Optimal control and CFD modeling for heat flux estimation of a baking process

doi:10.1016/j.compchemeng.2011.10.011

Computers & Chemical Engineering

Volume 38, 5 March 2012, Pages 139-150

https://doi.org/10.1016/j.compchemeng.2011.10.011 Get rights and content

Abstract

An optimal control method was combined with commercial CFD software for the optimization of heat flux during a baking process. The objective function was defined in terms of the transient heat flux, the temperature and the humidity of the baking product. The optimal control was achieved by the conjugate gradient method. The minimum energy consumption for a desired final baking product temperature and humidity was estimated. This study confirmed the relationship between the heat flux and baking product quality attributes and showed that optimal control models represent reliable tools to support decision making for complex processes such as baking. The proposed optimal control approach is general and could be easily extended to other processes.

Introduction

Energy consumption is of prime importance for industrial operations. The diminishing supply of non-renewable sources of energy and the negative impact on the environment caused by the intensive and uncontrolled use of energy are receiving more attention. Potential cost savings resulting from the reduction of the energy consumption can be obtained. The decrease of the energy consumption (optimal) should be achieved with minimal effect on the quality attributes of the final product (constraints). This can be viewed as an optimal control problem.

The optimal control approach is based on a set of partial differential equations (PDE) developed from first principles. Two families of methods are employed in PDE-constrained optimal control (Thorne, 2011):

Optimize-then-Discretize (OD) method, based on variational approach (Kameswaran & Biegler, 2006).
Discretize-then-Optimize (DO) method, based on Non-Linear Programming (NPL) solvers for optimization (Kameswaran & Biegler, 2006).

Collis and Heinkenschloss (2002) discussed and compared theoretically these two methods for steady-state advection diffusion problems with distributed control. Numerical tests have shown that the DO and the OD techniques give similar results. Differences between the two methods are associated with the problem formulation and the properties of the system of equations. The OD approach does not preserve the symmetry property and leads to asymptotically improved approximate solutions. The OD approach is less sensitive to the mesh properties and improves the approximation of the adjoint variable. The OD formulation allows for the integration of the optimal control technique with any Computational Fluid Dynamics (CFD) or Multiphysics software in a non-intrusive manner where modifications in the solver are not needed. The development is obtained in the continuous domain (time and space) and the discretization is carried out by the commercial software. The OD approach can also be used in inverse methods with models represented by PDE. Chen and Yang (2011) estimated the heat flux at the surface with an inverse method based on the adjoint variable method and the conjugate gradient algorithm.

The DO approach generates symmetrically linear systems that can be handled with the symmetrical iterative method, and a consistent expression between the discretized states, the discretized objective function and the discretized adjoint function are obtained. In this approach, the exact adjoint of the discretized state and the objective function are obtained. However, because of the explicit nature of the discretization of the PDE and the need for a link with NLP solvers, the DO approach is difficult to combine with commercial CFD software. Indeed most studies developed in this area are conducted with in-house codes.

The DO approach was applied to several chemical processes. Zhang, Hidajat, and Ray (2004) presented a simulation and multi-objective genetic optimization of the production of fructose syrup from glucose isomerization in packed bed reactors. Zavala and Biegler (2010) adopted the DO approach to obtain optimal strategies for tubular reactor operations. Estrada, Parodi, and Diaz (2009) used the DO approach to identify the parameters of an eutrophication model. The optimal control for diffusion, convection and reaction processes was obtained by combining spatial discretization and proper orthogonal decomposition producing reduced order models solved subsequently with NLP solvers (Li & Christofides, 2008).

Banga, Balsa-Canto, Moles, and Alonso (2003) reviewed optimization methods for thermal food processes and the DO approach. Recently, Luo, He, and Li (2008) and Alvarez-Vázquez, Fernández, and Muñoz-Sola (2008) presented theoretical results on the controllability of food processes and the existence and uniqueness of the optimal solution. Other applications include thermal sterilization (Alonso et al., 1998, Alvarez-Vázquez et al., 2004, Garcia et al., 2006, Simpson et al., 2007), drying (Luz et al., 2010, Temmerman et al., 2009, Wongrat et al., 2010) and cooking (Stigter, Scheerlinck, Nicolai, & Van Impe, 2001). Balsa-Canto, Alonso, and Banga (2002a), Balsa-Canto, Banga, and Alonso (2002b) presented a model reduction technique based on proper orthogonal decomposition to overcome the difficulty in solving PDE constrained problems and to minimize the computation time.

