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A network theory for the structured modelling of chemical processes
Chemical Engineering Science 57 (2002) 4099 – 4116
www.elsevier.com/locate/ces
A network theory for the structured modelling of chemical processes M. Mangolda;∗ , S. Motzb , E.D. Gillesa; b a Max-Planck-Institut
b Universit at
fur Dynamik komplexer technischer Systeme, Sandtorstra e 1, 39106 Magdeburg, Germany Stuttgart, Institut fur Systemdynamik und Regelungstechnik, Pfa(enwaldring 9, 70550 Stuttgart, Germany Received 19 November 2001; received in revised form 17 May 2002; accepted 2 June 2002
Abstract A structuring methodology for dynamic models of chemical engineering processes is presented. The main ideas of the methodology were outlined in a previous publication for the class of well-mixed systems. In this contribution, the methodology is extended to spatially distributed systems and to particulate processes. Furthermore, the structuring principle is used to make a conceptual link between the macroscopic world of process simulation and the microscopic world of molecular simulation. It is shown that a uniform structuring principle can be applied to the modularisation of most classes of chemical engineering models. The structuring principle can be used as a theoretical framework for the implementation of modular families of chemical engineering models in modern computer aided modelling tools. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Systematic modelling; Computer aided modelling; Mathematical modelling; Spatially distributed systems; Population balances; Dynamic simulation; Process control
1. Introduction Mathematical models of chemical processes have been widely accepted as a useful tool for the development of processes and control schemes. Today, many numerical algorithms and tools for the analysis of very complex models are available. However, the formulation of the models themselves is still an obstacle for the application of model-based techniques in industry. The development and implementation of new models is considered as a di:cult and expensive task. An important reason for this fact is the complexity and low reusability of process models as they are typically implemented today. Usually, models of chemical plants are formulated on a
structuring of the models on a level below the process unit have been recognised as important steps towards a more e:cient model development. In the past, several authors proposed general concepts for the structuring of process models (Ponton & Gawthrop, 1991; Perkins, Sargent, VDazquez-RomDan, & Cho, 1996; Marquardt, 1996; Preisig, 1996). The main interest in the past was in well-mixed systems in general or special subclasses thereof, like distillation processes or phase equilibrium models. Little attention has been given to distributed systems. Based on those general structuring concepts modelling tools and languages have been developed which permit a modular and highly structured model formulation. Examples are Omola (Andersson, 1990), ModDev (Jensen & Gani, 1999), Modeller (Westerweele, Preisig, & Weiss, 1999), ModKit (Bogusch & Marquardt, 2001), or ProMot (TrIankle, Zeitz, Ginkel, & Gilles, 2000). These modelling tools have provided a powerful framework for the development of model libraries based on simple,
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to keep the number of elements in the library =nite and to ensure compatibility between the modelling entities in the library, a theoretical concept is needed as a guideline for the model library developer. The Network Theory of Chemical Processes, which was proposed in a former publication (Gilles, 1998), oOers such a modelling concept. In Gilles (1998), the Network Theory was discussed for the class of well-mixed systems. The purpose of this article is to extend the ideas to other classes of chemical engineering processes. In the next section, the basic concepts of the Network Theory are described, as they were stated in (Gilles, 1998). It is shown that the same kind of structural description can be used to decompose a model on diOerent hierarchical levels of granularity. The molecular level, the level of storages of macroscopic thermodynamic quantities, the level of thermodynamic phases, and the level of process units are considered. The molecular level will be examined more closely. Interfaces between the molecular level and the macroscopic thermodynamic level will be discussed. The main part of the article will deal with the internal structuring of thermodynamic phases, as this level of description usually is the most complicated part in the development of chemical process models. The concept of Network Theory will be extended to models described by partial diOerential equations. Spatially distributed systems as well as dispersed phase systems and population balances are considered. Inside those systems, additional types of transport mechanisms occur which are not present in lumped systems. Their description requires the de=nition of new classes of structural modelling units. In addition, the treatment of boundary conditions must be included in the structuring process. The boundary conditions de=ne the interface between an distributed system and its environment. Therefore, a structured formulation of the boundary conditions is of special importance for the coupling of distributed systems with other systems and for the use of distributed modelling entities in a model library. 2. Basic concepts of network theory Plant models in current
elling tools consists in facilitating and supporting that second step of the model development. The basic idea is to provide pre-de=ned model formulations the modeller can use to solve his speci=c modelling task. The usability of a model library strongly depends not only on the amount of information contained but also on the way the information is structured. A library of process unit models is only of limited use. Considering the enormous number of existing and conceivable process units, the attempt to store models of all of them in a library is doomed to failure. On the other hand, models of diOerent process units often are very similar. The model formulations are based on a limited number of physical assumptions and phenomenological descriptions. First principle models consist of balance equations derived from conservation principles for energy, mass, momentum, and charge. The same formulations for diOusion, convection, heat or mass transfer are used again and again in many apparatus models. The vast range of diOerent process unit models mainly originates from the many possibilities how existing modelling approaches can be combined. Therefore, the aim to collect modelling knowledge in a library becomes much more attainable if pursued below the level of process units. Process units can be considered as an aggregation of interacting thermodynamic phases. The thermodynamic phases may serve as elementary modelling units, if only a certain class of systems, e.g. well-mixed systems, is considered (Perkins et al., 1996). However, a uniform treatment of lumped and distributed parameter systems requires a segregation of the phases into smaller units. A high amount of
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
generalised thermodynamic forces (Callen, 1985). Similar structuring principles have been proposed by other authors (Ponton & Gawthrop, 1991; Marquardt, Gerstlauer, & Gilles, 1993; Marquardt, 1996; Preisig, 1996; Weiss & Preisig, 2000). Components may be considered as devices in the terminology of Marquardt and co-workers or as systems in the terminology of Preisig and co-workers. Coupling elements are termed connections by Marquardt and Preisig. • Components and coupling elements can be de=ned on different hierarchical modelling levels. For example, a component may be a single thermodynamic phase or it may be a process unit which consists of several interacting phases. Consequently, systems of components and coupling elements can be aggregated to a single component on a higher level. On the other hand, elementary units on one level may be decomposed into systems of components and coupling units on a lower level, which gives a more detailed view on the system. The two basic ideas will subsequently be discussed in detail. 2.1. The potential-8ux-vector concept of bidirectional signal transfer When structuring a chemical process, a clear distinction has to be made between the direction of physical
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Fig. 1. Interconnection of two components.
When two components Ck and Cl interact via a coupling element CE(Ck ; Cl ) , the direction of
where NJ is the number of
(2)
Because the Jacobian @gi =@Ji is non-singular by de=nition, a single derivation in time of (2) su:ces to turn the model equations into a system of ordinary diOerential equations. Simplifying modelling assumptions tend to violate the proposed direction of signal transfer. Typical examples are equilibrium assumptions or assumptions of vanishing mass or heat transfer resistances. If such an assumption is made, the coupled components will be no longer independent, but will lose a degree of freedom. In Gilles (1998) this type of coupling is therefore termed rigid coupling. In the case of rigid coupling, it is no longer possible to assign unique directions of information
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level of process units liquid phase
phase boundary
vapor phase
level of phases
X=(U,V,mi )
+ + level of storages
molecule
intermolecular forces molecular level
Fig. 2. Levels of process structuring and examples for structural decomposition on the diOerent levels.
alternative formulation for the components at equilibrium, e.g. by the formulation of pseudo-homogeneous balance equations (Ponton & Gawthrop, 1991; Gilles, 1998). 2.2. The levels of process structuring The model of a chemical plant can be structured on different levels. This is illustrated by Fig. 2. The top level regarded here is the level of process units. On this level, process units like distillation columns, reactors, or heat exchangers are considered as the elementary components. They are connected by coupling elements such as pipes, valves, pumps etc. A major part of chemical engineering research has concentrated on the analysis and modelling of individual process units. Therefore, the elementary functional units on the level of process units are described by very complex models in many cases. In order to improve the structuring of the process models and to get simple and reusable modelling units, it is helpful to decompose the models of process units into smaller units. A process unit may be considered as a system of interacting components and coupling elements on a lower level. This lower level is the level of phases. A process unit usually contains several phases, e.g. a liquid and a vapour phase. Each phase is an elementary component on the level of phases. The phases interact by heat transfer, mass transfer, or transfer
of momentum. The corresponding coupling elements which describe this interaction are phase boundaries, membranes, diOusive layers, etc. The state of a thermodynamic phase is described by macroscopic thermodynamic quantities like energy, mass, and momentum. In order to decompose a phase into smaller units, the phase can be considered as a system of interacting storages of mass, energy, and momentum, which are the components on the next lower level—the level of storages. The components on the level of storages interact by transport processes like reaction, diOusion, and convection. Therefore, the purpose of coupling elements on the level of storages is to describe the
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
equations of state
X
(T,p, µi )
CE P[X]
+
J
+ level of storages
ε2
ε1
ε3
molecular level
Fig. 3. Interfaces between molecular simulation on the molecular level and process simulation on the level of storages for the example of a well-mixed homogeneous phase. The levels interact via the reactive coupling element CE and via the equations of state.
