Heat Field Characteristics of Solid Domain of Nozzle Flowmeter with Different Weld Structures

1. Introduction

In industrial production, serious safety accidents often occur due to weld issues[1,2]. One example is the accident of the large high-pressure steam pipeline "8-11" in Hubei Danyang, in which the weld defects on the accident nozzle expanded under the action of high temperature and high pressure, and local cracks occurred, resulting in steam leakage, creating a hidden danger of accident[3]. In order to increase the equipment safety, the heat flow field of the nozzle flowmeter under different weld structures needs to be analyzed.

In terms of the impact of the flow of the conveyed medium on the flowmeter's performance, Beck et al.[4] investigated the effect of local turbulence and velocity profiles on the accuracy of an electromagnetic flowmeter and found that the deviation in the accuracy of the flowmeter is disproportionate to the degree of local turbulence. Singh et al.[5] investigated the flow characteristic of a V-cone flowmeter under various Reynolds number and upstream disturbance conditions and discovered that the discharge coefficients are irrelevant to the Reynolds number and that upstream disturbances at farther locations do not affect flow coefficients. Choudhary et al.[6] explored the sound drift effect of ultrasonic flowmeter at different flow rates and different profiles, which is of great significance to improve the signal to noise ratio and reduce the error of flow rate calculation. Jepson et al.[7] found that the instrumentation design has a significant effect on the error due to upstream profile changes and proposed a theoretical model to minimize the error during instrumentation design. Shavrina et al.[8] investigated the effect of gas distribution asymmetry on the multiphase metering accuracy of the Coriolis flowmeter, and discovered that increases in gas volume fraction and decreases in mixing flow rate lead to increases in gas distribution asymmetry and that the gas distribution asymmetry affects the accuracy of the flowmeter. Stone et al.[9] performed an experimental and theoretical study of a viscous flowmeter in a steady and unsteady state. Bobovnik et al.[10] investigated the Coriolis flowmeter under various input velocity circumstances by numerical simulation to assess the effect of flow conditions on performance. Džemić et al.[11] explored the response of a turbine flowmeter to transitional flow regimes and found well response of the turbine flowmeter to accelerated flow and a slower response to decelerated flow.

Regarding structural influences as well as the improvement of measurement accuracy, Ren et al.[12] investigated the response and flow characteristics of the dual-rotor turbine flowmeter through visualization experiments, mathematical calculations and numerical simulations. The results show that the rotor located downstream responds to a smaller flow rate, which increases measurement range; the speed of the rotor located downstream has a compensating effect on the speed calculation, which improves the measurement accuracy. Shavrina[13] investigated the coupled fluid-solid methods for the Coriolis flowmeter and found that both the braces and the entire length of the tube should be considered for the numerical study and that the baseline Reynolds stress model has a higher accuracy than the shear stress turbulence model based on eddy viscosity. Beck et al.[14] explored the effect of short radius on the accuracy of electromagnetic flowmeters and found that laboratory calibration can improve the installation accuracy of electromagnetic flowmeters whether they are installed in a straight pipe or pipe with disturbances upstream. Jin et al.[15] investigated the effects of perforation form, plate thickness, equivalent diameter ratio and hole diameter on the discharge coefficients and pressure loss coefficients of the multi-perforated plate flowmeters. It was found that the perforated plate with a larger center hole diameter is better for liquid hydrogen measurement than the perforated plate with equal hole diameter holes and that the equivalent diameter ratio is the dominant factor affecting the cavitation of liquid hydrogen. Furthermore increasing plate thickness and equivalent diameter ratio improve the performance of the perforated plate. Costa et al.[16] proposed a new model for the Coriolis mass flowmeter for calibration according to the effect from room temperature to low temperatures. Hu et al.[17] simulated the fluid-solid coupling dynamics of a dual U-tube Coriolis flowmeter and found that the unbalanced structure of the dual U-tube leads to zero drift and that the drive current of the voice coil actuator or the pressure loss between the inlet and outlet can determine viscosity. Enz[18] found that any asymmetry in the position of the detector induces a phase shift, leading to incorrect measurements in a Coriolis meter. Anklin et al.[19] explored the effect of sound velocity on flow measurements of the Coriolis flowmeter and proposed a theory to improve the accuracy. Zhang et al.[20] proposed a theoretical flow correction factor that takes into account the actual velocity distribution to improve the accuracy of ultrasonic propagation in laminar and turbulent flows, and the correction factor matches well with the measured correction factor. Zheng et al.[21] presented a flow numerical integration method based on Gauss quadrature improvement for ultrasonic flowmeters, which significantly improves the measurement performance in disturbed flows. As for the thermal characteristics of the solid domain of flowmeters, only Zhang and Tong et al.[22-24] investigated the heat flow field and thermal effect of different wall temperatures, flow rates and medium with different temperatures on the same nozzle flowmeter. It was found that the heat transfer effect of the nozzle flowmeter decreases with increasing flow rate; when conveying high-temperature media, there is obvious temperature stratification near the inner wall.

