Empirical Mass and Kinetic Models for the Flash Evaporation of NaCl

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Cite This: Ind. Eng. Chem. Res. 2018, 57, 6115−6122

Empirical Mass and Kinetic Models for the Flash Evaporation of NaCl−Water Solution Yong Liu,*,† Qiong Luo,† Guodong Wang,† Siyuan Zhao,‡ Xianlong Li,† and Ping Na‡ †

School of Environmental Science and Engineering and ‡School of Chemical Engineering and Technology, Tianjin University, Tianjin 300350, China

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S Supporting Information *

ABSTRACT: This paper attempts to provide empirical mass and kinetic models for the flash evaporation of sodium chloride (NaCl) aqueous solution based on experimental phenomena. The models have nine parameters and six affecting factors including initial temperature, operation pressure, NaCl mass fraction, solution depth, evaporator diameter and time. On the basis of a large number of flash evaporation experimental data from various literatures, the mass model parameters were optimized and validated. After optimization of the model parameters with 283 sets of literature experimental data, the average relative error between the model values and the experimental data is about 5.7%. And a statistical method proved the mass model is well posed. The verification with other 215 sets of literature experimental data showed the mass model is in good agreement with flash evaporation phenomena, and the average relative error between the model values and the experimental data is about 8.3%. Then, the kinetic model of flash evaporation was obtained according to the empirical mass model. Finally, the analysis of these models indicated that the increase of initial temperature or evaporator diameter and the decrease of operating pressure are in favor of evaporation. Although the increase of solution depth can improve the evaporated mass, the corresponding evaporation efficiency will be slightly reduced. And the increase of salt content is having a detrimental effect on the evaporation of NaCl−water solution. In addition, the influence of salt content on evaporation at higher operating pressure is more obvious than that at lower operating pressure. The above results show that these models proposed in our work have high accuracy, wide practicability, and good rationality.

1. INTRODUCTION Flash evaporation belongs to an evaporation process with the intense heat and mass transfer phenomena. If the temperature of solution is higher than its boiling point, flash evaporation will happen. Flash evaporation will result in partial vaporization of solution and rapid decrease of solution temperature. According to the feeding state, flash evaporation can be divided into spray flash evaporation and static flash evaporation. Flash evaporation has attracted extensive attention because of its importance in chemical separation industry, materials industry, and energy industry. Nowadays, flash evaporation has been widely used in the field of crystallization,1−4 desalination5−7 and concentration.8 At present, flash evaporation process has been widely studied. Miyatake et al.9,10 used water solution to study the flash evaporation process. They found that static flash evaporation could be divided into two stages, such as fast evaporation stage and gradual evaporation stage according to the temperature drop rate. Peterson et al.11 used an interferometer and highspeed photography to observe the temperature field in the static flash evaporation of Fron-11. Popov et al.12 had used a fine wire thermocouple with 25 μm diameter to study the temperature profile near the interface of water at low © 2018 American Chemical Society

evaporation rate conditions. The authors found that there exists a temperature jump in the region near the water−vapor interface; moreover, the temperature jump at the interface increases with the increase of evaporation rate. Liu et al.13 used an infrared thermal imager and a high speed camera to investigate the flash evaporation of salt water droplets under vacuum conditions. The results showed that the evaporation rate decreases with the increase of salt content. Saury et al.14 had studied the flash evaporation process of a water film with the initial water height of 15 mm, superheat ranging from 1 to 35 K, and initial temperature from 40 to 74 °C. In 2005, Saury et al.15 had investigated the influence of initial solution height and depressurization rate on the flash evaporation of water. Zhao et al.16 had carried out many flash evaporation experiments with the initial NaCl mass fraction varying from 0 to 0.264 and superheat from 1.8 to 43.4 K. They found that both evaporation and steam-carrying effects influence the change of the equilibrium concentration of NaCl. Yan et al.17 Received: Revised: Accepted: Published: 6115

January 25, 2018 April 13, 2018 April 17, 2018 April 17, 2018 DOI: 10.1021/acs.iecr.8b00402 Ind. Eng. Chem. Res. 2018, 57, 6115−6122

