泄爆和氮气惰化耦合作用对氢-空气爆炸影响的实验研究

张凯; 杜赛枫; 陈昊; 郭进; 王金贵; 洪溢都

doi:10.11883/bzycj-2021-0459

泄爆和氮气惰化耦合作用对氢-空气爆炸影响的实验研究

doi: 10.11883/bzycj-2021-0459

福州大学环境与安全工程学院，福建福州 350116

基金项目: 国家自然科学基金(51904285); 福建省自然科学基金(2020J01505)

详细信息

作者简介:
张　凯（1996－　），男，硕士研究生，17318532395@163.com

通讯作者:
洪溢都（1989－　），男，博士，讲师，yidu.hong@fzu.edu.cn

中图分类号: O381; X932
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- 文章访问数: 344
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- 被引次数: 0
出版历程
- 收稿日期: 2021-11-08
- 修回日期: 2022-01-12
- 网络出版日期: 2022-11-14
- 刊出日期: 2022-12-08

Experiments on the effects of venting and nitrogen inerting on hydrogen-air explosions

College of Environment and Safety Engineering, Fuzhou University, Fuzhou 350116, Fujian, China

摘要

摘要: 为了解受限空间内不同氮气体积分数φ对氢-空气泄爆的影响，在高1 m的顶部开口容器中进行了实验。结果表明：当φ≤40%时，容器内部的最大压力峰值由外部爆炸造成；而当φ>40%时，内部最大压力峰值则由泄爆膜破裂引起；在所有实验中，都观察到内部压力的亥姆霍兹振荡，其振荡频率随φ的增加而降低；声学振荡仅出现在φ=25%, 30%时；容器内3个不同压力监测点（靠近泄爆口、容器中心和接近容器底端）的最大爆炸超压p_max都随着φ的增加而降低，且整体上最大的p_max始终在爆炸容器底部附近出现。但当φ>40%时，3个监测点间p_max的差异可忽略不计；外部最大爆炸超压也随φ的增加而减小，且不论其大小如何，均对内部压力曲线有显著影响。
- 氢气 /
- 泄爆 /
- 氮气惰化 /
- 超压 /
- 火焰
Abstract: Explosion venting and inerting are two commonly used explosion protective measures in hydrogen-based industries, both of them are effective in reducing the maximum explosion overpressure when used alone. However, the coupling effects of venting and inerting on hydrogen deflagrations have not been well understood. To this end, experiments were carried out in a 1 m high top-vented vessel with a cross-section area of 0.3 m×0.3 m to investigate the effects of nitrogen volume fraction (φ) in the range of 0 to 50% by volume on vented hydrogen-air explosions with a fixed equivalence ratio. The premixed hydrogen-nitrogen-air mixtures obtained according to Dolton’s law of partial pressure were ignited in the center of the vented container by an electric spark with an energy of about 500 mJ. A 0.75-m long transparent window was installed in the center of the vented container, through which the flame images in the container were recorded by a high-speed camera at 2 000 frames per second. The pressure-time histories within and outside the vented container were measured by four piezoresistive pressure sensors with a measuring range of 0–150 kPa. The experimental results reveal that φ significantly affects the vented deflagration of hydrogen-air mixtures. The pressure peak owing to the external explosion dominates the internal pressure-time histories when φ≤40% and that resulting from the rupture of vent cover becomes dominant for higher values of φ. Under the current experimental conditions, Helmholtz-type oscillations with a frequency decreasing with φ are always observed, and acoustic oscillations appear in the tests only for φ=25%, 30%. The maximum internal explosion overpressures (p_max) near the vent, at the center of the vessel, and near the bottom of the vessel decrease with increasing φ. Moreover, the highest overall p_max is obtained always near the bottom of the vessel. However, the difference of p_max between the three measuring points is negligible when φ is larger than 40%. The maximum external explosion overpressure decreases with increasing φ. In addition, significant effects of the external explosion on the internal pressure-time histories are observed in all tests, regardless of its explosion overpressure.
- hydrogen /
- explosion venting /
- nitrogen inerting /
- overpressure /
- flame