From the published literature, the majority of researchers have largely studied flowmeters under various flow circumstances and parameter designs. However, the impact of various weld structures on the heat flow field of the nozzle flowmeter has not been addressed. In this study, the numerical simulation is used to perform calculations on the impact of various weld structures on a nozzle flowmeter's heat flow field. Furthermore, the heat flow field distribution characteristics of the nozzle flowmeter are obtained for different weld structures, which provides guidance and reference for the welding of nozzle flowmeter in high-risk environments and has important significance for improving the safety of the whole transmission pipeline.

2. Calculation model and method

2.1 Flowmeter model

The calculation model consists of measuring tubes, weld metal and nozzle, which is exactly the same as that in the literature[23]. Figure 1 shows the flowmeter structures, and Table 1 shows the material attributes of each flowmeter structure.

(a) (b)

(a) Real products (b) Basic structure

Fig. 1 Overall structure[23]

Table 1 Material properties[25]

2.2 Numerical simulation method

The modeling was completed using Unigraphics (UG) software and the components are shown in Figure 2(a). The constituent components were divided using unstructured meshes. After the mesh independence test, the final number of elements were determined to be 63875, 1424041, 38623, and 63872 for the front measuring tube, eight slot nozzle, weld metal, and rear measuring tube, respectively, and the total number of elements was 1590411[23]. Figure 2(b) shows the calculation domain meshes.

The numerical calculation of the structural domain is done using the finite element method based on ANSYS Workbench[26-29]. The heat conduction equation to be solved[26] is:

[TeX:] \frac{\mathbf{\partial}}{\mathbf{\partial x}}\mathbf{(k}\frac{\mathbf{\partial T}}{\mathbf{\partial x}}\mathbf{) +}\frac{\mathbf{\partial}}{\mathbf{\partial y}}\mathbf{(k}\frac{\mathbf{\partial T}}{\mathbf{\partial y}}\mathbf{) + q = 0}$\frac{\partial }{\partial x}(k\frac{\partial T}{\partial x})+\frac{\partial }{\partial y}(k\frac{\partial T}{\partial y})+q=0$ (1)

where T is the temperature, and q is the heat source equation.

The setting of material properties for each component is shown in Table 1 above. In this paper, the temperature of the conveyed medium is 700℃, so the temperature of wall in contact with the medium is also set to 700℃. The outside walls of the flowmeter are set as the temperature wall, since the environment temperature of the flowmeter is 22℃, the temperature of the outside wall is also set to 22℃. The calculation model in this paper is a nozzle flowmeter after finishing welding, and the contact type between all parts is set to bonded. For the front measuring tube in contact with weld metal, the front measuring tube is set to contact bodies and the weld metal is set to target bodies. For the front measuring tube in contact with the eight-groove nozzle, the front measuring tube is set to contact bodies and the eight-groove nozzle to target bodies. For weld metal in contact with the rear measuring tube, the rear measuring tube is set to contact bodies and the weld metal to target bodies. For the eight-groove nozzle in contact with the rear measuring tube, the eight-groove nozzle is set to contact bodies and the rear measuring tube to target bodies. For weld metal in contact with the eight-groove nozzle, weld metal is set as contact bodies and eight-groove nozzle as target bodies.

