Dynamic simulation on a thermodynamic canonical basis

Transcription

1 Olaf Trygve Berglihn Dynamic simulation on a thermodynamic canonical basis Doctoral thesis for the degree of philosophiae doctor Trondheim, June 2010 Norwegian University of Science and Technology Faculty of Natural Sciences and Techology Department of Chemical Engineering

2 NTNU Norwegian University of Science and Technology Doctoral thesis for the degree of philosophiae doctor Faculty of Natural Sciences and Techology Department of Chemical Engineering c 2010 Olaf Trygve Berglihn. ISBN (printed version) ISBN (electronic version) ISSN Doctoral theses at NTNU, 2010:132 Printed by Tapir Uttrykk

3 Acknowledgments This thesis is a contribution to the field of chemical engineering and modeling from a thermodynamic perspective. But as in all aspects of science, this work would never have been possible without the work of others. And as such, I am humble to have been given the possibility of exploring the field from the elevated perspective standing on the shoulders of giants. First and foremost, I would like to thank my main supervisor Associate Professor Tore Haug-Warberg for having faith in me and opening my eyes to the world of pure thermodynamics, not obfuscated by mundane engineering and applied perspectives only. Professor Heinz Preisig and Professor Terje Hertzberg also deserve special thanks for their valuable input and critical comments. When this work started, the thermodynamic models used were tediously coded by hand. If not for the work of Ph.D. Bjørn Tore Løvfall on gradients of thermodynamic potentials, the development of the methods in this thesis would have been tremendously impeded by the implementation cost associated with the thermodynamic models. Thanks, Bjørn Tore, for our many debates and for help with proofreading the thesis. I would also like to thank my co-worker through these years of Ph.D.- studies, Jens Petter Strandberg for his good spirit and encouragement. Ph.D. Tarek Ali Yousef deserves his thanks for the many interesting and lengthy theoretical debates, and for help with proofreading the thesis. Finally I would like to thank my family, my spouse Berit, and especially my two small kids Una and Sondre for their patience and support through these years. Without you, this thesis would never have been completed. Financial support for this project was provided by The Norwegian Research Council, Statoil AS, Yara AS, and The Chemical Engineering Department, NTNU. i

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5 Summary In this thesis a general approach is outlined for the modeling and simulation of dynamic processes involving phase and reaction equilibria. A new methodology is presented that exploits the intrinsic structure of thermodynamic functions in their canonical form. The equations for the equilibrium sub-problems of the dynamic process are stated such as to get a mathematical problem with linear constraints. This implies iterating in the variables most suited for the problem. A methodology is given by an equation structure called the dynamic Newton-Lagrange-Euler formulation. The equation structure is a set of linear equations that when solved yield a simultaneous integration in time and iteration towards the equilibrium of each sub-problem. A software tool has been designed and implemented in order to automate the construction and updating of the dynamic Newton-Lagrange-Euler equations. Two case studies are shown that explore the described methodology. The simulation results are on par with the results found in the literature. This systematic approach to the process modeling of flow sheets on a canonical thermodynamic basis has some promising benefits compared to traditional approaches based on a single choice of iteration variables. The presented methodology and software design is very general and has the potential for wide application in dynamic simulation and the development of simulation software. iii

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7 Contents Acknowledgments Summary Nomenclature i iii xi 1 Introduction Organization of the thesis Contribution of this work Thermodynamic functions Euler functions and homogeneity Legendre transformations and Massieu functions Surfaces and manifolds Thermodynamic states and time Derivatives of thermodynamic functions on manifolds Modeling on a canonical basis Alternative formulations for solving phase equilibria The Newton-Lagrange-Euler formulation Canonical basis and linear constraints Dynamics Coupled minimization problems Dynamic systems equations - DAE-formulation The dynamic Newton-Lagrange-Euler formulation Coupled minimization problems - dynamic case Gradient prediction and differential index v

8 5 Simulating on canonical basis Building blocks The constraint matrix Q Accumulation, steady-state, and stream building blocks Specifications on intensive properties Norms and diagnostics Convergence properties Scaling The dynamic Newton-Lagrange-Euler simulator Design choices System components Thermodynamic functions and analytical derivatives Manifolds of Legendre transforms and Massieu functions Blocks and right hand sides Topology of blocks and connections Linear algebra UML-diagrams Sample code Case studies A two phase separator The Joule experiment Concluding remarks 89 A The Ruby numerical library RNum 97 A.1 Design and implementation A.2 Features A.3 Execution speed B Jacobian, gradient and Hessian of thermodynamic manifolds 101 B.1 S(U,V,N)-manifold B.2 F(H, V T,N)-manifold vi

9 List of Figures 2.1 Legendre transform as tangent envelope From model to manifold Transformation from a thermodynamic surface to a thermodynamic manifold Change in time of thermodynamic state by explicit vs. implicit integration Simple steady state process with mass connection Snamprogetti urea-process (with courtesy of Volker Siepmann) An isolated system of two gas tanks mimicking the Joule experiment of Connecting from a steady-state block Connecting from an accumulation block Vapor-liquid separator base case Vapor-liquid separator base case with initial t = 100s, reduced by a factor Vapor-liquid separator, with feed and product streams set to 1% at time t = 5000s Vapor-liquid separator, with both liquid product draw set to 300% at time t = 5000s Class diagram for the Manifold Class inheritance diagram for the different block types Class diagram for the Topology Components overview and interfaces Dynamic separator with feeding reservoir and product streams Block structure for the dynamic two phase separator Composition profiles for the separator case Temperature profile for the separator case vii

