变弥散系数和对流速度下有机溶质在多孔介质中一维运移的解析解

变弥散系数和对流速度下有机溶质在多孔介质中一维运移的解析解 Analytical solution for one dimensional transport of organic solute in porous media with variable dispersion coefficient and variable advection velocity 1 2 3 first-author 1 3 corresp 李江山，博士，研究员，从事污染场地工程修复和固体废弃物资源化研究；E-mail：jsli@whrsm.ac.cn 李江山，博士，研究员，从事污染场地工程修复和固体废弃物资源化研究；E-mail：jsli@whrsm.ac.cn jsli@whrsm.ac.cn 1 3 1 2 3 中国科学院武汉岩土力学研究所 岩土力学与工程国家重点实验室，湖北 武汉，430071 中国科学院武汉岩土力学研究所 岩土力学与工程国家重点实验室，湖北 武汉，430071 The State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China The State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China 中国科学院大学，北京，100049 中国科学院大学，北京，100049 The University of Chinese Academy of Sciences, Beijing 100049, China The University of Chinese Academy of Sciences, Beijing 100049, China 污染泥土科学与工程湖北省重点实验室，湖北 武汉，430071 污染泥土科学与工程湖北省重点实验室，湖北 武汉，430071 Key Laboratory of Contaminated Sludge and Soil Science and Engineering of Hubei Province, Wuhan 430071, China Key Laboratory of Contaminated Sludge and Soil Science and Engineering of Hubei Province, Wuhan 430071, China 伍锦花 为研究变弥散系数和对流速度对有机溶质在多孔介质中运移规律的影响，考虑有机溶质的降解作用，设定多孔介质的厚度为有限厚度，建立变弥散系数和对流速度下有机溶质一维运移的数学模型，并利用变量代换和分离变量法获得该数学模型的解析解。将本文所提解析解分别与试验测定结果、已有解析解和有限差分解进行对比分析，对所提解析解的正确性进行验证。基于所提解析解，分析非均质特性系数 δ 、降解半衰期 t 1/2 和Robin边界常数 α 对有机溶质一维运移过程的影响。研究结果表明：一方面，非均质特性系数 δ 的增大会在一定程度上加快有机溶质的一维运移过程；另一方面， δ 增大使得运移后期有机溶质的质量浓度减小；有机溶质质量浓度随降解半衰期 t 1/2 增大而增大，当 t 1/2 不超过800 a时，应考虑有机溶质的降解作用；Robin边界常数 α 主要影响出流边界附近有机溶质质量浓度的变化；在两种不同入流边界条件下，出流边界附近的有机溶质的质量浓度均随 α 增大而减小。 To investigate the effects of variable dispersion coefficient and variable advection velocity on the transport process of organic solutes in porous media, a mathematical model for one dimensional transport of organic solutes with variable dispersion coefficient and variable advection velocity was established, where the degradation of organic solutes was considered and the thickness of porous media was set as a finite thickness. The analytical solution for the mathematical model was obtained by using the methods of variable substitution and variable separation. By comparing the analytical solution proposed in this paper with the experimental results, the existing analytical solution as well as the finite difference solution, and the correctness of the proposed analytical solution was verified. Based on the proposed analytical solution, the effects of heterogeneous characteristic coefficient δ , degradation half-life t 1/2 and Robin boundary constant α on the one-dimensional transport process of organic solutes were analyzed. The results show that on one hand, the increase of heterogeneous characteristic coefficient δ will accelerate the one dimensional transport process of organic solutes to a certain extent, and on the other hand, it will reduce the mass concentration of organic solutes in the later stage. The mass concentration of organic solutes will increase with the increase of degradation half-life t 1/2 . When t 1/2 does not exceed 800 a, the degradation of organic solutes should be considered. The Robin boundary constant α mainly affects the variation of organic solutes mass concentration near the outflow boundary. Under two different inflow boundary conditions, the mass concentration of organic solutes near the outflow boundary will decrease with the increase of α . 多孔介质 一维运移 弥散系数 对流速度 有机溶质 解析解 porous media one-dimensional transport dispersion coefficient advection velocity organic solute analytical solution http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681949&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681931&type= 14.95016479 14.95016479 对于污染物、盐分等溶质，其在多孔介质(如岩土体)中的运移过程往往会受到对流、分子扩散、弥散、吸附等多种因素影响。为分析溶质在多孔介质中的一维运移规律，学者们通常采用对流-弥散方程来描述这一过程，并展开了大量的试验和理论研究 [ 1 1 - 14 14 1-14 ] 。 在溶质的对流-弥散方程中，弥散系数和对流速度是反映溶质一维运移速度的重要参数。在已有研究中，国内外学者假定弥散系数和对流速度在运移过程中保持不变，获得了一系列解析解 [ 15 15 - 23 23 15-23 ] 。SHACKELFORD [ 15 15 ] 通过假定弥散系数和对流速度为常数，求解得到了不同入流边界条件下溶质一维运移的解析解；ZISKIND等 [ 16 16 ] 研究了脉冲入流条件下溶质的一维运移问题，并基于拉普拉斯变换获得了溶质一维运移的解析解。余闯等 [ 18 18 ] 考虑到有机溶质在运移过程中会发生降解，对考虑降解情况下的一维运移问题进行了研究，并得到了相应解析解。张文杰等 [ 19 19 ] 考虑了底部边界条件对溶质运移行为的影响，推导得到了弥散系数和对流速度为常数时溶质一维运移解析解。XIE等 [ 20 20 , 22 22 ] 在研究中考虑了热扩散作用，并通过相关假定获得了溶质一维运移解析解。LI等 [ 21 21 ] 基于溶质在防渗墙体中一维运移的解析解，绘制了与溶质一维运移过程相关的设计图表。然而，多孔介质往往具有非均质性，且溶质在运移过程中会与多孔介质发生物理、化学反应等，这使得溶质在运移过程中的弥散系数和对流速度具有空间和时间相关性 [ 8 8 , 24 24 - 36 36 24-36 ] 。 近年来，许多学者对考虑变弥散系数和对流速度下溶质的一维运移问题展开了研究，并求解得到了不同情况下溶质一维运移的解析解 [ 8 8 , 24 24 - 32 32 24-32 ] 。YATES [ 24 24 ] 通过假定弥散系数与运移距离呈不同的函数关系，对常浓度和常通量入流边界条件下溶质的一维运移问题进行了求解；ZOPPOU等 [ 25 25 ] 假设对流速度与运移距离呈线性关系，获得了半无限下边界条件下溶质一维运移模型的解析解；CHEN等 [ 26 26 ] 假定弥散系数与运移距离呈双曲渐近函数关系，运用拉普拉斯变换得到了溶质一维运移的解析解；KUMAR等 [ 28 28 ] 假定弥散系数和对流速度分别与运移距离呈特定的函数关系，获得了不同入流边界下溶质一维运移的解析解；YADAV等 [ 29 29 ] 假定弥散系数和对流速度与运移距离呈指数函数关系，获得了变系数条件下对流-弥散方程的解析解；魏峰等 [ 8 8 ] 对脉冲入流条件下溶质的一维运移问题进行了研究，并通过拉普拉斯变换获得了相应的解析解；余闯等 [ 31 31 ] 考虑了弥散系数随时间的变化，采用同伦分析法获得了考虑降解条件时溶质一维运移的近似解。但以上有关溶质在多孔介质中一维运移过程的研究大多涉及半无限下边界，而实际工程中多孔介质厚度是有限的，因此，有必要对有限厚度边界下溶质的一维运移问题展开研究 [ 34 34 - 41 41 34-41 ] 。 对于考虑变系数条件下有限厚度多孔介质中溶质的一维运移问题，已有学者开展了相关试验和理论研究 [ 33 33 - 43 43 33-43 ] 。在试验研究方面，PANG等 [ 34 34 ] 通过在较短时间内注入溶质，对变弥散系数条件下溶质在土柱中的一维运移过程进行了研究；LEE等 [ 36 36 ] 对不同边界条件下钾离子在土体中的运移问题进行了研究，并采用数值模拟方法对钾离子的一维运移过程进行了分析。在理论研究方面，KUMAR等 [ 37 37 - 38 38 37-38 ] 考虑到弥散系数和对流速度随运移距离发生变化，获得了不同入流边界条件下溶质一维运移的解析解；徐江伟等 [ 40 40 - 41 41 40-41 ] 假定弥散系数与运移距离呈线性关系，建立了考虑降解作用时有机溶质的一维运移模型，并求得了相应的解析解；YU等 [ 43 43 ] 考虑到弥散系数与时间呈不同函数关系，给出了考虑降解作用的有机溶质一维运移解析解。然而，由于变系数条件下有机溶质在多孔介质中的一维运移问题较为复杂，上述解析解中有关有机溶质一维运移的机理并不完整，如未同时考虑对流和降解等因素对运移过程的影响。此外，已有研究大多假定溶质一维运移的出流边界为零浓度梯度边界或零浓度边界，并未考虑出流边界为Robin边界类型的情况 [ 19 19 , 44 44 - 47 47 44-47 ] 。