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压电材料可以制造成执行器或传感器等智能元件，广泛应用于国防工业与实际生活中。由于压电材料中力学与电学性质相互耦合，在SH波作用下压电材料中夹杂或圆孔等缺陷处的动应力集中及电场强度集中问题也比一般材料更复杂。近年来，许多学者对缺陷问题进行了研究，并取得了丰富的成果^[1-12]。X.F.Li等^[1]基于电磁材料弹性理论研究了径向非均匀性的压电压磁球壳的静态响应问题；时朋朋等^[2]利用分离变量法和Hilbert核奇异积分方程理论研究了功能梯度压电压磁双材料的周期界面裂纹问题；靳静等^[3]利用积分变换法和奇异积分方程技术研究了压电压磁双材料界面裂纹的二维断裂问；舒小平^[4-5]基于等效单层理论的位移场和电势场求解了正交压电复合材料层板在各类边界条件下的解析解；宋天舒等^[6-7]研究了双相压电介质中圆孔与界面裂纹相互作用的动力学问题。但是，以上工作中大部分是关于径向非均匀介质的静态响应问题的求解，对含圆孔的压电介质在SH波作用下的动态响应问题，目前仍未见报道。

1. 控制方程

含圆孔的全空间非均匀压电介质如图 1所示，已知其密度ρ(r)=ρ₁β²r^2(β-1)，其中ρ₁为常数，β为幂次。弹性常数、压电常数、介电常数分别为c₄₄、e₁₅、κ₁₁；圆孔内部可以形成电场，其压电常数为e₁₅^c，介电常数为κ₁₁^c。在直角坐标系中：r²=x²+y²，ρ(x, y)=ρ₁β²(x²+y²)^(β-1)。满足控制方程：

$\left\{ \begin{array}{l} {c_{44}}{\nabla ^2}w + {e_{15}}{\nabla ^2}\varphi + \rho \left( {x, y} \right){\omega ^2}w = 0\\ {e_{15}}{\nabla ^2}w - {\kappa _{11}}{\nabla ^2}\varphi = 0 \end{array} \right.$

(1)

图 1 含圆孔径向非均匀压电介质模型

Figure 1. Model of the radial inhomogeneous piezoelectric medium with a circular cavity

下载: 全尺寸图片幻灯片

式中：w和φ分别为压电材料的位移和电势，ω为SH波的圆频率。令φ=e₁₅(w+f)/κ₁₁，对式(1)化简得：

$\left\{ \begin{array}{l} {\nabla ^2}w + k_0^2{\beta ^2}{\left( {{x^2} + {y^2}} \right)^{\left( {\beta - 1} \right)}}w = 0\\ {\nabla ^2}f = 0 \end{array} \right.$

(2)

波数满足：

${k^2} = \rho {w^2}/{c^ * } = k_0^2{\beta ^2}{\left( {{x^2} + {y^2}} \right)^{\left( {\beta - 1} \right)}}$

(3)

式中：k为波数；k₀²=ρ₁ω²/c^*，c^*为压电介质的剪切波速，且c^*=c₄₄+e₁₅²/κ₁₁。

利用复变函数法，令z=x+iy, z=x-iy，在复平面(η, η)中控制方程可化为：

$\left\{ \begin{array}{l} \frac{{{\partial ^2}w}}{{\partial z\partial \bar z}} + \frac{1}{4}{\beta ^2}{\left( {z\bar z} \right)^{\beta - 1}}k_0^2w = 0\\ \frac{{{\partial ^2}f}}{{\partial z\partial \bar z}} = 0 \end{array} \right.$

(4)