(a) (b)

(a) Flowmeter structure (b) Solid field domain mesh

Fig. 2 Solid structure and flow field calculation domain schematic

2.3 Calculation scheme

Four weld structures (I-IV) are designed for the distance between weld and nozzle L₁, the width below weld L₂, the taper angle θ and opening L₃, with three parameter combinations for each weld structure, as shown in Figure 3. The designed metal welds presented on Figures 4-7 are consistent with those reported in literature[30].

(a) (b)

(a) Type I-III weld structure (b) Type IV weld structure

Fig. 3 Schematic of weld structures

(a) (b) (c)

Fig. 4 Cross section of type I weld structure

(a) (b) (c)

Fig. 5 Cross section of type II weld structure

(a) (b) (c)

Fig. 6 Cross section of type III weld structure

(a) (b) (c)

Fig. 7 Cross section of type IV weld structure

3. Analysis of results

The monitoring paths of the temperature field are shown in Figure 8[30]. Monitoring path 1 and path 2 are abbreviated as P₁ and P₂ in the following text.

Fig. 8 Monitoring path schematic [30]

3.1 Heat flow field of monitoring path

3.1.1 Heat flow distribution along the path P₁

Heat flux[31, 32], sometimes referred to as heat flux density or heat flow density, is the energy flow per unit time per unit area. The heat flow distribution of P₁ with different welds is shown in Figure 9. Overall, under each type of weld metal structure, there are four stages of total heat flow change of P₁: first, it quickly falls to stable and remains relatively stable; then it quickly falls again to stable and remains relatively stable; then it rises quickly to a steady value and keeps around; and finally, it rises rapidly again. It is also found that the symmetrical distribution of heat flow of P₁ is well characterized, and all of them reach the valley between 0.4L and 0.6L. As can be seen for a weld of type I structure, on the path section of 0L-0.11539L, the heat flow decreases rapidly from 2.712W/mm² to 0.831W/mm²; on the path section of 0.115L-0.384L, the heat flow remains relatively stable; on the path section of 0.385L-0.423L, the heat flow decreases rapidly from 0.833W/mm² to 0.234W/mm²; on the path section of 0.423L- 0.577L, the heat flow remains relatively stable; on the path section of 0.577L-0.615L, the heat flow rapidly increases from 0.234W/mm² to 0.819W/mm²; on the path section of 0.615L-0.885L, the heat flow remains relatively stable; on the path section of 0.885L-1.0L, the heat flow increases rapidly from 0.850W/mm² to 2.288W/mm². It is seen that the heat flow has the most variations of plunging, surging, and remaining stable on P₁, and the symmetry about the midpoint at 0.5L is extremely good.

For the type I structure, the minimum valley value of P₁ is 0.226W/mm², 0.230W/mm² and 0.234W/mm² under metal welds of type a, type b and type c, respectively. It is recognized that as L₁ increases, the valley value of the total heat flow of P₁ increases. In addition, the maximum variation amplitude of P₁ is 2.487W/mm², 2.332W/mm², and 2.287W/mm². As the L₁ increases, the maximum magnitude of the total heat flow of P₁ decreases. It is also found that compared to both ends of P₁, the heat flow change curves of type a, b, and c metal welds overlap more in the center section of P₁.

For metal welds of types a, b, and c under type II structure, the minimum valley value of P₁ is 0.226W/mm², 0.227W/mm², and 0.230W/mm², respectively. As the L₂ increases, the valley value of the total heat flow of P₁ also increases. Under the same conditions, the maximum magnitude of P₁ is 2.487W/mm², 2.257W/mm², and 2.272W/mm², respectively. Similarly to the type I structure, the heat flow curves under the type II structure are extremely coincident in the center section of P₁.