10 7.5 Pressure profile for the separator case Pressure profiles for the separator at selected t and from a model in gproms R Block structure for the Joule case Simulation results for the Joule case Mach number for the Joule case viii

11 List of Tables 3.1 Thermodynamic state functions Diagnostic norms Scaling of units for a typical vapor and liquid using the Peng- Robinson equation of state Initial conditions for separator simulation Description of the diagonal blocks for the Joule case Initial conditions for the Joule-case ix

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13 Nomenclature Greek letters µ Chemical potential α α γ i φ i ρ k Slack variable introduced for overspecified systems of equations. Vapor phase mole fraction Activity coefficient Fugacity coefficient Density Iteration counter Latin letters h H g H N c p t ˆF f ˆF l ˆF v Heat transfer coefficient Hessian Gradient Enthalpy The number of chemical components (species). Pressure Time Feed flow rate mole based Liquid flow rate mole based Vapor flow rate mole based xi

14 ˆN ˆV R M S L M A Q A a C v f(z) g h K K i l N p (l) S T U u V Flow rate mole-based Volumetric flow. The set of real numbers Thermodynamic manifold Thermodynamic surface Legendre transform Massieu function Coefficient matrix for the Newton-Lagrange-Euler equations Constraint matrix for balance equations Helmholtz energy Cross sectional area Valve constant. Valve characteristics, z being actuator position. Standard gravity Specific enthalpy Equilibrium constant Equilibrium constant for component i Liquid level Mole number Pressure at liquid outlet Entropy Temperature Inner energy Specific internal energy Volume xii

15 v V t x i y i z i Specific volume Total volume Mole fraction, liquid Mole fraction, vapor Mole fraction, feed Mathematical symbols ẋ x T Time differential, x/ t vector Transpose Supersrcipts Reference (ω) F L V Refers to the ω phase. Feed Liquid Vapor Subsrcipts k (i,j) Iteration variable Matrix element or sub block indices xiii

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17 Chapter 1 Introduction The study of how phenomena evolve over time in chemical unit operations give highly valued insights for chemical engineers. With the progress in computing powers of modern computers, processes of great complexity and detail can be modeled and simulated for use in design studies and control system design, optimization, production planning, operator training and more. A simulation study can roughly be split into the tasks of: 1. Physical description. 2. Mathematical description. 3. Equation manipulation and solving using a numerical method. 4. Programming. 5. Simulation and presentation. Preforming the above tasks involves many disciplines of science. Firstly, the physical insights of the process and the decision on which relevant phenomena to describe is paramount for the success of the simulation study. Secondly, knowledge of both mathematical modeling and numerical methods is required to formulate an algorithm that can be implemented. Programming requires yet another set of capabilities, especially for complex models and large systems. At last, if all the prior subtasks have been completed successfully, the simulation can be run and the results disseminated using physical and systems insight. For other than simple modeling and simulation, the proficiency required in all these disciplines exceeds the skills of the average undergraduate chem- 1

18 2 Introduction ical engineer. Even at graduate level, people mastering both modeling and simulation will usually belong to a specialized group of chemical engineers. Several initiatives have been taken to lessen the requirements on skills for engaging in modeling and simulation studies. Commercial packages such as Aspen Plus R and Hysys R, PRO-II R, ProSimPlus R and others offer the chemical engineer graphical user interfaces which hide most of the details of modeling and numerics for the user. Other tools, such as gproms R, Ascend [Allan, 1997] and Jacobian R, offer an input language that resembles the mathematical representation, and will automatically do equation rearrangement and solve the system numerically. However, none of the above tools are problem free. From an academic point of view, one would like to know what exactly is happening under the hood of the commercial simulators something that is often impossible due to the proprietary nature of these products. Moreover, the user should always be able to analyze the problems arising from the choice of equation structure and numerical strategy when the equation oriented tools are used. For most chemical engineering problems involving rigorous phase equilibria and reactions in dynamic simulation, the equations can be put into the form of an Differential Algebraic Equation system (DAE): f(ẋ,x,y,t) = 0. (1.1) Much work has been done over the last three decades to develop methods and codes for solving this class of problems. Some of the known tools are DASSL [Brenan et al., 1989] and various other derived implementations (Matlab R ode15s/ode15i, Sundials Suite IDA [Hindmarsh et al., 2005], etc). Though these codes show great merit, they still only solve a subset of problems (index 1-2), and depend on the equation system being well formed, and on an appropriate choice of initial values. An alternative to solving the complete DAE is to integrate each subprocess separately by a dynamic sequential-modular scheme as shown by Hlaváček [1977] and Hillestad and Hertzberg [1986]. The advantage of this method is that each sub-process can be developed and tested separately. However, care must be taken when choosing the calculation sequence, the handling of calculation loops by using tear streams, and controlling the step length for stiff systems. Instead of looking for general tools to solve specific problems, the motivation for this work started off with the opposite perspective: How can specific problems be solved with the simplest possible means, and without the loss of control over the implementation and the numerical strategy? The motivation for taking this perspective stems from the experience with sim-