引入变量ζ=z^β，ζ=z^β，控制方程可进一步转化为：

$\frac{{{\partial ^2}w}}{{\partial \zeta \partial \bar \zeta }} + \frac{1}{4}k_0^2w = 0$

(5)

本构方程为：

$\left\{ \begin{array}{l} {\tau _{rz}} = \left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right)\left( {\frac{{\partial w}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} + \frac{{\partial w}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right) + \frac{{e_{15}^2}}{{{\kappa _{11}}}}\left( {\frac{{\partial f}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} + \frac{{\partial f}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right)\\ {\tau _{\theta z}} = {\rm{i}}\left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right)\left( {\frac{{\partial w}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial w}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right) + {\rm{i}}\frac{{e_{15}^2}}{{{\kappa _{11}}}}\left( {\frac{{\partial f}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial f}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right)\\ {D_r} = - {e_{15}}\left( {\frac{{\partial f}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} + \frac{{\partial f}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right)\\ {D_\theta } = - {\rm{i}}{e_{15}}\left( {\frac{{\partial f}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial f}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right) \end{array} \right.$

(6)

式中：τ_rz和τ_θz分别为非均匀压电介质的径向应力和切向应力，D_r和D_θ分别为圆孔中电场的径向电位移和切向电位移。

2. 介质中的位移场

SH波散射过程中，入射波引起的压电材料位移wⁱⁿ表达式为：

${w^{{\rm{in}}}} = {w_0}\exp \left[ {\frac{{{\rm{i}}k}}{2}\left( {\zeta {{\rm{e}}^{ - {\rm{i}}{\alpha _0}}} + \bar \zeta {{\rm{e}}^{{\rm{i}}{\alpha _0}}}} \right)} \right]$

(7)

散射波引起的压电材料位移w^s表达式为：

${w^{\rm{s}}} = \frac{{\rm{i}}}{{2{c_{44}}\left( {1 + \lambda } \right)}}\sum\limits_{n = - \infty }^{ + \infty } {{A_n}H_n^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^n}}$

(8)

式中：H_n⁽¹⁾( $k\left| \zeta \right|$ )为n阶第一类Hankel函数，λ=e₁₅²/(c₄₄κ₁₁)，A_n为系数。

$\varphi = \frac{{{e_{15}}}}{{{\kappa _{11}}}}\left( {{w^{{\rm{in}}}} + {w^{\rm{s}}} + {f^{\rm{s}}}} \right)$

(9)

散射波引起的电场附加函数f^s表达式为：

${f^{\rm{s}}} = \sum\limits_{n = 1}^{ + \infty } {{B_n}{z^{ - n}}} + {C_n}{{\bar z}^{ - n}}$

(10)

式中：B_n和C_n为系数。由此得到：

$\left\{ \begin{array}{l} \tau _{rz}^{{\rm{in}}} = \frac{{{\rm{i}}k}}{2}\left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right)\beta {w_0}\left( {{z^{\beta - 1}}{{\rm{e}}^{{\rm{i}}\left( {\theta - {\alpha _0}} \right)}} + {{\bar z}^{\beta - 1}}{{\rm{e}}^{ - {\rm{i}}\left( {\theta - {\alpha _0}} \right)}}} \right)\\ \exp \left[ {\frac{{{\rm{i}}k}}{2}\left( {\zeta {{\rm{e}}^{ - {\alpha _0}}} + \bar \zeta {{\rm{e}}^{{\rm{i}}{\alpha _0}}}} \right)} \right]\\ \tau _{rz}^{\rm{s}} = \frac{{{\rm{i}}\beta k}}{4}\sum\limits_{n = - \infty }^{ + \infty } \\ {{A_n}\left[ {H_{n - 1}^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^{n - 1}}{z^{\beta - 1}}{{\rm{e}}^{{\rm{i}}\theta }} - H_{n + 1}^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^{n + 1}}{{\bar z}^{\beta - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right]} \\ \; - \frac{{e_{15}^2}}{{{\kappa _{11}}}}n\left( {\sum\limits_{n = 1}^{ + \infty } {{B_n}{z^{ - n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} } \right)\\ D_r^{\rm{s}} = {e_{15}}n\left( {\sum\limits_{n = 1}^{ + \infty } {{B_n}{z^{ - n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} } \right) \end{array} \right.$

