Sep 25, 1999 - steady-state solution that could be used in the design of distillation ..... In multicomponent distillation, it is assumed that all the resistance to ...
Dynamic Modelling of Plate Distillation Columns Based on Nonequilibrium Models by Pedro Martínez Conesa
A Thesis submitted in partial fulfilment of the requirement for the degree of M.Sc. in Process Systems Engineering of the University of London and the Diploma of Imperial College
Centre for Process Systems Engineering Department of Chemical Engineering and Chemical Technology Imperial College of Science, Technology and Medicine University of London
September 25th, 1999
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Abstract
A nonequilibrium model for the dynamic simulation of distillation columns has been devised. The model solves analytically the MERSHQ equations, neglecting the high flux correction factor matrices. Both assumptions of well-mixed flow and also plug flow have been considered. The well-mixed flow case has been compared with a model presented by Sivertsson (1998) in which the MaxwellStefan equations are solved numerically across the film. Dynamic simulations of the analytical model have been realised and they results have been compared with an equilibrium model. No considerable differences have been observed when the deviations from the steady state were small. A nonequilibrium averaged plug flow model based on an integrated steady state differential balance is also presented. This plug flow model may lead to more reasonable estimates of industrial column separation efficiencies. However, some difficulties have been observed in order to implement this model in gPROMS. Alternative solutions for this implementation, including a distributed unsteady state model, are proposed, but they have not been implemented.
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Acknowledgements First of all, the author wishes to acknowledge his supervisors J.D. Perkins and R.W.H. Sargent for their helpful suggestions and for having been so patient with my particular way of work. I am also indebted to my sponsors FUNDACIÓN REPSOL and DOW CHEMICAL IBERICA. Without their financial aid I could not have accomplished this thesis. On the other hand, I have also gained many debts of gratitude to my colleagues in college. Our never-ending discussions were very useful. Besides, I also want to thank my friends in London for their continuous support and the ones I still must have in Spain (despite I have not answered their e-mails for ages). Finally, I have to express my gratitude to my parents for their continuous encouragement and, special thanks to my fiancée, Sandra, for her understanding and infinite patience in the worst moments.
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Table of Contents
Table of Contents Abstract..............................................................................................................................i Acknowledgements ...........................................................................................................ii Table of Contents.............................................................................................................iii List of Figures..................................................................................................................iv List of Tables....................................................................................................................vi Nomenclature..................................................................................................................vii Chapter 1 ..........................................................................................................................1 Chapter 2 ..........................................................................................................................8 Chapter 3 ........................................................................................................................18 Chapter 4 ........................................................................................................................39 Chapter 5 ........................................................................................................................55 Chapter 6 ........................................................................................................................74 References .......................................................................................................................76 Appendix A......................................................................................................................79
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List of Figures
List of Figures Figure 3.1. Schematic diagram of the nonequilibrium tray model.........................................19 Figure 5.1. Liquid phase composition profiles in a 33-trays depropanizer column using a noneq. well-mixed model (ethane-1, propane-2, butane-3 and pentane-4). ................56 Figure 5.2. Temperature profiles in a 33-trays depropanizer column using a noneq. wellmixed model. ............................................................................................................56 Figure 5.3. Liquid and vapour molar flow profiles in a 33-trays depropanizer column using a noneq. well-mixed model. .........................................................................................57 Figure 5.4. Interface molar flow profiles in a 33-trays depropanizer column using a noneq. well-mixed model (ethane-1, propane-2, butane-3 and pentane-4).............................58 Figure 5.5. Interface heat flow profile in a 33-trays depropanizer column using a noneq. wellmixed model. ............................................................................................................58 Figure 5.6. Murphree efficiencies back-calculated for a 33-trays depropanizer column using a noneq. well-mixed model (ethane-1, propane-2, butane-3 and pentane-4). ................59 Figure 5.7. Average mass transfer coefficients for the vapour phase calculated for a 33-trays depropanizer column using a noneq. well-mixed model and two different thermodynamic packages. .................................................................................................................61 Figure 5.8. Heat transfer coefficients for the vapour phase calculated for a 33-trays depropanizer column using a noneq. well-mixed model and two different thermodynamic packages. .................................................................................................................61
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List of Figures
Figure 5.9. Component 1 (ethane) Murphree efficiencies back-calculated for a 33-trays depropanizer column using a noneq. well-mixed model and two different thermodynamic packages. .................................................................................................................62 Figure 5.10. Component 1 (ethane) Murphree efficiencies back-calculated for a 33-trays depropanizer column using an analytical solution and Sivertsson’s numerical approximation...........................................................................................................63 Figure 5.11. Comparison of the liquid phase composition profiles in a 33-trays depropanizer column using a noneq. well-mixed model and an equilibrium one (ethane-1, propane-2, butane-3 and pentane-4)...........................................................................................64 Figure 5.12. Comparison of the temperature profiles in a 33-trays depropanizer column using a noneq. well-mixed model and an equilibrium one (ethane-1, propane-2, butane-3 and pentane-4)................................................................................................................65 Figure 5.13. Dynamic results. 1% increase en the feed flow. Butane composition in the reboiler of a 33-trays depropanizer column. Comparison between noneq. and eq. models......................................................................................................................66 Figure 5.14. Dynamic results. 1% increase en the feed flow. Propane composition in the reflux drum of a 33-trays depropanizer column. Comparison between noneq. and eq. models......................................................................................................................66 Figure 5.15. Dynamic results. 1% increase en the feed flow. Distillate molar flow rate in a 33trays depropanizer colum. Comparison between noneq. and eq. models.....................67 Figure 5.16. Liquid phase compositions in a 23-trays depropanizer column using an averaged plug flow model (ethane-1, propane-2, butane-3 and pentane-4). ..............................69 Figure 5.17. Murphree efficiencies back-calculated for a 23-trays depropanizer column using an averaged plug flow model (ethane-1, propane-2, butane-3 and pentane-4). ...........70
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List of Tables
List of Tables Table 3.1. Nonequilibrium well-mixed model summary.......................................................31 Table 4.1. Number of variables in the implemented models. ...............................................40 Table 4.2. 33-trays depropanizer column feed conditions...................................................42 Table 4.3. 33-trays depropanizer column layout design. .....................................................43 Table 4.4. 23-trays depropanizer column layout design. .....................................................44 Table 4.5. Surrounding equipment layout design.................................................................44 Table 4.6. Physical Property models..................................................................................45
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Nomenclature
Nomenclature a
Interface area [m2].
Areboiler
Reboiler sectional area [m2].
Arefluxdrum
Reflux drum sectional area [m2].
Atray
Active bubbling area on tray [m2].
Aholes
Holes area on tray [m2].
B
Matrix function of the inverted product between binary mass transfer coefficients and the interface area [(m3/s)-1].
CP
Molar heat capacity [kJ/(mol·K)].
CV , tray
Orifice coefficient in dry pressure drop calculations [-].
CV
Friction factor for the head loss between the top of the column and the condenser [(mol/s)·(kg/(m·s2))1/2].
di
Driving force for mass diffusion for component i [-].
Di, j
Maxwell-Stefan binary diffusion coefficients between components i and j [m2/s].
Doi , j
Infinite dilution Maxwell-Stefan binary diffusion coefficients between components i and j [m2/s].
EMV
Murphree efficiency [-].
F
Liquid molar flow rate of a feed stream [mol/s].
FS
Superficial factor [(m/s)·(kg/m3)1/2].
g
Gravity constant [9.81 m/s2].
h
Height of clear liquid on a tray [m]. - vii -
Nomenclature
hf
Height of liquid on the froth of a tray [m].
hw
Height of the weir on a tray [m].
H ~ H
Molar enthalpy [kJ/mol].
ha
Product between the low flux heat transfer coefficient and the ni terface area
Partial molar enthalpy [kJ/mol]. [(kJ/(s·m2·K))·m2].
I
Unit tensor [-].
kai, j
Product between binary mass transfer coefficients and the interface area [(m/s)·m2].
L
Liquid molar flow rate across the tray [mol/s].
LFPW
Liquid flow of average liquid-flow-path-width [(m3/s)/m].
lw
Length of the weir in a tray [m].
l
Length of the film layer [m].
Le
Lewis adimensional number [-].
Mi
Molar holdup of component i [mol].
MW
Molecular weight [g/mol].
Ni
Molar flux of component i [mol/(m2·s)].
NTU
Number of transfer units [-].
nc
Number of components in a mixture [-].
np
Number of trays in a distillation column [-].
Q&
Heat supply [kJ/s].
R
Universal gas constant in Eq. 2.12 [8.314 J/(mol·K)].
T
Temperature [K].
tL
Liquid residence contact time on tray [s].
U
Internal energy holdup [kJ].
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Nomenclature
v
Molar volume of the mixture [m3/mol].
V
Vapour molar flow rate across the tray [mol/s].
Vtray
Volume available between trays [m3].
W&
Shaft work [kJ/s].
xi
Molar fraction of component i [-].
yi
Molar fraction of component i in a vapour phase [-].
zi
Molar fraction of component i in a feed stream [-].
z
Coordinate direction or position [m].
ZC
Liquid holdup on the tray [m3/m2].
ZL
Liquid flow path length on the tray [m].
Greek letters
α
Dry tray pressure drop friction factor coefficient [-].
β
Aeration factor [-].
γ
Coefficient of fugacity in Eq. 2.13 [-].
Γ
Thermodynamic factors matrix [-].
λ
Thermal conductivity [kJ/(s·m·K)].
µ
Viscosity [kg/(m·s]; Chemical potential in Eq. 2.12.
Ξ
Correction factors matrix for high mass transfer fluxes [-].
ΞH
Correction factor for high heat transfer fluxes [-].
ρ
Mass density [kg/(m3].
σ
Surface tension [kg/s2].
ΣV
Molecular diffusion volume [-].
φi
Molar flow of component i across the interface [mol/s].
φH
Heat flow across the interface [kJ/s]. - ix -
Nomenclature
Φi
Fugacity coefficients for component i [-].
Φ
Mass transfer rate factor matrix [-].
ΦH
Heat transfer rate factor [-].
Subscripts avg
Averaged value.
i
Component i property or parameter.
i , j, k
Component indices.
p
Stage indices.
D
Reflux drum.
DIST
Distillate stream.
R
Reboiler.
REF
Reflux stream.
tot
Overall phase property.
0
Quantity evaluated at position z = 0 .
l
Quantity evaluated at position z = l .
Superscripts F
Referring to a feed variable.
H
Referring to a heat related variable or parameter.
I
Referring to the interface.
L
Referring to liquid phase.
nbp
Referring to a physical property evaluated at the normal boiling point.
V
Referring to vapour phase.
T
Referring to total mixture.
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Nomenclature
Matrix Operations and Notation φ
Column matrix.
φ
Square matrix.
φ
−1
Inverse of a square matrix.
φ
Average value.
φ′
Adimensionalised variable.
φ0
Variable evaluated in position z = 0 .
φl
Variable evaluated in position z = l .
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Introduction
Chapter 1
Chapter 1 1 Introduction Distillation is considered to be the most important unit operation in the chemical process industry. It is also one of the highest energy consumers, reason why it is particularly convenient to be able to predict satisfactorily its performance. During the last 50 years, the use of computers and their continuously increasing capabilities has led to the parallel development of rigorous calculation procedures. These methods involve the definitions of the widely known MESH (Material balances, Equilibrium equations, Summation constraints and Heat or energy balance) equations. A whole chapter concerning the mathematical methods available to solve these equations is included in Kister (1992). Traditionally, all the rigorous methods were developed with the main objective of finding a steady-state solution that could be used in the design of distillation columns. In addition, several assumptions were made in order to reduce the complexity of the model. Considering that both phases achieved chemical equilibrium in each of the trays and quantifying the deviation from ideality with a parameter (stage efficiency) is, probably, the main assumption used. Several different definitions of the tray efficiency are available and there is no agreement on which definition is best (Krishnamurty and Taylor, 1985a). On the other hand, its value may also be characterised using different methods, either an empirical correlation or using a mechanistic model (King, 1980; Kister, 1992). Notwithstanding this, it is still common practice to assign constant values to the stage efficiencies throughout the column, instead of including them in the model equations. In order to come up with the inherent deficiencies of the equilibrium model and the use of tray efficiencies, a new approach was introduced by Krishnamurty and Taylor (1985abc). These -1-
Introduction
Chapter 1
authors claim that their model might be particularly useful to predict composition profiles in mixtures in which different species may exhibit different facilities for mass transfer. It is known as the nonequilibrium model, due to the fact that the mass transfer between the liquid and the vapour phase are modelled explicitly. The set of equations generated is known as the MERSHQ (Material balances, Energy balances, Rate equations, Summation constraints, Hydraulic equations and eQuilibrium equations). Apart from the rate equations, which are introduced to model the mass transfer, hydraulic equations need to be included, as they are necessary to compute the mass transfer coefficients. The computation of stage efficiencies can be avoided and, thus, the generality of the method is increased. In contrast, the number of equations needed to solve for each tray is augmented considerably. Dynamic simulation of distillation columns is also an important issue. There has been a continuous development of dynamic models, with particular emphasis on process operability (start-ups and changes in the throughput) and process control (Gani et al., 1986). Nevertheless, the basis of all dynamic simulations of distillation columns has traditionally been the equilibrium model and the MESH equations.
1.1 Literature review Distillation columns have been a matter of continuous research, due to its major relevance in the chemical processing industries. Different books about distillation equipment, its design and operation are available in the literature (King, 1980; Henley and Seader, 1981; Kister, 1989 and 1992). A review of the state of the art of sieve trays design was presented by Zuiderwerg (1982) and Barnicki and Davis (1989) have also published on this subject. Concerning the development of nonequilibrium stage models for multicomponent mixtures, they were first introduced in 1985 by Krishnamurty and Taylor (1985a). They proposed a new stage model in which material and energy balances for each phase could be solved simultaneously with heat and mass transfer models at the interface for multicomponent systems. They suggested that the liquid and vapour compositions used in the computation of the transfer coefficients should depend on the flow model applied. They recommended a wellmixed flow model for the liquid (using the leaving liquid composition) and a plug-flow model -2-
Introduction
Chapter 1
for the vapour (approximated by an averaged composition between the entering and the leaving vapour streams). The Newton method or one of its relatives is recommended to solve the model equations. In the second and third part of their paper (1985bc), they present the mass transfer rate equations, based on an exact solution of the Maxwell-Stefan equations, developed by Krishna and Standart (1976). These equations are presented in matrix form, where the mass transfer coefficients are expressed as a matrix, corrected by another matrix of correction factors. They propose the AIChE correlations of the number of transfer units (NTU) to compute the mass transfer coefficients. Powers et al. (1988) presented the fifth part of the series of papers started in 1985. They stated several simplifications in the nonequilibrium model, applicable when the mass transfer rate is low enough (the matrix of correction factors reduces to the identity matrix), and studied different initialisation procedures for these models. They compared their original initialisation method (linear interpolation for the temperature profile, constant molar overflow to compute the flows and one iteration of the Wank and Henke method for equilibrium staged models for the composition profiles) with several other methods. They concluded that the use of continuation methods (Newton homotopy and also a nonequilibrium continuation method) was particularly useful to initialise even extremely difficult cases. Seader (1989) states that this new rate-based approach could be the beginning of a new era in tray-type separator design and simulation. He also concludes recommending this new approach particularly for nonideal multicomponent mixtures, for nearly ideal multicomponent systems when good predictions of product distributions are desired for components other than the key components, and even for binary mixtures which might exhibit tray efficiencies that vary over the height of the column. In 1994, Taylor et al. presented their second generation nonequilibrium model. It is apparently the same model already published in 1985, with the novel addition of entrainment and weeping variables, as well as pressure drop equations. Due to the fact that the mass transfer coefficients are a function of the column type and tray internals layout, the authors
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Introduction
Chapter 1
have included an internal design mode that calculates the mechanical layout of the trays in every iteration, once the type of internals has been selected. All the theoretical background needed to deal with the multicomponent matrix systems generated in the development of nonequilibrium trays is included in the book written by Taylor and Krishna (1993). In 1992, Kooijman and Taylor released ChemSep, (Taylor et al., 1994; Kooijman and Taylor, 1998) a steady-state simulator to solve multicomponent separation process calculations. This software is supposed to be able to solve equilibrium models as well as nonequilibrium ones. The design mode described by Taylor et al. (1994) is also implemented in this software. Its user’s manual includes a complete review of the different methods available to compute the physical properties, equilibrium constants and transport properties. Kooijman and Taylor (1995a) reached a further step in their development of their nonequilibrium model. They implemented several flow models to the nonequilibrium equations, including plug flow models for the liquid and the vapour and also a dispersion model for the liquid phase. The compared the different models results, and concluded that the plug-plug flow (liquid and vapour, respectively) model and the plug-dispersion flow model were the combinations that lead to a better estimation of the separation efficiencies. The mixed-mixed and the plug-mixed flow models performances under-predicted them. In addition, they compared several different methods to compute the mass transfer coefficients, concluding that the method of Chan and Fair (Taylor and Krishna, 1993) was the best method available, although they pointed out some of its limitations, some of which could prevent the model from converging a solution. Apart from the work published by Taylor and co-workers, very few publications are available in this subject. One of the few exceptions is the work of Biardi and Grottoli (1989). They propose an alternative formulation for the nonequilibrium model (that they call real tray model), where the interfacial mass transfer flows are also computed but the definition of efficiency is maintained in their model equations. Concerning the dynamic simulation of distillation columns, Gani et al. (1986) published a generalised model, in which plate hydraulics are included, but the vapour holdup is neglected -4-
Introduction
Chapter 1
and stage efficiencies are used. The entrainment and weeping are properly modelled, so that the model may be used for the dynamic simulation of start-up procedures. In 1987, Choe and Luyben studied the importance of the vapour holdup and the pressure modelling in rigorous dynamic models. With respect to nonequilibrium models, a lack of publications in this area has been observed. Kooijman and Taylor (1995b) developed a nonequilibrium model for the dynamic simulation of separation processes. They compared a nonequilibrium model with two holdups (vapour and liquid) with an equilibrium one, also with two holdups. In addition, they also considered a nonequilibrium model with three holdups (liquid in the froth and downcomer lumped together, vapour in the froth and vapour above the froth). On the other hand, only the well-mixed flow model for the liquid and the vapour were considered in their simulations.
1.2 Previous work This work is a continuation of the thesis submitted by Sivertsson in 1998 for his M.Sc. degree. Sivertsson developed a nonequilibrium dynamic model for the simulation of tray distillation columns. He modelled the liquid holdup on the froth, and in one holdup he lumped together the vapour on and above the froth. The liquid holdup in the downcomer was neglected. Other relevant assumptions used in his model agreed with those presented in the original work of Krishnamurty and Taylor (1985abc). An exception is made in the treatment of the flow model for the liquid and vapour phases, where Sivertsson assumes that they are both perfectly mixed. In addition, the liquid phase is assumed to behave ideally, avoiding thus to include in the model the thermodynamic correction factors. With respect to the interface mass transfer, he proposed to solve the Maxwell-Stefan equations with a numerical approximation of the differential equations, instead of using the analytical solution that may be obtained for the case of a film model in steady-state conditions (proposed by Krishnamurty and Taylor). It should be noted that in the model proposed by Sivertsson several mistakes were found. The most relevant was in the calculation of the liquid mass transfer coefficient (when
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Introduction
Chapter 1
computing the liquid residence time), which in some cases could result in the prediction of values several orders of magnitude higher (4 times larger in the extractive column and 20 times larger in the depropanizer). Besides, this error led to the computation of higher mass transfer coefficients for the liquid than for the vapour. This misleading result, brought him to the erroneous conclusion that the relation between liquid and vapour mass transfer coefficients could also be applied to the heat transfer coefficients (for which the resistance to heat transfer is higher in vapour than in liquids, opposite to mass transfer behaviour). As a result, Sivertsson computed higher mass and heat transfer coefficients for the liquid phase that reduced the resistance in the films and, therefore, increased the efficiency of the separation. Sivertsson implemented his dynamic model in gPROMS (Process Systems Enterprise Ltd.), and prepared three case studies (binary distillation, extractive distillation and a depropanizer column). Then, he compared its dynamic behaviour with a nonequilibrium model with averaged mass transfer coefficients and with an equilibrium model (with two holdups, as in the nonequilibrium model). Sivertsson compared the initial response of both models and no relevant differences were observed. In contrast, he compared both models using modal analysis (state space formulation) and observed a considerable difference between the largest time constants in the equilibrium and nonequilibrium models Moreover, Sivertsson also failed to devise an efficient procedure to initialise his model, which implied that for every new case that might be tested, a long and tedious series of steps needed to be done to initialise the problem.
