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Study on the effects of different carrier gases on characteristics of supersaturated environment in the one/multi-section growth tube Yan Yu, Junchao Xu, Jun Zhang, Guangchuang Chen, and Hui Zhong Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b03153 • Publication Date (Web): 12 Jan 2018 Downloaded from http://pubs.acs.org on January 18, 2018
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Study on the effects of different carrier gases on characteristics of supersaturated environment in the one/multi-section growth tube Yan Yu, Junchao Xu, Jun Zhang*, Guangchuang Chen, Hui Zhong Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, Southeast University,
Nanjing 210096, Jiangsu Province, China
ABSTRACT: The supersaturation profiles in one/multi-section growth tube were predicted under the conditions of different carrier gases for two kinds of typical supersaturation achieving methods. The results show that Lewis number (Le) of the carrier gas is an important factor affecting the supersaturation profiles. For the one-section growth tube, the carrier gas with smaller Le is beneficial to higher level and flatter trend of supersaturation profiles for method 1, where supersaturated environment is achieved by a cool saturated flow into a warm-walled growth tube. However, for the carrier gas of larger Le, the higher level but less stable supersaturation profiles are presented from method 2, where a warm saturated flow into a cold-walled growth tube. With the carrier gas of large Le (>1) from method 1 and the carrier gas of small Le ( Tw1 , Tw1 > Tw2 ...Twn-1 > Twn for method 2. Tw1
Tw3
Tw2
...
Twn
Tin
Tout
Figure 2. Scheme of multi-section growth tube
3. THERMAL AND MASS DIFFUSION PROPERTIES OF CARRIER GASES The thermal diffusivity of water vapor is smaller than its mass diffusivity in carrier gases oxygen, nitrogen, carbon dioxide and sulfur dioxide and their Lewis numbers (Le) are 0.8, 0.79, 0.49 and 0.34 at 298K, respectively. So for comparison, the carrier gas with helium that has a large Lewis number (Le=2.12 at 298K) was also employed. In addition, air as a common gas in industry was considered and its Le is 0.85 at 298K. For the mixture of water vapor and air, the functions of thermal and mass diffusion depending on temperature were calculated23. Similarly, for the mixtures of water vapor and other carrier gases, according to the thermal physical properties of different gases, the functions are derived by relevant parameters and relations of the Matheson gas data book28. And the functions of mass diffusivity with temperature are derived on the basis of the Fuller relation29. Those functions for various gases are listed in Table 1. The unit of at and av is cm2/s. T is the gas flow temperature and its unit is K. 5
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Table 1 Functions of thermal and mass diffusivity coefficients for various gases
Function of thermal diffusivity coefficient
Function of mass diffusivity coefficient
αv,He = 3.92 × 10 −5 T 1.75
Carbon dioxide
αt,He = 0.0038T + 0.64 αt,Air = 0.0014T − 0.19 αt,N 2 = 10−1 (0.0057T+0.39) αt,O2 = 10−1 (0.0059T+0.36) αt,CO 2 = 10−1 (0.0044T − 0.27)
αv,CO2 = 9.85 × 10−6 T 1.75
Sulfur dioxide
αt,SO 2 = 10−1 (0.0022T − 0.077)
αv,SO 2 = 7.97 × 10−6 T 1.75
Gas Helium
Air Nitrogen Oxygen
αv,Air = 1.18 × 10−5 T 1.75 αv,N 2 = 1.23 × 10−5 T 1.75 αv,O 2 = 1.23 × 10−5 T 1.75
4. NUMERIC MODELING
The supersaturation profiles in the growth tube were predicted by a two-dimensional model of heat and mass transfer with variable physical parameters. For the cylindrical geometry, the values of the gas flow temperature, T, throughout the growth tube are obtained by solution of the partial differential equation24, r ∂T 1 ∂ ∂T 2U [1 − ( )2 ] (r ) (1) = α t (T ) R ∂z r ∂r ∂r
Where r and z are radial and axial coordinates, respectively; R is tube radius; U is average velocity of gaseous mixture; and at (T ) is thermal diffusivity with gas flow temperature23. Profiles of the water vapor partial pressure, Pv , are determined by equations analogous to Eq.(1) by replacing at (T ) with mass diffusivity with gas flow temperature23, av (T ) . Then the mass diffusion balance equation is described as follows: r ∂P 1 ∂ ∂Pv 2U [1 − ( )2 ] v = α v (T ) (r ) (2) R ∂z r ∂r ∂r
Axial thermal diffusion and other second-order effects such as Stefan flow are ignored. The inlet profiles of vapor pressure are assumed to be uniform; and at the tube wall the vapor is assumed to be saturated. And the saturation ratio, S2, defined as the ratio of the partial pressure of the condensing water vapor, Pv, to its equilibrium vapor pressure, PT0 , at the gas flow temperature, T: 6
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S = Pv / PT0 (3)
A finite element analysis method was adopted for calculation. Considering the axial symmetry of the growth tube, this tube was reduced to a two-dimensional domain that corresponds to a growth tube half-plane. The schematic representation of the model is exhibited in Figure 3. The domain of the growth tube was divided into X × Y cells that were defined as X radial and Y axial space intervals of size ∆r and ∆z ( ∆r = 3.75 × 10 −3 cm and ∆z = 0.12cm ). In each finite mesh, the values of flow temperature T (r , z ) , water vapor partial pressure Pv (r , z ) were obtained by Eq.(1) and Eq.(2). And the values of saturation ratio S ( r , z ) were calculated according to Eq.(3). In each cell, the values of gas flow temperature, water vapor partial pressure and saturation ratio were assumed to coincide with their values calculated from the analytical solution of the classical Graetz problem at the central point of the cell. ∆r , ∆z
