Research progress on acoustic impedance eduction technology of aeronautical liner
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摘要:
本文综述了航空声学环境下声衬声阻抗实验提取技术，介绍了多种方法的基本原理、实验布置、研究进展和优劣势。在低频平面波场下，介绍了阻抗管法、单模态法，以及目前主流的目标函数法和直接提取法，其中直接提取法因具有高效、准确的解析优势而受到认可，以上方法均受模态成分限制而存在频率上限。在高频多模态场下，介绍了原位测量法、三维目标函数法、多模态直接提取法、准三维直接提取法、镜像多模态直接提取法，这些方法解决了模态限制，可在不同程度上拓展频率上限。原位测量法的侵入性测量导致其误差和局限性大，三维目标函数法因需三维数值模拟和迭代而效率极低，多模态直接提取法因测点横向间距小而精度不足，准三维直接提取法因受最高阶模态限制而测点需求量大。镜像多模态直接提取法能较好地解决上述问题，它采用锯齿形阵列，基于声场周期性和镜像原理，先将物理阵列展开为多倍宽度等效声场中的对角映射阵列，再调用多模态直接提取法以多模态Prony算法进行声场分解，进而提取声阻抗。镜像多模态直接提取法打破了以往3 kHz的频率上限，能覆盖航空发动机风扇噪声0.2～10 kHz的主要频带，全频提取仅需1 min，比三维目标函数法的效率至少高3个量级，准确性和可靠性较高，且测点大幅减少，兼具宽频、高效、准确、测点少且无损的优势。
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关键词:
- 航空声衬 /
- 声阻抗提取技术 /
- Prony算法 /
- 镜像多模态直接提取法 /
- 高频三维多模态气动声场
Abstract:This paper provides an overview of the impedance eduction techniques of liners in aeroacoustic environments, including the basic principles, experimental arrangements, research progress and features of various methods. In low-frequency plane-wave fields, the impedance tube method, the single mode method, the current mainstream Objective Function Method (OFM) and Straightforward Method (SFM) are available, the latter of which is recognized for high efficiency and accuracy. But all these methods are confined to low-frequency ranges for the modal component limitation. In high-frequency multimodal fields, the in-situ method, the three-dimensional (3D) OFM, the Multimodal SFM (M-SFM), the quasi-3D SFM and the Mirror-based Multimodal SFM (MM-SFM) are developed. All of them address the modal limitation and thus can extend the upper frequency limit to different degrees. The in-situ method suffers from invasive tests resulting in significant errors and limitations. The 3D-OFM suffers from an iteration of 3D numerical simulations resulting in extremely low efficiency. The M-SFM suffers from small transverse spacing between adjoint probes so that its accuracy is not enough. The quasi-3D-SFM suffers from large demand for measuring probes depending on the highest-order mode. The MM-SFM can effectively solve the above problems. It adopts a zigzag array and unfolds it into a mapping diagonal array in a multiple-width equivalent sound field, based on the field periodicity and the mirror symmetry, and then invokes the multimodal Prony algorithm of the M-SFM to decompose the field, so as to educe the acoustic impedance finally. This method breaks the previous upper frequency limit of 3 kHz and can cover the main frequency range of 0.2−10 kHz for aeroengine fan noises; takes only 1 minute to conduct a full-frequency eduction which is at least 3 orders of magnitude more efficient than the 3D-OFM; is of high accuracy and reliability; and decreases the probe number significantly. Generally, it possesses advantages of broadband, efficient, accurate, fewer-probes and non-invasive tests.
