Solid Dissolution into a Vertical Falling Film under ... - ACS Publications

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Solid Dissolution into a Vertical Falling Film under Industrial-like Conditions Erik Karlsson,* Mathias Gourdon, and Lennart Vamling Department of Energy and Environment, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden ABSTRACT: During the evaporation of black liquor, a residual stream in pulp mills, scales form on heat transfer surfaces due to the crystallization of sodium carbonate and sodium sulfate salts. As a result, falling film evaporators need regular cleaning to remove these water-soluble scales, and therefore, knowledge about the dissolution process is important. In this work, dissolution of the aforementioned salts was tested experimentally in a pilot evaporator close to the industrial scale. Dissolution was diffusioncontrolled and could be described by film theory, where the concentration difference between the saturated wall and an assumed perfectly mixed bulk was the driving force of the process. The measured mass transfer coefficient could be predicted to within 30% accuracy using the Chilton−Colburn heat and mass transfer analogy together with a standard heat transfer correlation.

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INTRODUCTION Heat and mass transfer between a solid wall and falling liquid film is of interest in many industrial applications. For instance, falling film evaporation is often used to efficiently concentrate different liquids. Other applications of falling film technology are crystallizers and condensers. In this study, the dissolution of scales consisting of double salts of sodium carbonate and sodium sulfate are investigated. These scales are formed by crystallization fouling during the black liquor evaporation process, the most steam-consuming part of a pulp mill. Regular cleaning is needed to maintain operation. Because these salts are easily dissolved in water, the normal procedure for cleaning is to recirculate condensate or weak black liquor. Previous research has mainly focused on how to prevent fouling by optimizing operating conditions rather than on how to efficiently remove scales.1−3 Cleaning the evaporators’ surfaces is also an important part of the operation, but the cleaning process is based on trial and error due to a lack of fundamental knowledge.4 The aim of this work was to gain an understanding of the process of scale dissolution into a falling film of water and to determine how this dissolution can be modeled under industrial-like conditions. Emphasis was placed on making a simple and robust model that can be implemented in industry. The parameters of interest were the solvent temperature, flow rate, and inlet salt concentration. First, dissolution experiments were conducted on a relatively large scale, and then, a dissolution model was developed and used to fit a mass transport coefficient to the experiments. Another aim of this work was to compare the mass transport coefficient using the Chilton−Colburn analogy with heat transfer.

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into the liquid. In this case, the concentration at the solid− liquid interface is equal to the solubility of the substance. Thus, the dissolution rate can be described using a first-order expression: dm = kA(c i − cb) dt

The total mass of the solid, m, will decrease over time as mass dissolves and is transported away from the interface. The driving force is the difference in concentration between the solid−liquid interface, ci, and the bulk of the solvent, cb. A is the surface area of the solid−liquid interface, and k is the mass transfer coefficient. This type of expression was developed over 100 years ago.6,7 The dissolution process has become critical for modeling the release of drugs in the pharmaceutical industry.8 However, the dissolution process is also important for modeling the removal of foulants in the chemical and food processing industries9,10 and for understanding corrosion.11 For the dissolution of a solid into a falling liquid film, earlier studies have focused on processes in which the dissolved substance does not penetrate deeply into the falling film.12−15 This is only true for situations with short contact times, but this assumption makes it possible to simplify the boundary condition for the liquor−gas interface and thereby enables an analytical solution for laminar flow. Early studies12−15 provided solutions for Newtonian fluids, which were then complemented for non-Newtonian fluids.16,17 Improvements of these solutions to allow for deeper penetration of laminar flow is possible; however, because the problem is complex, only numerical solutions are possible.18,19 In many situations true laminar conditions are not possible. Stirba and Hurt20 found that falling films cannot be considered laminar for Reynolds numbers (Re ≡ 4Γ/μ) as low as 300, i.e., mixing and turbulence were present and significantly increased

THEORY

Solid Dissolution. The dissolution process, i.e., the mass transfer from a solid surface to a liquid, can be controlled by both reaction and diffusion.5 In the case of a salt, the reaction involves breaking bonds in the crystal lattice to generate free ions. When dissolution is diffusion-controlled, the limiting step is the diffusion of the substance from the solid−liquid interface © 2014 American Chemical Society

(1)

Received: Revised: Accepted: Published: 9478

February 10, 2014 April 28, 2014 May 11, 2014 May 11, 2014 dx.doi.org/10.1021/ie500581q | Ind. Eng. Chem. Res. 2014, 53, 9478−9487

