. As an alternative to the fitted Blasius equation proposed by Gutie´rrez, Talens (1991) proposed a modification of a explicit Colebrook equation (Zygrang and Sylvester, 1982):
1 e/d 2.51 e/d 13 "!4 log ! log # C 3.7 Re 3.7 Re D
,
Fig. 4. Apparent friction factors for dodecanol, diethoxylated dodecanol and DDB. Even though dodecanol 2OE is more viscid than DDB, it builds up resistance to frictional drag more slowly due to its higher surface tension.
(6)
where the film roughness e, was replaced by Ud. U is a function of the interfacial liquid velocity and d is the film thickness, which sometimes is directly taken as the value of the roughness. U was derived from experimental pressure drop data in annular flow taken in the same reactor originally used by Gutie´rrez for a set of three organic liquids (dodecylbenzene, dodecanol, and dodecanol 2OE, technical grade, supplied by KAO Corporation, Barbera´ del Valle´s, Spain) whose properties (Bromstrom, 1974; Talens et al., 1994) are given in Table 2. Assuming laminar film flow, the film thickness, shear stress, interfacial velocity and apparent friction factor were calculated by means of the
set of equations given in the Appendix. Dodecylbenzene and dodecanol had very similar surface tensions (0.026 and 0.028 N/m, respectively) while the dodecanol 2OE had a surface tension of 0.038 N/m. The estimated friction factors (Fig. 4) for the fluids with similar surface tension have the same general shape, with a maximum about the same Re . Gutierrez¨s fitting factors assume % that the friction factors along the reactor are 14 times higher than those given by Blasius equation for a smooth wall. Fig. 4 shows that this overestimates friction factors for Dodecylbenzene (white circles) by a factor of 2, and there is even more of an overestimation below Re " % 6000. For dodecanol 2OE (with a less deformable
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interface) resistance shows to build up more slowly. The existence of conditions for the onset of large disturbance waves (responsible for strong gas—liquid interactions) and of threshold wave heights required to cause additional roughness are described in the literature (Hewitt and Hall—Taylor, 1970; Andreussi and Zanelli, 1982; Takahama et al., 1983; Okada and Fujita, 1993). These facts can explain why there is a sharp increase in roughness, as the onset of large waves is an abrupt condition, and why the interface with the higher surface tension (less deformable) does not build up additional resistance so quickly. The data for dodecanol 2OE were discarded, and the correlation assumes a liquid surface tension of about 0.027 N/m. The friction factors were substituted in the left-hand side of the Zygrang and Sylvester equation, and the calculated film thickness was substituted in the righthand side of the equation. Then the apparent roughness factor U was calculated by iteration to fit the equation for each set of values. The results can be seen in Fig. 5. It must be stressed that this correlation assumes that pure dodecanol or dodecylbenzene and their mixtures with their sulphonation and sulphation product have approximately the same surface tension: in order for LABS to have surfactant properties in organic phase it must be forming a salt like, for instance, its Calcium or Isopropylamine salts (Porter, 1994). Dispersion is strong and regression coefficients are poor, but the numerical agreement with actual pressure drop data is better than with the Gutierrez ‘black-box’ fit. The following equations are represented as solid lines in Fig. 5: ¸n(U)"3.59!5.1» , (7) '* ¸n(U)"20.55» !0.93. (8) '* Eq. (7) is valid for interfacial liquid velocities above 0.175 m s\, and Eq. (8) is valid below that value. The contradiction in taking into account turbulent viscosity and assuming parabolic velocity profiles is easily justified
Fig. 6. Kinematic viscosity of DDB is about 5;10\ m s\, larger than the values of turbulent viscosity. However, the associated turbulent diffusivity is high when compared with molecular diffusivity.
for the systems and conditions involved in this particular case. While eddy viscosity is important enough to cause non-negligible turbulent diffusivities (Fig. 6), its value is still far smaller than the kinematic viscosity of the liquid, and therefore laminar velocity profiles can be assumed without significant error. 2.2. Heat and mass transfer equations The complete set of Eqs. [(A.1)—(A.14)] is listed in the Appendix together with details on the linearisation procedure. Essentially it includes the Yih and Liu (1983) turbulent viscosity model and its expansion for the calculation of turbulent diffusivities, using the MCCready— Hanratty equation for the gas—phase transfer coefficient. 2.3. Colour intensity It is possible to estimate the relative colour intensity (or the likelihood of by-products affected by higher temperatures) in different runs for the same reaction system under different conditions by solving the following integral (Talens et al., 1993): * B
e 2C1 dr dz.
