本文關(guān)鍵詞： 菲克定律 無(wú)限擴(kuò)散 有限擴(kuò)散 拉普拉斯變換 復(fù)變函數(shù)　出處：《數(shù)學(xué)的實(shí)踐與認(rèn)識(shí)》2017年01期 　論文類型：期刊論文

【摘要】：介紹了三維和一維擴(kuò)散下的菲克定律,以及兩類涉及到擴(kuò)散的實(shí)際問(wèn)題,即求擴(kuò)散粒子通過(guò)曲面的擴(kuò)散通量和求解擴(kuò)散粒子的濃度分布.通過(guò)拉普拉斯變換和復(fù)變函數(shù)相關(guān)數(shù)學(xué)理論,求解了菲克擴(kuò)散定律在無(wú)限長(zhǎng)介質(zhì)和有限長(zhǎng)介質(zhì)兩種非穩(wěn)態(tài)擴(kuò)散情況下的解.粒子在無(wú)限長(zhǎng)介質(zhì)中的非穩(wěn)態(tài)擴(kuò)散和濃度分布可通過(guò)方程φ(z,t)=Φ·erfc(z/2DT~(1/2))表示.方程為余補(bǔ)高斯誤差函數(shù).粒子在有限長(zhǎng)介質(zhì)中的非穩(wěn)態(tài)擴(kuò)散和濃度分布可通過(guò)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.該方程為無(wú)限加和形式,當(dāng)n≥100000時(shí),φ可以精確到小數(shù)點(diǎn)后6位,在方程的圖像上不再能觀察出由n的取值造成的誤差.從方程的圖像可得到粒子在擴(kuò)散介質(zhì)中達(dá)到飽和的時(shí)間或粒子擴(kuò)散到z=0處的時(shí)間等具有重要物理意義的參數(shù).
[Abstract]:The Fick's law under three dimensional and one dimensional diffusion and two kinds of practical problems related to diffusion are introduced. In other words, the diffusion flux of diffusion particles through the surface and the concentration distribution of diffusion particles can be solved by means of Laplace transformation and mathematical theory of complex function. The solution of Fick's diffusion law in two kinds of unsteady diffusion of infinite medium and finite length medium is solved. The unsteady diffusion and concentration distribution of particles in infinite medium can be obtained by the equation 蠁 ~ (z). The equation is a complementary Gao Si error function. The unsteady-state diffusion and concentration distribution of particles in a finite medium can be obtained by the equation 蠁 _ (z). Tr = 桅 桅 路4 / 蟺 鈭，

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