Modeling of forced flow-thermal gradient CVI: effects of control parameters and processing schemes on a densification

: A two-channel temperature-control infiltration reactor aiming at fabricating thicker and more uniform composites was designed. Unsteady numerical simulations containing hydrodynamics, mass transfer, heat transfer and pore structure evolution were carried out. By analyzing the effects of concentration, velocity and temperature on the porosity distribution, different densification schemes were proposed and compared. Results show that a preform of thickness 0.04 m and initial porosity 0.8 can be densified to a uniform and compact one with the maximum porosity of 0.1 within 300 hours through the three-step densification scheme. This scheme not only assures a low and uniform porosity over the whole preform but also avoids the huge pumping pressure which may cause the preform cracking.


Introduction
The ceramic matrix composites (CMCs) have continued to be of special concern that promises high safety and economic benefits in the fields of aeronautic and aerospace due to their excellent thermal and mechanical properties [1][2]. Its main fabrication technique is chemical vapor infiltration (CVI), whereby silicon carbide densification going on, the top hot side of the preform may be quite dense while other portions are still coarse. For instance, the coating thickness is about 8 μm at the cold side while 22-24 μm at the hot side [14]. This phenomenon gives rise to two problems.
One is that the preform cannot be thick. The densified preform reported in Refs. [13][14][15][16] are quite thin, only 0.64-1.27 cm. The other one is that as the infiltration progressed, the back pressure increased as the pores in the preform were filled by the deposited matrix, the back pressure increased from the initial 3.4 KPa to a final 170 KPa in Ref. [13]. The huge pump pressure may cause preform cracking [17].
In order to solve these problems, we designed a two-channel F-TGCVI reactor, which is expected to obtain a thicker and more uniform composite material. Since running timescale experiments to test this design or optimize the parameters can be very expensive [18], the numerical simulation might be an economically viable way.
In F-TGCVI, the control parameters, such as temperature, pressure, the gas composition and the flow rate, all affect chemical reaction and furtherly change the pore structure. The evolution of pore structure, in turn, will affect the species transport and temperature distribution. Therefore, it is not appropriate to simulate the infiltration process with an oversimplified mathematic model or assumption of a steady state of temperature and concentration [19][20]. In the present paper, a two-dimensional unsteady numerical simulation was carried out, in which hydrodynamics, mass transfer, heat transfer and pore structure evolution were considered. The effects of inlet reagent concentration, inlet gas velocity and temperature on the porosity distribution were analyzed and an improved three-step densification scheme was also given. 4 In the present paper, precursor methyltrichlorosilane (CH 3 SiCl 3 , MTS), carrier gas hydrogen and carbon fiber are adopted to fabricate C f /SiC composites through the two-channel F-TGCVI reactor. Since the complex relationships among the time-dependent variables, such as chemical reaction, concentration, temperature and porosity, will make it difficult to model the deposition in the preform, some necessary assumptions are considered:

Numerical scheme
 The pyrolysis of MTS is regarded as a first order reaction [21][22], and the reaction is simplified as CH 3 SiCl 3 → SiC (s) + 3HCl. The heat generation by the chemical reaction is neglected.
 The porosity is considered uniform in the porous preform before infiltration, and the model of the surface area distribution between and within bundles proposed in Ref. [19] is also applicable to this work.
 All of the gases are considered as incompressible ideal gas, and the variation of their certain properties, such as density, dynamic viscosity, diffusion coefficient with temperature are neglected.

Physical model
The physical model, as shown in Fig. 1, contains two rectangle channels (the horizontal one 0.1× 0.5 m 2 , the vertical one 0.1 × 0.25 m 2 ) and a 0.1 × 0.04 m 2 rectangle porous preform locating at the junction. A, B and C are velocity inlet or pressure outlet, reactor walls are thermal insulation and a certain temperature is assigned to the top and bottom edges of the preform (in practice, this can be achieved by the focused IR heating technology). Based on this two-channel F-TGCVI reactor, the flow control and the temperature control can be closely matched to optimize the deposition process, thus fabricating thick and uniform composite, simultaneously protecting the fiber preform from cracking.

Mass transport
According to mass conservation, chemical species transport through diffusion and convection can be expressed as [23]: where ε denotes the porosity of preform; c i is the concentration of species i, mol/m 3 ; t is the infiltration time, s; u is the velocity field, m/s; D eff denotes the effective diffusion coefficient, m 2 /s, eff = AB DD , τ is the tortuosity factor, 0.95 0 ( 0.03) / ( 0.03) [6], D AB is the Fick diffusion coefficient [24], -7 1.75 1/2 the subscript A, B represents MTS and H 2 ; T is the temperature, K; M is the molecular weight, kg/mol; p is the absolute pressure, atm; ∑V is the diffusion volume, a sum of atomic volumes in each molecule. R i is the reaction rate expression for the species i, mol/(m 3 ·s), for the reagent (MTS, i = 1), for example, R 1 can be expressed as: k s is the reaction rate constant, m/s, s 0 r = exp( R ) k k E T , k 0 is the preexponential factor, m/s, Er is the activation energy, J/mol, and R is the gas constant, J/(K·mol); c 1 is the concentration of MTS, mol/m 3 , initial MTS concentration c 1,0 = χp/RT, χ is the molar fraction of MTS; S v is the surface area per unit volume, 1/m, , r 0 and ε 0 are the initial pore radius (m) and porosity, respectively. The deposition of SiC (i = 2) causes the progressive plugging of the preform, therefore, the preform structural evolution can be written as: M 2 is the molecular weight of SiC, kg/mol; ρ 2 is the density of SiC, kg/m 3 .

