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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39256515
72219
10.1038/s41598-024-72219-z
Article
Numerical study on heat transfer and pressure performance of different suspension nozzles
Liu Zhihui lzh840319@163.com

12
Zhang Zhijian 13685754887@163.com

12
Zhang Jiahao 12
Chen Zhigang 12
1 https://ror.org/03fx09x73 grid.449642.9 0000 0004 1761 026X College of Mechanical and Energy Engineering, Shaoyang University, Shaoyang, 422000 China
2 https://ror.org/03fx09x73 grid.449642.9 0000 0004 1761 026X Key Laboratory of Hunan Province for Efficient Power System and Intelligent Manufacturing, Shaoyang University, Shaoyang, 422000 China
10 9 2024
10 9 2024
2024
14 2108423 5 2024
4 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
The drying process of the lithium battery pole pieces makes extensive use of the suspension nozzle. It is of great significance to study the heat transfer and pressure steady-state characteristics of the suspension nozzle and to select the appropriate nozzle structure for the production of pole pieces. Based on the SSTk-ω turbulence model, this article numerically simulates the impact jet process of suspension nozzles with slits, injection holes, and effusion holes. There is a qualitative and quantitative analysis of the distribution of their velocity field, temperature field, local Nusselt number, average Nusselt number, local pressure coefficient, and average pressure coefficient, and the comprehensive performance index of the nozzle is proposed. The results show that when the weight factor of heat transfer performance α is less than 21.61% and the weight factor of pressure performance β is more than 78.39%, the comprehensive performance of the traditional suspension nozzle with double slits is the best. As the α is increasing, the β is decreasing. The comprehensive performance of the suspension nozzle with effusion holes is the best. The turbulent intermittence, interaction between neighbouring jets, and edge effects affect the heat transfer and pressure uniformity of the suspension nozzle.

Keywords

Suspension nozzle
Impact jet
Heat transfer performance
Pressure performance
Subject terms

Engineering
Mechanical engineering
Hunan graduate research innovation projectQL20230277 QL20230278 Zhang Jiahao Shaoyang University graduate research innovation projectCX2023SY009 CX2023SY010 Zhang Jiahao Natural Science Foundation of Hunan Province of China2024JJ7494 Shaoyang key research and development project2023CG2009 issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Owing to its exceptional drying capabilities, the suspension nozzle has garnered increasing interest in the engineering community. It has found extensive use in the manufacturing domains of lithium battery pole piece drying, material forming1, cloth textile2, and electronics printing3. Figure 1 shows a pole piece drying system. An "air cushion" is created on the pole piece's surface by hot air that is released through the top and lower nozzle slits. It enables the pole piece to be suspended between the upper and lower nozzles and completes the heat transfer drying of the pole piece. The heat transfer process from the suspension nozzle to the pole piece is virtually an effective impact heat transfer process. The pole piece in the nozzle jet impact is prone to shaking, wrinkling, crimping, and other defects; these defects will affect the quality of lithium batteries, primarily because it is difficult to ensure the pole piece suspension stability and fast and uniform drying. Therefore, to increase the performance and lifespan of lithium batteries, it is crucial to investigate the heat transfer mechanism and pressure properties of the pole piece sheet under the influence of the suspension nozzle jet.Fig. 1 Pole piece drying system.

Many academics have examined the pressure properties and heat transfer of suspension nozzles. A forced convective heat transfer is created when the nozzle jet impinges on the pole piece's surface; the intensity of this transfer is measured by the Nussle number Nu. The distance between the nozzle and the pole piece, the jet angle, the nozzle's geometry, its length and width dimensions, and the pole piece's motion speed are all parameters that influence the pole piece's heat transfer process. Tepe A. Ü et al.4 discovered that when the space between the nozzle and pole piece gets smaller, there is an increase in heat transfer between the jet and pole piece. The distribution and size of the Nussle number on the target wall are directly correlated with the nozzle jet angle, as demonstrated by Afroz, F. et al.5,6. Additionally, the maximum Nussle number on the target wall decreases as the jet angle lowers. The impact of jet orifice shape on the heat transfer properties of suspension nozzles was examined by Caliskan, S., and Attalla, M. et al.7,8 Their findings demonstrated that at the target wall surface, the Nussle number of elliptical nozzles is higher than that of round nozzles, and the square nozzle's average heat transmission is 7.8% lower than that of the round nozzle. Fan Z, Wang L, Xu F, et al.9 investigation shows that the diffusion height of the jet is higher when a rectangular, non-equal nozzle is applied. The circular nozzle is more active in spreading. Li J, Zhao X, Zhang H, et al.10 comparing the Nussle number distribution at the target location for single and double jet jets operating under the identical heat transfer conditions, including the same flow rate and jet distance, They found that there is a significant difference between the Nusselt number for single-jet impingement and the Nusselt number for double-jet. The Nusselt number distribution is more inhomogeneous due to the two jets in the dual-jet condition, and the peak of the Nusselt number is larger in the dual-jet condition. Li, H. et al.11 created a nozzle model with various aspect ratios (Car) and discovered that the heat transfer uniformity grows as the Car increases and reaches its optimal value when it equals 4.17. Researchers Chattopadhyay, H. et al.12 showed that the shock wall's translational velocity has an impact on the heat transmission in a shock jet. The distribution of Nussle numbers on the wall surface was examined by Benmouhoub, D. et al.13 in relation to the ratio of wall travelling velocity to jet velocity (Rsj). The findings indicated that an increase in Rsj causes a decrease in the local Nussle number but an increase in the mean Nussle number. Liu Z, Zhang J, and Zhang Z14 also showed that the wall’s ability to convey heat is stronger the higher the Rsj.The above parameters work together to impact the effectiveness and efficiency of heat transfer from the suspension nozzle and establish the nozzle's heat transfer performance.