1.3 Present work summary In this thesis, the work accomplished by Sivertsson has been revised and improved. The fact that some of his results were contradictory suggested that the models could not have been satisfactorily implemented. His model development has been modified in several aspects, especially in how the rate equations are handled and solved. Instead of solving numerically the Maxwell-Stefan
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Introduction
Chapter 1
equations in the film models, their analytical solution (Taylor and Krishna, 1993) has been successfully implemented. On the other hand, the hypothesis of well-mixed liquid and vapour phases will be revised, and the possibility of including plug flow assumptions to model the liquid, the vapour or both phases will be considered. An averaged plug flow model has been implemented for both phases, while the viability of a distributed plug flow model has been considered. Concerning the implementation on gPROMS, it has been conveniently modified in order to minimise the number of variables defined (a 40% reduction has been attained). Therefore, the number of equations solved during the simulation has been reduced in the same proportion. Special attention has been dedicated to the initialisation procedure, which has been considerably improved. Now it is possible to initialise the model using linear composition profiles for the temperature, pressure and compositions, in a similar way to the traditional equilibrium model initialisation procedures. The main difficulty observed when initialising the models was found because of limitations of the physical property package that gPROMS uses.
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Conceptual Framework
Chapter 2
Chapter 2 2 Conceptual Framework In this chapter, an introduction to the fundamental concepts in which nonequilibrium theory is based is presented. A detailed development of the equations presented and the theory behind them may be found in Taylor and Krishna’s book (1993). The fundamentals of mass transfer will be reviewed first, followed by the development of the heat transfer equations
2.1 Mass Transfer Multicomponent mass transfer theory is based on the Maxwell-Stefan equations for multicomponent diffusion. These equations are derived from a sum of forces vs. rate of change of momentum balance, using molecular collision theory to predict the change of momentum. The generalised expression of the Maxwell-Stefan is nc
di = v ⋅ ∑ j =1
(
xi ⋅ N j − x j ⋅ N i ) Di, j
(2.1)
which relates the component driving force, d i , with the component compositions, xi , and fluxes, N i . The parameter v is the total molar volume of the mixture and Di, j is a proportionality factor known as Maxwell-Stefan binary diffusion coefficient. This equation is valid for all the components (nc) in the mixture, although only (nc-1) of them are linearly independent, due to the fact that the driving forces are directly related to the composition gradients (and the summation of the gradients of composition must add to zero). Moreover, this limitation also implies that for a couple of components in a mixture, the diffusion coefficients have to be the same, independently of their order. Thus,
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Conceptual Framework
Chapter 2
Di, j = D j,i
(2.2)
In multicomponent distillation, it is assumed that all the resistance to mass transfer is concentrated in a thin film, adjacent to the interface. The degree of mixing in both liquid and vapour phases is very high, removing all composition gradients in the bulk of the phases. At steady state, the component fluxes are constant, and the Maxwell-Stefan equations may be solved analytically in the boundary layer, provided that appropriate boundary conditions are given:
z = 0 x i = xi 0
(2.3)
z = l xi = xil On the other hand, the molar flux of component i may be substituted by the molar component flow across the interface, φ i , divided by the interfacial area, a , leading to the following expression: nc
di = v ⋅ ∑ j =1
(
xi ⋅ φ j − x j ⋅ φi ) Di, j ⋅ a
(2.4)
In the next two sections, a solution to these equations will be given for ideal gas mixtures and for nonideal fluids.
2.1.1 Ideal Gas Mixtures In a nc-component mixture, an exact analytical solution for a film model may be found if the diffusion coefficients are independent of the composition. The driving force in this case may be considered equal to the composition gradient, leading to the following expression: nc ( y ⋅ φ − y j ⋅ φi ) dyi = vV ⋅ ∑ i jV dz Di, j ⋅ a j =1
with the following boundary conditions:
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(2.5)
Conceptual Framework
Chapter 2
z = 0 yi = yi 0
(2.6)
z = l y i = y il For the sake of definiteness, the molar flux is considered to be positive when flowing from the liquid to the vapour. Therefore, the position z = 0 will correspond to the interface ( yiI ), and
z = l will correspond to the bulk vapour ( yVi ). An exact solution for every yi may be found, and its derivative may be evaluated in either of the two boundaries (0 or l ), allowing to compute the component flow through the film. It is recommended to evaluate this derivative in the bulk vapour, leading to the following expression for the flow of component i: 1 φ = V v
( )
⋅ BV l
−1
⋅ Ξl
V
⋅( y
0
−y
)+ y ⋅ ∑[ φ ] nc
l
l
j
(2.7)
j =1
V
where B l is a matrix function of the inverted product between binary mass transfer coefficients and the interface area, while Ξ l is a matrix of high flux correction factors. Once V
again, it should be noted that this expression is only valid for the first (nc-1) components, the flow of the last component may be calculated imposing the energy balance condition. V
B l may be calculated with the following expression: nc y Vi y Vk + ∑ kaV V k =1 kai ,k i , nc k ≠i BiV, j = V 1 1 − yi ⋅ kaV − kaV i ,nc i, j
, i = j , i ≠ j
(2.8)
where kaVi, j is the product of the mass transfer coefficient for the binary couple of components i and j and the total interfacial area. The reason why these two parameters are lumped together is because the available correlations in the literature usually allow computing this product, rather than each of them separately. The mass transfer coefficient is defined as the ratio between the binary Maxwell-Stefan diffusivity and the film thickness, leading to the following expression:
- 10 -
Conceptual Framework
Chapter 2
V
ka
V i, j
Di, j = ⋅a l
(2.9)
On the other hand, the high flux correction factor may be calculated from:
(
)[
(
) ]
Ξ l = Φ l ⋅ exp Φ l ⋅ exp Φ l − I V
V
V
V
−1
(2.10)
where Φ l is the mass transfer rate factor matrix: V
Φ Vi , j
= vV
nc φi φk v ⋅ V + ∑ V , i = kai, nc kk =≠1i kai, k 1 1 ⋅ − φ i ⋅ V − V , i ≠ kai, j kai,nc V
j j
(2.11)
2.1.2 Nonideal Fluid Mixtures In the case of a multicomponent film model for mass transfer in nonideal fluid mixtures, the procedure used to find a solution is very similar. In this case, the driving force is no longer proportional to the composition derivative, being now proportional to the chemical potential gradients. Notwithstanding this, these gradients may be related directly to the composition gradients if a thermodynamic factors matrix is defined: x dµ di = i ⋅ i R ⋅ T dz
nc−1
T ,P
= ∑ Γi , j ⋅ j =1
dx j dz
(2.12)
where the thermodynamic factors are defined as: Γi, j = δ i, j + xi ⋅
∂ ln γ i ∂xi
(2.13) T ,P ,Σ
Taylor and Krishna (1993) presented a complete review of how to compute this thermodynamic factors matrix using several different activity models. The resulting expression of the Maxwell-Stefan equations is as follows:
- 11 -
Conceptual Framework
Chapter 2
nc −1
∑Γ j =1
i,j
⋅
nc dx j ( x ⋅φ − x ⋅φ ) = v L ⋅ ∑ i jL j i dz Di , j ⋅ a j =1
(2.14)
This system of equations may also be solved analytically, but the solution obtained is not so straightforward to use. However, an approximate solution may be obtained if the coefficients Γi , j and Di, j are considered constant along the diffusion path. With this assumption, Eq. L
(2.14) represents a linear matrix differential equation, which can be solved with a solution exactly analogous to the one obtained for the ideal gas case. The boundary conditions are the same that apply for the ideal gases case:
z = 0 x i = xi 0
(2.15)
z = l xi = xil Considering that the flux has been defined positive from the liquid to the vapour, the position
z = 0 will correspond to the bulk liquid ( xiL ), and z = l will correspond to the interface ( x iI ). Again, the composition profile may be found and the derivative may be evaluated in either of the two boundaries ( 0 or l ). Nevertheless, it is still recommended to compute it at
z = 0 , which corresponds to the bulk liquid. The expression obtained for the flow of component i is:
( )
1 L φ = L ⋅ B0 v
−1
(
)
NC
[ ]
⋅ Γ0 ⋅ Ξ 0 ⋅ x 0 − x l + x 0 ⋅ ∑ φ j L
j =1
(2.16)
where nc xiL x kL + , i = j ∑ kaL L k =1 kai ,k i , nc k ≠i L Bi , j = L 1 1 − xi ⋅ ka L − ka L , i ≠ j i , nc i,j
(2.17)
kaiL, j is again defined as the product of the mass transfer coefficient for the binary couple of components i and j and the total interfacial area:
- 12 -
Conceptual Framework
Chapter 2
L
ka
L i, j
Di, j = ⋅a l
(2.18)
On the other hand, the high flux correction factor is calculated with a different expression, due to the fact that now it is evaluated at the other boundary. Besides, the matrix of thermodynamic correction factors appears also in this term:
[ ((Γ)
Ξ 0L = ( Γ0 ) ⋅ Φ L0 ⋅ exp −1
−1
0
) ]
⋅ Φ L0 − I
−1
(2.19)
where Φ l is the mass transfer rate factor matrix for the liquid phase: L
Φ i, j L
nc φk v L ⋅ φi + ∑ ka L , i = L ka i, nc k =1 i, k k≠i = v L ⋅ − φ i ⋅ 1 − 1 , i ≠ ka L ka L i , nc i,j
j j
(2.20)
2.1.3 Mass transfer coefficients In order to be able to compute the component flows across the interface, an additional piece of information is needed: the mass transfer coefficients. They may generally be calculated from an empirical model, the most common of which is the AIChE model. In this transport model, the product of the mass transfer coefficients and the interfacial area are evaluated together, as a function of the number of transfer units ( NTU ), with the expressions:
kaVi, j = NTUVi, j ⋅ ( V ⋅ vV )
(2.21)
kaiL, j = NTUiL, j ⋅ ( L ⋅ v L )
(2.22)
where V and L are the vapour and liquid molar flow rates in the tray. The number of transfer units may be calculated from the expressions (King, 1980):*
*
It should be pointed out that Krishnamurty and Taylor (1985abc) failed to translate correctly these
equations to the SI system of units in their original paper about nonequilibrium models. Correct - 13 -
Conceptual Framework
Chapter 2
NTUiV, j =
( 0.776 + 4.567 ⋅ h
w
− 0.2377 ⋅ FS + 104.8 ⋅ LFPW ) µV ρ V ⋅ DVi, j
NTUiL, j = 20314.732 ⋅ DiL, j ⋅ ( 0.213 ⋅ FS + 0.15)⋅ t L
(2.23)
(2.24)
where µ V and ρ V are the vapour viscosity and density, hw is the height of the weir, FS is the superficial factor, LFPW is the volumetric flow divided by the length of the weir ( l w ) and t L is the liquid phase residence time on the tray. The last three parameters also need to be computed, using the following expressions: Vavg ⋅ v V FS = A plate LFPW =
tL =
⋅ ρV
Lavg ⋅ v L lw
ZC ⋅ ZL L fpw
(2.25)
(2.26)
(2.27)
where Vavg and Lavg are the averaged vapour and liquid molar flow rates across the tray. The parameter Z L corresponds to the liquid flow path length and Z C is defined as the holdup on the tray, which can be computed using the following expression: ZC = 0.0254 ⋅ ( 1.65 + 7.4803 ⋅ hw + 96.6236 ⋅ LFPW − 0.5328 ⋅ FS )
(2.28)
2.1.3.1 Maxwell-Stefan binary diffusion coefficients Before proceeding any further, some attention should be paid to the way the diffusivity coefficients are going to be computed. For the case of ideal gas mixtures, it is recommended here to use the empirical correlation proposed by Fuller et al. (Reid et al., 1987):
expressions for the NTU and the parameters required may be found in King (1980) in U.S. units and in Taylor and Krishna (1993) in S.I. units. - 14 -
Conceptual Framework
Chapter 2
DVi, j = 1.0112 ⋅10 −2 ⋅
T
1. 75
P
⋅
(
1 1 + M W ,i M W , j 3
)
Σ V , j + 3 Σ V ,i
2
(2.29)
where T and P are the temperature (K) and pressure (Pa) of the system; M W ,i corresponds to the molecular weight of component i (g/mol), and Σ V ,i is defined as the molecular diffusion volume for component i, computed as a summation of atomic contributions. The authors of this correlation reported a 4% average absolute error. Besides, this correlation is recommended for systems at low pressure, and its value should be corrected for systems working at high pressure. The estimation of binary liquid diffusion coefficients is generally accomplished by calculating the value of the binary coefficients at infinite dilution, Dio, Lj , and correcting them for the effect of concentration. Several empirical correlations are also available in the literature. A method presented by Tyn and Calus (Reid et al., 1987) is one of the best known: D = 8.93 ⋅10 oL i, j
−16
(v ) ⋅ (v ) nbp j
nbp i
0 .267
0 .15
T ⋅ 0 .433 µj
σ nbp j ⋅ nbp σ i
(2.30)
where µ j is the viscosity of component j at the system temperature and pressure, while σ i and σ j are the surface tension of components i and j, respectively, both evaluated at their normal boiling point. The molar volumes should also be evaluated at their normal boiling points. Taylor and Krishna (1993) propose the following expression to compute the Maxwell-Stefan binary diffusion coefficients for multicomponent liquid mixtures as a function of the infinite dilution diffusivities: 1+ x j − xi o L 2 i, j
D =( D L i, j
)
- 15 -
⋅( D
1+ xi − x j o L 2 j ,i
)
(2.31)
Conceptual Framework
Chapter 2
2.2 Heat Transfer Starting from a differential energy balance, and assuming that the flux of energy across the interface must be continuous, and using again the film model analysis, with the following boundary conditions:
z = 0 T = T0
(2.32)
z = l T = Tl a solution for the heat flow across this film, φ H , may be found:
[
nc ~ φ H = ha ⋅ Ξ H ⋅ ( T0 − Tl ) + ∑ φ j ⋅ H j j =1
]
(2.33)
where ha is the product of the low flux heat transfer coefficient and the interfacial area, which ~ will again be computed together; H j corresponds to the partial molar enthalpy of component j; Ξ H is now the high flux correction factor, which may be calculated from the expression: ΞH =
ΦH exp ( Φ H ) − 1
(2.34)
where Φ H is the heat transfer rate factor, defined as:
∑( φ ⋅C ) nc
ΦH =
i
i =1
P ,i
ha
(2.35)
where CP, i is the molar heat capacity of species i.
2.2.1 Heat transfer coefficients In order to compute the product of the low flux heat transfer coefficient and the interfacial area, an analogy may be used between mass and heat transfer. One of the best known is the Chilton-Colburn analogy:
- 16 -
Conceptual Framework
Chapter 2
ha =
kaavg ⋅ CP ⋅ ( Le) v
2 3
(2.36)
Le is the adimensional Lewis number, which is defined as:
λ ⋅v Davg ⋅ CP
Le =
(2.37)
where λ is the heat conductivity. The Chilton-Colburn analogy may be used in the calculation of the vapour phase heat transfer coefficient. In the case of the liquid phase, a penetration model is recommended to compute this coefficient (Kooijman and Taylor, 1998): ha =
kaavg v
⋅ CP ⋅ Le
(2.38)
Again, it should be reminded that the ratio between the liquid and the vapour phase heat transfer coefficients has no relation at all with the relation between the mass transfer coefficients, and no analogy may be applied.
- 17 -
Model Development
Chapter 3
Chapter 3 3 Model Development In this chapter, a dynamic model for a complete distillation column is presented. A model for a nonequilibrium tray will be developed. The reboiler and the condenser and reflux drums models will also be presented.
3.1 Nonequilibrium tray model The model presented in this chapter is a modification of the model developed by Sivertsson (1998). His model was based in a previous dynamic one developed originally by R.W.H. Sargent. The key assumptions considered by Sivertsson have also been maintained: §
Only two holdups have been modelled (liquid in the froth and vapour on the froth and above lumped together). The liquid holdup in the downcomer has not been modelled, assuming that the transfer of liquid between the trays is instantaneous.
§
The two phases in the tray are in mechanical equilibrium (i.e. the pressure is constant).
§
The resistance to mass and energy transfer between the phases may be reduced to two film layers, located on each side of the interface. Thermodynamic equilibrium is assumed only at the interface.
§
Weeping and entrainment have not been included in the model equations.
§
The feed stream is introduced in a tray as a liquid (this assumption could be removed easily modifying the material balance equations). Furthermore, no product draw-offs have been included.
- 18 -
Model Development
Chapter 3
It should be noted that at this stage no assumptions have been made concerning the type of flow expected to find in the column trays. This issue will be revised thoroughly in the tray model development. A feed stream has been included for the liquid phase only, and heat supplies are possible in both phases. A schematic diagram of the nonequilibrium model presented is given in the following figure:
Lp-1 Xi,p-1 T Lp-1 Pp-1 HLp-1
Vp Yi,p T Vp Pp H Vp
VAPOUR Mass Flow
Feed Stream Fp Z i,p T Fp PFp HFp
Heat Flow Xi,p YIi,p QL
QV
XIi,p Yi,p
LIQUID Lp Xi,p
Vp+1 Yi,p+1 T Vp+1 Pp+1 HVp+1
T Lp Pp H Lp
Figure 3.1. Schematic diagram of the nonequilibrium tray model
- 19 -
Model Development
Chapter 3
3.1.1 Tray model equations The equations presented in this section will be valid for every tray in the column. For the sake of definiteness, the top tray will be numbered as 1, and the bottom tray will be numbered np. The forthcoming balances will be generalised for a tray p.
