A TOPOLOGY APPROACH TO MODELLING. Abstract

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1 A TOPOLOGY APPROACH TO MODELLING Heinz A Preisig Department of Chemical Engineering Norwegian University of Science and Engineering N-7491 Trondheim, Norway Heinz.Preisig@chemeng.ntnu.no Abstract Going back to basic physics, combine it with discrete mathematics and system theory, resulted in a powerful tool for generating dynamic process models that are consistent with the assumptions made, consistent with the basic physical laws, where appropriate, and are guaranteed structurally solvable. The user is left the freedom to map his view of the process in the model design process, thus has all the freedom he desires, whilst being strictly watched on the basic facts of physics and system theory. The last year s experience indicates a speedup of the model building and coding process of up to a factor of 10, but quite usually 5. Keywords: Computer-aided, modelling, DAE, process systems engineering Nomenclature n :: Vector of molar mass [mol] E :: Total energy ˆn a b :: Vector of molar mass flow from capacity (system) a to b [mol/s] ˆq a b :: Conductive heat flow from capacity (system) a to b [mol/s] ŵ a b :: Vector of work flow from capacity (system) a to b [mol/s] ñ s :: Vector of transposition rates in capacity (system) s [mol/s] x s :: Vector of fundamental state variables (component mass, energy) of capacity (system) s z a b :: Vector of flows from capacity (system) a to b [mol/s] v a b :: Vector of secondary flows from capacity (system) a to b [mol/s] r s :: Vector of transposition rates in capacity (system) s in terms of extent of reactions [mol/s] F :: Flow matrix (may be typed) [-] R :: Transposition matrix (may be typed) [-] S :: Selection matrix, selects from vectors elements to form a new vector [-] For the rest of the symbols see text. Heinz.Preisig@chemeng.ntnu.no web: preisig/ Modeller Project Models are omnipresent in today s process systems engineering activities. Almost all methods use models in one or the other way. Thus it is not surprising that the demand for models is increasing rapidly. The Modeller project has its roots in flow sheeting that is steady state process simulators, as models for flow sheet simulators were, and to a large degree still are, essentially generated manually. Flow sheeting simulators are programs that aim at simulation the process industry s dynamic plants and construct plant training simulators. A flow sheet is a graphical representation of a plant, which shows the different apparatuses required in the process to perform the different, individual tasks. Typical plant components are: reactors, heat exchangers, separation processes such as distillation, crystallisation, extractions, filters etc. The need for a programmed approach to constructing models for flow sheeting programs was recognised at the time the first flow sheeting programs were written in the Sixties and Seventies and several research groups have had efforts going that were aiming at generating a tool for doing so [1,2,4,6,7]. The Modeller is the result of our effort on this subject, though it is not anymore limited to flow sheeting. The modelling tool can in principle generate 1 of 8

2 models for any kind of problem, including dynamic simulation, optimisation and control design. The Modeller has, after the third generation [3, 5, 11], reached industrial standards and is currently used as the main model building program in a company specialising on building training simulators for the chemical industry. It consists of a set of context sensitive editors with the core module being the Modeller and the other modules being used to maintain data bases on support material such as different classes of equations (transport, reaction kinetics, phase transitions, equilibrium relations, statevariable transformations and physical properties or interfaces therefore), and species, reaction information (what species and what reactions). Currently it generates output in the form of software modules that slot into the in-house solver used in the mentioned company, but also Matlab s Differential Algebraic Equation solver and Modelica. Concepts The Modeller is built on implementing a physical view of the world. It constructs an abstract process representation in form of a topology with two levels of refinements. First a physical view of the space occupied by the process and its relevant environment is defined. This we call the physical topology. The first refinement can be seen as a colouring of the topology by adding the species that are present in the plant. Finally the second refinement adds the variables and equations describing the behaviour of the individual components of the topology. Physical Topology The first, and basic level consists of the PHYSICAL TOPOLOGY. It represents the plant as a network of PRIMITIVE CAPACITIES and CONNECTIONS. The capacities represent what one also often calls CON- TROL VOLUMES, namely parts of the space to which one assigns a common property, such as some uniform intensive properties characterising a physical phase. Generating this first level is crucial to the definition of the process model. Any following up step is limited by the structure of the physical topology. Thus it is critical to understand what it represents. plant Figure 1: Generating an abstract physical topology: split the plants volume into