Mathematical treatment of permeation for cylindrical geometry
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Fig. 2 - - ( a ) Extensive growth into ferrite of the M23C6carbide, confirmed by (b) the TEM diffraction pattern, and (c) dark-field image.
Gas phase permeation experiments have been performed for years by using the time lag method developed by Barrer. 1 In these experiments, a metallic sample separates two vacuum environments. At time t = 0, a constant pressure, P0, is placed on one side of the sample and the pressure rise on the output side is monitored as a function of time. The diffusivity and permeability are then related to the initial transient behavior and steady state behavior of the uptake curve. 2 Permeation experiments can also be performed by measuring the flux of gas through the sample. In flux experiments, the mathematics relating the diffusivity to the uptake curve has only been developed for planar geometry by Boes and Zuchner. 3 In this paper, we will discuss the case of cylindrical geometry, deriving an expression for the breakthrough time, tb, which will relate the time transient to the diffusivity. We will also discuss the errors in the diffusivity introduced by assuming the planar breakthrough time expression applies for samples of cylindrical geometry. To solve for the breakthrough time, we will need an expression for the flux of gas through the sample. First, the concentration of gas in the sample, C, found by solving Fick's second law for the following boundary conditions applied to the three sample geometries shown in Figure 1. t = 0 : C = 0 for 0
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