3.1.1.1 Material and energy balances First of all, the component material balances should be defined, one for each phase. However, in order to prevent the system of equations to be high-indexed, it is convenient to substitute the component balances for one of the phases by the total material balances. Therefore, the interface flows, φ i , only appear in one differential equation each. The convention adopted for the interface molar flow is that it is positive when flowing from the liquid to the vapour phase. A complete definition of all the symbols used in the following equations may be found in the nomenclature. dM iL, p dt
= −φi , p + L p−1 ⋅ xi, p−1 − L p ⋅ xi, p + Fp ⋅ zi, p
∀ i = 1,..., nc
(3.1)
T
dM i , p = Lp−1 ⋅ x i, p−1 − L p ⋅ xi , p + Vp+1 ⋅ y i, p +1 −V p ⋅ y i, p + Fp ⋅ zi , p dt
∀i = 1,..., nc (3.2)
where M iL, p and M iT, p are respectively the liquid and total molar holdups of component i in tray p. The component liquid and vapour compositions correspond to the leaving streams compositions. A summation constraint must be added for both of them: nc
∑x
i, p
=1
(3.3)
=1
(3.4)
i =1 nc
∑y
i ,p
i =1
As for the mass balances, only one of the phases heat balance should be used, and the total energy balance should substitute the other one, in order to prevent the interface heat flow φ H to appear in two differential equations. - 20 -
Model Development
Chapter 3
dU Lp dt dU Tp dt
= −φ pH + Lp −1 ⋅ H Lp−1 − Lp ⋅ H Lp + Fp ⋅ H pF + Q& pL
(3.5)
(
= Lp−1 ⋅ H Lp−1 − L p ⋅ H pL + V p+1 ⋅ H Vp+1 − V p ⋅ H Vp + Fp ⋅ H pF + Q& pL + Q& Vp
)
(3.6)
where U Lp and U Tp are respectively the liquid and total internal energy holdups in tray p; Q& is defined as the heat supplied externally to that tray (defined positive when heat is introduced in the tray). It should be pointed out that the compositions used to evaluate the enthalpies in Eqs. (3.5-6) should be the leaving compositions. On the other hand, the component molar holdups must be related to the total molar holdup for L V each phase, M tot , p and M tot , p , via the component molar fractions, which now should be the
average molar fractions on the tray: L M iL, p = xi, p ⋅ M tot ,p
∀i = 1,..., nc
(3.7)
M Vi, p = y i, p ⋅ M Vtot, p
∀i = 1,..., nc
(3.8)
The total component holdup on a tray may be defined as the sum of the component holdup in each of the phases:
M iT, p = M iL, p + M Vi, p
∀i = 1,..., nc
(3.9)
And the volume restriction must be satisfied by the two phases in the system, affecting thus the system pressure via the molar volumes, which are a function of the average composition, temperature and pressure of the system: L L V V M tot , p ⋅ v p + M tot , p ⋅ v p = Vtray, p
(3.10)
On the other hand, The energy holdups are related to the molar holdups, via the average molar enthalpy: L ( H pL − Pp ⋅ v pL ) U Lp = M tot ,p ⋅
(3.11)
L L V V U Tp = M tot , p ⋅ H p + M tot , p ⋅ H p − Pp ⋅ Vtray, p
(3.12)
- 21 -
Model Development
Chapter 3
The total energy holdup is again the summation of the phase energy holdups:
U Tp = U pL + U Vp
(3.13)
3.1.1.2 Interface equations Once the differential material end energy balances have been defined, as well as the relations between holdups, the interface mass and energy transfer needs to be properly modelled. The equations for the interface flow have been introduced in Chapter 2. However, before implementing these equations in the tray model, several assumptions may be introduced: §
First of all, the high flux correction factors matrices Ξ may be approximated by the identity matrix I . This is only true in the limit of low-mass transfer rates, but generally the mass transfer rates that can be achieved in distillation are low enough so that no errors of any consequence are introduced (Powers et al., 1988; Taylor and Krishna, 1993). This assumption is very significant, because the use of the correction factors would have implied the need of computing several complicated matrix operations. Moreover, the same may be applied for heat transfer, and the high flux correction factor Ξ H may also be dropped from the equation (Kooijman and Taylor, 1995a).
§
The liquid phase will also be treated as an ideal mixture, which is equivalent to substitute the matrix of thermodynamic factors Γ by the identity matrix I in the development for nonideal fluids presented in Section 2.1.2.
§
The explicit expressions for the computations of the flows presented in Chapter 2 (Eqs. 2.7 and 2.16) require the inversion of a matrix B (matrix of the inverse of the products between the mass transfer coefficient and the interface area). However, the inversion of matrices is not an operation that can be handled in gPROMS. Notwithstanding this, these two expressions may be rearranged (moving the second term of the right hand side of the equations to the left and multiplying by B both sides of the equation) so that no inverse of a matrix needs to be computed (Powers et al., 1988).
Thus, the final expressions of Eqs. (2.16) and (2.7) are as follows: - 22 -
Model Development
Chapter 3
nc −1
j =1
nc −1
j =1
[
] =
⋅ ( xi , p − xiI, p )
∀i = 1,..., ( nc − 1) (3.14)
[
] =
⋅ ( y iI, p − yi , p )
∀i = 1,..., ( nc − 1) (3.15)
L i , j, p
nc ⋅ φ j, p − x j, p ⋅ ∑ φk , p k =1
V i , j, p
nc ⋅ φ j, p − y j, p ⋅ ∑ φ k , p k =1
∑ B
∑ B
1 L vp
1 V vp
It should be pointed out that only (nc-1) of Eqs. (3.14-15) are linearly independent, and the B matrices have only been defined for (nc-1)2. Additional equations need to be added to obtain a completely determined system. These equations correspond to the summation constraints for the interface compositions: nc
∑x
I i, p
=1
(3.16)
I i, p
=1
(3.17)
i =1 nc
∑y i =1
The terms of the B matrices necessary to solve Eqs. (3.14-5) may be computed from Eqs. (2.17) and (2.8), respectively. In both cases, the product of the mass transfer coefficients and the interface area may be computed using the procedure described in Section 2.1.3, using the average compositions. Besides, the assumption of thermodynamic equilibrium in the interface introduces the additional equations: Φ iL, p I I yi , p = V ⋅ xi, p Φ i, p
∀i = 1,..., nc
(3.18)
where Φ i,L p ΦVi, p are the liquid and vapour phase fugacity coefficients. On the other hand, additional expressions must be provided to model the interface heat transfer. They are based on the interface heat flow equation defined in Section 2 (Eq. 2.30). Again, the heat transfer coefficients may be computed as explained in Section 2.2.1. Two equations may be written, one for the liquid film layer and one for the vapour. φ = ha H p
L p
⋅( T
L p
−T
)+ ∑ ( φ NC
I p
i, p
i =1
- 23 -
~L ⋅ H i, p
)
(3.19)
Model Development
Chapter 3
(
NC ~V H V I V φ p = ha p ⋅ ( Tp − T p ) + ∑ φ i, p ⋅ H i, p i =1
)
(3.20)
3.1.1.2.1 Flow models Depending on the flow model selected, the average mole fractions are calculated differently, allowing the computation of the mean mass transfer rates across the interface and the evaluation of mean molar and energy holdups. Mixed Flow It implies that the liquid and the vapour are both well-mixed, and a good estimation of the bulk compositions would be the leaving flows compositions: x i, p = x i, p
(3.21)
yi , p = yi, p
(3.22)
This assumption leads to a relatively simple system of equations that can be solved without any further consideration. The assumption of this flow pattern was also used by Sivertsson (1998). However, in some cases it may not be accurate to represent the flow patterns that can be achieved in a real distillation tray (Kister, 1992). As a result, the equivalent tray efficiencies predicted with the assumption of mixed flow may be lower than their real value. Plug flow model With the assumption of plug flow, either the liquid or the vapour move through the froth without mixing. This complicates the rate equations so that no exact solution of the rate equations is possible. Kooijman and Taylor (1995a) have proposed the computation of the difference of the average mole fractions as a function of the tray exiting composition and the matrix of the inverses of the product between the mass transfer coefficients and the interface areas. The starting point for the derivation of the plug flow model is a differential steady state mass balance in a vertical slice of the froth. Three main assumptions need to be made to solve this differential equation:
- 24 -
Model Development
Chapter 3
§
The matrix B , obtained as a function of the mass transfer coefficients, needs to be considered constant along the integration path.
§
The compositions of both phases at the interface also must be treated as a constant.
§
The molar flow across the tray and the molar volumes may also be considered constant at an average value.
The differential steady state mass balance that may be obtained for the vapour phase is as follows: d ( yiV ⋅V ) φ i = dz hf
(3.23)
where h f is the level of the froth. The derivative on the left hand side of this equation may be expanded, and the interface mass flow rate may be substituted using Eq. (2.7), neglecting the high flux correction factors matrix, leading to the following expression: y ⋅ V
V 1 1 dy dV +V ⋅ = ⋅ V h v dz dz f
( ) ⋅( y
V ⋅ B
−1
I
−y
V
)+ y ⋅ ∑ [ φ ] V
nc
j, p
j =1
(3.24)
Two terms in Eq. (3.24) may be simplified, in order to satisfy the overall differential mass balance, leading to the following expression after rearranging the total molar volume: 1 = h dz f
dy
V
( ) (
−1 BV I V ⋅ ⋅ y −y vV ⋅ V
)
(3.25)
The integration limits of this expression are: z = 0 yVi = yVi, IN
(3.26)
z=z y = y V i
V i
The solution of the differential equation Eq. (3.25), considering the vapour molar flow and the molar volume are constant (at an average value) along the integration path, leads to an
- 25 -
Model Development
Chapter 3
expression for the vapour flow composition as a function of the position in the froth and the entering stream composition:
(
y −y I
V
)
( )
− BV −1 z = exp V ⋅ v ⋅V hf
⋅
(y
I
− y IN V
)
(3.27)
This equation may be evaluated in the height of the froth, in order to obtain the composition of the leaving stream:
(
y − y OUT I
V
)
( ) (
− BV −1 ⋅ = exp V v ⋅V
y − y IN I
V
)
(3.28)
Nevertheless, the aim of this development is to obtain an expression for the mean vapour flow composition. Therefore, it is necessary to compute the average of Eq. (3.27):
(
y −y I
V
)
1 = h f
( )
− BV −1 z hf ⋅ exp V ⋅ ∫0 v ⋅V hf
dz ⋅ y I − yV IN
(
)
(3.29)
An exact solution of this integral may be derived. In addition, Eq. (3.28) may be substituted in it, in order to express the average composition as a function of the composition of the flow leaving the froth. The solution obtained is as follows:
(
y I − yV
)
( )
( )
( )
−1
−1
− BV −1 − BV −1 − BV −1 ⋅ y I − yV = exp −I ⋅ ⋅ exp V OUT vV ⋅V vV ⋅V v ⋅V
(
)
(3.30)
Due to the fact that the computation of the exponential of two matrices and several inverses are required to solve this equation, Kooijman and Taylor (1995a) propose this equivalent expression:
(
y −y I
V
)
( )
( )
−1
BV −1 BV −1 ⋅ y I − yV = exp V − I ⋅ V OUT v ⋅V v ⋅V
(
)
This last equation may be rearranged, leading to this also equivalent expression: - 26 -
(3.31)
Model Development
Chapter 3
(
)
( (( v
y I − y V = exp
V
⋅ V ⋅ BV
)
−1
)− I )⋅ ( v
V
)(
⋅V ⋅ BV ⋅
)
y I − y VOUT
(3.32)
which is more easy to solve, requiring only one exponential and one inverse of a matrix. On the other hand, Kooijman and Taylor propose the same approach to plug flow for the liquid phase, integrating along the liquid flow path. The main difference between two phases, is that the now the thermodynamic factors matrix defined for nonideal fluid mixtures has to be taken into account:
(
( )
( )
−1
B L −1 ⋅ Γ B L −1 ⋅ Γ L I − I ⋅ x − x = exp ⋅ x OUT − x L L v ⋅L v ⋅L L
I
)
(
)
(3.33)
which may also be rearranged as:
(x
L
−x
I
) = ( exp (( v
L
⋅L ⋅B
)
L −1
) )(
−1
(
)) (
)
⋅ Γ − I ⋅ Γ ⋅ v L ⋅ L ⋅ B ⋅ ⋅ x OUT − x (3.34) L
L
I
Notwithstanding this, ideal behaviour will be assumed as well for the liquid phase, and the thermodynamic factors matrix may be substituted by the identity matrix, resulting in the same expression for both phases. It should be pointed out that only (nc-1) of Eqs. (3.32) and (3.34) are linearly independent. Additional equations need to be added to obtain a completely determined system. These equations correspond to the summation constraints for the average compositions: nc
∑x
L i, p
=1
(3.35)
V i ,p
=1
(3.36)
i =1 nc
∑y i =1
Before proceeding any further, it should be noted that Kooijman and Taylor (1995a, 1998) defined in their work the ratio between the matrix B −1 and the product between the molar volume and molar flows as the NTU matrix. However, it should not be confused with the matrix formed with the binary NTUi, j computed using Eqs. (2.23) and (2.24). The author believes that this definition might be misleading, and thus its utilisation has been avoided.
- 27 -
Model Development
Chapter 3
Kooijman and Taylor (1995a) also proposed a correction factor for the compositions at the interface. They stated that at least the vapour interface mole fractions needed to be corrected They suggested that this correction should be a function of the degree of mixing in the liquid (and should be zero if the liquid is completely mixed). Finally, they proposed a correction for this interface composition that satisfied the conditions required, but lacked of theoretical basis. Therefore, and the fact that they don’t mention it in their ChemSep’s users manual (1998), this correction for the interface compositions has not been considered in this work.
3.1.1.3 Hydraulic equations Additional equations need to be added in order to model the outflow of liquid from the tray and the pressure drop. The liquid exit flow from the tray is supposed to be related to the clear height of liquid in the tray, via the Francis weir formula (Kister, 1992). Moreover, if there is not enough liquid on the tray to surpass the height of the weir, no outflow should occur:
hp =
L L M tot , p ⋅ vp Atray, p
3 l w, p h p − β p ⋅ hw, p 2 , h p > β p ⋅ hw, p 1.84 ⋅ L ⋅ v p βp Lp = 0 , h < β ⋅ h p p w,p
(3.37)
(3.38)
where β p is the tray aeration factor (Kister, 1992), which value is usually around 0.6. On the other hand, the total pressure drop between trays is equal to the dry tray pressure drop (which may be related to the vapour flow entering the tray) and the pressure drop through the aerated liquid. In addition, the dry tray pressure drop may be related to the flow of vapour entering the tray. As a result, the following expression may be derived and used to compute the incoming vapour flow:
- 28 -
Model Development
Chapter 3
V ⋅ vV (Pp+1 − Pp ) = α ⋅ ρ Vp ⋅ p+1 p Aholes, p V p+1 = 0
2 − ρ pL ⋅ g ⋅ hp , (Pp+1 − Pp ) > ρ pL ⋅ g ⋅ h p (3.39) , (Pp +1 − Pp ) < ρ pL ⋅ g ⋅ h p
where Aholes is the total tray area occupied by the holes and g is the gravity constant; α is the dry pressure drop friction factor coefficient, which can be calculated according to the following expression: α=
0.05 ⋅ g
(C )
2
(3.40)
V ,tray
where CV , tray is the tray orifice coefficient. Additional information to compute this parameter may be found in Kister (1992). It should be noted that, even though expression Eq. (3.39) might be solved explicitly for Vp+1 , it is not recommended because it would lead to the utilisation of a square root function that could be susceptible to produce a non-feasible solution.
3.1.1.4 Physical properties All the physical properties necessary in the development of the equations have not been considered as variables, because they are explicit function of the compositions, temperature and pressure. The only exception has been made for the enthalpies, because it is necessary to store their value in order to connect the different trays with streams.
H Lp = H ( T pL , Pp , xi , p )
(3.41)
H Vp = H ( T pV , Pp , yi, p )
(3.42)
H pL = H ( T pL , Pp , xi, p )
(3.43)
H Vp = H ( T pV , Pp , yi , p )
(3.44)
H Fp = H ( TpF , PpF , zi, p )
(3.45)
- 29 -
Model Development
Chapter 3
All the other physical properties (i.e. fugacity coefficients, molar volumes, and so on) will be evaluated every time they appear in any of the equations of the model.
3.1.1.5 Model summary In the nonequilibrium model presented, the equations and variables form a square system of index one. In the case of the well-mixed flow assumption, both the average and the leaving compositions in a stage are equal, and one variable may be used for both. The variables defined and the equations used are summarised in the following table:
Type
Dimension
Variables L i, p
Equations
V i, p
M ,M ,M
T i,p
Component molar holdups
nc
Total phase molar holdups
1
L V M tot , p , M tot, p
(3.3) and (3.4)
Bulk compositions
nc
xiL, p , yiV, p
(3.7), (3.8)
Int. compositions
nc
x iI, p . yiI, p U
L p,
U ,U
1
Interface molar flows
nc
φ i, p
1
φ
Interface heat flow Temperatures
L p ,
T
1
(3.15), (3.17) and (3.18)
T p
Energy holdups
V p
T L
(3.5), (3.6) and (3.7) (3.14) and (3.16)
H p V p
(3.1), (3.9) and (3.2)
(3.19) I p
T V
(3.11), (3.12) and (3.20)
(nc-1)2
Bp , Bp
(2.17) and (2.8)
Mass transfer coeff.
(nc2-nc)
kaiL, j, p , kaVi, j, p
(2.22) and (2.21)
Diffusion coefficients
(nc2-nc)
DiL, j , p , Dio, Lj , p , DVi, j , p
(2.31), (2.30) and (2.29)
Pressure
1
Pp
(3.10)
Hydraulic factors
1
FS, p , LFPW , p , t L , p , ZC , p
(2.25-8)
Heat transfer coeff.
1
ha Lp , ha Vp
(2.38) and (2.36)
Averaged diffusion coeff.
1
L V Davg , p , Davg , p ,
Not shown
Avgd. mass transfer coeff.
1
L V kaavg , p , kaavg, p
Not shown
Int. mass transfer matrix
- 30 -
Model Development
Chapter 3
Tray clear liquid level
1
hp
(3.37)
Liquid outflow
1
Lp
(3.38)
Vapour inflow
1
Vp+1
(3.39)
Dry tray ?P coeff.
1
αp
(3.40)
Vapour outflow
1
Vp
Degree of freedom
Physical properties
1
H Lp , H Vp , H pF
(3.41), (3.42) and (3.45)
Physical Properties (2)
nc
Tinbp ,p
Not shown
Table 3.1. Nonequilibrium well-mixed model summary.
(
)
Therefore, a total of np ⋅ 5 ⋅ (nc 2 − nc ) + 2 ⋅ (nc − 1)2 + 9 ⋅ nc + 28 variables and equations are defined to solve the system. The fact that the incoming flow is calculated in the model implies that the vapour outflow in the reboiler model must be a degree of freedom (as it will be calculated for the bottom tray equations). On the other hand, the vapour outflow in the top tray mast either be fixed or an equation for it will have to be added in the condenser. For the case of the plug flow averaged model assumption, the number of variables added is
np ⋅ (2 ⋅ nc + 2 ) , which correspond to the averaged molar fractions and enthalpies for the liquid and vapour, which now have to be included as their conditions are different from the leaving conditions. The corresponding equations that must be included in the system, In order to keep it squared, are Eqs. (3.34), (3.35), (3.32), (3.36), (3.43) and (3.44).
