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Kinetics of Materials - R. Balluff_ S. Allen_ W. Carter (Wiley_ 2005) Episode 13

Tham khảo tài liệu 'kinetics of materials - r. balluff_ s. allen_ w. carter (wiley_ 2005) episode 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | EXERCISES 527 20.4 Find an expression for the thickness of the growing B-rich 3-phase surface layer shown in Fig. 20.3 as a function of time. Details of the growth of this layer have been discussed in Section 20.1.2 and a strategy for determining the growth rate has been outlined. Assume constant diffusivities in both phases. Solution. Solutions to the diffusion equation in the a and 3 phases which match the boundary conditions are CB Cb _ srf x V4Ĩ 0t erf Az V4D0 C0B - C0B CB CBX _ 1 erf a 4DQt cb C BX 1 erf Al 4DQ where and A vt 20.86 20.87 The Stefan conditions at interfaces 1 and 2 are given by Eqs. 20.12 and 20.17 respectively. Substituting the appropriate relationships from those given above into these two equations then yields equations that can be solved simultaneously for Al and A2 Al CB CB e-Aị ííõi3 -CaB 20.88 e-Al Da 1 - erf Ai x w - B CB 20.89 Finally the layer thickness is given by X Xi X2 Al A2 Vt. 20.5 Using the scaling method find an expression for the diffusion-limited rate of growth of a cylindrical B-rich precipitate growing in an infinite Q-phase matrix. Assume the same boundary conditions as in the analysis in Section 20.2.1 Eqs. 20.37-20.39 for the growth of a spherical particle. Note that Fig. 20.6 which applied to the growth of a spherical particle in Section 20.2.1 will also apply. Use the scaling parameter r r 4DQt 1 2. You will need the integral 20.90 which has been tabulated 29 . Solution. Starting with the diffusion equation in cylindrical coordinates see Eq. 5.8 and using the scaling parameter to change variables the diffusion equation in 77-space becomes _ 20.91 drf rj dr This result along with the boundary conditions given by Eqs. 20.37-20.39 shows that the particle will grow parabolically according to R t r RV4 Dat 20.92 528 CHAPTER 20 GROWTH OF PHASES IN CONCENTRATION AND THERMAL FIELDS Integrating Eq. 20.91 once dcẵ . _ e-T 2 T 1- dri Tj where ai constant. Integrating again yields QOO a _ f -y2 dy Cb - Cb ai e Jn y Determining