The analysis of the impact of nozzle structure on the jet
To investigate the influence of different nozzle structures on jet states, nozzles with four different structures were designed for comparative analysis: a straight nozzle, a conical contraction nozzle, an arc contraction nozzle, and an expansion nozzle. Figure 1 shows schematic diagrams of the four nozzle structures, while Fig. 2 presents the simulation analysis results of jet states for different nozzle structures.
Schematic of nozzle structure. (a) Straight nozzle. (b) Conical contraction nozzle. (c) Arc contraction nozzle. (d) Expansion nozzle.
Based on the theory of continuum mechanics, and according to the laws of conservation of mass, momentum, and energy, the control equations for the motion of a continuous fluid are obtained.
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(1)
Continuity equation:
$$frac{{partial rho }}{{partial t}} + nabla cdot (rho u) = 0$$
(1)
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(2)
Momentum conservation equation:
$$rho left( {frac{{partial u}}{{partial t}} + u cdot nabla u} right) = – nabla p + nabla cdot left[ {mu (nabla u + nabla u^{T} )} right] + f_{b} + f_{{st}}$$
(2)
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(3)
Energy conservation equation:
$$rho C_{p} frac{{partial T}}{{partial t}} + nabla p + rho C_{p} u cdot nabla T = nabla cdot (knabla T)$$
(3)
where, u is the fluid velocity, ρ is the fluid density, p is pressure, µ is the fluid’s viscosity coefficient, fb is volumetric force, fst is the surface tension at the multiphase flow interface, Cp is the specific heat capacity at constant pressure, T is the fluid temperature, k is the thermal conductivity coefficient.
In this study, the liquid nitrogen jet is a non-submerged free jet, ejected from the nozzle into the air, involving gas-liquid two-phase flow. The VOF (Volume of Fluid) model is selected for interface tracking, and the VOF equation can be expressed as:
$$frac{1}{{rho _{k} }}left[ {frac{partial }{{partial t}}(alpha _{k} rho _{k} ) + nabla cdot (alpha _{n} rho _{k} nu _{k} ) = S_{{alpha _{k} }} + sumnolimits_{{p = 1}}^{n} {(m_{{pk}} – m_{{kp}} )} } right]$$
(4)
where, ρk is the density of phase k, vk is the velocity of phase k, Sαk is the source term, mpk is the mass transfer from phase p to k, mkp is the mass transfer from phase k to p, and αk is the volume fraction of phase k.
The jet is in a high turbulence state, and the standard k–ε turbulence model is selected to close the control equations. In the standard k–ε turbulence model, the transport equations for the turbulent kinetic energy k and the specific dissipation rate ε are expressed as follows:
$$left{ begin{gathered} frac{partial }{{partial t}}(rho k) + frac{partial }{{partial x_{i} }}(rho ku_{i} ) = frac{partial }{{partial x_{j} }}left[ {left( {mu + frac{{mu _{t} }}{{sigma _{k} }}} right)frac{{partial k}}{{partial x_{j} }}} right] hfill \ quad + G_{k} + G_{b} – rho varepsilon – Y_{M} + S_{k} hfill \ frac{partial }{{partial t}}(rho varepsilon ) + frac{partial }{{partial x_{i} }}(rho varepsilon u_{i} ) = frac{partial }{{partial x_{j} }}left[ {left( {mu + frac{{mu _{t} }}{{sigma _{varepsilon } }}} right)frac{{partial varepsilon }}{{partial x_{j} }}} right] hfill \ quad + C_{{1varepsilon }} frac{varepsilon }{k}left( {G_{k} + C_{{3varepsilon }} G_{b} } right) – C_{{2varepsilon }} rho frac{{varepsilon ^{2} }}{k} + S_{varepsilon } hfill \ end{gathered} right.$$
(5)
where, Gk is the turbulent kinetic energy generated by the gradient of mean velocity, Gb is the turbulent kinetic energy generated by buoyancy, YM is the dissipation occurring in turbulence, C1ε, C2ε, C3ε are constants, σk and σε are the turbulent Prandtl numbers for the k-equation and ε-equation, respectively, Sk and Sε are source terms, the turbulent viscosity coefficient (mu _{t} = sigma C_{mu } frac{{k^{2} }}{varepsilon }).