3.2 Reboiler model The column reboiler has been modelled as an internal reboiler. It receives the liquid outflow from the last tray of the column, and an additional external feed has also been considered. The key assumptions are that both phases are perfectly mixed and in thermodynamic equilibrium. The outgoing vapour stream flow that ascends towards the bottom tray is computed by the pressure-drop equation of this bottom tray. Moreover, the product stream flow is computed
- 31 -
Model Development
Chapter 3
as a function of the level in the reboiler, according to Francis equation, once the reboiler weir has been defined. The model presented here is very similar to the one used by Sivertsson (1998), with several minor modifications adapted from the gPROMS model library (1997). The material balance and energy equations used are very similar to the ones presented in the tray model. The heat supplied to the reboiler is the only degree of freedom.
dM i ,R = Lnc ⋅ xi, nc − LR ⋅ xi, R − VR ⋅ y i, R + FR ⋅ z i, R dt
∀i = 1,..., nc
dU R = Lnc ⋅ H ncL − LR ⋅ H RL − VR ⋅ H VR + FR ⋅ H RF + Q& reboiler dt
(3.46)
(3.47)
The summation constraints must also be satisfied in the reboiler. In addition, the total molar holdups may also be defined, and the internal energy may be related to them. nc
∑x
i, R
=1
(3.48)
i, R
=1
(3.49)
i =1 nc
∑y i =1
L V M i, R = xi, R ⋅ M tot , R + y i , R ⋅ M tot , R
∀i = 1,..., nc
L L V V U R = M tot ,R ⋅ H R + M tot, R ⋅ H R − PR ⋅ Vreboiler
(3.50) (3.51)
The volume available must be occupied by both phases, what will in turn fix the pressure of the reboiler. L L V V M tot ,R ⋅ vR + M tot , R ⋅ v R = Vreboiler
(3.52)
Finally, the assumption of thermodynamic equilibrium between phases leads to the following equilibrium relationships:
yi ,R
Φ Li, R = V ⋅ xi, R Φ i, R
∀i = 1,..., nc
(3.53)
Concerning the product flow, its value may be related to the level in the reboiler, using Francis equation. A height of the weir must be defined in this case. - 32 -
Model Development
Chapter 3
L L M tot ,R ⋅ vR hR = Areboiler
1.84 ⋅ l w ,R vRL LR =
3 2 h − β ⋅ h p R w ,R ⋅ , hR > β R ⋅ hw ,R β R 0 , hR < β R ⋅ hw , R
(3.54)
(3.55)
As with the tray model, the only physical properties that will be considered as variables will be the enthalpies:
H RL = H ( TR , PR , xi ,R )
(3.56)
H VR = H ( TR , PR , y i, R )
(3.57)
H RF = H ( TRF , PRF , zi, R )
(3.58)
3.3 Condenser and reflux drum models The condenser model presented here is an idealisation of a real physical one. The holdup has been neglected and it is assumed that the leaving flow has been condensed to its saturation point. First of all, the condenser inlet needs to be related with the first tray vapour outflow:
xi ,C = y i,1
∀i = 1,..., nc
LC = V1
(3.59) (3.60)
An energy balance on the unit allows to compute the amount of heat that need to be removed to condense the stream:
V1 ⋅ H CV,in − LC ⋅ H CL. out − Q& condenser = 0
(3.61)
The exit temperature is computed by performing a bubble point calculation in the leaving stream:
- 33 -
Model Development
Chapter 3
nc
∑y
i ,C
=1
(3.62)
i =1
Φ iL,C yi ,C = V Φ i ,C
⋅ xi,C
∀ i = 1,..., nc
(3.63)
It is assumed that there is no pressure drop through the condenser. Besides, the pressure in the condenser is a degree of freedom, and needs to be specified. PC ,out = PC ,in
(3.64)
On the other hand, due to the fact that in the tray model, the incoming and not the leaving vapour stream is computed, it is necessary to include an equation for the leaving stream in the first tray (which is the condenser inlet). This equation relates the pressure drop in the piping that connects the column with the condenser: V1 = CV ⋅ P1 − PC ,in
(3.65)
where CV is a friction factor coefficient. Finally, the enthalpies of the incoming vapour (at the condenser inlet pressure) and of the condensed liquid have also been considered as variables:
H CL ,out = H ( TC ,out , PC ,out , xi ,C )
(3.66)
H VC ,in = H ( TC ,in , PC ,in , y i,C )
(3.67)
The condensed liquid that leaves the condenser is sent to the reflux drum, where it is stored, from where it will be led back to the column (reflux stream) or it will form part of the top product (distillate stream). The reflux drum has been modelled in the same way as the reboiler, with the exception that now the leaving stream is the summation of the two outflows. The model equations are these:
dM i ,D = LC ⋅ x i,C − (LREF + LDIST ) ⋅ xi ,D + FD ⋅ z i, D dt
∀i = 1,..., nc
dU D = LC ⋅ H CL − (LREF + LDIST ) ⋅ H DL + FD ⋅ H DF − Q& refluxdrum dt - 34 -
(3.68)
(3.69)
Model Development
Chapter 3
nc
∑x
i, D
=1
(3.70)
=1
(3.71)
i =1 nc
∑y
i, D
i =1
L V M i, D = xi, D ⋅ M tot , D + y i , D ⋅ M tot , D
∀i = 1,..., nc
(3.72)
L L V V U D = M tot , D ⋅ H D + M tot ,D ⋅ H D − PD ⋅ Vrefluxdrum
(3.73)
L L V V M tot , D ⋅ vD + M tot, D ⋅ vD = Vrefluxdrum
(3.74)
L L M tot ,D ⋅ v R hD = Arefluxdrum
LDIST + LREF
1.84 ⋅ l w, D v DL =
3 hD − β D ⋅ hw, D 2 ⋅ , hD > β D ⋅ hw, D β D 0 , hD < β D ⋅ hw ,D
(3.75)
(3.76)
H DL = H (TD , PD , x i,D )
(3.77)
H VD = H ( TD , PD , yi, D )
(3.78)
H DF = H ( TDF , PDF , z i, D )
(3.79)
The only different equation is the addition of a relationship between the reflux and the distillate stream. Both streams are related by the reflux ratio, RR , which is a parameter that needs to be specified: LDIST = RR ⋅ LREF
(3.80)
Finally, it should be pointed out that the reflux stream, according to the equation presented so far, should be at a pressure smaller than the pressure in the first tray (where the reflux is supposed to enter the column). Thus, a simplified model of a pump has been defined, assuming that an isoenthalpic compression takes place. As expected, the flow, composition and enthalpy of the outlet stream will be the same of the inlet:
- 35 -
Model Development
Chapter 3
LPump = LREF
xi ,Pump = xi, D
(3.81)
∀i = 1,..., nc
(3.82)
L H Pump = H DL
(3.83)
The outlet pressure has been designed to be equal to the first tray pressure, neglecting any possible pressure drop in the piping, or due to changes in potential energy. PPump = P1
(3.84)
The energy consumed by the pump may be calculated, and the outlet temperature has to be calculated, in order to satisfy to isoenthalpic restriction:
W& Pump = ( PPump − PD )⋅ LPump ⋅ v LPump
(3.85)
L H Pump = H ( TPump, PPump, xi ,Pump )
(3.86)
3.4 Equilibrium tray model In order to compare the results of the nonequilibrium model, an equilibrium model has also been developed. This model also takes into account two holdups (liquid on the froth and vapour on the froth and above, lumped together), but it is more simple, due to the fact that the rate equations are not included. In this model, the mass and energy flows across the interface are not modelled and, thus, the number of differential variables necessary for the holdups is reduced to the half of them. The mass and energy balances are the only differential equations that appear in the model: dM i, p = L p−1 ⋅ x i, p−1 − L p ⋅ xi, p + Vp +1 ⋅ y i, p +1 − V p ⋅ yi, p + Fp ⋅ zi, p dt
∀ i = 1,..., nc (3.87)
dU p = Lp −1 ⋅ H pL−1 − L p ⋅ H Lp + V p+1 ⋅ H Vp+1 − Vp ⋅ H Vp + Fp ⋅ H pF + Q& p dt
- 36 -
(3.88)
Model Development
Chapter 3
Due to the fact that now the phase component molar holdups are not used as a differential variable, they don’t need to be defined. However, the total holdups must still be related to the phase total holdups using the component molar fraction, and the summation constraints for the compositions also apply: L V M i, p = x i, p ⋅ M tot , p + y i , p ⋅ M tot , p nc
∑x
i, p
∀i = 1,..., nc
(3.89)
=1
(3.90)
=1
(3.91)
i =1 nc
∑y
i, p
i =1
The total energy holdup may also be related to the phase total molar holdups, via the enthalpies: L L V V U p = M tot , p ⋅ H p + M tot, p ⋅ H p − Pp ⋅Vtray, p
(3.92)
The volume restriction also applies for the equilibrium case, which allows calculating the pressure of the system: L L V V M tot , p ⋅ v p + M tot, p ⋅ v p = Vtray, p
(3.93)
Thermal equilibrium has been assumed in the development of this model, but not thermodynamic equilibrium, which has been quantified using Murphree efficiencies. These efficiencies may be incorporated in the equilibrium equation (Kister, 1992):
yi , p
Φ iL, p = EMV , i, p ⋅ V ⋅ xi, p + ( 1 − EMV ,i, p ) ⋅ yi, p+1 Φ i ,p
∀i = 1,..., nc
(3.94)
Finally, the hydraulic equations have been modelled in the same way used for the nonequilibrium model, using Eqs. (3.37-40). The same applies to the definitions of the enthalpies as variables, using Eqs. (3.41-2) and (3.45) This equilibrium model must be connected to the same reboiler, condenser, reflux drum and pump models defined for the nonequilibrium model.
- 37 -
Model Development
Chapter 3
- 38 -
gPROMS implementation
Chapter 4
Chapter 4 4 gPROMS implementation The nonequilibrium model described in the previous section has been implemented in the gPROMS simulation software package. In this equation-oriented simulator, a model with the equations needed to characterise the dynamic behaviour of each of the items of equipment (trays, reboiler, etc.) is defined. Then, a higher level unit, i.e. the distillation column, which uses instances of the previously defined models, is created. In this higher level unit, the submodels are interconnected. Finally, a process is defined for this model, where all the parameters and degrees of freedom receive an assigned value, and an operating policy for the distillation column is specified.
4.1 Model implementation The total number of variables that are considered in this model depends on the number of components (nc) and on the number of trays (np). The number of variables defined in every model is given as a function of these two parameters in Table (4.1). As it can be seen in this table, the number of variables required in the nonequilibrium model is very high, compared to the number of variables required in the equilibrium one. Most of these variables are explicit functions of other variables, and could be eliminated from the model, but then the resulting equations would be far too complicated to be handled in an understandable way. The number of variables has been reduced with respect to the model developed by Sivertsson (1998). This reduction has been achieved mainly by including in the same model all the equations that apply for a tray, avoiding thus a considerable amount of connecting variables.
- 39 -
gPROMS implementation
Chapter 4
Number of variables Model Model variables
Physical Prop. Variables
Connecting variables
Dummy variables
3(nc)+9
3
Nc+4
1
np(nc+3)
np(2nc+8)
np(2nc+2)
np(nc+5)
np(2nc+8)
np(2(nc-1)2+ 2nc+2)
np[4nc+9]
3np
np(nc+5)
np
Condenser
2(nc)+3
3
Nc+5
1
Reflux Drum
3(nc)+8
3
Nc+4
1
Pump
2
1
2(nc)+7
1
Column
0
0
2(nc)+8
1
Reboiler Noneq. Tray
np[5·(nc2-nc)+
(well-mixed)
2(nc-1)2+8nc+25]
Noneq. Tray
np[5·(nc2-nc)+
(plug-plug) Equilibrium tray
2
2(nc-1) +10nc+25]
Table 4.1. Number of variables in the implemented models.
The number of variables could also be reduced with a different treatment of the physical properties. Sivertsson defined most of the physical properties of the system as variables. This implies that the number of times that the foreign object property package software is called is minimised. In contrast, the number of equations that are included in the system increases. Provided that the model proposed by Sivertsson had serious problems in the initialisation, a reduction in the number of equations might be a form to reduce the size of the model and perhaps facilitate its convergence. Which of these two possibilities (minimise the physical properties evaluations or reduce the size of the model) is best and how they affect the time needed to solve the problem has not been studied. However, it is quite straightforward to assume that the minimisation of the variables of the system (and thus the number of equations) should facilitate the convergence of the problem. On the other hand, the associated consequence of evaluating several times the same physical properties could lead to increased times of convergence, but in any case could lead to an increase in the complexity of the model or its solvability (the same properties are called several times). The way in which the equations are finally implemented may also be a source of nonnecessary complexity of the model. A good example of this idea could be the way in which
- 40 -
gPROMS implementation
Chapter 4
binary mass transfer coefficients and diffusion coefficients have been handled. According to Eq. (2.2), the order in which the components are expressed doesn’t affect the value of these parameters. Thus, it is only necessary to compute the half of them, using the original empirical equation. The other half may be related to its twin coefficient:
L L L kai, j = NTUi, j ⋅ ( L ⋅ v ) , i < j kaiL, j = 0.0 , i = j kaiL, j = ka Lj,i , i > j
(4.1)
With this approach, the complexity of the system of equations is reduced, and also the nonlinearities, while the number of variables and equations is kept constant. The diagonal terms of the resulting matrices are dummy variables, because they don’t need to be defined in the model. However, a suitable implementation in gPROMS implies that they should be defined as matrices and, thus, all the terms have to been defined (and an equation must be provided). On the other hand, the fact that that the terms below the diagonal are equal to the terms above could suggest that these variables could be eliminated from the model, reducing even more the number of equations needed. Nevertheless, this approach would reduce the understandability of the model equations and would make more difficult its debugging.
4.1.1 Physical properties A distillation column is an operation that requires the estimation of a considerable amount of physical properties. Reliable methods to estimate the fugacity coefficients and the enthalpies are of central importance. When nonequilibrium models are used, the number of physical property evaluations is increased, acquiring more relevance. In gPROMS, the physical properties may be defined explicitly by the user, creating his own models, or they can be calculated using physical property packages, linked to the main program using a foreign object interface. The Multiflash physical property package has been selected. This package requires the definition of an additional file, where a database and the methods used to compute the - 41 -
gPROMS implementation
Chapter 4
different properties must be specified. Different models may be used for each phase and for each of the following properties: 1) fugacity, 2) volume/density, 3) enthalpy/entropy, 4) viscosity, 5) thermal conductivity and 6) surface tension. Additional information concerning the models available may be found in Multiflash (1997) and in gPROMS (1998). An example of the Multiflash implementation file used in the simulations is given in Appendix A.1.
4.2 Benchmark problem: depropanizer In order to be able to test the gPROMS code generated, it is necessary to apply it to a particular case. The column internal layout needs to be specified properly, because it is necessary for the interface equations as well as for the hydraulic equations. The example chosen for these purposes has been an industrial scale depropanizer column. A feed mixture of ethane, propane, butane and pentane is introduced as a liquid in the 15th (out of 33) sieve trays. This column operates at 15 atm, with a reflux ratio of 2.5. A liquid feed stream is introduced in the 15th tray, with the following conditions:
Feed stream Stage
15
Pressure (kPa)
1519.875 298.0
Temperature (K) Molar flow (mol/s)
1000.0
Compositions (molar fraction) Ethane
0.1
Propane
0.3
n-Butane
0.5
n-Pentane
0.1
Table 4.2. 33-trays depropanizer column feed conditions.
- 42 -
gPROMS implementation
Chapter 4
In order to be able to implement a nonequilibrium model in this column it is necessary to specify the internal layout, needed to compute the mass transfer coefficients and also to solve the hydraulic equations. All the relevant information concerning the internals of this column may be found in Taylor and Krishna, 1993 (pp. 407-410). In the following table, only some key parameters are presented:
Column Layout Design Type of internals 2
Active Area (m ) 2
Holes Area (m ) 3
Rectification Section
Stripping Section
Trays 1 to 14
Trays 15 to 33
Sieve trays
Sieve trays
14.96
24.51
0.8813
Tray Volume(m )
7.48
Weir height(m)
0.0508
Weir length(m) Liquid flow path length (m)
17.6 0.7945
1.355 12.255 0.03734 22.97 1.037
Table 4.3. 33-trays depropanizer column layout design.
The data given in Table (4.3) corresponds to a column designed with a flooding factor of 70 %. It should be pointed out that the design presented above corresponds to a column with 5 flow passes per tray, leading to a high value for the weir length and a relatively small liquid flow path length. A copy of the gPROMS code generated to simulate the dynamic behaviour of this case is included in Appendix A.2. The internals data specified above were obtained using an average efficiency of ca. 60%. When plug flow is assumed, the average efficiencies may increase up to 100 %, and the depropanizer column with 33 trays becomes over-designed. Kooijman and Taylor (1995a) used a reduced version of this one. The internals for this new benchmark column are given in Table (4.4). The same feed stream may be used, provided the feed tray is relocated to number 10.
- 43 -
gPROMS implementation
Chapter 4
Apart from the data specified above, an aeration factor in the trays has been assigned a value of 0.6 (c.f. Eq. (3.38)). Besides, the tray orifice coefficient (needed to calculate the dry pressure drop friction factor, c.f. Eq. (3.40)) has been given a value of 0.7.
Column Layout Design
Rectification Section
Stripping Section
Trays 1 to 9
Trays 10 to 23
Type of internals 2
Active Area (m ) 2
Holes Area (m )
Sieve trays
Sieve trays
16.5
31.5
0.932
1.57
Tray Volume(m )
8.25
15.75
Weir height(m)
0.0508
3
Weir length(m)
0.0464
19.1
Liquid flow path length (m)
26.5
0.862
1.19
Table 4.4. 23-trays depropanizer column layout design.
The 33-trays depropanizer was also implemented by Sivertsson (1998). He also used the internals specifications presented above. In addition, Sivertsson designed the auxiliary equipment (reboiler and reflux drum). The layout proposed by him has been maintained here, in order to be able to compare the results obtained.
Surrounding Equipment Layout Design
Reboiler
Vessel Volume(m3) 2
Area (m )
Reflux Drum
25
14
25
14
Weir height(m)
0.7
0.7
Weir length(m)
5
3.74
Table 4.5. Surrounding equipment layout design.
- 44 -
gPROMS implementation
Chapter 4
Concerning the physical properties, two different combinations of models have been tested. The first package tested was the one used by Sivertsson. However, several problems found during the initialisation obliged to modify this set of models. An alternative model combination was devised and proved to work successfully for a wider range of conditions (c.f. Section 4.2.1).
Package 1 (Sivertsson)
Package 2 (recommended)
Liquid and Vapour phases
Liquid and Vapour phases
Fugacity
Redlich-Kwong Soave eos
Redlich-Kwong Soave eos
Volume/Density
Redlich-Kwong Soave eos
Lee-Kesler-Plocker eos
Enthaply/Entropy
Redlich-Kwong Soave eos
Lee-Kesler-Plocker eos
Viscosity
Lohrenz-Bray-Clark
Lohrenz-Bray-Clark
Thermal Conductivity
Chung-Lee-Starling
Chung-Lee-Starling
Macleod-Sudgen
Macleod-Sudgen
Physical Property
Surface Tension
Table 4.6. Physical Property models.
4.2.1 Initialisation Once the model has been satisfactorily implemented in gPROMS and all the errors or possible inconsistencies have been eliminated, it is necessary to initialise the program, so that dynamic simulations may be run. At time zero, the equations that compose the model reduce to an algebraic system of equations, that may be solved once the required initial conditions have been specified. This system of equations is usually solved using the Newton method or one of its relatives. Therefore, the convergence of the method is not guaranteed, and a good estimation of the final solution is necessary in order to ensure the convergence of the method. Nevertheless, the main problem observed when trying to initialise this particular case study was that the Multiflash package failed constantly when evaluating the physical properties. The Newton method (required to solve the initial algebraic system) uses an iterative procedure to find a suitable solution. If the starting guess introduced by the user is not good enough, the
- 45 -
gPROMS implementation
Chapter 4
path followed in the first iterations may possibly correspond to mixtures where the requested physical property is not feasible. To put it in simple words, gPROMS may call Multiflash to evaluate a vapour property, but the conditions given may correspond to a liquid mixture. This may happen when a solution has not been converged and, therefore, the model equations are not satisfied yet. In these situations, Multiflash may fail to provide gPROMS with suitable values for the partial derivatives of some properties, and thus, gPROMS can not calculate the Jacobian matrix properly and the method may not proceed satisfactorily. The nonequilibrium model was implemented firstly in the 33-trays depropanizer column. Provided that the assumption of well-mixed liquid and vapour phases leads to the simplest set of equations, it is the one that has been used in the initialisation. The plug flow assumption model may be initialised using the solution obtained in the previous case. On the other hand, the equilibrium model may be initialised using the same procedure devised for the nonequilibrium well-mixed phases model.