19 1.1. Organization of the thesis 3 ulating multiphase equilibrium problems with reaction, and the observation that most of the numerical cost associated with solving these systems stems from the calculation of thermodynamic properties and solving chemical and phase equilibria. In other words: Can the knowledge of the properties of thermodynamic functions and their equation structure lead to simpler or maybe better ways of constructing models for simulation studies? The condition of phase and reaction equilibrium is a frequently used assumption when modeling chemical phenomena. A reaction and phase equilibrium is characterized by a maximum in entropy, subject to given energy, volume and mass constraints. Equivalently, the equilibrium can be described by a minimum in internal energy (subject to given entropy, volume and mass constraints), or by a minimum of Gibbs free energy (subject to given constraints on temperature, pressure and mass). Many other formulations exist, using an extremum of a thermodynamic potential with accompanying constraints. The question that was raised above can partly be answered by investigating how the equilibrium condition and constraints can be described in a manner that gives the simplest possible system of equations to solve. This imposes a certain structure on the problem. The topic of this thesis is the analysis of the emerging structure and how to exploit this structure when modeling and simulating dynamic systems. 1.1 Organization of the thesis The thesis starts with a brief summary of the theory of thermodynamic functions in Chapter 2. Euler homogeneity and the Legendre transform are important concepts which forms the basis for defining a thermodynamic surface and a thermodynamic manifold, respectively. An analysis of the derivatives on the surface and the manifold is given, and this is related to integration of dynamic processes. Chapter 3 compares three different formulations of the phase equilibrium problem and introduces the Newton-Lagrange-Euler formulation. The selection of an appropriate thermodynamic potential to optimize and the problem constraints are discussed. Each thermodynamic potential has an accompanying set of variables which give the potential in the simplest form the canonical basis. To arrive at a simplest possible optimization problem, it is argued that a canonical basis should be chosen such that the constraints become linear. The term canonical variables is applied to this canonical basis. Each equilibrium problem can be viewed as a node. Nodes that exchange mass and energy compromise a process. A methodology for exchanging mass

20 4 Introduction and energy between different nodes is given in Chapter 4. The chapter starts by looking at the steady-state case, and progresses to the dynamic case by first reviewing the formulation of a differential algebraic equation set (DAE). Based on the idea of using canonical variables, a formulation is given that use the structure of the thermodynamic equations to state the dynamic case in the Newton-Lagrange-Euler formulation. The chapter concludes with a brief discussion on the DAE index problem. In Chapter 5 the dynamic Newton-Lagrange-Euler formulation is described in detail. The construction and update of the system of equations are shown for phase equilibrium, both with and without reaction equilibrium. It is also shown how to add constraints on intensive quantities, e.g. quantities that are not affected by system size; by fixing Lagrange multipliers. Normed quantities for step length algorithms and diagnostics are suggested, and the convergence properties are analyzed by means of an example. A simple scaling of the physical units is suggested for achieving good conditioning of the Newton-Lagrange-Euler equations, without having to rewrite the equations. A tool has been developed for automating the construction and updating of the Newton-Lagrange-Euler equations for an arbitrary set of nodes. Chapter 6 describes the design and implementation of this tool and gives an example of its use. Finally, the methodology presented in this thesis has been applied to two case studies given in Chapter 7. The first case study looks at a simple dynamic flash model found in the literature. The model has been reimplemented using the dynamic Newton-Lagrange-Euler method and the results compare favorably with the original paper. The second case study presents a model of the famous Joule experiment from 1843 with a blowdown from a high pressure reservoir to a lower pressure reservoir. Simulations are preformed for the Joule experiment both with and without reaction equilibrium. The model includes the restriction of choked flow in the valve between the two reservoirs. The results are discussed. 1.2 Contribution of this work The philosophy behind the perspective taken in this thesis builds, to a large extent, on the work by Haug-Warberg [1988], Brendsdal [1999] and Siepmann [2006]. The main new contributions lies in expanding on the basis from these authors and applying the results in the modeling and simulation of dynamic systems. The analysis of the canonical basis and distinguishing between explicit

21 1.2. Contribution of this work 5 and implicit thermodynamic functions guides the understanding of the challenges involved in the calculation of reaction and phase equilibria. The distinction is formalized by introducing the concept of a thermodynamic surface and a thermodynamic manifold. With the choices available for thermodynamic manifolds, a manifold can be chosen such that it best suits the model balance equations and constraints. This leads to the definition of the canonical basis that yields a linearly constrained problem: the canonical variables. A new methodology is developed for simultaneous integration and iteration of phase and reaction equilibrium problems based on canonical variables. This method is called the dynamic Newton-Lagrange-Euler formulation. The method is capable of simulating the complete process formed by a set of equilibrium nodes that exchange mass and energy. A framework for constructing and updating the equations for the dynamic Newton-Lagrange- Euler formulation is given. To explore the new methodology, a software tool has been designed and implemented to automate the construction and updating of the equations.

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23 Chapter 2 Thermodynamic functions Thermodynamic potentials are mathematical expressions that aim to describe the thermodynamic properties of a collection of interacting particles. The collection, or ensemble, is described by averaging the properties of the individual particles over time or space. The thermodynamic potential is a scalar function that take the form f : R Nc+2 R, where N c 1 is the number of chemical species in the mixture. A thermodynamic potential is a state function. This implies that the function value of f for a system depends only on its N c + 2 free variables, not on the way in which the system acquired the state. This chapter investigates the properties of thermodynamic potentials and the use of the Legendre transform to arrive at alternative thermodynamic potentials with different sets of free variables. The concepts of the thermodynamic surface and the thermodynamic manifold are introduced to distinguish the explicitly and implicitly defined thermodynamic potentials. Methods are shown for approximating the mapping from a thermodynamic manifold to another thermodynamic manifold or surface, and the relevance to integration in time on a thermodynamic manifold is discussed. 2.1 Euler functions and homogeneity The treatment of thermodynamic potentials is greatly simplified by the fact that they are linear along every straight line passing through the origin. The function f is Euler homogeneous in the variables x i,...,x n to degree 7