CI"
Fig. 5. The increased roughness vs the interfacial liquid velocity. Surface tension has a strong effect (Talens et al., 1996b).
(9)
This is done after the model has been solved and composition and temperature profiles over the length and width of the liquid film are available. Eq. (9) assumes that colour is formed in a first-order kinetic equation affecting the sulphonate, as suggested by Bromstro¨m (1974). For a given reaction, with constant frequency factor and activation energy, the integral CI will be proportional to the integration of the differential equation for the first order kinetic equation. Therefore, comparison between
F.I. Talens-Alesson/Chemical Engineering Science 54 (1999) 1871—1881
Fig. 7. Colour Integral as a function of SO fraction in the gas. The relative colour intensity predicted by the model is in agreement with observations by the industrialists: a strong decrease in colour is observed as the carrier gas provides stronger shear stress.
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Fig. 8. The a priori model presented here predicts the sharp increase in conversion, and the shift between control of mass transfer by the gas phase and control by the liquid phase.
the value CI for different runs will show under which conditions heat-induced side reactions are more likely (Fig. 7).
3. Model results The model retains the ability to predict sharp increases in conversion at the top of the reactor (Fig. 8). Higher conversions are initially linked to higher SO molar fraction in the gas (lower overall gas flow rate) while later there is a shift and the higher conversion is finally achieved with the lower SO molar fraction in the gas (higher overall gas flow rate). This means that at the top of the reactor mass transfer is controlled by the gas phase, and that in the lower part of the reactor it is the liquid phase which controls the mass transfer: the higher the turbulence induced by the gas on the liquid, the higher the amount of SO transferred into the film and the higher the conversion. Fig. 9 shows schematically the fast conversion region at the top of the reactor (linked with gas-phase control) and the slow conversion region at the bottom (linked with liquid-phase control). Fig. 10 shows a simulation at a given reactor length (0.425 m). Depletion of DDB near the interface is faster when the gas flow rate is lower. As turbulence is lower, convective effects are less important. Notice that the profiles in the sub-layer near the gas—liquid interface are almost linear. This linearity is also the case in Fig. 11. SO molar fraction in the gas is 4%. The effect between 0.175 and 0.425 m is interesting and indicative of the change in phase control on mass transfer: The viscosity increases with conversion and distance from the top of the reactor, and diffusion of DDB towards the interface becomes relevant as controlling step of the reaction. Figs. 10 and 11 confirm that turbulent diffusivity values are significant even though turbulent viscosity
Fig. 9. Schematic view of the falling film reactor reactor.
values are several orders of magnitude lower than the kinematic viscosity of DDB. The flat portion of the solid curve to the right-hand side gives the molecular diffusivity calculated by the Wilke—Chang equation. The right side gives the turbulent diffusivities obtained from the Yih—Liu and Cebeci equations. Fig. 12 gives some radial temperature profiles at different distances from the top of the reactor. The almost flat profiles indicate that heat removal at the reactor wall is a crucial factor. A strong initial temperature increase is observed. The temperature is initially predicted to be almost 400 K. Later, as differential conversion along the reactor becomes near zero, the temperature in the film almost equals that in the cooling jacket (298 K).
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Fig. 10. The height for this simulation of radial profiles is 0.425 m. Higher overall carrier gas rate leads to stronger renewal of the unreacted DDB near the gas—liquid interface. Note that in all cases there is a sublayer with almost linear variation of DDB concentration.
Fig. 12. Radial temperature profiles. The temperature increases strongly at the beginning, and afterwards heat dissipates faster.
Fig. 13. The model gives better predictions at higher shear stress. Predictions are poorer for lower shear stress, and this seems to be the essential defect of spatially averaged film thickness models. Fig. 11. The effect of conversion and associated increase of viscosity on the radial profiles, calculated at the local composition and temperature for each distance from the interface.