Industrial & Engineering Chemistry Research

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includes the diffusion coefficient DAB and diffusion film thickness δd. At higher Reynolds numbers (Re), δd will decrease because stronger turbulence results in stronger bulk mixing. Thus, k has a positive flow dependence. Both DAB and δd are temperature dependent, i.e., diffusivity increases for liquids and δd decreases (due to decreased viscosity) at higher temperatures. Thus, k also has a positive temperature dependence. For the film theory (and eq 2) to be valid, the major part of the film must consist of a mixed bulk, i.e., δd must be significantly thinner than δtot (Figure 1). The bulk concentration will increase significantly for long contact times; hence, the driving force for dissolution will decrease along the surface and approach zero when the bulk becomes saturated. Correlations by Nusselt27 for laminar flow and Brauer28 for turbulent flow can be used to estimate the total film thickness of a falling film:

the dissolution rate. This conclusion was also reached by Wasden and Dukler,21 who found with numerical solutions that waves greatly improved dissolution compared to a perfect laminar falling film. In contrast to dissolution, the related process of gas absorption into a falling liquid film has been studied more extensively.22 The situation under study in this work, i.e., dissolution in large-scale falling film evaporators, is even more complex, mainly because of long contact times (deep penetration) and the fact that the contact time between the solid and liquid changes with time (the area decreases as salts are dissolved). A preliminary investigation demonstrated that a first-order expression was the best way to model the dissolution process, and the dissolution was likely limited by diffusion and not breaking of the crystal lattice.23 An alternative mechanism for cleaning process equipment is mechanical removal, and the removal rate is then a function of the shear stress at the surface.24 However, mechanical removal is more relevant when removing foulants with low mechanical strength, e.g., soils, because soils are removed as particles released to the fluid rather than being dissolved into a solvent. Removal of soils are common in many industrial applications, such as the food industry.25 Solid Dissolution into a Liquid under Turbulent Conditions. If the dissolution process is controlled by diffusion and the flow is turbulent, convective mass transfer can be approximated by film theory.26 Here, all mass transfer resistance is assumed to exist in a thin film close to the dissolving solid surface, in which transport occurs entirely through molecular diffusion (Figure 1). This film will be called

⎛ 3v 2Re ⎞1/3 δtot,lam = ⎜ ⎟ ; ⎝ 4g ⎠

Re < 1600

⎛ v 2 ⎞1/3 δtot, turb = 0.208⎜ ⎟ Re8/15; ⎝g⎠

(3)

Re > 1600 (4)

Heat and Mass Transfer Analogy. Because heat transfer has been more extensively studied for falling films, that knowledge is valuable for predicting mass transfer. In this work, the Chilton−Colburn analogy was used to relate heat transfer to mass transfer. Chilton and Colburn defined a jfactor, which is equal for both mass and heat transfer:26 jd = jh

Sh Nu = ReSc1/3 RePr1/3

(5)

The analogy has been proven valid for 0.6 < Sc < 2500 and 0.6 < Pr < 100.26 For a vertical falling film, the Nusselt (Nu) and Sherwood (Sh) numbers are defined as h ⎛ v2 ⎞ ⎜ ⎟ kt ⎝ g ⎠

(6)

k ⎛ v2 ⎞ ⎜ ⎟ D⎝ g ⎠

(7)

1/3

Nu ≡

1/3

Sh ≡ Figure 1. Schematic representation of the concentration profile using film theory, where δd is a fictive film thickness between a solid surface and the bulk flow. The flow is in the z-direction, and δd should be considerably thinner than the total thickness of the falling film δtot.

The Reynolds number (Re) can be defined in two ways for a falling film (with and without multiplication by 4), and it was defined as Re = 4Γ/μ here. The other dimensionless numbers are standard and defined in the Nomenclature. The use of the heat and mass transfer analogy is most correct when the heat and mass transfer situations can be described by a corresponding set of boundary conditions. For mass transfer during solid dissolution, this implies that the analogous heat transfer situations should have no heat transfer at the liquid/gas interface on the falling film. Therefore, nonevaporative (sensible) heating is more appropriate than evaporative heat transfer. In this study, two different heat transfer correlations were used to predict heat transfer and were then converted to mass transfer using the analogy (eqs 5−7). Both correlations are from Schnabel and Schlünder.29 The first is for nonevaporative conditions with a constant wall temperature. The correlation consists of three parts: expressions for the laminar, turbulent, and transition regions:

the diffusion film to distinguish it from the falling film. When the dissolved substance has diffused through the diffusion film, it can be regarded as part of the bulk flow, and thus, the resistance to mass transfer in the x-direction can be neglected in the bulk flow. The concentration at the solid−liquid interface will be equal to the saturation concentration c* at the current temperature. The dissolution rate (or mass flux) can then be modeled as follows (analogous to eq 1): D ∂c Δc Jx = DAB ≈ DAB = AB Δc = k(c* − cb) ∂x Δx δd (2) k is a mass transfer coefficient in m/s; this coefficient accounts for diffusion from the solid surface to the bulk flow and thus 9479

dx.doi.org/10.1021/ie500581q | Ind. Eng. Chem. Res. 2014, 53, 9478−9487

Industrial & Engineering Chemistry Research Nu = max(Nulam , Nutrans , Nuturb)