24 8 Thermodynamic functions k if it satisfies the following relations F ˆ=f(X 1,...,X n,ξ n+1,...,ξ m ) =λ k f(x 1,...,x n,ξ n+1,...,ξ m ), (2.1) X i ˆ=λx i. (2.2) The function F is a scaling of the function f with the factor λ k when the variables x i are scaled by a factor λ. The remaining variables, or parameters, ξ i are not taking part in the scaling and are thus regarded as constants. A thermodynamic quantity is said to be extensive if the thermodynamic potential is Euler homogeneous to degree k = 1, and intensive if the state function is Euler homogeneous to degree k = 0. Scaling all extensive quantities by a factor λ will scale the potential accordingly. Examples of extensive quantities are mole number and volume, and generally any other quantity concerning the system size. Intensive quantities are independent of the system size, such as pressure and temperature. The total differential of F can be written as follows by substituting the total differential dx by dx = λdx + xdλ: ( ) ( ) F F df = dx + dξ (2.3) = X ξ ) xdλ + ( F X ξ ξ ( F X An alternative formulation using F = λ k f yields: X ) ξ ( ) F λdx + dξ. (2.4) ξ X df = kλ k 1 fdλ + λ k df (2.5) ( ) ( ) f f = kλ k 1 fdλ + λ k dx + λ k dξ. (2.6) x ξ ξ x By comparing equivalent differentials in the Equations (2.4) and (2.6), and using the definitions in equations (2.1) and (2.2), the following three properties are found: ( ) F dλ : X = kf. (2.7) X ξ ( ) ( ) F f dx : = λ k 1. (2.8) dξ : X ) ( F ξ ξ X = λ k ( f ξ x ) ξ. (2.9) x

25 2.2. Legendre transformations and Massieu functions 9 Using the result in equation (2.7) for k = 1, the general integral of F has the solution ( ) F F(X,ξ) = (df) ξ = dx = F X. (2.10) X ξ X This is the Euler integration on F and is described as Euler s first theorem. An implication of this result is that for a function of all extensive variables (k = 1) the curvature 2 f/( x x T ) is always zeros in the direction of x: f x x = f ( 2 f x x T ) x = 0. (2.11) Equation (2.11) forms a part of the background for developing the methods in this thesis, as will be discussed in Section Legendre transformations and Massieu functions A function f : x f(x) can be described by different functions, whose argument is the derivative of f, rather than the function argument x, by means of a Legendre transform. Given f, the new function f is defined as f : ξ f (ξ) = min x (f(x) ξx) (2.12) This is the general Legendre transform, L [Arnold, 1989]. For a convex function, the minimum f is found when ξ = f(x)/ x. In the general case in thermodynamics, the Legendre transform can be written as fi (ξ i,x j,...,x n ) ˆ= f i (x i,x j,...,x n ) ξ i x i ( ) f ξ i ˆ=. (2.13) x i x i,x k,...,x n The use of this transform is central in the manipulation of thermodynamic functions. The information content of the original function f is conserved under the transformation, such that it is possible to get back the original variables by means of an inverse Legendre transform. As an example, the function of internal energy U as a function of (S,V,N) can be subjected to a Legendre transform with the variable S, yielding the Helmholtz energy function A : (T,V,N) U TS (2.14) ( ) U where T ˆ= S V,N

26 10 Thermodynamic functions Energy H 1 H 2 U 1 H 3 U 2 U 3 V 1 V 2 V 3 Volume Figure 2.1: Legendre transform as tangent envelope. or with p, yielding the enthalpy function H : (S, p,n) U + pv (2.15) ( ) U where pˆ=. V The tangent envelope of the function being transformed gives the Legendre transform as is shown in Figure 2.1. Starting with U 1, the product of V 1 and the tangent slope p is subtracted. This yields H 1 = U 1 + pv 1. In addition to the Legendre transform, swapping a variable x i with the state function is a useful operation. The function resulting from this swapping is called a Massieu function of the variable x i. Massieu functions, M, are useful for expressing e.g. S(U,V,N) from U(S,V,N) and S(H,p,N) from H(S,p,N). The term Massieu function is somewhat loosely defined in the literature and is most commonly described as Legendre transforms starting from S, rather than U [Callen, 1985]. Thermodynamic models can be categorized into two classes, depending on the set of free variables used. Models using temperature, volume and mass as free variables are referred to as pressure explicit Equations of state, and can be viewed as a Helmholtz free energy surface A(T,V,N). Many equations of state are able to predict both liquid and vapor properties. Models using pressure instead of volume as a free variable are generally denoted Gibbs excess models, or activity models, and can be written as a Gibbs free energy surface G(T, p, N). Activity models typically describe liquid and solid properties, and are unable, at the same time, to predict the S,N