Fig. 13 leads to an interesting reflection on the limits of spatially averaged film thickness models. At higher shear stress (lower SO molar fractions) there is a better agree ment between experimental and predicted conversions. In practice, the Yih and Liu model predicts higher or lower turbulence near the gas—liquid interface as a consequence of gas shear stress, but not a radical change in the depth of the turbulent zone. However, actual films consist of a substrate film and waves, and changes in shear stress modify the thickness of the substrate film (Nencini and Andreussi, 1982). The figure suggests that at lower shear stress the fraction of liquid in the substrate film (and under non-turbulent flow conditions) has increased significantly and the result of this is that conversion is much poorer. On the other hand, for highly sheared films, a spatially averaged film thickness model seems to
behave well. However, the case studied is a A#B gives C reaction, with no significant parallel or subsequent reactions, and it is reasonable to assume that predictions from spatially averaged film thickness models will be less accurate when dealing with more complex reactions, such as polymerisation.
4. Conclusions The main conclusions which can be drawn, from the point of view of modelling strategies, are: (1) The Johnson and Crynes modification to the Gilliland and Sherwood correlation should be avoided. Using it would cause convective effects to be accounted for twice in the model: once in the turbulent diffusivity coefficients, and again implicitly in the Johnson and Crynes correlation.
F.I. Talens-Alesson/Chemical Engineering Science 54 (1999) 1871—1881
(2) Correlations for frictional drag between gas and liquid should be obtained for each specific reactor. Both general transportability problems of existing correlations and technical obstacles for data acquisition in organic films make advisable not to use equations from the literature, which are essentially derived for aqueous films. (3) A reasonable approach could be the one describe here, where interfacial effects are associated with a roughness factor. The user should pay attention to the fact that spatially averaged film thickness is an assumption, which seems to work better at high gas—liquid shear stresses.
Notation C D CI d D D R h % DH 0 K % K * k u
friction factor colour integral, dimensionless column inner diameter, m diffusivity coefficient at infinite dilution, m s\ turbulent diffusivity heat transfer coefficient, J m\ s\ reaction enthalpy, J kmol\ overall mass transfer coefficient, kmol m\ s\ liquid thermal conductivity, J s\ m\ K\ reaction kinetic coefficient, s\ average gas velocity in the liquid-free reactor core, m s\ u characteristic turbulence velocity, m s\, v species diffusion velocity, m s\ v interfacial liquid velocity, m s\ '* x molar fraction x average molar fraction y> non-dimensional distance to the wall: y(q g/o) /l U Pr Prandtl number Re gas Reynolds number: ou(d!2d) k\ Sc, Sc Schmidt number, turbulent Schmidt number R C mass flow rate per unit periphery, kg m\ s\ d film thickness, m e roughness, m k viscosity, kg m\ s\ l, l kinematic viscosity, kinematic turbulent viscos# ity, m s\ o liquid density, kg m\ * q ,q shear stress at the gas—liquid interface, shear % 5 stress at the wall U roughness enhancement factor
Acknowledgements The author wishes to acknowledge the insight gained through his co-operation with Prof. B.J. Azzopardi, Dr. J. Hills, and Mr. W. Clark, from the University of Nottingham, UK, as an advisor to EPSRC. Project GR/K
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77044. The author also wishes to mention the involvement of Drs. S. Esplugas-Vidal, C. Mans-Teixido, and J. Costa-Lo´pez, in his redundancy from the Department of Chemical Engineering of the University of Barcelona, Spain. References Agrawal, A.K., & Peckoever, R.S. (1980). Nonuniform grid generation for boundary-layer problems, Comput. Phys. Commun., 19, 171—178. Alekseenko, S.V., Nakoyakov, V.E., & Pokusave, B.G. (1994). ¼ave flow of liquid films, New York: Begell House. Asali, J.C., Hanratty, T.J., & Andreussi, P. (1985). Interfacial Drag and film height for vertical annular flow. A.I.Ch.E. J., 31, 895—902. Bromstro¨m, A. (1974). Rate of color formation in sulfonation of dodecylbenzene with gaseous surfur trioxide. JAOCS, 51, 507—508. Bromstro¨m, A. (1975). Mathematical model for simulating the sulphonation of dodecylbenzene with gaseous sulphur trioxide in an industrial reactor of Votator type. ¹rans. Instn. Chem. Engrs., 53, 29—33. Dabir, B., Riazi, M.R., & Davoudirad, H.R. (1996). Modelling of Falling Film