实际上，相比于零浓度梯度边界(Neumann边界)或零浓度边界条件(Dirichlet边界)，Robin边界更具一般性 [ 19 19 , 44 44 - 45 45 44-45 ] 。 为此，针对有机溶质在多孔介质中的一维运移问题，本文作者首先在上述研究基础上考虑有机溶质的降解作用，设定多孔介质的厚度为有限厚度，建立变弥散系数和对流速度下有机溶质一维运移的数学模型，并利用变量代换和分离变量法，获得该数学模型的解析解；然后，将本文所提解析解分别与试验测定结果、已有解析解和有限差分解进行对比分析，对所提解析解的正确性进行验证；最后，基于所提解析解，探讨相关参数变化对有机溶质一维运移过程的影响。 1 有机溶质一维运移的数学模型 1 对于变弥散系数和对流速度下有机溶质在多孔介质中的一维运移问题，参考JAISWAL等 [ 27 27 - 30 30 27-30 ] 的研究，在考虑线性吸附和一阶降解作用的条件下，有机溶质一维运移的对流-弥散方程可写为 <math display="block" id="M1"><mfrac><mrow><mo>∂</mo><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfrac><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mfrac><mrow><mo>∂</mo><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mo>-</mo><mi>v</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>-</mo><mi>λ</mi><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681175&type= 81.36466980 8.29733276 式中： <math id="M2"><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681752&type= 8.89000034 3.38666677 为多孔介质中有机溶质质量浓度； <math id="M3"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681178&type= 3.64066648 3.72533321 为因吸附作用而产生的阻滞因子； <math id="M4"><mi>D</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681182&type= 8.97466660 3.38666677 和 <math id="M5"><mi>v</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681200&type= 7.87400007 3.38666677 分别为有机溶质的弥散系数和对流速度； <math id="M6"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681202&type= 1.77800000 2.62466669 为有机溶质的降解速率， <math id="M7"><mi>λ</mi><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mn mathvariant="normal">2</mn><mo>/</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681197&type= 15.40933418 3.72533321 ， <math id="M8"><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681216&type= 4.06400013 3.72533321 为降解半衰期； <math id="M9"><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681219&type= 1.60866666 2.62466669 和 <math id="M10"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681205&type= 1.10066664 2.62466669 分别为运移距离和时间。 在溶质实际运移过程中，由于多孔介质的不均匀性，弥散系数和对流速度会随运移距离发生变化 [ 27 27 - 30 30 27-30 ] 。本文参考JAISWAL等 [ 27 27 - 30 30 27-30 , 37 37 - 38 38 37-38 ] 的研究结果，仅考虑弥散系数和对流速度随运移距离的变化。在一定厚度范围内，弥散系数和对流速度与运移距离的关系为 <math display="block" id="M11"><mi>D</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681221&type= 28.19400024 4.82600021 <math display="block" id="M12"><mi>v</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681225&type= 26.07733345 4.82600021 式中： <math id="M13"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681298&type= 1.77800000 2.62466669 为反映多孔介质中有机溶质一维运移的非均质特性系数； <math id="M14"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681230&type= 4.14866638 3.72533321 和 <math id="M15"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681212&type= 2.96333337 3.72533321 分别为入流边界处的弥散系数和对流速度。需指出的是， <math id="M16"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681298&type= 1.77800000 2.62466669 通常较小 [ 8 8 , 37 37 - 38 38 37-38 ] 。 将 式(2) 式(2) 和 式(3) 式(3) 代入 式(1) 式(1) ，则有： <math display="block" id="M17"><mfrac><mrow><mo>∂</mo><mi>C</mi></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfrac><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfrac><mrow><mo>∂</mo><mi>C</mi></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>z</mi><mo stretchy="false">)</mo><mi>C</mi></mrow></mfenced><mo>-</mo><mi>λ</mi><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681236&type= 78.82466125 8.46666718 根据上述关系式可知，在任意位置处，有机溶质一维运移的通量 <math id="M18"><mi>J</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681786&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681770&type= 8.38199997 3.38666677 为 <math display="block" id="M19"><mi>J</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfrac><mrow><mo>∂</mo><mi>C</mi></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>z</mi><mo stretchy="false">)</mo><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681258&type= 61.55266953 7.28133297 一般认为，溶质入流边界处的边界条件有浓度连续和通量连续两种形式 [ 11 11 , 15 15 , 17 17 , 27 27 - 30 30 27-30 ] 。考虑到入流边界处有机溶质的质量浓度会随时间发生变化 [ 43 43 , 48 48 - 49 49 48-49 ] ，基于浓度连续和通量连续的入流边界条件分别为 <math display="block" id="M20"><mi>C</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681318&type= 22.01333237 4.48733330 <math display="block" id="M21"><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>z</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681261&type= 56.47266769 10.24466610 式中： <math id="M22"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681264&type= 3.80999994 3.72533321 为入流边界处有机溶质的初始质量浓度； <math id="M23"><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681250&type= 5.67266655 3.38666677 为时间函数，反映了入流边界处有机溶质的质量浓度随时间变化， <math id="M24"><mi>f</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681251&type= 11.34533310 3.38666677 。 对于出流边界，设定多孔介质的厚度为 <math id="M25"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681272&type= 2.20133328 2.62466669 。采用更具一般性的Robin边界条件来描述出流边界条件，则有： <math display="block" id="M26"><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>z</mi><mo>=</mo><mi>L</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>α</mi><mi>C</mi><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681276&type= 33.95133591 8.89000034 式中： <math id="M27"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681257&type= 1.94733346 2.62466669 为大于0 m -1 的Robin边界常数， <math id="M28"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681257&type= 1.94733346 2.62466669 与出流边界处的环境有关 [ 19 19 , 44 44 - 47 47 44-47 ] 。