Structure Design and Dynamic Performance Analysis of Aerostatic Bearing with Variable Height Restrictor
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摘要: 为了使气浮支承的承载力动态可调,设计了一种可变节流高度气浮支承. 通过建立气浮支承计算流体动力学(Computational Fluid Dynamics,CFD)模型,利用CFD动网格技术来模拟小孔节流器的运动,研究小孔节流器的结构参数、运动参数及气浮支承的工作参数对可变节流高度气浮支承动态性能的影响. 结果表明:通过调节小孔节流器的节流高度可以明显改变气浮支承的承载力;在只考虑单一变量的前提下,气浮支承承载力的波动量随着小孔节流器的运动幅值、运动频率、节流高度、直径和气浮支承供气压强的增加而增加,但随着气膜厚度的增大而减小;当小孔节流器直径较小时,随着小孔节流器运动频率的增加,气浮支承动刚度的增幅很小,但当小孔节流器直径增大时,随着小孔节流器运动频率的增加,气浮支承动刚度的增幅会明显变大.Abstract: Aerostatic bearing widely used in precision manufacturing equipment because of the characteristics of no friction, no wear, no pollution and high stability. But with the ultra-precision machinery to the requirement of increasing the machining precision, loading capacity and stability of the aerostatic bearing greatly affect the machining precision of ultra-precision machine tools. The traditional surface throttling method is mainly to improve the static performance of the aerostatic bearing. Therefore, a piezoelectric active aerostatic bearing with variable height restrictor was designed to improve the loading capacity and reduce the micro vibration of the aerostatic bearing. The aerostatic bearing worked by controlling the output force of piezoelectric actuator to change throttling height of the orifice restrictor. The quarter gas domain model of the aerostatic bearing was established. The quarter gas domain model was meshed and boundary conditions were set. In the CFD (Computational Fluid Dynamics) software, the dynamic grid technology was used to simulate the movement of the orifice restrictor. The realizable k-ε turbulence model was used to simulate the dynamic characteristics of the aerostatic bearing with variable height restrictor. The influences of the structure parameters, movement parameters of the orifice restrictor and the working parameters of the aerostatic bearing on the dynamic performance of aerostatic bearing with variable height restrictor were analyzed by orthogonal analysis method. The adaptability of the computational model and the accuracy of the dynamic grid technology were verified by comparing with the simulation data of references. The results of transient simulation showed that the loading capacity of the aerostatic bearing can be obviously changed by adjusting the throttling height of the orifice restrictor by controlling the expansion of the center part of the flexure hinge. The motion state of the orifice restrictor had great influence on the dynamic characteristics of the aerostatic bearing with variable height restrictor. Under the premise of considering only a single variable, the fluctuation of the loading capacity of the aerostatic bearing with the orifice restrictor increased with the increase of the moving amplitude, moving frequency, throttling height, diameter and the supply pressure of the aerostatic bearing, but decreased with the increase of the gas film clearance. When the diameter of the orifice restrictor was small, the increase of the dynamic stiffness of aerostatic bearing with variable height restrictor was small with the increase of the moving frequency of the orifice restrictor. When the diameter of orifice restrictor was large, the increase of the dynamic stiffness of the aerostatic bearing with variable height restrictor was obviously large with the increase of the moving frequency of the orifice restrictor. The dynamic characteristics of the aerostatic bearing with variable height restrictor can be improved effectively by adopting appropriate control strategy.
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Keywords:
- aerostatic bearing /
- orifice restrictor /
- throttling height /
- loading capacity /
- dynamic stiffness
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由于气浮支承接近于零摩擦和发热低等特点而广泛应用于超精密机械中[1],而气浮支承有限的刚度、承载能力及动态性能限制了其发展[2-3]. 随着机电一体化技术的快速进步,压电主动控制技术已经成为提高气浮支承动态特性的有效方法[4-6].