Article

Industrial & Engineering Chemistry Research had conducted static flash evaporation experiments for NaCl− water solution with initial NaCl mass fraction from 0 to 0.26, initial height from 0.1 to 0.4 m, and superheat ranging from 1.8 to 49.5 °C. The results indicated that NEF (nonequilibrium function) with crystallization will reach a low equilibrium value. Wang et al.18 used aqueous NaCl solution as working fluid in their continuous flash evaporation experiment with superheat from 0.3 to 11.5 °C to study the bubble morphology and evaporation rate. Yang et al.19 had used salt water with the initial mass fraction of 0.05 to 0.26 to explore the flash evaporation phenomena under different conditions. They believed that when superheat and initial liquid level are same, the effect of salt mass fraction on evaporation mass is very weak, and the increase of initial liquid level can promote total evaporation mass but will slightly reduce evaporation efficiency, and evaporation mass will linearly increases with superheat. For the desalination of seawater or wastewater and the industrial crystallization of NaCl, the accurate control of evaporation process seriously affects the process efficiency and the product quality of salt. Therefore, evaporation kinetic model plays an important role in the process design of desalination and NaCl crystallization. However, nowadays, the evaporation kinetics with wide applicability and good accuracy is very rare. The well-known kinetic theory of gases (KTG) was often used to analyze the evaporation process for decades at molecular level.20 Hertz−Knudsen equation, based on the kinetic theory of gases, can be used to estimate evaporation flux with the interfacial temperatures and evaporation coefficient. On the basis of Hertz-Knudsen equation, in ANSYS Fluent software, an evaporation−condensation model with the coefficients of 0.1 was given. However, in the actual simulation process, the evaporation coefficient needs to be corrected in order to match the simulation results with the experimental phenomena. In fact, the evaporation coefficient of water from a large number of literature reports is different and ranges from 0.01 to 1.21 The reliability of the evaporation−condensation model for process simulation and design is often not high unless there is an accurate evaporation coefficient obtained from experiments. In order to avoid the uncertainty of evaporation coefficient, based on the transition probability concept of quantum mechanics, called statistical rate theory (SRT), Ward et al.22 had develop an expression to predict evaporation flux. The authors believed the new model values are better than those obtained by the KTG compared with experimental observations. But this model requires accurate vapor temperature and liquid temperature near the interface. Up to now, it is difficult to determine the temperature of interfacial gas−liquid phase.11,12 Therefore, the model is quite inconvenient for engineering application. Furthermore, at present the model is mainly used for single component case such as fresh water or carbon monoxide. For a multicomponent system such as NaCl−water solution, seawater, and wastewater, it would be difficult to truly predict evaporation flux using this new model. Some authors had used flash evaporation experimental data to present a correlation. On the basis of the experiments of fresh water and saline water (3.5% NaCl solution), under the conditions of initial temperature from 25 to 80 °C, superheat from 0.5 °C to l0 °C, and solution depth from 165 mm to 457 mm, Gopalakrishna et al.23 had develop an empirical correlation of the total mass of vapor released during flash evaporation. Owing to the narrow range of experimental conditions, the applicability of Gopalakrishna’s model should be not high. Through many flash evaporation

experiments, Saury et al.14 and Yang et al.19 had respectively proposed the formula of final evaporation mass. But these models cannot describe flash evaporation process. In order to provide a more accurate and widely applicable evaporation model for the flash evaporation of NaCl−water solution, based on the typical change curve of steam mass versus time, this paper tries to present empirical mass and kinetic models. These models include six influencing factors, such as initial temperature, operation pressure, solution depth, evaporator diameter, salt content, and time. Compared with Gopalakrishna’s model or the evaporation−condensation model, the empirical mass and kinetic models presented in this work are more consistent with the experimental phenomena. Finally, in this paper the influence of various factors on evaporation mass and rate will be analyzed under a certain condition. These new evaporation models should be an important tool for the simulation and design of the flash crystallization process of NaCl−water solution and the desalination process of seawater or wastewater.

