Study on Calculation of Diffusion Coefficient for Sublimation Diffusion of Disperse Dye Using Fourier Series

Park, Geon Yong

doi:10.1007/s12221-020-8760-z

Cited by 7 publications

(7 citation statements)

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“…In the previous two studies, the diffusion coefficients were obtained by two different equations. Equation (37) was deduced from a trigonometrical series and the diffusion coefficients DnormalF were calculated by the surface concentration C1, the steady-state diffusion length lv, and the total amount diffused from the first layer to the last layer, that is, lfalse∑i=1ntrueCi¯, in time t to reach an approximate steady-state condition for the diffusion of 190°C × 240 minutes, 180°C × 240 minutes, and 170°C × 300 minutes 13 …”

Section: Resultsmentioning

confidence: 99%

“…In another recent study, the solution technique for the diffusion equation using the Fourier series was used to study the calculation of the diffusion coefficient for the sublimation diffusion of disperse dye using the film-roll method. 13 When an approximate steady-state condition was set up, the diffusion coefficients were calculated by the surface concentration, the total amount of dye, and the steady-state diffusion length for the diffusion at time to reach an approximate steady-state condition. The correlation coefficient of the Arrhenius plot for the diffusion coefficients obtained from the steady-state concentration distribution was almost 1.…”

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Diffusion coefficient calculated by time-lag using the film-roll method

Park

2021

Textile Research Journal

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A method for determining the diffusion coefficient by time-lag using the film-roll method for the sublimation diffusion of disperse dye was proposed. A polyethylene terephthalate film-roll coated with dye paste was treated at 170–190°C for various times. A solution consisting of the sum of a steady-state solution and a transient solution was obtained by the homogeneous boundary value problem from a trigonometrical series. The boundary conditions of the steady-state first layer and the steady-state first layer amount of dye were determined from the steady-state concentration distribution. For various diffusion times, the steady-state first layer-passed total amounts of dye that passed through the first layer in the steady-state condition were obtained by subtracting the steady-state first layer amounts from the total amounts. The time-lag was calculated from the linear regression line for the plot of the steady-state first layer-passed total number(X) of positive values against time. The diffusion coefficient was calculated by the boundary conditions of the steady-state first layer and the time-lag. For diffusion at 170°C, 180°C, and 190°C, the correlations of the steady-state first layer-passed total amounts with respect to time were very linear and the reliability of the diffusion coefficients obtained by the time-lag was proved by the good linearity of the Arrhenius plot. The activation energy obtained was 36.8 kcal[Formula: see text]mol−1.

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Section: Resultsmentioning

confidence: 99%

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Diffusion coefficient calculated by time-lag using the film-roll method

Park

2021

Textile Research Journal

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“…The film roll coated by the paste was dried at 100 C for 10 min in a dryer. [13][14][15] Figure 1 shows the schematic image of the PET film roll.…”

Section: Preparation Of Paste and Film Rollmentioning

confidence: 99%

“…e Àst e Àz 2 dzdt (14) By changing the order of integration, the region of integration is changed from 0 t 1 and…”

Section: Constant Surface Concentrationmentioning

confidence: 99%

“…13 In another recent study, the diffusion coefficients were calculated by the steady-state surface concentration, the total amount of dye and the steady-state diffusion length for the diffusion at the time to reach a steady-state condition, and the correlation coefficient of the regression line for the Arrhenius plot was almost 1, by which the reliability of the diffusion coefficients was proved. 14 In the most recent study on the diffusion coefficient calculated by time lag using the film-roll method, the steady-state first layer-passed total amounts of dye that passed through the first layer in the steady-state condition were obtained by subtracting the steady-state first layer amount from the total amounts. The time lag was calculated from the regression line for the plot of the steady-state first layer-passed total amounts of positive values against time.…”

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Diffusion coefficient for diffusion of time-varying surface concentration by the film-roll method

Park

2022

Textile Research Journal

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This research on the sublimation diffusion of disperse dye in print paste to polyethylene terephthalate film was performed at 170–190°C for various times using a film-roll method. It was found that the second layer surface concentrations obtained from the concentration–distance curves were proportional to the square root of time. To determine the diffusion coefficient for the diffusion of the time-varying surface concentrations, the solution of the diffusion equation for the boundary condition of the surface concentration varying with the square root of time was deduced by applying the Laplace transform. Assuming that the time taken for a dye in paste to pass through the first layer is the delay time and the second layer surface is the starting point for diffusion, the diffusion coefficient and the delay time were determined by the proportional constant obtained from the plot of the second layer surface concentrations against the square root of the total time and by the total amount which is the amount that has passed through the second layer surface and is a primary function of time. The diffusion coefficients for the total time were similar to those for the diffusion time excluding the delay time, and their reliability was identified by Arrhenius plots. The activation energy for the diffusion coefficients obtained from the equation deduced by the complementary error function in this study was similar to that for the diffusion coefficients obtained by the complementary error function in three recent studies by about 27 kcal · mol−1.

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