The pressure coefficient (Cp) in the lithium battery pole piece heat exchange process is the ratio of the pole piece's surface pressure to the dynamic pressure at the nozzle outlet. When examining the suspension nozzle's pressure properties, the pressure coefficient is a crucial parameter to consider. Nozzle pressure properties are influenced by several causes and bear similarities to heat transfer characteristics. Nozzle pressure properties are similar to heat transfer characteristics and are influenced by a number of factors. They include the number of slits, the shape of the pole piece, the distance between the nozzle and the pole piece, the jet Reynolds number, and further information. The pressure coefficient distribution pattern of a single slit nozzle jet impinging on a horizontally shaped strip was examined by Ramezanpour, A., Baydar, E. et al.15–17 The pressure coefficient distribution law for a double-slit nozzle jet impinging on a horizontally shaped strip was studied by Abdel-Fattah, A. et al.18. The pressure coefficient distribution of a double-slit nozzle jet impinging on a sinusoidally shaped strip was examined by Liang, B., et al.19, and it was demonstrated that the pressure coefficient decreases as the sinusoidal amplitude A and the distance between the nozzle and the strip increase. Furthermore, Hung, J.Y. et al.20 reported that in suspension nozzle systems, the wall pressures are all positive, comparable to the heat transfer; the ratio between the wall pressure and the nozzle inlet pressure is almost 40%. Can Kang et al.21 studied the relationship between the jet's Reynolds number and the pressure coefficient and revealed that, while the nozzle suspension ability is enhanced as the Reynolds number increases, the pressure distribution's uniformity is reduced. The pressure properties of the suspension nozzle are determined by the combined influence of the aforementioned factors.

In conclusion, despite the fact that a large number of researchers have studied the heat transfer and pressure properties of nozzles in great detail, their work has primarily concentrated on the analysis and optimization of the same kind of suspension nozzles and has not included a thorough comparison of different types of nozzles. The present study involves the modelling of three distinct types of suspension nozzles: those with slits, injection holes, and dispersion holes. The heat transfer and pressure properties of these nozzles are examined, and a thorough performance index of the nozzles influenced by the weighting coefficients of the pressure performance and the heat transfer performance is suggested. The research findings have major implications for enhancing the performance and service life of lithium batteries and can be used as a guide for choosing extremely effective suspension nozzles in the production of lithium battery pole pieces.

Numerical Method

Suspension nozzle model

Three distinct suspension nozzle structure types are covered in this paper: the conventional double-slit suspension nozzle, the injection-holed suspension nozzle, and the effusion-holed suspension nozzle. The supplier of the three versions of suspension nozzles is Hunan Dali Intelligent Equipment Co., Ltd. Hot air enters the nozzle from the top. The suspension nozzle with injection holes not only emits air through double slits, but it also adds additional injection holes at the bottom, which can emit gas simultaneously. Additionally, the suspension nozzle with injection holes not only emits air through double slits, but it also allows for the partial discharge of air from the bottom of the injection holes. The structural parameters of these three nozzle types—such as the jet angle, slit width, separation spacing, and slit spacing—are all the same. The precise sizes are found in Table 1 and Fig. 2. All three of these suspension nozzles have the same inlet flow rate and set the jet Reynolds number to 6000 to guarantee the computation's accuracy and fairness. Table 1 Nozzle size and main structural parameters.

Parameter	Name	Value	
Lx	Range of nozzle influence in the x-direction	150 mm	
Ly	Range of nozzle influence in the y-direction	500 mm	
h	Separation spacing	6 mm	
b	Slit spacing	70 mm	
w	Slit width	1.5 mm	
α	Jet angle	45°	
d	Injection or effusion holes diameter	2 mm	
Ax	Injection or effusion holes are spaced in the x-direction	10 mm	
Ay	Injection or effusion holes are spaced in the y-direction	10 mm	

Fig. 2 Schematic diagram of suspension nozzle and target wall structure (a) Cross-section of traditional suspension nozzle with double slits (b) Cross-section of suspension nozzle with injection holes (c) Cross-section of suspension nozzle with effusion holes.