压电主动控制技术根据气浮支承所调控的参数不同可以分为两个方面,一方面是气膜形状控制:Colombo等[7]设计了一种主动控制气膜厚度的气浮支承,试验验证了该模型有良好的抗扰动能力;Aguirre等[8-9]设计了一种主动控制气膜锥度的气浮支承,建立了多物理场有限元模型,试验证明了模型对主动控制气膜锥度气浮支承优化设计的必要性;朱定玉[10]设计了一种压电主动控制气膜锥度气浮支承,并对其动刚度增强机制进行理论和试验研究,试验表明主动控制气膜锥度气浮支承有良好的动刚度性能. 另一方面是流量控制:Park等[11-12]设计了一种主动控制节流器,通过改变小孔节流器的节流面积来提高气浮支承的动刚度;Mizμmoto等[13]提出了一种压电主动固有小孔节流器,试验表明主动固有小孔节流器能增强气浮支承的静刚度,但对气浮支承动态性能的提升程度有限. 在现有的研究中,对于压电主动控制流量方面做的研究较少.
本文中在传统小孔节流气浮支承的基础上增加压电陶瓷促动器和柔性铰链,设计了一种基于流量控制的可变节流高度气浮支承,并通过Fluent软件仿真计算得出小孔节流器的结构参数、运动参数和气浮支承工作参数对可变节流高度气浮支承动态性能的影响规律,为主动控制气浮支承的设计提供参考.
1. 可变节流高度气浮支承结构设计及理论模型
1.1 可变节流高度气浮支承结构设计
可变节流高度气浮支承主要由上端盖、压电陶瓷促动器、促动器接头、柔性铰链、小孔节流器和止推板组成. 可变节流高度气浮支承结构示意图如图1所示,结构参数列于表1中. 柔性铰链工作时处于振动状态,因此需要具有良好的韧性和弹性,其材料选用弹簧钢(65Mn)[14],其他零件主要承受载荷,其材料选用铝合金[15]. 小孔节流器镶嵌在柔性铰链中心,柔性铰链、压电陶瓷促动器和促动器接头之间用螺栓连接,上端盖、柔性铰链和止推板之间用螺栓连接. 可变节流高度气浮支承的工作原理是高压气体由上端盖进气孔流入,经促动器接头流入小孔节流器,通过小孔节流器的节流作用,止推板和基座之间会产生1层气膜,从而产生向上的承载力. 压电陶瓷促动器在受到正弦激励电压后推动柔性铰链和小孔节流器一起做如图2所示的幅值为A、频率为f的正弦运动. 小孔节流器的运动会引起气浮支承质量流量变化,从而影响均压腔和气膜区域的压强分布,改变气浮支承的承载力.
表 1 气浮支承各项参数Table 1. Parameters of the aerostatic bearingParameters Specifications The outside diameter of flexure hinge, D1/mm 65 The diameter of the aerostatic bearing, D2/mm 100 The diameter of the orifice restrictor (the diameter of equalizing cavity), D3/mm 2/3/4/5 The diameter of orifice, D4/mm 0.2 The outer diameter of the annular slit, D5/mm 5.2 The thickness of flexure hinge, H1/ mm 2 The height of the orifice, H2/mm 0.5 The height of the orifice restrictor, H3/mm 6 The height of the circular slit, H4/mm 3 The throttling height (the height of equalizing cavity), ha/μm 10/15/20/25/30 The thickness of gas film, hf/μm 10/15/20/25/30 Supply pressure, Ps/MPa 0.2/0.3/0.4/0.5/0.6 Atmospheric pressure, Pa/MPa 0.1 1.2 可变节流高度气浮支承理论方程
可变节流高度气浮支承工作时的气体实际流动情况是十分复杂的,因此根据经典润滑理论[16]做出了如下假设:
(1) 润滑气体为牛顿流体;
(2) 润滑气体是单相、连续介质;
(3) 气浮支承内壁无热交换;
(4) 忽略惯性力的作用;
(5) 忽略阻尼腔质量流量的变化对气浮支承承载力的影响.
基于以上假设,可以将复杂的N-S方程运用一维分析方法来表示,对柱坐标系下的气体运动方程进行简化得式(1~3).
$$ \frac{{\partial p}}{{\partial r}} = \eta \frac{{{\partial ^2}{v_{\text{r}}}}}{{\partial {z^2}}} $$ (1) $$ \frac{{\partial p}}{{\partial \theta }} = 0 $$ (2) $$ \frac{{\partial p}}{{\partial z}} = 0 $$ (3) 式中:η为空气动力黏度,vr为气体在半径方向上的速度.