applied successfully, e.g. to the computation of vapour liquid equilibria of diOerent substances (Vrabec et al., 1995; Kriebel et al., 1998; Stoll et al. 2001). In the structuring principle of the network theory, the components on the molecular level are the states of motion of the molecules, i.e. their coordinates and velocities, and further molecular properties like size or polarity required to solve the equations of motion. The coupling elements describe the attracting and repulsing forces between the molecules, e.g. by potentials of the Lennard–Jones type. For the application of molecular simulation to real world problems, it is necessary to de=ne clear interfaces between the molecular level and the higher levels of process structuring. In the following, possible interfaces will be discussed from two points of view. The =rst viewpoint is a strictly thermodynamic one and is shown in Fig. 2. According to statistical thermodynamics, the states of all quantities on the macroscopic level can be obtained by averaging over the molecular states. Therefore, in theory it may be possible to gather all information on the level of storages by solving the equations of motion on the molecular level. In reality, this approach will hardly be viable because determining the states of all the molecules of a technical process is a far too complex task. Therefore, a slightly more pragmatic viewpoint will be presented subsequently and is shown in Fig. 3. The interaction between macroscopic process simulation and molecular simulation will be e:cient, if as many computations as possible are done on the macroscopic level and molecular simulation is only used if the available macroscopic information is not su:cient. By macroscopic balancing, it is always possible to derive diOerential equations for the extensive state variables required to describe the macroscopic state of a phase, e.g. the internal energy U , the volume V , and the masses mi , i = 1; : : : ; NC of all NC components. The right-hand side of those diOerential equations contain the sinks and sources of the phase, i.e. the behavioural description of the coupling
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elements between the macroscopic storages. Typically, the coupling elements depend on many parameters which describe the physical and chemical properties of the phase. The determination of the parameters is not possible by macroscopic balancing, but it may be possible to =nd some of them by molecular simulation. Therefore, the internal sinks and sources in a phase can be regarded as one type of interface between the molecular and the macroscopic level. For most applications, it is convenient to replace the extensive variables U and V by the intensive variables temperature T and pressure p. The transformation between the extensive variables U , V , mi and the intensive variables T , p, i de=nes the three equations of state @U @U @U T= ; p = ; = (3) i @S @V @mi V; mi
S; mi
S; V
(see e.g. (Callen, 1985)). The equations of state can be obtained by macroscopic measurements or by simulations on the molecular level. Therefore, the transformation between extensive and intensive variables is a second type of interface between the molecular level and the level of storages. 4. Structuring of well-mixed phases on the level of storages and on the level of phases The structured modelling of homogeneous well-mixed phases was treated in detail in Gilles (1998) and will only be summarised brie
(4)
J [U ](1) + J [U ](2) = 0;
(5)
1
If there are electrical eOects, then also the transformation from internal energy to electrical energy and vice versa will have to be taken into account. This requires the de=nition of additional coupling elements, but will not be treated here.
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where T (1) and T (2) are the temperatures in the two phases, and J [U ](1) and J [U ](2) are the
velocity distributions and momentum balances. Therefore, the spatially distributed phase will be divided into two subsystems, called the
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
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n jB
Ω(x,y,z) = 0
Fig. 4. One-dimensional example system.
• • •
• •
y
mass of component i, j[mi ] is the corresponding mass
Under these assumptions, the following equations hold for the one-dimensional example system: • Equation of continuity: @ @ = − (vz ): @t @z • Momentum balance (Euler equation): @vz @vz @p = −vz − : @t @z @z • Total mass balances at the system boundaries: 0 = vz (z1 ; t) − 0 = vz (z2 ; t) +
NC i=1 NC
(6)
(7)
j B [mi ](z1 ; t);
(8)
j B [mi ](z2 ; t):
(9)
i=1
• Component mass balances (i = 1; : : : ; NC − 1): @gi @gi @jD; i = −vz − + i M i r0 ; @t @z @z
x z Fig. 5. Three-dimensional example system. n: normal vector on the surface ! = 0 as the system boundary.
• Balance of internal energy: NC @u @ @vz @u = −vz − jQ + ; (13) jD; i hi − p @t @z @z @z i=1 NC − j[U ](z1 ; t) = −vz u|z1 ;t − jQ (z1 ; t) + jD; i hi |z1 ;t i=1
− pvz |z1 ;t ;
(14)
j[U ](z2 ; t) = −vz u|z2 ;t −
jQ (z2 ; t) +
NC
jD; i hi |z2 ;t
i=1
−pvz |z2 ;t :
(15)
In the above equations, p denotes the pressure, gi is the mass fraction of component i, i is the stoichiometric coe:cient of component i, Mi is its molar mass, and hi is its partial molar enthalpy. In Sections 5.1.1 and 5.1.2, the equations will be structured according to the methodology of the network theory. A general structure of the balance equations will be derived, which is also valid for other spatially distributed systems. This will be demonstrated by a second example, which is a rather general three-dimensional homogeneous system as shown in Fig. 5. The geometry of the system is de=ned by some function !(x; y; z), where !(x; y; z) ¡ 0
(16)
is the inner of the system, and (10)
!(x; y; z) = 0
(17)
− j B [mi ](z1 ; t) = −vz gi |z1 ;t − jD; i (z1 ; t);
(11)
is the system boundary. In the case of the one-dimensional example, ! can be de=ned as
j B [mi ](z2 ; t) = −vz gi |z2 ;t − jD; i (z2 ; t):
(12)
! = (z − z1 )(z − z2 ):
(18)
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In this notation, the vector @ @ @ (∇!)T ; ∇= n= ; ; ∇! @x @y @z
(19)
is a normal vector perpendicular to the surface of the system pointing outwards. The derivation of the balance equations of the threedimensional example system can be found in literature, e.g. (Bird, Stewart, & Lightfoot, 1960). Therefore, in Sections 5.1.1 and 5.1.2 only the results of the structuring process will be given for the three-dimensional example. 5.1.1. Structure of the 8uid-dynamic subsystem The
• Balance of momentum M in the interior of the system !(x; y; z) ¡ 0: @v = C [M ] + D [M ] + j[M ]: (21) @t • Balance of total mass on the system boundary !(x; y; z) = 0: 0 = %CB [m] + j B [m]:
(22)
• Balance of momentum on the system boundary !(x; y; z) = 0: 0 = BC [M ] + BD [M ] + j B [M ]:
(23)
The one-dimensional example may be seen as a special case of the general
(25) (26)
• Dispersive coupling element for the one-dimensional example: @p T D [M ] = 0; 0; − : (27) @z • External
NC
(28) j B [mi ];
(29)
i=1
j[M ] = 0:
(30)
It should be noted that in the one-dimensional example two boundary conditions of type (22) are used, but none of type (23). This point will be discussed in detail at the end of this section. Before, it will be shown that Eqs. (20)–(23) also hold for the three-dimensional example. The only diOerence between the one-dimensional and the three-dimensional example is in the behavioural description of the coupling elements. Balancing of mass and momentum leads to the following source terms: • Convective coupling element for the three-dimensional example: %C [m] = −∇(v);
(31)
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
C [M ] = −vT ∇T vT ;
(32)
%CB [m] = −vT n;
(33)
BC [M ] = −vvT n:
(34)
The three-dimensional convective terms are a rather straightforward extension of the one-dimensional case. • Dispersive coupling element for the three-dimensional example: NC 1 T D [M ] = −∇ pI + + (35) jD; i jD; i ; i i=1 T NC 1 B T D [M ] = − pI + + jD; i jD; i n: (36) i i=1
The =rst term in each of the bracketed expressions in Eqs. (35) and (36) already appeared in the one-dimensional example. The other two terms have been neglected so far. The second term describes transport of momentum caused by viscous forces; is the stress tensor (Bird et al., 1960). The third term describes transport of momentum due to dispersive
(37)
j B [m] = j B [m]T n;
(38)
j[M ] =
NC
i fi :
(39)
i=1
In contrast to one- and two-dimensional systems, the mass
(40)
Based on the above considerations, a graphical representation of a general
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Fig. 6. Graphical representation of the
state of the storages is the only input to the convective coupling element. The information exchange between storages and coupling elements is not meant to be local, i.e. not only states and
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M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
some ambiguity concerning states and
• Dispersive coupling elements D describe transport phenomena due to diOusion and heat conduction. • Reactive coupling elements R describe the change in the composition by chemical reaction processes. They are identical to the reactive coupling elements of well-mixed systems. • In the balance equation for the internal energy, conversion of other kinds of energy into internal energy has to be taken into account. Examples are the conversion of kinetic energy into internal energy which manifests itself as friction heat or the conversion of potential energy into internal energy caused by diOusive
(42)
• Balance of internal energy in the interior of the system !(x; y; z) ¡ 0: @u = %C [U ] + %D [U ] + %T [U ] + j[U ]: (43) @t • Balance of internal energy on the system boundary !(x; y; z) = 0: 0 = %CB [U ] + %DB [U ] + %TB [U ] + j B [U ]:
(44)
As in the case of the
%C [U ] = −vz
(47) (48)
In Eqs. (46) and (48), the negative sign holds for the left boundary z =z1 , and the positive sign is valid for the right boundary z = z2 .
