Regular Article
The stress characteristics of platefin structures at the different operation parameters of LNG heat exchanger
^{1}
School of Civil Engineering, Lanzhou University of Technology,
Lanzhou  PR China
^{2}
Sinopec Petroleum Engineering Zhongyuan Corporation,
Puyang  PR China
^{*} Corresponding author: mhq2014@sina.com
Received:
20
October
2017
Accepted:
5
March
2018
In this paper, the stresses of platefin structures at the different operation parameters were analyzed in actual operation process of LNG platefin heat exchanger based on finite element method and thermal elastic theory. Stress characteristics of platefin structures were investigated at the different operation parameters of that. The results show that the structural failure of platefin structures is mainly induced by the maximum shear stress at the brazing filler metal layer between plate and fin while by the maximum normal stress in the region of brazed joint near the fin side. And a crack would initiate in brazed joint near the fin side. The maximum normal stress is also main factor to result in the structural failure of platefin structures at the different temperature difference (between Natural Gas (NG) and Mixture Refrigerant (MR)), MR temperature and NG pressure of LNG heat exchanger. At the same time, the peak stresses obviously increase as the temperature difference, MR temperature and NG pressure increase. These results will provide some constructive instructions in the safe operation of LNG platefin heat exchanger in a largescale LNG coldbox.
© H. Ma et al., published by IFP Energies nouvelles, 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
PlateFin Heat Exchanger (PFHE), as a main heat exchange equipment, has been widely used in smallscale LNG plants while rarely in largescale LNG plants as main cryogenic heat exchanger in the world (Kuznetsov and Shamirzaev, 2007; Ligterink et al., 2012; Liu and Winterton, 1991). PEHE is consisted of platefin structures and usually installed in a LNG coldbox in largescale LNG plants. Therefore, the platefin structure is the key component in the PFHE which is the key equipment in a largescale LNG coldbox. Its security is affected by stress characteristics of platefin structures at the different operation conditions of LNG platefin heat exchanger in largescale LNG plants (Mizokami et al., 2013; Picard et al., 2006). However, the stress characteristics of platefin structures are still far from well understood at the different operation conditions of LNG platefin heat exchanger in largescale LNG plants.
At present, many studies have been done on stress characteristics of platefin structures in heat exchanger. But most of the investigations have been focused on residual stress of platefin structures in brazing process of Platefin heat exchanger which was made of stainless steel (Jiang et al., 2011a; Jiang et al., 2010; Jiang et al., 2011b; Jiang et al., 2011c; Jiang et al., 2012). At the same time, in our previous work, the stress characteristics of platefin structures were also analyzed in the cooldown or heatup process of LNG PFHE in a largescale LNG coldbox (Ma et al., 2016a; Ma et al., 2014) based on finite element method and thermal elastic theory. The results show that a crack initiates in the brazed joint in the cooldown or heatup process of LNG PFHE because that the stresses rapidly changes and consistently reach the peak value in this region. Meanwhile, the stress of platefin structures is obviously influenced by the temperature difference between Natural Gas (NG) and Mixture Refrigerant (MR) while the influence of the temperature drop or rise rate is slight as the temperature drop or rise rate of NG is the same as that of MR. In order to ensure the structural safety of platefin structures in actual operation process of LNG PFHE and guide the structural design, stress characteristics of platefin structures are analyzed at the different structure parameters based on finite element method and thermal elastic theory. The results show that the stresses of platefin structures is obviously impacted by the brazing seam thickness, fin thickness and fin distance and it is mainly induced by the direct stress in y direction(Ma et al., 2016b). In these investigations, the peak value of equivalent stress, the direct stresses along x, y and z directions and the shear stress on xy plane and yz plane were analyzed to explain the stress characteristics of platefin structures in LNG PFHE. In fact, the direct stresses along x, y and z directions and the shear stresses on xy plane and yz plane are distinct in different coordinate systems. According to material mechanics, the maximum normal stress (σ_{max}) and maximum shear stress (τ_{max}) are always consistent in different coordinate systems. Therefore, it is necessary to explain the stress characteristics of platefin structures in LNG PFHE. In other words, it is necessary that the maximum normal stress and maximum shear stress of platefin structures are analyzed in operation process of LNG PFHE based on the thermal elastic theory in order to investigate the stress characteristics of platefin structures in LNG PFHE. However, now it is lack for the analysis of the maximum normal stress and maximum shear stress of platefin structures in LNG PFHE.