由式(1)、(2)和(3)可知压力p与
$ \theta $ 和z没有关系,因此式(1)可以写为式(4).$$ \frac{{{\text{d}}p}}{{{\text{d}}r}} = \eta \frac{{{\partial ^2}{v_{\text{r}}}}}{{\partial {z^2}}} $$ (4) 根据质量守恒原理:
$$ {q_{{{\text{m}}_1}}} = {q_{{{\text{m}}_2}}} $$ (5) 式中:
$ {q_{{{\text{m}}_1}}} $ 为单位时间流入均压腔的质量流量,$ {q_{{{\text{m}}_2}}} $ 为单位时间流入气膜的质量流量.$$ {q_{{{\text{m}}_{\text{1}}}}} = 2{\text{π}}r\int_0^{{h_{\text{f}}} + {h_{\text{a}}} + \Delta {h_{\text{a}}}} {\rho {v_{\text{r}}}} {\text{d}}z $$ (6) $$ {q_{{{\text{m}}_{\text{2}}}}} = 2{\text π} r\int_0^{{h_{\text{f}}}} {\rho {v_{\text{r}}}} {\text{d}}z $$ (7) r方向速度表达式[17]如式(8)所示:
$$ {v_{\text{r}}} = - \frac{1}{{2\eta }}\frac{{{\text{d}}p}}{{{\text{d}}r}}\left( {h - z} \right)z $$ (8) 气体的状态方程:
$$ p = \rho {\rm R}T $$ (9) 式中:T为绝对温度,R为气体常数,ρ为空气密度.
将式(8)带入到式(6)和(7)中,整理可得质量流量:
$$ {q_{{{\text{m}}_{\text{1}}}}} = \frac{{{\text{π}}r\rho {{\left( {{h_{\text{f}}} + {h_{\text{a}}} + {{\Delta }}{h_{\text{a}}}} \right)}^3}}}{{6\eta }}\frac{{{\text{d}}p}}{{{\text{d}}r}} $$ (10) $$ {q_{{{\text{m}}_{\text{2}}}}} = \frac{{{\text{π}}r\rho {h_{\text{f}}}^3}}{{6\eta }}\frac{{{\text{d}}p}}{{{\text{d}}r}} $$ (11) 把式(9)带入到式(10)和(11)中,整理可得:
$$ p{\text{d}}p = \frac{{{\text{6}}\eta {\rm R}T{q_{{{\text{m}}_1}}}}}{{{\text{π}}{{\left( {{h_{\text{f}}} + {h_{\text{a}}} + \Delta {h_{\text{a}}}} \right)}^3}}}\frac{{{\text{d}}r}}{r} \quad ({R_1} \leqslant {r_1} \leqslant {R_2}) $$ (12) $$ p{\text{d}}p = \frac{{6\eta {\rm R}T{q_{{{\text{m}}_2}}}}}{{{\text{π}}{h_{\text{f}}}^3}}\frac{{{\text{d}}r}}{r} \quad ({R_2} \leqslant {r_2} \leqslant {R_3}) $$ (13) 式(12)和(13)分别在对应区段上积分得:
$$ {P_1}^2 - {P_{\text{b}}}^2 = \frac{{12\eta {\rm R}T{q_{{{\text{m}}_{\text{1}}}}}}}{{{\text{π}}{{\left( {{h_{\text{f}}} + {h_{\text{a}}} + \Delta {h_{\text{a}}}} \right)}^3}}}\ln \frac{{{r_1}}}{{{R_2}}} \quad ({R_1} \leqslant {r_1} \leqslant {R_2}) $$ (14) $$ {P_2}^2 - {P_{\text{a}}}^2 = \frac{{12\eta {\rm R}T{q_{{{\text{m}}_2}}}}}{{{\text{π}}{h_{\text{f}}}^3}}\ln \frac{{{r_2}}}{{{R_3}}} \quad ({R_2} \leqslant {r_2} \leqslant {R_3}) $$ (15) 式中:
$ {P_{\text{d}}} $ 为进气小孔出口处压强,$ {P_{\text{b}}} $ 为均压腔和气膜交界处压强,$ {R_1} $ 为小孔半径,$ {R_2} $ 为均压腔半径,$ {R_3} $ 为气浮支承半径.利用边界条件,当
$ {r_1}{\text{=}}{R_1} $ 时,$ {P_1}{\text{=}}{P_{\text{d}}} $ ,当$ {r_2}{\text{=}}{R_2} $ 时,$ {P_2}{\text{=}}{P_{\text{b}}} $ ,可以求得均压腔区域的压力:$$ \begin{aligned}{P_1} = &\sqrt {\frac{{12\eta {\rm R}T{q_{{{\text{m}}_{\text{1}}}}}}}{{{\text{π}}{{\left( {{h_{\text{f}}} + {h_{\text{a}}} + \Delta {h_{\text{a}}}} \right)}^3}}}\ln \frac{{{r_1}}}{{{R_2}}} + \frac{{12\eta {\rm R}T{q_{{{\text{m}}_{\text{2}}}}}}}{{{\text{π}}{h_{\text{f}}}^3}}\ln \frac{{{R_2}}}{{{R_3}}} + {P_{\text{a}}}^2} \\ &({R_1} \leqslant {r_1} \leqslant {R_2})\end{aligned} $$ (16) 气膜区域压力:
$$ {P_2} = \sqrt {\frac{{12\eta {\rm R}T{q_{{{\text{m}}_2}}}}}{{{\text{π}}{h_{\text{f}}}^3}}\ln \frac{{{r_2}}}{{{R_3}}} + {P_{\text{a}}}^2} \quad ({R_2} \leqslant {r_2} \leqslant {R_3}) $$ (17) 小孔节流器的运动方程:
$$ \Delta {h_{\text{a}}} = A\sin 2{\text{π}}ft $$ (18) 气浮支承的承载力等于均压腔的承载力和气膜的承载力之和减去环境压力,因此气浮支承的承载力方程如下所示:
$$ {W_1} = 2{\text{π}}\int_{{R_1}}^{{R_3}} {Pr{\text{d}}r} - {\text{π}}{R_3}^2{P_{\text{a}}} $$ (19) $$ {W_2} = \int_{{R_2}}^{{R_1}} {{P_1}(A,f,r,t){\text{d}}r} + \int_{{R_3}}^{{R_2}} {{P_2}\left( {A,f,r,t} \right){\text{d}}r} - {\text π} {R_3}^2{P_{\text{a}}} $$ (20) $$ \Delta W = {W_2} - {W_1} $$ (21) $$ k = \frac{{\Delta W}}{{\Delta {h_{\text{a}}}}} $$ (22) 式中:
$ {W_1} $ 为气浮支承稳态承载力,$ {W_2} $ 为气浮支承瞬态承载力,k为气浮支承动刚度.由式(20)和式(22)可见,气浮支承的承载力与小孔节流器运动幅值、运动频率和小孔节流器直径有相关性,气浮支承的动刚度与小孔节流器运动频率有相关性.
2. 可变节流高度气浮支承计算流体动力学(CFD)模型
2.1 可变节流高度气浮支承网格划分
由于圆形气浮支承具有对称性,为了提升计算速度仅计算1/4气域模型. 为了保证仿真计算精度,对气浮支承气域模型进行六面体网格划分,且网格总数不少于200万. 由于节流口和压力出口部分压力梯度很大,因此对节流小孔和均压腔流域进行了网格加密处理,气膜流域压力较小则网格划分相对稀疏. 可变节流高度气浮支承CFD模型如图3所示.
2.2 可变节流高度气浮支承边界条件设置
可变节流高度气浮支承气域模型可分为4个部分:小孔流域(fluid 1)、均压腔流域(fluid 2)、阻尼室(fluid 3)和气膜流域(fluid 4). 气域部分边界条件设置:小孔上表面设为压力入口(pressure-inlet),气膜外边界设为压力出口(pressure-outlet),左右侧为对称面(symmetry),均压腔上表面为动网格边界(move)和耦合面(interface 1、interface 2),其他边界均为壁面(wall). 假设气浮支承内部气体为理想气体,瞬态仿真计算运用realizable k-ε湍流模型,动网格边界运动形式采用用户自定义程序(User Defined Functions,UDF)进行定义,动网格及时更新方法采用动态铺层法,气浮支承边界条件示意图如图4所示.