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
• Dispersive coupling element for the one-dimensional example: %D [mi ] = − %DB [mi ]
@jD; i ; @z
(49)
= ∓jD; i ;
@ %D [U ] = − @z
(50)
jQ +
%DB [U ] = ∓ jQ +
NC
jD; i hi
i=1 NC
jD; i hi
;
(51)
:
(52)
In Eqs. (50) and (52), the negative and positive sign hold for the left and for the right boundary, respectively. The dispersive terms in the energy balance consist of a term for the heat conduction and additional terms for the enthalpy transport by dispersive
Because the chemical reaction occurs in the interior of the phase and because of the conservation of energy, the chemical reaction does not aOect directly the internal energy balance. • Energy transforming coupling element for the onedimensional example: %T [U ] = −p
@vz ; @z
%TB [U ] = ∓pvz :
• Dispersive coupling element for the three-dimensional example: %D [mi ] = −∇jD; i ; %DB [mi ] = −nT jD; i ; %D [U ] = −∇ jQ + %DB [U ] = −nT
(60)
NC
jQ +
(61)
jD; i hi
i=1 NC
;
jD; i hi
(62) :
(63)
i=1
i=1
%R [mi ] = i Mi r0 :
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(54) (55)
Here, the energy transforming coupling element describes a transformation from kinetic to internal energy. This can be shown by balancing the internal energy and the kinetic energy and by comparing the terms in the two resulting equations (Bird et al., 1960). The structure of the coupling elements of the threedimensional example diOers only slightly from the one-dimensional example. Balancing of component masses and internal energy results in the following expressions: • Convective coupling element for the three-dimensional example: %C [mi ] = −vT ∇T gi ;
(56)
%CB [mi ] = −nT vgi ;
(57)
%C [U ] = −vT ∇T u;
(58)
%CB [U ] = −nT vu:
(59)
• Reactive coupling element for the three-dimensional example: %R [mi ] = i r0 Mi :
(64)
• Energy transforming coupling element for the threedimensional example: NC %T [U ] = −p∇v − : ∇v + (∇ i jD; i ); (65) i=1
%TB [U ] = −nT (pv + v + jD; i i ):
(66)
The transforming coupling element of the threedimensional example is more complicated than that of the one-dimensional example, because also viscous forces and the potential energy i of each component i are taken into account. In Eq. (65), the =rst two terms describe transformation from kinetic to internal energy. The =rst term is equivalent to the transforming coupling element of the one-dimensional example. The second term de=nes irreversible viscous eOects. The colon in that term stands for a scalar tensor product as de=ned in Bird et al. (1960). The third term is due to dispersive transport in a potential =eld. • External
(67)
j B [mi ] = nT j[mi ];
(68)
j[U ] = 0;
(69)
j B [U ] = nT j[U ]:
(70)
The structure of the thermodynamic subsystem can be visualised by graphical symbols as shown in Fig. 7. The system contains spatially distributed storages for internal energy and for the masses of the diOerent species. The storages interact via the four types of coupling elements described above. In general, the coupling elements do not only depend on the states of the thermodynamic but also on those of the
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Fig. 8. Example for orthogonal coupling of distributed systems: Radial heat transfer between two one-dimensional heat conductors. Fig. 7. Graphical representation of the thermodynamic subsystem.
This completes the structuring of spatially distributed phases on the level of storages. A general structure has been derived which is valid for all kinds of spatially distributed systems, independent of the number of space dimensions. Special attention has been given to the boundary conditions. The formulation of the boundary conditions seems to be a crucial point in the modularisation of spatially distributed models. The principle of modularity requires that the coupling of a distributed phase with diOerent kinds of distributed or well-mixed systems must be possible without having to change the model of the phase. On the other hand, the type of boundary condition is clearly determined not only by the model of the inner of the phase but also by the assumptions on the interaction of the phase with its environment. In order to get a
Eigenberger (1997). The orthogonal coupling determines the
(71)
J [U ]I (z; t) = *(T II (z; t) − T I (z; t));
(72)
I
II
where T and T are the temperature pro=les of the two coupled phases, and J [U ]I and J [U ]II are the heat
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
of freedom also increases the diOerential index of the resulting system of model equations and makes its numerical solution more di:cult. The inner structure of orthogonal coupling elements may be much more complicated than that of coupling elements for well-mixed systems. Because orthogonal coupling elements are spatially distributed systems themselves, transport processes inside the coupling element, e.g. mass transport by surface diOusion, are possible. However, no special attention must be paid to that more complicated inner structure when coupling the element to other components. The case of normal coupling is more interesting, because it is not found in well-mixed systems. Possible phenomena are best discussed by using a simple example. In the following, the boundary conditions of the component mass balances of the one-dimensional example in Fig. 4 will be revisited. If Fick diOusion is assumed, Eqs. (11) and (12) simplify to @gi 0 = vgi |z1 ;t − Di − j B [mi ](z1 ; t); (73) @z z1 ;t 0 = −vgi |z2 ;t
@gi + Di − j B [mi ](z2 ; t): @z z2 ;t
(74)
The mass
j [mi ](z1 ; t) = in vgi; in :
(75)
For the coupling element at the outlet, one obtains j B [mi ](z2 ; t) = −vgi (z2 ; t):
(76)
Obviously, (75) and (76) in combination with (73) and (74) are identical to the well-known Danckwerts boundary conditions. This shows that the modularity principle proposed here leads to the usual set of model equations. But it may make the underlying model assumptions more transparent and may force the modeller to think about validity of a chosen type of boundary conditions.
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Example 2. Coupling of two distributed phases in series (Fig. 9 b). The second example treats the formulation of boundary conditions between two adjacent distributed systems which are often called inner boundary conditions. It is assumed that the composition changes continuously at the phase boundary. Therefore, the equations of the coupling element will read gi (z2I ; t) = gi (z1II ; t);
(77)
j B [mi ](z2I ; t) = −j B [mi ](z1II ; t):
(78)
In combination with the general boundary conditions (73) and (74) this is just a variant of the usual inner boundary conditions for this situation. In this case, a clear distinction between inputs and outputs of the coupling element is no longer possible. The normal coupling between two distributed systems can be regarded as a kind of rigid coupling, because (77) establishes a =xed relation between the states at both sides of the boundary. However, in contrast to well-mixed systems, this rigid coupling does not increase the diOerential index of the resulting system of equations and is uncritical with respect to numerical treatment. From a mathematical point of view, this may be explained by the fact that (73), (74), (77), (78) form a system of algebraic equations for the compositions and the mass
(79)
and gi (z2 ; t) = gi;∗2
(80)
with parameters gi;∗1 and gi;∗2 . Obviously, these boundary conditions of the Dirichlet type are a highly idealising model formulation. A physical justi=cation may be the assumption of perfectly working controllers for the composition at both ends of the distributed phase. The controllers have to adjust the mass
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Fig. 9. Examples for normal coupling: (a) coupling of distributed and well-mixed systems; (b) inner boundary conditions for two distributed systems in series; (c) boundary conditions of Dirichlet type.
The distributed phase responds to the prescribed boundary composition with mass
simplifying or idealising model assumptions like vanishing transport resistances. However, a change in the direction of information is less critical for distributed systems than it is for well-mixed systems. As long as all boundary equations are algebraic, the diOerential index of the resulting system of model equations is not increased and no special measures have to be taken in order to ensure numerical solvability. 6. Structuring of dispersed phase systems Dispersed phase systems describe processes, where one or more dispersed phases like, e.g. bubbles, drops or crystals, are embedded in a continuous medium. A suitable and commonly accepted approach to model such dispersed phase systems is the concept of population balances (Hulburt & Katz, 1964; Ramkrishna, 2000; Randolph & Larson, 1962). This population balance approach considers the collectivity of all entities in a dispersed phase as a population distributed with regard to some purposive particle properties. Particle properties might be, for example, the particle size or shape in crystallisation processes, the chain length to distinguish diOerent polymers or the volume and composition of drops within a liquid–liquid extraction column. In this section, the characteristic structure of dispersed phase systems on the level of storages and on the level of phases will be discussed. In this context, further classes of coupling elements have to be de=ned, in order to include characteristic phenomena of dispersed phase systems into the structuring framework of the Network Theory.
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
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processes are the formation and the disappearance of particles. The term %exchange in Eq. (81) is thus given by %exchange = %birth − %death ;
Fig. 10. Schematic sketch of a simple well-mixed dispersed phase system consisting of one dispersed phase, which is embedded in a continuous phase.