smaller volumes and introduce connections for extensive quantity transfer. This can be readily extended by introducing a hierarchy. Firstly, establishing a physical topology of a process is not an automatic process but a design process. It requires an in-depth understanding of the process being modelled. The process leading to the physical topology can be depicted as in figure 1. The user must first determine what shall be modelled, think about the environment and what affects the plant. This subdivision defines capacities and connections, thus implies a certain dynamic behaviour. This in turn links to time-scales, thus one needs to think about the time-scale range for which the model shall represent the behaviour of the realworld object. It is the WIDTH of the time-scales being modelled that implies complexity. The model will describe only those parts dynamically, which are modelled as capacities. The connections have no capacity. Relating the ability to store FUNDAMEN- TAL EXTENSIVE QUANTITY (= conserved quantities) with what comes in and out in terms of streams of affecting extensive quantities, gives a measure for the time scale in which the respective capacity operates. Computing the range is a mini-max calculation. The model is to match the purpose for which it is being used. Thus one has to know about the time-scale of the application of the model in order to decide how fine the GRANULARITY of the model 2 of 8

3 must be. Once one has decided on a granularity, one can refine the model by lumping too fine parts and subdivide too gross parts thus do model agglomeration or model refinement, respectively. The aim of this process must be to get a small distribution for the time constant ranges of the individual capacities. Species Topology The first refinement of the topology is to put the matter into the structured physical space. This is done by a couple of basic mechanisms: 1) injection of species into capacities, 2) injection of reactions into capacities, 3) constraining flows by introducing permeabilities for individual mass connections, 4) directionality of flow (bi-directional or unidirectional). 1) Injection of species: This introduces a species in a specific location. Often there is a natural point for the injection, namely the source for the species. In chemical plants this is mostly a feed tank or another resource. 2) Injection of reactions: This introduces potential reactions, that is, the reaction may take place, or one may say, the reaction is enabled. Injecting the reaction does not say anything about the conditions. It merely indicates that if a set of species is present, the reactants, than a set of other species, the products, may be generated through this transposition. 3) The permeabilities proved means to constrain flows, that is, a black-white description of a mass connection is given in which one states if or if not a species is transferred. One may think here of a absolutely selective membrane. 4) Directionality reduces the complexity, whilst constraining the descriptive power of the model significantly. In essence this last mechanism is not necessary, but reduces often the complexity. For example a feed tank that connects to a plant part such that no backwards flow can occur may be modelled in this way. These concepts are sufficient for the computation of the species distribution. For this purpose, one makes use of the typing of connections as one looks only at the connections that can transfer species, mass connections. The typing is usually done when defining the physical topology, but it may also be done later. The typing allows also a distinction of topologies. One can view coloured, or typed, networks and define for example mass transfer networks related to mass in general or species mass or any combination of species as a colour. One may do the same for other quantities such as heat or work, etc. The computation of the species distribution is then simply an extended colouring algorithm as one can find them in any discrete mathematics text that talks about graphs and their properties etc. Variable and Equation Topology Finally to each element in the topology a mathematical description is added. The basic dynamic description is given by a basic dynamic description of the individual capacities and the physical topology combined with the species topology. For each capacity a set of component mass balances and an energy balance as well as momentum balances can be generated when introducing the concept of INDUCED FLOWS. Latter are easiest explained on examples, namely mass flow induces energy flow as mass carries internal energy, potential energy and kinetic energy. Equally, mass flow induces momentum flow. Assigning symbols to molar component mass, n, total energy, E, conductive and radiation heat flow, ˆq, work flows, ŵ, one can write the component mass balances and the energy balance [11]. For the two {A,B} {A,B}&{A B} a work flow component mass flow conductive heat flow Figure 2: Two simple systems communicating component mass, work and conductive heat. Species A was injected in system a (marked with a * in the species set), is transferred to system b through the mass connection. In system b there is a possible reaction of A B. Thus species B is generated and again transferred back to system a through the mass connection. In addition work and convective heat is communicated through the respective connections. systems in Figure 2 one can write the component b 3 of 8