3. 仿真适应性分析
为了验证本文中基于realizable k-ε模型的数值仿真方法对可变节流高度气浮支承动态性能分析的适用性,运用相同气浮支承模型,将Ishibashi等[18]利用层流模型瞬态仿真计算结果和在realizable k-ε模型下瞬态仿真计算结果进行对比. 参考文献[18]中的气浮轴承结构为小孔节流无腔圆形气浮支承,其结构示意图如图5(a)所示. 主要参数包括:小孔半径r1=0.25 mm,气浮支承半径r2=5 mm,小孔高度h1=1 mm,气膜厚度h2=6 μm,供气压强Ps=0.49 MPa,出口压强Pa=0. 气膜下表面为动网格边界,动网格运动边界的运动幅值A=0.05 μm,运动频率f=125 Hz. realizable k-ε模型和层流仿真结果如图5(b)所示.
通过对比分析可知,层流模型和realizable k-ε模型的承载力仿真结果均呈正弦规律变化,且两者承载力波动量基本吻合,仅在静态承载力上存在约0.04 N的误差,证明了本文中所采用的计算模型和CFD动网格数值仿真方法在可变节流高度气浮支承动态性能分析上的可行性和有效性.
4. 仿真分析
利用Fluent软件通过上述气浮支承CFD模型和边界条件设置对可变节流高度气浮支承动态特性进行瞬态仿真,可以得到小孔节流器的结构参数、运动参数和气浮支承工作参数对可变节流高度气浮支承动态性能的影响规律.
4.1 小孔节流器运动幅值和供气压强对气浮支承承载力的影响
为了分析小孔节流器运动幅值和供气压强对气浮支承承载力的影响,取初始条件:气膜厚度hf=10 μm,运动频率f=1000 Hz,节流高度ha=10 μm,小孔节流器直径D3=2 mm,供气压强Ps=0.2/0.3/0.4/0.5/0.6 MPa,运动幅值A=5/7.5/10/12.5/15 μm,其他结构参数保持不变. 小孔节流器运动幅值和供气压强对承载力的影响曲线如图6(a~e)所示.
从图6(a~e)能够分析出:气浮支承承载力呈周期性规律变化,且在t=0~0.0005 s的承载力波动量均小于t=0.0005~0.001 s的承载力波动量. 当供气压强相同时,气浮支承的承载力波动量随着小孔节流器的运动幅值的增大而增大. 当运动幅值相同时,随着供气压强的增大,气浮支承的承载力波动量会明显增大.
4.2 小孔节流器运动幅值和膜厚对气浮支承承载力的影响
为了分析小孔节流器运动幅值和膜厚对气浮支承承载力的影响,取初始条件:小孔节流器初始位置ha=10 μm,运动频率f =1 000 Hz,供气压强Ps=0.6 MPa,小孔节流器直径D3=2 mm,运动幅值A=5/7.5/10/12.5/15 μm,气膜厚度hf=10/15/20/25/30 μm,其他结构参数均保持不变. 小孔节流器运动幅值和膜厚对承载力的影响曲线如图7(a~e)所示.
由图7(a~e)能够分析出:当小孔节流器运动幅值一定时,气浮支承的承载力波动量受气膜厚度的影响较大,并且随着气膜厚度的减小,气浮支承的承载力波动量会明显增大. 当气膜厚度一定时,气浮支承承载力的波动量随着运动幅值的增加而增大. 当膜厚hf=10 μm、t=0.00075 s时,由于运动幅值A=15 μm,此时均压腔高度减小为5 μm,气浮支承质量流量迅速减小,承载力也随之迅速减小,因此图7(d)和图7(e)中会出现承载力变化曲线相交的现象.
4.3 小孔节流器运动幅值和节流高度对气浮支承承载力的影响
为了分析小孔节流器运动幅值和节流高度对气浮支承承载力的影响,取初始条件:气膜厚度hf=10 μm,运动频率f=1 000 Hz,供气压强Ps=0.6 MPa,小孔节流器外径D3=2 mm,运动幅值A=5/7.5/10/12.5/15 μm,节流高度ha=10/15/20/25/30 μm,其他结构参数保持不变. 小孔节流器运动幅值和节流高度对气浮支承承载力的影响曲线如图8(a~e)所示.