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a metal rod and pushed inward until it reached the surface of the scales. The scale thickness was then measured from the outside of the evaporator using Vernier calipers. The estimated accuracy of the measurements was ±0.2 mm. Further details about the research apparatus and measurement equipment can be found in other publications by the authors.3,30 Experimental Design and Procedure. Each experiment consisted of two steps: first, a layer of salt (scales) was built up on the evaporator tube. Then, dissolution experiments were performed by carefully washing the tube. The experiments were designed to test the variation of two operational parameters: temperature (40−80 °C) and specific mass flow rate (0.1−0.5 kg/(m·s)), each at three different levels. Two different solvents were used: water and salt solutions (with the same salt composition as used during scale buildup). A total of 12 experiments were performed: 10 using deionized water, denoted W1−W10, and 2 using salt solutions, denoted S1 and S2. Scale Buildup. Scale buildup was achieved by evaporating a salt solution of sodium carbonate and sodium sulfate with a sodium carbonate mole fraction of approximately 0.87. The system was operated at atmospheric pressure on the solution side of the system, corresponding to 105−107 °C. The density of the salt solution was increased over the course of the evaporation to reach the first nucleation point with the first precipitation of crystals. Evaporation continued for 3−5 h, which typically corresponded to a scale buildup of 2−5 mm in the lower part of the evaporator. Initial scale thickness was measured, and the tube was inspected visually to determine the quality and distribution of the scales before starting the dissolution sequence. Dissolution. The solvent (or wash liquid) used in the dissolution experiments was first preheated to the desired temperature in a separate vessel. When the experiment began, the wash liquid was introduced into the evaporator and distributed using the same distributor as for evaporation, ensuring an even film. The flow rate was controlled to yield the desired specific mass flow rate. Video cameras were mounted at the three sight glasses, and the entire experiment was recorded to monitor the dissolution process. The video films supplied information on how the scales were dissolved and when the tube could be considered clean. To obtain the desired temperature, the hot tube was cooled with water on the inside prior to the dissolution experiment. During the dissolution experiment, the outgoing wash liquid was sampled in plastic containers of 1·10−3 m3 volume. The sampling time and temperature of the exiting wash liquid was recorded together with the mass and density of each sample. When the tube was visually determined to be clean, the flow of wash liquid was stopped and the experiment ended. The mass fraction of salt was measured by determining the dry solids mass fraction for a few of the samples. The density was correlated to the salt mass fraction and temperature, enabling the mass of salt to be calculated in all samples. The temperature of the wash liquid was experimentally limited to 90 °C. Additionally, because the salts have poor solubility below 40 °C, the range from 45 to 85 °C was investigated. The specific mass flow rate was limited to a maximum of 0.45 kg/(m·s) due to sputtering and a minimum of 0.1 kg/(m·s) due to insufficient surface wetting. Therefore,

(8)

where Nulam = 2.06Re−1/3

(9)

Nutrans = 0.0322Re1/5Pr 0.344

(10)

Nuturb = 0.0078Re 2/5Pr 0.344

(11)

However, most of the work on falling film heat transfer presented in the literature is based on measurements during evaporation. The second correlation is for evaporative heat transfer, which has been proven accurate for conditions with constant wall temperature: Nu = (Nulam 2 + Nuturb 2)1/2

(12)

where

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Nulam = 1.43Re−1/3

(13)

Nuturb = 0.0036Re 0.4Pr 0.65

(14)

EXPERIMENTAL SECTION Apparatus. The experimental portion of this work was performed in a pilot evaporator, and a schematic of the experimental setup is provided in Figure 2. The heat transfer

Figure 2. Flow sheet of the research evaporator.

surface consisted of a 60 mm diameter, 4.5 m long vertical tube with the falling liquid film on the outside. The scale of the experimental equipment was appropriate for obtaining industrially relevant results; however, these experiments were extremely resource intensive and time-consuming, which limited the number of possible parameter combinations that could be investigated. The evaporator tube could be visually inspected through three sight glasses at different vertical positions: 0.15, 1.9, and 3.9 m from the inlet. To measure the scale thickness before and during experiments, a device was constructed and installed at two different positions (1.8 and 3.8 m from the inlet). The device consisted of a small flat disc that was fixed on the end of 9480



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the investigated range of specific mass flow rates was 0.12−0.43 kg/(m·s).

distribution: the scale thicknesses at 0, 1.8, and 3.8 m were known from experimental measurements, and the total mass of scale was known from the total mass of dissolved salt. An iterative procedure was used for parameter fitting: first, the value of k was set and the best distribution was fitted. Then, a new value of k was set and a new initial distribution was found. It was then possible to say of the new value of k gave an overall better fit between the model and the experiment. This process was repeated to find the best value of k. Initial Scale Distribution. Because there was not a sufficient amount of experimental information to determine the complete initial scale distribution, the procedure described above was used to determine the scale distribution. However, the thickness was measured at 1.8 and 3.8 m, and the scales were visually examined prior to each dissolution experiment. At the inlet (0 m), the scale thickness was always zero because some initial heating was needed before evaporation and crystallization began during buildup. The distribution between the known positions varied in the experiments, and thus, it was not possible to find a common, model-independent method of distributing the scales. The typical appearance of scales is depicted in Figure 4. Visual observations indicated that the scales were typically thickest at the bottom, with no drastic changes in thickness between the two lowest observation points (1.8 and 3.8 m). Between 0 and 1.8 m, more drastic changes were observed, and here, the individual experiments displayed significant differences but also similarities. Therefore, the distribution can be divided into two different categories: plateau and exponentially declining; these categories are shown in Figure 5. Because the distribution determines the contact time, the two categories exhibited individual dissolution behaviors that could be determined from the outgoing salt concentration. For example, a plateau in the outgoing salt concentration can only be described by a plateau in the distribution, and therefore, this information is model-independent. Due to the complex nature of the crystallization fouling process, it was not possible to establish which parameters determined which of the two scale distributions was formed. In general, scales were formed from the bottom and grew upward. In the lowest part of the evaporator tube, there was a mixed cup (used for temperature measurements) where a thick ring of scales was formed, as shown in Figure 4. This ring was not sufficiently thin to be fully dissolved during the experiments;

