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Optimal control of leachate recirculation for anaerobic processes in landfills
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 |
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.
References:
[1] |
M. Bisi, F. Conforto and L. Desvillettes,
Quasi-steady-state approximation for reaction-diffusion equations, Bull. Inst. Math., 2 (2007), 823-850.
|
[2] |
J. Elderling, Normally hyperbolic invariant manifolds. The noncompact case, Atlantis Studies in Dynamical Systems, 2 (2013).
doi: 10.2991/978-94-6239-003-4. |
[3] |
J. R. G. Evans, M. J. Edirisinghe, J. K. Wright and J. Crank, On the removal of organic vehicle from moulded ceramic bodies, Proc. R. Soc. London A, 432 (1991), 321-340. Google Scholar |
[4] |
K. Feng and S. J. Lombardo,
Modeling of the pressure distribution in three-dimensional porous green bodies during binder removal, J. Am. Ceram. Soc., 86 (2003), 234-240.
doi: 10.1111/j.1151-2916.2003.tb00005.x. |
[5] |
N. Fenichel,
Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1972), 193-226.
doi: 10.1512/iumj.1972.21.21017. |
[6] |
N. Fenichel,
Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eqs., 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[7] |
S. J. Fraser,
The steady state and equilibrium approximations: A geometrical picture, J. Chem. Phys., 88 (1988), 4732-4738.
doi: 10.1063/1.454686. |
[8] |
R. M. German, Theory of thermal debinding, Int. J. Powder Metall., 23 (1987), 237-245. Google Scholar |
[9] |
K. Kim and Y. Yao,
The Patlak–Keller–Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602.
doi: 10.1137/110823584. |
[10] |
J. A. Lewis,
Binder removal from ceramics, Annual Rev. Mater. Sci., 27 (1997), 147-173.
doi: 10.1146/annurev.matsci.27.1.147. |
[11] |
L. C-K. Liau and C-C. Chiu,
Optimal heating strategies of polymer binder burnout process using dynamic optimization scheme, Ind. Che. Res., 44 (2005), 4586-4593.
doi: 10.1021/ie049143a. |
[12] |
L. C.-K. Liau, B. Peters, D. S. Krueger, A. Gordon, D. S. Viswanath and S. J. Lombardo,
Role of length scale on pressure increase and yield of poly(vinyl butyral)-barium titanate-platinum multilayer ceramic capacitors during binder burnout, J. Am. Ceram. Soc., 83 (2000), 2645-2653.
doi: 10.1111/j.1151-2916.2000.tb01609.x. |
[13] |
K. C. Liddell,
Shrinking core models in hydrometallurgy: What students are not being told about the pseudo-steady approximation, Hydromet., 79 (2005), 62-68.
doi: 10.1016/j.hydromet.2003.07.011. |
[14] |
J. D. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley, Hoboken, 2008.
doi: 10.1002/9780470287095. |
[15] |
S. J. Lombardo, Minimum time heating cycles for diffusion-controlled binder removal from ceramic green bodies, J. Amer. Ceram. Soc., 98 (2015), 57-65. Google Scholar |
[16] |
S. J. Lombardo and Z. C. Feng,
Pressure distribution during binder burnout in three-dimensional porous ceramic bodies with anisotropic permeability, J. Mat. Res., 17 (2002), 1434-1440.
doi: 10.1557/JMR.2002.0213. |
[17] |
S. J. Lombardo and Z. C. Feng,
Analytic method for the minimum time for binder removal from three-dimensional porous green bodies, J. Mat. Res., 18 (2003), 2717-2723.
doi: 10.1557/JMR.2003.0378. |
[18] |
S. J. Lombardo and D. G. Retzloff,
A process control algorithm for reaction-diffusion minimum time heating cycles for binder removal from green bodies, J. Amer. Ceram. Soc., 102 (2019), 1030-1040.