Baking operations are characterized by their high energy consumption (Le-bail et al., 2010). Energy optimization for this operation is important, but is constrained by the product quality attributes, mainly color (Maillard reactions and caramelization) and texture (softness and crispiness). There are a limited number of studies on baking product quality optimization. Baucour, Cronin, and Stynes (2003) developed process optimization strategies for the reduction of the variability in the product quality. Their approach considered a simplified energy model and only one decision variable (ambient temperature). Hadiyanto, Esveld, Boom, Van Straten, and Van Boxtel (2008a), Hadiyanto, Esveld, Boom, Van Straten, and Van Boxtel (2008b) estimated the optimal heat flux for a target product quality. The optimal heat flux was computed by two methods: continuous heat flux from the Hamiltonian minimization and the discrete heat flux from the control vector minimization. The initial value problem of coupled PDE was discretized into 108 ordinary differential equations. The optimal control problem was solved according to the Bryson's method. The objective function was limited to the product quality and did not contain an energy component.

The objective of the present work was to optimize the transient heat flux requirements of bread baking in a pilot oven and ultimately reduce the energy consumption of this operation. For baking processes, the major product quality attributes, brownness, color and crispness are associated with the humidity and temperature distribution inside the product during the baking (Zanoni et al., 1995a, Zanoni et al., 1995b). Consequently, the transient humidity and temperature at the center of the product were selected to represent target product quality as optimization constraints. The optimal heat flux thus should meet quality constraints while globally minimizing the energy consumption.

The heat flux optimization was performed according to the classical optimal control theory. The OD method, combined with COMSOL software, was selected to solve the PDE-constrained problem. The mathematical formulation presented in (Marcos, Boulet, Ousegui, & Moresoli, 2011) was adopted. The basis of the implementation was the computation of the gradient of the objective function. The gradient computation was performed with the help of the adjoint system and the sensitivity function for the PDE. The PDE equations corresponding to the sensitivity function and the adjoint function were solved with COMSOL. The gradient computation then served to build a sequential conjugate gradient minimization to estimate the optimal heat flux.

Section snippets

Baking model structure

The bread baking was considered as a coupled heat and mass transfer problem taking place in a multiphase porous media, with the following simplifying assumptions:

•
all thermophysical properties of the bread are constant (Hadiyanto et al., 2008a);
•
the gas phase consists of water vapor (Thorvaldsson & Janestad, 1999);
•
the convective heat transfer and mass transfer are neglected (Ousegui, Moresoli, Dostie, & Marcos, 2010);
•
the moisture removed from the bread surface to the air in the oven is water

Optimal control formulation

The objective of the control problem was to find a minimal transient heat flux Q for a desired baking product quality, temperature and humidity. The associated objective function was given as: $J_{a} (Q) = \frac{1}{2} \int_{0}^{t_{f}} (ω_{T} {(T (t, 0,0) - T_{pr} (t, 0,0))}^{2} + {(W (t, 0,0) - W_{pr} (t, 0,0))}^{2} + ω_{q} \cdot Q^{2}) d t$ where T_pr(t, 0, 0), W_pr(t, 0, 0) were the temperature and humidity at the center of the baking product, ω_T was the scaling factor and ω_q was the weighting factor for the energy target.