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DATA EVALUATION Model Implementation. MATLAB was used to model the dissolution process and fit the mass transport coefficient in eq 2. Dissolution was simulated by discretizing the process in both time and space. Time steps of 4−10 s were used, and the tube was divided into 500 cells in the z-direction with rotational symmetry (Figure 3). In each of these cells, density and

Figure 3. Tube was discretized in the z-direction into several cells; each cell had its own local scale thickness.

solubility were calculated at the current local temperature (the local temperature was linearly interpolated between the inlet and outlet temperatures at the current time step). These values were then combined with the mass fraction of salt in the cell (which was known from the mass balance) to obtain the concentration difference between the surface and bulk flow. Each cell had its own scale thickness, which decreased depending on the dissolution rate. Parameter Fitting. The mass transport coefficient was assumed to be constant for each experiment and was determined using a best-fit approach to the experimental data, i.e., k was adjusted together with the initial scale distribution along the tube to predict the outlet salt concentration and local scale thickness. The initial scale distribution was divided into 13 points that could be individually adjusted, and linear interpolation was used between the points. There were two restrictions on the scale

Figure 4. Typical appearance of the scales prior to the dissolution experiments. The left image is taken from the sight glass at 1.9 m in direction upward along the tube, with a scale thickness measurement device in the top right corner. The right image is taken from the sight glass at 3.9 m in direction downward along the tube, with a mixed cup located at the junction between the tube and the bottom of the shell. 9481



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The density of water was taken from the literature and contains all of the temperature dependence. The accuracy of the correlation was within 1%. Diffusivity is a difficult property to measure, and literature data for diffusivity is rare, especially because the solution used here contained a mixture of two salts. The diffusivity of NaHCO3 and Na2SO4 in water was measured by Vinograd and McBain,33 and the diffusivity of HCO3− and HCO32− in water was simulated by Zeebe.34 From these two studies, the diffusion coefficient was estimated to be 1·10−9 m2/s at 25 °C. Both studies indicated that diffusivity depends on the concentration of salt in the solution, but this dependence was not sufficiently strong to be considered important compared to the uncertainty in the diffusivity at zero concentration. The diffusion coefficient has a strong dependence on temperature, and the following relation was used:26

Figure 5. Scale distributions used in experiments W1 and W5 exemplifying the two different distribution categories: exponential and plateau.

D ABT1 D ABT2

thus, the outlet salt concentration never reached the inlet concentration. This phenomenon was modeled by adding a thick layer of scales in the lowest part of the tube (Figure 5). Physical Properties of the Fluid. When mass transfer coefficients were fitted, mean values between the inlet and outlet were used as described above. However, the physical properties and operational conditions at the inlet were used for the remaining analyses performed in this work. The values of viscosity (μ), density (ρ), specific heat (cp), and thermal conductivity (kt) are well-known for pure water. For the two experiments using a salt solution as the solvent, the viscosity was calculated from a correlation for sodium carbonate solutions by Abdulagatov et al.,31 whereas the specific heat and thermal conductivity were assumed to be the same as pure water. The density of the scales was measured to be 1800 kg/ m3. The solubility limit of the salt and the dependence of the solubility on temperature were taken from Shi and Rousseau.32 A correlation for the density of the salt solution as a function of the salt mass fraction (ω) and temperature was developed from our own measurements:

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RESULTS AND DISCUSSION Dissolution Behavior and Model Performance. Typical behaviors for the dissolution process using water (W5 and W6) and a salt solution (S2) as the solvent are shown in Figures 6, 7, and 8, respectively. In the left-hand figures, the outlet salt concentration is expressed as a mass fraction. The experimental values are presented as bars representing individual samples. The dashed line represents the solubility limit, which indicates the maximum possible amount of salt that can be dissolved in the wash liquid. The outlet concentration was initially higher than estimated by the model for all experiments. The presence of residual (saturated) salt solution in the inlet distributor prior to the dissolution experiment contributed to this behavior. During the cooling of the tube before the dissolution experiment, water condensed, leading to a thin saturated layer of liquid covering the scales, which also contributed to this behavior. In addition, the flow was not homogeneously distributed in the beginning until the falling film was fully developed around the circumference. Due to these uncertainties, the high initial concentrations were not considered when fitting the model. The salt mass fraction did not reach the inlet concentration in

(15a)

ρwater Vfactor(1 − ω)

(16)

where Tc is the critical temperature and n = 6 when using water as the solvent.

