Flow Field Simulation and Nozzle Parameter Optimization of Rust Removal Robot Based on High-Pressure Water Jet Technology

To enhance the safety and longevity of large ships, employing high-pressure water jet technology for rust removal is crucial in this article, the qualitative relationship among the key parameters of high-pressure water jet rust removal is analyzed theoretically. Secondly, the flow field simulation is carried out using Computational Fluid Dynamics (CFD) software to verify the relationship among the key parameters. Finally, the influence of different nozzle length-diameter ratios on jet characteristics is analyzed. The simulation results show that the nozzle end length is 0 mm and the nozzle diameter is 1mm for the straight-cone convergent nozzle with a shrinking angle of 30° when the diameter of the length-diameter ratio remains unchanged and the length of the nozzle end is changed. Simultaneously, the highest values are observed for the nozzle exit velocity and the impact force on the target surface. The impact range of target shear force takes the maximum value when the length-diameter ratio is 1. Further, by changing the length of the nozzle-end and the nozzle diameter in the length-diameter ratio, the influence on the jet characteristics can be concluded that the nozzle diameter has a greater influence on the nozzle exit velocity than the nozzle-end length. On this basis, the optimized nozzle parameters can be obtained i.e., the nozzle-end length is 1 mm and the nozzle diameter is 1 mm, such that the effective striking area of the jet is the largest and the striking effect is the best.


Introduction
As the ship equipment manufacturing industry rapidly advances, ship de-rusting plays an increasingly important role in ship building and repairing.The outer wall of the ship is soaked in seawater for a long time, resulting in rust attached to the outer wall of the ship.To improve the service cycle and safety of the ship, it must be cleaned regularly.The traditional cleaning is manual or machine sandblasting.This way of rust cleaning is not only low efficiency and waste resources, but also pollute the environment.The development of high-pressure water jet technology provides an efficient and environmentally friendly method for ship rust removal [1].
High-pressure water jet technology involves converting mechanical energy into water pressure energy using a supercharging device.Subsequently, this pressure energy is released through a smalldiameter nozzle, generating a high-speed jet beam with significant specific energy for processing various materials.The water jet has high speed and great energy, so it has a great striking force and destructive power [2,3].However, the existing studies only focus on the influence of a certain parameter on the rust removal efficiency of high-pressure water jet or the motor power, without actual simulation experiments on the specific parameters of the nozzle [4][5][6][7].
This paper employs Computational Fluid Dynamics (CFD) software to examine the flow dynamics of high-pressure water jets in the process of ship de-rusting.Initially, the accuracy of the simulation model and parameters is validated within the scheme.Subsequently, variations in the length-diameter ratio of the nozzle are explored through flow field simulations.Finally, the parameters of nozzle-end length and nozzle diameter with the best de-rusting effect is obtained, which provides the optimal parameters for future research on high pressure water jet de-rusting.

Analysis of Parameters Related to High-Pressure Water Jets
The high-pressure de-rusting water jet belongs to the high-pressure non-submerged jet structure.The jet stream ejected by the nozzle is arranged symmetrically with the central axis of the nozzle, and the conical distribution is divergent.The jet can be divided into the core segment, basic segment and divergent segment [8].
The calculation of jet force conforms to the law of momentum equation.The length of the core section can be calculated by the nozzle diameter.As the nozzle diameter increases, the length of the core section also increases [9].It can be obtained from Eq. ( 1), and the rust removal using high-pressure water jets occurs in close proximity to the target.Therefore, the simulation focuses exclusively on analyzing the core section.
( ) In Eq. ( 1): L y is the length of original section of jet, d z is the nozzle diameter.
The jet exhibits a characteristic where dynamic pressure remains nearly constant along the axial direction, and the dynamic pressure at the jet tip equals that at the nozzle outlet.Operating as an incompressible viscous fluid in a turbulent state, the high-pressure water jet adheres to the continuity, motion, and energy equations during flow.This adherence establishes a theoretical foundation for deriving subsequent formulas [10].

