一维对流-扩散试验各种边界条件及其统一形式解析解

Boundary conditions of one-dimensional convection-diffusion column tests and unified analytical solution

黏土系材料广泛应用于各种污染防治工程,扩散系数是其关键参数之一,实验室一般通过拟合一维土柱对流-扩散试验数据来确定土的扩散系数。探讨了常用的零浓度边界、零浓度梯度边界和半无限边界对应的试验条件,给出了各边界条件的统一形式,通过引入变换式和辅助问题,求解得到各边界条件下统一的有限厚度土层中对流-扩散-吸附解析解,分析了各解析解之间的差异和用于数据拟合产生的误差。结果表明,每一种底边界条件都与土柱试验底部的实际情况相对应,柯西边界可看作各种边界条件的统一形式;土柱未被击穿时各边界条件解析解无差别;土柱击穿后,越靠近土柱底面,各种边界条件解析解计算结果差异越大,底面处浓度值随柯西参数?/?的增大而减小,?/?越大,代表底边界处扩散在溶质迁移中所占比例越大,同时底部浓度梯度越大;算例中对底部为半无限边界的试验数据进行拟合,若采用零浓度边界和零浓度梯度边界,计算结果分别偏大15%和偏小9%,根据土柱底部的对流-扩散比例和浓度梯度具体情况选择相应边界条件下的解析解,是保证数据拟合正确性的基本要求。

Clay-based materials are widely used in practice of contaminant control. Diffusion coefficient is one of the most important parameters of clay. It is usually determined by fitting the data of one-dimensional convection-diffusion soil column tests. Zero concentration boundary, zero concentration-gradient boundary and semi-infinite boundary are discussed. A unified boundary condition is proposed. By transforming and introducing auxiliary problems, unified convection-diffusion-adsorption analytical solutions for three boundary conditions are acquired. Both difference between these analytical solutions and data-fitting errors are investigated. The results show that different boundaries should correspond to different conditions in the bottom of a column test. Cauchy boundary can be regarded as an unified form of all these boundaries. The calculated results corresponding to these boundaries are equivalent before breakthrough, however, the results become quite different in the bottom of the column after breakthrough. The concentration decreases as Cauchy parameter μ/λ increases. The bigger the Cauchy parameter μ/λ is, the more dominant role the diffusion effect plays in the solute transport process, at the same time, the bigger the concentration gradient at the bottom is. Fitting the data of a semi-infinite boundary test using analytical solutions of zero concentration boundary and zero concentration-gradient boundary will overestimate 15% and underestimate 9% of the result, respectively. It is essential for data fitting to choose an analytical solution according to convection-diffusion proportion and concentration-gradient at the bottom of a soil column.