For the type III structure, there is a large difference in the variation of heat flow of P₁ for type a, b, and c metal welds. In contrast to the other three structures (type I, type II, and type IV), the largest difference in heat flow curves is found in the center section of P₁. As θ increases, the composition of metal materials along P₁ is changed significantly: at the center section along P₁, the proportion of tin is the least in the type b, and that of type c is the most, wherein that of type a is in the middle. At the same time, due to the different characteristics of the different metals, resulting in different spans of valley region for each type III weld: type b weld has the smallest valley region, type a weld has the middle valley region and type c weld has the largest valley region. However, the change in θ also has little effect on the total heat flow valley and variation amplitude. Under the type a, b, and c metal welds, the corresponding minimum valley value of P₁ is 0.226W/mm², 0.227W/mm², and 0.244W/mm², and the corresponding maximum magnitude of P₁ is 2.487W/mm², 2.305W/mm² and 2.270W/mm², respectively.

For type IV structures, the minimum valley value of P₁ is 0.227W/mm², 0.228W/mm², and 0.237W/mm² under metal welds of type a, type b and type c. It is clear that as L₃ increases, the valley value of the total heat flow of P₁ also increases. Furthermore, the maximum magnitude of P₁ is 2.630W/mm², 2.534W/mm², and 2.376W/mm², respectively. As can be observed, as L₃ increases, the maximum magnitude of the total heat flow of P₁ decreases.

Overall, L₁, L₂, θ and L₃ have little effect on the valley value and the maximum variation amplitude of the total heat flow of P₁, but θ significantly affects the variation of the heat flow situation of P₁. Although L₁, L₂, and L₃ change, there is no significant change in the composition of metal materials of P₁, and, therefore, the heat flow curves under type I, type II and type IV structures overlap in the center section of P₁.

(a) (b) (c) (d)

(a) Type I weld structure (b) Type II weld structure (c) Type III weld structure (d) Type IV weld structure

Fig. 9 Total heat flow distribution of monitoring path 1

3.1.2 Heat flow distribution along the path P₂

The heat flow distribution of P₂ with different welds is shown in Figure 10. Under each weld of each structure, the total heat flow of P₂ shows an overall trend of increasing and then decreasing. Under type a, b, and c welds of type I structure, the maximum peak value of P₂ is 0.227W/mm², 0.237W/mm², and 0.247W/mm², and the maximum magnitude value of P₂ is 0.179W/mm², 0.186W/mm² and 0.195W/mm², respectively. It indicates that as L₁ increases, the peak and maximum magnitude of the total heat flow of P₂ increase. Under type a, b, and c welds of type II structure, the maximum peak value of P₂ is 0.227W/mm², 0.232W/mm², and 0.231W/mm², respectively, and L₂ has little effect on the total heat flow peak value of P₂. For the three same cases, the maximum magnitude of P₂ is 0.179W/mm², 0.183W/mm², and 0.182W/mm², respectively. Altogether, it is clear that compared to the variation in L₁, L₂ has less effect on the heat flow of P₂. The curves overlap more under the type II structure than under the type I structure.

For the type III structure, there is a significant difference in the heat flow curves of P₂ of type a, b, and c metal welds. As seen in Figure 9(c), there are small peaks and valleys in the heat flow curves of 0.1L-0.3L path of type a and b welds, while the heat flow curves of type c weld shows a monotonically increasing trend. In addition, the largest difference is that the maximum peak of P₂ for type c weld is higher than for types a and b. Under type a, b, and c welds of type III structure, the maximum peak value of P₂ is 0.226W/mm², 0.227W/mm², and 0.286W/mm², respectively, and the maximum magnitude of P₂ is 0.179W/mm², 0.171W/mm² and 0.258W/mm². It indicates that θ significantly affects the peak and variation amplitude of P₂, and the peak value and variation amplitude of the total heat flow are the largest when the θ is 90°.