Fluid model and governing equations

The interior structure of the nozzle is simplified during the fluid modelling process, and just the airflow from the slits and holes is examined in the suspension nozzle flow field model. The pressure state and heat transfer inside the flow field will stabilise over time. The steady-state flow is calculated by the Reynolds-Averaged Navier–Stokes (RANS) equations and the descriptions of each item are as follows22:1 ∂ui∂xi=0

2 ρ∂uiuj∂xj=-∂P∂xi+∂∂xjμ(∂ui∂xj+∂uj∂xi)-ρui′uj′¯

3 ρuj∂T∂xj=∂∂xjμPr∂T∂xj-ρT′uj′¯

where T, P, and u denote the time-averaged temperature, pressure, and velocity of the fluid, Pr denote the Prandtl number of the air, ρui′uj′¯, ρT′uj′¯ denote the turbulent stress and turbulent heat flow density. The SSTk-ω equation, which is widely used in the field of narrow-channel flow fields and heat transfer23, is chosen to model the turbulent stress in Eq. (4) and the turbulent heat flow density in Eq. (5)24.4 ∂ρk∂t+∂ρkui∂xi=∂Γk∂k∂xi∂xj+Gk-Yk+Sk

5 ∂ρω∂t+∂ρωui∂xi=∂Γω∂ω∂xi∂xj+Gω-Yω+Sω

where Γk, Γω represent the effective diffusion terms of k and ω, indicating the diffusivity of k and ω. Gk is the turbulent kinetic energy produced by the laminar velocity gradient. Gω is generated by the ω equation. Yk,Yω represent the divergent terms of k and ω. Sk and Sω are user-defined25. The SSTk-ω turbulence model was solved by SIMPLEC pressure–velocity coupling equations using Ansys Fluent 2022 software.

The gas flow into the suspension nozzle is regulated by Eq. (6):6 Re=ρujwμ

where ρ denotes fluid density, uj denotes jet velocity, w denotes slit width, and μ represents the fluid viscosity.

The target wall Nu is computed using Eqs. (7) and (8) to describe the target wall's heat transfer intensity:7 Nu=φwλ

8 φ=qTj-Tw

where φ denotes the convective heat transfer coefficient, λ denotes the target wall thermal conductivity, q denotes the convective heat flow density, Tj, Tw represent jet temperature and target wall temperature.

According to Eq. (9), the pressure coefficient of the target wall is calculated to represent the pressure distribution of the target wall.9 Cp=2Pρuj2

where P represents the turbulent kinetic energy.

Model boundary condition

The model boundary conditions are configured as shown in Fig. 3, with the inlet turbulence intensity TI set to 5%26 and the slit exit configured as a velocity inlet boundary condition with a uniform velocity distribution;10 TI=2k3uj2

where k represents the local pressure at the target wall.Fig. 3 Model boundary condition.

The location where the other air flows outward is designated as the pressure outlet; the bottom surface of the model is designated as the target wall; the surface with injection or effusion holes is designated as an insulated wall with a heat flux Φ of 0; the injection holes are designated as the boundary condition of the uniform velocity inlet in the suspension nozzle model with injection holes; and the effusion holes are designated as the boundary condition of the pressure outlet in the suspension nozzle model with effusion holes. Table 2 below lists the exact boundary condition settings. Table 2 Model boundary condition setting.

Boundary condition	Description	
left slit (velocity inlet 1)	ux=0.707uj, uy=0, uz=-0.707uj, TI=5%, T=Tj=313K,Re=6000	
right slit (velocity inlet 2)	ux=-0.707uj, uy=0, uz=-0.707uj, TI=5%, T=Tj=313K,Re=6000	
injection holes (velocity inlet 3)	ux=0, uy=0, uz=-uj, T=Tj=313K,Re=6000	
effusion holes (pressure outlet)	P=0	
insulated wall	Φ=0	
target wall	T=Tw=298K	
pressure outlet	P=0	

Model meshing and validation

Figure 4 shows the meshing of three different types of suspension nozzle computational domains using tetrahedral grids. Figure 4a shows the computational domain meshing of traditional suspension nozzle with double slits, and Fig. 4b shows the computational domain meshing of suspension nozzles with injection holes or effusion holes. In the study, the three suspension nozzle flow field models were first meshed by Ansys Fluent 2022 software27, and the tetrahedral mesh was used to discretize the calculated fluid domain. In order to better capture the various physical parameters at the velocity inlet in the fluid domain, a mesh local encryption technique is used at the velocity inlet. Similarly, at the interface between fluid and solid, because the velocity gradient between fluid and solid is too large, it is necessary to set a certain number of boundary layers so as to better capture various physical characteristics parameters on the target wall, insulated wall, or flow field and ensure calculation convergence. Therefore, mesh encryption was also performed on the target wall and the insulated wall. The SSTk-ω turbulence model requires y+<1, so a boundary layer mesh needs to be added near the wall to ensure convergence of the calculations. y+ is defined as shown in Eqs. (11) and (12)28.11 y=y+μuτρ

12 y+=yuτρμ

where y represents the distance from the first node closest to the wall, μ represents the fluid viscosity, uτ represents the shear velocity, ρ represents the fluid density.Fig. 4 Creation of mesh in the computational domain (a) traditional suspension nozzle with double slits (b) suspension nozzles with injection holes or effusion holes.

