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 应用科技  2018, Vol. 45 Issue (5): 108-113  DOI: 10.11991/yykj.201712012 0

### 引用本文

GUO Yu, YAO Shiwei, LI Bangming, et al. Calculation of the leak rate of through-wall crack in pressurized water reactor nuclear power plant[J]. Applied Science and Technology, 2018, 45(5), 108-113. DOI: 10.11991/yykj.201712012.

### 文章历史

Calculation of the leak rate of through-wall crack in pressurized water reactor nuclear power plant
GUO Yu, YAO Shiwei, LI Bangming, LI Shaodan, LI Yong
Science and Technology on Thermal Energy and Power Lab. Wuhan Second Ship Design and Research Institute, Wuhan 430205, China
Abstract: Based on the analysis of leak before break of pressurized water reactor pipeline in nuclear power plant, the through-wall crack leak rate was calculated and analyzed. The flow characteristics of crack channel were analyzed by numerical simulation, and the influence of crack morphology on the leakage rate of crack was studied. A calculation procedure for the crack leakage rate in pressurized water reactor pipeline was established based on the Henry-Fauske critical flow theory. The calculation results were compared with the experimental data, showing that the computation error of the program was within 30%. The influence of different parameters on leakage rate was studied on the basis of the leak rate calculation program, which are instructive on leak rate of through-wall crack in pressurized water reactor.
Keywords: pressurized water reactor    through-wall crack    leak before break    leak rate calculation    crack morphology    crack opening displacement    roughness    critical flow

1984年由美国电力研究院发起的PICEP项目对LEAK-01程序进行了修正，基于Henry均相非平衡临界流模型，形成可用于预测裂纹泄漏率的程序PICEP。此后，美国核管会通过IPIRG项目资助了管道泄漏相关研究，开发出用于预测管道泄漏率的SQUIRT程序。SQUIRT程序使用Henry-Fauske临界流理论，考虑了液体流过裂纹时的非均匀相变过程[4]。随着我国核电发展对相关技术的需求，国内相关单位开展了一系列关于泄漏率预测的研究，形成了诸如PICLES、CLR和COLROPC等程序[5-8]。由于贯穿裂纹内的临界流动特性及其不规则形状对裂纹流道内的流体流动有直接影响，也决定了泄漏率的大小，然而关于其影响关系尚缺乏深入分析。

1 泄漏率计算模型

1.1 裂纹几何形貌描述

 $\mu = \left\{ \begin{array}{l}{\mu _L},\;0 < \displaystyle\frac{\delta }{{{\mu _L}}} < 0.1\\{\mu _L} +V \displaystyle\frac{{{\mu _G} - {\mu _L}}}{{4.9}}\left( {\displaystyle\frac{\delta }{{{\mu _L}}} - 0.1} \right),\;0.1 < \frac{\delta }{{{\mu _L}}} < 5\\{\mu _G},\;\displaystyle\frac{\delta }{{{\mu _L}}} > 5\end{array} \right.$

 ${n_t} = \left\{ \begin{array}{l} {n_{tL}},\;0 <\displaystyle \frac{\delta }{{{\mu _L}}} < 0.1 \hfill \\ {n_{tL}} + \displaystyle\frac{{{n_{tL}}}}{{4.9}}\left( {\frac{\delta }{{{\mu _L}}} - 0.1} \right),\;0.1 < \frac{\delta }{{{\mu _L}}} < 5 \hfill \\ 0,\;\displaystyle\frac{\delta }{{{\mu _L}}} > 5 \hfill \\ \end{array} \right.$
1.2 贯穿裂纹临界流动模型

 $G_c^2 = {\left[ {\frac{{x{v_g}}}{{\gamma p}} - \left( {{v_g} - {v_l}} \right)N\frac{{{\rm{d}}{x_e}}}{{{\rm{d}}p}}} \right]^{ - 1}}$ (1)

 $x = N{x_e}\left\{ {1 - \exp \left[ { - B\left( {L/{D_{\rm{H}}} - 12} \right)} \right]} \right\}$ (2)

 N = \left\{ \begin{aligned}& 20{x_e} ,\;{x_e} < 0.05 \\& 1.0 ,\;{x_e} \geqslant 0.05 \\ \end{aligned} \right.