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0 引言
飞机穿越含有过冷水滴的云层时，过冷水滴会撞击于飞机表面并冻结成冰，进而严重威胁飞行安全[1]。粒径大于50 μm的过冷水滴被称为“过冷大水滴”(Supercooled Large Droplet, SLD)，其引发结冰导致的后果更加严重[2]。鉴于过冷大水滴结冰的严重危害，美国联邦航空管理局(Federal Aviation Administration, FAA)于2014年扩展了FAR 25部原有附录C“结冰气象条件”，增加了附录O“SLD结冰气象条件”，这意味着某些型号的飞机将面临更加严苛的适航认证要求[3]。
结冰风洞试验是飞机结冰适航取证的重要手段之一。在飞机过冷大水滴结冰适航取证日益增长的需求牵引下，目前世界范围内的主要结冰风洞均在大力发展过冷大水滴云雾模拟能力，以期实现对附录O“SLD结冰气象条件”的覆盖。其中，过冷大水滴粒径测量方法是结冰风洞过冷大水滴云雾模拟能力的关键组成部分[4]，亟需开展相关研究，为过冷大水滴云雾模拟能力发展奠定技术基础。
在附录O的SLD条件滴谱中，云雾液滴粒径范围大约为5~2229 μm[5]，如此宽的粒径范围给准确测量带来极大挑战。针对结冰风洞过冷大水滴云雾液滴粒径测量问题，美国NASA格林中心IRT风洞配套了多台粒径测量设备，采用分段测量方法对结冰风洞试验段液滴粒径分布进行测量[6-10]，其中，适用于小粒径范围的有前向散射分光测量仪FSSP(2~47 μm)和云雾水滴探头CDP(2~50 μm)，适用于大粒径范围的有光学阵列测量仪OAP-230X(15~930 μm)、OAP-230Y(50~1500 μm)、云雾成像探头CIP(15~450 μm)以及降雨成像探头PIP(15~930 μm)。意大利CIRA结冰风洞采用相位多普勒粒子分析仪PDPA，对试验段的SLD颗粒群进行了测量[11]。加拿大NRC结冰风洞开展了大粒径喷嘴和小粒径喷嘴的组合喷射工作，使用Malvern粒度分析仪测量了粒径分布特性，与附录O的FZDZ(冻细雨)曲线吻合较好[12]。
在国内，中国空气动力研究与发展中心3 m×2 m结冰风洞正在发展SLD结冰云雾模拟能力，但目前仅配套了一种机载式设备(双通道机载式相位多普勒干涉仪，PDI-FPDR)和两种地面测量设备(Malvern粒度分析仪、欧美克DP-02激光粒度分析仪)。其中，Malvern粒度分析仪和欧美克DP-02激光粒度分析仪无法在低温、高湿的结冰云雾环境中使用，仅PDI-FPDR可以实现风洞内云雾液滴粒径的测量。
PDI-FPDR是新一代LDV/PDPA系统，广泛应用于结冰风洞液滴粒径测量[13-15]。经实验验证，在液滴中值体积直径(Median Volume Diameter, MVD)小于50 μm的结冰云雾条件下，PDI-FPDR的适用性较好[16-18]，但能否准确测量大粒径的云雾参数还需作进一步研究。
本文通过标准液滴流发生器产生特定尺寸的大粒径液滴流，使用PDI-FPDR对液滴粒径进行测量，评估PDI-FPDR的液滴粒径测量不确定度，进而采用PDI-FPDR和Malvern粒度分析仪同时测量真实大液滴喷雾的粒径特征参数，对比评估PDI-FPDR的大液滴喷雾测量能力。
1 实验仪器
1.1 双通道机载式相位多普勒干涉仪
双通道机载式相位多普勒干涉仪(Phase Doppler Interferometer Flight Probe Dual Range, PDI-FPDR)，是一款能满足自然结冰云雾参数测量需求的机载式测量设备，基于相位多普勒干涉方法同时实现液滴粒径和液滴速率的测量。该设备由光学发射探头、光学接收探头、ASA信号处理器以及AIMS系统软件平台组成[19]。粒径测量基本原理如图 1所示。
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图 1 PDI-FPDR粒径测量的基本原理Fig. 1 The basic principle for droplet size measurement of PDI-FPDR激光器产生的激光束通过带有频移装置的分光器后，分成两束等强度的蓝光和绿光。两束光经透镜聚焦后，在目标测量体内形成明暗相间的干涉条纹。当云雾粒子穿过条纹区域时，会依次散射出光强随时间变化的一列散射光波，即“多普勒信号”。光线以不同角度入射到粒子球体内，在空间内途经的光程不同，这意味着抵达3个并排放置的光电探测器存在时间差，光电探测器接收的散射光存在相位差[20]。3个探测器的相位差存在如下关系：
(1) 
(2) 利用相位差可求得液滴粒径：
(3) 