Under type a, b, and c welds of type IV structure, the maximum peak value of P₂ is 0.228W/mm², 0.230W/mm², and 0.247W/mm², respectively, and the maximum magnitude of P₂ is 0.171W/mm², 0.174W/mm² and 0.232W/mm², respectively. It appears that as L₃ increases, the peak value and maximum magnitude of the total heat flow of P₂ also increase and they are much bigger than those of type a and b for type c weld.

(a) (b) (c) (d)

(a) Type I weld structure (b) Type II weld structure (c) Type III weld structure (d) Type IV weld structure

Fig. 10 Total heat flow distribution of monitoring path 2

3.1.3 Heat flow distribution along the path P₁ in ThE X direction

The heat flow distribution of P₁ in the X direction with different welds is shown in Figure 11. In the cases of all welds in this study, the heat flow distribution of P₁ in the X direction shows a decreasing trend and decreases abruptly at the ends of P₁, and varies gently between 0.2L and 0.8L. As can be seen in the type a weld of type I structure, on the path section of 0L-0.192L, the heat flow decreases abruptly from 2.634W/mm² to 0.196W/mm²; on the path section of 0.192L-0.808L, the heat flow varies from 0.196W/mm² to -0.179W/mm², on the path section of 0.808L-1.0L, the heat flow decreases abruptly from -0.179W/mm² to -2.271W/mm²; and on the path section of 0.808L-1.0L, the heat flow decreases abruptly from -0.179W/mm² to -2.271W/mm². Compared with the variation curves under each weld, the curves have small differences and a high degree of overlap. It can be observed that the variation of the structural parameters has a small effect on the heat flow distribution of P₁ in the X direction. Also, compared with Figure 8 and Figure 10, on both ends of P₁, the absolute value of heat flux on P₁ in the X direction is slightly smaller than the heat flux on P₁, and the changing trend is the same. It indicates that heat energy is conducted mainly in the X direction on both ends of P₁.

(a) (b) (c) (d)

(a) Type I weld structure (b) Type II weld structure (c) Type III weld structure (d) Type IV weld structure

Fig. 11 Heat flow distribution of monitoring path 1 in the X direction

3.1.4 Heat flow distribution along the path P₂ in THE X direction

The heat flow distribution of P₂ in the X direction with different welds is shown in Figure 12. Compared with Figure 9 and Figure 11, the heat flux of P₂ is much larger than the absolute value of the heat of P₂ in the X direction. It shows that the main direction of heat conduction is not the X direction on P₂. It is also found that under all welds of all types of structures, the values of heat flux in all path sections are extremely small except near 0.2L; there is a clear change in the curves only near 0.2L. Different metal characteristics have a significant effect on heat flow. There are two different metal materials, structural steel and tin, on the path section near 0.2L, and the width of the same metal near 0.2L is smaller, and the other metal materials cause more influence, which ultimately results in the peaks and valleys of the X direction heat flow on P₂ appearing near 0.2L.

For type a, b, and c metal welds of type I structure, the minimum valley value of the heat flow of P₂ in the X direction is -0.044W/mm², -0.143W/mm² and -0.094W/mm², respectively, and the maximum magnitude of the heat flow in the X direction of P₂ is 0.067W/mm², 0.164W/mm² and 0.124W/mm². Clearly, L₁ affects both the valley and maximum magnitude of the heat flow of P₂ in the X direction. For type II, type III, and type IV structures, the heat flow curves have large differences on the path section near 0.2L, with varying degrees of peaks or valleys. In summary, L₁, L₂, θ and L₃ all have some effect on the peak, valley, and maximum magnitude of the heat flow of P₂ in the X direction.

(a) (b) (c) (d)

(a) Type I weld structure (b) Type II weld structure (c) Type III weld structure (d) Type IV weld structure

Fig. 12 Heat flow distribution of monitoring path 2 in the X direction

3.1.5 Heat flow distribution along the path P₁ in the Y direction

The heat flow distribution of P₁ in the Y direction with different welds is shown in Figure 13. With the same characteristics as the heat flow distribution of P₁ in the X direction, it varies gently from 0.2L to 0.8L, and the heat flux is close to 0. More heat conduction in the Y direction is only at the ends of P₁. Altogether, the Y direction is not the main direction of heat conduction of P₁. In addition, comparing the heat flow curves under each weld, it is found that there is a significant difference in the heat flow distribution between the two ends of P₁ in the Y direction. The variations in L₁, L₂, θ and L₃ mainly affect the heat flow distribution at both ends of P₁ in the Y direction.