With the dimensionless variables (3.1) became: $J (Q^{*}) = 1$

Adjoint problem

The adjoint problem was obtained by first transforming Eq. (3.5) in Eq. (3.6) with the adjoint variable λ = [λ_v,λ_w,λ_θ] and subsequently computing the gradient G. By using (3.7), the following inner product was zero and disappeared. ${(λ, A_{X} (δ X))}_{E^{3}} = 0$

The adjoint operator $A_{X}^{*}$ satisfied the following condition: ${(λ, A_{X} (δ X))}_{E^{3}} = {(A_{X}^{*} (λ), δ X)}_{E^{3}} + boundary terms = 0$

By writing the boundary terms as ${(c, δ Q^{*})}_{L^{2}}$ (4.2) was given as: $0 = {(A_{X}^{*} (λ), δ X)}_{E^{3}} + {(c, δ Q^{*})}_{L^{2}}$

Eq. (3.5) was given as: $δ J (δ Q^{*}) = {(i (X) (X - X_{\exp}) δ (τ, 0,0), δ X (δ Q^{*}))}_{E^{3}} + ω (δ$

Descent step

The vector δQ* = −G is a descent direction for the functional J. The descent step β in the direction δQ* was computed by the following equation. $J (Q^{*} + β δ Q^{*}) = \frac{1}{2} \int_{0}^{τ_{f}} [{(θ + β δ θ - θ_{\exp})}^{2} + {(W + β δ W - W_{\exp})}^{2} + ω \cdot {(Q^{*} + β δ Q^{*})}^{2}] d τ$

The minimum of the functional J with respect to β(dJ/dβ = 0) gave the descent step: $β = - \frac{\int_{0}^{τ_{f}} (δ θ (θ - θ_{\exp}) + δ W (W - W_{\exp}) + ω δ Q^{*} Q^{*}) d τ}{\int_{0}^{τ_{f}} ({(δ θ)}^{2} + {(δ W)}^{2} + ω δ Q^{*^{2}}) d τ}$

The flux Q*⁽ⁿ⁺¹⁾ was updated by the expression Q*⁽ⁿ⁺¹⁾ = Q*⁽ⁿ⁾ + βδQ*.

Mathematical properties of optimal control problem

For linear quadratic PDE problems, the existence and the uniqueness of the optimal solution can be proved for convex domain and regularity assumptions of the control variable. However, for non-linear PDE-constrained optimization problems, the existence and the uniqueness of the optimal solution are quite difficult to establish. The present PDE-constrained optimization has a quadratic objective function and the PDE is non linear (and defined as distribution equation i.e. Dirac function in the

Validation

Experimental baking product humidity and temperature were obtained from Hadiyanto et al. (2008a). In industrial baking oven operations, the oven chamber is generally preheated before the introduction of the products. Such conditions were obtained by setting the wall temperature at 70 ̊C which represents an initial heat flux Q*₀ = 1 (using Formula of Q* in sub Section 2.2). Four different values of the weighting factor, ω (1 × 10⁻⁰³, 5 × 10⁻⁰³, 1 × 10⁻⁰² and 2 × 10⁻⁰²), were considered and compared in the

Conclusion

An optimal control method, combined with commercial CFD software, was successful for the optimization of the heat flux at the surface of a baking product subjected to final temperature and humidity content of the product as specifications for the quality constraints. An objective function was defined in terms of the transient heat flux, the temperature and the humidity properties of the baking product. The proposed method consisted of solving three PDE systems (direct, adjoint and sensitivity)

Acknowledgements

This project is part of the R&D program of the NSERC Chair in Industrial Energy Efficiency established in 2006 at “Université de Sherbrooke”. The authors acknowledge the support of the Natural Sciences & Engineering Research Council of Canada, Hydro Québec, Rio Tinto Alcan and CANMET Energy Technology Center.

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Optimal control and CFD modeling for heat flux estimation of a baking process

Abstract

Introduction

Section snippets

Baking model structure

Optimal control formulation

Adjoint problem

Descent step

Mathematical properties of optimal control problem

Validation

Conclusion

Acknowledgements

Computers & Chemical Engineering

Journal of Differential Equations

Journal of Food Engineering

Journal of Food Engineering

Trends in Food Science & Technology

Journal of Food Engineering

Applied Mathematical Modelling

Journal of Food Engineering

Computers and Chemical Engineering

Journal of Food Engineering

Food and Bioproducts Processing

Computers and Chemical Engineering

Journal of Food Engineering

Computers and Chemical Engineering

Food and Bioproducts Processing

Journal of Food Engineering

Food Control