Reactors. Chem. Engng. Sci., 51, 2553—2558. Davis, E.J., Van Ouwerkerk, M., & Venkatesh, S. (1979). An Analysis of the falling film gas—liquid reactor. Chem. Engng Sci., 34, 539—550. Dicoi, O., & Canavas, C. (1993). German Patent DE 41 28 827 A 1. Gilliland, E.R., & Sherwood, T.K. (1934). Diffusion of vapors into air streams. Ind. Engng Chem., 26, 516—523. Gutie´rrez, J., Mans, C., & Costa, J. (1988). Improved mathematical model for a falling film sulfonation reactor. Ind. Engng Chem. Res., 27, 1701—1707. Haumer, J. (1970). Determination of optimal conditions for sulfonation of alkylbenzenes with SO gas (Czech). »eda »yzk. Potravin. Prum., 21, 115—137. Henstock, W.H., & Hanratty, T.J. (1976). The Interfacial drag and the height of the wall layer in annular flows. A.I.Ch.E. J., 22, 990—1000. Hewitt, G.F., & Hall-Taylor, N.S. (1990). Annular two-phase flow, Oxford: Pergamon Press. Hurlbert, R.C., Knott, R.F., & Cheney, H.A. (1967). Apparatus for small scale sulfonation with SO . Soap Chemi. Specialties, 88—94 and 100. Johnson, G.R., & Crynes, B.L. (1974). Modelling of a thin-film sulfur trioxide sulfonation reactor. Ind. Engng Chem. Process Des. Dev., 13, 6—14. Ka´lnay de Rivas, E. (1972). On the use of nonuniform grids in finitedifference equations. J. Comput. Phys., 10, 202—210. Lamourelle, A.P., & Sandall, O.C. (1972). Gas absorption into a turbulent liquid. Chem. Engng Sci., 27, 1035—1043. Lapidus (1962). Digital computations for chemical engineers. New York: McGraw-Hill. Liles, D.R., & Mahaffy, J.H. (1984). ¹RAC-PF1/MOD1 An advanced best estimate computer program for pressurised water reactor thermal-hydraulic analysis. Los Alamos National Laboratory Report. Mann, R., Knysh, P., & Allan, J.C. (1982). Exothermic gas absorption with complex reaction: sulfonation and discoloration in the absorption of sulfur trioxide in dodecylbenzene. In J. Wei & K.C. Georgakis (Eds.), ACS Simposium series 196, (pp. 441—456). McCready, M.J., & Hanratty, T.J. (1984). Gas transfer at water surfaces, (Vol. 283). D. Reidel Dordrecht, Holland: Publishing Company. Nencini, F., & Andreussi, P. (1982). Studies of the behavior of disturbance waves in annular two-phase flow. Can. J. Chem. Engng, 60, 459—465. Okada, O., & Fujita, H. (1993). Behavior of liquid films and droplets in the non-equilibrium region of a downward annular mist flow (comparison of porous and central nozzle mixing methods). Int. J. Multiphase Flow, 19, 79—89.
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Owen, D.F.G., & Hewitt, G.F. (1987). An improved annular two-phase flow model. 3rd International Conference on Multiphase Flow (BHRA), The Hague. Porter, M.R. (1994). Handbook of surfactants (pp. 138—143). Bury St. Edmunds: Blackie Academic and Professional. Takahama, H., Okada, O., Fujita, H., & Mizuno, A. (1983). Study on annular mist flow in pipe (2nd. report, Behavior of water film in the non-equilibrium region of downward annular mist flow with low water flow rate). Bull. HSME, 26, 2091—2099. Talens, F.I. (1991). Fluid dynamics and transport in wetted wall columns (Spa.) Ph.D. thesis, University of Barcelona, Spain. Talens, F.I., Gutie´rrez, J.M., & Mans, C. (1993). Prediction of colour and other quality parameters in falling film reactors. ¹enside, 30, 331—335. Talens, F.I., Gutie´rrez, J.M., & Mans, C. (1994). Effect of surface tension and viscosity on the efficiency in wetted wall reactors. ¹enside, 31, 6—8. Talens, F.I., Hreczuch, W., Bekierz, G., & Szymanowski, J. (1987). Ethoxylation of alcohols in falling film reactor, J. Disp. Sci. ¹echnol., 18, 423—433. Talens, F.I., Pereira, G., Chenlo, F., & Vazquez, G. (1996a). Device for research into falling film wave structures under shear stress. A.I.Ch.E. J., 42, 3293—3295. Talens, F.I., Gutie´rrez, J., & Mans, C. (1996b). An improved falling-film reactor for viscous liquids. JAOCS, 73, 857—862. Taylor, R., & Krishna, R. (1993). Multicomponent mass transfer. New York: Wiley. Theofanous, T.G., & Amarasooriya, W.H. (1992). Physical benchmarking exercise. In: Hewitt, G.F., Delhaye, J.M., & Zuber, N. (Eds.), Multiphase science and technology (Vol. 6, pp. 2—12). Washington, DC: Hemisphere. Wesselingh, J.A., & Krishna, R. (1990). Mass transfer. Chichester: Ellis Hurwood. Yao, S.C., & Sylvester, N.D. (1987). A Mechanistic model of two-phase annular-mist flow in vertical pipes. A.I.Ch.E. J., 33, 1008—1012. Yih, S., & Liu, J. (1982). Prediction of heat transfer in turbulent falling liquid films with or without interfacial shear. A.I.Ch.E. J., 29, 903—909. Zygrang, D.J., & Sylvester, N.D. (1982). Explicit approximations to the solution of Colebrook’s friction factor equation. A.I.Ch.E. J., 28, 514—515.