4 mass balances: dn a := ˆn a b, (1) dn b := ˆn a b + ñ b. (2) With the hats indicating flows and the notation a b giving the direction of the flow, namely from system a to system b. The direction indicated is to be seen as reference coordinate system against which the flow of the individual components are measured. The same applies to all the other quantities. The graphs are thus directed. For the energy balances we can write: de a := Ê a b ˆq a b ŵ a b ŵ(ˆn a b ), (3) de b := Ê a b + ˆq a b + ŵ a b + ŵ(ˆn a b ). (4) Notice that the energy balances are given in their pure form, thus no transformation of any kind has been applied. Also, there are induced flows to be observed, here volume work flows ŵ(ˆn a b ) that are induced by mass flow besides the already mentioned energy flow, Ê a b. The energy E is including internal, kinetic and potential energy. Because no state variable transformations have been yet applied, the reaction term is not appearing in any form in the energy balance of system b. It should also be noted that the directionality reflects into the balances and it is easy to see that one can map all balances into a block matrix equation: dx := Fz + Rr. (5) Where x is the vector of fundamental quantities, a stack of the component mass and energy for the two systems. The vector z is the stack of flows and the vector r the vector of transpositions, here reactions. The matrix F is the flow matrix and contains only information about the graph thus blocks of 1 and +1, whilst the matrix R is a block matrix with the blocks being the stoichiometric matrices for the respective systems. The flows and the reactions introduce secondary state variables such as concentrations and the PO- TENTIALS temperature, pressure and chemical potential. These secondary state variables MUST be the result of a mapping from the primary, fundamental state x. Whilst this seems obvious, the MOD- ELLER program, which was the result of three consecutive PhD thesises [3, 5, 11], is the first program that actually enforces these mappings. This has been introduced in the third thesis by Westerweele. The algebraic equations listed below must be assigned to the flows and reactions: z := ] [S [ ] z(y,p z,s z ) v, (6) S yv y v(y,p z ) r := r(y,p r ), (7) Finally the secondary states and the properties are extended. y := y(y,x,p y ), (8) p j := p j (y,p p ) ; j {z,r,y, p,i}, (9) The resulting equations must generate a completely defined set otherwise the system is not proper [9]. Whilst some of these equations are not explicit, it is in most cases possible to arrange them in a lower triagonal form. The analysis can be facilitated by a bi-partite graph analysis. The concept of LOCAL- ITY is essential: Connections are local to the two connected systems, whilst the rest of the algebraic part of the model is local to the individual system. The analysis is thus limited always to single or pairs of systems. In the case of our implementation in MODELLER, this analysis is done on-line as the equations are being selected from a pre-defined set of lists. The relative complex representation of the flow equations allows for the use of transformations first. It is quite common that, for example, mass flow is given in terms of volumetric flow. Time-scale assumptions The ability to make time-scale assumptions in the form of fast transfer, fast kinetics and small capacities (all reaching the respective limit of infinity or zero) and resolve the resulting structural problems in the differential algebraic model is unique to the current implementation of the MODELLER. The index problem is resolved through model reduction, which utilises the fact that the balance equations are linear. The model reduction reduces consequently to null-space computations of submatrices of the flow matrix and the transposition matrix. For details the reader is referred to [8, 10, 11]. 4 of 8

5 Implementation The current implementation of the MODELLER has the following functions: Context sensitive editor for manipulating a hierarchical physical topology A B Refinement with species topology Refinement with variable and equation topology On-line check on consistency of equation set Definition of simulation models On-line implementation of time-scale assumptions with automatic model reduction to index 1 problems Generation of code for matlab s DAE solver (MathWorks), e-modeller (Protomation BV, NL), Modellica (Dynasim). Besides the brief example to follow, the interested reader is referred to the thesis of Westerweele [11], which can be found on the web page of the author. The current implementation is limited to lumped systems, component mass and enthalpy, thus constant pressure system. A project to extend into distributed systems is currently on its way. Further, efforts are taken to make the input to the equation topology harder, that is, a context sensitive editor checking on the validity of thermodynamic relations is now defined as a new project. A Brief Example For the purpose of demonstrating some of the features of the discussed representation, we look at the equations for a very simple plant Figure 3. The first step is to suggest a physical topology. We use a simple concept by assuming the behaviour of an ideally stirred tank reactor for the jacket (J) and the contents of the reactor (F). The feed and product tanks as well