The jet simulation diagram of different structure nozzles. (a) Straight nozzle. (b) Conical contraction nozzle. (c) Arc contraction nozzle. (d) Expansion nozzle.
Computational fluid dynamics software ANSYS FLUENT was used for the simulation calculation of liquid nitrogen jet. The computational domain was defined by the combination of a liquid nitrogen nozzle and its downstream jet region. Owing to the axisymmetric nature of the round jet, a two-dimensional modeling approach was adopted by extracting a representative cross-section along the jet axis. The domain was treated as an axisymmetric system and constructed accordingly using a rotationally symmetric plane. To ensure adequate spatial development of the jet flow, a large flow domain measuring 80 mm in length and 18 mm in width was employed. The upper boundary and both lateral sides of the jet region were defined as adiabatic, no-slip wall boundaries to represent thermal and mechanical insulation. Owing to the regular geometry of the entire computational region, a structured mesh was utilized for spatial discretization. Mesh refinement was applied around the nozzle exit and the central axis to capture detailed flow features and enhance numerical accuracy. The geometric model consists of the liquid nitrogen nozzle and the liquid nitrogen jet domain, forming the computational domain. Since the geometric model is a circular jet, any cross-section passing through the axis is taken for two-dimensional modeling. Assuming the jet domain is axisymmetric, modeling can be done using an axisymmetric plane. The jet domain is set as a large spatial domain of 80 mm×18 mm to ensure the full development of the jet flow field, with the top surface and the left and right sides of the jet domain as adiabatic no-slip wall boundaries. All computational domain spaces are of regular shape, allowing for spatial discretization using structured grids, and the nozzle and the area around the axis of the jet domain are refined.
In the simulation analysis model, the total length of the nozzle is 10 mm, and the length of the variable diameter part is 2 mm.The inlet pressure is 0.4 MPa. To ensure the consistency of the outlet size, the diameter of the nozzle outlet is set to 2 mm. The inlet diameter is determined by back-calculating from the outlet diameter and the nozzle length. As can be seen from the velocity contour map in Fig. 2, the expansion nozzle has a significant effect on the length of the zone of flow establishment, reducing the length of the zone of flow establishment. This type of nozzle structure is not suitable for use, as the other three structures do not significantly affect the jet state. To further analyze the impact of different nozzle structures on the jet state, it is necessary to analyze the distribution of velocity and temperature along the jet path.
Figure 3 is a comparative diagram of the jet velocity and temperature distribution for nozzles with different structures. Figure 3a shows the velocity distribution of jets from nozzles with different structures, where the expansion nozzle exhibits a rapid decay in velocity with a shorter length of the zone of flow establishment, and there is no significant difference in the zone of flow establishment among the straight nozzle, the conical contraction nozzle, and the arc contraction nozzle. From region I in the diagram, it can be seen that the conical contraction nozzle and the arc contraction nozzle have a rising phase at the beginning of the jet, which is absent in both the straight and expansion nozzles, suggesting that this is caused by the contraction structure, but the maintenance length is not significantly different from that of the straight nozzle. Figure 3b shows the temperature distribution of jets from nozzles with different structures, which follows the same pattern as the velocity distribution, with the expansion nozzle having the fastest temperature decay rate among the four types of nozzles. Although the conical contraction nozzle and the arc contraction nozzle have a rising phase in their velocity distribution, it does not affect the temperature distribution of the jet. The straight nozzle maintains stable velocity and temperature in the zone of flow establishment, without any significant abrupt changes.
Comparison of jet velocity and temperature of different structure nozzles. (a) Velocity distribution diagram. (b) Temperature distribution diagram.
Comprehensive consideration of the velocity and temperature distribution patterns of the four nozzles, as well as the feasibility of the actual manufacturing process, compared to the other three types of nozzles, the straight nozzle has obvious advantages, so the nozzle of the liquid nitrogen internal spray turning tool can adopt the straight nozzle structure.
Optimization of the outlet elbow angle
In the manufacturing process of cutting tools, to ensure the accuracy of the spray position, a bend pipe structure will inevitably appear at the outlet of the transmission channel. Due to the high pressure of liquid nitrogen in the internal channel of the tool, there is a significant pressure change when it is ejected compared to the atmospheric pressure at the outlet. The pressure difference between the inside and outside of the spray outlet leads to the cavitation effect at the outlet elbow, affecting the jet speed and the volume fraction of liquid nitrogen, as shown in Fig. 4.