In this work, the direct coupling method was adopted to simulate the stress characteristics of platefin structures at the different operation parameters in LNG PFHE. The distributions of the maximum normal stress (σ_{max}), maximum shear stress (τ_{max}) and equivalent stress (σ_{r}) in platefin structures were analyzed in actual operation process. And the influence of operation parameters was investigated for the peak value of maximum normal stress and maximum shear stress in the concentration region of stress in the total LNG platefin structures.
2. Finite element model analysis
2.1. The model description
In the LNG PFHE, Platefin structures mainly consist of fins, plate (Carey and Mandrusiak, 1986; Manglik and Bergles, 1995), as shown in Figure 1. Table 1 is the detailed structural parameters of platefin structures in LNG PFHE of a LNG coldbox in largescale LNG plants. Figure 1 shows that the structural shape of platefin structures is cyclical repeatability and complexity in LNG PFHE. In order to simplify the analysis model in the numerical simulation, some issues would be assumed. In this paper, the influence of layers number is ignored for stress characteristics of platefin structures. The flow of NG is assumed to be alternating and counterflow with mixture refrigerant MR, as shown in Figure 1. At the same time, the change of fluid temperature is ignored because that is very small in a very short length (L) along the length direction of platefin structures. Therefore, only one unit of platefin structures which consists of four layers plate and fin, was analyzed in a very short length (L) at the different operation parameters in LNG heat exchanger, as dotted lines shown in Figure 1. Figure 2a shows the simplified model of platefin structures. Figure 2b, c display the local structure of region 2 and regions 4, 5 and 6 in Figure 2a, respectively.
Fig. 1 The diagram of Platefin structures. 
Structural parameters (From the bottom plate to top is from No. 1 to No. 5).
Fig. 2 The simplified platefin structure model and local structure. 
2.2. Material properties
According to the literature (Ma et al., 2016b), the materials of platefin structures are treated as isotropic and linear elastic. At the same time, In order to get fine results, the expansion coefficient and elastic modulus are assumed to be temperature dependent for the AL3003 and the AL4004 (The fin and plate is assumed to be AL3003 and the brazing material is AL4004). Table 2 is the detailed material properties of AL3003 and AL4004 used in the simulations.
Material parameters used in FEM simulation.
2.3. Analysis strategy
In ANSYS software, the thermalmechanical coupling analysis for stress mainly includes the sequentially coupled and directly coupled (Khan et al., 2011; Zhao et al., 2012). For the sequentially coupled, the heat transfer process is firstly calculated by the ANSYS software. The temperature distribution calculated by that is used as the body loads in the mechanics analysis. For the directly coupled, the thermal and mechanical will be simultaneously calculated in the ANSYS software. In this work, the interaction between temperature field and strain field is considered because of the relatively large deformation on platefin structures. Therefore, the SOLID5 element for both plates and fins is adopted and the directly coupling method based on the thermal elastic theory is used to analyze the stress characteristics of platefin structures at the different operation parameters in LNG PFHE.
The convective heat transfer boundary is applied between the NG or MR and the wall of platefin structures to simulate the heat transfer process with phase change between the NG or MR and the wall of platefin structures in LNG PFHE. The cyclical symmetry boundary on the leftmost and rightmost surfaces is applied to simulate the periodicity and symmetry of platefin structures in LNG PFHE. The NG and MR operating pressure is adopted to simulate the interaction between the NG or MR and the wall of platefin structures.
At the same time, the computational domain is discretized into hexahedral finite control volumes using a mesh so as to decrease the grid numbers and improve the computation efficiency. The influence of the number on mesh cells for the simulation results is also done according to the method in literature(Ma et al., 2016a). The total number of mesh cells used in the paper was 141 000 according to the calculation accuracy and efficiency. ANSYS Parametric Design Language was adopted to program the above simulation process. And the analysis method used in this paper is also validated by the experimental data in literature (Ma et al., 2014).
According to the von Mises yield criterion, the plastic yield failure of platefin structures may occur in operation process when the equivalent stress (von Mises stress) is more than the allowable stress. In other words, the initial cracking damage may be happened at the location of platefin structures which the equivalent stress reaches peak value. Therefore, in this paper, the equivalent stress of platefin structures is also be analyze at the different operation parameters in LNG heat exchanger.
3. Results and discussion
3.1. Stress distribution