1.3 贯穿裂纹流体流动阻力模型

 ${p_o} - {p_c} = {p_e} + {p_a} + {p_f} + {p_k} + {p_{aa}}$ (3)

 ${p_e} = \frac{{G_o^2{v_{lo}}}}{{2C_D^2}}$

 ${p_f} = \left( {f\frac{L}{{{D_H}}}} \right)\frac{{G_c^2}}{2}\left[ {\left( {1 - \bar x} \right){{\bar v}_l} + \bar x\;{{\bar v}_g}} \right]$ (4)

 ${p_f} = f\left( {\frac{L}{{{D_H}}} - 12} \right)\frac{{G_c^2}}{2}\left[ {\left( {1 - \bar x} \right){{\bar v}_l} + \bar x\;{{\bar v}_g}} \right] + 12f\frac{{{{\bar G}_o}^2}}{2}{v_{lo}}$

 ${A_o}{G_o} = {A_c}{G_c}$

 ${p_f} = f\left( {\frac{L}{{{D_H}}} - 12} \right)\frac{{G_c^2}}{2}\left[ {\left( {1 - \bar x} \right){{\bar v}_l} + \bar x\;{{\bar v}_g}} \right] + 6f{\left( {\frac{{{A_c}}}{{{A_o}}}} \right)^2}G_o^2{v_{lo}}$

 $f = {\left[ {{C_1}\log \left( {\frac{{{D_H}}}{{2\mu }}} \right) + {C_2}} \right]^{ - 2}}$

 $\begin{array}{l}f = {\left[ {{C_1}\log \left( {{D_{\rm{H}}}/\left( {2k} \right)} \right) + {C_2}} \right]^{ - 2}} = \\\;\;\;\;\;\;\;\;\left[ {{C_1}\log \left( {{D_{\rm{H}}}/k} \right) + {C_2} - {C_1}\log 2} \right]\end{array}$

 ${D_{\rm{H}}} = \frac{{4 \times {\text{面积}}}}{{\text{湿周}}}$

 ${p_k} = \left( {{e_v}} \right)\frac{{G_c^2}}{2}\left[ {\left( {1 - \bar x} \right){{\bar v}_l} + \bar x\;{{\bar v}_g}} \right]$

 ${e_v} = e\left[ L \right]$

$\overline G$ 是平均质量流量，即

 $\overline G = \frac{{{A_o}{G_o} + {A_c}{G_c}}}{{{A_o} + {A_c}}}$

 ${p_a} = \overline G{_T^2}\left[ {\left( {1 - {x_c}} \right){v_{lc}} + {x_c}{v_{gc}} - {v_{lo}}} \right]$

 $\begin{array}{l}{p_{aa}} = \displaystyle\frac{{G_c^2{v_{lo}}}}{2}\left[ {{{\left( {\frac{{{A_c}}}{{{A_i}}}} \right)}^2} - {{\left( {\displaystyle\frac{{{A_c}}}{{{A_o}}}} \right)}^2}} \right] + \\ \;\;\;\;\;\;\;\;\;\displaystyle\frac{{G_c^2}}{2}\left[ {\left( {1 - x} \right){v_{lc}} + x{v_{gc}}} \right] \times \left[ {1 - {{\left( {\frac{{{A_c}}}{{{A_i}}}} \right)}^2}} \right] \\ \end{array}$

2 泄漏率计算流程

 $\mathit{\Phi }\left( {G_c^2,{p_c}} \right) = G_c^2 - \left[ {\frac{{{x_c}{v_{gc}}}}{{\gamma {p_c}}} - \left( {{v_{gc}} - {v_{lc}}} \right)N\frac{{{\rm{d}}{x_E}}}{{{\rm{d}}p}}} \right]_c^{ - 1}$
 $\mathit{\Omega }\left( {G_c^2,{p_c}} \right) = \Delta {p_{co}} + G_c^2\left( {{B_{{p_e}}} + {B_{{p_a}}} + {B_{{p_f}}} + {B_{{p_k}}} + {B_{{p_{aa}}}}} \right)$