In the following, the simple example system depicted in Fig. 10 will be considered. This system consists of one well-mixed dispersed phase, which is coupled by mass and energy exchange
and
F(e; t = 0) = F0 (e):
(81) (82)
The terms on the right-hand side of this population balance (81) account for diOerent population phenomena that act on the dispersed phase. In particular, these terms describe • convective transport terms −∇e (vint F) in direction of each particle property. In this expression, ∇e denotes the vector (@=@e1 ; @=@e2 ; : : : ; @=@eN ) of the partial diOerential operators with respect to each particle property. The internal velocities vint; ei with which the particles move in direction of each particle property ei are summarised by the vector vint . Considering the particle size as a particle property ei , this convective transport would account for the particle growth with the growth rate given by the internal velocity vint; ei . Such convective transport terms will in general be associated with exchange
(83)
where %birth and %death account for the newly formed particles and for the particles vanishing from the population, respectively. Examples for such phenomena are the thermodynamically driven nucleation or the dissolution of particles in crystallisation processes (Gerstlauer, MitroviDc, Motz, & Gilles, 2001; Gerstlauer, Motz, MitroviDc, & Gilles, 2002). • sink and source terms %intra due to intra particle phenomena. Such intra particle phenomena are the coalescence and fragmentation of particles. Thus, the term %intra can be subdivided according to the diOerent phenomena into two parts: %intra = %coal + %frag :
(84)
An important characteristic of such internal phenomena is, that they do not change overall conservative quantities of the whole dispersed phase like, e.g. the overall mass of all particles within the dispersed phase. The change of the number density function F by %coal and %frag thus results in a redistribution of the material within the dispersed phase with respect to the particle properties. Examples for intra particle phenomena are the aggregation of nano-particles in precipitation processes (Hounslow et al., 1999) or the attrition of crystals (Gerstlauer et al., 2001; Gerstlauer et al., 2002). The required boundary and initial conditions (82) for the population balance (81) have to be chosen depending on the considered process and the de=ned particle properties. In modelling crystallisation processes, for example, the crystal length L is an important property, for which the boundary condition F(L=0; t)=0 has to hold, since crystals with zero length are not possible (Gerstlauer et al., 2001; Gerstlauer et al., 2002). 6.1. Structure of dispersed phases on the level of storages Using the population balance (81), the structure of a general dispersed phase in terms of the Network Theory can be determined by subdividing all population phenomena into phase internal phenomena %intra and phenomena that describe an exchange with other phases. Within this subsection these external phenomena will be summarised by the general
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M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
Fig. 12. Graphical representation of the coupling of well-mixed continuous and dispersed phases.
6.2. Coupling of dispersed phases Fig. 11. Graphical representation of a dispersed phase on the level of storages.
see Fig. 11. As can be seen in Fig. 11, there are two possible coupling elements %coal and %frag , which can aOect the storage component, i.e. the number density function F(e; t). Those phenomena of coalescence and fragmentation build a new class of coupling elements, since they describe transport processes between points in the entire domain !e spanned by the de=ned property coordinates. These particle interactions, which are typical for dispersed phase system, have thus in general to be modelled by integrals over the particle population. Considering, for example, coalescence of drops in a liquid–liquid extraction column, the coupling element accounting for binary coalescence can be given by ∞ %coal = −F(V; t) /coal (V; V˜ )F(V˜ ; t) d V˜ +
1 2
0
0
V
/coal (V − V˜ ; V˜ )F(V − V˜ ; t)F(V˜ ; t) d V˜ (87)
with the rate /coal (V; V˜ ), which de=nes the probability of the coalescence of two drops with Volume V and V˜ (Gerstlauer, 1999). The binary dispersion of a drop as an example for particle fragmentation can be formulated as %frag = −/frag (V )F(V; t) ∞ P(V; V˜ )/frag (V˜ )F(V˜ ; t) d V˜ ; + V
(88)
where /frag (V ) is the dispersion rate of the drops depending on the drop volume V and P(V; V˜ ) is a probability density function that describes the connection between the volumes of the original, dispersed drop and the drops resulting from this fragmentation event. This concludes the structuring of well-mixed dispersed phases on the level of storages. An extension to spatially distributed dispersed phases described by a number density function F(e; x; y; z; t), which additionally depends on the three space coordinates x; y; z is straightforward, since such a storage distributed over the physical and a property space domain can easily be coupled with a
This subsection deals with the coupling of dispersed phases with other phases. Especially the coupling of dispersed phases with a continuous phase, see Fig. 10, is very important, since in chemical engineering dispersed phases are always embedded in a continuous medium. The simplest case, which will be considered in the following, is the one depicted in Fig. 10, i.e. the coupling of one well-mixed dispersed phase with an also well-mixed continuous phase. To establish this connection on the level of phases, appropriate coupling elements have to be de=ned, in order to calculate the exchange
for the calculation of a general
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
dispersed phase, respectively, since the population phenomena, which eOect an exchange
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Furthermore, a separation of a spatially distributed phase into two subsystems proves to be advantageous. The =rst subsystem stands for
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The structuring concept proposed here is a useful theoretical framework for the development of modular families of chemical engineering models. It has already been applied successfully to modelling of =xed bed reactors (Mangold, 2000), reactive distillation columns (Mohl, 2001), and fuel cell systems (Zhou, Schultz, Peglow, & Sundmacher, 2001) within the modelling tool ProMot (TrIankle et al., 2000). A closely related structuring principle has been applied to biological models of metabolic processes (Ginkel, Kremling, TrIankle, Gilles, & Zeitz, 2000). Acknowledgements The authors thank Prof. Hasse and Dr. Vrabec, Stuttgart, for their helpful contributions to the section on molecular modelling. The =nancial support of Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 412, University of Stuttgart, is gratefully acknowledged. References Andersson, M. (1990). Omola—An object-oriented language for model representation. Ph.D. thesis, Lund Institute of Technology. Bauer, M., & Eigenberger, G. (1999). A concept of multi-scale modeling of bubble columns and loop reactors. Chemical Engineering Science, 54, 5109–5117. Bauer, M., & Eigenberger, G. (2001). Multiscale modeling of hydrodynamics, mass transfer and reaction in bubble column reactors. Chemical Engineering Science, 56, 1067–1074. Berry, R. S., Rice, S. A., & Ross, J. (2000). Physical chemistry. Oxford: Oxford University Press. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (1960). Transport phenomena. New York: Wiley. Bogusch, R., & Marquardt, W. (2001). Computer-aided process modeling with ModKit. Computers & Chemical Engineering, 25, 963–995. Callen, H. B. (1985). Thermodynamics and an introduction to thermostatistics. New York: Wiley. Dieterich, E. E., & Eigenberger, G. (1997). The ModuSim concept for modular modeling and simulation in chemical engineering. Computers & Chemical Engineering, 21(Suppl.), S805–S809. Gerstlauer, A. (1999). Derivation and reduction of population models for the example of liquid–liquid extraction. Ph.D. thesis, University of Stuttgart (in German). Gerstlauer, A., Hierlemann, M., & Marquardt, W. (1993). On the representation of balance equations in knowledge-based process modeling tool. In 11th congress on Chemical Engineering, Chemical Equipments, Design and Automation. Gerstlauer, A., MitroviDc, A., Motz, S., & Gilles, E. D. (2001). A population model for crystallization processes using two independent particle properties. Chemical Engineering Science, 56(7), 2553–2565. Gerstlauer, A., Motz, S., MitroviDc, A., & Gilles, E. D. (2002). Development, analysis and validation of population models for continuous and batch crystallizers. Chemical Engineering Science, accepted for publication. Gilles, E. D. (1998). Network theory for chemical processes. Chemical Engineering Technology, 21, 121–132. Ginkel, M., Kremling, A., TrIankle, F., Gilles, E. D., & Zeitz, M. (2000). Application of the process modeling tool ProMot to the modeling of metabolic networks. In I. Troch, & F. Breitenecker (Eds.), IMACS symposium on mathematical modelling.
