Optimization of isothermal-isobaric chemical vapor infiltration

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Suppl.,pp. S713-Sl76. 1998 0 1998 Elsevier Science Ltd. All rinhts reserved Printed in &eat Britain 0098-1354198 $19.00 + 0.00

chcm. Engng Vol. 22,

Pergamon PII: SOO98-1354(98)00145-S

Optimization of Isothermal-Isobaric Chemical Vapor Infiltration Vishak

Sampath

and Srinivas

Palanki’

Department of Chemical Engineering Florida A & M University - Florida State University Tallahassee, FL 32310. U.S.A. Abstract This study addresses the issue of determining optimal operating conditions for isothermal isobaric chemical vapor infiltration for obtaining products of maximum densitication. The optimal conditions am determined by a conventional steepest descent technique which guarantee at least a local minimum for a cost function related to the residual porosity. This work shows that the optimal operating temperature is not necessarily a flat profile. 0 1998 Elsevier Science Ltd. All rights reserved. Keywords: Isothermal chemical vapor infiltration, Optimization, Densiflcation. Introduction Chemical vapor infiltration (CVI) is a novel method of manufacturing advanced ceramics which have important applications in the aerospace, nuclear, chemical and electrical industries. Lackey and Starr (1990) have provided an exhaustive review on different aspects of CVI. The simplest and most primitive type of CVI is the isothermal-isobaric CVI (ICVI) in which there is no spatial variation of temperature or pressure in the processing environment. Reactant transport takes place by diffusion alone. Since ICVI can be performed over a large range of temperatures, it is necessary to determine what temperature results in the best product. Current industrial practice is to Rnd the operating temperature via expensive experimentation (Naslain & Langlais 1986). In this paper, we utilize a mathematical model, developed by considering the transport phenomena in an ICVI process, to calculate the optimal operating strategy. It is shown that this optimal operating strategy does not necessarily entail maintenance of a constant temperature. For our model, the optimal temperature is a temporal

mass transfer resistance at the surface of substrate. The initial porosity, ~0, of the substrate is uniform. The temperature of the substrate is kept uniform spatially, thus conforming to the condition of isothermality, but it may vary temporally. The diffusion rate is fast, relative to the deposition rate of the solid aggregate, so that a pseudosteady state condition may be assumed. This condition is generally valid since the time scale for deposition is typically around five orders of magnitude larger than that for diffusion. The pyrolytic reaction, which occurs at the walls of the pores, is assumed to be of first order. Based on the above assumptions and conditions, the mathematical model is given as follows. 1. Equation ofcontinuity (simplified from McAllister and Wolf, 1991)

subject to

pn$le.

dC

Mathematical Model There is a wide body of literature available on the mathematical modeling of isothermal CVI (Chung, McCoy, Smith & Cagliostro 1993, Currier 1990, Moene, Dekker, Makkee, Schoonman & Moulijn 1994, Tai & Chou 1989). The models differ on the basis of the geometry considered, expressions used for parameters and other assumptions. The following model chosen for this study is simple so as to facilitate the determination of optimal conditions. The substrate is considered to be a rectangular slab of finite thickness, 2L, in the direction, z. This substrate is kept in an environment of constant precursor concentration, Co. Gas transport in the pores takes place only in one direction (i.e. z) by pure dimion. There is no ’Author to whom

and

=0

C(z=

AL) =Co

(2)

-xi r=O

all correspondence should be addressed.

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2. Equation for rate of change ofpotvsig

t%

-

at =

-Ek(E,T)C

(3)

subject to &(t = 0) = Ec

(4)

where 2

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and

The term enclosed in square brackets is the surface area per unit pore volume (Melkote & Jensen 1989). D is the effective diffusivity and k is the reaction rate constant. T, n/r,, Pd, E and R respectively denote temperature, molecular weight of the aggregate, density of the aggregate, activation energy of the pyrolytic reaction and the universal gas constant. DO and so are the initial effective diffusivity and initial surface area per unit volume respectively. The continuity equation (1) may be solved analytically and used in eq.(3) to yield

minimum time. However, it would be more convenient to consider a tixed-time optimization problem. The solution to this problem would answer the question of determining the maximum den&cation possible for a given processing time. We know that, for any instant of time, porosity at the center (ec) is always maximum and porosity at the surface (Ed) is always minimum. Therefore, we need to consider only the center and surface porosities for the optimization problem, which may be posed as M$n J = “t’(& -

Ed);+

+

b(E,

-

Ed)&

(6)

subject to d& c-

1

h’&i

&It=0 = Eo

dt --pdk

cash cash (5) cash d&

d-

where z now represents the dimensionless distance, t, from the center. Eq.5 along with the initial condition E = EOmay then be used for the simulation studies. Simulation Studies Dynamics of the System