2. EMPIRICAL MASS MODEL For comparison of the evaporated mass in various literatures and model optimization, the evaporated mass per unit volume of solution will be used in this study. The evaporated mass per unit volume of solution can be calculated using the following formula. m mv = (1) AH On the basis of the flash evaporation experiments in the literatures,14,24 the typical change curves of vapor mass versus time are shown in Figure 1. According to these phenomena, the relationship between vapor mass and time is in good agreement with exponential function (eq 2). m v = m vf (1 − exp(−p2 t ))

(2)

On the basis of fresh water and saline water flash evaporation experiments, Gopalakrishna et al.23 had proposed a dimensionless equation of final evaporation mass like eq 3, which is mainly related to solution depth, superheat, salt content, and solution properties including density, specific heat, viscosity, thermal conductivity, vaporization heat. m vf = 0.8867ρl Ja Ja T 0.05 Pr −0.05 (ΔP /H )−0.05 (1 + c)0.06 (3)

Through various experimental studies on pool brine flash evaporation, Yang et al.19 had also obtained a more simple correlation formula of final evaporation mass, which can be written as eq 4. m vf = 0.914ρl Ja 0.95 Pr −0.24 (H /D)−0.06

(4)

According to pool water flash evaporation experiments under various initial temperature and operation pressure, Saury et al.14,15 had found final evaporation mass can be well expressed as the following equation. m vf = ρl CpΔT /h

(5) 14

When solution depth was 15 mm, Saury et al. found the final evaporation mass under various conditions is proportional to superheat, and the value of ρlCp/h is a constant of about 1.558 kg·m−3·K−1, indicating that solution temperature and operating pressure have weak effect on the value of ρlCp/h even 6116

DOI: 10.1021/acs.iecr.8b00402 Ind. Eng. Chem. Res. 2018, 57, 6115−6122

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Industrial & Engineering Chemistry Research

the above viewpoint about flashing time, p2 in eq 2 can be empirically represented as a power function like eq 7. p2 = a5Te a6ΔT a7H a8Da9

(7)

On the basis of eqs 2, 6, and 7, the evaporation mass per unit volume of solution at t moment can be written as m v = a1ΔT a2H a3Da4 (1 − exp(−a5Te a6ΔT a7H a8Da9t )

(8)

3. EVAPORATION MASS CALCULATION METHODS Saury et al.14,15 estimated the evaporated mass under various time using a simple formula like eq 9. ⎛ 1 + CpT /h ⎞ ⎟ m = ρl AH ⎜⎜1 − 1 + CpT0/h ⎟⎠ ⎝

(9)

or eq 10 ⎧ ⎡ C ⎤⎫ m = ρl AH ⎨1 − exp⎢ − v (T0 − T )⎥⎬ ⎣ ⎦⎭ ⎩ h

(10)

The above two equations both were neglecting the second order terms during the formula deduction, which leads to some errors. In order to estimate evaporation mass between i and i + 1 moments more accurately, difference equation of eq 11 was used in this work. miCpTi − (mi − Δmi + 1)CpTi + 1 = Δmi + 1h

Figure 1. Change of the evaporated mass of experiments and eq 8 with time during water flash evaporation: (a) H = 0.06 m;24 (b−d) H = 0.015 m.14

(11)

Rearranging eq 11, we can get miCp(Ti − Ti + 1)

Δmi + 1 =

though the initial temperature ranges from about 40 °C to about 74 °C and the operation pressure ranges from 5000 to 20000 Pa. Later, Saury et al.15 examined the effect of solution depth, ranging from 25 mm to 275 mm, on the final evaporation mass through many flash evaporation experiments of water at 5000 Pa. They found that dmf/dH > 0 and d2mf/ dH2 < 0. For the flash evaporation of NaCl−water solution, Yang et al.19 had also found the similar phenomena mentioned by Saury et al.14,15 According to the law of mass transfer, it is clear that the cross section area of evaporator has an important influence on evaporation. According to the study by Yang et al.,19 the effect of salt concentration on evaporation is mainly reflected in superheat. Therefore, based on the analysis above, an empirical formula of final evaporation mass per unit volume of solution is mainly related to superheat, solution depth, and evaporator dimeter and can be simply expressed as a powder function like eq 6. m vf = a1ΔT a2H a3Da4

i = 0, 1, 2, ...

h − CpTi + 1

i = 0, 1, 2, ... (12)