当出流边界处存在纯水快速流动时，可认为出流边界处质量浓度为0 mg/L，此时 <math id="M29"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681897&type= 1.94733346 2.62466669 趋近于正无穷大，对应边界(Robin边界条件)可退化为Dirichlet边界条件 [ 12 12 ] ；当出流边界处为不透水层，出流边界处水流速度为0 m/s时，出流边界内、外的质量浓度相等，此时 <math id="M30"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681787&type= 13.29266548 3.04800010 ，对应边界(Robin边界条件)可退化为Neumann边界条件 [ 12 12 ] 。 在初始时刻，假定多孔介质中有机溶质的质量浓度为 <math id="M31"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681292&type= 7.70466709 4.48733330 ，则有： <math display="block" id="M32"><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681294&type= 20.40466690 4.48733330 式( 1 1 )~( 9 9 )为有限厚度多孔介质中有机溶质一维运移的数学模型，该数学模型考虑了多孔介质对有机溶质的吸附作用、有机溶质的降解作用、入流边界处有机溶质质量浓度的变化以及弥散系数和对流速度随运移距离的变化等因素，因而更加接近实际情况，也更具普遍性。 2 数学模型的解析解 1 为获得本文所建立的有机溶质一维运移数学模型的解析解，首先对控制方程的变量进行变量代换。令 <math id="M33"><mi>Z</mi><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681768&type= 18.71133423 3.38666677 (以下推导和分析中均假定 <math id="M34"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681298&type= 1.77800000 2.62466669 为非零常数)，将其代入 式(4) 式(4) ，则有： <math display="block" id="M35"><mfrac><mrow><mo>∂</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfrac><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681281&type= 42.07933044 8.72066593 <math display="block" id="M36"><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>δ</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>Z</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>δ</mi><mo>+</mo><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mi>λ</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfrac><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681284&type= 62.31467056 8.72066593 相应的初始条件和边界条件可写为 <math display="block" id="M37"><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mn mathvariant="normal">0</mn><mo>≤</mo><mi>Z</mi><mo>≤</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681286&type= 37.76133347 4.48733330 <math display="block" id="M38"><mi>C</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681318&type= 22.01333237 4.48733330 <math display="block" id="M39"><mfenced open="[" close="]" separators="|"><mrow><mi>C</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mi>δ</mi><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>Z</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>Z</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681320&type= 54.61000061 10.24466610 <math display="block" id="M40"><msub><mrow><mfenced open="" close="|" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>Z</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>Z</mi><mo>=</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mo>=</mo><mo>-</mo><mi>ϕ</mi><mo>⋅</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681322&type= 39.36999893 9.56733322 式中： <math id="M41"><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>L</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681324&type= 20.48933220 4.48733330 ； <math id="M42"><mi>ϕ</mi><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>L</mi><mo stretchy="false">)</mo><mo>/</mo><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681326&type= 20.65866661 3.38666677 。 为获得 式(10) 式(10) 的解析解，参考文献[ 48 48 ]中的求解方法，令 <math display="block" id="M43"><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681328&type= 37.08399963 3.38666677 式中： <math id="M44"><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681330&type= 9.22866726 3.38666677 和 <math id="M45"><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681332&type= 6.34999990 3.38666677 均为待定函数。 将 式(15) 式(15) 代入 式(10) 式(10) ，则 式(10) 式(10) 可转化为如下控制方程式： <math display="block" id="M46"><mi>A</mi><mfrac><mrow><msup><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mn mathvariant="normal">2</mn><mi>B</mi><mfrac><mrow><mi mathvariant="normal">d</mi><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">d</mi><mi>Z</mi></mrow></mfrac><mo>-</mo><mi>η</mi><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681334&type= 48.42933273 8.29733276 <math display="block" id="M47"><mi>A</mi><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mn mathvariant="normal">2</mn><mi>B</mi><mfrac><mrow><mo>∂</mo><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>Z</mi></mrow></mfrac><mo>-</mo><mi>η</mi><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681336&type= 81.02600098 8.29733276 式中： <math id="M48"><mi>A</mi><mo>=</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681338&type= 17.94933319 4.06400013 ； <math id="M49"><mi>B</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>δ</mi><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681340&type= 32.76599884 4.82600021 ； <math id="M50"><mi>η</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>δ</mi><mo>+</mo><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mi>λ</mi><mo stretchy="false">)</mo><mo>/</mo><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681342&type= 25.65399933 4.48733330 。 