In liquefied heat exchanger of largescale LNG coldbox, the NG is cooled from 210 K to 150 K and become liquid from the gas by condensation heat transfer at the NG pressure P_{NG} = 7.1 MPa and the MR is heated from 155 K to 215 K and vaporized by the boiling heat transfer at the MR pressure P_{MR} = 0.4 MPa. According to the design condition, the heat transfer coefficient between NG and the wall of platefin structures is 1500 W/(m^{2 }· K) at the NG temperature T_{NG} = 155 K and the NG pressure P_{NG} = 7.1 MPa; the MR heat transfer coefficient between NG and the wall of platefin structures is 1000 W/(m^{2 }· K) at the MR temperature T_{MR} = 150 K and the MR pressure P_{MR} = 0.4 MPa. Therefore, the stress distribution of platefin structures in a very short length near the NG entrance of the above liquefied heat exchanger is analyzed at the above parameters of the actual operation process.
For the sake of a better understanding, four reference paths were selected, as marked in Figure 2. Figures 3–8 present the distributions of the maximum normal stress, maximum shear stress and equivalent stress along the four reference paths in platefin structures at temperature difference ΔT = 5 K between NG and MR, heat transfer coefficient h_{NG} = 1.5 kW/(m^{2 }· K), h_{MR} = 1 kW/(m^{2 }· K), fluid pressure P_{NG} = 7.1 MPa, P_{MR} = 0.4 MPa. Figures 3 and 4 exhibit the distribution of the maximum normal stress, maximum shear stress and equivalent stress along paths 1 and 2, respectively. Path 1 and 2 is located at the interface between the brazing filler metal and the plate and between the brazing filler metal and the fin, respectively. It can be seen clearly that the maximum normal stress, maximum shear stress and equivalent stress are symmetrically distributed. In Figure 3, they reach the peak values in brazed joint which is the small transitional arcs on the two terminals of the brazing seam (regions 1 and 2 in Fig. 2). The same conclusion can be also drawn for path 2 in Figure 4, but the maximum normal stress, maximum shear stress and equivalent stress is much larger in regions 1 and 2 along the path 2 than that along the path 1. At the same time, the maximum normal stress is obviously larger than the maximum shear stress in the region of brazed joint near the fin side, while the maximum normal stress is obviously smaller than the maximum shear stress in the brazing filler metal layer. Therefore, the main reason of structural damage is the maximum shear stress at the brazing filler metal layer between plate and fin while is the maximum normal stress in the region of brazed joint near the fin side.
Figure 5 depicts the stress distribution of path 3. The results show that the stresses distribution of maximum normal stress, maximum shear stress and equivalent stress are very complex along the path 3. the stress gradient of brazed joint (region 2 in Fig. 2), brazing seam (region 6 in Fig. 2) and rectangular region (region 5 in Fig. 2) are much larger, and the maximum normal stress, maximum shear stress and equivalent stress reach the peak value in these regions due to the stress concentration and the mismatching thermal expansion coefficients between AL4004 filler metal and AL3003 base metal. But the peak stress in region 2 is larger than that in regions 5 and 6. In order to further analyze the stress distribution in regions 2 and 6, the stress distribution of the local position on in regions 2 and 6 (region 2 in Fig. 2b and region 6 in Fig. 2c) are presented in Figures 6 and 7. Figure 6 depicts the distributions of the maximum normal stress, maximum shear stress and equivalent stress in the local position of region 6. The results show that the maximum normal stress, maximum shear stress and equivalent stress are much larger in the brazing seam between plate and fin than that in the adjacent region due to the mismatching thermal expansion coefficients between AL4004 filler metal and AL3003 base metal. Figure 7 presents the distribution of the maximum normal stress, maximum shear stress and equivalent stress in the local position of region 2. The results also indicate that the peak stress occur in brazed joint near the fin side.
Figure 8 exhibits the distributions of the maximum normal stress, maximum shear stress and equivalent stress along path 4. These results are also obtained that the maximum normal stress, maximum shear stress and equivalent stress reach the peak value, and the stress gradient is much larger in brazed joint (regions 3 and 4 in Fig. 2) due to the stress concentration and the mismatching thermal expansion coefficients between AL4004 filler metal and AL3003 base metal.
To sum up, in the actual operating processes, the maximum normal stress and maximum shear stress along four paths of platefin structure are consistent with the equivalent stress based on the Von Mises yield criterion and reach the peak value in the brazed joint (regions 1, 2, 3 and 4 in Fig. 2), and the stress is larger in brazed joint near the fin side than that in the adjacent region. The brazing seam between the fin and the plate also suffer the larger stress than that in the adjacent region due to the stress concentration and the mismatching thermal expansion coefficients between AL4004 filler metal and AL3003 base metal. This means that the crack failure would occur in the area of brazed joint next to the fin side firstly. At the same time, the structural damage is mainly induced by the maximum shear stress at the brazing filler metal layer between plate and fin while by the maximum normal stress in the region of brazed joint near the fin side. For platefin heat exchangers in a large LNG coldbox, when the platefin structure is served under extreme conditions (low temperature and high pressure), the high stress in brazed joint next to the fin side would induce stress crack and result in failure, eventually.