where q is calculated from experimental pressure drop % data, it is possible to obtain a value of the correction factor U which can be correlated to an estimated interfacial velocity
Appendix A
B>"Sc\ C (log Sc)G\ (A.10) G G with C "34.96, C "28.97, C "13.95, C "6.33, C "!1.186 Mass transport equations
A.1. Calculation of average film thickness, roughness correction factor average interfacial liquid velocity The initial value for the iterative calculation of the film thickness d can be obtained from
3Ck . g o ! * By combining d"
(A.1)
(A.4)
A.2. Simulation model Mass transfer coefficient in the gas phase is taken as (McCready and Hanratty, 1984) k %"0.8 Sc\ , (A.5) u where the turbulent velocity u is defined as q u " % . (A.6) o % Turbulent viscosity in the liquid phase (Yih and Liu, 1982):
l q #"!0.5#.5 1#.64 (y>) l q 5 !y>(q/q ) 5 ; 1!exp A>
,
(A.2)
and
1 (Ud)/d 2;2.51 (Ud)/d 13 "!4 log ! log # C 3.7 Re 3.7 Re D
,
(A.3)
(A.7)
q q * (y>/d>) (A.8) "1! q q #q 5 % * Turbulent diffusivity in the liquid phase follows the Cebeci equation (see Yih and Liu, 1982) l 1!exp (!y>(q/q ) /A>) 5 Sc " R " R D 1!exp (!y>(q/q ) /B>) R 5 with A>"25.1 and
l 8
(A.9)
*C * *C "" " (D #D ) "" !kC C , (A.11) "" R *y "" 1- *z *y
*C * *C 1-" l (D #D ) 1- !kC C . 8 *z 1R "" 1- *y *y
o g d q d C" * ! ! % 3k 2k
g y q y v " ! dy! # % . '* l 2 k
(A.12)
The equations are solved considering that the curvature of the film is negligible, and that radial symmetry exists. The solution involves linearisation by finite differences using the Laasonen implicit forms for the first- and second-order derivatives in Eqs. (16) and (17) (Lapidus, 1962). This guarantees that the solution is always convergent. The assumption of linear behaviour may be
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a source of error. To obtain a better result the computational grid was made non-uniform by using a transformation function which converts a coordinate a made non-dimensional by dividing the coordinate x by the magnitude ¸ of the dimension in the domain into a variable q. This guarantees that no additional errors are introduced in the calculations. The transformation equation chosen (Eiseman, cited by Agrawal and Peckover, 1980) was q"0.5(atanh ma/atanhm)#0.5a. The regrouping parameter m, which must be between 0 and 1, was taken 0.001 in the radial direction and 0.9999 in the axial direction after considering the stability of the equations. This particular form of Eiseman transformation equation ensures that the reorganisation of the computational grid does not leave an unacceptably low number of points in the domain region where variations of the
properties are mild. The resulting linear equations are solved by the Thomas method (Lapidus, 1962). A.3. Heat transport equations The heat transport equations follow the Prandtl analogy, and are equivalent to the ones used for mass transfer. v 8
*oC ¹ * *¹ N " !K #(DH) kC C , (A.13) * 0 "" 1- *z *y *y
h %"8 Pr\ . (A.14) u The model is solved for dodecylbenzenesulphonic acid (DDBS), SO , and ¹. Dodecylbenzene (DDB) is cal culated from the overall mass balance.