Simulations were carried out to study the dynamics of the process and identify the key issues for the formulation of the optimiition problem. Since the model is symmetric about the t = 0 axis, only the region 0 < z < 1 needs to be considered. The solution was calculated numerically using a Forward Euler Finite Difference method. A stopping condition of surface porosity, Ed < 0.01, was used because, it is assumed that further diffusion into the pores would then not be practically possible unless pressure gradients are applied. It was seen that porosity, which was uniform initially, became increasingly non-uniform during the process. This is due to the concentration gradient set up in the substrate by diflirsion resistance. At any given time, porosity was minimum at the surface and maximum at the center. In order to minimize this non-uniformity of porosity, it is necessary to maintain a lower operating temperature. This will, however, increase the processing time significantly. Ideally, the process, with low operating temperatures, would take iniinite time to produce a final product with perfect uniformity. Thus, it becomes necessary to address the issue of determining the best possible porosity distribution that can be obtained in a minimum span of time by manipulating temperature. Optimization Prvbiem The optimization objective is to calculate the optimal temperature which gives maximum densification in

dt

Md --pdk

E&=0 = eo

where J is the performance index to be minimized with respect to 2’. 7 and p are adjustable weights. &d is the desired final porosity, which is taken to be 0.01 (at which diffusion is assumed to stop). Solution to the Optimization Problem The above optimization problem has been solved by using a steepest descent technique (Press, Teukolsky, Vetterling & Flamrery 1992) since it is known that this technique guarantees at least a local minimum. The various steps involved are as follows.

Choose a fixed final time, tf . Make two initial guesses for the temperature profiles (P and Tnvl) and compute the correaponding costs (P and P-r). Calculate the gradient, V J, of the cost by backward differencing, i.e. v J = J” - J”-’ T” _ Tn-1

Update the guess for temperature, T”+l, by using the steepest descent formula T”+r = T” - aV J. 0: is the step size to be taken in the descent direction, -V J. This is determined by a simple line search. The two newest temperature and cost profiles are stored and the previous two steps are repeated until (Ybecomes less than some prescribed tolerance. At this stage, the cost is very close to the local optimum and the corresponding temperature protie may be considered to be the optimum.

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Discussion It is seen from Fig. (1) that the optimal operating temperature for each case shown is not a constant but a profile. The profiles suggest that, initially, the temperature should be increased to hasten the rate of deposition since porosity is uniform and precursor concentration is almost uniform. After some time, the temperature should be reduced to prevent excessive deposition at the surface (which would then lead to premature closure of the entrance, thus leading to an undesirable hollow core). Processing time strongly infhrences the profile. As expected, the average operating temperature reduces as the processing time is increased. Also, the profile gets progressively flatter as the processing time is increased. Initial conditions also affect the profile but to a lesser extent.

p E co &E &d E* Y pd

Weight on cost, dimensionless Porosity, dimensionless Initial porosity, dimensionless Porosity at center of slab, dimensionless Desired porosity, dimensionless Porosity at surface, dimensionless Weight on cost, dimensionless Density of the aggregate, kg/m3

References Chung, G.-Y., McCoy, B., Smith, J. & Cagliostro, D. ( 1993), ‘Chemical vapor infiltration: Dispersed and graded depositions for ceramic composites’, AIChE Journal 39( 1I), 1834.

Fig. (2) shows a comparison of the optimal and suboptimal (constant) temperature profiles for the processing time of 30 days. It is seen that the higher temperature (1100 K) leads to a more non-uniform porosity distribution. The lower temperature (1050 K) results in a more uniform distribution but the overall porosity of the substrate is higher. (The surface, with a porosity of 0.05 has still not “closed”.) The optimal profile has a surface porosity close to that of the higher temperature and a center porosity close to that of the lower temperature. The average porosity due to the implementation of this profile is the least of the three cases shown.

Currier, R. (1990), ‘Overlap model for chemical vapor infiltration of fibrous yarns’, Journal ofthe Ameri-

Notation

Moene, R., Dekker, J., Makkee, M., Schoonman, J. & Moulijn, J. (1994), ‘Evaluation of isothermal chemical vapor infiltration with langmuirhinshelwood type kinetics’, Journal of the ElectrvchemicalSociety 141(l), 282.

c co D(E) E J WE, T) ko L Md

Dimensionless concentration Bulk concentration, kmole/m3 Effective diffusivity, m2/s Activation energy of reaction, k J/kmole Performance index, dimensionless First order reaction rate constant, 1/s Pre-exponential factor, l/s Slab thickness, m Molecular weight of the aggregate, kglkmole

R c90

Universal gas constant, k J/( kmole . K) Initial surface area per unit volume, m2/m3

T t tf i

Operating temperature, K Time, s Final time, s Dimensionless distance

can Ceramic Sociefy 73(8), 2274.

Lackey, W. & Starr, T. (1990), Fiber-Reinforced

Ceramic Composites: Materials, Pnxessing and Technology, Noyes Publications, chapter Fabrica-

tion of Fiber-Reinforced Ceramic Composites by Chemical Vapor Infiltration: Processing, Structure and Properties. Melkote, R. & Jensen, K. (1989), ‘Gas diffision in random-fiber substrates’, AIChE Journal 35( 12), 1942.

Naslain, R. & Langlais, F. (1986), Tailoring Multiphase and Composite Ceramics, Vol. 20, Plenum Press, New York, NY, chapter CVD-Processing of Ceramic-Ceramic Composite Materials, p. 145. Press, W., Teukolsky, S., Vetterling, W. & Fhnmerv, B. ( 1992). Numerical Recipes ii Fortran: The A-rt of Scientific Computing, 2 edn, Cambridge University Press. Tai, N.-H. & Chou, T.-W. (1989), ‘Analytical modeling of chemical vapor infiltration in fabrication of ceramic composites’, Journal of the American Ceramic Society 72(3), 4 14.

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Figure 2: Difference between optimal and sub-optimal temperature profiles for a fixed processing time (30 days).