Hounslow, M. J., Mumtaz, H. S., Collier, A. P., Barrick, J. P., & Bramley, A. S. (1999). Aggregation during precipitation—putting the pieces of the puzzle together. In Industrial crystallization 1999. Institute of Chemical Engineers. Hulburt, H. M., & Katz, S. (1964). Some problems in particle technology—a statistical mechanical formulation. Chemical Engineering Science, 19, 555–574. Jensen, A. K., & Gani, R. (1999). A computer aided modeling system. Computers & Chemical Engineering, 23(Suppl.), S673–S678. Kriebel, C., Mecke, M., Winkelmann, J., Vrabec, J., & Fischer, J. (1998). An equation of state for dipolar lennard-jones molecules and its application to refrigerants. Fluid Phase Equilibria, 142, 15–32. Mangold, M. (2000). Nonlinear analysis and technical application of circulating reaction fronts. Ph.D. thesis, University of Stuttgart (in German). Marquardt, W. (1996). Trends in computer aided modeling. Computers & Chemical Engineering, 20, 591–609. Marquardt, W., Gerstlauer, A., & Gilles, E. D. (1993). Modeling and representation of complex objects: A chemical engineering perspective. In Proceedings of the sixth international conference on industrial and engineering applications of artiBcial intelligence and expert systems, Edinburgh, Scotland. Mattson, S. E., & SIoderlind, G. (1993). Index reduction in diOerential-algebraic equations using dummy derivatives. SIAM Journal of Science and Computers, 14, 677–692. Mohl, K. D. (2001). Nonlinear dynamics of reactive distillation processes. Ph.D. thesis, University of Stuttgart (in German). Perkins, J. D., Sargent, R. W. H., VDazquez-RomDan, R., & Cho, J. H. (1996). Computer generation of process models. Computers & Chemical Engineering, 20, 635–639. Ponton, J. W., & Gawthrop, P. J. (1991). Systematic construction of dynamic models for phase equilibrium processes. Computers & Chemical Engineering, 15, 803–808. Preisig, H. A. (1996). Computer-aided modelling: Two paradigms on control. Computers & Chemical Engineering (Suppl.), S981–S986. Ramkrishna, D. (2000). Population balances, theory and application to particulate systems in engineering. New York: Academic Press. Randolph, A. D., & Larson, M. (1962). Transient and steady state size distribution in continuous mixed suspension crystallizers. A.I.Ch.E. Journal, 8, 639–645. Reuss, M., Schmalzriedt, S., & Jenne, M. (2000). Application of computational
www.elsevier.com/locate/ces
A network theory for the structured modelling of chemical processes M. Mangolda;∗ , S. Motzb , E.D. Gillesa; b a Max-Planck-Institut
b Universit at
fur Dynamik komplexer technischer Systeme, Sandtorstra e 1, 39106 Magdeburg, Germany Stuttgart, Institut fur Systemdynamik und Regelungstechnik, Pfa(enwaldring 9, 70550 Stuttgart, Germany Received 19 November 2001; received in revised form 17 May 2002; accepted 2 June 2002
Abstract A structuring methodology for dynamic models of chemical engineering processes is presented. The main ideas of the methodology were outlined in a previous publication for the class of well-mixed systems. In this contribution, the methodology is extended to spatially distributed systems and to particulate processes. Furthermore, the structuring principle is used to make a conceptual link between the macroscopic world of process simulation and the microscopic world of molecular simulation. It is shown that a uniform structuring principle can be applied to the modularisation of most classes of chemical engineering models. The structuring principle can be used as a theoretical framework for the implementation of modular families of chemical engineering models in modern computer aided modelling tools. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Systematic modelling; Computer aided modelling; Mathematical modelling; Spatially distributed systems; Population balances; Dynamic simulation; Process control
1. Introduction Mathematical models of chemical processes have been widely accepted as a useful tool for the development of processes and control schemes. Today, many numerical algorithms and tools for the analysis of very complex models are available. However, the formulation of the models themselves is still an obstacle for the application of model-based techniques in industry. The development and implementation of new models is considered as a di:cult and expensive task. An important reason for this fact is the complexity and low reusability of process models as they are typically implemented today. Usually, models of chemical plants are formulated on a
structuring of the models on a level below the process unit have been recognised as important steps towards a more e:cient model development. In the past, several authors proposed general concepts for the structuring of process models (Ponton & Gawthrop, 1991; Perkins, Sargent, VDazquez-RomDan, & Cho, 1996; Marquardt, 1996; Preisig, 1996). The main interest in the past was in well-mixed systems in general or special subclasses thereof, like distillation processes or phase equilibrium models. Little attention has been given to distributed systems. Based on those general structuring concepts modelling tools and languages have been developed which permit a modular and highly structured model formulation. Examples are Omola (Andersson, 1990), ModDev (Jensen & Gani, 1999), Modeller (Westerweele, Preisig, & Weiss, 1999), ModKit (Bogusch & Marquardt, 2001), or ProMot (TrIankle, Zeitz, Ginkel, & Gilles, 2000). These modelling tools have provided a powerful framework for the development of model libraries based on simple,
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 3 7 2 - X
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to keep the number of elements in the library =nite and to ensure compatibility between the modelling entities in the library, a theoretical concept is needed as a guideline for the model library developer. The Network Theory of Chemical Processes, which was proposed in a former publication (Gilles, 1998), oOers such a modelling concept. In Gilles (1998), the Network Theory was discussed for the class of well-mixed systems. The purpose of this article is to extend the ideas to other classes of chemical engineering processes. In the next section, the basic concepts of the Network Theory are described, as they were stated in (Gilles, 1998). It is shown that the same kind of structural description can be used to decompose a model on diOerent hierarchical levels of granularity. The molecular level, the level of storages of macroscopic thermodynamic quantities, the level of thermodynamic phases, and the level of process units are considered. The molecular level will be examined more closely. Interfaces between the molecular level and the macroscopic thermodynamic level will be discussed. The main part of the article will deal with the internal structuring of thermodynamic phases, as this level of description usually is the most complicated part in the development of chemical process models. The concept of Network Theory will be extended to models described by partial diOerential equations. Spatially distributed systems as well as dispersed phase systems and population balances are considered. Inside those systems, additional types of transport mechanisms occur which are not present in lumped systems. Their description requires the de=nition of new classes of structural modelling units. In addition, the treatment of boundary conditions must be included in the structuring process. The boundary conditions de=ne the interface between an distributed system and its environment. Therefore, a structured formulation of the boundary conditions is of special importance for the coupling of distributed systems with other systems and for the use of distributed modelling entities in a model library. 2. Basic concepts of network theory Plant models in current
elling tools consists in facilitating and supporting that second step of the model development. The basic idea is to provide pre-de=ned model formulations the modeller can use to solve his speci=c modelling task. The usability of a model library strongly depends not only on the amount of information contained but also on the way the information is structured. A library of process unit models is only of limited use. Considering the enormous number of existing and conceivable process units, the attempt to store models of all of them in a library is doomed to failure. On the other hand, models of diOerent process units often are very similar. The model formulations are based on a limited number of physical assumptions and phenomenological descriptions. First principle models consist of balance equations derived from conservation principles for energy, mass, momentum, and charge. The same formulations for diOusion, convection, heat or mass transfer are used again and again in many apparatus models. The vast range of diOerent process unit models mainly originates from the many possibilities how existing modelling approaches can be combined. Therefore, the aim to collect modelling knowledge in a library becomes much more attainable if pursued below the level of process units. Process units can be considered as an aggregation of interacting thermodynamic phases. The thermodynamic phases may serve as elementary modelling units, if only a certain class of systems, e.g. well-mixed systems, is considered (Perkins et al., 1996). However, a uniform treatment of lumped and distributed parameter systems requires a segregation of the phases into smaller units. A high amount of
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
generalised thermodynamic forces (Callen, 1985). Similar structuring principles have been proposed by other authors (Ponton & Gawthrop, 1991; Marquardt, Gerstlauer, & Gilles, 1993; Marquardt, 1996; Preisig, 1996; Weiss & Preisig, 2000). Components may be considered as devices in the terminology of Marquardt and co-workers or as systems in the terminology of Preisig and co-workers. Coupling elements are termed connections by Marquardt and Preisig. • Components and coupling elements can be de=ned on different hierarchical modelling levels. For example, a component may be a single thermodynamic phase or it may be a process unit which consists of several interacting phases. Consequently, systems of components and coupling elements can be aggregated to a single component on a higher level. On the other hand, elementary units on one level may be decomposed into systems of components and coupling units on a lower level, which gives a more detailed view on the system. The two basic ideas will subsequently be discussed in detail. 2.1. The potential-8ux-vector concept of bidirectional signal transfer When structuring a chemical process, a clear distinction has to be made between the direction of physical
4101
Fig. 1. Interconnection of two components.
When two components Ck and Cl interact via a coupling element CE(Ck ; Cl ) , the direction of
where NJ is the number of
(2)
Because the Jacobian @gi =@Ji is non-singular by de=nition, a single derivation in time of (2) su:ces to turn the model equations into a system of ordinary diOerential equations. Simplifying modelling assumptions tend to violate the proposed direction of signal transfer. Typical examples are equilibrium assumptions or assumptions of vanishing mass or heat transfer resistances. If such an assumption is made, the coupled components will be no longer independent, but will lose a degree of freedom. In Gilles (1998) this type of coupling is therefore termed rigid coupling. In the case of rigid coupling, it is no longer possible to assign unique directions of information
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M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
level of process units liquid phase
phase boundary
vapor phase
level of phases
X=(U,V,mi )
+ + level of storages
molecule
intermolecular forces molecular level
Fig. 2. Levels of process structuring and examples for structural decomposition on the diOerent levels.