Nozzle Diameter
The nozzle diameter is contingent on both the jet pressure and the flow rate.In theory, a higher jet pressure corresponds to a larger nozzle diameter, resulting in heightened cleaning efficiency under constant conditions.However, considering factors like the flow rate and power loss of the water jet, and assuming constant pressure and flow rate in the high-pressure chamber jet, the nozzle diameter reaches its maximum value.Eq. ( 2) is employed to determine the nozzle diameter, highlighting its direct influence on the maximum velocity of the jet at the nozzle exit.
In Eq. (2): d is the nozzle orifice diameter with the unit being mm.q is the flow rate with the unit being L/min.p is the jet pressure with the unit being MPa.β is the hydraulic coefficient (determined by the system structure).n is the number of nozzle holes.η2 is the nozzle structure coefficient (determined by the nozzle type).

Jet Speed
In general, the outlet velocity of the nozzle is used to represent the jet velocity.For continuous jets, the Bernoulli's equation is applied between the two points inside and outside the nozzle exit section.
Neglecting the difference between the heights of the two locations, Eq. ( 3) can be rewritten as According to the principle of water jet, the densities before and after the jet are constant values.Thus, setting ρa1=ρa2 leads to In Eq. ( 5): The equation of continuity between the two end-points of the fluid flow can be expressed as Let the nozzle be a common round pipe nozzle, then a A can be expressed as With the assumptions ρa1=ρa2, the jet flow rate at v can be obtained In Eq. ( 7): at v is the jet flow rate with the unit being m/ s .a p is the jet pressure with the unit being MPa. 1 a P and 2 a P are nozzle internal and external static pressure.da1and da2 are diameter of nozzle front section and end.

Establishment and Analysis of The Flow Field Simulation Model
The flow field efflux characteristic of the high-pressure water jet is that when high-pressure water is ejected from the nozzle, it combines with air in a short time, and then the efflux characteristic will change dramatically, resulting in its instability.When the fluidic characteristics cannot be determined by traditional numerical calculation, it is expected to use CFD software for finite element analysis of the fluidic characteristics.

Selection of Nozzle
The first step is to determine the shape of the nozzle.In the realm of high-pressure water jet rust removal research, straight-cone convergent nozzles are predominantly utilized [11,12].This type of nozzle exhibits favorable characteristics in terms of velocity distribution concentration.The water jet expelled from it showcases more consistent clustering features compared to other nozzle types.For the purposes of this paper, we opt for the straight-cone convergent nozzle, specifically denoted as the A-type nozzle.The crucial parameters are outlined in Fig 1, where the shrinkage Angle (α) is 30°and the lengthdiameter ratio (l/d) is 2. It can be seen from Ref. [13] that the rust layer and paint layer can be easily removed when the water jet pressure reaches 180 MPa.Therefore, the water jet pressure of the high-pressure water jet for rust removal is generally 200 ~ 250 MPa.Considering real-world operational conditions and practical experimentation, the theoretical flow rate of the pump in this paper is 30 L/min and the jet pressure is set to 200 MPa.
On the base of consulting relevant data, the hydraulic coefficient β is 0.658, the nozzle structure coefficient η2 is 1.1 according to the nozzle type, then the diameter of each nozzle can be determined as 0.978 mm by Eq (2).In this way, the diameter of each nozzle in the simulation is set to be 1 mm.

Selection and Boundary Setting of Flow Field Model
The physical model employed for simulating the flow field of high-pressure water jets falls under the category of a multiphase flow model.Upon ejection from the nozzle, the high-pressure water swiftly integrates with the surrounding air during the diffusion process.This leads to cavitation.The energy exchange between the water and air leads to a big difference between the speed of the water and air.In this situation, it is suitable to use the mixture model to simulate the multiphase flow field.There are three phases in Mixture models.The main phase (the first term) is selected as water, the secondary phase (the second term) is selected as water vapor, and the final phase (third term) is selected as air.The turbulence model is selected as the Realizable k-ε model.As shown in Fig 2, the flow field is meshed and the boundary conditions are set.The model for the high-pressure water jet flow field is a cylinder measuring 50 mm in diameter and 100 mm in height.The meshes are divided and the local meshes at the nozzle are encrypted.The jet was pure water jet, the static pressure at the entrance was set as 200MPa, the injection ambient pressure was set as standard atmospheric pressure (0.1MPa), the environmental medium outside the nozzle outlet was air, and all the solid boundary mechanical conditions were set as no slip condition.