as the source and sink of the two streams serving the jacket, are not of interest and are modelled as infinitely large capacities, thus thermodynamic reservoirs (Figure 4). Also, we limit the representation to component mass and energy. Notice that the figure indicates a two-level hierarchy in the R Figure 3: A simple reactor installation with two feed tanks, a jacketed stirred tank reactor and a product tank H C ˆn C J ˆn J H J R A ˆn A F ˆq J F F P P ˆn F P B ˆn B F Figure 4: Assuming an ideally stirred tank behaviour for the reactor tank contents and the jacket contents, and only being interested in the reactor and not the feed as well as the product tank, the topology is rather simple. representation, as the reactor is shown as a subnetwork. In a next step, we introduce the "chemistry" adding the species A and S to tank (reservoir) A, B and S to tank (reservoir) B and the cooling fluid K to reservoir C. In the fluid phase of the reactor we introduce a chemical reaction in which A and B are reacting to form species D. The topology can be typed generating coloured topologies. If we choose to show the mass domains we only have to delete all the heat-flow connections and, if they would be present, the work-flow connections, to find two of them (Figure 5). For the heat-flow domains we delete all mass flows and, if they would be present, the remaining work flows (Figure 6). With this simple structure, we only have to generate the equations for the two lumps: jacket and fluid con- 5 of 8

6 H C ˆn C J ˆn J H J A ˆn A F F P ˆn F P B ˆn B F Figure 5: The two mass-flow domains give the massflow matrix F m J ˆq J F Figure 6: The heat-flow domains with the reservoirs being primitive domains. This graph yields the convective heat-flow matrix F q tents. Before we do that so, we introduce a set of early assumptions, which are quite commonly, not to say nearly always, made in such systems. The assumptions are related to the energy balance. Let us write a generic energy balance for an arbitrary network: de = F m Ê + F q ˆq + F m ŵ(ˆn)+f w ŵ, = F m ( Ê + ŵ(ˆn) ) + F q ˆq + F w ŵ. We see clearly the effect of defining induced flows, here volumetric work flow, which is induced by the mass flow. The total energy (E) is the sum of internal (U), kinetic (K) and potential (P) energy: F E := U + K + P. The effect of kinetic and potential energy, in the capacities as well as in the mass-flow streams, are negligible which gives rise to the definition of enthalpy H := U + pv, which leads to a simpler representation dh = F m Ĥ + F q ˆq + F w ŵ. Finally, before we can generate the equations we need to determine the component-mass-flow domains. These depend on the assumptions of bi-directional or uni-directional mass flows. In the case we have here, it would be natural to assume uni-directional flow, meaning that one a priori eliminates the possibility that the mass stream flows in negative direction, that is, opposite the arrows direction. The difference is seen quickly: cap. reac. uni-dir. bi-dir. A {A,S} {A,B,D,S} B {B,S} {A,B,D,S} F {A + B D} {A,B,D,S} {A,B,D,S} P {A,B,D,S} {A,B,D,S} J {K} {K} Now it is straightforward to generate the component mass balances and the energy balances for the two capacities J and F. ˆn A F [ dnf ] ˆn B F dn := F m J ˆn F P + Rñ F, ˆn C J ˆn J H For uni-directional flows, the mass-flow matrix F m is: The two mass domains stand out clearly as the two blocks (lower left, upper right) are zero. Also notice the compactness of the representation. There is no unnecessary information included, for example only those species are being included that are actually present. There are also no "cut-equations" as they are often used in simulation software such as Modelica and bondgraph programs. The stoichiometric matrix is also easily found R := Finally one would define the expressions for the unidirectional mass flows (select second option in equa- 6 of 8

7 tion 6) and the 2nd order reaction kinetics: n a b := c a ˆV a b, r F := k(t F )c F,A c F,B. The kinetic constant could be a function of the temperature as indicated, which usually is modelled with an Arrhenius equation. Both these define concentrations, which are secondary state variables that need to be linked back to the primary state. For the arbitrary system a these are the equations: c a := n a V a, V a := [1,...,1]n a ρ 1 a. with ρ being the molar density. The energy balances are: Ĥ A F [ ] Ĥ B F EF = F m Ĥ F P + F q ˆq E J J F. Ĥ C J Ĥ J H The mass flow matrix does here only refer to the total flows (thus the modified symbol F m instead of F m ): [ ] F m := The conductive heat-flow matrix is: [ ] 1 F q :=. 