27 2.3. Surfaces and manifolds 11 Modeling (T, V, N) or (T, p, N) as independent variables Surfaces Helmholtz (A) or Gibbs (G) free energy Manifolds Legendre transforms on (A) or (G) Figure 2.2: From model to manifold properties of a vapor phase. The Helmholtz (A) or Gibbs (G) free energy surfaces can form the basis of thermodynamic manifolds which have variables that are more convenient for a given modeling problem. Depending on the constraints of the system being studied, a set of Legendre transforms can be applied in order to arrive at a manifold which has the model constrained variables as its free variables. The different set of manifold variables will be described further in Section 3.3. The choice of free variables in the thermodynamic state function thus dictates the possible independent set of variables of the thermodynamic surface, and further transformation to manifolds with different sets of independent variables is possible through a series of Legendre transforms. A schematic of the path from a thermodynamic model to a thermodynamic manifold is shown in Figure Surfaces and manifolds The following terminology will be used to describe thermodynamic potentials and the transformations of these: Thermodynamic surface: The explicit thermodynamic potential, a state function f : R Nc+2 R, forms a thermodynamic surface S in R Nc+2+1. Thermodynamic manifold: The thermodynamic surface S formed by f can be modeled on to an Euclidean space M by Legendre transforms

28 12 Thermodynamic functions or Massieu functions. The space M forms a thermodynamic manifold in R Nc+2+1. According to the definitions above, the Helmholtz energy A(T, V, N) is a thermodynamic surface. From this surface the entropy S can be obtained as a Massieu function of (U,V,N): using that A : (T,V,N) A(T,V,N) (2.16) S : (U,V,N) 1 T (U A) = 1 (U + pv µn) (2.17) T U = A TS = A + T ( ) A p = V ( ) A µ = N T,V T,N ( ) A T V,N (2.18) (2.19) (2.20) Given the coordinates [A,T,V,N] on this thermodynamic surface S and the partial derivatives of A at this point, the function value S(U,V,N) can be calculated and the point [S,U,V,N] on the thermodynamic manifold M is found. Now, assume changes in the internal energy, volume and mole numbers from [U 0,V 0,N 0 ] to the new values [U 1,V 1,N 1 ]. To calculate the new value of the entropy S 1 on the manifold M, the quantities 1 T 1, p 1 and µ 1 must be found in order to evaluate the function (2.17). Equation (2.18) must therefore be iterated on T 1 until the value for U equals U 1. Then, the value of S 1 can be calculated. In other words, a point on the surface S can be mapped to a point on the manifold M. Only a single point and its local derivatives on the manifold are known. New points on the manifold can only be found by first finding a new point on the surface S. If gradients of A at [T 0,V 0,N 0 ] are known, the gradients of S for the corresponding values of [U 0,V 0,N 0 ] can be calculated by the implicit function theorem. For small changes [ U, V, N], the gradients can be assumed constant, and the change in S can be approximated by By calculating the Jacobian ds 1 T 0 (du + p 0 dv + µ 0 N). (2.21) J T0,V 0,N 0 = ([T,V,N]T ) ([U,V,N]) T=T 0,V =V 0,N=N 0 (2.22)

29 2.3. Surfaces and manifolds 13 a linear approximation of the update is given by T U V J T0,V 0,N 0 V (2.23) N N The transformation from a surface to a manifold will be illustrated by a trivial example. The graphical representation of a thermodynamic surface and the associated manifold will in general span N c +2+1 dimensions, which are hard to illustrate graphically. For the sake of the illustration the scope is therefore limited to a surface of one single variable, A : T A(T). A transformation that will give entropy S as a function of the internal energy U is given by: A : T A(T) (2.24) U = A + TS = A T A T (2.25) S = A T (2.26) (2.27) The mapping from U to S can not be written explicitly. However, given a point [A,T] and its derivative A/ T, the corresponding [ S,U] can be calculated. In order to have more information than a single point in the S U-plane, derivative information can be used to extrapolate along a second order Taylor polynomial. Figure 2.3 shows a plot of the Helmholtz energy surface using ideal gas law. The points marked with circle, square and triangle represent three states in the Helmholtz-temperature, and the states in the entropy-internal energy plane are given by the corresponding symbol. The solid curve on the left represents the explicit surface. The dashed lines on the right lines show the approximations of the manifold by Taylor-polynomials from the given points. Assume a model is to be constructed of a closed container with fixed mass and volume, subjected to an external heat source. A typical dynamic equation to describe the flow of energy to the system is given by U = ˆQ = h(t s T) (2.28) where h is a heat transfer coefficient, T is the temperature in the tank, and T s is the temperature of the heat source. If the heat flow equation is integrated from t 0 to t 1 using an explicit scheme, the change in internal energy is given by U = h(t s T t0 ) t. This yields the update in internal energy

30 14 Thermodynamic functions A(T) surface S(U) Manifolds A N Explicit mapping S N Implicit mapping T U N Figure 2.3: Transformation from a thermodynamic surface to a thermodynamic manifold. The points marked by a circle, a square and a rectangle on the thermodynamic surface (a line) in the A T plane to the left are mapped to the corresponding points in the S U plane to the right. Dashed lines represent approximations of the manifold by second order Taylor polynomials. The mapping from a point in A T to S U is explicit. The reverse mapping is implicit.