Simulation of the Flow Field
The simulation of the flow field involves conveying high-pressure water to the nozzle via a pipeline, followed by the expulsion of the water through the nozzle.The water jet interacts with the target surface through the air medium, simulating the process of ship rust removal.Fig. 3 depicts the velocity cloud diagram of the flow field section at a target distance of 100 mm.It can be seen from Fig. 3 that the maximum velocity of the nozzle exit is 632 m/s.When the jet pressure p is 200 MPa, the theoretical velocity of the jet can be calculated as 633.14 m/s according to Eq. ( 9).The maximum velocity of the nozzle outlet obtained by simulation is slightly different from the theoretical calculation result, but it belongs to the error range.The comparison results demonstrate the correctness of the simulation result of the water jet and the effectiveness of the selected hydraulic model.When the high-pressure water jet acts on the surface of the object, the surface of the object is subject to great impact force.When the shear stress reaches a critical value, the surface of the object will produce cracks.With the expansion of cracks, the target surface will produce greater stress concentration.This process will cause part of the material to be stripped from the body, and the cleaning effect can occur.Simply put, it is that the shear force strikes to produce cracks and the striking force is removed.Therefore, the impact pressure of jet and the shear force of target surface are the main parameters to evaluate the impact effect of water jet.By referring to literature [14], it can be seen that the rust layer and paint layer can be easily removed when the jet striking pressure reaches 1.7x108Pa and the target shear force reaches 6.0x105Pa.At this time, the force area on the target surface is the effective striking area of the jet.
When a high-pressure water jet strikes an object's surface, the impact force comprises the total force of the jet.However, this force alone does not adequately convey the jet's destructive capability.The pressure is used to directly express the destructive ability of jet flow, namely, the force of jet flow applied to the unit area of the target surface, which is generally called the jet impact pressure.Combined with Fig. 4 and Fig. 6, it can be seen that the impact pressure on the target surface presents a Gaussian distribution, that is, the impact pressure near the center is the largest, and then decreases as the position moves away from the center.The data on both sides presents an axisymmetrical distribution, which is caused by the energy exchange between the air and the water beam that starts from the boundary to the inside.
As can be seen from Fig. 5 and Fig. 6, the main contour feature of shear stress on the target surface is a circular ring in the central impact area, in the form of a bimodal curve.In other words, the shear force at the center of impact is 0, and then increases sharply.After reaching the maximum shear force, the value of shear force becomes smaller and smaller with the increasing distance from the impact center.

Influence of Different Nozzle length-diameter Ratio on The Jet Characteristics
Simulation of the jet flow field reveals that the velocity at the nozzle is influenced by nozzle size, subsequently impacting the efficiency of rust removal.The nozzle with good performance, suitable material and matchup with the pump unit can be designed to improve the efficiency of jet cleaning.Therefore, the length-diameter ratio is changed on the basis of A-type nozzle, and the four types of different length-diameter ratio nozzles, i.e., A0 (l/d=0), A1 (l/d=1), A2 (l/d=2) and A3 (l/d=3) are designed as shown in Fig. 7.It is evident from the figure that the high-pressure water jet flow bundles in all four cases share a similar shape throughout the entire flow field, with the maximum velocity occurring at the nozzle-end.However, it is observed that the A0 nozzle, characterized by a length-diameter ratio of 0, attains a maximum velocity of 693 m/s.Meanwhile, the A1, A2, and A3 nozzles, each possessing length-diameter ratios of 1, 2, and 3 respectively, exhibit velocities of approximately 630 m/s.This is because the increasing length of the end of the nozzle makes the water jet energy dissipate faster here.Therefore, whether the length of the tail of the nozzle exists or not both affects the flow velocity of the  By further comparing the simulated target surface hitting pressure clouds in Fig. 9, it can be seen that A0 nozzle has the largest striking force and the widest striking range.Furthermore, we observe a decrease in impact pressure on the target as the length-diameter ratio at the nozzle end increases.
Referring to Fig. 10, the shear force nebulae on the target surface for nozzles with varying lengthdiameter ratios reveal that the striking range of target surface shear force is maximized at a lengthdiameter ratio of 1, indicating the largest effective striking area.For length-diameter ratios of 2 and 3, the maximum shear force on the target surface isn't uniformly distributed on the ring, except at the center.However, the entire ring area remains within the effective striking range.

