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在风洞试验时,通过红外热图技术测量非金属模型表面瞬态温度Tw,求解一维瞬态热传导方程时,如果模型表面热流为常数,那么可得:
$$ \frac{{{T_w} - {T_i}}}{{{T_{aw}} - {T_i}}} = 1 - \exp \left( {{\; \beta ^2}} \right)erfc \beta $$ (1) 式中:Tw为模型表面温度,单位是K;Ti为模型表面初始温度,单位是K;Taw为绝热温度,单位是K。当公式(1)左边各项已知时,便可求出β值,从而得到模型表面对流换热系数h,即:
$$ h = \beta \sqrt {\frac{{\rho ck}}{t}} $$ (2) 式中:h为模型表面对流换热系数,单位是W/(m2·K);t为模型加热时间,单位是s;k、ρ、c分别为模型材料热传导系数、密度和比热容,其中模型材料热传导系数,单位是W/(m·K),模型材料密度,单位是kg/m3;模型材料比热容,单位是J/(kg·K)。从公式(2)可知,模型表面对流换热系数h是模型加热时间t的函数,如何确定模型加热时间,需要考虑多方面因素。
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在求解一维瞬态热传导方程时,常常假设模型表面热流为常数。那么,在什么条件下模型表面热流为常数这个假设成立呢?采用隐式差分格式,求解一维瞬态热传导方程,通过改变模型表面对流换热系数,比较模型表面温差。对目前稀薄流地面试验所测量的热流量值来看,模型表面对流换热系数的范围一般为1~150 W/(m2·K),取模型加热时间为1~6 s。所获得的结果如图1所示。从图中可以看出,模型加热时间越短,壁面定常热流假设的近似性越好;模型表面对流换热系数越低,壁面定常热流假设的近似性越好。
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在热流测量中,常常采用一维半无限大假设。如果模型在自由流中的加热时间满足
$$ t \leqslant 0.2{b^2}/\alpha $$ (3) 一维半无限大假设成立。式中:b为模型壁厚,单位是m;α为热扩散系数,单位是m2/s。对试验使用的厚0.5 mm不锈钢薄壁模型材料,其热扩散系数为4.9 mm2/s,由公式(3)确定的试验时间要小于0.01 s,这在风洞试验中很难实现;对试验采用的某种环氧树脂绝热模型材料,其热扩散系数为2.79×e−7 m2/s,该模型控制舵处最薄,为4.5 mm,对应的模型加热时间应该小于14.52 s。
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Srtarner研究了半球圆柱驻点热流的侧向导热误差[7-9]:
$$ {E_c} = \frac{{8kt}}{{\rho {c}{D^2}}} $$ (4) 式中:Ec是侧向导热热流与入射热流的比值;t是模型加热时间,单位是s;D是半球圆柱模型的直径,单位是m;k是导热系数,单位是W/(m·K);ρ是薄壁材料密度,单位是kg/m3;c是比热容,单位是J/(kg·K)。
根据公式(4),侧向导热误差并不是热流的函数,只与模型材料热物性参数、半球圆柱体直径和模型加热时间有关。由于侧向传热误差与模型加热时间成正比,所以模型加热时间不能取得太长。
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在推算模型表面热流时,常常假设模型为一维半无限介质,实际上模型厚度不可能为半无限大,那么模型壁厚在什么条件下才满足一维半无限假设呢?考虑模型热渗透厚度z,它表示在时刻t模型内部热传导为零的最小深度。如果表面热流为阶跃函数,通过对一维瞬态热传导方程作LAPLACE反变换,可得
$$ {\textit{z}} \geqslant 4\sqrt {\alpha t} $$ (5) 式中:t是模型加热时间,单位是s;α为热扩散系数,单位是m2/s。当满足公式(5)时,模型表面温度、表面热流的测量误差不超过百分之一。试验中常把公式(5)作为模型壁厚选取的准则。对热扩散系数为2.79×e−7m2/s的某种环氧树脂绝热模型材料,如果试验加热时间取1 s,模型表面热渗透深度为2 mm左右;如果试验加热时间取2 s,热渗透深度为3 mm左右。
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在公式(1)中,通过对流换热系数h对Tw、Taw、Ti、(ρck)1/2、t这五个参数求偏导数,并使用误差传递的泰勒级数法,可得对流换热系数随机不确定度为[10]:
$$ \frac{{\Delta h}}{h} = \pm \frac{1}{h}{\left\{ {\sum\limits_{i = 1}^5 {{{\left[ {{{\left( {\frac{{\partial h}}{{\partial {P_i}}}} \right)}^2} \cdot {{\left( {\Delta {P_i}} \right)}^2}} \right]}^2}} } \right\}^{\frac{1}{2}}} $$ (6) 式中:Δh为h的绝对不确定度;ΔPi为五个参数Pi的绝对不确定度。
把这五个偏导数求出后,代入公式(1),经过整理,可得对流换热系数的随机不确定度为:
$$ \begin{split} \dfrac{{\Delta h}}{h} \!\!=\!\! {\left\{\!\!\!\! \begin{array}{l} {\left[ {\left( {\dfrac{{Q{T_w}}}{{2\left(\!\! {{T_{aw}} - {T_w}} \! \right)}}}\! \right) \cdot \left(\!\! {\dfrac{{\Delta {T_w}}}{{{T_w}}}}\! \right)} \! \right]^2} \!\!+\!\! {\left[ {\left(\!\! {\dfrac{{Q(1 \!-\!\! Z){T_{aw}}}}{{2\left(\!\! {{T_{aw}} \!-\! {T_w}} \right)}}} \! \right) \cdot \left( \! {\dfrac{{\Delta {T_{aw}}}}{{{T_{aw}}}}} \! \right)}\! \right]^2}{\rm{ \!+\!\!\! }}\\ {\left[ {\left(\! {\dfrac{{QZ{T_i}}}{{2\left(\! {{T_{aw}} - {T_w}} \! \right)}}}\! \right) \cdot \left( \!{\dfrac{{\Delta {T_i}}}{{{T_i}}}} \right)} \right]^2} \!+\! \!{\left[\! {\dfrac{{\Delta {{(\rho ck)}^{1/2}}}}{{{{(\rho ck)}^{1/2}}}}} \right]^2} \!+ \!{\left[ {0.5 \cdot \dfrac{{\Delta t}}{t}} \right]^2} \end{array} \!\!\!\!\!\! \right\}^{\frac{1}{2}}} \end{split} $$ (7) 其中,