(a) (b) (c) (d)

(a) Type I weld structure (b) Type II weld structure (c) Type III weld structure (d) Type IV weld structure

Fig. 13 Heat flow distribution of monitoring path 1 in the Y direction

3.1.6 Heat flow distribution along the path P₂ in the Y direction

The heat flow distribution of P₂ in the Y direction with different welds is shown in Figure 14. There is a maximum on the path section near 0.2L, while all other path sections have minimal heat flow values. For type a, b and c welds of type I structure, the maximum peak value of the heat flow of P₂ in the Y direction is 0.025W/mm², 0.010W/mm² and 0.008W/mm²; for type a, b and c welds of type II structure, the maximum peak value of the heat flow of P₂ in the Y direction is 0.025W/mm², 0.015W/mm² and 0.003W/mm²; for type a, b and c welds of type III structure, the maximum peak value of the heat flow of P₂ in the Y direction is 0.025W/mm², 0.015W/mm² and 0.003W/mm². It shows that the peak of heat flow of P₂ in the Y direction decreases as L₁, L₂, and θ increase. For type a, b and c welds of type IV structures, the maximum peak value of the heat flow field of P₂ in the Y direction is 0.101W/mm², 0.007W/mm² and 0.011W/mm². The maximum peak value of type a weld of type IV structure is much larger than that of other welds, and L₃ significantly affects the heat flow peak of P₂ in the Y direction. It is clear that the lower half of the type a weld of type IV structure is "▽" type, the width of the tin is narrower, the influence of the nearby metal is greater and leads to the maximum peak of heat flow of P₂ in the Y direction under this weld.

(a) (b) (c) (d)

(a) Type I weld structure (b) Type II weld structure (c) Type III weld structure (d) Type IV weld structure

Fig. 14 Heat flow distribution of monitoring path 2 in the Y direction

3.1.7 Heat flow distribution along the path P₁ in the Z direction

The heat flow distribution of P₁ in the Z direction with different welds is shown in Figure 15. It is found that on the path section of 0.2L-0.8L, the heat flow of P₁ in the Z direction has the same trend as the heat flow of P₁ and the values are close to each other. The type a weld of type I structure can be observed as an example. On the path section of 0.192L-0.385L, the total heat flow changes smoothly from 0.778W/mm² to 0.833W/mm², and the heat flow in the Z direction changes smoothly from 0.748W/mm² to 0.823W/mm². On the path section of 0.385L-0.423L, the total heat flow decreases rapidly from 0.833W/mm² to 0.234W/mm², and the heat flow in the Z direction decreases rapidly from 0.823W/mm² to 0.234W/mm². On the path section of 0.423L-0.577L, the total heat flow varies smoothly from 0.233W/mm² to 0.234W/mm², and the heat flow in the Z direction varies smoothly from 0.233W/mm² to 0.234W/mm². On the path section of 0.577L-0.615L, the total heat flow increased rapidly from 0.234W W/mm² to 0.819W/mm², and the heat flow in the Z direction increased rapidly from 0.234W/mm² to 0.808W/mm². On the path section of 0.615L-0.808L, the total heat flow varies smoothly from 0.819W/mm² to 0.798W/mm², and the heat flow in the Z direction varies smoothly from 0.808W/mm² to 0.776W/mm². Clearly, the total heat flow on P₁ has exactly the same trend as that in the Z direction, and the total heat flow is slightly larger than that in the Z direction. In other words, for the 0.2L-0.8L path section of P₁, the main direction of heat conduction is the Z direction.

