应用拉普拉斯变换和留数法求解常见非稳态扩散情况下的菲克定律
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摘要
介绍了三维和一维扩散下的菲克定律,以及两类涉及到扩散的实际问题,即求扩散粒子通过曲面的扩散通量和求解扩散粒子的浓度分布.通过拉普拉斯变换和复变函数相关数学理论,求解了菲克扩散定律在无限长介质和有限长介质两种非稳态扩散情况下的解.粒子在无限长介质中的非稳态扩散和浓度分布可通过方程φ(z,t)=Φ·erfc(z/2DT~(1/2))表示.方程为余补高斯误差函数.粒子在有限长介质中的非稳态扩散和浓度分布可通过方程φ(z,t)=Φ+Φ·4/π∑_(n=1)~(+∞)((-1)~n)/(2n-1)cos[z/L(n-1/2)π]e~((D_t)/(L~2)(n-1/2)~2π~2)表示.该方程为无限加和形式,当n≥100000时,φ可以精确到小数点后6位,在方程的图像上不再能观察出由n的取值造成的误差.从方程的图像可得到粒子在扩散介质中达到饱和的时间或粒子扩散到z=0处的时间等具有重要物理意义的参数.
This article described the Fick's Law of three-dimensional and one-dimensional diffusion,and two types of practical issues related to diffusion,which were solving diffusion fluxes of particles transferred through a surface and solving concentration distribution of diffusion particles.In this paper,the solutions of Fick diffusion law in unsteady situations,which were infinite medium and finite medium,were obtained by application of mathematical theory such as Laplace transform and complex functions.Unsteady diffusion and concentration distribution of particles in infinite media could be expressed by the equationφ(z,t) = Φ·erfc(z/2DT~(1/2)) which was complementary Gaussian error function.Unsteady diffusion and concentration distribution of particles in finite media could be described by equation ofφ(z,t) = Φ + Φ·4/π ∑_(n=1)~(+∞)(-1)~n/2n-1 cos[z/L(n-1/2)π]e~(-D_t/L~2(n-1/2)~2π~2)which was in form of unlimited plus.When n ≥ 100000,φ could be accurate to six decimal,and the error caused by n was no longer able to observe on image of the equation.Important parameters which had physical significant could be obtained by considering images of the kinetic equations,such as saturation time of particles in diffusion media and of particles reached the position of z = 0.
This article described the Fick's Law of three-dimensional and one-dimensional diffusion,and two types of practical issues related to diffusion,which were solving diffusion fluxes of particles transferred through a surface and solving concentration distribution of diffusion particles.In this paper,the solutions of Fick diffusion law in unsteady situations,which were infinite medium and finite medium,were obtained by application of mathematical theory such as Laplace transform and complex functions.Unsteady diffusion and concentration distribution of particles in infinite media could be expressed by the equationφ(z,t) = Φ·erfc(z/2DT~(1/2)) which was complementary Gaussian error function.Unsteady diffusion and concentration distribution of particles in finite media could be described by equation ofφ(z,t) = Φ + Φ·4/π ∑_(n=1)~(+∞)(-1)~n/2n-1 cos[z/L(n-1/2)π]e~(-D_t/L~2(n-1/2)~2π~2)which was in form of unlimited plus.When n ≥ 100000,φ could be accurate to six decimal,and the error caused by n was no longer able to observe on image of the equation.Important parameters which had physical significant could be obtained by considering images of the kinetic equations,such as saturation time of particles in diffusion media and of particles reached the position of z = 0.
引文
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[3]周杰荣,龚玮.一个反应-扩散方程组解的整体存在与有限时刻爆破[J].大学数学,2007,04:53-56.
[4]张鉴清.电化学测试技术[M].化学工业出版,北京,2010.
[5]曹楚南.腐蚀电化学原理[M].化学工业出版,北京,2008.
[6]胡玲,徐芸芸,吴瑾.Fick第二定律的应用研究现状与展望[J].河北建筑科技学院学报,2006,22(3):50-52.
[7]Wind M M,Lenderink H J W.A capacitance study of pseudo-fickian diffusion in glassy polymer coatings[J].Progress in Organic Coatings,2006,28(4):239-242.
[8]Hu J M,et al.Determination of water uptake and diffusion of Cl~-ion in epoxy primer on aluminum alloys in NaCl solution by electrochemical impedance spectroscopy[J].Acta Phys-chim Sin,2003,46(4):273-276.
[9]Zhang J T,et al.Studies of impedance models and water transport behaviors of polypropylene coated metals in NaCl solution[J].Prog Org Coat,2004,49(4):293-298.
[10]Bellucci F,Nicodemo L,Monetta T,Kloppers M J,Latanision R M.A study of corrosion initiation on polyimide coatings[J].Corrosion science,1992,33(8):1203-1206.
[11]汪德新.数学物理方法[M].华中科技大学出版社,武汉,2001.
[12]郝晓辉,李宝凤.关于图的拟拉普拉斯谱半径[J].数学的实践与认识,2008(4):158.160.
[13]沈克精.拉普拉斯(Laplace)变换的一个新应用[J].数学的实践与认识,1991(1):50-55.
[14]郝晓辉,张利军.连通图拟拉普拉斯矩阵的最大特征值[J].数学的实践与认识,2009(7):178-181.
[15]谭沈阳,张学华.H-型群上多重次拉普拉斯算子的基本解及其估计[J].数学的实践与认识,2013(7):241-246.
[16]张兰.Rayleigh型p-拉普拉斯方程周期解的存在性和唯一性[J].数学的实践与认识,2013(20):222-226.
[17]贾会才,刘瑞芳.关于拉普拉斯谱半径的一个不等式[J].数学的实践与认识,2011(2):206-209.
[18]卢世芳.完全多部图K_(a_1n_1,a_2n_2,…,a_sn_s)的拉普拉斯谱[J].数学的实践与认识,2014(5):275-279.
[19]G.M.Fichetnholz,吴亲仁,陆秀丽,丁寿田.数学分析原理[M].高等教育出版社,北京,2013.