r=R
α v (r , z ),α t (r , z ), T (r , z ), Pv (r , z ), S (r , z )
r z
r =0
Figure 3. Scheme of the model of heat and mass transfer
5. RESULTS AND DISCUSSION 5.1Supersaturation profiles in the one-section growth tube
To discuss the calculated results briefly, all the distributions of saturation ratio presented in the following figures are the mean values of different radial positions. Fig.4 provides the distribution of S achieved by method 1 with different carrier gases. It could be found that the distribution of S with carrier gas He shows an obvious difference with others. Its S is always smaller than 1 under this condition, and decreases firstly then increases with 7
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the distance away from the inlet of the growth tube. For the carrier gases of SO2, CO2, O2, N2 and air, the distribution of S shows the trend of first increasing and then decreasing along the flow. However, the increasing rate is different among them. S in the carrier gas SO2 shows the fastest increasing and the slowest increasing in the carrier gas air. And the decreasing rate of S is also related to the composition of carrier gases. The decrease of S with the carrier gas SO2 is flatter. In addition, S shows a similar distribution among the carrier gases of N2, O2 and air. For carrier gases of SO2, CO2, N2, O2 and air, S is always more than 1 in this situation. However, the supersaturation level reveals a big difference among them, and the highest S are 2.2611, 1.8836, 1.4819, 1.4770 and 1.3621, respectively. Moreover, the smaller the peak of S is, the closer it is to the entrance. Actually it could be noted that the achievement of supersaturated environment depends on thermal and mass transfer between the water and flow. In method 1, the water evaporates from the hot water to the cold gas flow, leading to the increase of the partial pressure of water vapor Pv; the transfer of heat between water and gas flow makes the flow temperature T increase, it means that the increase of PT0 . For carries gas He, its Le is larger than 1, implying that T increases quickly in the early stage. And there is an exponential relationship between T and PT0 30. Thus the increase of PT0 is faster than that of Pv in the early stage, resulting in the decrease of S. Subsequently slow
increasing of T is presented, but Pv always increases because of more water evaporation into the gas flow. So S begins to improve after the nadir, but finally it is lower than 1 due to a bad characteristic in the mass transfer process. For other five gases, their values of Le are less than 1. So distribution of S shows an opposite 8
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change to that with He. For SO2, its Le is only 0.34 and it means that the water evaporates into the flow more easily. Thus S with it shows a quick increase in the early stage and keeps a high and stable level finally.
Tin=298K,Tw=333K
2.5
SO2 O2
CO2 Air
N2 He
2.0
Saturation ratio
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1.5
1.0
0
25
50
75
100
125
Axial location of the growth tube/cm
Figure 4. Distribution of saturation ratio by different carrier gases from Method 1
The distribution of S achieved by method 2 with different carrier gases is shown in Fig.5. It could be seen that for the carrier gases of He, air, O2 and N2, the distribution of S shows the trend of first increasing and then decreasing along the flow. For He (Le>1), the peak of supersaturation reaches near the entrance of the growth tube. Afterwards it drops sharply and then maintains a constant value at last. For carrier gases of air, O2 and N2, the distribution of S presents the similar but more stable trend, compared with method 1. The carrier gas He has a larger value of Le. So the process of cooling the flow is much faster than water vapor diffusing to the walls. Thus less time is necessary for the appearance of peak value and the high supersaturation level is easier to be obtained. For carrier gases of air, N2 and O2, their values of Le range from 0.5 to 1, indicating that they have relatively faster heat transfer and slower water vapor diffusion. In this case, the decreasing flow temperature causes PT0 reducing 9
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more rapid. Thus the region of supersaturation can also be realized although their values of Le are less than 1. It is worth noticing that supersaturation cannot be obtained in this method with the carrier gases CO2 and SO2. The distribution of S has been declining during the whole process. In method 2, heat and water vapor transmit from the flow to the cold wet wall at the same time. In addition, the two carrier gases have small Le (Le1) has the most evident advantage of achieving and promoting the degree of supersaturation form method 2. It also can be seen that supersaturation can be obtained from both methods with carrier gases air, N2 and O2 (0.5