Schematic of outlet elbow cavitation.
The effects of cavitation caused by different bend angles at the outlet on the speed, volume fraction, and temperature of liquid nitrogen were analyzed using a simulation approach. Due to the occurrence of secondary flow phenomena at the outlet bend, the RNG k-ε turbulence model is more accurate in simulating transient flows and curved streamlines compared to the standard k-ε model21. Therefore, the RNG k-ε turbulence model was employed for simulation analysis. The expression for the RNG k-ε turbulence model is as follows:
$$left{ begin{gathered} frac{partial }{{partial t}}(rho k) + frac{partial }{{partial x_{i} }}(rho ku_{i} ) = frac{partial }{{partial x_{j} }}left( {alpha _{k} mu _{{eff}} frac{{partial k}}{{partial x_{j} }}} right) hfill \ quad + G_{k} + G_{b} – rho varepsilon = Y_{M} + S_{k} hfill \ frac{partial }{{partial t}}(rho varepsilon ) + frac{partial }{{partial x_{i} }}(rho varepsilon u_{i} ) = frac{partial }{{partial x_{j} }}left( {alpha _{varepsilon } mu _{{eff}} frac{{partial varepsilon }}{{partial x_{j} }}} right) + C_{{1varepsilon }} frac{varepsilon }{k}(G_{k} + C_{{3varepsilon }} G_{b} ) hfill \ quad – C_{{2varepsilon }} rho frac{{varepsilon ^{2} }}{k} + S_{varepsilon } – frac{{C_{mu } rho eta ^{3} (1 – frac{eta }{{eta _{0} }})}}{{1 + beta eta ^{3} }}frac{{varepsilon ^{2} }}{k} hfill \ end{gathered} right.$$
(6)
where, Gk is the turbulence kinetic energy produced by the mean velocity gradient, Gb is the turbulence kinetic energy generated by buoyancy, YM is the dissipation produced in turbulence, Sk and Sε are source terms, αk and αε are the turbulent Prandtl numbers for the k and ε equations respectively, C1ε, C2ε, C3ε, Cµ, β, η0 are constants, η is the dimensionless strain rate, µt is the turbulent viscosity.
The Zwart–Gerber–Belamri22 model is selected as the cavitation model, and its expression is as follows:
$$left{ begin{gathered} R_{e} = F_{{vap}} frac{{3alpha _{{nuc}} (1 – alpha _{v} )rho _{v} }}{{r_{b} }}sqrt {frac{2}{3}frac{{P_{{sat}} – P_{0} }}{{rho _{l} }}} quad P_{0} < P_{{sat}} hfill \ R_{c} = F_{{cond}} frac{{3alpha _{v} rho _{v} }}{{r_{b} }}sqrt {frac{2}{3}frac{{P_{0} – P_{{sat}} }}{{rho _{l} }}} quad P_{0} < P_{{sat}} hfill \ end{gathered} right.$$
(7)
where, Re is the evaporation term, Fvap is the evaporation coefficient, Rc is the condensation term, Fcond is the condensation coefficient, αv is the gas phase volume fraction, αnuc is the nucleation site volume fraction, ρv and ρl is the gas and liquid phase densities respectively, rb is the bubble diameter, P0 is the actual pressure, Psat is the saturation pressure.
To analyze the impact of changes in the bend angle on parameters such as the velocity, volume fraction, and temperature of liquid nitrogen, a comparative analysis was conducted with the angle range set from 30° to 160°, and the step interval set at 10°. The pipe diameter was set to 2 mm, with a single segment length of 10 mm, using structured mesh. The inlet and outlet of the computational domain were set as constant pressure boundary conditions, with the inlet absolute pressure set at 0.4 MPa, and the inlet temperature undercooling issue considered, setting the undercooling degree at 10 K, with an inlet temperature of 80 K. The outlet pressure of the liquid nitrogen was set to one standard atmosphere, with a temperature of 300 K. The walls of the bend were set as adiabatic surfaces. The meshing method employed was MultiZone, with a grid size of 200 μm. Figure 5 shows the trend of changes in the velocity, volume fraction, and temperature of liquid nitrogen under different bend angles.