(4) 式中：d为液滴粒径；fR为接收器透镜的焦距；δ为干涉条纹间距；k为尺寸斜率因子；λ为散射光的空间波长；S为光电探测器之间的距离。
PDI-FPDR包含Channel 1(Small size range)和Channel 2(Large size range)两个测量通道，即小粒径通道和大粒径通道，对应的粒径测量范围分别为0.5~100.0 μm和20.0~1000.0 μm。3 m×2 m结冰风洞使用小粒径通道进行了云雾参数校测[18-19](如图 2所示，图 2中的仪器有3个白色探头，上方为光学发射探头，下方两探头为光学接收探头)，结果表明其能准确测量粒径小于100.0 μm的小尺寸液滴。有文献指出基于相位多普勒干涉方法的PDI设备对于球形液滴适用性较好[13, 21]，对于非球形液滴则存在测量误差；而大尺寸液滴在运动过程中很容易发生变形，因而PDI-FPDR虽有大粒径通道，仍需对其测量能力进行评估，以正确认识PDI-FPDR测量过冷大水滴的可靠度。
1.2 标准液滴流发生器
图 3所示为标准液滴流发生器。其工作原理为：推动微流注射泵，液滴发生器中的液体从喷嘴喷出形成圆柱射流；受高频电压驱动的压电陶瓷振动，并将振动传播给圆柱射流；通过控制驱动电压的频率与幅度，使圆柱射流在瑞利破碎模式下破碎成粒径、速度相同的单分散液滴流。标准液滴流发生器产生液滴的粒径取决于微孔片孔径、注射泵流量、破碎频率等3个参数；调整这些参数至合适范围，可产生粒径大于100 μm的大液滴流[22]。
1.3 Malvern粒度分析仪
Malvern粒度分析仪由英国Spraytec公司研制，广泛应用于喷雾液滴粒径测量[13-14]，如图 4所示。其主要技术参数为：粒径范围0.1~2000.0 μm，采集频率最高10 kHz。工作原理为：液滴通过分析仪采样区域时，引发发射器所产生光束的散射，接收器测量散射光强度并反演液滴粒径信息。散射光强度由安装于接收器内呈半圆形排列的一系列光电二极管来测量。一套曲线拟合程序将散射光强度的累积数密度转换为喷雾液滴粒径分布函数[23]。
前期，3 m×2 m结冰风洞对Malvern粒度分析仪进行了标定，可以认为该设备测量喷雾具有一定可靠性。本文利用该设备与PDI-FPDR开展对比研究。
2 实验方法和数据处理
2.1 PDI-FPDR标准液滴流测量不确定度评估
以标准液滴流发生器产生特定尺寸的大粒径液滴流，使用PDI-FPDR对液滴粒径进行测量，将测量结果和标准液滴粒径进行对比，评估PDI-FPDR的液滴粒径测量不确定度。如图 5所示，将PDI-FPDR置于液滴流发生器下方，使两个光学接收探头中心距离液滴流发生器喷嘴出口150 mm，调整PDI-FPDR水平位置(左右和前后)，使液滴下落时经过PDI-FPDR光学采样区域中心。标准液滴流发生器参数设定及产生的标准液滴粒径如表 1所示。
表 1 标准液滴流发生器参数及液滴粒径Table 1 Parameters of droplet flow generator and droplet size序号 微孔片孔径dm/μm 信号发生频率f/kHz 注射泵流量Q/(mL·min-1) 标准液滴粒径d0/μm 1 60 32.44 1.5 116.0 2 60 19.39 1.8 150.5 3 100 7.67 1.5 189.0 4 100 6.45 1.5 202.0 5 100 3.49 1.4 240.5 液滴流发生器产生的液滴串下落稳定后，打开PDI-FPDR小粒径通道，使用配套的AMIS采集系统对液滴粒径进行采集测量，采集时长15 s，连续采集3次，将平均值作为测量结果。小粒径通道采集完毕后，切换至PDI-FPDR大粒径通道，重复以上采集测量步骤。
以相对误差γ1衡量PDI-FPDR对大尺寸液滴的测量准确度：
(5) 式中：d0表示液滴流发生器产生的标准液滴粒径；dP表示使用PDI-FPDR测量得到的液滴粒径。
以偏差ν1衡量PDI-FPDR对大尺寸液滴的测量精度：
(6) 式中：d′P表示PDI-FPDR在15 s采集过程中任意时刻的液滴粒径测量值；dP表示PDI-FPDR在15 s采集过程中液滴粒径测量的平均值。
2.2 PDI-FPDR大液滴喷雾测量能力评估
通过3 m×2 m结冰风洞的98818型喷嘴产生大粒径喷雾场，使用PDI-FPDR和Malvern粒度分析仪(后文简称Malvern)同时测量喷雾粒径特征参数，对比测量结果以评估PDI-FPDR大液滴喷雾粒径测量能力。如图 6所示，在喷嘴测试台上进行实验，将喷嘴竖直放置，喷口向下。喷口距离PDI-FPDR和Malvern测点分别为500和300 mm，调整两测量仪器水平位置(左右和前后)，使喷雾整体位于PDI-FPDR和Malvern光学采样区域中心。喷嘴供水供气条件设定如表 2所示。