4.2.1.1 Nonequilibrium well-mixed model Before proceeding any further, the degrees of freedom in the resulting system should be considered. Mainly, there are two degrees of freedom available in the whole column: reboiler heat duty and pressure in the condenser. However, during the initialisation, it proved to be more suitable to leave the reboiler heat duty as a free variable, while the pressure in the reboiler was fixed instead. On the other hand, the piping that connects the top of the column with the condenser has not been designed, and no estimate of the pressure drop is available. Therefore, the friction factor in Eq. (3.65) is unknown, reason why this equation has been eliminated. This new degree of freedom created has been assigned to the first tray leaving vapour stream, which may be easily calculated for the steady state solution for a fixed reflux ratio. Once the column is satisfactorily initialised, the original degrees of freedom may be restored and the model may run dynamically. On the other hand, it should also be noted that the model has been initialised for a steady state particular solution. This is due to the fact that the assumption that no weeping takes place would not be acceptable when the column is started from dry conditions. In addition, the
- 46 -
gPROMS implementation
Chapter 4
correlation used to compute the liquid flow was obtained under the assumption that the tray was working in the froth regime. Therefore, the results of a simulation of the model out of this regime should not be taken as reliable. Nevertheless, it could be an acceptable way to bring the model to an acceptable initial steady state solution. In any case, this approach has not been considered. In 1998, Sivertsson solved his initialisation problem for the depropanizer in a very tortuous way. He tried to start with a very simple model and then added gradually more complex parts. His first model consisted of 33 non-connected trays, with all the inlet streams specified manually (and scaled) and the tray spacing was increased (to increase the holdup). The mass transfer coefficients were considered constant (calculated with an additional program) and no feed stream was defined yet. Secondly, the real tray model was used, but with constant mass transfer coefficients and no connection between trays. In a third model, he added the mass transfer coefficients calculations and, in a fourth, the dynamic equations were implemented (only steady state equations had been implemented so far) and some trays were linked, creating three different blocks of trays. A fifth model was used to connect two of the blocks and a sixth to connect the remaining two. The final model was used to connect also the reboiler and the condenser, and leading to the final nonequilibrium dynamic model. Due to the fact that the method used by Sivertsson was not practical and very difficult to reproduce for different case studies, an alternative initialisation procedure was devised. Powers et al. (1988) performed a serious study concerning the initialisation of nonequilibrium model. They presented different initialisation procedures and applied them to several cases, comparing the results obtained. They recommended the use of continuation methods as a valuable aid in the convergence of the model. They presented the Newton homotopy method and also a nonequilibrium continuation method. The philosophy of these methods consists of finding an easily converging solution, and then bringing it to the real solution using a parameter. In the nonequilibrium continuation method that they presented, they apply this to the equilibrium constant and to the enthalpies. Nevertheless, a suitable initial guess is still necessary. On the other hand, Kooijman and Taylor (1998) also presented a procedure to obtain an initial guess for the nonequilibrium model, which is the same that they used to solve their equilibrium models. - 47 -
gPROMS implementation
Chapter 4
All these continuation methods have proved to be very useful in the convergence of distillation column models, particularly for highly nonideal systems. However, the case of the depropanizer column is a fairly ideal system and, despite it has a considerable number of trays and components, it should converge without the aid of a continuation method. The main objective when trying to devise a suitable initialisation procedure for the nonequilibrium model has been to keep it simple and straightforward, so that it may be used as a general procedure applicable to different systems, and not only to a particular case. Nevertheless, it should be noted that for very difficult problems, like highly nonideal systems or columns with side streams, a customised initialisation procedure might be advantageous. Initialisation procedure The procedure proposed here consists of estimating the top and bottom conditions (composition, temperature and pressure). Secondly, an estimate of the vapour and liquid enthalpies for these two streams is needed, as well as an estimate of the ratio of fugacity coefficients (K-values). Then, the whole column may be initialised using linear profiles for all the main variables. On the other hand, the internal liquid and vapour flows may be initialised assuming constant molar overflow, so that the flows are constant throughout the column (except at the tray where the feed is introduced). Finally, the mass and energy holdups may be calculated provided a level in every tray (and also in the reboiler and reflux drum) is specified. Once the initial guess has been specified, it may be used to converge an initial solution for the nonequilibrium model. The steps followed to find the initial solution are as follows: Step 1 In the first step, an initial guess is obtained for all the model variables. The real nonequilibrium model has been used as starting point, and one equation has been removed for every variable that is specified instead of calculated. In the reboiler, the following variables are specified: product molar flow, (nc-1) liquid compositions, pressure and temperature. The clear liquid level is fixed to be a 6% higher than the product between the aeration factor and the weir height. Besides, the mixture liquid and
- 48 -
gPROMS implementation
Chapter 4
vapour enthalpies, as well as the K-values for every component must be specified. These values may be calculated easily using a separate gPROMS file, or they may be found elsewhere. With the information given above, the remaining variables may be calculated using the following equations: the fourth liquid composition is obtained with Eq. (3.48); the vapour compositions are solved with Eq. (3.53); the total liquid holdup with Eq. (3.54); the total vapour holdup with Eq. (3.52); the total component holdups with Eq. (3.50); and the energy holdup with Eq. (3.51). In this first trial, the heat in the reboiler also needs to be specified a value, because the differential equations are still not included. In addition, the streams that link the reboiler with the bottom tray are not specified, because they will be fixed when specifying the tray conditions. The same procedure used for the reboiler applies for the reflux drum, due to the fact that both units share virtually the same model equations. On the other hand, the condenser model is kept almost completely unmodified. Only the temperature has been fixed, avoiding the dew point calculation. The same K-values used in the reflux drum have been used. Finally, the pump exit stream temperature has been fixed, so that it is not necessary to call Multiflash to evaluate the enthalpy of this stream. Once that both the reboiler and the reflux drum have been specified, their values have been used to fix the tray conditions. Due to the fact that the equations needed to implement the linear profiles need some data from different models, they must be implemented in the higher level model (i.e. the column). Liquid temperature, pressure, (nc-1) liquid compositions, liquid and vapour enthalpies and K-values have been assigned values corresponding to a linear profile between the top and the bottom of the column. The clear liquid levels have been specified to be 1.45 times higher than the product between the aeration factor and the weir height in the rectification section, and 2 times higher in the stripping section. These values have been obtained from an average value of the levels in the initial point obtained by Sivertsson (1998). Additional equations have been added to ensure that the temperature of the liquid, vapour and the interface are equal, and that the compositions in the bulk and in the interface are also equal. The K-values are used to compute the vapour compositions, and the summation constraint in the liquid for the last component composition. Eqs. (3.7-13) allow computing all the molar and energy holdups. Concerning the mass and heat transfer, both - 49 -
gPROMS implementation
Chapter 4
rates have been set to zero. The infinite dilution liquid diffusion coefficients have been set to a fixed value, and also the liquid and vapour mass and heat transfer coefficients, because their evaluation requires the use of Multiflash. The superficial factor and the flow-path-width flow have also been set to a value. Step 2 The same model structure is used in this second step as the one used in the first step. However, now the Multiflash foreign object package may be called because reasonable composition, temperature and pressure profiles have been defined in the previous step. So, the enthalpies and the K-values are now calculated using Multiflash. In addition, all the transport variables (diffusivities and mass transfer coefficients) may be evaluated properly, provided Multiflash does not fail. The interface mass transfer flow is still fixed to zero, and the interface and bulk compositions are still maintained equal. The same applies for the temperatures, considered equal for both phases and the interface. The interface heat transfer flow has also been set to zero. Step 3 (optional) In this step, the solution of the previous step is used to solve the real tray model of the column. The differential material and energy balances equations may be implemented, as well as the hydraulic equations. The interface transfer equations have also been included, but the heat transfer coefficients have been set to a very high value, so that the resistance to heat transfer is minimal. Nevertheless, this step should be implemented only if the direct change to step 4 fails. In case both mass and heat transfer may be implemented in the same step, there is no need to accomplish it separately. Moreover, it should be noted that now the reboiler heat duty may be calculated with the differential energy equation, and it does not need to be specified. Step 4 This is the first step with the real tray model implemented. The heat transfer coefficients are properly calculated, and the temperature gradient between phases may be properly modelled. Starting from the previous step point, where the mass transfer across the interface was already included, the complete model may be solved satisfactorily.
- 50 -
gPROMS implementation
Chapter 4
Step 5 (optional) In case that the thermodynamic method used in the previous stages is not the method desired in the final implemented model, it may be the right time to change the thermodynamic properties specification file, in order to implement more suitable or rigorous thermodynamic models. Step 6 In this last step, the calculated value of the reboiler heat duty may be assigned, leaving the reboiler pressure as a variable, which will be calculated. On the other hand, the equivalent friction factor for the top of the column piping may be calculated, and Eq. (3.65) may be included in the model again. Now, the model is ready to be run dynamically.
4.2.1.2 Nonequilibrium plug-flow model The plug-flow model was successfully implemented for the 23-trays column. As mentioned above, the reason why a smaller column is necessary is because the plug flow models predict higher efficiencies, so that a column with 33 trays would be over-designed. For this reason, the 23-tray column was implemented. This smaller depropanizer was initialised directly form the initial point obtained for the 33-tray one, changing the parameter concerning the internals design and the feed tray location. Once this 23-tray column was initially solved for the well-mixed flow case, its solution may be used as initial point for the plug-flow model. In this model, the equations for the computation of the average compositions need to be added. Besides, it is necessary to define an additional matrix variable to compute the ratio between the B matrix and the product of the average molar flow and the molar volume. Due to the fact that the correction factors matrix may not be computed in gPROMS (exponentials and inverses of matrices may not be handled), it is necessary to export this matrix and calculate the correction one in MATLAB. An EXCEL spreadsheet has been used to write both matrices in understandable forms for every program. The procedure used in the initialisation is as follows:
- 51 -
gPROMS implementation
Chapter 4
Step 1 In the well-mixed flow case code, the equations and variables needed to compute the modified B matrix and the average compositions are introduced. The correction factors matrix is defined as a parameter, due to the fact that it is specified externally, and its value will not change during the simulation. In this first iteration, the first modified B matrix is defined, and the correction factors are specified as an identity matrix. The average composition equations are not used yet, and their value is set to the computed value in the well-mixed flow case. Step 2 With the computed modified B matrix from the previous gPROMS initialisation, a suitable expression of the modified B matrices for every tray and phase have been generated in EXCEL and exported to MATLAB, where the correction factors matrices have been calculated. Step 3 Again, EXCEL is used to create the gPROMS understandable expression of the correction factors and they are pasted in the gPROMS simulation file. Moreover, now the holdup and interface transfer equations may be modified, so that the used compositions for these calculations are the average ones, and not the leaving compositions. This new model is initialised again, obtaining a new modified B matrix. At this point, several problems when converging this step were found. They were solved by broadening the margin of variables like the composition and the molar holdups. gPROMS, in its iterative procedure might compute directions that are physically infeasible, but it needs to be allowed to move in that direction, which will be corrected in successive iterations. Step 4 Check if the two last correction factors matrices calculated are equal. If not, return to step 2. This presented procedure achieved convergence in three iterations, obtaining finally a steadystate solution of the plug-plug flow model. This model may be run dynamically, keeping the correction factors matrix constant.
- 52 -
gPROMS implementation
Chapter 4
4.2.1.3 Outstanding issues In the procedure shown above, some minor modifications needed to be done either to ensure the convergence or to formulate the problem in gPROMS. First of all, the viscosity physical property model failed when evaluating the infinite dilution liquid diffusion coefficient (c.f. Eq. (2.30)). This viscosity should be evaluated for pure component at the system temperature and pressure. However, for the component ethane in the first tray, Multiflash detected an unstable phase and failed to compute the liquid viscosity. Therefore, the composition of the mixture was used instead. Sivertsson (1998) also must have faced this problem, because in his gPROMS code he computed the pure component viscosity, but at its normal boiling point and not at the system pressure. Besides, Sivertsson changed the conditions at which he evaluated some of the other properties needed in the infinite dilution diffusivity calculations. On the other hand, it was impossible to compute the liquid and vapour partial molar enthalpies needed for the second term on the right hand side of Eqs. (3.19-20). It had to be approximated as the product between the summation of interface flows and the enthalpy of the mixture.
4.2.1.4 Concluding remarks The procedure presented in the previous section proved to work successfully for the depropanizer column with 33 trays, assuming a well-mixed flow model. The nonequilibrium model was initialised and dynamic simulations may be realised with that initial point. In case that problems were found when trying to initialise a different case study, the recommendation would be to start the modifications trying to use different physical properties models, and not trying to devise different intermediate models. In case of highly nonideal systems, it might be useful to study carefully which is the expected profile along the column, and trying to implement it instead of the linear profile. Moreover, an estimate of an intermediate tray composition might be guessed, and two (or more) different slopes might be used in the linear profiles specified. However, if all the above was not enough, then a continuation method might be useful to initialise it.
- 53 -
gPROMS implementation
Chapter 4
Concerning the plug-flow model implemented, there is no guarantee that the correction factors matrix is going to be constant over the simulation. It should be checked under which circumstances this assumption is valid (small perturbations, etc.). In case the matrix should be corrected during the simulation, the process may be stopped, the matrix updated and the simulation continued, but this procedure does not seem to be recommendable and, in these cases, an alternative solution procedure should be devised.
- 54 -
Results
Chapter 5
Chapter 5 5 Results In this section, the results obtained from the initialisation and dynamic simulations of the different models developed in the previous chapters are presented. Firstly, the steady state solution for the nonequilibrium well-mixed model will be presented. Straightaway, its results will be compared with the ones obtained for the corrected nonequilibrium model presented by Sivertsson. Additional analysis of the influence of the thermodynamic models selected will be achieved. Secondly, this nonequilibrium model will be evaluated against an equilibrium model solved for the same conditions. Besides, dynamic simulations of the equilibrium and nonequilibrium model will be compared. Afterwards, the results for the averaged plug flow will be presented, and some discussion concerning its applicability in dynamic simulations will be included.
5.1 Nonequilibrium well-mixed model The well-mixed model with the analytical solution of the Maxwell-Stefan equations implemented was satisfactory initialised using the procedure described in Chapter 4. The 33trays depropanizer column was used, so that the results obtained may be compared with the results published by Taylor and Krishna (1993). In the following figures, the liquid phase composition, the temperature and the internal flows have been plotted for the initial steady state solution. The profiles obtained are virtually identical to those presented by Taylor. The liquid phase composition profiles (see Fig. (5.1)) correspond to typical profiles in multicomponent separation. The kinks observed for the feed tray (number 15) may also be observed in the results of Taylor and Krishna (p.409, 1993). The same holds for the
- 55 -
Results
Chapter 5
temperature profiles. In Fig. (5.2), the temperature of the liquid, the interface and the vapour has been plotted. 1.00 Composition (molar fraction)
0.90 0.80 0.70 0.60
X(1) X(2) X(3) X(4)
0.50 0.40 0.30 0.20 0.10 0.00 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.1. Liquid phase composition profiles in a 33-trays depropanizer column using a noneq. well-mixed model (ethane-1, propane-2, butane-3 and pentane-4).
390 380
Temperature (K)
370 360 350
Tliq Tint
340
Tvap
330 320 310 300 290 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.2. Temperature profiles in a 33-trays depropanizer column using a noneq. well-mixed model.
- 56 -
Results
Chapter 5
As it can be seen in the previous figure, there is almost no difference between the liquid and the interface temperature (with exception of the feed tray). Therefore, the resistance to heat transfer is concentrated in the vapour phase boundary layer, which is the only one that exhibits a temperature gradient. Concerning the internal molar flows, they can be seen in Fig. (5.3). The profile observed is just the expected when a sub-cooled feed stream is introduced in the column. 2500 2300
Molar flows (K)
2100 1900 1700 L V
1500 1300 1100 900 700 500 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.3. Liquid and vapour molar flow profiles in a 33-trays depropanizer column using a noneq. wellmixed model.
When nonequilibrium models are used, the mass and heat transfer flow rates across the interface are also relevant variables. They have been illustrated in Fig. (5.4-5). The behaviour observed in these profiles correlates perfectly with the changes observed in the temperature and composition profiles. Therefore, high values for the transfer flows at a certain stage indicate steep changes in the composition or temperature at that particular point.
- 57 -
Results
Chapter 5
Interface molar flows (mol/s)
150 100 50 Phi(1)
0
Phi(2) Phi(3) Phi(4)
-50 -100 -150 -200 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.4. Interface molar flow profiles in a 33-trays depropanizer column using a noneq. well-mixed model (ethane-1, propane-2, butane-3 and pentane-4).
0
Interface heat flow (kJ/s)
-200 -400 -600 -800 -1000 -1200 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.5. Interface heat flow profile in a 33-trays depropanizer column using a noneq. well-mixed model.
In order to be able to compare the results obtained with this nonequilibrium model and the ones that may be obtained with classical (equilibrium) models, the equivalent Murphree
- 58 -
Results
Chapter 5
separation efficiency for every component has been computed. This efficiency may be computed if a bubble point calculation is performed to the leaving liquid compositions. The results obtained are shown in Fig. (5.6).
Calculted Murphree Efficiencies (-)
1.00 0.90 0.80 0.70 0.60
Emv1 Emv2 Emv3 Emv4
0.50 0.40 0.30 0.20 0.10 0.00 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.6. Murphree efficiencies back-calculated for a 33-trays depropanizer column using a noneq. wellmixed model (ethane-1, propane-2, butane-3 and pentane-4).
According to the results displayed in the previous figure, it may be seen that separation efficiencies may lie out of the traditional bounds of zero and one. In fact, despite the average efficiency of the system for the mixed-flow model is around 0.61, local variations of these efficiencies dropped to a value of –3.8 for propane in the feed tray and +2.2 for ethane in the ninth tray. In addition, it should be noted that the efficiencies presented here are very similar to the ones presented by Taylor and Krishna (p. 411, 1993), with exception of two points: (1) the efficiency of propane in the feed tray has been predicted to be –3.8, while Taylor predicts a value also out of boundaries, but on the other side (i. e. a high positive value, out of the scale); and (2) the efficiency of butane in the tray 33 has been predicted to be slightly higher than one, while Taylor predicted a value smaller than zero. No apparent reason has been found for these discrepancies and, in contrast, the agreement of the other data seems to be excellent.
- 59 -
Results
Chapter 5
5.1.1 Effect of the thermodynamic model All the results presented in the previous section have been obtained with the thermodynamic package 2 (c.f. Table (4.6)). This thermodynamic package proved to be more robust and less likely to fail when initialising the column. Once the column has been converged in steady state, the set of thermodynamic properties may be changed easily and the model converged again. This is the procedure followed to converge the nonequilibrium model with the package 1 (c.f. Table (4.6)). The initial steady state solution obtained with these two thermodynamic models has been compared and the results are presented in this section. Only minor differences have been observed between both solutions. The thermodynamic package 1 predicts slightly lower efficiencies, with an average value obtained of 0.59. The way in which the thermodynamic models may affect the results is via the transfer coefficients and also via the physical properties evaluated throughout the model (mainly the enthalpy and the fugacity coefficients. In this case, the same thermodynamic model (Redlich-KwongSoave) has been used for the computation of the fugacity coefficients, but different models have been used for the enthalpies and the molar volumes (RKS in pack. 1 and Lee-KeslerPlocker in Pack. 2). All the other transport properties have been computed using the same models. The mass and heat transfer coefficients calculated using both packages are very similar for the rectification section, but package 1 calculates higher coefficients in the stripping section. These increased values should increase the efficiency of the separation, but in fact it is reduced. Therefore, the main source of the difference between both cases seems to be the computation of the enthalpies and the molar volumes used in the molar and energy balances and in the calculation of the holdups.
- 60 -
Results
Chapter 5
Avg. mass transfer coeff. (m^3/s)
12 10 8 6
Pack-2_kaV Pack-1_kaV
4 2 0 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.7. Average mass transfer coefficients for the vapour phase calculated for a 33-trays depropanizer
Heat transfer coeff. (kJ/(s*K))
column using a noneq. well-mixed model and two different thermodynamic packages.
600 500 400 300 Pack-2_haV Pack-1_haV
200 100 0 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler) Figure 5.8. Heat transfer coefficients for the vapour phase calculated for a 33-trays depropanizer column using a noneq. well-mixed model and two different thermodynamic packages.
The effects of the change in the thermodynamic models are noticeable only in the calculated separation efficiency for component 1 (ethane, see Fig. (5.9)). The calculated value is the - 61 -
Results
Chapter 5
same throughout the column (where the computed efficiency is close to 0.6) except for the five trays above the feed, where this efficiency exhibits a prominent nonlinear behaviour. In these five trays, package 1 underpredicts the value of the efficiency for these components. No other significant differences may be observed in the results for the other components. 2.50
Component 1 Efficiency (-)
2.25 2.00 1.75 1.50 Pack-2 Pack-1
1.25 1.00 0.75 0.50 0.25 0.00 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.9. Component 1 (ethane) Murphree efficiencies back-calculated for a 33-trays depropanizer column using a noneq. well-mixed model and two different thermodynamic packages.