对于控制方程 式(16) 式(16) ，根据其形式可确定 <math id="M51"><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681709&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681708&type= 6.34999990 3.38666677 的通解为 <math display="block" id="M52"><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681307&type= 47.15933609 4.48733330 式中： <math id="M53"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681310&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681309&type= 2.70933342 3.72533321 和 <math id="M54"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681311&type= 2.70933342 3.72533321 为待定系数； <math id="M55"><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mroot><mrow><mrow><mrow><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow><mo>+</mo><mi>η</mi><mo>/</mo><mi>A</mi></mrow><mrow/></mroot><mo>-</mo><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mi>A</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681313&type= 38.26933289 5.58799982 ； <math id="M56"><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo>-</mo><mroot><mrow><mrow><mrow><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow><mo>+</mo><mi>η</mi><mo>/</mo><mi>A</mi></mrow><mrow/></mroot><mo>-</mo><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mi>A</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681315&type= 40.30133438 5.58799982 。 当入流边界为浓度连续条件时，根据边界条件 式(12) 式(12) 和 式(14) 式(14) ，可确定待定系数 <math id="M57"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681316&type= 2.70933342 3.72533321 和 <math id="M58"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681345&type= 2.70933342 3.72533321 的表达式为 <math display="block" id="M59"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681346&type= 63.41533279 8.21266651 <math display="block" id="M60"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681349&type= 63.41533279 8.21266651 当入流边界为通量连续条件时，根据边界条件 式(13) 式(13) 和 式(14) 式(14) ，可确定待定系数 <math id="M61"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681351&type= 2.70933342 3.72533321 和 <math id="M62"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681353&type= 2.70933342 3.72533321 的表达式为 <math display="block" id="M63"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>/</mo><mfenced open="[" close="" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>δ</mi><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>⋅</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681363&type= 66.12466431 4.74133301 <math display="block" id="M64"><mfenced open="" close="]" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>δ</mi><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681365&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681364&type= 75.69200134 4.74133301 <math display="block" id="M65"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>/</mo><mfenced open="[" close="" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>δ</mi><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681355&type= 67.14067078 4.74133301 <math display="block" id="M66"><mfenced open="" close="]" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>δ</mi><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681368&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681367&type= 75.69200134 4.74133301 采用分离变量法对 式(17) 式(17) 进行求解。根据文献[ 23 23 ]中的求解方法，可假定 <math id="M67"><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681638&type= 9.22866726 3.38666677 的形式为 <math display="block" id="M68"><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi>Z</mi><mo stretchy="false">)</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681370&type= 46.73600006 8.72066593 式中： <math id="M69"><mi>a</mi><mo>=</mo><mo>-</mo><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mi>A</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681372&type= 13.37733269 3.13266683 ； <math id="M70"><mi>m</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo>,</mo><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681374&type= 19.13466644 3.13266683 ； <math id="M71"><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681376&type= 8.89000034 4.48733330 和 <math id="M72"><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681358&type= 7.70466709 4.48733330 均为与 <math id="M73"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681534&type= 2.70933342 2.62466669 有关的待定函数。 与此同时，基于 式(23) 式(23) 的函数形式，利用级数变换可假定： <math display="block" id="M74"><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi>Z</mi><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681379&type= 40.80933380 8.72066593 式中： <math id="M75"><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681361&type= 4.14866638 3.72533321 为与 <math id="M76"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681534&type= 2.70933342 2.62466669 有关的待定常数。 将 式(23) 式(23) 和 式(24) 式(24) 代入 式(17) 式(17) ，并对控制变量进行变量分离，可得： <math display="block" id="M77"><msubsup><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><msup><mrow><mi>f</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681395&type= 38.86199951 4.82600021 <math display="block" id="M78"><msubsup><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mi>η</mi><mo>/</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681383&type= 50.37666702 4.82600021 式中： <math id="M79"><msubsup><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681397&type= 7.70466709 4.82600021 和 <math id="M80"><msup><mrow><mi>f</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681385&type= 6.85799980 3.72533321 分别为 <math id="M81"><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681400&type= 7.70466709 4.48733330 和 <math id="M82"><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681388&type= 5.67266655 3.38666677 的一阶导数； <math id="M83"><msubsup><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681390&type= 8.89000034 4.82600021 为 <math id="M84"><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681406&type= 8.89000034 4.48733330 的二阶导数； <math id="M85"><msub><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681694&type= 3.89466691 4.48733330 为与 <math id="M86"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681534&type= 2.70933342 2.62466669 有关的待定常数。 