Vfactor = 0.8341ω3 + 0.5579ω 2 + 0.0267ω + 1.000 ρsalt =

⎛ T − T2 ⎞n =⎜ c ⎟ ⎝ Tc − T1 ⎠

(15b)

Figure 6. Model fitting of the outlet salt concentration (left) and scale thickness measurements (right) for experiment W5. 9482



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Figure 7. Model fitting of the outlet salt concentration (left) and scale thickness measurements (right) for experiment W6.

Figure 8. Model fitting of the outlet salt concentration (left) and scale thickness measurements (right) for experiment S2 with salt solution as the solvent.

the end because of the thick ring of scales at the bottom of the evaporator, as shown in Figure 4. For experiments with salt solution as the solvent (Figure 8), as expected, the rate of dissolution was observed to be significantly lower because the driving force was decreased by the high salt concentration in the bulk flow. The measured and modeled values for local scale thickness are shown in the right-hand figures of Figures 6, 7, and 8. Measurements of scale thickness and visual observations revealed that the scales started to dissolve at the inlet, where the concentration difference was the highest. Thus, the rate of decrease in thickness was higher at 1.8 m than at 3.8 m from the inlet. A clean front, i.e., the edge between the scaled surface and clean surface, was observed to move downward over time as more salt dissolved. This front moved faster when the scale thickness was lower. As the front moved, the contact time between the scales and solvent decreased, and thus, the outlet concentration also decreased. Therefore, the plateau at t = 1.5− 3 min in Figure 6 (left) for experiment W5 can be explained by the plateau at 1.1 m in the initial scale distribution shown in Figure 5; the contact time will remain constant when the clean front has reached the thicker scales at the plateau, resulting in a constant outlet concentration until the plateau is dissolved. The mass transfer coefficient was fitted to all experiments; however, the agreement between the model and experimental data varied. The best fits were achieved for the experiments that had a plateau distribution, as, for example, W5 in Figure 6.

However, some experiments had an uneven distribution of scales around the circumference, as, for example, W6 in Figure 7, which led to larger uncertainties. To determine the uncertainties and model performance, a higher and lower value for the mass transport coefficient were fitted (red and blue lines in Figures 6, 7, and 8). For experiment W5, it was not possible to perfectly predict both the outlet salt concentration and the two local scale thickness measurements, as shown in Figure 6. The model did not agree with the two intermediate scale thickness measurements at 3.8 m. The model displayed better agreement with the scale thickness measurements for experiment W6. However, here it was not possible to fit the outgoing concentration (Figure 7). The initial thickness at 1.8 m was higher than expected compared to the total amount of scales dissolved from the tube. If the initial thickness at 1.8 m is allowed to be lower, then a good fit for both the outgoing concentration and thickness measurement at 3.8 m can be achieved (using the same value of k). This discrepancy can be explained by an uneven distribution of scales around the circumference, i.e., the scale thickness was higher only where the measuring device was located. Because there are significant deviations between the model and experimental data, one can ask if a more advanced model would provide a better fit. However, there were no clear patterns in the deviations that could motivate a different model. For example, the deviations did not correlate with the Reynolds number. For the scale thickness measurements, the model 9483



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Table 1. Experimental Conditions and Results for Experiments with Water (W) and Salt Solution (S) as Solvents Exp. #

Tavg,in (°C)a

Tavg,out (°C)a

Γ (kg/ms)

W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 S1 S2

40 76 45 84 82 63 84 63 76 60 80 76

53 76 48 81 78 75 84 72 78 57 81 80

0.15 0.14 0.16 0.12 0.29 0.29 0.4 0.42 0.29 0.2 0.33 0.19

μ ×103 (Pa·s) k × 106 (m/s) 0.655 0.374 0.598 0.338 0.346 0.447 0.338 0.447 0.374 0.467 0.678 0.624

50 70 37 80 90 70 110 90 90 50 70 80

± ± ± ± ± ± ± ± ± ± ± ±

15 20 10 15 10 15 15 15 20 15 25 15

D × 109 (m2/s)b δtot × 105 (m)c δd × 105 (m)d 1.3 2.6 1.4 3.0 2.9 2.0 3.0 2.0 2.6 1.9 2.8 2.6

31 26 31 24 37 38 44 46 37 31 39 31

2.6 3.7 3.9 3.8 3.2 2.9 2.8 2.2 2.9 3.8 4.0 3.2

δd (% of δtot)

τr (Pa)e

8 14 12 16 9 8 6 5 8 12 10 10

3.04 2.45 3.01 2.25 3.53 3.67 4.18 4.47 3.57 3.03 4.35 3.35

Time-averaged. bDiffusivity according to eq 16. cFalling film thickness according to eqs 3 and 4. dCalculated according to eq 2. eWall shear stress, τr, calculated as τr = ρgδtot.