$$ {Q = \frac{{\sqrt \pi Z}}{{\beta (\; \beta \sqrt \pi Z - 1)}}} $$ (8) $$ {Z = \dfrac{{{T_{\alpha w}} - {T_w}}}{{{T_{\alpha w}} - {T_i}}} = {{\rm exp}(\beta ^2)} \cdot erfc (\beta )} $$ (9) 由公式(7)可以看出,在一定的试验条件下,对流换热系数的随机不确定度是模型表面温度的函数。在马赫数17.44、总温962 K、总压1.577 MPa、攻角30°的试验条件下,某升力体模型对流换热系数不确定度与模型表面温度的关系如图2所示[11]。从图中可以看出,对流换热系数测量误差是随模型表面温度变化的,在超出40~200 ℃温度范围时,测量的对流换热系数具有较大的误差。而对于给定的试验条件和选定的模型材料,模型表面温度主要取决于模型加热时间。因此合理选择模型加热时间,控制模型表面温升在一定范围内,可提高红外热图测量精度。文中使用的红外热像仪工作波段为2.5~5.1 μm,采用碲镉汞探测器,测温范围为5~500 ℃,帧频为5~115 Hz,灰度分辨率为14 bit,图像分辨率为640×512。
Measurement of mid-low order of magnitude of heat transfer rate using infrared thermography in rarefied flow
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摘要: 在分析高高空热流测量需求基础上,针对高超声速飞行器稀薄气流地面试验热流量值小等特点,从满足模型壁面定常热流假设和一维半无限大假设条件、减小试验模型侧向导热误差和控制试验模型表面温升等方面分析了试验模型加热时间对热流测量的影响。其次选择较低热扩散系数模型绝热材料、采用瞬变平面热源法提高试验模型材料热物性参数标定精度、采用漫反射补偿等提高发射率测量精度等手段,提高中低量值热流测量精度。最后,在利用薄壁量热法获得模型表面热流时,测量MW/m2量级热流是把热电偶焊接在试验模型内壁面,而用红外热图及测量几kW/m2到几百kW/m2量级热流是测量模型外壁面热流,为了对这三者结果进行比较,在马赫数 Ma 为 12、试验总压 P0 为4.2 MPa、试验总温 T0 为700 K的试验状态下,用热电偶与红外热图同时测量了双锥薄壁模型不同点的热流,结果表明:红外测热结果与热电偶测量外壁面结果更接近,热电偶布置在外壁面位置所获得的热流大于布置在内壁面位置所获得的热流,模型加热时间对不同量值热流测量的影响是不同的。Abstract: Based on the analysis of demand for measuring heat transfer of hypersonic vehicle at high attitude, aimed at the characteristic of mid-low order of magnitude of heat transfer rate at ground test for simulating rarefied flow, various factors of the influence on measurement accuracy of mid-low order of magnitude of heat transfer were analyzed. Firstly, the influence of model heating time on heat transfer measurement was analyzed from reducing lateral error of experiment model to controlling temperature rise of model surface, which was based on a steady heat transfer of model surface and one-dimensional and semi-infinite hypothesis. Secondly, model adiabat materials with low thermal diffusivity were put to use. Transient Plane Source Method was introduced so as to calibrate thermal physical parameter(density, specific heat ratio and thermal conductivity) of model material with high accuracy. Kinds of method, such as diffusing compensation and phase-locked method, were adopted to raise measurement accuracy of model surface emittance. Finally, when using thin-wall calorimetry to obtain model surface heat transfer rate, thermocouple was welded on the inner surface of test model to measure heat transfer rate such as MW/m2 order of magnitude, while one was done on the outer surface of model to measure that with several kW/m2 to several hundred kW/m2 by thermocouple or by infrared thermography. In order to compare these results, on the experimental condition of Mach number 12, total pressure 4.2 MPa and total temperature 700 K, heat transfer rate at different points of the double cone thin-wall model was measured by thermocouple and infrared thermography simultaneously. The results indicated that the heat transfer rate measured with infrared thermometry was approaching to that done with thermocouple on the outer surface of model, that heat transfer rate by thermocouple on the outer surface of model was greater than that by thermocouple on the inner surface of test model, and that the influence of model heating time on the measurement of heat transfer rate with different order of magnitude was different.