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图 6 PDI-FPDR与Malvern测量喷雾Fig. 6 Spray was measured by PDI-FPDR and Malvern simultaneously表 2 喷嘴供水供气条件Table 2 Water pressure and air pressure of nozzle序号 水压pw/MPa 气压pa/MPa 序号 水压pw/MPa 气压pa/MPa 1 0.05 0.02 10 0.05 0.05 2 0.10 0.02 11 0.10 0.05 3 0.15 0.02 12 0.15 0.05 4 0.20 0.02 13 0.20 0.05 5 0.30 0.02 14 0.30 0.05 6 0.40 0.02 15 0.40 0.05 7 0.50 0.02 16 0.50 0.05 8 0.60 0.02 17 0.60 0.05 9 0.70 0.02 18 0.70 0.05 实验开始后，调节供水供气压力至指定值，开始喷雾。喷雾稳定后，同时使用PDI-FPDR和Malvern的采集系统对喷雾进行采集测量。PDI-FPDR采集时长15 s，连续采集3次，选择通过率(Validation)最高的一次进行数据处理和分析。Malvern采集时长45 s，选择某稳定时段的测量结果进行数据处理和分析。
以相对误差γ2衡量PDI-FPDR与Malvern测量结果的贴合程度：
(7) 式中：D表示中值体积直径(Median Volume Diameter, MVD)，定义为小于该直径的液滴体积与大于该直径的液滴体积相等[24]；DP、DM分别表示相同工况下使用PDI-FPDR和Malvern测量得到的D值。
3 结果分析
3.1 小粒径通道液滴粒径测量结果
使用PDI-FPDR小粒径通道测量标准液滴流的结果如表 3所示。可以看出，对于粒径116.0 μm的液滴，小粒径通道测量准确度高，相对误差低，仅为-1.2%，但正负偏差的绝对值均在10 μm左右；对于粒径150.5 μm的液滴，小粒径通道测量正负偏差减小，但粒径测量值偏低，相对误差达到-21.7%；当液滴粒径大于189.0 μm后，小粒径通道测量值完全偏离液滴真实粒径，相对误差高达-70%左右。
表 3 PDI-FPDR小粒径通道液滴粒径测量结果Table 3 Measurement results of droplet size in PDI-FPDR small size range channel序号 标准液滴粒径d0/μm 小粒径通道测量值dP/μm 相对误差γ1/% 最大正偏差ν1, plus/μm 最大负偏差ν1, minus/μm 1 116.0 114.6 -1.2 13.9 -9.1 2 150.5 117.9 -21.7 3.6 -5.4 3 189.0 51.3 -72.8 1.2 -1.8 4 202.0 56.5 -72.0 1.5 -2.0 5 240.5 72.6 -69.8 1.9 -2.1 图 7显示了小粒径通道测量值偏离液滴真实粒径的程度。粒径在100.0~150.0 μm区间内，小粒径通道的测量结果比较接近标准液滴粒径，当粒径达到189.0 μm后，小粒径通道的测量结果远小于标准液滴粒径。图 8显示了采集过程中最大值和最小值偏离平均值的程度，正负误差较小，说明粒径测量结果离散程度较低，分布比较集中，精度较好。
从测量结果看，虽然小粒径通道性能表现稳定，但是测量大尺寸球形液滴误差很大，主要原因是大液滴的粒径已经超出了小粒径通道量程。因此，PDI-FPDR小粒径通道无法用于结冰风洞过冷大水滴的测量。同时，过冷大水滴云雾中也包含部分粒径小于100.0 μm的液滴，若采用分段测量方法, 以小粒径通道测量这部分液滴，则会使整个测量结果混乱，因为小粒径通道识别的小液滴实际上可能是大液滴也可能是小液滴。因此，PDI-FPDR小粒径通道不适用于过冷大水滴粒径测量。
3.2 大粒径通道液滴粒径测量结果
使用PDI-FPDR大粒径通道测量标准液滴流的结果如表 4所示。可以看出，对于粒径116.0 μm的液滴，大粒径通道测量准确度高，相对误差低，仅为0.3%，正负偏差较大；当液滴粒径大于150.0 μm时，大粒径通道测量相对误差约在5.0%~15.0%，准确度较高，但正负偏差很大。