It is found from Figure 15 that the difference in the heat flow curves of P₁ in the Z direction is small for type I, type II and type IV weld structures. Compared with the heat flow curves in other three weld structures, it is found that the difference of the heat flow curves at type III structure is the largest in the middle section of the path, especially the valley region spans differently. It indicates that only the change in θ has a greater effect on the heat flow of P₁ in the Z direction.

(a) (b) (c) (d)

(a) Type I weld structure (b) Type II weld structure (c) Type III weld structure (d) Type IV weld structure

Fig. 15 Heat flow distribution of monitoring path 1 in the Z direction

3.1.8 Heat flow distribution along the path P₂ in the Z direction

The heat flow distribution of P₂ in the Z direction with different welds is shown in Figure 16. By comparing Figure 16 and Figure 10, it is found that the heat flow trend of P₂ in the Z direction under the same weld is close to the total heat flow trend of P₂, and the value of heat flow in the Z direction is slightly smaller than the total heat flow. It is easy to see that the Z direction is the main conduction direction of heat flow on P₂.

For type a, b and c nozzle flowmeters of type I structure with inner wall temperature of 700°C, the maximum peak value of the heat flow of P₂ in the Z direction is 0.227W/mm², 0.237W/mm² and 0.247W/mm². Obviously, as L₁ increases, the peak of heat flow of P₂ in the Z direction increases. For type II welds, the maximum peak value of heat flow of P₂ in the Z direction is 0.227W/mm², 0.232W/mm² and 0.231W/mm² for type a, b and c, respectively. To sum up, L₁ and L₂ do not significantly influence the heat flow of P₂ in the Z direction: the overlap of the heat flow curves of P₂ in the Z direction is high for both type I and type II.

For type III welds, the maximum peak value of heat flow of P₂ in the Z direction is 0.227W/mm², 0.227W/mm² and 0.286W/mm² for type a, b and c, respectively. The maximum peak of type c weld is much higher than that of types a and b welds. For the type IV welds, the maximum peak value of heat flow of P₂ in the Z direction is 0.228 W/mm², 0.230 W/mm², and 0.246 W/mm², respectively. The maximum peak of type c weld of type IV structure is also much larger than that of types a and b. In addition, the heat flow change curves of P₂ in the Z direction for type c welds of type III and type IV structures are more different from those of types a and b. Taken together, θ and L₃ effects on the heat flow of P₂ in the Z direction are more significant than those of L₁ and L₂.

(a) (b) (c) (d)

(a) Type I weld structure (b) Type II weld structure (c) Type III weld structure (d) Type IV weld structure

Fig. 16 Heat flow distribution of monitoring path 2 in the Z direction

3.2 Heat flow field of solid domain

The total heat flow field of the flowmeter for type I weld is shown in Figure 17. It is clear that the heat flow inside the nozzle flowmeter gradually expands towards the inlet and outlet. Furthermore, the upstream pressure port, downstream pressure port, and inlet and outlet of the type I nozzle flowmeter have a large heat flow area. The maximum value of heat flow occurs at the outlet of all type I nozzle flowmeters with the value of 10W/mm². The type a, b and c metal welds of the type I structure all exhibit naturally higher heat flow at the inlet, and outlet with the pressure ports. It is evident that L₁ has less effect on the internal heat flow.

(a) (b) (c)

(a) Type a weld (b) Type b weld (c) Type c weld

Fig. 17 Total heat flow field in cross section of type I flowmeter (W/mm²)

The total heat flow field of the flowmeter for type II welds is shown in Figure 18. Same as the flowmeter for type I welds, the heat flow gradually expands to the inlet and outlet, and larger heat flow regions appear in all the same areas. The maximum value of heat flow occurs at the outlet of all type II nozzle flowmeters with the value of 10W/mm². The type a, b and c metal welds of the type II structure all exhibit naturally higher heat flow at the inlet and outlet with the pressure ports. Clearly, L₂ has less effect on the internal heat flow distribution.