Therefore, the total evaporation mass at i + 1 moments can be determined using eq 13. i

mi + 1 =

∑ Δmj + 1 (13)

j=0

For the evaporated data in the literatures, eq 9 was first used to obtain temperature values, and then eqs 11−13 were used to calculated the evaporated mass at various time. According to eq 10, the evaporated mass in the literature15 was corrected using the similar methods. On the basis of the temperature at various time, the evaporated mass in the literature 24,27,28 was calculated using eqs 11−13. The experimental data in the literature14,15,19,24,27,28 were obtained using scale reading method with the relative error of less than 5%. Finally, the evaporated mass per unit volume of solution was calculated according to eq 1. These literature experimental data are shown in Tables S1−S4. 14,19

(6)

Relaxation time, defined as the time from nonequilibrium state to equilibrium state, also known as flashing time or bubble time, plays an important role in tuning the boiling regimes.25 Through a large number of flash experiments with water in an evaporator, Guo et al.26 believed flashing time is mainly affected by superheat and solution depth. Miyatake et al.9,10 thought that flashing time is related to superheat and equilibrium temperature. Gopalakrishna et al.23 believed flashing time is related to superheat, solution depth, and solution properties. As stated earlier, the cross section area of evaporator affects evaporation rate and must affect flashing time. On the basis of

4. RESULTS AND DISCUSSION 4.1. Model Optimization and Verification. 4.1.1. Model Parameter Optimization. By use of the methods mentioned earlier, 283 sets of literature experimental data (shown in Tables S1−S3) under various conditions are shown in Figure 1 and Figure 2, respectively. Among them, the literature14 provided 156 sets of experimental data for fresh water, 49 sets of experimental data for aqueous salt solution were obtained from the literature, 19 and other 78 sets of 6117

DOI: 10.1021/acs.iecr.8b00402 Ind. Eng. Chem. Res. 2018, 57, 6115−6122

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Industrial & Engineering Chemistry Research Table 1. Statistics of Equation 8

a

Mp

M − Mp

Rea

p2

F

10 × FT

9

274

765

0.997

9695

19.1

Re = sum of residuals squares.

and the number of experimental data sets, respectively, and p2 is equal to (1 − sum of residuals squares)/(sum of experimental data squares). F can be calculated with the expression p2/Mp/ ((1 − p2)/(M − Mp)), and FT is the F value under the corresponding freedom degree with a significant level of 5%.29 It is generally believed that the model is well posed when p2 is greater than 0.9 and F is greater than 10 times of FT.29 According to Table 1, it is clear that the model statistics meet the above requirements, so the mass model of eq 8 and its parameters are quite nice. 4.1.2. Model Verification. The previous analysis shows that the proposed mass model has good regression effects. In order to show its applicability, this section will use other experimental data to verify the model. The flash evaporation experiments in the literature27 and the literature15 both were carried out using fresh water, but the former was square flash chamber of 0.125 m × 0.13 m, the initial solution height was 0.05 m, and the latter was cylindrical chamber with a diameter of 0.299m, whose solution level ranged from about 0.025 m to about 0.275 m. The flash evaporation experiments for NaCl−water solution in the literature19,28 were run 20 s and the experimental systems were same, but the experimental conditions are different. All of the experimental data used here were not included in the previous optimization process. Among them, 59 sets of experimental data were obtained from the literature,15 57 sets of experimental data were obtained from the literature,27 and 30 sets of experimental data were obtained from the literature.19 69 sets of experimental data were obtained from the literature.28 The number of the sets of experimental data is 215 in total (shown in Table S4). Figure 4a shows the experimental data and the calculated data by eq 8. The average relative error between the experimental data and the model values calculated by eq 8 is about 8.3%. The slope of the linear function and R2 are 1.027 and 0.967 (Figure 4a), respectively. This result proves that the proposed mass model is very nice and has good practicability. In order to compare with the effects of other existing evaporation models, Gopalakrishna’s formula23 and the evaporation−condensation model with the coefficient of 0.1, which has been used in Fluent software, both were used to estimate the evaporation mass under the literature conditions. For the evaporation−condensation model, numerical method was used to calculate the evaporation mass per unit volume of solution with the time step of 1.0 × 10−4 s. It was found that the errors of the calculated results can be ignored when the time step is less than or equal to 0.0001 s. Especially noted, the condensation effect was ignored during the numerical calculation of the evaporation−condensation model because the evaporation rate under those conditions was far greater than the condensation rate. The corresponding results calculated by the two evaporation models are shown in parts b and c Figure 4, respectively. Obviously, the linear relationships between the experimental data and the model values are very poor compared with Figure 4a. Figure 4b indicates the slope and R2 are 0.778 and 0.722, respectively. The average relative error between the experimental data and the calculated values by Gopalakrishna’s