根据 式(25) 式(25) ，可得 <math id="M87"><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681411&type= 7.70466709 4.48733330 的表达式为 <math display="block" id="M88"><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>-</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>τ</mi><mo stretchy="false">)</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681415&type= 69.08799744 7.95866632 式中： <math id="M89"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681419&type= 1.43933344 2.62466669 为与时间 t 对应的虚拟变量； <math id="M90"><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681422&type= 4.31799984 3.72533321 为与 <math id="M91"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681534&type= 2.70933342 2.62466669 有关的待定常数。 根据 式(26) 式(26) 可知， <math id="M92"><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681631&type= 8.89000034 4.48733330 的表达式存在两种情况。 当 <math id="M93"><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681444&type= 26.07733345 4.82600021 时，有： <math display="block" id="M94"><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681432&type= 55.37200165 4.48733330 当 <math id="M95"><mrow><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mi>A</mi></mrow></mrow><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681435&type= 26.75466537 4.48733330 时，有： <math display="block" id="M96"><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681437&type= 51.56200027 4.48733330 式中： <math id="M97"><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mroot><mrow><mfenced open="|" close="|" separators="|"><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681454&type= 34.45933533 6.43466663 ； <math id="M98"><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681469&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681456&type= 3.64066648 3.72533321 、 <math id="M99"><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681458&type= 4.23333359 3.72533321 和 <math id="M100"><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681474&type= 4.14866638 3.72533321 为与 <math id="M101"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681534&type= 2.70933342 2.62466669 有关的待定常数。 同样，由于入流边界条件的不同，待定系数 <math id="M102"><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681479&type= 4.23333359 3.72533321 和 <math id="M103"><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681483&type= 4.14866638 3.72533321 的确定也存在两种情况。 当入流边界为浓度连续条件时，则根据边界条件 式(12) 式(12) 可知，无论是当 <math id="M104"><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681466&type= 26.07733345 4.82600021 时，还是当 <math id="M105"><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681499&type= 26.07733345 4.82600021 时，待定系数 <math id="M106"><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681502&type= 9.31333351 3.72533321 。 此时，根据边界条件 式(14) 式(14) ，可确定 <math id="M107"><msub><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681507&type= 3.89466691 4.48733330 的超越方程为 <math display="block" id="M108"><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo>+</mo><mi>a</mi><mo stretchy="false">)</mo><mtext>  </mtext><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo><</mo><mn mathvariant="normal">0</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681493&type= 63.07666779 4.82600021 <math display="block" id="M109"><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo>+</mo><mi>a</mi><mo stretchy="false">)</mo><mtext>  </mtext><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681511&type= 61.29866791 4.82600021 由于 <math id="M110"><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681530&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681529&type= 7.70466709 4.48733330 和 <math id="M111"><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681631&type= 8.89000034 4.48733330 中均存在与 <math id="M112"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681534&type= 2.70933342 2.62466669 有关的待定常数，为便于求解，这里可令 <math id="M113"><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681537&type= 9.39799976 3.72533321 ，从而确定待定函数 <math id="M114"><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681631&type= 8.89000034 4.48733330 的表达式。 当入流边界为通量连续条件时，则根据边界条件 式(13) 式(13) 和 式(14) 式(14) 可确定 <math id="M115"><msub><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681694&type= 3.89466691 4.48733330 的超越方程为 <math display="block" id="M116"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mfrac><mrow><mi>δ</mi><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mi>δ</mi><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mi>a</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681562&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681561&type= 46.05867004 9.39799976 式中： <math id="M117"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681550&type= 4.48733330 3.72533321 和 <math id="M118"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681554&type= 4.48733330 3.72533321 为与 <math id="M119"><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681556&type= 3.64066648 3.72533321 相关的系数，其表达式与 <math id="M120"><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681631&type= 8.89000034 4.48733330 的函数形式有关。 