alternative formulation for the components at equilibrium, e.g. by the formulation of pseudo-homogeneous balance equations (Ponton & Gawthrop, 1991; Gilles, 1998). 2.2. The levels of process structuring The model of a chemical plant can be structured on different levels. This is illustrated by Fig. 2. The top level regarded here is the level of process units. On this level, process units like distillation columns, reactors, or heat exchangers are considered as the elementary components. They are connected by coupling elements such as pipes, valves, pumps etc. A major part of chemical engineering research has concentrated on the analysis and modelling of individual process units. Therefore, the elementary functional units on the level of process units are described by very complex models in many cases. In order to improve the structuring of the process models and to get simple and reusable modelling units, it is helpful to decompose the models of process units into smaller units. A process unit may be considered as a system of interacting components and coupling elements on a lower level. This lower level is the level of phases. A process unit usually contains several phases, e.g. a liquid and a vapour phase. Each phase is an elementary component on the level of phases. The phases interact by heat transfer, mass transfer, or transfer
of momentum. The corresponding coupling elements which describe this interaction are phase boundaries, membranes, diOusive layers, etc. The state of a thermodynamic phase is described by macroscopic thermodynamic quantities like energy, mass, and momentum. In order to decompose a phase into smaller units, the phase can be considered as a system of interacting storages of mass, energy, and momentum, which are the components on the next lower level—the level of storages. The components on the level of storages interact by transport processes like reaction, diOusion, and convection. Therefore, the purpose of coupling elements on the level of storages is to describe the
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
equations of state
X
(T,p, µi )
CE P[X]
+
J
+ level of storages
ε2
ε1
ε3
molecular level
Fig. 3. Interfaces between molecular simulation on the molecular level and process simulation on the level of storages for the example of a well-mixed homogeneous phase. The levels interact via the reactive coupling element CE and via the equations of state.
applied successfully, e.g. to the computation of vapour liquid equilibria of diOerent substances (Vrabec et al., 1995; Kriebel et al., 1998; Stoll et al. 2001). In the structuring principle of the network theory, the components on the molecular level are the states of motion of the molecules, i.e. their coordinates and velocities, and further molecular properties like size or polarity required to solve the equations of motion. The coupling elements describe the attracting and repulsing forces between the molecules, e.g. by potentials of the Lennard–Jones type. For the application of molecular simulation to real world problems, it is necessary to de=ne clear interfaces between the molecular level and the higher levels of process structuring. In the following, possible interfaces will be discussed from two points of view. The =rst viewpoint is a strictly thermodynamic one and is shown in Fig. 2. According to statistical thermodynamics, the states of all quantities on the macroscopic level can be obtained by averaging over the molecular states. Therefore, in theory it may be possible to gather all information on the level of storages by solving the equations of motion on the molecular level. In reality, this approach will hardly be viable because determining the states of all the molecules of a technical process is a far too complex task. Therefore, a slightly more pragmatic viewpoint will be presented subsequently and is shown in Fig. 3. The interaction between macroscopic process simulation and molecular simulation will be e:cient, if as many computations as possible are done on the macroscopic level and molecular simulation is only used if the available macroscopic information is not su:cient. By macroscopic balancing, it is always possible to derive diOerential equations for the extensive state variables required to describe the macroscopic state of a phase, e.g. the internal energy U , the volume V , and the masses mi , i = 1; : : : ; NC of all NC components. The right-hand side of those diOerential equations contain the sinks and sources of the phase, i.e. the behavioural description of the coupling
4103
elements between the macroscopic storages. Typically, the coupling elements depend on many parameters which describe the physical and chemical properties of the phase. The determination of the parameters is not possible by macroscopic balancing, but it may be possible to =nd some of them by molecular simulation. Therefore, the internal sinks and sources in a phase can be regarded as one type of interface between the molecular and the macroscopic level. For most applications, it is convenient to replace the extensive variables U and V by the intensive variables temperature T and pressure p. The transformation between the extensive variables U , V , mi and the intensive variables T , p, i de=nes the three equations of state @U @U @U T= ; p = ; = (3) i @S @V @mi V; mi
S; mi
S; V
(see e.g. (Callen, 1985)). The equations of state can be obtained by macroscopic measurements or by simulations on the molecular level. Therefore, the transformation between extensive and intensive variables is a second type of interface between the molecular level and the level of storages. 4. Structuring of well-mixed phases on the level of storages and on the level of phases The structured modelling of homogeneous well-mixed phases was treated in detail in Gilles (1998) and will only be summarised brie
(4)
J [U ](1) + J [U ](2) = 0;
(5)
1
If there are electrical eOects, then also the transformation from internal energy to electrical energy and vice versa will have to be taken into account. This requires the de=nition of additional coupling elements, but will not be treated here.
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where T (1) and T (2) are the temperatures in the two phases, and J [U ](1) and J [U ](2) are the
velocity distributions and momentum balances. Therefore, the spatially distributed phase will be divided into two subsystems, called the
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
4105
n jB
Ω(x,y,z) = 0
Fig. 4. One-dimensional example system.
• • •
• •
y
mass of component i, j[mi ] is the corresponding mass
Under these assumptions, the following equations hold for the one-dimensional example system: • Equation of continuity: @ @ = − (vz ): @t @z • Momentum balance (Euler equation): @vz @vz @p = −vz − : @t @z @z • Total mass balances at the system boundaries: 0 = vz (z1 ; t) − 0 = vz (z2 ; t) +
NC i=1 NC
(6)
(7)
j B [mi ](z1 ; t);
(8)
j B [mi ](z2 ; t):
(9)
i=1
• Component mass balances (i = 1; : : : ; NC − 1): @gi @gi @jD; i = −vz − + i M i r0 ; @t @z @z
x z Fig. 5. Three-dimensional example system. n: normal vector on the surface ! = 0 as the system boundary.
• Balance of internal energy: NC @u @ @vz @u = −vz − jQ + ; (13) jD; i hi − p @t @z @z @z i=1 NC − j[U ](z1 ; t) = −vz u|z1 ;t − jQ (z1 ; t) + jD; i hi |z1 ;t i=1
− pvz |z1 ;t ;
(14)
j[U ](z2 ; t) = −vz u|z2 ;t −
jQ (z2 ; t) +
NC
jD; i hi |z2 ;t
i=1
−pvz |z2 ;t :
(15)
In the above equations, p denotes the pressure, gi is the mass fraction of component i, i is the stoichiometric coe:cient of component i, Mi is its molar mass, and hi is its partial molar enthalpy. In Sections 5.1.1 and 5.1.2, the equations will be structured according to the methodology of the network theory. A general structure of the balance equations will be derived, which is also valid for other spatially distributed systems. This will be demonstrated by a second example, which is a rather general three-dimensional homogeneous system as shown in Fig. 5. The geometry of the system is de=ned by some function !(x; y; z), where !(x; y; z) ¡ 0
(16)
is the inner of the system, and (10)
!(x; y; z) = 0
(17)
− j B [mi ](z1 ; t) = −vz gi |z1 ;t − jD; i (z1 ; t);
(11)
is the system boundary. In the case of the one-dimensional example, ! can be de=ned as
j B [mi ](z2 ; t) = −vz gi |z2 ;t − jD; i (z2 ; t):
(12)
! = (z − z1 )(z − z2 ):
(18)
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In this notation, the vector @ @ @ (∇!)T ; ∇= n= ; ; ∇! @x @y @z
(19)
is a normal vector perpendicular to the surface of the system pointing outwards. The derivation of the balance equations of the threedimensional example system can be found in literature, e.g. (Bird, Stewart, & Lightfoot, 1960). Therefore, in Sections 5.1.1 and 5.1.2 only the results of the structuring process will be given for the three-dimensional example. 5.1.1. Structure of the 8uid-dynamic subsystem The
• Balance of momentum M in the interior of the system !(x; y; z) ¡ 0: @v = C [M ] + D [M ] + j[M ]: (21) @t • Balance of total mass on the system boundary !(x; y; z) = 0: 0 = %CB [m] + j B [m]:
(22)
• Balance of momentum on the system boundary !(x; y; z) = 0: 0 = BC [M ] + BD [M ] + j B [M ]:
(23)
The one-dimensional example may be seen as a special case of the general
(25) (26)
• Dispersive coupling element for the one-dimensional example: @p T D [M ] = 0; 0; − : (27) @z • External
NC
(28) j B [mi ];
(29)
i=1
j[M ] = 0:
(30)
It should be noted that in the one-dimensional example two boundary conditions of type (22) are used, but none of type (23). This point will be discussed in detail at the end of this section. Before, it will be shown that Eqs. (20)–(23) also hold for the three-dimensional example. The only diOerence between the one-dimensional and the three-dimensional example is in the behavioural description of the coupling elements. Balancing of mass and momentum leads to the following source terms: • Convective coupling element for the three-dimensional example: %C [m] = −∇(v);
(31)
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
C [M ] = −vT ∇T vT ;
(32)
%CB [m] = −vT n;
(33)
BC [M ] = −vvT n:
(34)
The three-dimensional convective terms are a rather straightforward extension of the one-dimensional case. • Dispersive coupling element for the three-dimensional example: NC 1 T D [M ] = −∇ pI + + (35) jD; i jD; i ; i i=1 T NC 1 B T D [M ] = − pI + + jD; i jD; i n: (36) i i=1
The =rst term in each of the bracketed expressions in Eqs. (35) and (36) already appeared in the one-dimensional example. The other two terms have been neglected so far. The second term describes transport of momentum caused by viscous forces; is the stress tensor (Bird et al., 1960). The third term describes transport of momentum due to dispersive
(37)
j B [m] = j B [m]T n;
(38)
j[M ] =
NC
i fi :
(39)
i=1
In contrast to one- and two-dimensional systems, the mass
(40)
Based on the above considerations, a graphical representation of a general
4107
Fig. 6. Graphical representation of the
state of the storages is the only input to the convective coupling element. The information exchange between storages and coupling elements is not meant to be local, i.e. not only states and
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M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
some ambiguity concerning states and
• Dispersive coupling elements D describe transport phenomena due to diOusion and heat conduction. • Reactive coupling elements R describe the change in the composition by chemical reaction processes. They are identical to the reactive coupling elements of well-mixed systems. • In the balance equation for the internal energy, conversion of other kinds of energy into internal energy has to be taken into account. Examples are the conversion of kinetic energy into internal energy which manifests itself as friction heat or the conversion of potential energy into internal energy caused by diOusive
(42)
• Balance of internal energy in the interior of the system !(x; y; z) ¡ 0: @u = %C [U ] + %D [U ] + %T [U ] + j[U ]: (43) @t • Balance of internal energy on the system boundary !(x; y; z) = 0: 0 = %CB [U ] + %DB [U ] + %TB [U ] + j B [U ]:
(44)
As in the case of the
%C [U ] = −vz
(47) (48)
In Eqs. (46) and (48), the negative sign holds for the left boundary z =z1 , and the positive sign is valid for the right boundary z = z2 .