Figure 2. Experimental data from literature19 and the values estimated by eq 8 under various conditions for NaCl−water solution when t = 20 s.

experimental data for fresh water were obtained from the literature.24 On the basis of eq 8 and 283 sets of literature experimental data, the nine parameters were obtained using nonlinear optimization techniques. According to the physical meaning of parameters, the values of a1 and a5 must be greater than 0. On the basis of the basic law of evaporation,9,10,14,15,19,24 the values of a2, a4, a6, and a9 should be larger than 0, the values of a3, a7, and a8 should be less than 0; moreover, a3 should be in the range of −1 to 0 according the flash evaporation phenomena from the literature.15,19 With these restrictive conditions of the model parameters, after optimization, the optimal values of a1− a9 are 2.05, 9.72 × 10−1, −5.54 × 10−2, 8.61 × 10−2, 1.00 × 10+4, 1.95, −2.16 × 10−1, −1.09 × 10−2, and 7.85, respectively. The values calculated by eq 8 are shown in Figures 1 and 2, respectively. The overall fitting effect of the experimental data14,19,24 is shown in Figure 3. mv,E and mv,C in Figure 3

Figure 3. Overall fitting effect of the experimental data.14,19,24

represent the experimental data and the model values, respectively. There is a good linear relationship between the experimental data and the model values, and the slope and the regression index (R2) are 0.997 and 0.989 (seen in Figure 3), respectively. The average relative error between 283 sets of experimental values and the calculated data is about 5.7%. The results indicated that the values calculated by eq 8 are in very good agreement with the experimental data. In addition, a statistical method has been used to analyze the well posedness of eq 8, and the model statistics are shown in Table 1. Mp and M are the number of the model parameters 6118

DOI: 10.1021/acs.iecr.8b00402 Ind. Eng. Chem. Res. 2018, 57, 6115−6122

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Figure 4. Experimental data15,19,27,28 and the model values: (a) the model of eq 8; (b) Gopalakrishna’s formula; (c) the evaporation−condensation model.

model is about 38.5%. Figure 4c shows the slope and R2 are 0.845 and 0.851, respectively. The average relative error between the experimental data and the calculated values by the evaporation−condensation model is about 27.3%. Obviously, the average relative errors of the two evaporation models largely exceed the allowable maximum error of about 20% in chemical engineering design industry. However, the empirical mass model proposed in this work fully meets the error requirements. In comparison, the effect of the evaporation−condensation model seems to be better than that of Gopalakrishna’s model. The reason may be related to the narrower scope of Gopalakrishna’s experiment conditions.23 In the process of model calculation, it was found that for Gopalakrishna’s model, the effect of liquid level on evaporation is contrary to the literature experimental phenomena.15,19 4.2. Analysis of Mass and Kinetic Models. According to eq 8 and its parameter values, the empirical mass model for the flash evaporation of water or NaCl−water solution can be expressed as follows:

ΔT = T0 − Te

(17)

On the basis of eqs 14−17, the effects of initial temperature, operation pressure, NaCl mass fraction, solution depth, evaporator diameter, and time on the evaporation mass and rate of NaCl solution under various conditions can be simulated. The results are shown in Figures 5−9, respectively.

m v = 2.05ΔT 0.972H −0.0554D0.0861 (1 − exp( −104Te1.95ΔT −0.216H −0.109D7.85t )

Figure 5. Effect of temperature on evaporation mass and rate under the conditions of P = 15 000 Pa, c = 0.1, H = 0.1 m, and D = 0.15 m.