Fig. 3 Stress distribution of path 1. 
Fig. 4 Stress distribution of path 2. 
Fig. 5 Stress distribution of path 3. 
Fig. 6 Stress distribution of the local position in region 6. 
Fig. 7 Stress distribution of the local position in region 2. 
Fig. 8 Stress distribution of path 4. 
3.2. The influence of operation parameters for stress
In Section 3.1, it is stated that the maximum normal stress, maximum shear stress and equivalent stress reach the peak value in the brazed joint of platefin structures next to the fin side due to the stress concentration and the mismatching thermal expansion coefficients between AL4004 filler metal and AL3003 base metal. This is likely to lead to a crack failure in these regions. So it is necessary to investigate the stress characteristics of platefin structures at the different operation parameters in LNG heat exchanger. in this section, the peak value of maximum normal stress, maximum shear stress and equivalent stress in the stress concentration region of the total platefin structures are investigated at the different operation parameters. The peak value of maximum normal stress, maximum shear stress and equivalent stress represented as (σ_{max})_{max}, and (τ_{max})_{max,} and (σ_{r})_{max}, respectively. In the following content, the peak value of maximum normal stress, maximum shear stress and equivalent stress will be analyzed at the different operation parameters (for example, heat transfer coefficient, fluid temperature and operation pressure) in LNG heat exchanger.
3.2.1. The influence of heat transfer coefficient
The heat transfer performance of heat exchanger directly impact the temperature distribution field of platefin structures. The different temperature distribution would induce the different stress characteristics of platefin structures in LNG platefin heat exchangers. The heat transfer coefficient between the MR or NG and the wall of platefin structures is the key parameters to describe the heat transfer performance of LNG heat exchanger. The relation between the stress and the heat transfer coefficient is analyzed in order to investigate the stress characteristics of platefin structures at the different heat transfer performance of heat exchanger. Figure 9 presents the data on the peak value of maximum normal stress, maximum shear stress and equivalent stress as a function of the heat transfer coefficient between the NG and the wall of platefin structures at temperature difference ΔT = 5 K between NG and MR, heat transfer coefficient h_{MR} = 1 kW/(m^{2 }· K), fluid pressure P_{NG} = 7.1 MPa, P_{MR} = 0.4 MPa and MR temperature T_{MR} = 150 K. The results indicate that the peak value of maximum normal stress, maximum shear stress and equivalent stress increase slightly as the heat transfer coefficient between the NG and the wall of platefin structures increases.
Figure 10 depicts the relation between the peak stress and the heat transfer coefficient between the MR and the wall of platefin structures at temperature difference ΔT = 5 K between NG and MR, heat transfer coefficient h_{NG} = 1.5 kW/(m^{2 }· K), fluid pressure P_{NG} = 7.1 MPa, P_{MR} = 0.4 MPa and MR temperature T_{MR} = 150 K. The results indicate that the peak stress of maximum normal stress, maximum shear stress and equivalent stress decrease slightly with the increase of heat transfer coefficient between the MR and the wall of platefin structures. This is because the temperature of platefin structures reduce slightly with the increase of heat transfer coefficient between the MR and the wall of platefin structures and the mismatching of thermal expansion coefficients between AL4004 filler metal and AL3003 base metal reduce with the reduction of temperature.
In other words, the peak value of maximum normal stress, maximum shear stress and equivalent stress is slightly impacted by the heat transfer coefficient between the MR or NG and the wall of platefin structures and decrease slightly with the increase of heat transfer coefficient between the MR and the wall of platefin structures because that the mismatch of thermal expansion coefficients between AL4004 filler metal and AL3003 base metal reduce with the reduction of temperature of platefin structures.
Fig. 9 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. heat transfer coefficient between the NG and the wall of platefin structures. 
Fig. 10 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. heat transfer coefficient between the MR and the wall of platefin structures. 
3.2.2. The influence of fluid temperature