31 2.4. Thermodynamic states and time 15 for the container. Since only a point in the S U-plane is known, S 1 or T 1 must be found by iteration on T till the value of A and A/ T is found that match the updated U 1, via Equation (2.25). However, extrapolation along the Manifold given by the Taylor-expansion from the point [ S 0,U 0 ] (confer with Figure 2.3), the update to [ S 1,U 1 ] can be approximated without iteration, and the new temperature T 1 is given by the inverse gradient ( S/ U) 1 at [ S 1,U 1 ]. With basic knowledge of thermodynamics, this problem could be solved using the isochoric heat capacity C v = ( ) U T, which is usually available V,N as explicit expressions if ideal gas behaviour can be assumed. The above case was chosen to allow for a simple and comprehensible illustration of the surface manifold usage and properties. For other modeling tasks, the surface must in general be represented in R Nc+2+1. The strategy for updating the variables of the surface from the Legendre-transformed variables is then non-trivial. 2.4 Thermodynamic states and time There is no time associated with thermodynamic state functions. Even though the name is thermodynamics, time is not one of the function variables, as the state is independent of the history of the system and how it has been prepared. The dynamics behavior of the system must therefore be explained by other factors, such as chemical reactions, flow relations, external forcing and control systems. While time is definitely relevant when looking at collisions and other interactions between a small set of particles, thermodynamic state functions attempt only to describe the average behavior of a large number of particles. However, the validity of the averaging becomes dubious when looking at systems on a tiny scale; i.e. comparable to the mean free path of the particles. As has been shown in this chapter, a set of Legendre transforms and Massieu functions are required to model a thermodynamic surface on to a thermodynamic manifold which has variables coinciding with the dynamic model equations. The surface manifold transformations, and the model example from the foregoing, assumes that the temperature is constant over the time period t = t 1 t 0. This is obviously incorrect unless the interval is made infinitesimally small. As the energy of the container increases, so does the temperature. When integrating the dynamic Equation (2.28) with an explicit step, the state gradient (here the temperature) is assumed constant over the time interval. Confer Figure 2.4. The explicit step is equivalent to

32 16 Thermodynamic functions State s i s 0 s 1 s e t 0 t 1 Time Figure 2.4: Change in time and thermodynamic state by explicit integration along the path s 0 to s e vs. implicit integration along the path s 0 to s i. The true state at t 1 is s 1. For the monotonous function, the true state is bracketed by the two methods. moving from state s 0 to s e. If instead the temperature at the end of the interval is used, which is found by iteration, the transition will be along the gradient s 0 to s i. Both the implicit and the explicit integration strategies fail to track the true dynamics represented by the solid line, arriving at the state s 1, unless the whole surface-to-manifold mapping or iteration can be bypassed by expressing the surface in the variables of the dynamic system directly. Unfortunately, this is impossible for most systems. 2.5 Derivatives of thermodynamic functions on manifolds Derivatives of the thermodynamic surface are required to calculate coordinates on a thermodynamic manifold, as shown in Sections 2.2 and 2.3. Calculating the Jacobian at a thermodynamic manifold M (from M to a thermodynamic surface S) involves second order gradients of S. Solving for thermodynamic equilibrium with Newton-type methods in the coordinates of M also requires second order derivatives on M. Without the access to analytical derivatives of the surface, numerical perturbations are required for approximating them (see Nocedal [1999]). The detailed derivations for the Jacobian, gradient and Hessian for the manifolds used in this thesis are shown in appendix B, and only the principle will be shown here for brevity. The formulation is inspired by Haug-Warberg

33 2.5. Derivatives of thermodynamic functions on manifolds 17 [2006]. Given an explicit thermodynamic function f and its transform f found by a combination of Legendre transforms and Massieu functions, the gradient g and Hessian are given by H g ˆ= f x H ˆ= g g = x T x T. (2.29) x. (2.30) x T Since dimx = dimx, the Hessian with respect to the untransformed variables x can be written as which leads to g x T = g x T x x T. (2.31) H = g x T ( ) x 1. (2.32) This establishes a route from the derivatives of the thermodynamic surface, to the local derivatives of the manifold, and a Jacobian for mapping a point on the manifold back to the surface. The Hessian H is a second order derivative, and it follows that it must be symmetric. As shown in the appendix B, the Hessian can be written as a congruence product H = αlal T. (2.33) x T

34 18 Thermodynamic functions

35 Chapter 3 Modeling on a canonical basis The term Canonical denotes the characteristic of being in its standard form, usually also the simplest form. In thermodynamics, a canonical basis gives a complete description of the thermodynamic state with no loss of information; the function U : (S,V,N) U(S,V,N) contains the same information as A : (T,V,N) A(T,V,N) [Callen, 1985]. In the previous chapter, methods for mapping from one canonical basis to another by the use of the Legendre transform and the Massieu function were shown. This chapter will focus on the selection of sets of variables when modeling phase and reaction equilibria such that the representation is as simple as possible. First, a review of alternative formulations for solving phase equilibria is given. It will be shown that the problem can be stated as a constrained minimization of Gibbs Free Energy and how this formulation can solve simultaneous phase and reaction equilibria. This leads to a general iteration scheme for solving the equilbria using any thermodynamic state function: The Newton-Lagrange-Euler formulation. The rest of this chapter deals with the selection of a proper canonical basis, such that the constrained minimization problem associated with the phase and reaction equilibria will be linearly constrained. A canonical basis that achieves this property will be defined as the canonical variables of the problem. 19