(11)

式中：上标“in”、“s”分别表示物理量与入射波、反射波相关。圆孔内部存在电场，满足方程：

$\frac{{{\partial ^2}{f^{\rm{c}}}}}{{\partial z\partial \bar z}} = 0$

(12)

式中：f^c为圆孔内部的电场附加函数。求解式(12)可得：

${f^{\rm{c}}} = \sum\limits_{n = 0}^{ + \infty } {{D_n}{z^n}} + \sum\limits_{n = 1}^{ + \infty } {{E_n}{{\bar z}^n}}$

(13)

式中：D_n和E_n为系数。由此可得：

$\left\{ \begin{array}{l} \tau _{rz}^{\rm{c}} = 0\\ {\varphi ^{\rm{c}}} = \frac{{e_{15}^{\rm{c}}}}{{\kappa _{11}^{\rm{c}}}}{f^{\rm{c}}}\\ D_r^{\rm{c}} = - e_{15}^{\rm{c}}n\left( {\sum\limits_{n = 0}^{ + \infty } {{D_n}{z^{n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}} + \sum\limits_{n = 1}^{ + \infty } {{E_n}{{\bar z}^{n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} } \right) \end{array} \right.$

(14)

式中：上标“c”表示物理量与圆孔中空气形成的电场相关。

3. 边界条件与定解方程

圆孔处的边界条件为：

$\begin{array}{*{20}{c}} {{\tau _{rz}} = \tau _{rz}^{{\rm{in}}} + \tau _{rz}^{\rm{s}} = \tau _{rz}^{\rm{c}} = 0}&{\varphi = {\varphi ^{\rm{c}}}}&{D_r^{\rm{s}} = D_r^{\rm{c}}} \end{array}$

(15)

利用以上边界条件式(15)建立关于A_n、B_n、C_n、D_n、E_n的方程组：

$\begin{array}{l} {\xi ^{\left( 1 \right)}} = \sum\limits_{n = - \infty }^{ + \infty } {{A_n}\xi _n^{\left( {11} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{B_n}\xi _n^{\left( {12} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}\xi _n^{\left( {13} \right)}} \\ {\xi ^{\left( 2 \right)}} = \sum\limits_{n = - \infty }^{ + \infty } {{A_n}\xi _n^{\left( {21} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{B_n}\xi _n^{\left( {22} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}\xi _n^{\left( {23} \right)}} + \sum\limits_{n = 0}^{ + \infty } {{D_n}\xi _n^{\left( {24} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{E_n}\xi _n^{\left( {25} \right)}} \\ {\xi ^{\left( 3 \right)}} = \sum\limits_{n = 1}^{ + \infty } {{B_n}\xi _n^{\left( {32} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}\xi _n^{\left( {33} \right)}} + \sum\limits_{n = 0}^{ + \infty } {{D_n}\xi _n^{\left( {34} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{E_n}\xi _n^{\left( {35} \right)}} \end{array}$

(16)