(a) (b) (c)

(a) Type a weld (b) Type b weld (c) Type c weld

Fig. 18 Total heat flow field in cross section of type II flowmeter (W/mm²)

The total heat flow field of the flowmeter for type III weld is shown in Figure 19. Similarly to the previous two structures, the flowmeter heat flow of Structure III welds also extends to both sides, and larger heat flow regions also appear at the upstream and downstream pressure ports and at the inlet and outlet of the type III nozzle flowmeters. The maximum value of heat flow occurs at the outlet of the nozzle flowmeter of type III structure with the value of 10W/mm². The type a, b and c metal welds of type III structure all show naturally higher heat flow at the inlet and outlet with the pressure ports. This shows that θ has less effect on the internal heat flow distribution.

(a) (b) (c)

(a) Type a weld (b) Type b weld (c) Type c weld

Fig. 19 Total heat flow field in cross section of type III flowmeter (W/mm²)

The total heat flow field of the flowmeter for type IV weld is shown in Figure 20. As can be seen, the heat flow inside the type IV nozzle flowmeter also gradually expands towards the inlet and outlet. Moreover, there are large heat flow areas at the upstream and downstream pressure ports and the inlet and outlet of the Type IV nozzle flowmeter. The maximum value of heat flow occurs at the inlet and outlet of all type IV nozzle flowmeters with the value of 10W/mm². Furthermore, type a, b and c welds of type IV structure all exhibit naturally higher heat flow at the inlet and outlet with the pressure ports. It can be seen that L₃ has a small effect on the internal heat flow distribution.

(a) (b) (c)

(a) Type a weld (b) Type b weld (c) Type c weld

Fig. 20 Total heat flow field in cross section of type IV flowmeter (W/mm²)

3.3 Discussion

The study is basic research, which grasps the heat flow field characteristics under different weld structures, provides a reference for the structural optimization of the weld, and has certain significance for improving the welding safety of nozzle flowmeters and pipeline safety. In this study, initially, only the finite element method was used to predict the thermal field characteristics in the solid domain of the nozzle flowmeter. However, due to the differences in different models, the accuracy of the calculation results is to be verified in future work. In addition, this paper only explores the thermal flow field characteristics when the metal material is tin. The influence of other materials on the thermal flow field characteristics is also to be examined in future work.

4. Conclusion

Under different welds, on the path from upstream to downstream (i.e., P₁), the total heat flow changes mostly show the trend of plummeting, rising and remaining stable, and the symmetrical distribution of heat flow is very clear, all reaching the valley value between 0.4L and 0.6L. The distance between the weld and nozzle L₁, the width below weld L₂, the taper angle θ and the taper opening L₃ have a small effect on the valley and the maximum variation amplitude of the total heat flow in this path, but there is a significant effect of θ on the middle part of the path.

On the path from the inner to the outer wall (i.e., P₂), the total heat flow changes in the overall trend, first increasing and then decreasing. Compared to L₁ and L₂, θ and L₃ significantly affect the peak and maximum variation amplitude of the total heat flow on this path: the peak value and the maximum amplitude of the total heat flow are much larger for type c welds of types III and IV than for types a and b.

The heat energy at both ends of the path is mainly conducted in the X direction along P₁, and the heat energy on the 0.2L-0.8L path section is mainly conducted in the Z direction. The Z direction along P₂ is the main conduction direction of heat flow.

At P₁, L₁ and L₂ as well as θ and L₃ have a smaller influence on the heat flow distribution in the X, Y and Z directions and a greater influence only on the heat flow field in the Y direction at the two ends of the path. L₁ and L₂, as well as θ and L₃ significantly affect the heat flow distributions in the X and Y directions of the near 0.2L path section at P₂.

The heat flow inside the nozzle flowmeter gradually expands toward the inlet and outlet, and larger heat flow regions appear at both the upstream and downstream pressure ports as well as the inlet and outlet. L₁ and L₂ as well as θ and L₃ have less effect on the heat flow distribution inside the nozzle flow meter.

5. Acknowledgment

The research was financially supported by the Science and Technology Project of Quzhou (No.2023K256) and Research Grants Program of Department of Education of Zhejiang Province (No. Y202455709).