当 <math id="M121"><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681589&type= 26.07733345 4.82600021 时，有： <math display="block" id="M122"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo>+</mo><mi>a</mi><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681579&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681594&type= 58.50466156 4.48733330 <math display="block" id="M123"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo>+</mo><mi>a</mi><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681596&type= 58.50466156 4.48733330 当 <math id="M124"><mrow><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mi>A</mi></mrow></mrow><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681599&type= 26.75466537 4.48733330 时，有： <math display="block" id="M125"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo>+</mo><mi>a</mi><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681601&type= 54.69466782 4.48733330 <math display="block" id="M126"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo>+</mo><mi>a</mi><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681604&type= 54.69466782 4.48733330 当 <math id="M127"><msub><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681694&type= 3.89466691 4.48733330 确定后， <math id="M128"><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681624&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681612&type= 3.64066648 3.72533321 即可确定。同样，令 <math id="M129"><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681615&type= 9.39799976 3.72533321 ，则 <math id="M130"><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>/</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681617&type= 19.81200027 3.72533321 ，从而可确定待定函数 <math id="M131"><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681631&type= 8.89000034 4.48733330 的表达式。 考虑到 式(28) 式(28) 中包含的双曲正余弦函数并非周期性函数，因而无论对于哪种入流边界条件，满足 <math id="M132"><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681636&type= 26.07733345 4.82600021 条件下的 <math id="M133"><msub><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681694&type= 3.89466691 4.48733330 数量有限。根据上述关系式， <math id="M134"><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681638&type= 9.22866726 3.38666677 可写为 <math display="block" id="M135"><mi>w</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>c</mi><mi>o</mi><mi>s</mi><mi>h</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi>Z</mi><mo>-</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>t</mi><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681642&type= 75.94599915 9.31333351 <math display="block" id="M136"><mo stretchy="false">[</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>-</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>τ</mi><mo stretchy="false">)</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo stretchy="false">]</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>∞</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">[</mo><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>+</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681665&type= 71.20466614 9.31333351 <math display="block" id="M137"><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mi>a</mi><mi>Z</mi><mo>-</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>-</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>τ</mi><mo stretchy="false">)</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681670&type= 80.17933655 7.87400007 式中： <math id="M138"><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681674&type= 4.40266657 3.72533321 为满足 <math id="M139"><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mi>η</mi><mo stretchy="false">)</mo><mo>/</mo><mi>A</mi><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681693&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681680&type= 26.07733345 4.82600021 条件时 <math id="M140"><msub><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681694&type= 3.89466691 4.48733330 的数量；当入流边界为浓度连续条件时， <math id="M141"><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681686&type= 9.31333351 3.72533321 ；当入流边界为通量连续条件时， <math id="M142"><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>/</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681698&type= 19.81200027 3.72533321 。 根据初始条件以及 <math id="M143"><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681709&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681708&type= 6.34999990 3.38666677 的表达式，可确定 <math id="M144"><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681705&type= 4.31799984 3.72533321 的表达式为 <math display="block" id="M145"><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msubsup><mrow><mo stretchy="false">[</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>-</mo><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mi>a</mi><mi>Z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>Z</mi></mrow></mrow><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msubsup><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>Z</mi></mrow></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681714&type= 60.79066849 17.01799965 同理，对于 <math id="M146"><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681716&type= 4.14866638 3.72533321 ，可确定其表达式为 <math display="block" id="M147"><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msubsup><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mi>a</mi><mi>Z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>Z</mi></mrow></mrow><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msubsup><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>Z</mi></mrow></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681727&type= 47.32866669 17.01799965 由此可得到 <math id="M148"><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681717&type= 9.39799976 3.38666677 的表达式为 <math display="block" id="M149"><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681734&type= 60.53666687 4.48733330 <math