 ${B_{{p_e}}} = \frac{{{v_{lo}}}}{{2C_D^2}}$
 ${B_{{p_a}}} = \left( {1 - {x_c}} \right){v_{lc}} + {x_c}{v_{gc}} - {v_{lo}}$
 ${B_{{P_f}}} = \frac{f}{2}\left( {\frac{L}{{{D_H}}} - 12} \right)\left[ {\left( {1 - \bar x} \right){{\bar v}_l} + \bar x\;{{\bar v}_g}} \right] + 6f{\left( {\frac{{{A_c}}}{{{A_o}}}} \right)^2}{v_{lo}}$
 ${B_{{P_k}}} = \frac{{\left( {{e_v}} \right)}}{2}\left[ {\left( {1 - \bar x} \right){{\bar v}_l} + \bar x\;{{\bar v}_g}} \right]$
 $\begin{array}{l}{B_{{P_{aa}}}} = \displaystyle\frac{{{v_{lo}}}}{2}\left[ {{{\left( {\frac{{{A_c}}}{{{A_i}}}} \right)}^2} - {{\left( {\frac{{{A_c}}}{{{A_o}}}} \right)}^2}} \right] + \displaystyle\frac{{\left[ {\left( {1 - x} \right){v_{lc}} + x{v_{gc}}} \right]}}{2} \times \left[ {1 - {{\left( {\displaystyle\frac{{{A_c}}}{{{A_i}}}} \right)}^2}} \right] \\ \end{array}$

 ${\left[ {G_c^2} \right]_{n + 1}} = {\left[ {G_c^2} \right]_n} + {h_G}$
 ${\left[ {{p_c}} \right]_{n + 1}} = {\left[ {{p_c}} \right]_n} + {h_p}$

 ${h_G} = - \left| {\begin{array}{*{20}{c}}{{\mathit{\Phi }_n}}&{{{\left( {\displaystyle\frac{{\partial \mathit{\Phi }}}{{\partial {p_c}}}} \right)}_n}}\\{{\mathit{\Omega }_n}}&{{{\left( {\displaystyle\frac{{\partial \mathit{\Omega }}}{{\partial {p_c}}}} \right)}_n}}\end{array}} \right|/{\left( {\displaystyle\frac{{\partial \left( {\mathit{\Phi },\mathit{\Omega }} \right)}}{{\partial \left( {G_c^2,{p_c}} \right)}}} \right)_n}$
 ${h_p} = - \left| {\begin{array}{*{20}{c}}{{{\left( {\displaystyle\frac{{\partial \mathit{\Phi }}}{{\partial G_c^2}}} \right)}_n}}&{{\mathit{\Phi }_n}}\\{{{\left( {\displaystyle\frac{{\partial \mathit{\Omega }}}{{\partial G_c^2}}} \right)}_n}}&{{\mathit{\Omega }_n}}\end{array}} \right|/{\left( {\frac{{\partial \left( {\mathit{\Phi },\mathit{\Omega }} \right)}}{{\partial \left( {G_c^2,{p_c}} \right)}}} \right)_n}$

 ${\left. {\frac{{\partial \left( {\mathit{\Phi },\mathit{\Omega }}\right)}}{{\partial \left( {G_c^2,{p_c}} \right)}}} \right|_n} = {\left[ {\frac{{\partial \mathit{\Phi }}}{{\partial G_c^2}}\frac{{\partial \mathit{\Omega }}}{{\partial {p_c}}} - \frac{{\partial \mathit{\Phi }}}{{\partial {p_c}}}\frac{{\partial \mathit{\Omega }}}{{\partial G_c^2}}} \right]_n}$

3 计算结果与实验值对比

4 裂纹参数对泄漏率的影响