Parameter Optimization and Result Analysis
It can be seen that when the length-diameter ratio is 0, i.e., the nozzle end length is 0, the A0 nozzle has the maximum hitting pressure on the target surface.The A1 nozzle with the aspect ratio of 1 has the largest shear force and the widest effective force area.The change of nozzle end length and nozzle diameter will affect the energy loss of water jet.On the other hand, it will also affect the convergence of water jet.To comprehensively assess the combined impact of optimal end length and nozzle diameter on the high-pressure water jet, we modified the nozzle diameter while keeping the end length constant for the previously mentioned four A-type nozzles.Specifically, we varied the diameters to 0.8 mm, 1.0 mm, and 1.2 mm while maintaining all other conditions unchanged.This allowed for a more thorough analysis of the overall influence of the nozzle on the high-pressure water jet.The influence of different nozzle outlet diameters on high-pressure water jet characteristics is simulated by the flow field.
A comparative test is designed in which the horizontal factor is the end-length of the nozzle and the vertical factor is the diameter of the nozzle.The comparative experimental results of the design are shown in Table 1.

Table 1. Comparative experimental design table
Nozzle end length(mm) Figure 11.Distribution of maximum impact force and maximum shear force on 12 experimental target surfaces Through simulation, the distribution diagram of maximum impact force and maximum shear force on the target surface can be obtained as shown in Fig. 11.On the broken line of maximum impact force on the target surface, when L0=0 and L1=1, no matter the diameter d1=0.8mm or d3=1.2mm, the maximum impact force of the jet on the target surface is smaller than that when d2=1mm, and when L2=2 and L3=3, In all cases, the highest impact force on the target plane occurs when d1 is set to 0.8mm.Subsequently, the maximum impact force gradually diminishes with an increase in diameter.Concurrently, the trend in the maximum shear force on the target plane, as depicted by the broken line, mirrors that of the maximum impact force.
Further, it can be seen from Fig 11 that the maximum impact force of group 2, group 5 and group 7 is above 1.8x108Pa on the broken line of the maximum impact force on the target surface, which means these experimental groups are the effective impact force groups on the target surface.On the broken line of the maximum shear force on the target surface, it can be seen that the maximum shear force of the first group, the second group, the third group, the fourth group, the fifth group, the seventh group, the eighth group, the tenth group and the eleventh group are all above 6x105Pa, that is, these experimental groups are the effective shear force groups on the target surface.It can be further seen that the groups meeting the above two critical values are group 2, group 5 and group 7.In the scatter diagram of the impact force on the target surface with the nozzle parameter L2=2 and d2=1 in Fig. 12.The estimation calculation method is adopted to divide the sample point areas with the impact force exceeding 1.8x108Pa on the target surface, and then find out the corresponding horizontal axis coordinate positions.The two points on the horizontal axis coordinate are taken as diameter, and the calculated circular area is the area of the effective impact force.Similarly, in the scatter diagram of  13, the estimation calculation method is adopted to divide the sample point areas where the target surface shear force exceeds 6x105Pa, and then find out the corresponding horizontal axis coordinate positions.The two points on the horizontal axis coordinate are taken as diametrically, and the calculated circular area is the effective shear force striking area.The calculation results of 12 groups are shown in Table 2  The experimental results in Table 2 reveal that the nozzle end length is 0 and the nozzle diameter changes from 0.8mm to 1mm, the change is 25% and the jet velocity increases by 553%; when the nozzle diameter changes from 1mm to 1.2mm, the change is 25% and the jet velocity decreases by 32%.When the nozzle end length is 1 mm and the nozzle diameter changes from 0.8mm to 1mm, the jet velocity increases by 550%; when the nozzle diameter changes from 1mm to 1.2mm, the jet velocity decreases by 25%.It's evident that when the nozzle diameter is below 1mm, the influence of nozzle diameter variation on jet velocity is greater than that when the nozzle diameter is more than 1mm.In contrast, when the nozzle diameter is 1, the nozzle end length changes from 1mm to 2mm, the change is 100%, and the jet velocity only increases by 0.80%; when the nozzle end length changes from 2mm to 3mm, the change is 100%, and the jet velocity only increases by 0.16%, so the velocity change is negligible.
It can be concluded that the change of nozzle diameter has a greater impact on its exit velocity than the change of end length.The ultimate purpose of the optimization experiment is to obtain the best striking effect, that is, the effective striking area of the jet's striking pressure on the target surface and the effective striking area of the shear force on the target surface are maximum.It can also be seen from Table 2 that the effective striking area of the target surface pressure and the effective striking area of the target surface shear force in group L1d2 is 2.29e-01mm 2 and 3.16e-04mm 2 , both of which are the largest values in all experimental groups.Hence, at an inlet pressure of 200MPa, a target distance of 100mm, and a contraction angle α of 30°, the optimal simulation outcome is observed in the L1d2 group.This implies that the most effective striking effect is achieved when the nozzle end length is 1mm and the nozzle diameter is 1mm.