doi: 10.1111/jace.15964. |
[19] |
S. J. Lombardo and D. G. Retzloff,
Reaction-permeability optimum time heating policy via process control for debinding green ceramic components, Adv. Appl. Ceram., 119 (2020), 150-157.
doi: 10.1080/17436753.2019.1707393. |
[20] |
L. A. Segel and M. Slemrod,
The quasi-steady-state assumption in biochemistry: A case study in perturbation, SIAM Rev., 31 (1989), 446-477.
doi: 10.1137/1031091. |
[21] |
I. Stakgold, K. B. Bischoff and V. V. Gokhale,
Validity of the pseudo-steady-state approximation, Int. J. Engng. Sci., 21 (1983), 537-542.
doi: 10.1016/0020-7225(83)90101-5. |
[22] |
G. Y. Stangle and I. A. Aksay,
Simultaneous momentum, heat and mass transfer with chemical reaction in a disordered porous medium: Application to binder removal from a ceramic green body, Chem. Eng. Sci., 45 (1990), 1719-1731.
doi: 10.1016/0009-2509(90)87050-3. |
[23] |
T. A. Turányi, S. Tomlin and M. J. Pilling, On the error of the quasi-steady-state approximation, J. Phys. Chem., 97 (1993), 63-172. Google Scholar |
[24] |
D-S. Tsai,
Pressure buildup and internal stresses during binder burnout: numerical analysis, AIChE J., 37 (1991), 547-554.
doi: 10.1002/aic.690370408. |
[25] |
J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Clarendon Press, Oxford, 2007.
![]() |
show all references
References:
[1] |
M. Bisi, F. Conforto and L. Desvillettes,
Quasi-steady-state approximation for reaction-diffusion equations, Bull. Inst. Math., 2 (2007), 823-850.
|
[2] |
J. Elderling, Normally hyperbolic invariant manifolds. The noncompact case, Atlantis Studies in Dynamical Systems, 2 (2013).
doi: 10.2991/978-94-6239-003-4. |
[3] |
J. R. G. Evans, M. J. Edirisinghe, J. K. Wright and J. Crank, On the removal of organic vehicle from moulded ceramic bodies, Proc. R. Soc. London A, 432 (1991), 321-340. Google Scholar |
[4] |
K. Feng and S. J. Lombardo,
Modeling of the pressure distribution in three-dimensional porous green bodies during binder removal, J. Am. Ceram. Soc., 86 (2003), 234-240.
doi: 10.1111/j.1151-2916.2003.tb00005.x. |
[5] |
N. Fenichel,
Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1972), 193-226.
doi: 10.1512/iumj.1972.21.21017. |
[6] |
N. Fenichel,
Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eqs., 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[7] |
S. J. Fraser,
The steady state and equilibrium approximations: A geometrical picture, J. Chem. Phys., 88 (1988), 4732-4738.
doi: 10.1063/1.454686. |
[8] |
R. M. German, Theory of thermal debinding, Int. J. Powder Metall., 23 (1987), 237-245. Google Scholar |
[9] |
K. Kim and Y. Yao,
The Patlak–Keller–Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602.
doi: 10.1137/110823584. |
[10] |
J. A. Lewis,
Binder removal from ceramics, Annual Rev. Mater. Sci., 27 (1997), 147-173.
doi: 10.1146/annurev.matsci.27.1.147. |
[11] |
L. C-K. Liau and C-C. Chiu,
Optimal heating strategies of polymer binder burnout process using dynamic optimization scheme, Ind. Che. Res., 44 (2005), 4586-4593.
doi: 10.1021/ie049143a. |
[12] |
L. C.-K. Liau, B. Peters, D. S. Krueger, A. Gordon, D. S. Viswanath and S. J. Lombardo,
Role of length scale on pressure increase and yield of poly(vinyl butyral)-barium titanate-platinum multilayer ceramic capacitors during binder burnout, J. Am. Ceram. Soc., 83 (2000), 2645-2653.