Figure 5 shows the influence of three different numbers of meshes in the fluid domain of the traditional suspension nozzle with double slits on Nu on the line y/w=3.33 of the target wall. The Fig. 5 (a) shows that Nu does not vary much when the number of meshes is increased to 1,980,000. As a result, 1,980,000 meshes are employed in the computational solution method that follows for the traditional suspension nozzle with double slits fluid domain. Figure 5 (b) shows the target wall y/w=3.33 position y+ curve. All y+ values on the straight line are less than 1.Fig. 5 Mesh independence of the traditional suspension nozzle with double slits validate (a) The Nu distribution on the line y/w=3.33 at the target wall (b) Target wall y/w=3.33 position y+ curve. Origin 2022, available at https://www.originlab.com/, disposes of the data.

Similarly, Fig. 6 shows the influence of three different number of meshes in the fluid domain of the suspension nozzle with injection holes on Nu on the line y/w=3.33 of the target wall. The Fig. 6a shows that Nu does not vary much when the number of meshes is increased to 4,110,000. As a result, 4,110,000 meshes are employed in the computational solution method that follows for the suspension nozzle with injection holes fluid domain. Figure 6b shows the target wall y/w=3.33 position y+ curve. All y+ values on the straight line are less than 1.Fig. 6 Mesh independence of the suspension nozzle with injection holes validate (a) The Nu distribution on the line y/w=3.33 at the target wall (b) Target wall y/w=3.33 position y+ curve. Origin 2022, available at https://www.originlab.com/, disposes of the data.

Since the size, structure, and shape of the fluid domain of the suspension nozzle with effusion holes are the same as those of the suspension nozzle with injection holes, the mesh encryption and the addition of the boundary layer mesh area are also the same, with only the difference of the boundary conditions (velocity inlet for the injection holes and pressure outlet for the effusion holes). Therefore, the number of meshes used for the numerical calculation of the fluid domain of the suspension nozzle with effusion holes for subsequent simulations is the same as the number of meshes used for the calculation of the fluid domain of the suspension nozzle with injection holes, which is 4,110,000 meshes.

Experimental verification

Create an unrestricted single-slit impact jet model and compare the numerical simulation results with published experimental data29. The experimental conditions are: Re=5200,h/w=4,α=90∘,Tj=343K,Tw=308K. The comparative results are displayed in Fig. 7, where it is possible to guarantee the prediction accuracy of the model developed in this research because the numerical Nu curve of the target wall computed by simulation essentially follows the same trend as the Nu data curve obtained experimentally.Fig. 7 Comparison of simulation calculation results and experiment. Origin 2022, available at https://www.originlab.com/, disposes of the data.

Evaluation index

The uniformity of heat transfer and pressure of the suspension nozzle was evaluated using the metrics mentioned in the literature30, calculated as follows:13 Nu¯(x)=1Ly∫-Ly2Ly2Nu(x,y)dy

14 Nu¯=1Lx∫-Lx2Lx2Nu¯(x)dx

15 σNu(x)=1Ly∫-Ly2Ly2[Nu(x,y)-Nu¯(x)]2dy

16 σNu=1Lx∫-Lx2Lx2σNu(x)dx

17 Cp¯(x)=1Ly∫-Ly2Ly2Cp(x,y)dy

18 Cp¯=1Lx∫-Lx2Lx2Cp¯(x)dx

19 σCp(x)=1Ly∫-Ly2Ly2[Cp(x,y)-Cp¯(x)]2dy

20 σCp=1Lx∫-Lx2Lx2σCp(x)dx

where Nu(x,y) represents the Nu value at the coordinate position of the target wall (x,y), Nu¯(x) represents the average value of Nu in the x-direction of the target wall, Nu¯ represents the average Nu value of the entire target wall, σNu(x) represents the Nu standard deviation in the x-direction of the target wall, σNu represents the standard deviation Nu value of the entire target wall. Cp(x,y) represents the Cp value at the coordinate position of the target wall (x,y),Cp¯(x) represents the average value of Cp in the x-direction of the target wall, Cp¯ represents the average Cp value of the entire target wall, σCp(x) represents the Cp standard deviation in the x-direction of the target wall, σCp represents the standard deviation Cp value of the entire target wall.

Standard deviation is an indicator to measure the size of data fluctuation, the smaller the standard deviation is, the closer the data is to the average value, the smaller the data fluctuation is, and the better the stability is. Therefore, the smaller the σNu value of the target wall is, the better the heat transfer stability of the nozzle is; the Nu¯ of the target wall represents the heat transfer performance of the nozzle, and the larger the Nu¯ value is, the stronger the heat transfer performance of the nozzle is.

