On the Mathematical Analysis and Optimization of Chemical Vapor Infiltration in Materials Science

Ditkowski, Adi; Gottlieb, David; Sheldon, Brian W.

doi:10.1051/m2an:2000144

Cited by 6 publications

(7 citation statements)

References 10 publications

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“…Though the equations in this paper are quite different than the ones in [5,7], the analysis done in this work was highly influenced by these manuscripts.…”

Section: Dynamic Optimal Acquisition and Retention Of A Monopolistmentioning

confidence: 98%

Mathematical analysis of the optimizing acquisition and retention over time problem

Ditkowski¹

2008

ESAIM: M2AN

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Abstract. While making informed decisions regarding investments in customer retention and acquisition becomes a pressing managerial issue, formal models and analysis, which may provide insight into this topic, are still scarce. In this study we examine two dynamic models for optimal acquisition and retention models of a monopoly, the total cost and the cost per customer models. These models are analytically analyzed using classical, direct, methods and asymptotic expansions (for the total cost model). In order to numerically simulated the models, an innovative numerical method was developed for solving ODE systems with initial/final value problems.

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“…Though the equations in this paper are quite different than the ones in [5,7], the analysis done in this work was highly influenced by these manuscripts.…”

Section: Dynamic Optimal Acquisition and Retention Of A Monopolistmentioning

confidence: 98%

Mathematical analysis of the optimizing acquisition and retention over time problem

Ditkowski¹

2008

ESAIM: M2AN

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“…Note that ␣ 2 is dimensionless and that ␤ has units of inverse time. (16). (5) has been removed in Eq.…”

Section: Formulationmentioning

confidence: 99%

“…(5) has been removed in Eq. (16) and (17) is subject to the initial condition 4 Transforming ⑀ to simplifies Eq.…”

Section: Formulationmentioning

confidence: 99%

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Optimization of chemical vapor infiltration with simultaneous powder formation

2000

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A key difficulty in isothermal, isobaric chemical vapor infiltration is the long processing times that are typically required. With this in mind, it was important to minimize infiltration times. This optimization problem was addressed here, using a relatively simple model for dilute gases. The results provided useful asymptotic expressions for the minimum time and corresponding conditions. These approximations were quantitatively accurate for most cases of interest, where relatively uniform infiltration was required. They also provided useful quantitative insight in cases where less uniformity was required. The effects of homogeneous nucleation were also investigated. This does not effect the governing equations for infiltration of a porous body; however, powder formation can restrict the range of permissible infiltration conditions. This was analyzed for the case of carbon infiltration from methane.

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“…Approaches have been performed in direction of optimising the CVI process (Chang et al, 1998;Ditkowski et al, 2000), but only one overall chemical reaction is treated and the underlying differential equations are based on steady-state assumptions both for the concentrations as well as for the transport coefficients. However, no mathematical optimisation techniques are used.…”

Section: Introductionmentioning

confidence: 99%

Modelling chemical vapour infiltration of pyrolytic carbon as moving boundary problem

Langhoff¹,

Schnack²

2008

Chemical Engineering Science

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Chemical vapour infiltration (CVI) of pyrolytic carbon is described as a moving boundary problem to determine the evolution of the pyrolytic carbon layer in space and time. Derived from real geometries, a one-dimensional single pore model is developed yielding a nonlinear coupled system of partial differential equations for the concentrations of the gas phase species and the height of the carbon layer within cylindrical pores. The evolution of the moving boundary of the gas phase domain is governed by a non-differentiable minimisation condition. Additionally, a CVI reactor model to describe the infiltration of several cylindrical pores within a porous substrate is presented on the basis of the single pore model. Both models are new in that they combine the following features: (i) derivation of the equations rigorously taking into account the temporal change of the gas phase, (ii) the explicit construction of the position of the gas--solid interface, (iii) the influence of the local curvature of the carbon layer on its growth velocity, and (iv) modelling of chemical kinetics using a reduced reaction scheme with intermediate gas phase species and several surface reactions. The models are solved numerically using a staggered strongly decoupled scheme with implicit Euler time integration. The results allow the identification of process conditions and geometries for which a complete infiltration of the pores or the whole substrate is achieved. For low pressures, the predictions of the CVI reactor model are in agreement with the available experimental data.

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On the Mathematical Analysis and Optimization of Chemical Vapor Infiltration in Materials Science

Cited by 6 publications

References 10 publications

Mathematical analysis of the optimizing acquisition and retention over time problem

Mathematical analysis of the optimizing acquisition and retention over time problem

Optimization of chemical vapor infiltration with simultaneous powder formation

Modelling chemical vapour infiltration of pyrolytic carbon as moving boundary problem

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