Variation of liquid nitrogen velocity, volume fraction and temperature for different angles.
Analysis of the velocity, liquid nitrogen volume fraction, and temperature from different bend angles shows that the bend angle has almost no effect on the fluid temperature. Due to the presence of liquid nitrogen, the temperature from the inlet fluid side to the outlet air side is maintained at 77 K. As the bend angle increases, the velocity and volume fraction of liquid nitrogen at the outlet exhibit a trend that is nearly linear, with the bend angle having the most significant impact on the volume fraction of liquid nitrogen, the maximum difference exceeding 20%. This indicates that increasing the bend angle can effectively reduce the effects of secondary flow and cavitation on liquid nitrogen. Based on the analysis, to maintain a high initial velocity and volume fraction of liquid nitrogen during injection, the outlet of liquid nitrogen injection should use as obtuse an angle as possible, to minimize the impact on velocity and liquid nitrogen volume fraction.
Study on the thermal insulation method of the main transmission channel
The process of transferring liquid nitrogen from the supply device to the tool necessitates the use of vacuum tubes to prevent vaporization. Vacuum tubes are thicker than ordinary tubes and possess a greater resistance to bending. To avoid interference with machine tool components during tool use, or even the breaking of adapters, the liquid nitrogen inlet is positioned at the tail of the tool holder. However, this increases the length of the liquid nitrogen’s transmission path, exacerbating the heat exchange between the nitrogen and the inner walls during its transmission within the tool body. At this point, the liquid nitrogen enters a uniformly heated channel. If the external heat input through the walls is substantial enough, saturated boiling may occur, leading to a continuous increase in dryness, and the flow pattern of the liquid may transition from bubbly flow to annular flow, to mist flow, and finally to a unidirectional gas state23,24. Disregarding the heat transfer along the length of the pipe, this process can be viewed as one-dimensional steady-state heat transfer through the cylindrical wall, with the heat Q transferred to the liquid nitrogen through the wall from the outside as:
$$Q = frac{{T_{0} – T_{{ln 2}} }}{{frac{1}{{2pi r_{1} lalpha _{1} }} + frac{1}{{2pi llambda }}ln frac{{r_{2} }}{{r_{1} }} + frac{1}{{2pi r_{2} lalpha _{2} }}}}$$
(8)
where, r1 is the inner diameter of the liquid nitrogen transmission channel in the tool handle, r2 is the equivalent outer diameter of the tool handle, l is the length of the tool handle, λ is the thermal conductivity of the tool material, α1 is the combined natural convection-radiation heat transfer coefficient of the air outside the tool wall; α2 is the forced convection heat transfer coefficient of the two-phase flow of liquid nitrogen inside the pipeline, T0 is the ambient temperature, and Tln2 is the temperature of the liquid nitrogen.
Therefore, the only method to reduce the external heat input is to increase the equivalent thermal resistance of the cylinder wall. Installing insulation material within the cylinder wall is an effective way to increase its equivalent thermal resistance. After the addition of insulation material, the heat transfer expression is as shown in Eq. (9).
$$Q = frac{{T_{0} – T_{{ln 2}} }}{{frac{1}{{2pi r_{1} lalpha _{1} }} + frac{1}{{2pi llambda }}ln frac{{r_{3} }}{{r_{1} }} + frac{1}{{2pi llambda }}ln frac{{r_{2} }}{{r_{3} }} + frac{1}{{2pi r_{2} lalpha _{2} }}}}$$
(9)
where, r3 is the inner diameter of the insulation sleeve, and λ1 is the thermal conductivity coefficient of the insulating material.
As shown by Eq. (9), the methods to increase the thermal resistance of insulation materials include selecting insulation materials with lower thermal conductivity, increasing the thickness of the insulation material walls, and decreasing the heat transfer area. Once the insulation material is selected, the thermal conductivity becomes a fixed value, and the insulation performance can only be improved by increasing the thickness of the insulation material walls and reducing the heat transfer area. Figure 6 is a simulation analysis result graph for increasing the thickness of the insulation material cylinder wall after fixing the thermal conductivity of the insulation material, where the thermal conductivity of the insulation material is set to 0.24 W/(m·℃), selecting different insulation material wall thicknesses for insulation performance comparison.