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
• Dispersive coupling element for the one-dimensional example: %D [mi ] = − %DB [mi ]
@jD; i ; @z
(49)
= ∓jD; i ;
@ %D [U ] = − @z
(50)
jQ +
%DB [U ] = ∓ jQ +
NC
jD; i hi
i=1 NC
jD; i hi
;
(51)
:
(52)
In Eqs. (50) and (52), the negative and positive sign hold for the left and for the right boundary, respectively. The dispersive terms in the energy balance consist of a term for the heat conduction and additional terms for the enthalpy transport by dispersive
Because the chemical reaction occurs in the interior of the phase and because of the conservation of energy, the chemical reaction does not aOect directly the internal energy balance. • Energy transforming coupling element for the onedimensional example: %T [U ] = −p
@vz ; @z
%TB [U ] = ∓pvz :
• Dispersive coupling element for the three-dimensional example: %D [mi ] = −∇jD; i ; %DB [mi ] = −nT jD; i ; %D [U ] = −∇ jQ + %DB [U ] = −nT
(60)
NC
jQ +
(61)
jD; i hi
i=1 NC
;
jD; i hi
(62) :
(63)
i=1
i=1
%R [mi ] = i Mi r0 :
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(54) (55)
Here, the energy transforming coupling element describes a transformation from kinetic to internal energy. This can be shown by balancing the internal energy and the kinetic energy and by comparing the terms in the two resulting equations (Bird et al., 1960). The structure of the coupling elements of the threedimensional example diOers only slightly from the one-dimensional example. Balancing of component masses and internal energy results in the following expressions: • Convective coupling element for the three-dimensional example: %C [mi ] = −vT ∇T gi ;
(56)
%CB [mi ] = −nT vgi ;
(57)
%C [U ] = −vT ∇T u;
(58)
%CB [U ] = −nT vu:
(59)
• Reactive coupling element for the three-dimensional example: %R [mi ] = i r0 Mi :
(64)
• Energy transforming coupling element for the threedimensional example: NC %T [U ] = −p∇v − : ∇v + (∇ i jD; i ); (65) i=1
%TB [U ] = −nT (pv + v + jD; i i ):
(66)
The transforming coupling element of the threedimensional example is more complicated than that of the one-dimensional example, because also viscous forces and the potential energy i of each component i are taken into account. In Eq. (65), the =rst two terms describe transformation from kinetic to internal energy. The =rst term is equivalent to the transforming coupling element of the one-dimensional example. The second term de=nes irreversible viscous eOects. The colon in that term stands for a scalar tensor product as de=ned in Bird et al. (1960). The third term is due to dispersive transport in a potential =eld. • External
(67)
j B [mi ] = nT j[mi ];
(68)
j[U ] = 0;
(69)
j B [U ] = nT j[U ]:
(70)
The structure of the thermodynamic subsystem can be visualised by graphical symbols as shown in Fig. 7. The system contains spatially distributed storages for internal energy and for the masses of the diOerent species. The storages interact via the four types of coupling elements described above. In general, the coupling elements do not only depend on the states of the thermodynamic but also on those of the
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Fig. 8. Example for orthogonal coupling of distributed systems: Radial heat transfer between two one-dimensional heat conductors. Fig. 7. Graphical representation of the thermodynamic subsystem.
This completes the structuring of spatially distributed phases on the level of storages. A general structure has been derived which is valid for all kinds of spatially distributed systems, independent of the number of space dimensions. Special attention has been given to the boundary conditions. The formulation of the boundary conditions seems to be a crucial point in the modularisation of spatially distributed models. The principle of modularity requires that the coupling of a distributed phase with diOerent kinds of distributed or well-mixed systems must be possible without having to change the model of the phase. On the other hand, the type of boundary condition is clearly determined not only by the model of the inner of the phase but also by the assumptions on the interaction of the phase with its environment. In order to get a
Eigenberger (1997). The orthogonal coupling determines the
(71)
J [U ]I (z; t) = *(T II (z; t) − T I (z; t));
(72)
I
II
where T and T are the temperature pro=les of the two coupled phases, and J [U ]I and J [U ]II are the heat
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
of freedom also increases the diOerential index of the resulting system of model equations and makes its numerical solution more di:cult. The inner structure of orthogonal coupling elements may be much more complicated than that of coupling elements for well-mixed systems. Because orthogonal coupling elements are spatially distributed systems themselves, transport processes inside the coupling element, e.g. mass transport by surface diOusion, are possible. However, no special attention must be paid to that more complicated inner structure when coupling the element to other components. The case of normal coupling is more interesting, because it is not found in well-mixed systems. Possible phenomena are best discussed by using a simple example. In the following, the boundary conditions of the component mass balances of the one-dimensional example in Fig. 4 will be revisited. If Fick diOusion is assumed, Eqs. (11) and (12) simplify to @gi 0 = vgi |z1 ;t − Di − j B [mi ](z1 ; t); (73) @z z1 ;t 0 = −vgi |z2 ;t
@gi + Di − j B [mi ](z2 ; t): @z z2 ;t
(74)
The mass
j [mi ](z1 ; t) = in vgi; in :
(75)
For the coupling element at the outlet, one obtains j B [mi ](z2 ; t) = −vgi (z2 ; t):
(76)
Obviously, (75) and (76) in combination with (73) and (74) are identical to the well-known Danckwerts boundary conditions. This shows that the modularity principle proposed here leads to the usual set of model equations. But it may make the underlying model assumptions more transparent and may force the modeller to think about validity of a chosen type of boundary conditions.
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Example 2. Coupling of two distributed phases in series (Fig. 9 b). The second example treats the formulation of boundary conditions between two adjacent distributed systems which are often called inner boundary conditions. It is assumed that the composition changes continuously at the phase boundary. Therefore, the equations of the coupling element will read gi (z2I ; t) = gi (z1II ; t);
(77)
j B [mi ](z2I ; t) = −j B [mi ](z1II ; t):
(78)
In combination with the general boundary conditions (73) and (74) this is just a variant of the usual inner boundary conditions for this situation. In this case, a clear distinction between inputs and outputs of the coupling element is no longer possible. The normal coupling between two distributed systems can be regarded as a kind of rigid coupling, because (77) establishes a =xed relation between the states at both sides of the boundary. However, in contrast to well-mixed systems, this rigid coupling does not increase the diOerential index of the resulting system of equations and is uncritical with respect to numerical treatment. From a mathematical point of view, this may be explained by the fact that (73), (74), (77), (78) form a system of algebraic equations for the compositions and the mass
(79)
and gi (z2 ; t) = gi;∗2
(80)
with parameters gi;∗1 and gi;∗2 . Obviously, these boundary conditions of the Dirichlet type are a highly idealising model formulation. A physical justi=cation may be the assumption of perfectly working controllers for the composition at both ends of the distributed phase. The controllers have to adjust the mass
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Fig. 9. Examples for normal coupling: (a) coupling of distributed and well-mixed systems; (b) inner boundary conditions for two distributed systems in series; (c) boundary conditions of Dirichlet type.