The variety of the temperature difference between NG and MR will also result in the different stress state in various design or operation conditions. In this work, the influence of the temperature difference between NG and MR is analyzed for the maximum normal stress, maximum shear stress and equivalent stress of platefin structures in a very short length near the NG entrance of liquefied heat exchanger. Figure 11 shows the peak value of maximum normal stress, maximum shear stress and equivalent stress vs. the temperature difference between NG and MR for heat transfer coefficient h_{NG} = 1.5 kW/(m^{2 }· K), h_{MR} = 1 kW/(m^{2 }· K), fluid pressure P_{NG} = 7.1 MPa, P_{MR} = 0.4 MPa and MR temperature T_{MR} = 150 K. The peak value of maximum normal stress, maximum shear stress and equivalent stress increase steadily with the increase of the temperature difference between NG and MR, and are obviously impacted by the temperature difference between NG and MR. This is because the thermal expansion coefficients of AL3003 base metal do not match with that of AL4004 filler metal and the large temperature difference between NG and MR results in the increase of temperature gradient in the platefin structure. The peak value of maximum normal stress is always larger than that of maximum shear stress at the different temperature between NG and MR. For main cryogenic heat exchanger in a large LNG coldbox, the boiling heat transfer and the condensation heat transfer are performed in the passage of the MR and the NG, respectively. Due to the complexity and instability of the boiling heat transfer and the condensation heat transfer, the temperature difference between NG and MR will vary greatly if the operations are not reasonable for the main cryogenic heat exchanger. This will result in fatigue failure in the brazed joint of platefin structures in LNG heat exchanger. And the fatigue failure of platefin structures is mainly induced by the maximum normal stress at the different temperature between NG and MR.
At the same time, the stress state of the platefin structures are also different along the length direction of platefin structures because the NG and MR temperature are various along the length direction of platefin structures from the NG entrance to the NG exit. It is necessary to analyze the influence of the NG or MR temperature for the stress characteristics of platefin structures along the length direction of liquefied heat exchanger from the NG entrance to the NG exit. Figure 12 depicts the relation between the peak stress and the MR temperature of the different position along the length direction of liquefied heat exchanger. Figure 12 presents the plots of peak value of maximum normal stress, maximum shear stress and equivalent stress vs. MR temperature. The plots are drawn at temperature difference ΔT = 5 K between NG and MR, heat transfer coefficient h_{NG} = 1.5 kW/(m^{2 }· K), h_{MR} = 1 kW/(m^{2 }· K) and fluid pressure P_{NG} = 7.1 MPa, P_{MR} = 0.4 MPa. The results show that the peak value of maximum normal stress, maximum shear stress and equivalent stress obviously increases as the MR temperature increases. This is because the mismatch of thermal expansion coefficients between AL4004 filler metal and AL3003 base metal is more obvious in high MR temperature than that in lower MR temperature. At the same time, the peak value of maximum normal stress is always larger than that of maximum shear stress at the different MR temperature.
Fig. 11 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. temperature difference. 
Fig. 12 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. MR temperature. 
3.2.3. The influence of operation pressure
The stress of platefin structures is different for the different operation pressures condition. In order to analyze the stress characteristics of platefin structures at the different operation pressure condition, the relation between the peak stress and the MR or NG pressure is simulated in this section. Figure 13 depicts the relation between the peak value of maximum normal stress, maximum shear stress and equivalent stress and MR pressure at NG pressure P_{NG} = 7.1 MPa, temperature difference ΔT = 5 K between NG and MR, MR temperature T_{MR} = 150 K and heat transfer coefficient h_{NG} = 1.5 kW/(m^{2 }· K), h_{MR} = 1 kW/(m^{2 }· K). The results show that the peak value of maximum normal stress, maximum shear stress and equivalent stress increases slightly with the increase of MR pressure, Meanwhile, the peak value of maximum normal stress is also larger than that of maximum shear stress at the different MR pressure.
The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. NG pressure at MR pressure P_{MR} = 0.4 MPa, temperature difference ΔT = 5 K between NG and MR, MR temperature T_{MR} = 150 K and heat transfer coefficient h_{NG} = 1.5 kW/(m^{2 }· K), h_{MR} = 1 kW/(m^{2 }· K) are presented in Figure 14. The results show that the peak value of maximum normal stress, maximum shear stress and equivalent stress increases with the increase of NG pressure, and the increase rate is obvious. At the different NG pressure, the peak value of maximum normal stress is also larger than that of maximum shear stress for the whole platefin structures in LNG heat exchanger. In other words, the influence of NG pressure for the stress characteristics of platefin structures in LNG heat exchanger is induced by the maximum normal stress.
Fig. 13 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs.MR pressures. 
Fig. 14 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. NG pressures. 
4. Conclusion
In this work, the stress characteristics of platefin structures in LNG heat exchanger were investigated at the different operation parameters condition. The distribution of maximum normal stress, maximum shear stress and equivalent stress is obtained in this paper. The following conclusions can be drawn from this study:

the results obtained by numerical simulation indicate that the maximum normal stress, maximum shear stress and equivalent stress reach the peak values at brazed joint near the fin side. The brazing seam between the fin and the plate also suffer larger stress than that in the adjacent region. The structural failure of platefin structures in LNG heat exchanger is mainly induced by the maximum shear stress at the brazing filler metal layer between plate and fin while by the maximum normal stress in the region of brazed joint near the fin side;

the peak value of maximum normal stress, maximum shear stress and equivalent stress are obviously impacted by the temperature difference between NG and MR in the actual operation process of LNG heat exchanger. And the structural failure of platefin structures in LNG heat exchanger is finally induced by the maximum normal stress. In order to ensure the safe and reliable operation of platefin structures, the temperature difference between NG and MR needs to be reasonably controlled in main cryogenic heat exchanger which is used in a largescale LNG coldbox;

the peak value of maximum normal stress, maximum shear stress and equivalent stress is obviously impacted by the NG pressure while influence of MR pressure is slight in the actual operation process. At the same time, the maximum normal stress is also main factor to result in the structural failure of platefin structures at the different NG pressure of LNG heat exchanger.
Acknowledgment
The authors are grateful for the support of the Natural Science Foundation of Gansu Province.
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All Tables
All Figures
Fig. 1 The diagram of Platefin structures. 

In the text 
Fig. 2 The simplified platefin structure model and local structure. 

In the text 
Fig. 3 Stress distribution of path 1. 

In the text 
Fig. 4 Stress distribution of path 2. 

In the text 
Fig. 5 Stress distribution of path 3. 

In the text 
Fig. 6 Stress distribution of the local position in region 6. 

In the text 
Fig. 7 Stress distribution of the local position in region 2. 

In the text 
Fig. 8 Stress distribution of path 4. 

In the text 
Fig. 9 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. heat transfer coefficient between the NG and the wall of platefin structures. 

In the text 
Fig. 10 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. heat transfer coefficient between the MR and the wall of platefin structures. 

In the text 
Fig. 11 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. temperature difference. 

In the text 
Fig. 12 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. MR temperature. 

In the text 
Fig. 13 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs.MR pressures. 

In the text 
Fig. 14 The peak value of maximum normal stress, maximum shear stress and equivalent stress vs. NG pressures. 

In the text 