Definition ZNu=Nu¯σNu, ZNu is the comprehensive heat transfer performance index of the suspension nozzle; the larger the value of ZNu, the better the heat transfer performance of the nozzle. Similarly, define ZCp=Cp¯σCp, ZCp is the comprehensive pressure performance index of the suspension nozzle; the larger the value of ZCp, the better the pressure performance of the nozzle.

Results and discussion

Uniformity analysis

Figure 8 shows the heat transfer and pressure uniformity curves for the target wall in the x-direction. The positions of the injection or effusion holes are shown in the figure by the black dashed lines. The figure illustrates that the maximum σNu(x) of the suspension nozzle with injection holes occurs at a position near the stationary point of the injection holes, while the traditional suspension nozzle with double slits and the suspension nozzle with effusion holes' maximum σNu(x) occurs at a position close to the slit's stationary point. In the impinging jet stagnation region, suspension nozzles of any type exhibit poor Nu and Cp homogeneity, while in the impinging jet wall jet region, they exhibit poor Nu homogeneity but good Cp homogeneity, with a σCp(x)-value that is almost equal to zero. Through the above phenomenon, we can infer that the factors affecting the uniformity of suspension nozzles Nu and Cp are different, and different types of suspension nozzles have different factors affecting their Nu and Cp homogeneity.Fig. 8 The heat transfer and pressure uniformity curves for the target wall in the x-direction. (a) σNu(x) curve of the target wall (b) σCp(x) curve of the target wall. Origin 2022, available at https://www.originlab.com/, disposes of the data.

The factors affecting the Nu uniformity of traditional suspension nozzles with double slits and suspension nozzle with effusion holes are turbulent intermittence and edge effects. Figure 9a shows the Nu curves and the Cp curves along the y-direction of the traditional suspension nozzle with double slits at the position near the slit stationing point. As can be observed, there is an uneven oscillation in the Nu while y/w is modest (in the middle of the suspension nozzle) and a dramatic decrease in the Nu when y/w increases to a specific value (near the edge of the suspension nozzle). While turbulent intermittence is the reason for the erratic fluctuation of Nu in the middle of the suspension nozzle, edge effects are responsible for the sudden decrease of Nu near the nozzle's edge. Dairay, T. et al.'s31 findings, which revealed a high level of temporal and geographical intermittency in the events connected to substantial heat transport, can support this phenomenon. The factors affecting the Nu uniformity of suspension nozzles with injection holes are not only turbulent intermittence and edge effects, but also the interaction between adjacent jets. Figure 9b shows the Nu curves and the Cp curves along the y-direction of the suspension nozzle with injection holes at the position near the stagnation point of the injection holes. Compared to the traditional suspension nozzle with double slits and the suspension nozzle with effusion holes, it is evident that the Nu in the middle of the suspension nozzle appears to oscillate periodically due to the jet ejected from the injection holes. These oscillations are more frequent and intense. The characteristics of periodic oscillation have also been observed in some literatures32–34. On the other hand, because of edge effects, the Nu periodic oscillations are much less pronounced at the suspension nozzle edge position.Fig. 9 Nu and Cp varying with the y-direction coordinate (a) The position near the slit stationing point of the traditional suspension nozzle with double slits (b) The position near the stagnation point of the injection holes of the suspension nozzle with injection holes. Origin 2022, available at https://www.originlab.com/, disposes of the data.

The factors affecting the Cp uniformity of traditional suspension nozzles with double slits and suspension nozzles with effusion holes are turbulent intermittence and edge effects, and the factors affecting the Cp uniformity of suspension nozzles with injection holes are edge effects and interaction between adjacent jets. Figure 9 (a) exhibits a Cp distribution curve of the traditional suspension nozzle with double slits. In the middle of the nozzle, Cp produces irregular oscillations due to intermittent turbulence. Additionally, the Cp decreases sharply at the nozzle's edge position due to edge effects, and the final Cp decreases to 0 due to the existence of the y-direction pressure outlet boundary condition, resulting in the inhomogeneity of the target wall's Cp. In Fig. 9b, the interaction between adjacent jets also leads to periodic oscillations in Cp, reducing the homogeneity of Cp at the target wall.

The interaction between adjacent jets has a greater effect on Nu homogeneity than turbulent intermittence and edge effects on Nu homogeneity, while edge effects have a greater effect on Cp homogeneity than the interaction between adjacent jets and turbulent intermittence on Cp homogeneity. Out of the three types of suspension nozzles, the one with effusion holes has superior Nu and Cp uniformity because exhaust gases can be excluded from the effusion holes, which reduces the characteristic velocity in the fluid domain and reduces the edge effects caused by the velocity field. The suspension nozzle with injection holes has a stronger interaction between adjacent jets due to the presence of injection holes, which improves the Nu and Cp values near the location of the stationing point of the injection holes and worsens the homogeneity of Nu and Cp.