a

Figure 9. Experimental mass transfer coefficient as a function of temperature (left) and specific mass flow rate (right). The error bars for k are the uncertainty intervals listed in Table 1. T was calculated as the average of Tavg,in and Tavg,out, where the error bars (left) show the span of Tavg,in and Tavg,out. The two filled points are the experiments with salt solution as the solvent.

possible to fit the remaining experiments to the model; W2 and W9 exhibited the highest uncertainties, and W3−W5 exhibited the lowest. Because the salt was a mixture of sodium carbonate and sodium sulfate, experiments W4−W8 were analyzed to determine the composition of both solutions used during buildup and the outgoing salt from the dissolution experiments. All samples had a salt composition with a sodium carbonate mole fraction of 0.87 ± 0.01, which means that all experiments had the same crystal species and no shift in composition that might influence the results. The number of comparable studies is limited, especially for long contact times (deep penetration). However, the dissolution of benzoic acid into a falling film was studied by Kramers and Kreyger,14 who obtained results on the same order of magnitude as this study. Benzoic acid was also studied by Tamas et al.,35 who obtained similar results; however, that study involved dissolution from tablets. An important assumption in the present study is that the dissolution can be modeled using film theory, which assumes sufficient mixing of the bulk phase by waves or other disturbances. The lowest Reynolds number is 916 in experiment W1. Stirba and Hurt20 have found that the dissolution rate was significantly improved by mixing and turbulence at low

displayed similar deviations at both 1.8 and 3.8 m, i.e., there was no clear over- or underestimation of the dissolution rate for a specific contact time. Most of the differences between the model and experiments can likely be explained by uncertainties in the experimental data, as it is always challenging to achieve precise control of all parameters when conducting large-scale experiments. Mass Transfer Results. Results for the fitted mass transfer coefficients for all experiments are listed in Table 1 along with the operational conditions. As previously mentioned, there were uncertainties in the fitting of the mass transfer coefficient; these uncertainties are reflected by the uncertainty interval listed in Table 1. The temperature fluctuated during the experiments due to difficulties in cooling the large equipment. These fluctuations are reflected by the differences in the inlet and outlet temperatures. To improve the model performance and account for the temperature fluctuations, the experiments were also fitted using a temperature-dependent mass transfer coefficient. However, no clear improvement was achieved compared to a constant mass transfer coefficient. Furthermore, the error created using a constant coefficient at varying temperatures was within the uncertainty interval noted above. The result from experiment S1 did not fit well with the model; thus, the value of k should be considered with caution. It was 9484



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Table 2. Comparing Experimental Data with Predictions by Heat Transfer Correlations from Schnabel and Schlünder29 using the Chilton−Colburn Analogy Exp.

S&S evaporation correlation

S&S heating correlation

Exp. #

Re

Pr

Sc

Sh

Nu

Sh

dev from exp (%)

Nu

Sh

dev from exp (%)

