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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (5): 588-592.  DOI: 10.11908/j.issn.0253-374x.2018.05.004 0

### 引用本文

LI Jingpei, YAO Mingbo. Analytical Solution for Sulfate Diffusion Reaction in Circular Concrete Piles[J]. Journal of Tongji University (Natural Science), 2018, 46(5): 588-592. DOI: 10.11908/j.issn.0253-374x.2018.05.004

### 文章历史

1. 同济大学 土木工程学院，上海 200092;
2. 同济大学 岩土及地下工程教育部重点实验室，上海 200092

Analytical Solution for Sulfate Diffusion Reaction in Circular Concrete Piles
LI Jingpei 1,2, YAO Mingbo 1,2
1. College of Civil Engineering, Tongji University, Shanghai 200092, China;
2. Key Laboratory of Geotechnical and Underground Engineering of the Ministry of Education, Tongji University, Shanghai 200092, China
Abstract: Based on the Fick's second law, a diffusion reaction equation of sulfate in concrete piles was established. With the method of separation of variables and Danckwerts, according to initial conditions and boundary conditions, the diffusion reaction equation was deduced. Furthermore, considering the effect of pore filling and crack on diffusion coefficient, an effective diffusion coefficient model was proposed to show the variation of the diffusion coefficient during the degradation process of concrete piles. Sulfate concentration profiles were obtained by the solution and the effective diffusion coefficient model, which agrees well with the experiment data to verify the validity of the proposed analytical solution. Case study shows that the inhibition effect of pore filling on sulfate diffusion is remarkable. The development of the crack promotes the diffusion reaction of sulfate. With water-cement ratio decreasing, the invasion depth of sulfate decreases, as well as the concentration of sulfate in concrete piles.
Key words: concrete pile    sulfate    diffusion reaction    crack    pore filling

1 径向扩散反应方程解析解

 $\frac{{\partial \rho }}{{\partial t}} = {D_{{\rm{eff}}}}\left( {\frac{{{\partial ^2}\rho }}{{\partial {r^2}}} + {r^{ - 1}}\frac{{\partial \rho }}{{\partial r}}} \right) - v\rho$ (1)

 $\left\{ \begin{array}{l} \rho \left( {r,0} \right) = {\rho _0},0 \le r < {r_0}\\ \rho \left( {{r_0},t} \right) = {\rho _{\rm{s}}},t > 0 \end{array} \right.$ (2)

 $\frac{{\partial \rho }}{{\partial t}} = {D_{{\rm{eff}}}}\left( {\frac{{{\partial ^2}\rho }}{{\partial {r^2}}} + {r^{ - 1}}\frac{{\partial \rho }}{{\partial r}}} \right)$ (3)

 ${\rho _1}\left( {r,t} \right) = {\rho _2}\left( {r,t} \right) + {\rho _{\rm{s}}}$ (4)

 $\frac{{\partial {\rho ^2}}}{{\partial t}} = {D_{{\rm{eff}}}}\left( {\frac{{{\partial ^2}{\rho _2}}}{{\partial {r^2}}} + {r^{ - 1}}\frac{{\partial {\rho _2}}}{{\partial r}}} \right)$ (5)

 $\left\{ \begin{array}{l} {\rho _2}\left( {r,0} \right) = {\rho _0} - {\rho _{\rm{s}}},0 \le r < {r_0}\\ {\rho _2}\left( {{r_0},t} \right) = 0,t > 0 \end{array} \right.$ (6)

ρ2=T(t)R(r)，T(t)是关于t的函数，R(r)是关于r的函数，代入式(5)可得

 $\frac{{T'}}{{{D_{{\rm{eff}}}}T}} = \frac{{\Delta R}}{R}$ (7)

 $\left\{ \begin{array}{l} \frac{{{\rm{d}}T\left( t \right)}}{{{\rm{d}}t}} + {D_{{\rm{eff}}}}{\beta ^2}T\left( t \right) = 0\\ \frac{{{{\rm{d}}^2}R\left( r \right)}}{{{\rm{d}}{r^2}}} + {r^{ - 1}}\frac{{{\rm{d}}R\left( r \right)}}{{{\rm{d}}r}} + {\beta ^2}R\left( r \right) = 0 \end{array} \right.$ (8)

 $T\left( t \right) = {c_1}{{\rm{e}}^{ - {\beta ^2}{D_{{\rm{ef}}{{\rm{f}}^t}}}}}$ (9)