36 20 Modeling on a canonical basis 3.1 Alternative formulations for solving phase equilibria Take as an example the steady-state flash drum. A common method for solving for the equilibrium composition at constant temperature and pressure is the well established Rachford-Rice equations, also known as the K-value method [Biegler, 1997]. A mass balance is written on the mass flow rates ˆF over the drum for each component. The mole fraction of component i in the vapor phase, y i, is given by the scalar K-function value K i multiplied with the liquid mole fraction x i. Note that the K-function is a composite function of the activity coefficient function γ i, the reference fugacity function f 0 i and the fugacity function φ i, and thus depends on the composition of both liquid and vapor phases, the temperature, and the pressure. The Rachford-Rice equations are given by ˆF f = ˆF l + ˆF v. (3.1) z i F f = F v y i + F l x i (3.2) y i = K i x i (3.3) K i = γ i(x,t)fi 0(T,p). (3.4) φ i (y,t)p yi x i = 0 (3.5) which after introduction of the vapor face fraction α and some substitution leads to α = V (3.6) F f(α) = y i x i = (K i 1)z i = 0. (3.7) 1 + (K i 1)α The problem can be solved by iteration with the Newton formula ( ) f 1 α k+1 = α k f. (3.8) α After converging the equation with respect to the vapor fraction α, the K i values must be updated, and the Newton iteration run repeatedly until no changes in the compositions x i, y i and the K i are observed. The Rachford- Rice method works well for mildly non-ideal mixtures, but may require very frequent calls to the thermodynamic functions for other cases. This is especially true when the enthalpy balance needs to be incorporated. Also,

37 3.1. Alternative formulations for solving phase equilibria 21 the method becomes less intuitive for multi-phase systems, and difficult to extend to reacting mixtures. An alternative method for solving the flash drum problem is to formulate it as an optimization problem with N c components and α and β phases. The objective is to minimize the Gibbs free energy G, at which the equilibrium is found: N (α) i min,n (β) i The solution is found when G (α) i dn (α) i G T,p (N (α) i,n (β) i ) i = 1,...,N c (3.9) N (α) i + G(β) i subject to + N (β) i = N i. (3.10) dn (β) i µ (α) i dn (α) i + µ (β) i dn (β) i = 0. dg = i N (α) i N (β) = i i (3.11) Here µ i is known as the chemical potential of component i. The mass balance constraint can be written as dn (α) i = dn (β) i. (3.12) Introducing G as the second order derivatives of the Gibbs free energy and n as the vector of mole numbers N i, the resulting Newton-Raphson formula can then be written as µ (α) + G (α) n (α) = µ (β) + G (β) n (β) (3.13) n (α) = n (β) (3.14) Defining µ = µ (α) µ (β) and H = G (α) +G (β), leads to the general Newton update formula: n (α) = H 1 µ (3.15) The Newton-Raphson iteration has second order convergence and shows good behavior in non-ideal mixtures and close to critical points. But having fairly accurate starting points is a prerequisite, and finding these is often a difficult task. Additionally, calculating µ and H requires first and second order derivatives of G. A third formulation, closely linked to the Newton-Raphson scheme is the technique of Lagrange multipliers as known from optimization textbooks [Nocedal, 1999]. The mass balance constraint is here combined with the

38 22 Modeling on a canonical basis function to be minimized in the Lagrangian function, L. The flash drum is generalized to the multi-phase equilibrium case with ω number of phases and N c chemical components. The problem can now be restated as min G T,p = n (α),n (β),...,n (ω) ω i=1 G (i) T,p subject to ω δ = n 0 n (i) = 0. (3.16) Introducing the Lagrangian function and some derivatives i=1 L(n,λ) = G λ δ (3.17) ( ) g (i) G (i) = n (i) (3.18) T,p ( ) H (i) 2 G (i) = n (i) n (i)t (3.19) T,p g (1) + λ L [n T,λ] T =. (3.20) g (ω) + λ δ H (1) I 2 L H (2) I A = [n T,λ][n T,λ] T =..... (3.21) H (ω) I I I I 0 The mass balance constraint is linear and can be solved for λ directly by adding λ to all but the N c last rows of A. The Newton updating scheme becomes H (1) I n (1) g (1) H (2) I n (2) g (2)..... =.. (3.22) H (ω) I } I I {{ I 0 } A k n (ω) } λ {{ } x g (ω) } 0 {{ } b k

39 3.1. Alternative formulations for solving phase equilibria 23 For the one- and two-phase case, the matrix A k is easily inverted: ( ) 1 ( ) H I 0 I = (3.23) I 0 I H H 1 I H 2 I I I 0 1 = (H (1) + H (2) ) 1 (H (1) + H (2) ) 1 (H(1) + H (2) ) 1 (H (1) + H (2) ) 1 I (H (1) + H (2) ) 1 H 1 (H (1) + H (2) ) 1 H 1 I H 1 (H (1) + H (2) ) 1 H 1 (H (1) + H (2) ) 1 H 2 (I (H (1) + H (2) ) 1 H 1 ) (3.24) But as the number of phases increase, so does the filling in of sub diagonal elements, making the possible advantage of a block based inverse less obvious in comparison to readily available methods of linear algebra packages [Anderson et al., 1999]. The mass balance constraint in (3.16) can be generalized to chemically reacting systems by introducing conservation on atoms instead of chemical components. For the sample system 2NO 2 = N 2 O 4, the balances on oxygen and nitrogen are linearly dependent, and a row-reduced constraint matrix Q = ( 1 2 ) N/O balance (3.25) replaces identity I for the non-reacting system. The equality constraint then reads ( ) ω δ = Q n 0 n (i) = 0. (3.26) and Equation (3.22) becomes H (1) H (2) Q T Q T.... H (ω) Q T i=1 n (1) n (2). n (ω) } Q Q {{ Q 0 }} λ {{ } A k x = g (1) g (2). g (ω) } Q0 {{ } b k (3.27) This implies that, at the converged solution where H n = 0, that g = Q T λ or in other words that µ NO2 = λ µ N2 O 4 = 2λ