doi: 10.1111/j.1151-2916.2000.tb01609.x. |
[13] |
K. C. Liddell,
Shrinking core models in hydrometallurgy: What students are not being told about the pseudo-steady approximation, Hydromet., 79 (2005), 62-68.
doi: 10.1016/j.hydromet.2003.07.011. |
[14] |
J. D. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley, Hoboken, 2008.
doi: 10.1002/9780470287095. |
[15] |
S. J. Lombardo, Minimum time heating cycles for diffusion-controlled binder removal from ceramic green bodies, J. Amer. Ceram. Soc., 98 (2015), 57-65. Google Scholar |
[16] |
S. J. Lombardo and Z. C. Feng,
Pressure distribution during binder burnout in three-dimensional porous ceramic bodies with anisotropic permeability, J. Mat. Res., 17 (2002), 1434-1440.
doi: 10.1557/JMR.2002.0213. |
[17] |
S. J. Lombardo and Z. C. Feng,
Analytic method for the minimum time for binder removal from three-dimensional porous green bodies, J. Mat. Res., 18 (2003), 2717-2723.
doi: 10.1557/JMR.2003.0378. |
[18] |
S. J. Lombardo and D. G. Retzloff,
A process control algorithm for reaction-diffusion minimum time heating cycles for binder removal from green bodies, J. Amer. Ceram. Soc., 102 (2019), 1030-1040.
doi: 10.1111/jace.15964. |
[19] |
S. J. Lombardo and D. G. Retzloff,
Reaction-permeability optimum time heating policy via process control for debinding green ceramic components, Adv. Appl. Ceram., 119 (2020), 150-157.
doi: 10.1080/17436753.2019.1707393. |
[20] |
L. A. Segel and M. Slemrod,
The quasi-steady-state assumption in biochemistry: A case study in perturbation, SIAM Rev., 31 (1989), 446-477.
doi: 10.1137/1031091. |
[21] |
I. Stakgold, K. B. Bischoff and V. V. Gokhale,
Validity of the pseudo-steady-state approximation, Int. J. Engng. Sci., 21 (1983), 537-542.
doi: 10.1016/0020-7225(83)90101-5. |
[22] |
G. Y. Stangle and I. A. Aksay,
Simultaneous momentum, heat and mass transfer with chemical reaction in a disordered porous medium: Application to binder removal from a ceramic green body, Chem. Eng. Sci., 45 (1990), 1719-1731.
doi: 10.1016/0009-2509(90)87050-3. |
[23] |
T. A. Turányi, S. Tomlin and M. J. Pilling, On the error of the quasi-steady-state approximation, J. Phys. Chem., 97 (1993), 63-172. Google Scholar |
[24] |
D-S. Tsai,
Pressure buildup and internal stresses during binder burnout: numerical analysis, AIChE J., 37 (1991), 547-554.
doi: 10.1002/aic.690370408. |
[25] |
J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Clarendon Press, Oxford, 2007.
![]() |







Name | Value | Description |
300.0 K | initial temperature | |
ambient gas pressure | ||
8.314 J/mol-K | ideal gas constant | |
molecular weight of gas | ||
activation energy | ||
pre-exponential factor | ||
viscosity of gas | ||
specific surface | ||
tortuosity | ||
0.4 | initial binder volume fraction | |
0.5 | ceramic volume fraction | |
polymer density | ||
0.001–0.1 m | green body length |
Name | Value | Description |
300.0 K | initial temperature | |
ambient gas pressure | ||
8.314 J/mol-K | ideal gas constant | |
molecular weight of gas | ||
activation energy | ||
pre-exponential factor | ||
viscosity of gas | ||
specific surface | ||
tortuosity | ||
0.4 | initial binder volume fraction | |
0.5 | ceramic volume fraction | |
polymer density | ||
0.001–0.1 m | green body length |
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