 $R\left( r \right) = {A_0}{{\rm{J}}_0}\left( {\beta r} \right) + {B_0}{{\rm{Y}}_0}\left( {\beta r} \right)$ (10)

 ${\rho _2}\left( {r,t} \right) = T\left( t \right)R\left( r \right) = \left[ {A{{\rm{J}}_0}\left( {\beta r} \right) + B{{\rm{Y}}_0}\left( {\beta r} \right)} \right]{{\rm{e}}^{ - {\beta ^2}{D_{{\rm{ef}}{{\rm{f}}^t}}}}}$

 $\left\{ \begin{array}{l} A{{\rm{J}}_0}\left( {\beta r} \right) + B{{\rm{Y}}_0}\left( {\beta r} \right) = {\rho _0} - {\rho _{\rm{s}}}\\ A{{\rm{J}}_0}\left( {\beta {r_0}} \right) + B{{\rm{Y}}_0}\left( {\beta {r_0}} \right) = 0 \end{array} \right.$ (11)

 ${\rho _2}\left( {r,t} \right) = \sum\limits_{n = 1}^\infty {{{\left( {{\rho _2}} \right)}_n}} = \sum\limits_{n = 1}^\infty {{A_n}{{\rm{e}}^{ - {D_{{\rm{eff}}}}\beta _n^2t}}{{\rm{J}}_0}\left( {{\mu _n}r/{r_0}} \right)}$ (12)

 $\begin{array}{l} \int_0^{{r_0}} {r\left( {{\rho _0} - {\rho _{\rm{s}}}} \right){{\rm{J}}_0}\left( {{\mu _m}r/{r_0}} \right){\rm{d}}r} = \\ \;\;\;\;\;\;\;\sum\limits_{n = 1}^\infty {{A_n}\int_0^{{r_0}} {r{{\rm{J}}_0}\left( {{\mu _n}r/{r_0}} \right){{\rm{J}}_0}\left( {{\mu _m}r/{r_0}} \right){\rm{d}}r} } = \\ \;\;\;\;\;\;\;{A_m}r_0^2{J_0}\left( {{\mu _m}} \right)/2 \end{array}$ (13)

 ${A_m} = \frac{{2\left( {{\rho _0} - {\rho _{\rm{s}}}} \right)}}{{{\beta _m}{r_0}{{\rm{J}}_1}\left( {{\beta _m}{r_0}} \right)}},m = 1,2, \cdots$ (14)

 $\begin{array}{l} {\rho _2}\left( {r,t} \right) = \sum\limits_{n = 1}^\infty {{{\left( {{\rho _2}} \right)}_n}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;2\left( {{\rho _0} - {\rho _{\rm{s}}}} \right)\sum\limits_{n = 1}^\infty {\frac{{{{\rm{J}}_0}\left( {{\mu _n}r/{r_0}} \right)}}{{{\beta _n}{r_0}{{\rm{J}}_1}\left( {{\beta _n}{r_0}} \right)}}{{\rm{e}}^{ - {D_{{\rm{eff}}}}\beta _n^2t}}} \end{array}$ (15)

 $\begin{array}{l} {\rho _1} = {\rho _2}\left( {r,t} \right) + {\rho _{\rm{s}}} = \\ \;\;\;\;\;\;\;{\rho _{\rm{s}}} + 2\left( {{\rho _0} - {\rho _{\rm{s}}}} \right)\sum\limits_{n = 1}^\infty {\frac{{{{\rm{J}}_0}\left( {{\mu _n}r/{r_0}} \right)}}{{{\beta _n}{r_0}{{\rm{J}}_1}\left( {{\beta _n}{r_0}} \right)}}{{\rm{e}}^{ - {D_{{\rm{eff}}}}\beta _n^2t}}} \end{array}$ (16)

 $\rho = {\rho _{\rm{s}}} + 2\left( {{\rho _0} - {\rho _{\rm{s}}}} \right)\sum\limits_{n = 1}^\infty {\frac{{{{\rm{J}}_0}\left( {{\mu _n}r/{r_0}} \right)}}{{{\beta _n}{r_0}{{\rm{J}}_1}\left( {{\beta _n}{r_0}} \right)}}F\left( t \right)}$ (17)