由图8(a~e)能够分析出:当小孔节流器的节流高度不变时,气浮支承承载力的波动量随运动幅值的增加而增加. 当小孔节流器的运动幅值不变时,节流器的节流高度由10 μm增至30 μm,气浮支承承载力的初始位置会由326 N增至340 N,而气浮支承的承载力波动量随着节流高度的增大而减小.
4.4 小孔节流器直径和运动频率对气浮支承承载力的影响
为探究小孔节流器直径和运动频率对气浮支承承载力的影响,取初始条件:节流高度ha=10 μm,气膜厚度hf=10 μm,运动幅值A=10 μm,供气压强Ps=0.6 MPa,小孔节流器直径D3=2/3/4/5 mm,运动频率f=100/1000/5 000/10000 Hz,其他结构参数均保持不变. 小孔节流器直径和运动频率对气浮支承承载力的影响曲线如图9(a~d)所示.
由图9(a~d)能够分析出:当运动频率不变时,小孔节流器的直径由2 mm增大至5 mm,气浮支承质量流量随之增大,气浮支承承载力初始位置由297 N增大至315 N. 当小孔节流器直径D3=2 mm时,随着运动频率的增加,气浮支承的承载力波动量增幅几乎为0. 当小孔节流器直径增加至D3=5 mm时,气浮支承的承载力波动量随着运动频率的增加而明显增加,且最大波动量达到190.9 N.
4.5 小孔节流器直径和运动频率对气浮支承动刚度的影响
为了探究小孔节流器直径和运动频率对气浮支承动刚度的影响,取初始条件:节流高度ha=10 μm,气膜厚度hf=10 μm,运动幅值A=10 μm,入口压力Ps=0.6 MPa,小孔节流器直径D3=2/3/4/5 mm,运动频率f为100/500/1000/5000/10000 Hz,其他结构参数保持不变. 小孔节流器的直径和运动频率对气浮支承动刚度的影响曲线如图10所示.
分析图10结果可得出:当小孔节流器运动频率f为100~1000 Hz时,增加小孔节流器直径对气浮支承的动刚度几乎无影响;但当小孔节流器运动频率f为1000~10000 Hz时,增加小孔节流器直径会使气浮支承的动刚度得到明显提升.
5. 结论
a. 对于可变节流高度气浮支承,可以通过调节节流高度来改变气浮支承的承载力.
b. 当可变节流高度气浮支承的其他参数一定时,增加小孔节流器的运动幅值会使气浮支承的承载力波动量明显增大;但当气浮支承供气压强减小时,小孔节流器的运动幅值对气浮支承承载力的影响随之减弱.
c. 当可变节流高度气浮支承的其他参数一定时,增加小孔节流器的运动幅值会使气浮支承的承载力波动量明显增大;但当气膜厚度逐渐增加时,小孔节流器的运动幅值对气浮支承的承载力影响随之减弱.
d. 当可变节流高度气浮支承的其他参数一定时,增加小孔节流器的运动幅值会使气浮支承的承载力波动量明显增大;但当节流高度增加时,小孔节流器的运动幅值对气浮支承的承载力影响随之减弱.
e. 当气浮支承小孔节流器直径较小时,随着小孔节流器运动频率的增加,气浮支承的承载力波动量和动刚度的增幅不明显;但当气浮支承小孔节流器直径增大时,气浮支承的承载力波动量和动刚度会随着小孔节流器的运动频率的增加而明显提升.
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表 1 气浮支承各项参数
Table 1 Parameters of the aerostatic bearing
Parameters Specifications The outside diameter of flexure hinge, D1/mm 65 The diameter of the aerostatic bearing, D2/mm 100 The diameter of the orifice restrictor (the diameter of equalizing cavity), D3/mm 2/3/4/5 The diameter of orifice, D4/mm 0.2 The outer diameter of the annular slit, D5/mm 5.2 The thickness of flexure hinge, H1/ mm 2 The height of the orifice, H2/mm 0.5 The height of the orifice restrictor, H3/mm 6 The height of the circular slit, H4/mm 3 The throttling height (the height of equalizing cavity), ha/μm 10/15/20/25/30 The thickness of gas film, hf/μm 10/15/20/25/30 Supply pressure, Ps/MPa 0.2/0.3/0.4/0.5/0.6 Atmospheric pressure, Pa/MPa 0.1 -
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