Tell a friend about us, add a link to this page, or visit the webmaster's page for free fun content. Figure 10. Figure 11. Gp() is a function of variable p and a special solution as the form of equation (5)16 Cx tG x Dt (), = Gp 4 (5) Equation (6) is obtained by differentiating p with respect to t p Choose a web site to get translated content where available and see local events and offers. 0000007175 00000 n The diffusion coefficients for the sublimation diffusion of disperse dye were calculated by the values of Cx/C1 and x/4Dt in equation (36). Thus, it was expected that the diffusion coefficients calculated by the equation of complementary error function adopted in this experiment were reliable. Access to society journal content varies across our titles. Figure 8.1 compares these two techniques and the resulting dopant profiles. Plot of Ci¯/C1 versus x/4Dt for the diffusion at 170°C and 180°C for 3 h. In the case of 180°C, the values of X and the inter-layer diffusion coefficients for five layers are presented in Table 2. Plot of Ci¯/C1 versus x/4Dt for the diffusion at 190°C for 3 h (insert: graph of Y=erfc(X)). When the disperse dye in paste was treated at 170°C–190°C for 3–4 h, the profile can be obtained from plotting the mean dye concentration of the ith layer (Ci¯) versus distance (x). If C(x,t) satisfies the diffusion equation of equation (1), then the partial derivatives of C(x,t), that is, equation (24), is also a solution of the diffusion equation, However, the Gaussian kernel S(x,t) of equation (25) is the fundamental solution of the diffusion equation with a special initial condition, By the property of translation invariance for the diffusion equation and its solutions, if S(x,t) is a solution of equation (1), then the function S(x−y,t) of equation (26) is also a solution for any fixed number y, S(x−y,t) has the form of a time-varying Gaussian function of equation (27), where σ=Dt and μ=y, A solution of the diffusion equation can be also obtained by taking the convolution of the Gaussian kernel S(x,t) with f(x) of the initial condition. error-function diffusion: translation. respectively. The top of roll was fixed firmly by a thin aluminium tape and cotton yarn with glass rod for dividing layers.10,11. Table 1. In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid - Wikipedia 0000009494 00000 n The coefficient of determination (r2), that is, the variation in Cx, is 0.994. Find out about Lean Library here, If you have access to journal via a society or associations, read the instructions below.

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