表 4 PDI-FPDR大粒径通道液滴粒径测量结果Table 4 Measurement results of droplet size in PDI-FPDR large size range channel序号 标准液滴粒径d0/μm 大粒径通道测量值dP/μm 相对误差γ1/% 最大正偏差ν1, plus/μm 最大负偏差ν1, minus/μm 1 116.0 116.4 0.3 11.1 -26.9 2 150.5 128.6 -14.6 37.9 -15.1 3 189.0 205.0 8.5 56.5 -60.5 4 202.0 189.6 -6.1 29.2 -32.1 5 240.5 228.3 -5.1 50.2 -39.8 图 9显示了大粒径通道测量值偏离液滴真实粒径的程度。可以看出，虽然有时偏大、有时偏小，但大粒径通道的测量结果比较接近标准液滴粒径。图 10显示采集过程中的最大值和最小值明显偏离平均值，说明粒径测量结果离散程度较高，分布很分散，精度较差。
从测量结果看，大粒径通道对大尺寸液滴测量的准确度较好，但与小粒径通道相比，精度较差，粒径测量值分布很分散，因此大粒径通道测量过冷大水滴粒径会引入误差，可靠度很低。分析认为，大水滴在运动过程中会发生变形与破碎[25]，对基于相位多普勒干涉方法的PDI设备形成干扰，从而影响测量结果的可靠性。
3.3 喷雾测量能力评估结果
图 11给出了PDI-FPDR大小通道与Malvern测量的不同水气压条件下喷雾的中值体积直径D。可以看出：随着喷嘴水压增大，Malvern和大粒径通道测量的D增大，而随着喷嘴气压增大，D会减小；小粒径通道测量结果曲线没有反映出喷嘴水气压对D的影响。
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图 11 PDI-FPDR与Malvern喷雾MVD测量对比Fig. 11 Comparison of MVD measurements between PDI-FPDR and Malvern在pa=0.02 MPa条件下，Malvern的测量值近似分布于75.0~225.0 μm之间；在pa=0.05 MPa条件下，近似分布于38.0~75.0 μm之间。在pa=0.02 MPa条件下，PDI-FPDR大粒径通道测量值近似分布于125.0~280.0 μm之间；在pa=0.05 MPa条件下，近似分布于100.0~175.0 μm之间。在pa为0.02 MPa和0.05 MPa条件下，小粒径通道测量值近似分布于50.0~75.0 μm之间。综上可知，大粒径通道测量结果与Malvern相比偏大。小粒径通道测量D≤75.0 μm的喷雾与Malvern结果比较接近；随着喷雾真实D值增大，小粒径通道测量值仅在75.0 μm左右浮动，与Malvern测量结果的偏差也随之增大。
图 12给出了PDI-FPDR大小通道与Malvern测量喷雾中值体积直径D的相对误差(DP和DM分别表示PDI-FPDR大小通道与Malvern测量的喷雾中值体积直径)。可以看出：D接近75 μm时，小粒径通道测量效果较好，误差很小；D>75.0 μm后，小通道测量结果逐渐偏离喷雾真实值；与Malvern相比，大粒径通道测量值一直偏大，且相对误差有随着D增大而递减的趋势；在pa=0.05 MPa、pw=0.05 MPa条件下，相对误差最大达154.7%。
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图 12 PDI-FPDR与Malvern喷雾MVD测量相对误差Fig. 12 Comparison of relative error measurement between PDI-FPDR and Malvern需要说明的是：根据前文标准液滴流的测量结果，大通道测量准确度在15%以内，至少在100.0~240.0 μm范围内具有一定可靠性，而该通道测量喷雾中值体积直径D始终偏大，其原因可能是标准液滴流下落速度慢，而喷雾液滴运动速度快并发生变形，显著影响测量结果，造成测量的差异。
图 13给出了pa=0.02 MPa、pw=0.20 MPa条件下PDI-FPDR小粒径通道与Malvern测量喷雾液滴粒径分布的对比。可以看出：PDI-FPDR小粒径通道和Malvern测得的喷雾液滴粒径分布都具有显著的单峰特征，其峰值对应的液滴粒径分别为50.5和96.0 μm；Malvern测量结果中有近50%的液滴粒径在100.0 μm以上，而小粒径通道测得的最大液滴粒径仅为140.0 μm，这进一步验证了小粒径通道会将大液滴识别为小液滴，从而造成测量误差。