Diagram of insulation channel temperature distribution. (a) Wall thickness 1 mm. (b) Wall thickness 1.5 mm. (c) Wall thickness 2 mm. (d) Wall thickness 2.5 mm.
From the simulation results in Fig. 6, it can be observed that as the thickness of the insulation material wall increases from 1 mm to 2.5 mm, with each increment of 0.5 mm, the external wall temperatures of the tool body are − 127.04℃, − 104.30℃, − 85.89℃, and − 70.59℃, respectively. The temperature of the tool body’s external wall does not increase linearly with the increase in wall thickness; instead, the decrement in temperature difference is observed to trend downwards, with a diminished increase in insulation performance, indicating that the improvement in insulation performance diminishes with thicker walls. Therefore, simply increasing the thickness of the wall has limited benefits for enhancing insulation performance. Moreover, due to the dimensional constraints of the tool channels, excessive thickness in the insulation material cylinder wall would reduce the space for liquid nitrogen flow, making it impractical to only increase the thickness of the insulation sleeve. Hence, the method of reducing the heat transfer area is adopted to improve the insulation capability of the insulation sleeve. By grooving the surface of the insulation sleeve to form a “dumbbell” structure, creating cavities between the inner surface of the tool body and the outer surface of the insulation sleeve, the method of heat transfer is altered, and the efficiency of heat conduction is reduced. Figure 7a presents the improved insulation material structure diagram, and Fig. 7b shows the heat transfer simulation result. In the simulation, the thermal conductivity of the insulation material is set to 0.24 W/(m·℃), with the medium inside the cavity being air, having a thermal conductivity of 0.0267 W/(m·℃). With wall thicknesses of 2 mm at both ends and 1 mm in the middle, the external wall temperature of the tool body is only − 27.30 ℃, which, compared to using a 2 mm thick insulation sleeve without cavities (with an external wall temperature of − 85.89 ℃), shows a significant improvement in insulation capability. Therefore, the insulation sleeve for the main liquid nitrogen transport channel of the tool will adopt a “dumbbell” type structure.
Diagram of improved thermal insulation channel structure and temperature distribution. (a) Thermal insulation structure. (b) Heat transfer simulation results.
Simulation analysis of the thermal insulation performance of a liquid nitrogen internal spray cooling turning tool
To investigate the impact of insulation material within the liquid nitrogen transmission channel on the temperature field of the tool, a simulation is employed to compare the temperature distribution patterns with and without insulation. Given that the temperature at various points of the tool only changes with spatial position and not with time when liquid nitrogen is in a stable spraying state, steady-state heat transfer analysis is utilized for computation. In view of the complex geometry of the cutting tool model, a hybrid meshing strategy combining structured and unstructured grids was adopted, with an element size of 1 mm. The inner wall of the liquid nitrogen transmission channel within the tool is set as a cold source, considering heat transfer between the liquid nitrogen and the tool body, as well as between the tool body and air, with the cold source temperature at − 196 °C and air temperature at 20 °C. The simulation results are presented in Fig. 8a.
The ridge area on the upper surface of the tool holder is selected, and the temperature distribution data from the head to the end of the tool are extracted to draw a temperature distribution curve, as shown in Fig. 8b. When no insulation structure is set on the tool handle, the temperature of the outer surface of the tool is low and distributed evenly, close to the temperature of liquid nitrogen. This indicates that when liquid nitrogen passes through the tool in the transmission channel, it directly exchanges heat with the outer wall, and the temperature starts to lose from the tool holder. At this time, the tool’s outer surface is prone to frost, and due to the heat loss in the channel, when liquid nitrogen is sprayed near the tool tip, the liquid volume fraction inevitably decreases, affecting the cooling efficiency of liquid nitrogen. Conversely, when an insulation structure is set up, the thermal exchange of liquid nitrogen with the environment is significantly suppressed at tool holder, resulting in a gradient temperature distribution in the tool holder with the insulation structure. This characteristic is due to the thermal conduction from the head of the tool holder, indicating that liquid nitrogen does not directly exchange heat with the external environment in the tool holder, thereby suppressing the temperature loss of liquid nitrogen in the transmission channel.
Temperature distribution with and without insulation. (a) Simulation results. (b) Temperature distribution in the extracted area.