式中：

$\left\{ \begin{array}{l} \xi _n^{\left( {11} \right)} =\\ \frac{{{\rm{i}}\beta k}}{4}\left[ {H_{n - 1}^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^{n - 1}}{z^{\beta - 1}}{{\rm{e}}^{{\rm{i}}\theta }} - H_{n + 1}^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^{n + 1}}{{\bar z}^{\beta - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right]\\ \xi _n^{\left( {12} \right)} = - \frac{{e_{15}^2}}{{{\kappa _{11}}}}n{z^{ - n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}\;\;\;\;\;\xi _n^{\left( {13} \right)} = - \frac{{e_{15}^2}}{{{\kappa _{11}}}}n{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}\;\;\;\;\\ \xi _n^{\left( {21} \right)} = \frac{{{\rm{i}}{e_{15}}}}{{2{c_{44}}{\kappa _{11}}\left( {1 + \lambda } \right)}}H_n^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)^n}\\ \xi _n^{\left( {22} \right)} = \frac{{{e_{15}}}}{{{\kappa _{11}}}}{z^{ - n}}\;\;\;\;\;\;\xi _n^{\left( {23} \right)} = \frac{{{e_{15}}}}{{{\kappa _{11}}}}{{\bar z}^{ - n}}\;\;\;\;\;\;\;\xi _n^{\left( {24} \right)} = - \frac{{e_{15}^{\rm{c}}}}{{\kappa _{11}^{\rm{c}}}}{z^n}\;\;\;\;\;\;\\\ xi _n^{\left( {25} \right)} = - \frac{{e_{15}^{\rm{c}}}}{{\kappa _{11}^{\rm{c}}}}{{\bar z}^n}\\ \xi _n^{\left( {32} \right)} = {e_{15}}n{z^{ - n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}\;\;\;\;\;\;\xi _n^{\left( {33} \right)} = {e_{15}}n{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}\\ \xi _n^{\left( {34} \right)} = e_{15}^{\rm{c}}n{z^{n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}\;\;\;\;\;\;\xi _n^{\left( {35} \right)} = e_{15}^{\rm{c}}n{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}\\ {\xi ^{\left( 1 \right)}} = - \frac{{{\rm{i}}k}}{2}\left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right)\beta {w_0}\left( {{z^{\beta - 1}}{{\rm{e}}^{{\rm{i}}\left( {\theta - {\alpha _0}} \right)}} + {{\bar z}^{\beta - 1}}{{\rm{e}}^{ - {\rm{i}}\left( {\theta - {\alpha _0}} \right)}}} \right)\\ \exp \left[ {\frac{{{\rm{i}}k}}{2}\left( {\zeta {{\rm{e}}^{ - {\rm{i}}{\alpha _0}}} + \bar \zeta {{\rm{e}}^{{\rm{i}}{\alpha _0}}}} \right)} \right]\\ {\xi ^{\left( 2 \right)}} = - \frac{{{e_{15}}}}{{{\kappa _{11}}}}{w_0}\exp \left[ {\frac{{{\rm{i}}k}}{2}\left( {\zeta {{\rm{e}}^{ - {\rm{i}}{\alpha _0}}} + \bar \zeta {{\rm{e}}^{{\rm{i}}{\alpha _0}}}} \right)} \right]\;\;\;\;\;\;\;{\xi ^{\left( 3 \right)}} = 0 \end{array} \right.$

(17)

将式(16)取有限截断项，等式两边同时乘以e^-imθ(m=0, ±1, ±2, ±3, …)，从(-π, π)进行积分得到多元一次方程组，从而求解出未知系数A_n、B_n、C_n、D_n、E_n。

4. 动应力集中系数与电场强度系数

根据文献[11]可知，动应力集中系数τ_θz^*(dynamic stress concentration factor, DSCF)和电场强度集中系数E_θ^*(electric field intensity concentration factor, EFICF)表达式分别为：

$\begin{array}{*{20}{c}} {\tau _{\theta z}^ * = \left| {{\tau _{\theta z}}/{\tau _0}} \right|}&{E_\theta ^ * = \left| {{E_\theta }/{E_0}} \right|} \end{array}$

(18)

式中：

$\begin{array}{*{20}{c}} {{\tau _0} = {\rm{i}}k\left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right){w_0}}&{{E_0} = \frac{{k{e_{15}}{w_0}}}{{{\kappa _{11}}}}}&{{E_\theta } = - {\rm{i}}\left( {\frac{{\partial \varphi }}{{\partial \eta }}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial \varphi }}{{\partial \bar \eta }}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right)} \end{array}$