To learn more about how three elements—turbulent intermittence, edge effects, and interaction between neighbouring jets—affect Nu and Cp, the Nu and Cp cloud plots of three different types of suspension nozzles are given in Fig. 10 and Fig. 11, where (a) is the traditional suspension nozzle with double slits, (b) is the suspension nozzle with injection holes, and (c) is the suspension nozzle with effusion holes. The middle region of the suspension nozzle with injection holes, Nu and Cp, is smaller than the edge region, as depicted in regions A and B of the figure, as can be observed. In comparison to region B, region A has a lower value. The primary reason for this is that the middle region of the exhaust gas is primarily discharged along the x direction, while the edge of the region is primarily discharged along the y direction. When the exhaust gas is discharged in the direction of the x, the slit jet's role is diminished, resulting in the middle region of the slit jet being weaker than the edge of the region. The shape of the high Nu and Cp region created by the injection holes is observed to be nearly circular in the middle region and nearly crescent-shaped in the edge region. This phenomenon is caused by edge effects, which are influenced by the y-direction exhaust gas discharge from the edge region. For the three different types of nozzles, Nu and Cp are not homogeneous at the nozzle slit injection region because of turbulent intermittence.Fig. 10 Nu for three types of suspension nozzles. (a) traditional suspension nozzle with double slits (b) suspension nozzle with injection holes (c) suspension nozzle with effusion holes. Ansys Fluent 2022, available at https://www.ansys.com/zh-cn/products/fluids/ansys-fluent, and Tecplot 360 EX 2023, available at https://tecplot.com/products/tecplot-360/, simulate and post-process all pictures.

Fig. 11 Cp for three types of suspension nozzles. (a) traditional suspension nozzle with double slits (b) suspension nozzle with injection holes (c) suspension nozzle with effusion nozzles. Ansys Fluent 2022, available at https://www.ansys.com/zh-cn/products/fluids/ansys-fluent, and Tecplot 360 EX 2023, available at https://tecplot.com/products/tecplot-360/, simulate and post-process all pictures.

To examine how three elements—turbulent intermittence, edge effects, and interaction between neighbouring jets—affect the velocity field, the velocity clouds for two cuts, y/w=3.33 and y/w=163.33, of three different nozzle types are shown in Fig. 12. The figure illustrates that, regardless of the suspension nozzle type, the velocity at the nozzle's edge position (y/w=163.33) is higher than that at the nozzle's centre position (y/w=3.33) in the stagnation region. This is primarily because the nozzle's edge position is close to the pressure outlet boundary condition, where exhaust gases are primarily discharged in the y-direction. In contrast, the nozzle centre position (y/w=3.33) in the wall jet zone has a higher velocity than the nozzle edge position (y/w=163.33), primarily because the exhaust gas in the centre region is primarily released in the x direction. Suspension nozzles with injection holes have higher velocities because gas is injected through them; on the other hand, exhaust gases are expelled from suspension nozzles with effusion holes, which cause higher velocities at the position of the effusion holes. As can be seen from the figure, the mutual jet effect between the slits and the injection holes causes the velocity in the red boxed region of the suspension nozzle with injection holes to be smaller than that of the traditional suspension nozzle with double slits and the suspension nozzle with effusion holes. At the centre of the suspension nozzle, the gas ejected from the injection hole is discharged in the x-direction, weakening the effect produced by the slit jet. The figure shows that at the centre of the suspension nozzle, the gas emitted from the jet hole directly impacts the target wall and then spreads out around it. However, at the edge position, the gas emitted from the jet hole does not impact the target wall because of the edge effect. The pressure outlet position is near the suspension nozzle's edge position. The cumulative action of the cross-flow in the y-direction affects the gas injected from the injection hole before it reaches the target wall, deflecting and discharging the gas in y-direction. This phenomenon can also be seen in the velocity cloud of the suspension nozzle with injection holes in the x/w=3.33 plane in Fig. 13.Fig. 12 Velocity cloud in plane y/w=3.33 and y/w=163.33 for three types of suspension nozzles. (a) plane y/w=3.33 of traditional suspension nozzle with double slits. (b) plane y/w=163.33 of traditional suspension nozzle with double slits. (c) plane y/w=3.33 of suspension nozzle with injection holes. (d) plane y/w=163.33 of suspension nozzle with injection holes. (e) plane y/w=3.33 of suspension nozzle with effusion holes. (f) plane y/w=163.33 of suspension nozzle with effusion holes. Ansys Fluent 2022, available at https://www.ansys.com/zh-cn/products/fluids/ansys-fluent, and Tecplot 360 EX 2023, available at https://tecplot.com/products/tecplot-360/, simulate and post-process all pictures.

Fig. 13 Velocity cloud in plane x/w=3.33 for suspension nozzle with injection holes. Ansys Fluent 2022, available at https://www.ansys.com/zh-cn/products/fluids/ansys-fluent, and Tecplot 360 EX 2023, available at https://tecplot.com/products/tecplot-360/, simulate and post-process all pictures.

