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TMPE model: Multiphase model with equilibrium approach
TMPE here stands for Temperature, Moisture content and Pressure with Equilibrium approach. Due to implication of equilibrium assumption, the governing equations can be simplied and are clipped together. This eliminates the need for solving the mass conservation of vapor phase. In equation governing the moisture content, evaporation rate is replaced directly by water vapor equation and the total gas pressure is expressed from mass conservation of air. Hypothesis: The assumptions enlisted in previous model are common for this model too, except points 3 and 4. Other hypothesis used in this model are:
1. There is equilibrium between vapor and liquid water (which is basic for this model).
2. The combined mass uxes for liquid water and vapor at boundary nodes are replaced by convective mass loss of water vapor.
3. Gases obey Dalton’s law of partial pressure and ideal gas law.
Model validation with known properties
With the availability of mathematical formulations for models with dierent hypothesis, validation of them is carried out using available key parameters that are presented in table 2.4. These models are validated by comparing simulated trends with experimental ones. This can help to understand how well the model is formulated and if there are any con icts in their simulation. It is one of the foremost steps for selecting proper mathematical model that will be used in future for solving the inverse problem. The key parameters listed in table are obtained from literature [9, 10, 13] as average or eective values since these variables are presented as functions.
Diusive model
Simulated temperature and moisture content proles are plotted for sensor positions at x? = [0, 0.25, 0.50, 0.75, 1] in gure 2.11. Temperature proles seem to validate the heating process inside the medium. A stagnant temperature curve near 100C and a slight increase in its value after baking time of 600 s for bottom sensor location is observed. This variation mimics the phase of dough’s transformation as observed in measurements, even though there is some dierence in magnitude. The maximum temperature reached at the end of baking is around 110C whereas in reality it is about 130C. For other sensor locations, a gradual increase in temperature is spotted.
Inverse procedure and considerations
A dynamic physical process can be modeled using mathematical tools likes ordinary or partial dierential equations and rarely fractional dierential equations. These equations contain dependent and independent variables and knowledge about them is mandatory to compete the model. Usually, the dependent variables are state variables that can be measured and are quantities of interest. Independent variables are the physical constants. For instance, consider a researcher who is trying to understand the thermal behavior of a metal rod that has been heated for a short time. Assuming unidirectional heat ow without any source or sink, the following partial dierential equations can be used to solve the problem mathematically. In the above problem, the independent or state variable is temperature (T; T1 &T0) while other parameters or variables (which can be functions of state variables) such as density (), specic heat capacity (cp), thermal conductivity (k) and boundary conditions (q&hq) are considered as dependent variables or constants. These dependent variables are physical properties of the metal and process parameters which are presumed to be known beforehand. The heating process can be simulated by solving the above equations (3.1) with aid of any numerical (like nite dierence, nite volume, nite element methods, etc.) or analytical method. In such situations, the mysterious parameters are measured through either experimentation methods or inverse problem. The term `Parameter estimation’ has not been frequently used earlier rather it was termed as nonlinear least square or regression problems.
Carl Friedreich Gauss, one of the greatest mathematicians, is rst to demonstrate parameter estimation using non-linear least square method for calculating orbital elements of the planets during late 1790s [1, 2]. Later in 1960s, due to great demand in space exploration program, attention towards inverse problem has been increased and extensively used for characterizing the thermal shield in space re-entry vehicle [3]. The inverse problem is diversied in the eld of medicine/biology [4{7], applied mathematics [8], wave propagation/signal processing [9{11], optics [12, 13], heat transfer [14, 15] and many others. The essentials of inverse problems are a mathematical model and accurate measurements of state variable of focus. The unknown parameters are approximated with an optimization algorithm that minimizes the Euclidean distance between the measured and computed variables. Usually in heat transfer problem, the parameter and/or function focused to be estimated are unknown boundary conditions like surface heat ux, thermal properties like conductivity, source terms.
Sensitivity or Jacobian matrix
Sensitivity of a parameter is dened as how much does the state or measured variable varies with respect to a small variation in the unknown parameter. Mathematically, sensitivity is expressed as the rst derivative of the measured variable with respect to the unknown or estimable parameter. It is expressed as: Xp = @T(x; t; p) @p (3.2).
where p is set of unknown parameters. Objective function is the most important component in an optimization problem. In case of gradient based optimization, the basic requirement for the excepted solution is that the derivative of the objective function must be as minimum as possible. This derivative is product of sensitivity that is stated above and dierence between the measured and simulated variables.
Table of contents :
Contents
List of Tables
List of Figures
Introduction
1 Heat and mass transfer during food baking
1.1 Physical phenomena
1.1.1 Major phenomena
1.1.2 Commonly used ratios
1.2 Modeling approaches
1.2.1 Transport mechanisms
1.2.2 Models from literature
1.3 Required physical properties
1.4 Inverse procedure for properties determination
1.4.1 Note about function estimation
List of symbols
2 Cake baking: Experimental and modeling developments
2.1 Experimental considerations
2.1.1 Moisture content measuring techniques
2.1.2 Preparation of dough
2.1.3 Experimental set-up
2.1.4 Typical baking test
2.1.5 Incoming heat
2.1.6 Measurement of thermophysical properties
2.2 Development of heat and mass transport models
2.2.1 Diusive model
2.2.2 TMPN model: Multiphase model with non-equilibrium approach
2.2.3 TMPE model: Multiphase model with equilibrium approach
2.3 Model validation with known properties
2.3.1 Diusive model
2.3.2 TMPN model
2.3.3 TMPE model
2.4 Conclusive remarks
List of symbols
3 Selection of the appropriate model for physical properties estimation
3.1 Inverse procedure and considerations
3.1.1 Sensitivity or Jacobian matrix
3.1.2 Objective funciton
3.1.3 Parameterization of a function
3.2 Sensitivity analysis
3.3 Correlation analysis
3.4 Design of experiments
3.4.1 D-Optimality
3.4.2 E-Optimality
3.5 Numerical Analysis
3.5.1 TM Model
3.5.2 TMPN Model
3.5.3 TMPE Model
3.6 Conclusive remarks
List of symbols
4 Estimation of physical properties from experimentation
4.1 Introduction
4.2 Parameter estimation
4.2.1 TM Model
4.2.2 TMPN Model
4.2.3 TMPE Model
4.2.4 Extension of inverse solution
4.2.5 Literature
4.3 Function estimation
4.3.1 Mathematical Modeling
4.3.2 Sensitivity analysis
4.3.3 Analysis on inverse solution
4.3.4 Functionality
4.3.5 Literature
4.4 Note about one way coupling
4.5 Conclusive remarks
List of symbols
Bibliography
Conclusion
A Heatux estimation
A.1 Governing equations
A.1.1 Variational problem
A.1.2 Adjoint problem
A.2 Inverse problem procedure
A.3 Conjugate gradient algorithm
A.4 Inverse solutions
B Steps of Non-dimensionalization
B.1 Diusive model
B.2 TMPN model
B.3 TMPE model
B.4 TMPE model for function estimation