Heat transfer
According to local thermal equilibrium, heat transfer equation in the porous domain is [26]: where (ρC p ) eff is the effective volumetric heat capacity, consisting of porous medium term and gas mixture term, , k eff is the effective thermal conductivity, W/(m·K), eff p =1 k k k , the 'irradiative conductivity' [27][28] is added to the k p ; Q is the heat source, W/m. Heat transfer equation in the free domain is similar except ruling out the porous medium term.

Numerical solver
The COMSOL Multiphysics 5.5 was adopted to conduct the present two-dimension  Table 1.   (Fig. 2c). These may mainly attribute to the deposition, that is, the increase of infiltration time or inlet MTS concentration results in more deposition, which will raise the thermal conductivity and induce more heat transferring upward from the bottom high-temperature zone (Fig. 3b). Given that the precursor flows downwards, the upward heat gives rise to the moving upward of the largest slope position.   Furtherly, we checked the MTS concentration and temperature distribution on the line l v to find more information about the effect of inlet velocity. As shown in Fig. 5a, the increase of inlet velocity can improve the MTS concentration on the whole line l v , but more obvious at the upside (top portion of the preform). Inlet velocity also has positive effect on the temperature distribution, but in a tiny degree (Fig. 5b). To sum up, the increase of inlet velocity will accelerate the deposition by increasing the concentration of MTS in the preform, but it is neither effective nor economic due to the large H 2 /MTS molar ratio. Hence, a moderate inlet velocity will be better for guaranteeing a forced-flow and an economic densification.

The effect of temperature
The effect of temperature was also investigated. Under conditions of A in C out, inlet MTS concentration 0.186 mol/m 3 and inlet velocity 0.0127 m/s, a number of cases with different top-bottom temperature pairs were simulated. Fig. 6 presents four typical cases: in the case one T top = 900 K and T bottom = 1150 K, the bottom preform was firstly densified (Fig. 6a) due to the excessively low top temperature; things are changed in the case of T top = 1100 K, T bottom = 1200 K. The bottom temperature is still higher than the top temperature, but the top temperature 1100 K is high enough for reaction, thus densification mainly occurs at the top portion (Fig. 6b) where the precursor concentration is higher; when T bottom remains at 1200 K, but the T top is lowered to 1050 K. The porosity distribution over the whole preform notably changes once again. As shown in Fig. 6c, the most distinguished porosity decrease occurs at the middle portion; when the preform is kept iso-thermal, 1000 K, the densification over the whole preform is quite uniform, but the deposition is very slow (Fig. 6d). In the cases of Fig. 6b and Fig. 6c, the only difference is the top temperature, i.e., 1100 K in the case b and 1050 K in the case c, but this small temperature difference causes quite different porosity distribution. Initially, as shown by Fig.7, the porosity difference of the two cases is quite small in magnitude (about 0.003) at t = 1 h. The distribution trend, however, is already quite evident. That is, in the case b, porosity at the top portion of the preform is smaller due to the higher top temperature (more intense reaction); while reaction at the top portion in the case c is less intense, thus more precursors reaching to the middle-lower portion where the temperature is higher, as a result, the middle-lower portion of the preform has the lowest porosity (Fig. 7). The initial porosity distribution will strongly affect the successive distribution of some parameters, such as surface area per unit volume, MTS concentration and temperature. According to the relationship between surface area per unit volume and porosity [19], the decrease of porosity will cause the surface area to increase first and then decrease (Fig. 8). Therefore, the porosity distribution trend in Fig. 7 will lead to a different surface area distribution in the two cases, i.e., larger surface area at the top portion in the case b while larger surface area at the middle portion in the case c, as shown in Fig. 9a and Fig. 9d. Likewise, the MTS concentration also closely relates to the porosity distribution: in the case b, the porosity at the top portion of the preform drops rapidly, which means a gradually plugging at the top portion and results in more precursors staying at the top portion (Fig. 9b); while lower top temperature and coarser top preform in the case c induce a higher MTS concentration through the whole preform (Fig. 9e). In terms of the temperature, as explained in section 3.1, the porosity relates to the thermal conductivity and furtherly affects the temperature distribution. In the case b, the thermal conductivity at the top portion is higher, corresponding to the smaller temperature gradient at the top portion (Fig. 9c), while in the case c, the smaller temperature gradient occurs near the middle portion (Fig. 9f).
As we should remember, the temporal distribution of porosity is proportional to the reaction rate R 1 , more specifically, it is proportional to the surface area, MTS concentration and temperature ( 1 That is to say, the initial parameters bring about an initial porosity distribution trend, this initial porosity distribution furtherly induces a corresponding parameter change, which in turn, will enhance the porosity distribution trend.