5.2 Sivertsson well-mixed model In his previous year thesis, Sivertsson proposed an alternative method to solve the MaxwellStefan equations. Instead of using the analytical solution of Eq. (2.5) he claimed that a numerical solution might be used. First of all, he rearranged this expression, adimensionalising it and expressing it as a function of the mass transfer rate matrix: dy =Φ⋅y z d l
- 62 -
(5.1)
Results
Chapter 5
Then, Sivertsson proposed that this differential equation could be approximated using increments for the film, with a global truncation error of Ο(l 2 ) . Thus, he used the following expressions to compute the mass transfer flows:
(y
I
(x
L
−y
V
)= Φ
V
)
L
−x =Φ I
(y ⋅ (x ⋅
+y 2
I
V
L
+ xI 2
)
(5.2)
)
(5.3)
These two equations may be implemented in the nonequilibrium model, substituting Eqs. (3.14-17). In addition, instead of the matrices B L and BV , now the matrices Φ L and Φ V should be calculated. No major differences have been found between these two models, with exception of the efficiency computed for component 1 (ethane). As it can be observed in Fig. (5.10), the numerical approximation of the Maxwell-Stefan equations overpredicts the efficiency of this component in two of the trays, just above the feed. 2.50 Component 1 Efficiency (-)
2.25 2.00 1.75 1.50 Exact Sivert.
1.25 1.00 0.75 0.50 0.25 0.00 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.10. Component 1 (ethane) Murphree efficiencies back-calculated for a 33-trays depropanizer column using an analytical solution and Sivertsson’s numerical approximation.
- 63 -
Results
Chapter 5
5.3 Comparison with the equilibrium model The steady state initial solution obtained for the nonequilibrium analytical model has been compared with the solution obtained for an equilibrium model. Due to the fact that the equilibrium models can not calculate the separation efficiency, it needs to be specified externally. A fixed value of 0.6 has been specified in the equilibrium model for all the trays and components. The results obtained are very similar to the ones already shown for the nonequilibrium model. In the following figures (5.11-12), the liquid compositions for both models have been plotted, as well as the temperature profiles. 1.00 Composition (molar fraction)
0.90 0.80 0.70
EQ.X1
0.60
EQ.X2 EQ.X3 EQ.X4 NEQ11.X1 NEQ11.X2
0.50 0.40
NEQ11.X3 NEQ11.X4
0.30 0.20 0.10 0.00 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.11. Comparison of the liquid phase composition profiles in a 33-trays depropanizer column using a noneq. well-mixed model and an equilibrium one (ethane-1, propane-2, butane-3 and pentane-4).
- 64 -
Results
Chapter 5
390.00 380.00
Temperature (K)
370.00 360.00 350.00 EQ.T NEQ11.Tint
340.00 330.00 320.00 310.00 300.00 290.00 0
5
10
15
20
25
30
Tray (0:reflux drum, 34:reboiler)
Figure 5.12. Comparison of the temperature profiles in a 33-trays depropanizer column using a noneq. wellmixed model and an equilibrium one (ethane-1, propane-2, butane-3 and pentane-4).
5.4 Dynamic results Starting from the initial point presented in Section 5.1, dynamic simulations have been run for the 33-trays depropanizer column. The dynamic behaviour observed for the nonequilibrium model is the same observed for the equilibrium one. Only minor differences may be observed. In the following figures, a disturbance of +1% has been made to the feed flow after 6 minutes of simulation. The composition of butane in the bottom product has been plotted in Fig. (5.13). In this case, inverse response may be observed for both models, with no fundamental differences between them. A new steady state is reached after 10 hours with the same time constant.
- 65 -
Results
Chapter 5
0.8310 0.8310
0.8300
0.8305 0.8300
Reboiler.X(3) (-)
0.8295
0.8290
0.8290 0.8285 0.8280 0.8275
0.8280
NEQ
0.8270 0
0.5
1
1.5
2
EQ
0.8270 0.8260 0.8250 0.8240 0
2
4
6
8
10
Time (h)
Figure 5.13. Dynamic results. 1% increase en the feed flow. Butane composition in the reboiler of a 33trays depropanizer column. Comparison between noneq. and eq. models.
0.7450 0.7445 RefluxDrum.X(2) (-)
0.7440 0.7435 NEQ
0.7450
0.7430
0.7445
EQ
0.7440
0.7425
0.7435 0.7430 0.7425
0.7420
0.7420 0.7415 0.7410
0.7415
0.7405 0
0.5
1
1.5
2
0.7410 0.7405 0
2
4
6
Time (h)
8
10
Figure 5.14. Dynamic results. 1% increase en the feed flow. Propane composition in the reflux drum of a 33-trays depropanizer column. Comparison between noneq. and eq. models.
- 66 -
Results
Chapter 5
In Fig. (5.14), the composition of propane in the distillate product has been illustrated. Again, the dynamic response of both models is virtually the same, only differenced by the offset, which is inherited from the steady state solution. No further differences have been observed between both models for any of the other variables of the system, with the exception of the distillate molar flow (see Fig. (5.15)). It has been observed that, despite both models start with an initial outflow of 400 mol/s, an offset is generated during the simulation, probably due to a change in the efficiency of the separation in the nonequilibrium model. Nevertheless, this small difference observed is not enough to consider that there are inherent differences between both models. With the results presented so far, the use of modal analysis techniques to linearise the models has been considered to be useless. The rationale behind these techniques assumes that close to a certain point, the behaviour of a system may be reproduced using a linear state space model. Provided that the results obtained with the real model already reflect this situation, the linearised model could only confirm what is already evident.
RefluxDrum.Fout(2) (mol/s)
400.5 399.5 398.5
400.5
NEQ
400.0
EQ
399.5
397.5
399.0
398.5
396.5 398.0 0
0.5
1
1.5
2
395.5 0
2
4
6
8
10
Time (h)
Figure 5.15. Dynamic results. 1% increase en the feed flow. Distillate molar flow rate in a 33-trays depropanizer colum. Comparison between noneq. and eq. models..
- 67 -
Results
Chapter 5
On the other hand, a relevant issue that has not been confirmed with the results obtained is how far from an initial steady state would both models behave essentially equally. Once proved that close to the steady state point, both models behave very similarly, it could happen that this behaviour could be different if the magnitude of the disturbance is increased to some extent. In particular, it was observed that the nonequilibrium model could not converge a solution when the disturbances in the column were too high. For instance, a step change in the reboiler heat duty of a 5% could not be modelled with the nonequilibrium model. The integration failed shortly after the disturbance was applied. On the other hand, the same disturbance was applied in the equilibrium model, and it could be integrated and a new steady-state was reached. Finally, it should be considered that the comparison realised was accomplished for a fairly ideal system. Therefore, in a highly nonideal case, the behaviour of the different components could be more different, and this could affect the separation efficiency. Moreover, in order to study nonlinear systems, the thermodynamic factors matrix should be considered and included in the model equations.
5.5 Plug flow model The nonequilibrium model with the assumption that both phases have a plug flow concentration profile has also been initialised using the procedure described in Chapter 4. However, due to the fact that with the plug flow assumption, the separation efficiency may be increased considerably, the depropanizer column with 23 trays has been used in this case. The composition profile obtained for this case is very similar to the one obtained with the mixed-flow model, as it can be seen in Fig. (5.16).
- 68 -
Results
Chapter 5
1.00 Composition (molar fraction)
0.90 0.80 0.70 0.60
X1 X2 X3 X4
0.50 0.40 0.30 0.20 0.10 0.00 0
5
10
15
20
Tray (0:reflux drum, 34:reboiler)
Figure 5.16. Liquid phase compositions in a 23-trays depropanizer column using an averaged plug flow model (ethane-1, propane-2, butane-3 and pentane-4).
Despite the composition profile obtained is very similar to the one obtained for the mixed flow model, the computation of the Murphree efficiencies shows that the degree of separation is much higher in the plug flow model. These separations efficiencies have been plotted in Fig. (5.17). It may be observed that the mean separation efficiency has increased to a value of 1.0, approximately. The efficiency profiles shown in this figure may be compared with the efficiencies calculated by Kooijman and Taylor (1995a). They also obtained average efficiencies in the vicinity of 1.0, but in their results, some kinks may be observed in the feed tray (no. 10) which are not evident in the results presented here. In addition, the decaying trend in the efficiency of ethane as the bottom of the column is approached was calculated as an increasing tendency in the results of Kooijman and Taylor.
- 69 -
Results
Chapter 5
3.00
Murphree efficiency (-)
2.50 2.00 1.50
Emv1 Emv2 Emv3
1.00
Emv4
0.50 0.00 -0.50 -1.00 0
5
10
15
20
Tray (0:reflux drum, 34:reboiler)
Figure 5.17. Murphree efficiencies back-calculated for a 23-trays depropanizer column using an averaged plug flow model (ethane-1, propane-2, butane-3 and pentane-4).
Taking into account the mixture of components present in the depropanizer column and the operating conditions, the computed values of 1.0 for the efficiencies in this column are much more likely to occur than the results computed with the mixed flow model assumptions. Therefore, it seems that for this particular industrial column, the well-mixed assumptions are wrong and produce misleading results. On the other hand, the method used to implement the plug flow model brings an exact solution only at the initial point, due to the fact that several recursive iterations have been made to find the correct matrix used in the calculation of the average compositions. The assumption that this initially calculated correction matrix may be considered invariant during a dynamic simulation should depend on how far from that initial point the system is moved. It has been considered to be beyond the purpose of this project to evaluate quantitatively the suitability and/or the range of applicability of this assumption. In case it was found that this correction matrix varies considerably during the dynamic simulation, a method to update its value should be devised.
- 70 -
Results
Chapter 5
Apart from the fact that this matrix may vary over time due to a change in the system conditions, it should also been taken into account that this matrix was derived starting from the assumptions that a steady state differential equation could be applied over a slice of the froth and an average interface mass flow could be used. In case that these assumptions were not acceptable, an alternative solution of the problem would be applying a distributed model on every tray, and solving the unsteady state differential equation with a finite differences method. Assuming that the flow is bidimensional and considering that the molar flows across the tray are constant (at an average value), an unsteady differential balance in a differential volume may be derived: 1 ∂ ∂y L ∂x vi = − V ⋅ i − ⋅ i ∂t Atray ⋅ h f ∂z ′y Atray ⋅ h f ∂z ′x
(5.4)
where z ′x and z ′y are the adimensionalised coordinate directions. The molar density may be decomposed in two equations, as a function of the liquid and the vapour compositions: L M tot ∂( xi ) L ∂x ⋅ =− ⋅ i − φˆi L L V V M tot ⋅ v + M tot ⋅ v ∂t Atray ⋅ h f ∂z ′x
(5.5)
M Vtot ∂( yi ) V ∂y ⋅ =− ⋅ i + φˆi L L V V M tot ⋅ v + M tot ⋅ v ∂t Atray ⋅ h f ∂z ′y
(5.6)
where φˆi is the interface molar flow per unit volume. These two last equations may be implemented in gPROMS using the PARTIAL statement, and may be integrated numerically, in order to obtain a distributed profile of the composition of both phases. In order to accomplish this, the interface molar flow, the interface compositions, the temperature at the interface and also the interface heat flow should be defined also as distributed value. Suitable boundary conditions for the partial derivatives with respect to z should also be defined (the entering compositions on each tray). Thus, the number of variables that have to be defined to solve this distributed system for every tray is the same as the averaged plug flow model plus
(5 ⋅ nc + 2 ) ⋅ nzx ⋅ nzy , where
nzx ⋅ nzy is the size of the grid defined for every tray. For a
mesh defined with 5x5 elements on every tray, the resulting model for the depropanizer with - 71 -
Results
Chapter 5
23 trays is composed by 18,000 variables, resulting in a model that is still solvable, at least in the initialisation. However, the number of points necessary to solve this problem with a minimum precision has not been determined. This distributed system has been unsuccessfully implemented in gPROMS. A copy of the model is included in Appendix A.3. The model equations have been written, but the model could not be converged. A singularity was observed in the bottom left corner of the tray, where a boundary condition fixes the liquid composition and another one fixes the vapour composition, leading possibly to a nonsolvable high index problem. A possible solution to this problem would be to define a grid for the liquid and vapour compositions, and another one with vertices in the centres of the previous one to compute the cells interface flows. In any case, the correct implementation of a distributed plug flow was beyond the scope of this thesis, and will be left for future work.
5.6 High flux correction matrix assumption Another issue that has been revised is the effect of the high flux correction matrix that is obtained in the development of the analytical solution of the Maxwell-Stefan equations. These correction factors should be computed using Eq. (2.10) for the vapour phase (where it is computed at the end of the integration limits) and using Eq. (2.19) for liquid phase (where it is computed at the beginning of the integration limit. In addition, the thermodynamic factor matrix was dropped from the model, assuming that the behaviour of this phase will be ideal. Nevertheless, the fact that in both films the fluxes are computed in different boundaries of the integration implies that the expressions used have to be different. In the development of the model, these high flux correction matrices were dropped, assuming that in distillation columns the fluxes are always low enough to admit that assumption. However, once the model has been converged, it is possible to check the value of this correction matrix, and this assumption may be evaluated. This has been done for the depropanizer column with 23 trays, and under the assumption of plug flow for both phases. The value of this high flux correction matrix has been computed for every tray. The matrix operations needed have been calculated using Matlab. It has been found that in the liquid, this - 72 -
Results
Chapter 5
matrix has the biggest coefficients in the tray no. 2, while for the vapour the highest coefficients were found in the tray no. 20. In any case, the assumption that this matrix may be substituted in the model by the identity matrix is fully justified for this case.
L p =2
- 6.6E - 05 - 1.0E - 04 1.012 = 3.3E - 04 1.013 1.7E - 04 8.1E - 04 3.6E - 04 1.013
V p =20
9.6E - 05 4.3E - 05 1.005 = 5.6E - 03 1.009 1.5E - 03 - 4.4E - 03 - 2.6E - 03 1.005
Ξ
Ξ
- 73 -
Conclusions
Chapter 6
Chapter 6 6 Conclusions In the development of this M.Sc. thesis, the nonequilibrium model developed by Sivertsson (1998) was used as starting point. Several corrections had to be made, concerning the AIChE model for the calculation of the mass transfer coefficients. The heat transfer coefficients equations had also to be revised. In addition, instead of the numerical solution proposed by Sivertsson, the analytical solution of the Maxwell-Stefan equations proposed by Taylor was implemented in the model. Both solution methods were compared and no remarkable differences were observed between both solution methods for the benchmark problem used. However, the problem considered, a depropanizer column, is a fairly ideal system and it has not been tested for a nonideal system. In both cases, the tray was assumed to be perfectly mixed, with no gradients of composition throughout it. Dynamic results of the nonequilibrium model using the well-mixed assumption have been obtained and compared with an equilibrium model. No differences were observed for small disturbances, and it was observed that the nonequilibrium model could not be converged for increased disturbances. On the other hand, an averaged plug flow model was solved for a 23 trays depropanizer column, observing that the predicted efficiencies were higher that the predicted for the wellmixed model. In order to implement the plug flow model it is necessary to use Matlab to calculate the matrix functions that appear. Therefore, this method is not valid for dynamic simulations if the matrix computed changes during the simulation. As a result, it should be evaluated to which extent does this matrix change if the conditions of the system do. In case it had to be updated during time, a suitable method to accomplish this should be devised. One
- 74 -
Conclusions
Chapter 6
possibility would be to design a foreign object subroutine that could calculate exponentials and inverses of matrix, which could be called from gPROMS and the required operations were executed. Another suitable solution would be writing a foreign object subroutine that simply links gPROMS to Matlab, using it to compute the required calculations, and bringing back the result to gPROMS. Notwithstanding this, it should also be noted that the averaged plug flow presented was obtained under the assumption that the differential across the froth on the tray was at steadystate. In case this assumption was considered not to be valid, it would be necessary to include the differential unsteady balance in the model, and it should be solved calculating the distributed profiles of composition on the trays. These calculations may be accomplished using the superior modelling capabilities of gPROMS, but the resulting model could not be implemented, and it is left for future work.
- 75 -
Conclusions
References
References
Agarwai, S. and Taylor, R., (1994). Distillation Column Design Using a Nonequilibrium Model, Ind. Eng. Chem. Res., 33, pp. 2631-2636. Amundson, N.R., (1966). Mathematical Methods in Chemical Engineering. Matrices and their Applications. Prentice Hall. Biardi, G. and Grottoli, M.G., (1989). Development of a New Model for Real Trays Distillation Column, Comput. Chem. Eng., 13(4/5), pp. 441-449. Bird, R.B., Stewart, W.E. and Lightfoot, E.N., (1960). Transport Phenomena. John Wiley and Sons. Cameron, I.T. et al., (1986). A Generalized Model for Distillation Columns-II, Comput. Chem. Eng., 10(3), pp. 199-211. Choe, Y.S. and Luyben, W.L., (1987). Rigorous Dynamic Models of Distillation Column, Ind. Eng. Chem. Res., 26, pp. 2158-2161. Gani, R. et al., (1986). A Generalized Model for Distillation Columns-I, Comput. Chem. Eng., 10(3), pp. 181-198. gPROMS (1997). The gPROMS Model Library. Process Systems Enterprise Ltd. gPROMS (1998). Advanced User’s Guide. Process Systems Enterprise Ltd. Henley, E.J. and Seader, J.D., (1981). Equilibrium-Stage Separation Operations in Chemical Engineering. John Wiley and Sons.
- 76 -
References
Holland, C.D. and Liapis, A.I., (1983). Computer Methods for Solving Dynamic Separation Problems Multicomponent Distillation. McGraw-Hill. Holland, C.D., (1963). Multicomponent Distillation. Prentice Hall. King, C.J., (1980). Separation Processes. 2nd edition, McGraw-Hill. Kister, H.Z., (1989). Distillation Operation. McGraw-Hill. Kister, H.Z., (1992). Distillation Design. McGraw-Hill. Kooijman, H.A. and Taylor, R., (1995a). Modelling Mass Transfer in Multicomponent Distillation, Chem. Eng. J., 57(2), pp. 177-188. Kooijman, H.A. and Taylor, R., (1995b). A Nonequilibrium Model for Dynamic Simulation of Tray Distillation Columns, AIChE J., 41(8), pp. 1852-1863. Kooijman, H.A. and Taylor, R., (1998). The ChemSep Book. Copyright 1998, www.clarkson.edu/~chengweb/faculty/taylor/chemsep/chemsep.html. Krishna, R. and Standart, G.L., (1976). A Multicomponent Film Model Incorporating a General Matrix Method of Solution to the Maxwell-Stefan Equations, AIChE J., 22(2), pp. 383-389. Krishnamurty R. and Taylor, R., (1985a). A Nonequilibrium Stage Model of Multicomponent Separation Processes-Part I, AIChEJ, 31(3), pp. 449-456. Krishnamurty R. and Taylor, R., (1985b). A Nonequilibrium Stage Model of Multicomponent Separation Processes-Part II, AIChEJ, 31(3), pp. 446-465. Krishnamurty R. and Taylor, R., (1985c). A Nonequilibrium Stage Model of Multicomponent Separation Processes-Part III, AIChEJ, 31(12), pp. 1973-1985. Multiflash (1997). Multiflash Command Reference. Infochem Computer Services Ltd. Powers, M.F. et al., (1988). A Nonequilibrium Stage Model of Multicomponent Separation Processes-Part V, Comput. Chem. Eng., 12(12), pp. 1229-1241. Reid, R.C. et al., (1987). The Properties of Gases and Liquids. 4th edition, McGraw-Hill.