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图 13 小粒径通道与Malvern液滴粒径分布测量对比Fig. 13 Comparison of droplet size distribution measurement between small size range channel and Malvern需要说明的是：在标准液滴流测量中，小粒径通道测得的最大粒径为117.9 μm，更大的液滴则被测为50.0~70.0 μm左右(如表 3所示)；而在图 13中，小粒径通道测出了140.0 μm左右的液滴，其原因可能是PDI-FPDR在测量过程中要进行自身修正。
图 14给出了pa=0.02 MPa、pw=0.20 MPa条件下PDI-FPDR大粒径通道与Malvern测量喷雾液滴粒径分布的对比。可以看出，PDI-FPDR大粒径通道和Malvern测得的液滴粒径分布也存在差异，主要表现在大粒径通道测出的粒径在0~100.0 μm范围的液滴数比Malvern少，测得的最大液滴粒径仅为624.5 μm，而Malvern测出的液滴在450.0~1143.0 μm范围内均有分布。大粒径通道测出的粒径在100.0~350.0 μm的液滴数明显多于Malvern，导致该通道中值体积直径D的测量值偏大。
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图 14 大粒径通道与Malvern液滴粒径分布测量对比Fig. 14 Comparison of droplet size distribution measurement between large size range channel and Malvern4 结论
本文采用标准液滴流发生器评估了PDI-FPDR的液滴粒径测量不确定度，进一步采用Malvern对比分析了PDI-FPDR的大液滴喷雾测量能力，得到以下结论：
1) 对于标准大粒径液滴流，小粒径通道无法实现准确测量，其测量结果显著小于真实液滴粒径；对于大粒径喷雾，小粒径通道不适合测量中值体积直径(MVD)大于75.0 μm的喷雾，尽管实际喷雾的MVD值不断增大，而该通道测量值仅在75.0 μm左右浮动。
2) 对于标准大粒径液滴流，大粒径通道可以较准确地实现大尺寸液滴的测量，但精度较差；对于大粒径喷雾，大粒径通道测得的喷雾MVD与Malvern相比偏大。
3) 小粒径通道虽然能准确测量过冷大水滴云雾中的小液滴，但也会将云雾中的大液滴识别为小液滴，造成测量结果混乱；大粒径通道应用于过冷大水滴粒径测量，MVD测量值将大于实际值，测得的液滴粒径分布与实际分布也会存在差异。
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图 8 NASA Langley研究中心的GFIT流管实验台[90]
Fig. 8 GFIT flow duct setup in NASA Langley research center[90]
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图 16 锯齿阵列与对角阵列的镜像映射关系[56]
Fig. 16 Mirror-based mapping from the zigzag array to the diagonal array[56]
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图 17 MM-SFM法（36个测点，$ {\text{Δ}} \mathit{x} $ = 22 mm, $ {\text{Δ}} \mathit{z} $ = 25.5 mm, $ \mathcal{M}\mathit{N} $ = 10）在75 m/s平均流速、50和
5000 次随机噪声扰动（最大幅度为声压级±0.5 dB/相位±1°）下提取声衬声阻抗的均值和不确定度[56]Fig. 17 Averaged educed liner impedance and uncertainties of the MM-SFM under 50 and
5000 random perturbations （with maximum errors of ±0.5 dB in SPL and ±1° in phase） using 36 probes, $ {\text{Δ}} \mathit{x} $ = 22 mm, $ {\text{Δ}} \mathit{z} $ = 25.5 mm and $ \mathcal{M}\mathit{N} $ = 10 at ${M a}$ = 0.22[56] -
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