The two-step densification scheme
According to Fig. 6a and Fig. 6b, a two-step densification scheme which has the possibility to simply densify the preform was proposed: The gas mixture first flows from A to C (Fig. 10a) to densify a half of the preform, and then the gas flows form C to B (Fig. 10b) to finish the densification of the remaining half. Cases with different parameters were tried, and table 2 lists a viable one. In this case, porosity at the upper portion quickly drops due to the suitable temperature and fresh precursors, and the other portion of the preform is also densified to some degree with the aid of high bottom temperature 1200 K (Fig. 11a). In the next step, the top and bottom temperature arrangement is reversed. The gas mixture flows from C to B, and the lower portion of the preform is densified (Fig. 11b).  As can be noticed, the infiltration time in the two steps are short and the final porosity is rather high (Fig. 11b). Actually, cases with longer infiltration time were tried but they all failed. The reason lies in the high reaction rate at a specific region.
When the gas flows from A to C, the upper left corner of the preform has the thinnest concentration boundary layer, i.e., the highest concentration. Moreover, the temperature there (1100 K) is suitable for the pyrolysis, thus the overlay of high concentration and temperature giving rise to an extremely low porosity zone at the upper left corner (Fig. 12a). The porosity of somewhere at the upper left corner decreases to zero when the total infiltration time exceeds 90 hours. As a result, the simulation cannot go on even if the maximum porosity over the entire preform is still as high as 0.5 (Fig. 12b).

The three-step densification scheme
To avoid the overlay of high concentration and high temperature, a three-step densification scheme was proposed: In the first step the gas flows from A to C (Fig.   13a), and deposition mainly occurs at the bottom portion where temperature is high; in the middle step the gas still flows from A to C (Fig. 13b), but the top and bottom temperature is adjusted so that the middle portion can be densified; due to the plugging of most of the preform, if we still pump all the gas out from the outlet C, the huge stress may cause preform cracking [17], thus in the final step the gas mixture flows from A to B (Fig. 13c), and precursors diffuse into the top portion to densify the remaining coarse top preform.   (Fig. 13d), and porosity near the bottom preform decreases significantly as expected (Fig. 14a). In the next step, with higher top and bottom temperature, the reaction zone moves upward, the precursor is almost depleted at the middle portion ( Fig. 13e), and the middle portion of the preform is densified (Fig. 14b). In these two steps, as the lower part of the preform first gets plugged, the porosity gradient induces an oscillatory flow in the preform (Fig. 13a-b). In the final step, the precursor diffuses into the top preform (Fig. 13f) and the remaining coarse top preform is also densified ( Fig. 14c) The three-step densification scheme significantly lowers the porosity of the whole preform, but the uniformity still needs to be improved. In the first two steps the deposition zone gradually moves from the bottom to the top. There is an obvious stratification and porosity in each layer is quite uniform, as shown by Fig. 15a and Fig.   15b. In the step III, however, the unwanted impermeable skin arises at the upper left corner of the preform while porosity under the right surface is still as high as 0.19 ( Fig. 15c). Likewise, the two blue high-porosity strips near the bottom preform are also very eye-catching. Three measures were taken to improve the three-step densification scheme. 1 (Fig.16). The detail parameters of improved three-step densification scheme are given in table 4. As can be seen from Fig. 17, the densification situation is significantly improved, i.e., the porosity over whole preform is not only lower but also more uniform (Fig. 17b).

Conclusions
In the present paper a two-channel temperature-control infiltration reactor aiming at fabricating thicker and more uniform composites was designed. The effects of concentration, velocity and temperature on the porosity distribution were first analyzed, based on which different densification schemes were proposed and compared. By adopting improving measures, an optimal densification scheme was given.
(1) The densification rate can be accelerated by increasing the inlet MTS concentration (MTS molar ratio) or inlet velocity (total gas flow), while the former is more effective. The deposition behavior is mainly determined by the top and bottom temperature, a small temperature difference can dramatically change the porosity distribution. Temperature, concentration and velocity determine the porosity evolution, the porosity evolution, in turn, also affects the distribution of these parameters.
(2) A three-step densification scheme based on the two-channel reactor is proposed: In the first step the gas flows through the preform whose top temperature is not suitable for reaction, and deposition mainly occurring at the high-temperature bottom portion; in the middle step the top and bottom temperature is adjusted to assure certain deposition occurring at the middle portion; in the final step, the two outlets (pump) are all open, and a linearly increasing temperature is applied to the top edge of the preform, whereby the coarse top portion is furtherly densified. This scheme avoids the unwanted sealing of preform by separating the high precursor concentration region from the high temperature region. Besides, in the step III a part of the gas is pumped out from the outlet C, and the main flow doesn't go through the dense preform, which not only assures the precursor supply in the preform but also protects the fiber preform from cracking due to the huge pumping pressure.