In conclusion, Table 3 illustrates the Nu and Cp standard deviation values for the three different types of suspension nozzles. The smaller the value, the better the performance. From the table, it can be seen that the suspension nozzle with effusion holes has the best Nu and Cp uniformity, the traditional suspension nozzle with double slits has the second best, and the suspension nozzle with injection holes has the worst Nu and Cp uniformity. Table 3 Nu and Cp standard deviation values.

Suspension nozzle type	σNu	σCp	
traditional suspension nozzle with double slits	2.868	0.064	
suspension nozzle with injection holes	4.609	0.083	
suspension nozzle with effusion holes	2.729	0.058	

Average performance analysis

Different types of suspension nozzles do not have the same average Nu and Cp performance at the target wall. From Fig. 14a, we can observe that the nozzle type affects the Nu average performance mainly in the stagnation region. In the range -10<x/w<10, the suspension nozzle with injection holes has the largest average value of Nu, where the maximum value is located near the position of the stationary point of the injection holes, which has the strongest heat transfer. The suspension nozzle with effusion holes’s performance in heat transport comes second. The traditional suspension nozzle with double slits has the lowest Nu average value and the weakest heat transfer. And in the range -22.3<x/w<-10 and 10<x/w<22.3, the suspension nozzle with effusion holes has the largest average value of Nu, Traditional suspension nozzles with double slits has the second largest Nu average value, the suspension nozzle with injection holes has the smallest average value of Nu.Fig. 14 Nuavg(x) and Cpavg(x) on the target wall (a) Nuavg(x) (b) Cpavg(x). Origin 2022, available at https://www.originlab.com/, disposes of the data.

Figure 14b, we can observe that the influence of nozzle type on Cp average performance is also mainly reflected in the retention area. The suspension nozzle with injection holes has the largest average value of Cp in the stagnation region, with the maximum value located near the stationary position of the injection holes, followed by the traditional suspension nozzle with double slits, and the suspension nozzle with effusion holes has the smallest average value of Cp.

Differences in heat transfer performance are due to changes in the temperature field in the region. Figure 15 provides temperature cloud plots for three different suspension nozzles at the plane y/w=3.33. The figure shows that the suspension nozzle with injection holes has a higher temperature in the area near the injection holes but a lower temperature in the red-boxed region (-22.3<x/w<-10 and 10<x/w<22.3) than the traditional suspension nozzle with double slits, while the suspension nozzle with effusion holes has a higher temperature in the red-boxed area (-22.3<x/w<-10 and 10<x/w<22.3) than the traditional suspension nozzle with double slits. Figure 16 provides velocity vector diagram for three different suspension nozzles at the plane y/w=3.33. The airflow from the injection holes weakens the slit jet, resulting in a smaller recirculation size in the red-boxed region and lower temperatures in the region, which ultimately leads to weak heat transfer performance in the region. while due to the existence of effusion holes, the size of the recirculation in the red-boxed region increased and raised temperatures in the region, which ultimately led to good heat transfer performance in the region.Fig. 15 Temperature cloud plots for three different suspension nozzles at the plane y/w = 3.33. (a) traditional suspension nozzle with double slits (b) suspension nozzle with injection holes (c) suspension nozzle with effusion nozzles. Ansys Fluent 2022, available at https://www.ansys.com/zh-cn/products/fluids/ansys-fluent, and Tecplot 360 EX 2023, available at https://tecplot.com/products/tecplot-360/, simulate and post-process all pictures.

Fig. 16 Velocity vector diagram for three different suspension nozzles at the plane y/w=3.33.(a) traditional suspension nozzle with double slits (b) suspension nozzle with injection holes (c) suspension nozzle with effusion nozzles. Ansys Fluent 2022, available at https://www.ansys.com/zh-cn/products/fluids/ansys-fluent, and Tecplot 360 EX 2023, available at https://tecplot.com/products/tecplot-360/, simulate and post-process all pictures.

The effect of nozzle type on the average Cp performance is also centred on the stagnation region, which has a significant effect on the overall performance. For suspension nozzles with injection holes, the existence of injection holes increases the pressure at the target wall close to the injection hole region, which indirectly raises the target wall Cp average. And when a suspension nozzle has effusion holes, the exhaust gases can be released from the holes, which lowers the average target wall Cp value.

Above all, Table 4 illustrates the Nu and Cp average values for the three different types of suspension nozzles. The bigger the value, the better the performance. From the table, it can be seen that the suspension nozzle with injection holes has the biggest Nu average value, the suspension nozzle with effusion nozzles has the second biggest, and the traditional suspension nozzle with double slits has the smallest Nu average value. The suspension nozzle with injection holes has the largest average value of Cp, followed by the traditional suspension nozzle with double slits. The suspension nozzle with effusion holes has the smallest average value of Cp. Table 4 Nu and Cp average values.