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图 5 (a)用热电偶测量模型外壁面热流与红外测量比较;(b)用热电偶测量模型内壁面热流与红外测量比较
Figure 5. (a) Comparison of heat transfer rate on the outer surface of model measured with thermocouple and infrared thermography; (b) Comparison of heat transfer rate on the inner surface of test model measured with thermocouple and infrared thermography
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[1] Yang Yanguang, Li Ming, Li Zhonghua, et al. Technique study of aerodynamic force/heating measurement on hypersonic vehicle across different flow region [J]. Acta Aerodynamica Sinica, 2016, 34(1): 5-13. (in Chinese) doi: 10.7638/kqdlxxb-2015.0149 [2] Allegre Jean, Dubreuilh X Heriard , Raffin M. Measurement of aerodynamic heat rates by infrared thermographic technique at rarefied flow conditions [J]. Progress in Astronautics and Aeronautics, 1989, 117: 157-167. [3] Westby M F. Heat Transfer Measurements Using Infra-Red Thermography in Rarefield Flows[M]//New Trends in Instrumentation for Hypersonic Research. Dordrecht: Springer, 1993: 503-512 [4] Li Ming, Yang Yanguang, Zhu Zhiwei. Experiment of the characteristic of aerodynamic heatingon CAV using infrared thermograpy [J]. Infrared and Laser Engineering, 2013, 42(2): 285-289. (in Chinese) doi: 10.3969/j.issn.1007-2276.2013.02.002 [5] Mao Yufeng, Li Yunze, Wang Jing, et al. Heat-flux measurement technology based on infrared thermography [J]. Infrared and Laser Engineering, 2016, 45(8): 0804001. (in Chinese) doi: 10.3788/IRLA201645.0804001 [6] Geng Zihai, Cai Jinsheng, Zhang Weiguo, et al. Application of infrared imaging and Michel model in rotor airfoil transition characteristics [J]. Infrared and Laser Engineering, 2019, 48(8): 0803002. (in Chinese) doi: 10.3788/IRLA201948.0803002 [7] Starner K E. Use of thin-skinned calorimeters for high heat flux arc jet measurements [J]. ISA Trans, 1967, 7: 22. [8] Kidd C T. Lateral heat conduction effects on heat-transfer measurements with the thin-skin technique [J]. ISA Transactions, 1987, 26(3): 7-18. [9] Kidd C T. Thin-skin technique heat-transfer measurement errors due to heat conduction into thermocouple wires [J]. ISA Transactions, 1985, 24(2): 1-9. [10] Bynum D S, Hube F K, Key C M, et al. Measurements and mapping of aerodynamic heating in VKF tunnel B with an infrared camera, ADA033116[R].USA: US Department of Defense, 1977. [11] Zeng Xuejun, Li Ming, Liu Taikui, et al. The character research of aerodynamic heating on lifting body model using infrared thermographic technique [J]. Acta Aerodynamica Sinica, 2004, 22(4): 494-498. (in Chinese) doi: 10.3969/j.issn.0258-1825.2004.04.024 [12] Liu Wei, Shi Haodong, Jiang Huilin, et al. Infrared polarization properties of targets with rough surface [J]. Chinese Optics, 2020, 13(3): 459-471. (in Chinese) doi: 10.3788/CO.2019-0123 [13] Huang Liang, Li Mingxuan, Lv Hengyi, et al. Large-range and high-resolution temperature measurement system for satellite-borne infrared detector [J]. Optics and Precision Engineering, 2019, 27(11): 2315-2320. (in Chinese) doi: 10.3788/OPE.20192711.2315 [14] Yao Yuming, Song Baoan, Xiao Chuanfu, et al. Optical non-uniformity test of transparent infrared chalcogenide film and influencing factors [J]. Optics and Precision Engineering, 2020, 28(5): 1005-1011. (in Chinese) doi: 10.3788/OPE.20202805.1005 [15] Li Yihan, Hu Haiyang, Wang Qiang. Radiative transmission property of infrared window in hypersonic vehicle [J]. Infrared and Laser Engineering, 2020, 49(4): 0404002. (in Chinese) doi: 10.3788/IRLA202049.0404002 [16] Yu Xiaoya, Liu Lituo, Li Rui, et al. Measurements of absolute radiative emissions for supersonic reentry [J]. Chinese Optics, 2020, 13(1): 87-94. (in Chinese) doi: 10.3788/co.20201301.0087