5. 算例分析

当β=1时，本文模型退化为均匀压电介质模型。为对本文方法进行验证，采用与文献[7]中相同的参数，求解得到动应力系数τ_θz^*沿圆孔周边的分布情况，如图 2所示。可以看出，计算结果与文献[7]中结果吻合较好，说明本文方法精确可行。以下取κ₁₁/κ₁₁^c=1000进行建模，分析各参数对动应力集中系数及电场强度系数的影响。

图 2 方法验证(与文献[7]比较)

Figure 2. Verification of the present method (compared with reference [7])

下载: 全尺寸图片幻灯片

图 3给出了SH波以不同角度(α₀)入射时圆孔周围动应力系数的变化情况。由图 3可知：SH波垂直入射时，τ_θz^*达到最大值3.8；SH波水平入射时，τ_θz^*最大值约为均匀压电介质的2~3倍。由此可见，入射角度α₀对非均匀介质具有一定的影响。

图 3 SH波入射角度不同时动应力集中系数的变化

Figure 3. Varition of DSCF around the circular cavity edge by SH-wave with different incident angles

下载: 全尺寸图片幻灯片

图 4给出了SH波水平入射时圆孔周边动应力集中系数随波数ka的变化情况。图 4显示：τ_θz^*随波数ka的增大而减小，SH波低频入射时，τ_θz^*的最大值约为高频入射时的2倍。

图 4 SH波水平入射时圆孔周边动应力集中系数随波数ka的变化情况

Figure 4. DSCF around circular cavity edge vs.ka by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 5给出了SH波垂直入射时圆孔周边动应力集中系数随波数ka的变化情况。图 5显示：τ_θz^*随波数ka增大而减小，与图 4中规律相同，但图 5中τ_θz^*的最大值比图 4中约大18%。由图 3~5可知，SH波低频垂直入射对径向非均匀压电介质破坏较大，在工程中应该对这种情况引起注意。

图 5 SH波垂直入射时圆孔周边动应力集中系数随波数ka的变化情况

Figure 5. DSCF around circular cavity edgevs.ka by vertical SH-wave

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图 6给出了SH波水平入射时圆孔周边动应力集中系数随λ的变化情况。图 6显示：压电参数λ对τ_θz^*几乎没有影响。图 7给出了SH波水平入射时圆孔周边动应力集中系数随β的变化情况。由图 7可知，τ_θz^*随幂次β的增大而增大，当β=4时，τ_θz^*达到最大值3.2，约为均匀压电材料τ_θz^*最大值的2倍，因此工程中应该合理调整参数，避免幂次β过大。

图 6 SH波水平入射时圆孔周边动应力集中系数随λ变化情况

Figure 6. DSCF around circular cavity edge vs. λ by horizontal SH-wave

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图 7 SH波水平入射时圆孔周边动应力集中系数随幂次β的变化情况

Figure 7. DSCF around circular cavity edge vs. β by horizontal SH-wave

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图 8给出了SH波水平入射时圆孔θ=π/2处动应力集中系数随波数ka的变化情况。图 8显示：τ_θz^*随ka值的增大而减小，下降率约为1.1%。不同压电参数λ条件下得到的τ_θz^*曲线几乎完全重合，说明λ对τ_θz^*几乎没有影响，与图 6中的结论一致。

图 8 SH波垂直入射时圆孔θ=π/2处动应力集中系数随ka的变化

Figure 8. DSCF around circular cavity edge vs. ka by vertical SH-wave

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图 9给出了SH波高频入射时圆孔周边电场强度系数随SH波入射角度的变化情况。由图 9可知：入射角度对E_θ^*最大值的影响不大；斜入射时，E_θ^*达到最大值3.1。