- 77 -
References
Seader, J.D., (1989). The Rate-based Approach for Modeling Staged Separations, Chem. Eng. Prog., October 1989, pp. 41-49. Sivertsson, P., (1998). Dynamic Modelling of Plate Distillation Columns Based on Nonequilibrium Models. M. Sc. Thesis, Chalmers University of Technology, Skogestad, S. and Morari, M., (1988). Understanding the Dynamic Behaviour of Distillation Columns, Ind. Eng. Chem. Res., 27, pp. 1848-1862. Taylor, R. and Krishna, R., (1993). Multicomponent Mass Transfer. John Wiley and Sons. Taylor, R. et al., (1994). A Second Generation Nonequilibrium, Model for Computer Simulation of Multicomponent Separation Processes, Comput. Chem. Eng, 18(3), pp. 205-217. Walas, S.M., (1985). Phase Equilibria in Chemical Engineering. Butterworth Publishers. Zuiderwerg, F.J., (1982). Sieve Trays: aview on the state of the art, Chem. Eng. Sci., 37(10), pp. 1441-1464.
- 78 -
Appendix A
Appendix A A.1 Thermodynamic package #2 (depropanizer) #The mfl-file containing all the information about the models and databanks #that needs to be supplied to Multiflash #File name: deprop2.mfl #as written by Sivertsson remove all ; # Definition of data bank for pure component properties #dippr corresponds to a data compilation from AIChE puredata dippr ; # Definition of components components ethane propane n-butane n-pentane ; # Definition of binary interaction parameters data #oilandgas provides BIP values for components typical of oil and gas mixtures #for cubic eos bipdata oilandgas ; # Definition of Thermodynamic and Transport models #Lee-Kesler-Plocker eos (accurate predictions of enthalpy and density) model meoslkp lkp lkp ; #Peng-Robinson eos model meospr pr ; #Redlich-Kwong-Soave eos model meosrks rks rks ; #UNIFAC liquid activity model using Peng-Robinson eos model munifac unifac vle meospr #Pedersen viscosity model (suitable for liquids and gases) recommended for #oil and gas applications model mviscpdv pdv pdv ; #Lohrenz-Bray-Clark viscosity model (suitable for liquids and gases) #recommended for oil and gas applications
- 79 -
Appendix A
model mvisclbc lbc lbc meospr; #Liquid Mixing rule viscosity model model mvisclmr lvs1 #Vapour Mixing rule viscosity model model mviscvmr vvs1 #Chung-Lee-Starling thermal conductivity model model mcondcls cls meosrks ; #Macleod-Sudgen surface tension method model mtensmcs mcs meosrks ; #phase descriptor # Fugacity Volume Enthalpy Viscosity Conduct SurfTens pd pdliq liquid meosrks meoslkp meoslkp mvisclbc mcondcls mtensmcs ; pd pdvap vapour meosrks meoslkp meoslkp mvisclbc mcondcls mtensmcs ; #---------------------------------------------------------------------------
A.2 Nonequilibrium well-mixed model (depropanizer 33 trays) #File name case_1_dyn.gPROMS #gPROMS input file for a base case column #This is an initialisation file for a nonequilibrium tray model #nonequilibrium model using analytical soution #well-mixed flow assumption #Depropanizer case Components C2-C3-C4-C5 # Pressure 15 bar # Number of Trays 33 # #Property package
Feed Location tray 15. Multiflash configured in file #deprop2
DECLARE TYPE Fraction MolarAmount MolarRate MolarRatePhi EnergyAmount EnergyRate MolarEnergy Temperature Tempnbp Pressure NoType Length
= = = = = = = = = = = =
0.25 1.0 0.1 0.1 -1E2 1.0 -10 330 330 1519 1.0 1.0
: : : : : : : : : : : :
0.0 -1E-3 -1E-5 -1E10 -1E20 -1E10 -1E10 100 100 1000 -1E20 -1E-3
: : : : : : : : : : : :
1.0 1E10 1E5 1E10 1E20 1E10 1E10 450 800 2000 1E20 1E2
STREAM
- 80 -
UNIT UNIT UNIT UNIT UNIT UNIT UNIT UNIT UNIT
= = = = = = = = =
"mol" "mol/s" "mol/s" "kJ" "kJ/s" "kJ/mol" "K" "K" "kPa"
UNIT = "m"
Appendix A
MainStream PStream
IS IS
MolarRate,Fraction,Pressure,Temperature,MolarEnergy Pressure
END #Declare #------------------------------------------------------------------------------Model reboiler_lib # model library with modifications PARAMETER NoComp Vvessel Area WHeight WLength Beta PhysProp1 Ffeed Zfeed Pfeed Tfeed VARIABLE M X T P H MTOT UTOT Fin Zin Pin Tin Hin Fout Hfeed Level QReb tfactor
AS AS AS AS AS AS AS AS AS AS AS AS AS AS AS AS AS
AS INTEGER AS REAL AS REAL AS REAL AS REAL AS REAL AS FOREIGN_OBJECT "Multiflash" AS REAL AS ARRAY(NoComp) OF REAL AS REAL AS REAL
ARRAY(NoComp) OF ARRAY(2,NoComp) OF
ARRAY(2) ARRAY(2)
OF OF
ARRAY(NoComp)
OF
ARRAY(2)
OF
MolarAmount Fraction Temperature Pressure MolarEnergy MolarAmount EnergyAmount MolarRate Fraction Pressure Temperature MolarEnergy MolarRate MolarEnergy Length EnergyRate NoType
STREAM LiquidInlet : Fin, Zin, Pin, Tin, Hin LiquidOutlet : Fout(1), X(1,), P, T, H(1) VapourOutlet : Fout(2), X(2,), P, T, H(2)
AS MainStream AS MainStream AS MainStream
EQUATION #---------------------Balances------------------------------#Component Molar Balances in Phase 1 and 2 FOR i:= 1 TO NoComp DO $M(i)=Ffeed*Zfeed(i)+Fin*Zin(i)-SIGMA(Fout*X(,i)) ; END #Heat Balance in the Mixer
- 81 -
Appendix A
$UTOT=Ffeed*Hfeed+Fin*Hin-SIGMA(Fout*H)+QReb ; #---------------------Definitions---------------------------SIGMA(X(1,))-1.0=0.0 ; SIGMA(X(2,))-1.0=0.0 ; FOR i:= 1 TO NoComp DO M(i)-SIGMA(X(,i)*MTOT)=0.0 ; END UTOT-SIGMA(MTOT*H)+P*Vvessel=0.0 ; #---------------------Hydrodynamics-------------------------IF Level < Beta*WHeight THEN Fout(1)=0.0 ; ELSE Fout(1)-1.84*WLength*(ABS((Level-Beta*WHeight)/Beta))^1.5/ PhysProp1.LiquidVolume(T,(P*1000),X(1,))=0.0 ; END #---------------------Geometry------------------------------MTOT(1)*PhysProp1.LiquidVolume(T,(P*1000),X(1,))+ MTOT(2)*PhysProp1.VapourVolume(T,(P*1000),X(2,))-Vvessel=0.0 ; Level-MTOT(1)*PhysProp1.LiquidVolume(T,(P*1000),X(1,))/Area=0.0 ; #---------------------PhysProp------------------------------X(2,)*PhysProp1.VapourFugacityCoefficients(T,(P*1000),X(2,))X(1,)*PhysProp1.LiquidFugacityCoefficients(T,(P*1000),X(1,))=0.0 ; H(1)=(PhysProp1.LiquidEnthalpy(T,(P*1000),X(1,))/1000) ; H(2)=(PhysProp1.VapourEnthalpy(T,(P*1000),X(2,))/1000) ; Hfeed=(PhysProp1.LiquidEnthalpy(Tfeed,(Pfeed*1000),Zfeed)/1000) ; END #model reboiler_lib #--------------------------------------------------------------------------MODEL noneq_tray PARAMETER NoComp Vtray WHeight WLength Zl PlateArea HolesArea Cv Beta g PhysProp1 Ffeed Zfeed Pfeed
AS AS AS AS AS AS AS AS AS AS AS AS AS ARRAY(NoComp) AS
INTEGER REAL REAL REAL REAL REAL REAL REAL REAL REAL DEFAULT 9.81 FOREIGN_OBJECT "Multiflash" REAL OF REAL REAL
- 82 -
Appendix A
Tfeed Qtray DiffVol MW Lambda Xp
AS AS AS AS AS AS
VARIABLE Mtotphase AS Mtotcomp AS Mcomp AS X AS T AS Xint AS Tint AS Uoverall AS Utot AS Fin AS Zin AS Pin AS Tin AS Hin AS Fout AS P AS H AS Hfeed AS Level AS B AS phi AS phiheat AS Favg AS ka AS D AS Dinf AS ha AS Tnbp AS Fs AS Lfpw AS tL AS Zc AS kaavg AS Davg AS Alpha AS tfactor AS #to compute murphree Emv AS ymv AS Tmv AS
STREAM LiquidInlet VapourInlet LiquidOutlet VapourOutlet
: : : :
ARRAY(2) ARRAY(NoComp) ARRAY(NoComp) ARRAY(2) ARRAY(NoComp,NoComp)
OF OF OF OF OF
ARRAY(2) ARRAY(Nocomp) ARRAY(2,Nocomp) ARRAY(2,NoComp) ARRAY(2) ARRAY(2,NoComp)
REAL REAL REAL REAL REAL REAL
OF OF OF OF OF OF
MolarAmount MolarAmount MolarAmount Fraction Temperature Fraction Temperature EnergyAmount ARRAY(2) OF EnergyAmount ARRAY(2) OF MolarRate ARRAY(2,NoComp) OF Fraction ARRAY(2) OF Pressure ARRAY(2) OF Temperature ARRAY(2) OF MolarEnergy ARRAY(2) OF MolarRate Pressure ARRAY(2) OF MolarEnergy MolarEnergy Length ARRAY(2,(Nocomp-1),(Nocomp-1)) OF Notype ARRAY(NoComp) OF MolarRatePhi EnergyRate ARRAY(2) OF MolarRate ARRAY(2,NoComp,NoComp) OF NoType ARRAY(2,NoComp,NoComp) OF NoType ARRAY(NoComp,NoComp) OF NoType ARRAY(2) OF NoType ARRAY(NoComp) OF Tempnbp NoType NoType NoType NoType ARRAY(2) OF NoType ARRAY(2) OF NoType NoType NoType efficiencies ARRAY(NoComp) OF NoType ARRAY(NoComp) OF Fraction Temperature
Fin(1), Zin(1,), Pin(1), Fin(2), Zin(2,), Pin(2), Fout(1), X(1,), P, T(1), Fout(2), X(2,), P, T(2),
- 83 -
Tin(1), Hin(1) AS MainStream Tin(2), Hin(2) AS MainStream H(1) AS MainStream H(2) AS MainStream
Appendix A
RefForPress
: P
AS PStream
EQUATION #---------------------------Balances------------------------------# Comp. Molar Balance in the Liquid and Comp. Overall Molar Balance FOR i := 1 TO NoComp DO $Mcomp(1,i)=-phi(i)+Fin(1)*Zin(1,i)-Fout(1)*X(1,i)+ (1+0.05*tfactor)*Ffeed*Zfeed(i) ; $Mtotcomp(i)=SIGMA(Fin*Zin(,i))-SIGMA(Fout*X(,i))+ (1+0.05*tfactor)*Ffeed*Zfeed(i) ; $Mcomp(1,i)=-phi(i)+Fin(1)*Zin(1,i)-Fout(1)*X(1,i)+Ffeed*Zfeed(i) ; $Mtotcomp(i)=SIGMA(Fin*Zin(,i))-SIGMA(Fout*X(,i))+Ffeed*Zfeed(i) ;
# # # #
END
# # # #
# Heat Balance in the Liquid and Overall Heat Balance $Utot(1)=-phiHeat+Fin(1)*Hin(1)-Fout(1)*H(1)+ (1+0.05*tfactor)*Ffeed*Hfeed+Qtray(1) ; $Uoverall=SIGMA(Fin*Hin)-SIGMA(Fout*H)+ (1+0.05*tfactor)*Ffeed*Hfeed+SIGMA(Qtray) ; $Utot(1)=-phiHeat+Fin(1)*Hin(1)-Fout(1)*H(1)+Ffeed*Hfeed+Qtray(1) ; $Uoverall=SIGMA(Fin*Hin)-SIGMA(Fout*H)+Ffeed*Hfeed+SIGMA(Qtray) ;
#------------------------Definitions------------------------------# Capacity relations SIGMA(X(1,))-1.0=0.0 ; SIGMA(X(2,))-1.0=0.0 ; FOR i:= 1 TO NoComp DO Mtotcomp(i)-SIGMA(X(,i)*Mtotphase)=0.0 ; Mcomp(1,i)-X(1,i)*Mtotphase(1)=0.0 ; Mcomp(2,i)-X(2,i)*Mtotphase(2)=0.0 ; END Uoverall-SIGMA(Mtotphase*H)+P*Vtray=0.0 ; Utot(1)-Mtotphase(1)*H(1)+ P*Mtotphase(1)*PhysProp1.LiquidVolume(T(1),(P*1000),X(1,))=0.0 ; Uoverall-SIGMA(Utot)=0.0 ; #--------------------Interphase Transfer-----------------------#Mass Transfer FOR i:=1 TO (NoComp-1) DO SIGMA(B(1,i,)*(Phi(1:(NoComp-1))-X(1,1:(NoComp-1))*SIGMA(Phi)))(1/PhysProp1.LiquidVolume(T(1),(P*1000),X(1,)))*(X(1,i)-Xint(1,i)) =0.0 ; #[(mol/s)*((m/s)*m2)] SIGMA(B(2,i,)*(Phi(1:(NoComp-1))-X(2,1:(NoComp-1))*SIGMA(Phi)))(1/PhysProp1.VapourVolume(T(2),(P*1000),X(2,)))*(Xint(2,i)-X(2,i)) =0.0 ; #[(mol/s)*((m/s)*m2)] END SIGMA(Xint(1,))-1.0=0.0 ; SIGMA(Xint(2,))-1.0=0.0 ; FOR i:=1 TO 2 DO
- 84 -
Appendix A
FOR j:=1 TO (NoComp-1) DO FOR k:= 1 TO (NoComp-1) DO IF j k THEN B(i,j,k)+X(i,j)*((1/ka(i,j,k))-(1/ka(i,j,NoComp))) =0.0 ; #[((m/s)*m2)^(-1)] ELSE B(i,j,k)-(X(i,j)/ka(i,j,NoComp))(SIGMA(X(i,)/ka(i,j,))-X(i,j)) =0.0 ; #[((m/s)*m2)^(-1)] END END END END #Liquid Diffusivities FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i = j THEN Dinf(i,j)=0.0 ; ELSE #outstanding issue 1 Dinf(i,j)*(PhysProp1.LiquidVolume(Tnbp(i),101325.0,Xp(,i)))^0.433* PhysProp1.LiquidViscosity(T(1),(P*1000),X(1,)) -8.93E-16*(PhysProp1.LiquidVolume(Tnbp(j),101325.0,Xp(,j)))^0.267* T(1)* ((PhysProp1.SurfaceTension(Tnbp(j),101325.0,Xp(,j),Xp(,j)))/ (PhysProp1.SurfaceTension(Tnbp(i),101325.0,Xp(,i),Xp(,i))) )^0.15 =0.0 ; #[m2/s] END END END FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i < j THEN D(1,i,j)-Dinf(i,j)^((1+X(1,j)-X(1,i))/2)* Dinf(j,i)^((1+X(1,i)-X(1,j))/2)=0.0 ; #[m2/s] ELSE IF i > j THEN D(1,i,j)-D(1,j,i)=0.0 ; #[m2/s] ELSE D(1,i,j)=0.0 ; #[m2/s] END END END END Davg(1)-(SIGMA(D(1,,)))/(NoComp^2-NoComp)=0.0 ; #[m2/s] #Vapour Diffusivities FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i < j THEN
- 85 -
Appendix A
D(2,i,j)-0.010111627*(T(2)^1.75)*SQRT((1/MW(i)+1/MW(j))) /(P*1000*(DiffVol(i)^(1/3)+DiffVol(j)^(1/3))^2)=0.0 ; #[m2/s] ELSE IF i > j THEN D(2,i,j)-D(2,j,i)=0.0 ; #[m2/s] ELSE D(2,i,j)=0.0 ; #[m2/s] END END END END Davg(2)-(SIGMA(D(2,,)))/(NoComp^2-NoComp)=0.0 ; #[m2/s] #Factors Fs-(Favg(2)*(PhysProp1.VapourVolume(T(2),(P*1000),X(2,)))/PlateArea) *SQRT(PhysProp1.VapourDensity(T(2),(P*1000),X(2,)))=0.0 ; Lfpw-(Favg(1)*(PhysProp1.LiquidVolume(T(1),(P*1000),X(1,)))/WLength) =0.0 ; tL-Zc*Zl/Lfpw=0.0 ; Zc-0.0254*(1.65+7.4803*WHeight+96.6236*Lfpw-0.5328*Fs)=0.0 ; #Liquid Mass Transfer Coefficients FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i < j THEN ka(1,i,j)-20314.732*SQRT(D(1,i,j))*(0.21313*Fs+0.15)*tL* (Favg(1)*PhysProp1.LiquidVolume(T(1),(P*1000),X(1,)))=0.0 ; #[(m/s)*m2] ELSE IF i > j THEN ka(1,i,j)-ka(1,j,i)=0.0 ; #[(m/s)*m2] ELSE ka(1,i,j)-1.0=0.0 ; #[(m/s)*m2] END END END END kaavg(1)-(SIGMA(ka(1,,))-NoComp)/(NoComp^2-NoComp)=0.0 ; #[(m/s)*m2] #Vapour Mass Transfer Coefficients FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i < j THEN ka(2,i,j)-((0.776+4.567*WHeight-0.2377*Fs+104.8366*Lfpw)/ (SQRT(PhysProp1.VapourViscosity(T(2),(P*1000),X(2,))/ (PhysProp1.VapourDensity(T(2),(P*1000),X(2,))*D(2,i,j)))))* (Favg(2)*PhysProp1.VapourVolume(T(2),(P*1000),X(2,)))=0.0 ; #[(m/s)*m2] ELSE IF i> j THEN
- 86 -
Appendix A