display="block" id="M150"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>∞</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi>Z</mi><mo>-</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>-</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>τ</mi><mo stretchy="false">)</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo stretchy="false">]</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681721&type= 77.30065918 8.72066593 根据 式(5) 式(5) 和 式(40) 式(40) 可得通量 <math id="M151"><mi>J</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681740&type= 8.89000034 3.38666677 的表达式为 <math display="block" id="M152"><mi>J</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>Z</mi></mrow></msup><mo stretchy="false">[</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681737&type= 63.58466721 4.82600021 <math display="block" id="M153"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>δ</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>δ</mi><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681758&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681745&type= 66.20932770 4.48733330 <math display="block" id="M154"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>∞</mi></mrow></munderover></mstyle><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>Z</mi></mrow></msup><mo stretchy="false">[</mo><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>δ</mi><msubsup><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>'</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>-</mo><mi>a</mi><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>δ</mi><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>⋅</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681762&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681760&type= 61.46799850 8.72066593 <math display="block" id="M155"><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mi>a</mi><mi>Z</mi><mo>-</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>-</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>β</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>τ</mi><mo stretchy="false">)</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681747&type= 63.16133118 7.95866632 当 <math id="M156"><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681749&type= 9.39799976 3.38666677 确定以后，将 <math id="M157"><mi>Z</mi><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>δ</mi><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681768&type= 18.71133423 3.38666677 代入 式(40) 式(40) 和 式(41) 式(41) 即可确定 <math id="M158"><mi>C</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681752&type= 8.89000034 3.38666677 和 <math id="M159"><mi>J</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681786&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681770&type= 8.38199997 3.38666677 的表达式。 上述解为有机溶质在多孔介质中一维运移的解析解，该解析解可反映有机溶质的一维运移过程。 3 解析解的验证 1 3.1 　与试验测定结果对比 2 为研究溶质在土体介质中的运移规律，曾兴等 [ 11 11 ] 以氯离子作为示踪剂在高岭土中开展了土柱试验，探究了氯离子在土体介质中的一维运移过程。尽管本文所考虑的溶质为有机溶质，但当忽略有机溶质的降解时，有机溶质和无机溶质的运移规律是基本一致的。因此，可将本文所提一维运移解析解与曾兴等 [ 11 11 ] 所开展试验的测定结果进行对比。在曾兴等 [ 11 11 ] 开展的2组(A1组和A2组)土柱试验中，入流边界条件为通量连续条件，出流边界条件为Neumann边界条件，即本文出流边界条件中的 <math id="M160"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681787&type= 13.29266548 3.04800010 。同时，入流边界处溶质(氯离子)的质量浓度保持恒定，即 <math id="M161"><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681774&type= 5.67266655 3.38666677 始终为1。此外，由于该土柱试验忽略了弥散系数和对流速度随运移距离的变化，因而对比分析时取 <math id="M162"><mi>δ</mi><mo>=</mo><mn mathvariant="normal">1.0</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">6</mn></mrow></msup><mtext> </mtext><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681791&type= 25.23066711 3.04800010 ，其余相关参数见 表1 表1 。 试验组别 运移距离 L /m 对流速度 v 0 /(m·s -1 ) 弥散系数 D 0 /(m 2 ·s -1 ) 阻滞因子 R d 时间 t /d A1组 0.091 9 3.674×10 -7 1.018×10 -9 1.031 1.00 A2组 0.053 6 5.306×10 -8 7.332×10 -10 1.084 2.39 图1 图1 所示为本文所提解析解的氯离子相对质量浓度计算结果和文献[ 11 11 ]中测定结果的对比情况。从 图1 图1 可以看出，本文所提解析解计算得到的氯离子相对质量浓度分布与文献[ 11 11 ]测定得到的氯离子相对质量浓度分布基本吻合，这在一定程度上验证了本文所提解析解的正确性。 3.2 　与已有的解析解对比 2 KUMAR等 [ 37 37 ] 对非均质多孔介质中溶质的一维运移问题进行了研究，并获得了溶质一维运移的解析解。为进一步验证本文所提解析解的正确性，下面将本文解析解与KUMAR等 [ 37 37 ] 所得解析解进行对比。在KUMAR等 [ 37 37 ] 所提解析解中，未考虑溶质类型对运移过程的影响，因而并未考虑降解作用。此外，为获得相应的解析解，KUMAR等 [ 37 37 ] 假定入流边界处的质量浓度保持恒定，初始溶质质量浓度为0 mg·L -1 ，忽略溶质运移过程中的阻滞作用，并考虑出流边界条件为Neumann 边界条件。因此，本文以入流边界为浓度连续条件为例展开对比分析，设定 <math id="M163"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681798&type= 11.51466751 3.04800010 ， <math id="M164"><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681802&type= 10.75266743 3.38666677 ， <math id="M165"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681816&type= 8.80533314 3.72533321 ， <math id="M166"><mi>α</mi><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681819&type= 13.29266548 3.04800010 ， <math id="M167"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">g</mi><mo>⋅</mo><msup><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681823&type= 25.14599991 4.82600021 。