The distributed phase responds to the prescribed boundary composition with mass
simplifying or idealising model assumptions like vanishing transport resistances. However, a change in the direction of information is less critical for distributed systems than it is for well-mixed systems. As long as all boundary equations are algebraic, the diOerential index of the resulting system of model equations is not increased and no special measures have to be taken in order to ensure numerical solvability. 6. Structuring of dispersed phase systems Dispersed phase systems describe processes, where one or more dispersed phases like, e.g. bubbles, drops or crystals, are embedded in a continuous medium. A suitable and commonly accepted approach to model such dispersed phase systems is the concept of population balances (Hulburt & Katz, 1964; Ramkrishna, 2000; Randolph & Larson, 1962). This population balance approach considers the collectivity of all entities in a dispersed phase as a population distributed with regard to some purposive particle properties. Particle properties might be, for example, the particle size or shape in crystallisation processes, the chain length to distinguish diOerent polymers or the volume and composition of drops within a liquid–liquid extraction column. In this section, the characteristic structure of dispersed phase systems on the level of storages and on the level of phases will be discussed. In this context, further classes of coupling elements have to be de=ned, in order to include characteristic phenomena of dispersed phase systems into the structuring framework of the Network Theory.
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
4113
processes are the formation and the disappearance of particles. The term %exchange in Eq. (81) is thus given by %exchange = %birth − %death ;
Fig. 10. Schematic sketch of a simple well-mixed dispersed phase system consisting of one dispersed phase, which is embedded in a continuous phase.
In the following, the simple example system depicted in Fig. 10 will be considered. This system consists of one well-mixed dispersed phase, which is coupled by mass and energy exchange
and
F(e; t = 0) = F0 (e):
(81) (82)
The terms on the right-hand side of this population balance (81) account for diOerent population phenomena that act on the dispersed phase. In particular, these terms describe • convective transport terms −∇e (vint F) in direction of each particle property. In this expression, ∇e denotes the vector (@=@e1 ; @=@e2 ; : : : ; @=@eN ) of the partial diOerential operators with respect to each particle property. The internal velocities vint; ei with which the particles move in direction of each particle property ei are summarised by the vector vint . Considering the particle size as a particle property ei , this convective transport would account for the particle growth with the growth rate given by the internal velocity vint; ei . Such convective transport terms will in general be associated with exchange
(83)
where %birth and %death account for the newly formed particles and for the particles vanishing from the population, respectively. Examples for such phenomena are the thermodynamically driven nucleation or the dissolution of particles in crystallisation processes (Gerstlauer, MitroviDc, Motz, & Gilles, 2001; Gerstlauer, Motz, MitroviDc, & Gilles, 2002). • sink and source terms %intra due to intra particle phenomena. Such intra particle phenomena are the coalescence and fragmentation of particles. Thus, the term %intra can be subdivided according to the diOerent phenomena into two parts: %intra = %coal + %frag :
(84)
An important characteristic of such internal phenomena is, that they do not change overall conservative quantities of the whole dispersed phase like, e.g. the overall mass of all particles within the dispersed phase. The change of the number density function F by %coal and %frag thus results in a redistribution of the material within the dispersed phase with respect to the particle properties. Examples for intra particle phenomena are the aggregation of nano-particles in precipitation processes (Hounslow et al., 1999) or the attrition of crystals (Gerstlauer et al., 2001; Gerstlauer et al., 2002). The required boundary and initial conditions (82) for the population balance (81) have to be chosen depending on the considered process and the de=ned particle properties. In modelling crystallisation processes, for example, the crystal length L is an important property, for which the boundary condition F(L=0; t)=0 has to hold, since crystals with zero length are not possible (Gerstlauer et al., 2001; Gerstlauer et al., 2002). 6.1. Structure of dispersed phases on the level of storages Using the population balance (81), the structure of a general dispersed phase in terms of the Network Theory can be determined by subdividing all population phenomena into phase internal phenomena %intra and phenomena that describe an exchange with other phases. Within this subsection these external phenomena will be summarised by the general
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M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
Fig. 12. Graphical representation of the coupling of well-mixed continuous and dispersed phases.
6.2. Coupling of dispersed phases Fig. 11. Graphical representation of a dispersed phase on the level of storages.
see Fig. 11. As can be seen in Fig. 11, there are two possible coupling elements %coal and %frag , which can aOect the storage component, i.e. the number density function F(e; t). Those phenomena of coalescence and fragmentation build a new class of coupling elements, since they describe transport processes between points in the entire domain !e spanned by the de=ned property coordinates. These particle interactions, which are typical for dispersed phase system, have thus in general to be modelled by integrals over the particle population. Considering, for example, coalescence of drops in a liquid–liquid extraction column, the coupling element accounting for binary coalescence can be given by ∞ %coal = −F(V; t) /coal (V; V˜ )F(V˜ ; t) d V˜ +
1 2
0
0
V
/coal (V − V˜ ; V˜ )F(V − V˜ ; t)F(V˜ ; t) d V˜ (87)
with the rate /coal (V; V˜ ), which de=nes the probability of the coalescence of two drops with Volume V and V˜ (Gerstlauer, 1999). The binary dispersion of a drop as an example for particle fragmentation can be formulated as %frag = −/frag (V )F(V; t) ∞ P(V; V˜ )/frag (V˜ )F(V˜ ; t) d V˜ ; + V
(88)
where /frag (V ) is the dispersion rate of the drops depending on the drop volume V and P(V; V˜ ) is a probability density function that describes the connection between the volumes of the original, dispersed drop and the drops resulting from this fragmentation event. This concludes the structuring of well-mixed dispersed phases on the level of storages. An extension to spatially distributed dispersed phases described by a number density function F(e; x; y; z; t), which additionally depends on the three space coordinates x; y; z is straightforward, since such a storage distributed over the physical and a property space domain can easily be coupled with a
This subsection deals with the coupling of dispersed phases with other phases. Especially the coupling of dispersed phases with a continuous phase, see Fig. 10, is very important, since in chemical engineering dispersed phases are always embedded in a continuous medium. The simplest case, which will be considered in the following, is the one depicted in Fig. 10, i.e. the coupling of one well-mixed dispersed phase with an also well-mixed continuous phase. To establish this connection on the level of phases, appropriate coupling elements have to be de=ned, in order to calculate the exchange
for the calculation of a general
M. Mangold et al. / Chemical Engineering Science 57 (2002) 4099 – 4116
dispersed phase, respectively, since the population phenomena, which eOect an exchange
4115
Furthermore, a separation of a spatially distributed phase into two subsystems proves to be advantageous. The =rst subsystem stands for
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The structuring concept proposed here is a useful theoretical framework for the development of modular families of chemical engineering models. It has already been applied successfully to modelling of =xed bed reactors (Mangold, 2000), reactive distillation columns (Mohl, 2001), and fuel cell systems (Zhou, Schultz, Peglow, & Sundmacher, 2001) within the modelling tool ProMot (TrIankle et al., 2000). A closely related structuring principle has been applied to biological models of metabolic processes (Ginkel, Kremling, TrIankle, Gilles, & Zeitz, 2000). Acknowledgements The authors thank Prof. Hasse and Dr. Vrabec, Stuttgart, for their helpful contributions to the section on molecular modelling. The =nancial support of Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 412, University of Stuttgart, is gratefully acknowledged. References Andersson, M. (1990). Omola—An object-oriented language for model representation. Ph.D. thesis, Lund Institute of Technology. Bauer, M., & Eigenberger, G. (1999). A concept of multi-scale modeling of bubble columns and loop reactors. Chemical Engineering Science, 54, 5109–5117. Bauer, M., & Eigenberger, G. (2001). Multiscale modeling of hydrodynamics, mass transfer and reaction in bubble column reactors. Chemical Engineering Science, 56, 1067–1074. Berry, R. S., Rice, S. A., & Ross, J. (2000). Physical chemistry. Oxford: Oxford University Press. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (1960). Transport phenomena. New York: Wiley. Bogusch, R., & Marquardt, W. (2001). Computer-aided process modeling with ModKit. Computers & Chemical Engineering, 25, 963–995. Callen, H. B. (1985). Thermodynamics and an introduction to thermostatistics. New York: Wiley. Dieterich, E. E., & Eigenberger, G. (1997). The ModuSim concept for modular modeling and simulation in chemical engineering. Computers & Chemical Engineering, 21(Suppl.), S805–S809. Gerstlauer, A. (1999). Derivation and reduction of population models for the example of liquid–liquid extraction. Ph.D. thesis, University of Stuttgart (in German). Gerstlauer, A., Hierlemann, M., & Marquardt, W. (1993). On the representation of balance equations in knowledge-based process modeling tool. In 11th congress on Chemical Engineering, Chemical Equipments, Design and Automation. Gerstlauer, A., MitroviDc, A., Motz, S., & Gilles, E. D. (2001). A population model for crystallization processes using two independent particle properties. Chemical Engineering Science, 56(7), 2553–2565. Gerstlauer, A., Motz, S., MitroviDc, A., & Gilles, E. D. (2002). Development, analysis and validation of population models for continuous and batch crystallizers. Chemical Engineering Science, accepted for publication. Gilles, E. D. (1998). Network theory for chemical processes. Chemical Engineering Technology, 21, 121–132. Ginkel, M., Kremling, A., TrIankle, F., Gilles, E. D., & Zeitz, M. (2000). Application of the process modeling tool ProMot to the modeling of metabolic networks. In I. Troch, & F. Breitenecker (Eds.), IMACS symposium on mathematical modelling.
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