# American Institute of Mathematical Sciences

January  2022, 27(1): 103-140. doi: 10.3934/dcdsb.2021034

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

 1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA 2 Department of Mechanical and Aerospace Engineering, and Department of Chemical Engineering, Columbia, MO 65211, USA 3 Department of Chemical Engineering, University of Missouri, Columbia, MO 65211, USA

* Corresponding author: Stephen J. Lombardo

Received  September 2020 Revised  November 2020 Published  January 2022 Early access  February 2021

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.

Citation: Carmen Chicone, Stephen J. Lombardo, David G. Retzloff. Modeling, approximation, and time optimal temperature control for binder removal from ceramics. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 103-140. doi: 10.3934/dcdsb.2021034
##### References:

show all references

##### References:
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
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)
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$^3$
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
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
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
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
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
 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
 [1] Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185 [2] Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553 [3] Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations & Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042 [4] Ting Kang, Qimin Zhang, Haiyan Wang. Optimal control of an avian influenza model with multiple time delays in state and control variables. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4147-4171. doi: 10.3934/dcdsb.2020278 [5] Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021, 11 (3) : 555-578. doi: 10.3934/mcrf.2021012 [6] Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 [7] Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021, 8 (4) : 381-402. doi: 10.3934/jdg.2021013 [8] Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159 [9] Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021 [10] Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082 [11] Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211 [12] Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279 [13] Mario Ohlberger, Ben Schweizer. Modelling of interfaces in unsaturated porous media. Conference Publications, 2007, 2007 (Special) : 794-803. doi: 10.3934/proc.2007.2007.794 [14] Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519 [15] Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241 [16] Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control & Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007 [17] Changjun Yu, Shuxuan Su, Yanqin Bai. On the optimal control problems with characteristic time control constraints. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021021 [18] María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks & Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004 [19] Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations & Control Theory, 2020, 9 (2) : 375-398. doi: 10.3934/eect.2020010 [20] Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644

2020 Impact Factor: 1.327