Summary
The multiphase flow model is selected for the flow field simulation.By comparing the simulation results with the theoretical outcomes, we validate the accuracy of the water jet simulation and the efficacy of the chosen turbulence model.
By changing the length-diameter ratio of the nozzle-end, the influence of the nozzle length-diameter ratio on jet characteristics is analyzed.The flow field velocity cloud maps and the target surface impact force cloud maps of the four types of nozzles (A0 (l/d=0), A1 (l/d =1), A2 (l/d =2) and A3 (l/d =3)) can be obtained through the simulation.It is found that A0 nozzles with aspect ratio of 0 have the largest velocity, while A1, A2 and A3 nozzles with aspect ratio of 1, 2 and 3 have little difference in velocity, indicating whether the nozzle end affects the size of jet velocity.The cloud diagram of the impact force on the target surface of the nozzle with different length-diameter ratio is further analyzed.It is found that the impact pressure on the target surface diminishes as the length-diameter ratio at the nozzle's end increases.Additionally, the kinetic energy loss of the jet amplifies with an increase in the length of the nozzle's end.When the aspect ratio is 1 (l/d=1), the impact range of the shear force on the target surface is the largest.
The influence of different nozzle diameters on jet characteristics is analyzed when the terminal length in length-diameter ratio is changed and other conditions are unchanged.The diameter of the nozzle is set to be 0.8 mm, 1.0 mm and 1.2 mm, respectively.The characteristics of the jet under the flow field after changing the nozzle parameters are compared by simulation.It can be seen from the experimental results that the influence of the diameter in the nozzle length-diameter ratio is much higher than that of the nozzle end-length.It is evident that the nozzle achieves its optimal striking effect when the nozzle end length is 1mm and the nozzle diameter is 1mm.This specific combination yields the largest effective striking areas for both target surface pressure and target surface shear force.

1 a P and 2 aP
are nozzle internal and external static pressure, respectively.1 a v and 2 a v are average fluid flow rates inside and outside the nozzle.

Figure 1 .
Figure 1.Design parameters of A-type straight-cone convergent nozzle.It can be seen from Ref.[13] that the rust layer and paint layer can be easily removed when the water jet pressure reaches 180 MPa.Therefore, the water jet pressure of the high-pressure water jet for rust removal is generally 200 ~ 250 MPa.Considering real-world operational conditions and practical experimentation, the theoretical flow rate of the pump in this paper is 30 L/min and the jet pressure is set to 200 MPa.On the base of consulting relevant data, the hydraulic coefficient β is 0.658, the nozzle structure coefficient η2 is 1.1 according to the nozzle type, then the diameter of each nozzle can be determined as 0.978 mm by Eq(2).In this way, the diameter of each nozzle in the simulation is set to be 1 mm.