对于其余参数，为便于分析，设定 <math id="M168"><mi>δ</mi><mo>=</mo><mn mathvariant="normal">0.2</mn><mtext> </mtext><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681826&type= 15.83266735 3.04800010 ， <math id="M169"><msub><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1.0</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">9</mn></mrow></msup><mtext> </mtext><mi mathvariant="normal">m</mi><mo>⋅</mo><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681830&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681828&type= 29.97200012 4.14866638 ， <math id="M170"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">4.1</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">10</mn></mrow></msup><mtext> </mtext><mi mathvariant="normal">m</mi><mo>⋅</mo><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681832&type= 32.17333603 4.14866638 ， <math id="M171"><mi>L</mi><mo>=</mo><mn mathvariant="normal">1.0</mn><mtext> </mtext><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681810&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681835&type= 13.80066681 2.70933342 ， <math id="M172"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1.0</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">g</mi><mo>⋅</mo><msup><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681838&type= 24.12999916 4.82600021 。 图2 图2 所示为不同运移距离处本文解析解和文献[ 37 37 ]中解析解计算得到的有机溶质质量浓度对比。从 图2 图2 可以看出：两种解析解所表征的不同运移距离处溶质质量浓度随时间的变化规律一致，这既进一步验证了本文解析解的正确性，也说明了本文所提解析解可用于计算考虑弥散系数和对流速度随运移距离变化下溶质的一维运移问题。 3.3 　与有限差分解对比 2 有限差分法是用于溶质一维运移过程计算的常用数值方法 [ 48 48 , 50 50 - 52 52 50-52 ] 。为验证本文所提解析解的正确性，下面参考SAVOVIĆ等 [ 51 51 - 52 52 51-52 ] 所采用的有限差分法，将本文所提解析解的计算结果与有限差分解的计算结果进行对比。为展开对比分析，假定 <math id="M173"><msub><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0.5</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">g</mi><mo>⋅</mo><msup><mrow><mi mathvariant="normal">L</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681847&type= 27.94000053 4.82600021 ，并设定时间函数 <math id="M174"><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>γ</mi><mi>t</mi></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681858&type= 23.79133224 3.72533321 ，以考虑入流边界处有机溶质质量浓度随时间的变化， <math id="M175"><mi>γ</mi><mo>=</mo><mn mathvariant="normal">2.0</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">10</mn></mrow></msup><mtext> </mtext><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681862&type= 24.55333328 3.72533321 。有机溶质一维运移的计算参数如 表2 表2 所示。 运移距离 L /m 对流速度 v 0 /(m·s -1 ) 弥散系数 D 0 /(m 2 ·s -1 ) 阻滞因子 R d 降解半衰期 t 1/2 /a 初始质量浓度 C 0 /(mg·L -1 ) Robin边界常数 α /m -1 非均质特性系数 δ /m -1 2.0 7.52×10 -10 1.62×10 -10 2.34 200 10 2.0 0.1 图3 图3 所示为有机溶质质量浓度本文解析解计算结果和有限差分解计算结果的对比情况。从 图3 图3 可以看出，在入流边界处为通量连续条件和浓度连续条件下，两种解所表征的有机溶质相对质量浓度随运移距离的分布规律以及有机溶质质量浓度随时间的变化规律均一致，验证了本文所提解析解的正确性。 (a) 有机溶质相对质量浓度随运移距离的分布；(b) 有机溶质质量浓度随时间的变化 4 参数分析 1 为研究变弥散系数和对流速度下有机溶质在多孔介质中的一维运移规律，以3.3节中所采用的计算参数作为基准计算参数，对非均质特性系数 δ 、降解半衰期 t 1/2 以及Robin边界常数 α 等参数展开分析。 4.1 　非均质特性系数 2 图4 图4 所示为不同非均质特性系数 δ 下有机溶质质量浓度随运移距离的分布曲线。从 图4 图4 可以看出，有机溶质质量浓度随运移距离增大而减小。在 t =50 a时，两种入流边界条件下靠近出流边界处有机溶质质量浓度均随 δ 增大而增大，而靠近入流边界处有机溶质质量浓度均随 δ 增大而减小。此外，根据最终浓度分布曲线可以看出，两种入流边界条件下相同位置处有机溶质的最终浓度均随 δ 增大而减小。 (a) 浓度连续条件；(b) 通量连续条件 考虑到两种入流边界条件下有机溶质的运移规律是基本相似的，下面以入流边界为浓度连续条件为例，绘制 z / L =0.2和 z / L =0.8处不同非均质特性系数 δ 下有机溶质质量浓度随时间的变化曲线，如 图5 图5 所示。从 图5 图5 可以看出：在运移初期，不同位置处有机溶质质量浓度均随 δ 增大呈现增大趋势；但在运移后期，不同位置处有机溶质质量浓度均随 δ 增大而减小。值得注意的是，无论在运移初期，还是在运移后期，相比于 z / L =0.2处， z / L =0.8处有机溶质质量浓度受 δ 的影响更为显著。与此同时，由于入流边界处有机溶质质量浓度随时间呈指数函数形式减小，不同位置处有机溶质质量浓度随时间均出现了先增大后减小的变化规律。 (a) z / L =0.2；(b) z / L =0.8 对于不同非均质特性系数 δ 下有机溶质的一维运移规律，分析认为， δ 增大会使弥散系数和对流速度随运移距离增大而增大。一方面， δ 增大会在一定程度上加快有机溶质的一维运移过程；另一方面，有机溶质在运移过程中会趋于溶质通量平衡(不同运移距离处的通量 <math id="M176"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681857&type= 2.11666679 2.62466669 相等)，因而 δ 增大还会使得运移后期有机溶质的质量浓度减小。 4.2 　降解半衰期 2 为研究降解作用对有机溶质一维运移过程的影响，下面以入流边界为通量连续条件为例，绘制不同降解半衰期 t 1/2 下有机溶质质量浓度随运移距离的分布曲线以及有机溶质质量浓度随时间的变化曲线，分别如 图6 图6 和 图7 图7 所示。从 图6 图6 可以看出，有机溶质的降解作用使得不同位置处有机溶质的质量浓度降低，有机溶质质量浓度随 t 1/2 增大而增大。从 图7 图7 可知，不同位置处有机溶质的质量浓度随降解半衰期 t 1/2 增大而增大，且与靠近入流边界处( z / L =0.2)相比，降解作用对靠近出流边界处( z / L =0.8)有机溶质质量浓度的影响更大。当 t =300 a时，相比于不考虑降解作用时有机溶质的质量浓度， t 1/2 =200、400、800和1 600 a时靠近出流边界处有机溶质的质量浓度分别降低了39.5%、22.6%、12.2%和6.3%。可以看出，当 t 1/2 大于1 600 a时，降解作用对运移过程的影响较小，但当 t 1/2 小于800 a时，降解作用会对运移过程产生较大影响。此外，相比于靠近出流边界处有机溶质质量浓度的变化，靠近入流边界处有机溶质的质量浓度更早达到峰值。 (a) z / L =0.2；(b) z / L =0.8 上述分析表明，降解作用会对有机溶质的运移过程产生较大影响，有机溶质的质量浓度会随降解半衰期 t 1/2 增大而增大。结合不同 t 1/2 下有机溶质质量浓度曲线的变化规律可知，当 t 1/2 不超过800 a时，降解作用对有机溶质一维运移过程的影响相对较大，应考虑有机溶质的降解作用；当 t 1/2 大于1 600 a时，降解作用对有机溶质一维运移过程的影响相对较小，可忽略有机溶质的降解作用。 4.3 　 Robin 边界常数 2 为探究出流边界条件对有机溶质一维运移过程的影响，考虑通量连续与浓度连续两种入流边界条件，得到不同Robin边界常数 α 下有机溶质质量浓度随运移距离的分布曲线，如 图8 图8 所示。从 图8 图8 可以看出，Robin边界常数 α 会对出流边界附近有机溶质的运移规律产生较大影响。在两种入流边界条件下， α 越大，出流边界附近有机溶质的质量浓度均越低，且当 α 较大(如 α= 100 0 m -1 )时，出流边界处有机溶质质量浓度接近于0 mg/L。与此同时，对比两种入流边界条件可知，在相同Robin边界常数 <math id="M177"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681897&type= 1.94733346 2.62466669 下，浓度连续条件下有机溶质的一维运移过程更快，相同位置处有机溶质的质量浓度更高。 Robin边界常数 α 的取值与出流边界处的环境有关 [ 19 19 , 44 44 - 47 47 44-47 ] ，因而 α 的变化反映了出流边界环境对有机溶质一维运移过程的影响。根据上述分析可知，出流边界环境会对出流边界附近有机溶质的质量浓度产生较大影响，出流边界附近有机溶质的质量浓度会随 α 增大而减小。对于多孔介质中有机溶质的运移过程，应结合实际情况选择合适的出流边界条件。 5 结论 1 1) 非均质特性系数 δ 增大使得弥散系数和对流速度随运移距离增大而增大。因此，一方面， δ 增大会在一定程度上加快有机溶质的一维运移过程；但另一方面，有机溶质在运移过程中会趋于溶质通量平衡，因而 δ 增大还会使运移后期有机溶质的质量浓度减小。 2) 降解作用对有机溶质的运移过程产生较大影响，有机溶质的质量浓度随降解半衰期 t 1/2 增大而增大。当 t 1/2 不超过800 a时，应考虑有机溶质的降解作用；当 t 1/2 大于1 600 a时，可忽略有机溶质的降解作用。 3) Robin边界常数 α 主要影响出流边界附近有机溶质质量浓度的变化，在浓度连续和通量连续两种入流边界条件下，出流边界附近有机溶质的质量浓度均会随 α 增大而减小。对于多孔介质中有机溶质的一维运移过程，应结合实际情况选择合适的出流边界条件。 参考文献 ACAR Y B , HAIDER L . 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Analytical solution for one dimensional transport of organic solute in porous media with variable dispersion coefficient and variable advection velocity[J]. Journal of Central South University(Science and Technology), 2023, 54(4): 1518-1530. 国家重点研发计划项目(2019YFC1804002)；国家自然科学基金资助项目(51625903，42177163) Project(2019YFC1804002) supported by the National Key Research and Development Program of China; Projects(51625903, 42177163) supported by the National Natural Science Foundation of China 2022-06-14 2022-07-28 2023-04-26 2023-05-22 37692359 中南大学学报(自然科学版) 4 54