W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 S1 S2

916 1499 1071 1420 3349 2595 4734 3758 3105 1711 1947 1218

4.3 2.3 3.9 2.1 2.2 2.8 2.1 2.8 2.3 3.0 4.2 3.9

507 149 424 115 122 228 115 228 149 252 215 220

1.36 0.67 0.87 0.61 0.73 0.97 0.84 1.25 0.86 0.75 0.83 0.99

0.21 0.17 0.20 0.17 0.18 0.19 0.19 0.21 0.18 0.19 0.22 0.20

1.00 0.68 0.95 0.63 0.69 0.84 0.73 0.91 0.74 0.82 0.82 0.77

−26 2 10 3 −5 −13 −13 −27 −14 9 −1 −22

0.21 0.20 0.21 0.18 0.26 0.26 0.30 0.30 0.26 0.22 0.27 0.21

1.04 0.70 1.06 0.70 1.01 1.12 1.13 1.30 1.04 0.98 0.98 0.82

−24 16 14 15 39 16 35 4 21 30 18 −17

Reynolds numbers, and the film could not be considered laminar even with Reynolds numbers as low as 300. This indicates that the mixing of the bulk flow was sufficient for film theory to be used. The thickness of the fictive diffusion film can be calculated from the mass transfer coefficient using the estimated value for the diffusion coefficient; these results are shown in Table 1. This thickness corresponds to 5−16% of the total film (estimated from eqs 3 and 4), and the thickness generally decreased for higher Reynolds numbers. If the thickness of the diffusion film can be assumed constant throughout the entire range of concentrations, the contact time, i.e., the distance along the surface, should not change the value of the mass transfer coefficient k. This also means that k should not be influenced by a high concentration of salt in the incoming solvent, which was tested in experiments S1 and S2. The mass fractions of salt for S1 and S2 were 0.153 and 0.126, respectively. In S1, k was lower than for similar conditions with pure water (70 μm/s compared to 90 μm/s in W5); however, because the result from this experiment was unsatisfying, as discussed above, the low value might be explained by other factors. S2 had better agreement with the other experiments (80 μm/s compared to 80 μm/s in W4 or 90 μm/s in W5). Thus, no evidence was found to reject the film theory hypothesis under the conditions tested. The two parameters that were varied in the different experiments were the temperature and specific mass flow rate. The dependence of the mass transfer coefficient on these two parameters is shown in Figure 9; the mass transfer coefficient is positively correlated with both parameters. The positive flow dependence indicates that turbulent conditions are present, and this result further supports the assumption that film theory is a reasonable approximation. The strong temperature dependence can be explained by the combined effect of the increase in diffusivity and the decrease in viscosity. The decreased viscosity decreases the thickness of the diffusion film, and this decreased thickness in combination with increased diffusivity results in increased mass transfer at higher temperatures. It would have been interesting to investigate a wider range of temperatures and specific mass flow rates; however, such an investigation was not possible with the current experimental setup. The positive dependence of the mass transfer coefficient on the specific mass flow rate may also indicate mechanical removal, in which case the “dissolution” (or rather removal rate) should be related to wall shear stress (Table 1). Thus, the

mechanical removal rate for water and salt solutions that exert equal shear stress on the wall should be equal. The effect of mechanical removal can be analyzed by comparing experiment S2 with W2 and W5 because S2 had conditions (T, Γ, and τr) in between the two water experiments. The mass transport coefficient was 80 μm/s for S2 and 70 and 90 μm/s for W2 and W5, respectively, i.e., the salt solution exhibited no deviation. As the driving force (concentration difference) is significantly lower for a salt solution, these experiments had a significantly lower dissolution rates, even though the shear stress was equal or higher to the water case. This result means that the removal was completely dominated by dissolution and that mechanical removal had only a marginal effect, if any. Heat and Mass Transfer Analogy. The analogy with heat transfer is a way of validating and predicting the mass transfer coefficient. The heat transfer coefficient for vertical falling films has been extensively studied for water, and many correlations exist, mainly for evaporative conditions. In Table 2, the experimental data are compared to predictions from the Chilton−Colburn analogy in combination with heat transfer correlations by Schnabel and Schlünder29 (eqs 8−14). Even though the Sherwood number is a measure of the mass transfer in dimensionless form, this number is also dependent on the diffusivity and viscosity of the fluid. These quantities are more dependent on temperature than the mass transfer coefficient. This means that Sh will increase while k will decrease at higher Sc (and lower temperatures). Experimental and predicted Sherwood numbers are shown in Figure 10 to obtain a better idea of which heat transfer correlation provides the most accurate predictions. In general, the correlation for nonevaporative conditions overpredicts the mass transfer coefficient, whereas the correlation for evaporative conditions underpredicts the mass transfer coefficient at high Schmidt numbers. In theory, the physics behind the correlation for nonevaporative conditions is more analogous to the dissolution process because it does not include gas phase interactions. However, the correlation for evaporative conditions displays slightly better agreement with the experimental data in general, as all predictions are within 30% accuracy. To examine the performance of the evaporation correlation in more detail, the experimental data are shown as a function of the Reynolds number together with the correlation in Figure 11. Because all data points have different Schmidt numbers, the predictions are plotted as a line for every hundred Sc between 100 and 500 to cover the span of the experimental data. The 9485



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thickness of the fictive diffusion film corresponded to 5−15% of the total film thickness, (3) the mass transfer coefficient could be predicted from the heat transfer coefficient using the Chilton−Colburn analogy, (4) the measured mass transfer coefficient was predicted to within 30% accuracy using the Chilton−Colburn analogy together with the evaporative heat transfer correlation by Schnabel and Schlünder,29 and (5) in accordance with the predictions, the mass transport coefficient showed a positive dependence on both the specific mass flow rate (or Reynolds number) and temperature. In addition to a greater understanding of the dissolution process in falling films, these findings can help optimize the cleaning of industrial evaporators, which will in turn increase equipment performance and availability.

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AUTHOR INFORMATION

Corresponding Author

*E. Karlsson. E-mail: [email protected].

Figure 10. Comparison of experimental data and predictions from the heat transfer correlations by Schnabel and Schlünder.29

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was cofunded by the Swedish Energy Agency, Valmet Power AB, Troëdssons research foundation, Bo Rydins research foundation, and Chalmers Energy Initiative.

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Figure 11. Experimental data together with predictions from the evaporative heat transfer correlation by Schnabel and Schlünder.29 The labels indicate the Schmidt number at each experimental point, with colors corresponding to the closest Sc prediction, displayed as lines.

heat transfer correlation predicts the flow regime to be turbulent (because the Sherwood number increases with increasing Reynolds numbers) but close to the transition between laminar and turbulent flow. To summarize, the simplicity of the film theory model allows for industrial implementation, for example, to simulate evaporator cleaning, with only one input parameter for the dissolution, i.e., a mass transfer coefficient. If unknown, this mass transfer coefficient can be estimated from the Chilton− Colburn analogy in combination with the evaporative heat transfer correlation by Schnabel and Schlünder.29