Figure 3 .
Figure 3. Velocity cloud map of the flow field screenshot.When the high-pressure water jet acts on the surface of the object, the surface of the object is subject to great impact force.When the shear stress reaches a critical value, the surface of the object will produce cracks.With the expansion of cracks, the target surface will produce greater stress concentration.This process will cause part of the material to be stripped from the body, and the cleaning effect can occur.Simply put, it is that the shear force strikes to produce cracks and the striking force is removed.Therefore, the impact pressure of jet and the shear force of target surface are the main parameters to evaluate the impact effect of water jet.By referring to literature[14], it can be seen that the rust layer and paint layer can be easily removed when the jet striking pressure reaches 1.7x108Pa and the target shear force reaches 6.0x105Pa.At this time, the force area on the target surface is the effective striking area of the jet.When a high-pressure water jet strikes an object's surface, the impact force comprises the total force of the jet.However, this force alone does not adequately convey the jet's destructive capability.The pressure is used to directly express the destructive ability of jet flow, namely, the force of jet flow applied to the unit area of the target surface, which is generally called the jet impact pressure.Combined with Fig.4and Fig.6, it can be seen that the impact pressure on the target surface presents a Gaussian distribution, that is, the impact pressure near the center is the largest, and then decreases as the position moves away from the center.The data on both sides presents an axisymmetrical distribution, which is caused by the energy exchange between the air and the water beam that starts from the boundary to the inside.As can be seen from Fig.5and Fig.6, the main contour feature of shear stress on the target surface is a circular ring in the central impact area, in the form of a bimodal curve.In other words, the shear force at the center of impact is 0, and then increases sharply.After reaching the maximum shear force, the value of shear force becomes smaller and smaller with the increasing distance from the impact center.

Figure 4 .
Figure 4. Target surface strike force cloud

Figure 7 .
Figure 7. 4 types of A-nozzles with different length-diameter ratios.

1stFigure 8 .
Figure 8. Flow field velocity clouds for different length-diameter ratios of nozzles.Let the above-mentioned pressure p is 200 MPa and other conditions remain unchanged, the flow field of the above types of nozzles is simulated.The velocity clouds of various nozzles' flow fields are depicted in Fig. 8.Among them, (a) is the velocity cloud diagram of A0 nozzle in the flow field, (b) is the velocity cloud diagram of A1 nozzle in the flow field, (c) is the velocity cloud diagram of A2 nozzle in the flow field, (d) is the velocity cloud diagram of A3 nozzle in the flow field.It is evident from the figure that the high-pressure water jet flow bundles in all four cases share a similar shape throughout the entire flow field, with the maximum velocity occurring at the nozzle-end.However, it is observed that the A0 nozzle, characterized by a length-diameter ratio of 0, attains a maximum velocity of 693 m/s.

Figure 9 .
Figure 9. Strike force cloud of target surface with different length-diameter ratios nozzle.

Figure 10 .
Figure 10.Shear force cloud of target surface with different length-diameter ratios nozzle.By further comparing the simulated target surface hitting pressure clouds in Fig.9, it can be seen that A0 nozzle has the largest striking force and the widest striking range.Furthermore, we observe a decrease in impact pressure on the target as the length-diameter ratio at the nozzle end increases.Referring to Fig.10, the shear force nebulae on the target surface for nozzles with varying lengthdiameter ratios reveal that the striking range of target surface shear force is maximized at a lengthdiameter ratio of 1, indicating the largest effective striking area.For length-diameter ratios of 2 and 3, the maximum shear force on the target surface isn't uniformly distributed on the ring, except at the center.However, the entire ring area remains within the effective striking range.

Figure 12 .Figure 13 .
Figure 12.Scatter diagram of the impact force on the target surface with nozzle parameter L1=1 and d2=2

Table 2 .
below.Table of experimental results