图 9 SH波以不同角度入射时圆孔周边电场强度系数的变化情况

Figure 9. Variation of EFICF around circular cavity edge by SH-wave with different incident angles

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图 10给出了SH波水平入射时圆孔周边电场强度系数随λ的变化情况。由图 10可知，E_θ^*随压电参数λ的增大而减小，当λ=0.2时，E_θ^*达到最大值6.2，因此工程中需要注意λ取值较小的情况。

图 10 SH波水平入射时圆孔周边电场强度系数随λ的变化情况

Figure 10. EFICF around circular cavity edge vs. λ by horizontal SH-wave

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图 11给出了SH波水平入射时圆孔周边电场强度系数随β的变化情况。由图 11可知，E_θ^*随幂次β增大而增大，与图 7中τ_θz^*变化规律一致。当β=4时，E_θ^*达到最大值3.1。

图 11 SH波水平入射时圆孔周边电场强度系数随β变化情况

Figure 11. EFICF around circular cavity edge vs. β by horizontal SH-wave

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图 12给出了SH波水平入射时圆孔θ=π/2处电场强度系数随波数ka的变化情况。由图 12可知：当ka < 0.2时，E_θ^*无明确的变化规律；当ka>0.2时，E_θ^*随ka的增大而减小。

图 12 SH波水平入射时圆孔θ=π/2处电场强度系数随波数ka变化情况

Figure 12. EFICF at the circular cavity edge vs. ka by horizontal SH-wave

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6. 结论

利用复变函数理论，对径向非均匀压电介质中圆孔对SH波的散射问题进行了研究。结果表明，SH波低频垂直入射对径向非均匀压电介质破坏较大；高频入射时，压电参数λ对τ_θz^*几乎没影响，但E_θ^*随λ的减小而增大，τ_θz^*与E_θ^*均随幂次β的增大而增大。另外，SH波水平入射时，τ_θz^*随ka的减小而增大，当ka>0.2时，E_θ^*也随ka的减小而增大。在实际工程中应该对这些规律引起注意，以避免非均匀压电介质发生破坏。

图 1 矩形泄放容器

Figure 1. Rectangular vented vessel

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图 2 当φ=20%时容器内外火焰演化过程

Figure 2. Flame evolution inside and outside the vessel for φ=20%

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图 3 当φ=0%时点火后火焰锋面的位置和速度

Figure 3. Flame front locations and velocities after ignition for φ=0%

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图 4 容器内部压力曲线

Figure 4. Internal pressure curves of the vessel

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图 5 容器最大内部超压和氮气体积分数的关系

Figure 5. Relations between maximum internal overpressures of the vessel and nitrogen addition ratios

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图 6 当φ=0%时容器内外压力曲线

Figure 6. Internal and external pressure curves of the vessel for φ=0

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图 7 当φ=20%, 40%时容器外部压力曲线

Figure 7. External pressure curves of the vessel for φ=20%, 40%

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图 8 当φ=40%时容器外部火焰演变过程

Figure 8. External flame evolution of the vessel for φ=40%

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图 9 容器最大外部超压与氮气体积分数的关系

Figure 9. Relations between maximum external overpressures of the vessel and nitrogen addition ratios

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图 10 不同氮气体积分数下最大外部超压峰值时的火球

Figure 10. Fireballs at maximum external overpressure under different nitrogen addition ratios

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图 11 容器外部爆炸对内部压力的影响

Figure 11. Effects of external explosion on internal pressure of the vessel

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1. 控制方程
2. 介质中的位移场
3. 边界条件与定解方程
4. 动应力集中系数与电场强度系数
5. 算例分析
6. 结论

泄爆和氮气惰化耦合作用对氢-空气爆炸影响的实验研究

doi: 10.11883/bzycj-2021-0459

作者简介: 张 凯（1996－ ），男，硕士研究生，17318532395@163.com

通讯作者: 洪溢都（1989－ ），男，博士，讲师，yidu.hong@fzu.edu.cn

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