利用拉普拉斯变换模拟渗灌材料吸水特性曲线

为了定量研究渗灌复合材料吸水特性与其组分之间的关系,该文利用拉普拉斯变换求解渗灌复合材料的吸水微分方程,同时模拟吸水率曲线,应用传递函数系数建立渗灌复合材料的吸水性与其试验组分之间的定量关系。设置10组比例不同的组分制备而成的渗灌复合材料进行一次和二次吸水试验,另外6组比例不同的组分制备而成复合材料进行一次吸水试验进行验证。结果表明:基于拉普拉斯变换求解的渗灌材料吸水曲线模拟值与实测值的相对误差为0.04%~7.17%,均方根误差0.0717,说明利用拉普拉斯变换对吸水曲线进行表征具有很好的稳定性、适用性及精确性;传递函数系数与材料组分之间的定量关系可以用指数函数来表达,应用指数函数计算获得的传递函数的模拟值与实测值的均方根误差≤0.5,经验证,最大相对误差为4.01%,表明传递函数系数与复合材料试验组分间指数函数关系的可靠性。该研究对于后续的渗灌材料的研制及其导水特性研究具有指导意义。

Simulation on water-absorption characteristic curve of infiltrating irrigation materials through Laplace-transform

Abstract: Infiltrating irrigation technology is an effective water-saving irrigation technology. As a new irrigation methods, it is getting a lot of attention. A kind of infiltrating irrigation material composed of montmorillonite (MMT) and polyacrylamide (PAM) developed recently has showed good self-adjustment property for soil moisture, and its water-absorption performance can be used to evaluate the sensitive degree that material responses to the external environment. This study aimed to simulating the water-absorption performance curve using the Laplace-transform method, which transformed a problem of solving differential equations into a problem of solving algebraic equation. Meanwhile, the model had also been used to explore the numerical relationship between water-absorbing capacity and material components. PAM and sodium montmorillonite (Na-MMT) was used as experimental material, circular material cube (4 g for each, 10 mm in height, 5 mm in radius) was for samples. Ten groups testing components of composite materials were designed including the mass proportions of MMT: PAM of 2.5:1, 5:1, 7.5:1, 10:1, 12.5:1, 15:1, 17.5:1, 20:1, 22.5:1, and 25:1. The samples made of the ratios were carried out the first and secondary water absorption performance test. In addition, another six groups with different proportions of composite materials (MMT: PAM 4:1, 14:1, 24:1, 40:1, 50:1, 60:1) were carried out the first water absorption test to verify the model. The water absorbing performance curve of composite was simulated as followed: 1) Describing the curves of water-absorption rate with algebraic equation rather than differential equation, which was to calculate the transfer function depending on the Laplace limit theorem; 2) Coding the equation in M file editor of MATLAB software; 3) Inputting the adjusted experimental values of water-absorption rate in the Matlab. Then, the system's transfer function was identified by the step response curve, and parameters of model were automatically formed, thereby the transfer function coefficients of water-absorption curves of samples during the first and second water-asborption test were obtained. Then, the water-absorption curves were simulated with the obtained parameters. The results showed that: the relative error between the simulated and measured water-absorption rate were 0.04%-7.17% and root mean square error (RMSE) was <0.0717, indicating that the water-absorption rate curves characterized by using Laplace transform is effective and accurate. Quantitative relationship between transfer function coefficient indicating the water absorbing capacity of material and material components followed exponential function. The RMSE between simulated transfer function coefficient values with the exponential function and the measured one was less than 0.5. In the validation test, the maximum relative error between the simulated and measured transfer function coefficient values was 4.01%. It suggested the reliability of the exponential function used for describing the relationship between water-absorption capacity of the compound material and its ratios. This study is valuble for design of infiltration irrigation materials and researchers on the water conductivity features of infiltration irrigation materials.