Suspension nozzle type	Nu¯	Cp¯	
traditional suspension nozzle with double slits	9.247	0.220	
suspension nozzle with injection holes	12.660	0.277	
suspension nozzle with effusion holes	9.381	0.196	

Comprehensive performance analysis

The Nu and Cp comprehensive performances represent how much average performance can be obtained on the target wall at the cost of the Nu and Cp uniformity. Table 5 shows the values of the comprehensive heat transfer performance index ZNu and the comprehensive pressure performance index Zcp of three different types of suspension nozzles. The bigger the value, the better the performance. Table 5 ZNu and Zcp values for three different types of suspension nozzles.

Suspension nozzle type	ZNu	Zcp	
traditional suspension nozzle with double slits	3.224	3.438	
suspension nozzle with injection holes	2.747	3.337	
suspension nozzle with effusion holes	3.438	3.379	

The suspension nozzle's comprehensive performance index is derived from the linear combination of its comprehensive pressure performance index and comprehensive heat transfer performance index. The nozzle's comprehensive performance index Z is ultimately determined by adjusting its weighting factors α and β in accordance with its preference settings for comprehensive heat transfer and comprehensive pressure performances. This expression is shown in Eqs. (21) and (22), and the larger the value of the index Z, the better the nozzle's performance is represented.21 α+β=1

22 Z=αZNu+βZCp

Figure 17 illustrates the link between the weight factors α and β of the comprehensive heat transfer and pressure performance and the comprehensive performance index Z of three types of suspension nozzles. When the weight factor of heat transfer performance is less than 21.61% and the weight factor of pressure performance is more than 78.39%, the comprehensive performance of the traditional suspension nozzle with double slits is the best. As the weight factor of heat transfer performance is increasing, the weight factor of pressure performance is decreasing. The comprehensive performance of the suspension nozzle with effusion holes is better than that of the traditional suspension nozzle with double slits and the suspension nozzle with injection holes. Moreover, the optimal range of α and β of the suspension nozzles with effusion holes is larger than that of the traditional suspension nozzle with double slits, so it can be concluded that the comprehensive performance of the suspension nozzles with effusion holes is the best, followed by the traditional suspension nozzle with double slits, and the comprehensive performance of the suspension nozzles with injection holes is the worst.Fig. 17 Z varying with weighting factors α and β for three types of suspension nozzles.

Conclusion

This study presents the results of simulation calculations to study the pressure characteristics and heat transfer for the three types of suspension nozzles. Numerous data indications were acquired, including the local and average Nussle numbers, the local and average pressure coefficients, and so forth. The comprehensive heat transfer performance index ZNu and the comprehensive pressure performance index Zcp of the suspension nozzle are proposed, and based on the weighting factor occupied by these two properties, the comprehensive performance index Z of the suspension nozzle is innovatively proposed. A detailed comparative analysis of three different types of suspension nozzles in terms of heat transfer uniformity, heat transfer averaging, pressure uniformity, and pressure averaging has been carried out using these indicators, and their comprehensive performance has been discussed. The main conclusions are summarized below:(1) A comprehensive nozzle performance index Z is proposed that combines the heat transfer and pressure performance of the nozzle. The results show that when the weight factor of heat transfer performance is less than 21.61% and the weight factor of pressure performance is more than 78.39%, the comprehensive performance of the traditional suspension nozzle with double slits is the best. As the weight factor of heat transfer performance is increasing, the weight factor of pressure performance is decreasing. The comprehensive performance of the suspension nozzle with effusion holes is better than that of the traditional suspension nozzle with double slits and the suspension nozzle with injection holes.

(2) The causes of the differences in suspension nozzle heat transfer and pressure uniformity and average performance were investigated. The main factors affecting the differences in suspension nozzle heat transfer and pressure uniformity are turbulent intermittence, interaction between neighbouring jets, and edge effects. The main factors affecting the nozzle heat transfer and pressure averaging performance are the size of the recirculation, the impact of the jet on the target wall, and so on.

(3) The comprehensive performance of the suspension nozzles with effusion holes is the best, followed by the traditional suspension nozzle with double slits, and the comprehensive performance of the suspension nozzles with injection holes is the worst. The optimal range of α and β of the suspension nozzles with effusion holes is larger than that of the traditional suspension nozzle with double slits.

Author contributions

Zhihui Liu was responsible for all aspects of the paper to ensure the accuracy and completeness of the paper, scheme design, etc. Zhijian Zhang was responsible for writing the first draft of the paper, numerical simulation, experimental comparison, etc. Jiahao Zhang was responsible for the picture processing and data analysis of the paper. Zhigang Chen was responsible for the supplement and improvement of the paper.

Funding

This work was supported by [Hunan graduate research innovation project and Shaoyang University graduate research innovation project] (Grant numbers [QL20230277], [CX2023SY009], [QL20230278] and [CX2023SY010]). The authors would also like to acknowledge financial support from the Natural Science Foundation of Hunan Province of China and the Shaoyang key research and development project. (Grant numbers 2024JJ7494 and 2023CG2009).

Data availability

All data has been provided in the manuscript.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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