40 24 Modeling on a canonical basis which in turn implies 2µ NO2 = µ N2 O 4. Inerts can be introduced in the n-vector to inhibit equilibrium reaction on some species, and different component lists for each phase can be used by introducing a projection onto a common basis [Brendsdal, 1999, Siepmann, 2006]. The minimization of Gibbs Free energy for solving a general, multiphase system was first proposed by White et al. [1958], and the methods have later been extended and explored by several authors [Castillo and Grossmann, 1981, Seider and Widagdo, 1996]. Computation of thermodynamic equilibrium by the iteration scheme shown in Equations (3.22) and (3.27) has been reviewed and explored by Haug-Warberg [1988] and Brendsdal [1999]. Although the minimization of Gibbs energy is a frequently encountered methodology in the literature, it is seldom used in commercial simulators where the K-value method is still dominates. Few publications exist using minimization of other energy functions. Two examples are Brendsdal [1999] and Castier [2009]. Note that phase stability is not considered here. Stability criterions and algorithm is thoroughly described by Haug-Warberg [1988]. 3.2 The Newton-Lagrange-Euler formulation The properties of first order Euler-homogeneous functions were described in Section 2.1, and Equation (2.11) stated that the Hessians H (i) k at x (i) k are singular in the direction x (i) k. Using this, the product H(i) x can be replaced by H (i) x. Equation (3.27) can then be rewritten as H (1) H (2) Q T Q T.... H (ω) Q T n (1) n (2). n (ω) } Q Q {{ Q 0 λ }}{{} A k x = g (1) g (2). g (ω) } Q0 {{ } b k. (3.28) The form given in Equation (3.27) will be denoted as the The Newton- Lagrange-Euler-formulation. This result is convenient as it enables the direct solution for the new variables and Lagrange-multipliers, without using a delta-form. Again, at the solution where all Hn = 0, then λ = g. In vicinity of the solution,

41 3.3. Canonical basis and linear constraints 25 the Lagrange multipliers give a second order approximation of the k + 1 gradient. 3.3 Canonical basis and linear constraints The Newton-Lagrange formulations (3.22) and (3.27) were stated for a system constrained to constant temperature and pressure, and by mass balance on component or atom basis. Other systems have other constraints. A closed container could be modeled at constant temperature and volume, or with constant volume and balance on energy and mass, or even with constraints on the chemical potentials, rather than mass, for membrane systems. The minimization problem associated with the choice of constraints follows from the basic equations for equilibrium as shown by Callen [1985]. At equilibrium, the combined system and reservoir/surroundings (r) is at minimum total energy when d(u + U (r) ) = 0 Energy conservation (3.29) d 2 (U + U (r) ) = d 2 U > 0 Minimum condition (3.30) This is the general form of the equilibrium conditions. Note that the minimum condition can be stated on U only, since the second order total differential with respect to the reservoir is zero: d 2 U (r) = j k 2 U (r) X (r) j X (r) k dx (r) j dx (r) k = 0. (3.31) Different formulations of the minimization problem can be chosen based on the system constraints. The most commonly used are stated here (from Callen [1985]): Helmholtz potential minimum principle. The equilibrium value of any unconstrained internal parameter in a system in diathermal contact with a heat reservoir minimizes the Helmholtz potential over the manifold of states for which T = T (r). Enthalpy minimum principle. The equilibrium value of any unconstrained internal parameter in a system in contact with a pressure reservoir minimizes the enthalpy over the manifold of states of constant pressure (equal to that of the pressure reservoir).

42 26 Modeling on a canonical basis Gibbs potential minimum principle. The equilibrium value of any unconstrained internal parameter in a system in contact with a thermal and a pressure reservoir minimizes the Gibbs potential at constant temperature and pressure (equal to those of the respective reservoirs). And this is further generalized to systems characterized by other extensive parameters in addition to the volume and mole numbers to yield The general minimum principle for Legendre transforms of the energy. The equilibrium value of any unconstrained internal parameter in a system in contact wth a set of reservoirs (with intensive parameters P (r) 1,P (r) 2,...) minimizes the thermodynamic potential L P1,P 2,...[U] at constant P 1,P 2,... (equal to P (r) 1,P (r) 2,... ). For other thermodynamic functions, involving variable substitution, e.g. Massieu-functions, the extremal condition can be reversed. Hence the equilibrium is found for the entropy function S when d 2 S > 0. In general, going from one formulation to another, the associated congruence transform of the Hessian will reveal whether the new function will have a minimum or a maximum at the equilibrium. If there is a sign change on the eigenvalues in the congruence transform, the extremal condition flips. Some transforms of the Hessians are shown in appendix B. This can be generalized even further to state the equilibrium problem in any Massieu function and set of Legendre transforms of a thermodynamic state function. Finding the extremal value (maximum or minimum, respectively) of these functions will give the equilibrium value of any unconstrained internal parameter. If the internal parameter is also the system constrained variable the extremal value problem will be a linearly constrained problem. This result should be emphasized, as it is of major importance: Given the constrained variables of the problem, seek the thermodynamic state function function that will have these constrained variables as free variables. The optimization problem of finding the equilibrium will then have linear constraints. Expressing the optimization problem in other variables than that of the explicit thermodynamic state function require the use of the Legendre transform and the Massieu function to model a manifold from an explicit thermodynamic surface. The functions from the set of possible transforms