Article Contents
Article Contents

Modeling, approximation, and time optimal temperature control for binder removal from ceramics

• * Corresponding author: Stephen J. Lombardo
• The process of binder removal from green ceramic components-a reaction-gas transport problem in porous media-has been analyzed with a number of mathematical techniques: 1) non-dimensionalization of the governing decomposition-reaction ordinary differential equation (ODE) and of the reaction gas-permeability partial differential equation (PDE); 2) development of a pseudo steady state approximation (PSSA) for the PDE, including error analysis via $L^2$ norm and singular perturbation methods; 3) derivation and analysis of a discrete model approximation; and 4) development of a time optimal control strategy to minimize processing time with temperature and pressure constraints. Theoretical analyses indicate the conditions under which the PSSA and discrete models are viable approximations. Numerical results indicate that under a range of conditions corresponding to practical binder burnout conditions, utilization of the optimal temperature protocol leads to shorter cycle times as compared to typical industrial practice.

Mathematics Subject Classification: Primary: 35Q92, 35Q93, 93C95; Secondary: 34E15.

 Citation:

• Figure 1.  Top panel: Centerline pressure versus temperature for linear temperature rise test case $\beta = 5.6 \times 10^{-4}$ K/s (that is $T(t) = \beta t+300$) and green body length $L = 0.01$ m using numerical experiments with PDE model (5), PSSA (46), and three-station model (64). Bottom three panels: Relative centerline PSSA error for green body lengths (from top to bottom) $0.001$ m, $0.01$ m and $0.1$ m

Figure 2.  Numerically approximated graphs of $L^2$ norms of difference between $\epsilon \rho$ and corresponding PSSA versus temperature in physical units ([$\rho \sqrt{L/2}$]) = mol/m$^{5/2}$ for the linear temperature rise (57) test case where $\beta = 5.6\times 10^{-4}$ and green body lengths $L = 0.001, 0.01, 0.1$ m (depicted in order from bottom to top). The $L^2$ error is obtained using finite-difference approximation (MacCormack's method) of the solution of the model equations (5) and numerical quadrature for the $L^2$ norm, and the theoretical bound is computed likewise from the solution of the ODE (55)

Figure 3.  Numerically approximated graphs of the absolute value of the quality function and the absolute difference between $u: = \epsilon \rho$ and the corresponding PSSA versus temperature (both computed at the centerline) for the linear temperature rise (57) test case where $\beta = 5.6\times 10^{-4}$ and three green body lengths $L = 0.001, 0.01, 0.1$ m (depicted in order from bottom to top). The solution of the model equations (5) is approximated using MacCormack's method, and the quality function is computed using (44). Note: To benchmark the values, $\rho_0 = 40$ mol/m$^3$

Figure 4.  Graphs of numerical approximations of pressure versus temperature for the linear temperature rise test case $\beta = 5.6 \times 10^{-4}$ K/s (that is $T(t) = \beta t+300$) are depicted for a solid cylinder of height $0.03$ cm and radius $0.005$ cm using the PDE model (5) and corresponding PSSA at the geometric center computed using one (PSSA$_1$), two (PSSA$_2$), and ten (PSSA$_{10}$) of its (transport scaled) power series representation (29) with post process conversion to corresponding physical units

Figure 5.  Top panel depicts optimal temperature protocol versus time for green body length $0.01$ m, maximum temperature rise rate of 10 K per minute and maximum gas pressure $200\, 000$ Pa for PDE model, PSSA, and three-station model. Bottom panel shows pressure versus temperature during the optimal temperature protocol for the three models

Figure 6.  Top panel depicts for the three-station model an approximation of the time-optimal temperature protocol for a green body of length $0.01$ m for maximum temperature rise rates $0.015$, $0.2$, $0.5$, $10.0$ K per minute and maximum allowed gas pressure $200\,000$ Pa. Bottom panel depicts corresponding pressure versus temperature

Figure 7.  Top panel depicts a comparison of the time optimal temperature protocols for the solid cylinder of height 3 cm and radius 0.5 cm as computed using the PDE model and ten terms of the series expansion representation (29) of its corresponding PSSA for the case of maximum heating rate 10 K per minute and maximum allowed gas pressure $150\,000$ Pa, where the maximum is computed at the geometric center of the cylinder. Bottom panel depicts corresponding geometric center gas pressure versus temperature

Table 1.  Model Parameters

 Name Value Description $T_0$ 300.0 K initial temperature $P_0$ $1.0\times 10^5$ Pa ambient gas pressure $R$ 8.314 J/mol-K ideal gas constant $M$ $4.4\times 10^{-2}$ kg/mol molecular weight of gas $E$ $2.22\times 10^5$ J/mol activation energy $A$ $1.67 \times 10^{16}$ /s pre-exponential factor $\mu$ $2.5\times 10^{-5}$ Pa-s viscosity of gas $S$ $6\times 10^6$/m specific surface $k$ $5$ tortuosity $\epsilon_{20}$ 0.4 initial binder volume fraction $\epsilon_3$ 0.5 ceramic volume fraction $\rho_2$ $1.0\times 10^3$ kg/m$^3$ polymer density $L$ 0.001–0.1 m green body length
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