(14)

Then, the evaporation kinetic model can be easily obtained after taking the time derivative of eq 14, which can be written as follows:

On the basis of Figure 5, it is clear that when other factors are fixed, the higher the initial temperature is, the greater the final evaporation mass is (Figure 5a). In addition, at the same operating time, the evaporation rate increases with initial temperature (Figure 5b). This result is consistent with the experimental phenomena of the literature.14,19,24 The reason is that when other conditions are fixed, the increase of initial temperature directly leads to the increase of superheat. According to Figure 6, when other influencing factors are fixed, the decrease of operating pressure will lead to a larger final evaporation mass (Figure 6a). The reason is that the decrease of pressure leads to the decrease of the saturation temperature of solution, which causes the increase of the superheat. This phenomenon is why the evaporation of brine solution is usually operated at lower pressure and is also consistent with a large number of literature experiments.14,15 For evaporation rate, the effect of pressure is shown in Figure 6b. In the earlier stage of evaporation, the evaporation rate curve for 10 000 Pa is intersected with the rate curve of 15 000 Pa (Figure 6b). However, when operating pressure is 20 000 Pa, the rate curve is always below those for P = 10 000 Pa and P = 15 000 Pa in the time range of 0−1s (Figure 6b).

r = (2.05 × 104)Te1.95ΔT 0.756H −0.0663D7.94 exp( −104Te1.95ΔT −0.216H −0.0109D7.85t )

(15)

On the basis of eqs 14 and 15, the influence of various affecting factors on evaporation mass and rate can be analyzed. In order to investigate the effect of operating pressure and salt content on superheat, a formula for calculating the boiling point under various conditions is needed. On the basis of the boiling point diagram of NaCl−water solution under different salt concentration and pressure conditions,30 the Antoine equation of water,31 the boiling point of salt water solution at atmospheric pressure,32 and Duhring’s rule, the boiling point of NaCl−water solution can be expressed as follows: Te =

1657.46(1.00 + 0.32c − 1.60c 2 + 5.79c 3) − 227.02 7.07406 − log(P /1000) − 93.37c + 571.27c 2 − 1836.59c 3

(16)

Then the superheat can be calculated using eq 17. 6119

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Figure 8. Effect of evaporator dimeter on evaporation mass and rate under the conditions of T0 = 75 °C, P = 15 000 Pa, c = 0.1, and H = 0.1m.

Figure 6. Effect of pressure on evaporation mass and rate under the conditions of T0 = 75 °C, c = 0.1, H = 0.1 m, and D = 0.15 m.

From Figure 7, the increase of solution depth is beneficial to the final evaporation mass and the evaporation rate when other

rate, eventually resulting in the cross phenomenon among the rate curves. Figure 9 shows that salt mass fraction has influence on evaporation mass and rate. Moreover, operating pressure will obviously affect the influence of salt mass fraction on evaporation. When P = 5000 Pa, salt mass fraction, such as 0, 0.1 and 0.2, has a weak impact on evaporation mass and rate. After 5 s, the final evaporation mass is about 0.1322, 0.1317, and 0.1282 kg, respectively. The ratios of final evaporation mass for the salt mass fraction of 0 and 0.1 to that of the salt mass fraction of 0.2 are 1.031 and 1.027, respectively. And the evaporation rate curves almost coincide with each other. However, when operating pressure is 30 000 Pa, the final evaporation mass is distinctly different. After 5 s, the corresponding final evaporation mass for the salt mass fraction of 0, 0.1, and 0.2 is 0.0223, 0.0187, and 0.0112 kg, respectively. The corresponding ratios calculated using the similar method mentioned above are 1.991 and 1.670, respectively. It is very clear that salt concentration has a great influence on evaporation amount when operating pressure is higher. Of course, high operation pressure is not conducive to solution evaporation (Figure 6). The reason for this phenomenon should be related to the effect of pressure and salt concentration on the boiling point of NaCl−water solution.