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CONCLUSIONS The dissolution of sodium scales in an industrial-like pilot falling film evaporator was measured, and the following key findings were obtained: (1) the mass transfer was diffusioncontrolled and could be described by film theory, where the concentration difference between the wall (saturated) and the bulk flow (assumed mixed) was the driving force, (2) for the studied conditions (approximately 1000 < Re < 5000), the

NOMENCLATURE A = surface area of the solid−liquid interface (m2) c = concentration (kgsalt/m3) cp = specific heat (W/(m2·K)) D = diffusion coefficient (m2/s) g = gravitational acceleration (m/s2) h = heat transfer coefficient (W/(m2·K)) i = index (−) J = mass flux or dissolution rate (kg/(m2·s)) j = Chiltion−Colburn j-factor (−) k = mass transport coefficient (m/s) kt = thermal conductivity (W/(m·K)) m = mass (kg) n = number (−) r = radial coordinate (m) T = temperature (°C) t = time (s) u = velocity (m/s) x = coordinate (m) z = coordinate (m) Γ = specific mass flow rate of liquid per unit width (kg/(m· s)) δ = film thickness (m) μ = dynamic viscosity (Pa·s) ν = kinematic viscosity (m2/s) ρ = density (kg/m3) τr = wall shear stress (Pa) ω = mass fraction of salt (kg/kg) Nu = Nusselt number Sh = Sherwood number Pr ≡ (cpμ/kt) = Prandtl number Re ≡ (4Γ/μ) = Reynolds number Sc ≡ (μ/ρD) = Schmidt number

Subscripts

* = saturated avg = average

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(20) Stirba, C.; Hurt, D. M. Turbulence in falling liquid films. AlChE J. 1955, 1, 178−184. (21) Wasden, F. K.; Dukler, A. A numerical study of mass transfer in free falling wavy films. AlChE J. 1990, 36, 1379−1390. (22) Killion, J. D.; Garimella, S. A critical review of models of coupled heat and mass transfer in falling-film absorption. Int. J. Refrig. 2001, 24, 755−797. (23) Karlsson, E.; Broberg, A.; Åkesjö, A.; Gourdon, M. Dissolution rate of sodium salt scales in falling film evaporators. Presented at the Heat Exchanger Fouling and Cleaning Conference, Budapest, June 9−14, 2013. (24) Fuchs, E.; Boye, A.; Stoye, H.; Mauermann, M.; Majschak, J.-P. Influence of the film flow characteristic on the cleaning behavior. Presented at the Heat Exchanger Fouling and Cleaning Conference, Budapest, June 9−14, 2013. (25) Plett, E. A. Cleaning of Fouled Surfaces. In Fouling and Cleaning in Food Processing; University of Wisconsin, Madison, WI, 1985; pp 288−311. (26) Welty, J. R.; Wicks, C. E.; Rorrer, G.; Wilson, R. E. Fundamentals of momentum, heat, and mass transfer; John Wiley & Sons Ltd: New York, NY, 2009. (27) Nusselt, W. Die oberflächenkondensation des wasserdampfes. Z. Ver. Dtsch. Ing. 1916, 60, 541−546. (28) Brauer, H. Strömung und Wärmeübergang bei Rieselfilmen; VDIVerlag: Düsseldorf, 1956. (29) Schnabel, G.; Schlünder, E. U. Wärmeübergang von senkrechten Wänden an nichtsiedende und siedende Rieselfilme. Verfahrenstechnik 1980, 14, 79−83. (30) Johansson, M. Heat Transfer and Hydrodynamics in Falling Film Evaporation of Black Liquor. Ph.D. Dissertation, Chalmers University of Technology, Göteborg, Sweden, 2008. (31) Abdulagatov, I. M.; Azizov, N. D.; Zeinalova, A. B. Density, apparent and partial molar volumes, and viscosity of aqueous Na2CO3 solutions at high temperatures and high pressures. Z. Phys. Chem. 2007, 221, 963−1000. (32) Shi, B.; Rousseau, R. W. Crystal properties and nucleation kinetics from aqueous solutions of Na2CO3 and Na2SO4. Ind. Eng. Chem. Res. 2001, 40, 1541−1547. (33) Vinograd, J. R.; McBain, J. W. Diffusion of electrolytes and of the ions in their mixtures. J. Am. Chem. Soc. 1941, 63, 2008−2015. (34) Zeebe, R. E. On the molecular diffusion coefficients of dissolved CO2, HCO3 and CO3 and their dependence on isotopic mass. Geochim. Cosmochim. Acta 2011, 75, 2483−2498. (35) Tamas, A.; Martagiu, R.; Minea, R. Experimental Determination of Mass Transfer Coefficients in Dissolution Processes. Chem. Bull.“Politeh.” Univ. Timisoara 2007, 52, 133−138.

b = bulk c = critical d = diffusion h = heat transfer i = interface in = relate to inflow lam = laminar flow out = relate to outflow tot = total trans = laminar-turbulent transition turb = turbulent flow x = coordinate (m) z = coordinate (m)

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REFERENCES

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