0 The heat transfer is modelled with Newton s law of cooling: ˆq J F := p J F (T F T J ). The temperature is to be computed from the relation: H := T T ref c p (T ). Where c p (T ) is the specific heat capacity, which is the partial derivative of the enthalpy with respect to the temperature and the "parameter" p J F is the product of the overall heat transfer coefficient and the heat transfer area. These equations give the main relations. The little being left out should be easy to fill in by the reader and complete the description. The attentive reader will also notice that the equation set defines a bi-partite graph, which can be used to establish what must be given in order to be able to integrate the equations. In the view of the limited space this is left to the reader, but given the initial conditions for the primary state variables being the molar masses in the capacities J and F and the parameters (kinetic constant, density etc.), a DAE solver will be readily able to solve these equations. A Thought on the Side If one looks into the current contents of education programs in process engineering, than one finds that there is very little education on the structuring mechanism as it underlies this analysis. In most cases it is hidden away or in the best cases hinted such as "making a pseudo-steady state assumption". It is astonishing that whilst any model is requiring going through this process, the required thinking pattern is currently not addressed explicitly in most of our teaching programs. Also: whilst electrical and mechanical engineers typically deal with scalar or 3-dimensional spaces for the primitive model components, the chemical engineers primitive elements are usually of higher dimension, namely of # of mass components + 1 for energy + possibly 3 for momentum. Thus it is surprising that it is chemical engineering that has the smallest amount of education in multidimensional spaces when comparing the three disciplines. Conclusions Models are not unique items. For every plant one may design different models. It is the user, who is asked to provide his view of how the physical space of the plant and its affecting environment by splitting the identified domain into a set of primitive systems that have capacity to store mass, energy etc, and connections that communicate extensive quantities between pairs of neighbouring primitive systems. The result is an abstraction of the control volume concept into a graph with the vertices representing the capacities and the arcs representing the 7 of 8

8 connections. Adding the colouring, where colours are species and type of extensive quantity, the main body of information about the process model is captured, namely the model granularity, the interactions and the relevant extensive quantities. After having defined the granularity and interaction pattern of the physical - chemical - biological entity, one only needs to fill in the mechanisms of (i) transfer of extensive quantity, (ii) chemical (biological) kinetic and phase transitions, (iii) state variable transformations and (iv) physical properties all as a function of the fundamental state, latter being the vector of conserved quantities. The graph, thus, contains all this latter information and is, as a picture, very well suited for communicating the process model properties in the large. We have not only been using this concept for the Modeller, but use it also for teaching and any kind of discussion, where the properties of a model are of relevance. Today, models are build with little analysis as indicated above and probably are linking to the main problems in modelling dynamic processes. The MODELLER is the first program of its kind that guarantees the generation of STRUCTURALLY SOLVABLE SIMULATION PROBLEMS, namely DIF- FERENTIAL ALGEBRAIC EQUATIONS OF INDEX 1 The key to this achievement is to not substitute the algebraic part but treat the problem in the fundamental space of the conserved quantities, which is linear in the essential quantities. The project has reached industrial standard by proving its efficiency in the construction of training simulators and other simulation models. A 10-fold increase in efficiency is not a-normal, but 3-5 times quicker is almost always achieved. Whilst the tool is still somewhat experimental and suffers of some shortcomings, these problems will sequentially be removed in the follow-up projects. References [1] J Bieszczad. A framework for the language and logic of computer-aided phenomena-based process modeling. PhD thesis, [2] J A Krogh. Generation of problem specific simulation models with an integrated computer aided system. PhD thesis, Technical University of Denmark, Kastrup, Denmark, [3] T Y Lee. The Development of an Object- Oriented Environment for the Modelling of Physical, Chemical and Biological Systems. PhD thesis, Texas A & M University, College Station, TX, USA, [4] W Marquar. Dynamic process simulation recent progress and future challenges. In Dynamic Process Simulation Recent Progress and Future Challenges, New York, CPC IV, CACHE-AIChE Publications. [5] A Z Mehrabani. Computer aided modelling of physical-cyhemical-biological systems. PhD thesis, University of New South Wales, Sydney, Australia, [6] H I Moe. Dynamic process simulation: studies on modeling and indes reduction. PhD thesis, Norwegian University of Science and Technology, Trondheim, Norway, [7] J D Perkins, R W H Sargent, R Vasquez- Roman, and J H Cho. Computer generation of process models. Computers & Chemical Engineering, 20: , [8] H A Preisig. On concentration control of fast reactions in slowly-mixed plants with slow inputs. pages FP 06 6, [9] H A Preisig. Modelling: Compartmental networks and topologies. In Modelling: Compartmental Networks and Topologies. ESCAPE 14, [10] H A Preisig and M R Westerweele. Effect of time-scale assumptions on process models and their reconciliation. In Effect of Time-Scale Assumptions on Process Models and Their Reconciliation. ESCAPE 13, [11] M R Westerweele. Five Steps for Building Consistent Dynamic Process Models and Their Implementation in the Computer Tool MOD- ELLER. PhD thesis, TU Eindhoven, Eindhoven, The Netherlands, ISBN of 8