Figure 7. Effect of solution depth on evaporation mass and rate under the conditions of T0 = 75 °C, P = 15 000 Pa, c = 0.1, and D = 0.15 m.

factors are kept constant. This phenomenon is in agreement with the results of the literature experiments.15,19,24 The fundamental reason is that higher solution depth leads to more heat energy, which can be used to vaporize more of solution at a certain saturation temperature. When solution depth is 0.1 m, 0.2 m, and 0.3 m, respectively, the corresponding final evaporation mass is 0.0654, 0.1259, and 0.1846 kg. The ratios of the final evaporation mass for 0.2 m and 0.3 m to that of 0.1m are 1.925 (less than 2) and 2.823 (less than 3), respectively. Clearly, the increase of solution depth will slightly reduce evaporation efficiency. Figure 8 a indicates that when other conditions are constant, if evaporator diameter is large, the final evaporation mass will be high. The reason is that big evaporator diameter will result in large solution volume; similar to the effect of liquid depth, it must have more energy for the vaporization of solution. Compared with Figure 7a, the increase of evaporator diameter can not only increase evaporation mass, but also shorten flashing time. That is why people often want to increase the diameter of evaporator. There is a cross phenomenon between the corresponding evaporation rate curves under different diameters (Figure 8b). This is because the larger the diameter is, the higher the evaporation efficiency is. The higher evaporation efficiency leads to a rapid decrease in evaporation

5. CONCLUSIONS On the basis of the typical flash evaporation phenomena and experimental data of fresh water and NaCl−water solution in many literature studies, this paper presents an empirical evaporation mass and kinetic models for the flash evaporation of NaCl−water solution. The results proved that this mass model and its parameters are well posed and can be used to accurately simulate the evaporation mass of aqueous NaCl solution. For 498 sets of literature flash evaporation experimental data, the average relative error between the model values and the experimental data is less than about 8.3%. The calculation accuracy of the models proposed in our work is much better than that of Gopalakrishna’s model or the evaporation−condensation model. The model analysis shows that the model is well consistent with the basic laws of the flash evaporation of water and NaCl−water solution. The increase of initial solution temperature, solution depth, and evaporator diameter can improve vapor mass. But the increase of liquid depth will reduce evaporation efficiency. However, the increase of the diameter of evaporator is beneficial to the rapid and highly efficient flash evaporation of solution. The increase of 6120

DOI: 10.1021/acs.iecr.8b00402 Ind. Eng. Chem. Res. 2018, 57, 6115−6122

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Industrial & Engineering Chemistry Research Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant 51478308), the Scientific Research Special Fund of Marine Public Welfare Industry (Grant 201405008), and the Natural Science Foundation of Tianjin (Grant 14JCYBJC23300).

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NOMENCLATURE A = horizontal cross-sectional area, m2 a1−a9 = correlation parameters c = salt mass fraction Cp = isobaric specific heat capacity, J·kg−1·°C−1 Cv = equivalent specific heat capacity, J·kg−1·°C−1 D = evaporator diameter, m H = initial height of liquid, m h = latent heat of vaporization, J·kg−1 Ja = Jakob number (=CpΔT/h) JaT = ρl CpΔT/ρvh Pr = Prandtl number of the liquid m = mass, kg mv = evaporated mass per unit volume, kg·m−3 mf = final evaporation mass, kg mfv = final evaporation mass, kg·m−3 P = operation pressure, Pa T = solution temperature, °C T0 = initial solution temperature, °C Te = saturation temperature, °C t = time, s r = evaporation rate per unit volume of solution, kg·m−3·s−1 R = evaporation rate, kg·s−1

Greek Symbols

■

Figure 9. Effect of salt content on evaporation mass and rate at various pressure when T0 = 75 °C, H = 0.1 m, and D = 0.15 m: (a) P = 5000 Pa; (b) P = 15 000 Pa; (c) P = 30 000 Pa.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.8b00402. Literature evaporation mass under various conditions (PDF)

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REFERENCES

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operating pressure and salt content will be detrimental to solution flash evaporation. In addition, the operating pressure has some influence on the effect of salt concentration on evaporation mass and rate.

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ρl = solution density, kg·m−3 ρv = vapor density, kg·m−3 Δmi+1 = steam mass between i and i + 1 moments, kg ΔP = pressure drop, Pa ΔT = initial superheat, °C

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Yong Liu: 0000-0002-6551-1608 6121

DOI: 10.1021/acs.iecr.8b00402 Ind. Eng. Chem. Res. 2018, 57, 6115−6122

Article

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DOI: 10.1021/acs.iecr.8b00402 Ind. Eng. Chem. Res. 2018, 57, 6115−6122