 $F\left( t \right) = \frac{v}{{{D_{{\rm{eff}}}}\beta _n^2 + v}} + \frac{{{D_{{\rm{eff}}}}\beta _n^2}}{{{D_{{\rm{eff}}}}\beta _n^2 + v}}{{\rm{e}}^{ - \left( {{D_{{\rm{eff}}}}\beta _n^2 + v} \right)t}}$ (18)

2 硫酸盐有效扩散系数

 ${D_{{\rm{eff}}}} = \frac{{{S_{\rm{c}}}{D_{\rm{c}}} + {S_{\rm{0}}}D}}{{{S_{\rm{c}}} + {S_{\rm{0}}}}}$ (19)

 $\varepsilon = \max \left[ {{\varphi _{\rm{c}}}\left( {\frac{{{m_{\rm{w}}}/{m_{\rm{c}}} - 0.36\alpha }}{{{m_{\rm{w}}}/{m_{\rm{c}}} + 0.32}}} \right),0} \right]$ (20)

 $\alpha = 1 - 0.5\left[ {{{\left( {1 + 1.67\tau } \right)}^{ - 0.6}} + {{\left( {1 + 0.29\tau } \right)}^{ - 0.48}}} \right]$ (21)

 ${D_{\rm{c}}} = \kappa b_{\rm{c}}^2,{b_{\rm{c}}} < {b_{{\rm{crit}}}}$ (22)
 ${D_{\rm{c}}} = \kappa {b_{{\rm{crit}}}}{b_{\rm{c}}},{b_{\rm{c}}} \ge {b_{{\rm{crit}}}}$ (23)

 ${D_{\rm{c}}} = \left\{ \begin{array}{l} \kappa b_{\rm{c}}^2,0 < {b_{\rm{c}}} < {b_{{\rm{crit}}}}\\ \kappa {b_{{\rm{crit}}}}{b_{\rm{c}}},{b_{{\rm{crit}}}} \le {b_{\rm{c}}} < 400\;{\rm{ \mathit{ μ} m}}\\ {D_{{\rm{free}}}} = {10^{ - 9}}{m^2} \cdot {{\rm{s}}^{ - 1}},{b_{\rm{c}}} \ge 400\;{\rm{ \mathit{ μ} m}} \end{array} \right.$ (24)

 ${S_0} = {\rm{ \mathsf{ π} }}{r_0} - {b_{\rm{c}}}$ (25)

 ${D_{{\rm{eff}}}} = D + \frac{{{b_{\rm{c}}}\left( {{D_{\rm{c}}} - D} \right)}}{{{\rm{ \mathsf{ π} }}{r_0}}}$ (26)
3 模型验证与分析

3.1 硫酸盐扩散反应解析验证

 图 1 硫酸盐质量分数分布理论解析解与试验数据的比较 Fig.1 Comparison of sulfate mass fraction distribution between analytical results and experimental data
3.2 影响因素分析 3.2.1 孔隙和裂缝宽度

 图 2 裂缝宽度对有效扩散系数的影响 Fig.2 Effect of crack width on effective diffusion coefficient

 图 3 考虑孔隙填充和裂缝影响的硫酸盐质量分数分布 Fig.3 Sulfate mass fraction distribution considering the effect of pore filling and crack
3.2.2 水灰比

 图 4 水灰比对混凝土桩中硫酸盐质量分数分布的影响 Fig.4 Effect of water-cement ratio on sulfate mass fraction distribution in concrete piles
4 结论

(1) 本文以硫酸盐侵蚀混凝土桩的微观机理为基础，通过Fick第二定律建立了硫酸盐在圆形混凝土桩中的扩散反应方程.根据初始条件和边界条件，采用分离变量法和Danckwerts法求解出扩散反应方程解析解.

(2) 本文提出了混凝土劣化全过程有效扩散系数模型，综合考虑了侵蚀产物孔隙填充和侵蚀产物膨胀导致混凝土开裂的2种侵蚀作用对混凝土有效扩散系数的影响.侵蚀产物对孔隙的填充阻碍了硫酸盐的扩散，膨胀产物使混凝土开裂又加速了硫酸盐的扩散侵入.考虑混凝土的孔隙填充与损伤开裂对硫酸盐的扩散影响可以更加准确地反映硫酸盐在混凝土中的侵蚀过程.

(3) 混凝土水灰比的设计对混凝土抗硫酸盐侵蚀影响显著.水灰比越小，混凝土中的硫酸盐质量分数分布越低，硫酸盐的侵入深度也越浅，有效地延缓了硫酸盐对混凝土的侵蚀.

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