柱坐标系下冻土内部热流侵蚀相变问题的求解

针对冻结工程灾变过程中渗水孔隙等灾害源在地下水热流侵蚀作用下逐渐扩展的过程进行了研究,在一定假设条件下建立了柱坐标系下基于对流换热边界条件(第三类边界条件)的相变传热数学模型。在对原偏微分方程无量纲化处理后,采用了对数形式分布的热积分平衡方法(HBIM)求解;对于同一问题,在变量代换的基础上采用了基于乘方定律格式的有限差分数值方法进行了求解。对于侵蚀相变位置的求解,两种方法的计算结果吻合良好,在计算的时间范围内,两者最大偏差不超过1%。研究结果表明:史蒂芬数St、毕渥数Bi、以及过冷系数φ为侵蚀相变界面位置随时间变化规律的主要影响因素;在不同参数组合情况下,侵蚀相变位置随时间均呈现出近似线性变化规律;在其他参数不变的条件下,St,Bi增大一倍时,侵蚀相变速率均增加一倍左右,而φ增加一倍时,侵蚀相变速率仅减小了5%。进而说明在实际工程中,降低冻土的平均温度能够减缓侵蚀相变速率,但作用十分有限;相反,降低水流温度或者减缓水流的流速等措施则能够有效减缓水流侵蚀速率。利用本文结果预测实际冻结工程灾变过程时,需考虑土壤性质的不同以及地下水温度、对流换热系数等随孔径变化带来的影响,可将由本文方法得到的侵蚀相变速率视为冻结工程灾害发生时孔隙发展速度的上限值。

Solutions to phase change problem due to internal thermal erosion of frozen soil

To solve the thermal erosion problem of frozen soil due to gradual expansion of the seepage pores during the process of freezing engineering catastrophe,a corresponding mathematical model of phase change heat transfer based on the convective heat transfer boundary condition(the third type boundary condition) in the cylindrical coordinate system was established under certain assumptions.The heat balance integral method(HBIM) was adopted to solve the nondimensionalized partial differential equation and the problem was also numerically solved for verification via a finite-difference procedure facilitated by coordinate transformations after the equation was dispersed using power law scheme.For the solution of the erosion phase change position,the calculation results of the two methods were in good agreement.The maximum deviation between the two methods was less than 1% in the calculated time range.The results show that Stephen number St,Biot number Bi,and sub-cooling coefficient φ are the main factors influencing the moving velocity of phase change position.The dimensionless phase change position almost linearly increases with time under the condition of different parameter combinations.When the St and Bi are doubled,the velocity of phase change position is doubled under the condition that other parameters are constant.While the velocity of phase change position is only reduced by 5% when φ is doubled.It is indicated that in engineering practice,reducing the average temperature of frozen soil can slow down the rate of phase change,but it is of limited effectiveness.On the contrary,the measures such as lowering the temperature of water flow or slowing the flow rate of water flow can effectively slow down the phase change rate.When the results are adopted to predict the actual freezing engineering disasters,it is necessary to consider the difference of soil properties and the influence of groundwater temperature and convective heat transfer coefficient changing with the pore size.The phase change erosion rate obtained by the method can be regarded as the upper limit of the erosion speed in engineering practice.