ka(2,i,j)-ka(2,j,i)=0.0 ; #[(m/s)*m2] ELSE ka(2,i,j)-1.0=0.0 ; #[(m/s)*m2] END END END END kaavg(2)-(SIGMA(ka(2,,))-NoComp)/(NoComp^2-NoComp)=0.0 ; #[(m/s)*m2] #Heat Transfer # outstanding issue 2 phiheat-ha(1)*(T(1)-Tint)SIGMA(phi)*(PhysProp1.LiquidEnthalpy(T(1),(P*1000),X(1,))/1000)=0.0 ; phiheat-ha(2)*(Tint-T(2))SIGMA(phi)*(PhysProp1.VapourEnthalpy(T(2),(P*1000),X(2,))/1000)=0.0 ; #Heat Transfer Coefficients ha(1)-(kaavg(1)/PhysProp1.LiquidVolume(T(1),(P*1000),X(1,)))* (PhysProp1.LiquidHeatCapacity(T(1),(P*1000),X(1,))/1000)* SQRT(ABS( ((PhysProp1.LiquidConductivity(T(1),(P*1000),X(1,))/1000)* PhysProp1.LiquidVolume(T(1),(P*1000),X(1,)) )/ (Davg(1)*(PhysProp1.LiquidHeatCapacity(T(1),(P*1000),X(1,))/1000)) )) =0.0 ; #[(kJ/(s*m2*K))*m2] ha(2)-(kaavg(2)/PhysProp1.VapourVolume(T(2),(P*1000),X(2,)))* (PhysProp1.VapourHeatCapacity(T(2),(P*1000),X(2,))/1000)* (ABS(((PhysProp1.VapourConductivity(T(2),(P*1000),X(2,))/1000)* PhysProp1.VapourVolume(T(2),(P*1000),X(2,)))/(Davg(2) *(PhysProp1.VapourHeatCapacity(T(2),(P*1000),X(2,))/1000)) ) )^(2/3) =0.0 ; #[(kJ/(s*m2*K))*m2] #----------------------Hydrodynamics--------------------------IF Level < Beta*WHeight THEN Fout(1)=0.0 ; ELSE Fout(1)-1.84*WLength*ABS(((Level-Beta*WHeight)/Beta))^1.5/ PhysProp1.LiquidVolume(T(1),(P*1000),X(1,))=0.0 ; END IF ((Pin(2)-P)*1000g*PhysProp1.LiquidDensity(T(1),(P*1000),X(1,))*Level ) > 0.0 THEN (Pin(2)-P)*1000( (alpha*PhysProp1.VapourDensity(T(2),(P*1000),X(2,))* (Fin(2)*PhysProp1.VapourVolume(T(2),(P*1000),X(2,))/HolesArea)^2) +g*PhysProp1.LiquidDensity(T(1),(P*1000),X(1,))*Level ) =0.0 ; ELSE Fin(2)=0.0 ; END alpha=0.05*g/(Cv^2) ; #---------------------Geometry------------------------------Mtotphase(1)*PhysProp1.LiquidVolume(T(1),(P*1000),X(1,))+ Mtotphase(2)*PhysProp1.VapourVolume(T(2),(P*1000),X(2,))-Vtray=0.0 ; Level-Mtotphase(1)*PhysProp1.LiquidVolume(T(1),(P*1000),X(1,))/
- 87 -
Appendix A
PlateArea=0.0 ; #---------------------PhysProp------------------------------Xint(2,)*PhysProp1.VapourFugacityCoefficients(Tint,(P*1000),Xint(2,))Xint(1,)*PhysProp1.LiquidFugacityCoefficients(Tint,(P*1000),Xint(1,)) =0.0 ; H(1)=(PhysProp1.LiquidEnthalpy(T(1),(P*1000),X(1,))/1000) ; H(2)=(PhysProp1.VapourEnthalpy(T(2),(P*1000),X(2,))/1000) ; Hfeed=(PhysProp1.LiquidEnthalpy(Tfeed,(Pfeed*1000),Zfeed)/1000) ; Tnbp=PhysProp1.BoilingPoint ; #[K] Favg(1)=(Fin(1)+Fout(1))/2 ; Favg(2)=(Fin(2)+Fout(2))/2 ; #Computation of Murphree Efficiencies SIGMA(ymv) =1.0 ; ymv*PhysProp1.VapourFugacityCoefficients(Tmv,(P*1000),ymv)X(1,)*PhysProp1.LiquidFugacityCoefficients(Tmv,(P*1000),X(1,)) =0.0 ; Emv-(X(2,)-Zin(2,))/(ymv-Zin(2,)) =0.0 ; END #MODEL SIMPLE_NONEQ_TRAY #--------------------------------------------------------------------------Model simple_condenser #based in Sivertsson model, with modifications PARAMETER NoComp Cv PhysProp1
AS AS AS
INTEGER REAL FOREIGN_OBJECT "Multiflash"
VARIABLE #added to take into acount the differences between vapour leaving the first tray # and vapour arriving to the condenser. P AS Pressure H AS MolarEnergy # Fin AS MolarRate Xin AS ARRAY(NoComp) OF Fraction Pin AS Pressure Tin AS Temperature Hin AS MolarEnergy Fout AS MolarRate Xout AS ARRAY(NoComp) OF Fraction Pout AS Pressure Tout AS Temperature Hout AS MolarEnergy Y AS ARRAY(NoComp) OF Fraction Qcond AS EnergyRate tfactor AS NoType STREAM
- 88 -
Appendix A
VapourInlet : Fin, Xin, P, Tin, H LiquidOutlet : Fout, Xout, Pout, Tout, Hout
AS MainStream AS MainStream
EQUATION #---------------------Definitions---------------------------FOR i:= 1 TO NoComp DO Xout(i)-Xin(i)=0.0 ; END Fout-Fin=0.0 ; #Heat Balance Fin*Hin-Fout*Hout-QCond=0.0 ; Pout-Pin=0.0 ; (Fin)^2-(Cv^2)*(P-Pin)*1000=0.0 ; Fin=1400 ;
#
#--------------------BubblePoint Calculations---------------SIGMA(Y)-1.0=0.0 ; Y*PhysProp1.VapourFugacityCoefficients(Tout,(Pout*1000),Y)Xout*PhysProp1.LiquidFugacityCoefficients(Tout,(Pout*1000),Xout) =0.0 ; #---------------------PhysProp------------------------------Hout=(PhysProp1.LiquidEnthalpy(Tout,(Pout*1000),Xout)/1000) ; Hin=(PhysProp1.VapourEnthalpy(Tin,(Pin*1000),Xin)/1000) ; END #model simple_condenser
#--------------------------------------------------------------------------Model simple_reflux_drum #based in Sivertsson model, with modifications PARAMETER NoComp Vvessel Area RefluxRatio WHeight WLength Beta PhysProp1 Ffeed Zfeed Pfeed Tfeed QRefluxDrum
VARIABLE M X P
AS INTEGER AS REAL AS REAL AS REAL AS REAL AS REAL AS REAL AS FOREIGN_OBJECT "Multiflash" AS REAL AS ARRAY(NoComp) OF REAL AS REAL AS REAL AS REAL
AS ARRAY(NoComp) OF MolarAmount AS ARRAY(2,NoComp) OF Fraction AS Pressure
- 89 -
Appendix A
T H MTOT UTOT Fin Zin Pin Tin Hin Fout Hfeed Level tfactor
AS AS AS AS AS AS AS AS AS AS AS AS AS
ARRAY(2) ARRAY(2)
ARRAY(NoComp)
ARRAY(2)
Temperature OF MolarEnergy OF MolarAmount EnergyAmount MolarRate OF Fraction Pressure Temperature MolarEnergy OF MolarRate MolarEnergy Length NoType
STREAM LiquidInlet : Fin, Zin, Pin, Tin, Hin Reflux : Fout(1), X(1,), P, T, H(1) Distillate : Fout(2), X(1,), P, T, H(1)
AS MainStream AS MainStream AS MainStream
EQUATION #---------------------Balances------------------------------#Component Molar Balances in Phase 1 and 2 FOR i:= 1 TO NoComp DO $M(i)=Ffeed*Zfeed(i)+Fin*Zin(i)-SIGMA(Fout)*X(1,i) ; END #Heat Balance $UTOT=Ffeed*Hfeed+Fin*Hin-SIGMA(Fout)*H(1)-QRefluxDrum ; #---------------------Definitions---------------------------SIGMA(X(1,))-1.0=0.0 ; SIGMA(X(2,))-1.0=0.0 ; FOR i:= 1 TO NoComp DO M(i)-SIGMA(X(,i)*MTOT)=0.0 ; END UTOT-SIGMA(MTOT*H)+P*Vvessel=0.0 ; Fout(2)*RefluxRatio-Fout(1)=0.0 ; #---------------------Hydrodynamics-------------------------IF Level < Beta*WHeight THEN SIGMA(Fout)=0.0 ; ELSE SIGMA(Fout)-1.84*WLength*(ABS((Level-Beta*WHeight)/Beta))^1.5/ PhysProp1.LiquidVolume(T,(P*1000),X(1,))=0.0 ; END #---------------------Geometry------------------------------MTOT(1)*PhysProp1.LiquidVolume(T,(P*1000),X(1,))+ MTOT(2)*PhysProp1.VapourVolume(T,(P*1000),X(2,))-Vvessel=0.0 ; Level-MTOT(1)*PhysProp1.LiquidVolume(T,(P*1000),X(1,))/Area=0.0 ; #---------------------PhysProp-------------------------------
- 90 -
Appendix A
X(2,)*PhysProp1.VapourFugacityCoefficients(T,(P*1000),X(2,))X(1,)*PhysProp1.LiquidFugacityCoefficients(T,(P*1000),X(1,))=0.0 ; H(1)=(PhysProp1.LiquidEnthalpy(T,(P*1000),X(1,))/1000) ; H(2)=(PhysProp1.VapourEnthalpy(T,(P*1000),X(2,))/1000) ; Hfeed=(PhysProp1.liquidenthalpy(Tfeed,(Pfeed*1000),Zfeed)/1000) ; END #model simple_reflux_drum #--------------------------------------------------------------------------Model simple_pump #based in Sivertsson model, with modifications PARAMETER NoComp PhysProp1 VARIABLE Fin Zin Pin Tin Hin Fout Zout Tout Pout Hout P Wpump tfactor
AS AS
AS AS ARRAY(NoComp) AS AS AS AS AS ARRAY(NoComp) AS AS AS AS AS AS
Integer FOREIGN_OBJECT "Multiflash"
OF
OF
MolarRate Fraction Pressure Temperature MolarEnergy MolarRate Fraction Temperature Pressure MolarEnergy Pressure EnergyRate NoType
STREAM LiquidInlet : Fin, Zin, Pin, Tin, Hin AS MainStream LiquidOutlet : Fout, Zout, Pout, Tout, Hout AS MainStream RefForPress : P AS PStream EQUATION #---------------------Balances------------------------------Fout-Fin=0.0 ; Hout-Hin=0.0 ; Zout-Zin=0.0 ; Pout-P=0.0 ; Wpump-(Pout-Pin)*Fout* PhysProp1.LiquidVolume((Tout+Tin)/2,(((Pout+Pin)/2)*1000),Zout) =0.0 ; #---------------------PhysProp------------------------------Hout=(PhysProp1.LiquidEnthalpy(Tout,(Pout*1000),Zout)/1000) ; END #model simple_pump
- 91 -
Appendix A
#--------------------------------------------------------------------------Model column PARAMETER NoComp NoTrays PhysProp1 VARIABLE Fout Xout Pout Tout Hout tfactor
AS AS AS AS AS AS
AS AS AS
INTEGER INTEGER FOREIGN_OBJECT "Multiflash"
ARRAY(2) OF MolarRate ARRAY(2,NoComp) OF Fraction ARRAY(2) OF Pressure ARRAY(2) OF Temperature ARRAY(2) OF MolarEnergy NoType
UNIT Tray Condenser RefluxDrum Reboiler Pump
AS ARRAY(NoTrays) AS AS AS AS
STREAM Distillate Bottoms
: Fout(2), Xout(2,), Pout(2), Tout(2), Hout(2) AS MainStream : Fout(1), Xout(1,), Pout(1), Tout(1), Hout(1) AS MainStream
OF noneq_tray simple_condenser simple_reflux_drum reboiler_lib simple_pump
EQUATION #---------------------Connections------------------------------RefluxDrum.LiquidInlet RefluxDrum.Reflux Pump.LiquidOutlet Tray(1).RefForPress FOR i:= 1 TO NoTrays-1 DO Tray(i).LiquidOutlet Tray(i+1).VapourOutlet END
IS IS IS IS
Condenser.LiquidOutlet ; Pump.LiquidInlet ; Tray(1).LiquidInlet ; Pump.RefForPress ;
IS IS
Tray(i+1).LiquidInlet ; Tray(i).VapourInlet ;
Condenser.VapourInlet IS Reboiler.LiquidInlet IS Tray(NoTrays).VapourInlet IS
Tray(1).VapourOutlet ; Tray(NoTrays).LiquidOutlet ; Reboiler.VapourOutlet ;
Bottoms Distillate
Reboiler.LiquidOutlet ; RefluxDrum.Distillate ;
IS IS
Reboiler.tfactor=tfactor ; FOR i:= 1 TO NoTrays DO Tray(i).tfactor=tfactor ; END Condenser.tfactor=tfactor ; RefluxDrum.tfactor=tfactor ;
- 92 -
Appendix A
Pump.tfactor=tfactor ; END #model column #--------------------------------------------------------------------------PROCESS DC PARAMETER PhysProp1 UNIT C101
AS
FOREIGN_OBJECT "Multiflash"
AS
Column
MONITOR C101.REFLUXDRUM.P ; C101.REFLUXDRUM.T ; C101.REFLUXDRUM.X(*) ; C101.REFLUXDRUM.FOUT(*) ; C101.REFLUXDRUM.LEVEL ; C101.TRAY(5).P ; C101.TRAY(5).T(*) ; C101.TRAY(5).X(*) ; C101.TRAY(5).FOUT(*) ; C101.TRAY(5).LEVEL ; C101.TRAY(5).EMV(*) ; C101.TRAY(15).P ; C101.TRAY(15).T(*) ; C101.TRAY(15).X(*) ; C101.TRAY(15).FOUT(*) ; C101.TRAY(15).LEVEL ; C101.TRAY(15).EMV(*) ; C101.TRAY(28).P ; C101.TRAY(28).T(*) ; C101.TRAY(28).X(*) ; C101.TRAY(28).FOUT(*) ; C101.TRAY(28).LEVEL ; C101.TRAY(28).EMV(*) ; C101.REBOILER.P ; C101.REBOILER.T ; C101.REBOILER.X(*) ; C101.REBOILER.FOUT(*) ;C101.REBOILER.LEVEL ; # # # #
C101.TRAY(*).PHI(*) ; C101.TRAY(*).PHIHEAT ; C101.TRAY(*).KAAVG(*) ; C101.TRAY(*).HA(*) ;
SET #
PhysProp1 := 'depropanizer.mfl'; PhysProp1 := 'deprop2.mfl'; WITHIN C101 DO NoComp := PhysProp1.NumberOfComponents; NoTrays := 33; END WITHIN C101.Reboiler DO NoComp := C101.NoComp ; Vvessel := 25.0 ; #UNIT = "m3" Area := 25 ; #UNIT = "m2" WHeight := 0.7 ; #UNIT = "m" WLength := 5 ; #UNIT = "m" Beta := 0.8 ; Ffeed := 0 ; #UNIT = "mol/s" Zfeed := [0.1,0.3,0.5,0.1] ; Pfeed := 15*101.325 ; #UNIT = "kPa" Tfeed := 298 ; #UNIT = "K"
- 93 -
Appendix A
END FOR i:= 1 TO 14 DO WITHIN C101.Tray(i) DO NoComp := C101.NoComp ; Vtray := 7.48 ; #UNIT = "m3" WHeight := 0.0508 ; #UNIT = "m" WLength := 17.6 ; #UNIT = "m" Zl := 0.7945; #UNIT = "m" PlateArea := 14.96 ; #UNIT = "m2" HolesArea := 0.8813 ; #UNIT = "m2" Cv := 0.7 ; Beta := 0.6 ; DiffVol := [45.66, 66.18, 86.7, 107.22] ; MW := [30.0694, 44.0962, 58.1230, 72.1498] ; #UNIT = "g/mol" Lambda := [6.96E-5, 2.33E-5] ; Xp := [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1] ; END END FOR i:= 15 TO C101.NoTrays DO WITHIN C101.Tray(i) DO NoComp := C101.NoComp ; Vtray := 12.255 ; #UNIT = "m3" WHeight := 0.03734 ; #UNIT = "m" WLength := 22.97 ; #UNIT = "m" Zl := 1.037 ; #UNIT = "m" PlateArea := 24.51 ; #UNIT = "m2" HolesArea := 1.355 ; #UNIT = "m2" Cv := 0.7 ; Beta := 0.6 ; DiffVol := [45.66, 66.18, 86.7, 107.22] ; MW := [30.0694, 44.0962, 58.1230, 72.1498] ; #UNIT = "g/mol" Lambda := [6.96E-2, 2.33E-2] ; Xp := [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1] ; END END FOR i:=1 TO 14 DO WITHIN C101.Tray(i) DO Ffeed := 0 ; #UNIT = "mol/s" Zfeed := [0.1,0.3,0.5,0.1] ; Pfeed := 15*101.325 ; #UNIT = "kPa" Tfeed := 298 ; #UNIT = "K" QTray := [0.0, 0.0] ; END END WITHIN C101.Tray(15) DO
- 94 -
Appendix A
Ffeed Zfeed Pfeed Tfeed QTray END
:= := := := :=
1000 ; #UNIT = "mol/s [0.1,0.3,0.5,0.1] ; 15*101.325 ; #UNIT = "kPa" 298 ; #UNIT = "K" [0.0, 0.0] ;
FOR i:=16 TO 33 DO WITHIN C101.Tray(i) DO Ffeed := 0 ; #UNIT = "mol/s" Zfeed := [0.1,0.3,0.5,0.1] ; Pfeed := 15*101.325 ; #UNIT = "kPa" Tfeed := 298 ; #UNIT = "K" QTray := [0.0, 0.0] ; END END WITHIN C101.Condenser DO NoComp := C101.NoComp ; Cv := 38.9041146 ; END WITHIN C101.RefluxDrum DO NoComp := C101.NoComp ; Vvessel := 14.0 ; #UNIT = "m3" RefluxRatio := 2.5 ; Area := 14 ; #UNIT = "m2" WHeight := 0.7 ; #UNIT = "m" WLength := 3.74 ; #UNIT = "m" Beta := 0.8 ; Ffeed := 0 ; #UNIT = "mol/s" Zfeed := [0.1,0.3,0.5,0.1] ; Pfeed := 15*101.325 ; #UNIT = "kPa" Tfeed := 298 ; #UNIT = "K" QrefluxDrum := 0.0 ; END WITHIN C101.Pump DO NoComp := C101.NoComp ; END # End Set -----------------------------------------ASSIGN C101.tfactor := 0.0 ; WITHIN C101.Reboiler DO QReb := 2.682255893270197E+04 ; #UNIT = "kJ/s" # P := 1532.0 ; #UNIT = "kPa" #15.12 atm END
WITHIN C101.Condenser DO
- 95 -
Appendix A
Pin := 1519.875 ; #UNIT = "kPa" END
# 15 atm
# End Assign --------------------------------------------------PRESET RESTORE "Steady_State_solution_neq_v1" INITIAL STEADY_STATE SOLUTIONPARAMETERS blockdecomposition := on ; absoluteaccuracy := 1E-6 ; outputlevel := 1 ReportingInterval := 60 ; Diagnostics := Off ; RelativeAccuracy := 1E-5 ; GPLOT := ON ;
# # # # # # # #
;
SCHEDULE SAVE "Steady_State_solution_neq_v1" SEQUENCE CONTINUE FOR 360 RESET C101.tfactor := 0.2 ; END CONTINUE FOR 86400 END
END #Process
A.3 Distributed plug flow nonequilibrium model MODEL distributed_pf_tray PARAMETER NoComp Vtray WHeight WLength Zl PlateArea HolesArea
AS AS AS AS AS AS AS
INTEGER REAL REAL REAL REAL REAL REAL
- 96 -
Appendix A
Cv Beta g PhysProp1 Ffeed Zfeed Pfeed Tfeed Qtray DiffVol MW Lambda Xp
AS AS AS AS AS AS AS AS AS AS AS AS AS
ARRAY(NoComp)
OF
ARRAY(2) ARRAY(NoComp) ARRAY(NoComp) ARRAY(2) ARRAY(NoComp,NoComp)
OF OF OF OF OF
REAL REAL REAL DEFAULT 9.81 FOREIGN_OBJECT "Multiflash" REAL REAL REAL REAL REAL REAL REAL REAL REAL
DISTRIBUTION_DOMAIN PFZx AS ( 0 : 1 ) PFZy AS ( 0 : 1 ) VARIABLE Mtotphase AS Mtotcomp AS Mcomp AS X AS T AS Uoverall AS Utot AS Fin AS Zin AS Pin AS Tin AS Hin AS Fout AS P AS H AS Hfeed AS Level AS B AS phi AS phiheat AS Favg AS ka AS D AS Dinf AS ha AS Tnbp AS Fs AS Lfpw AS tL AS Zc AS kaavg AS Davg AS Alpha AS tfactor AS #to compute murphree Emv AS
ARRAY(2) ARRAY(Nocomp) ARRAY(2,Nocomp) ARRAY(2,NoComp) ARRAY(2)
OF OF OF OF OF
MolarAmount MolarAmount MolarAmount Fraction Temperature EnergyAmount ARRAY(2) OF EnergyAmount ARRAY(2) OF MolarRate ARRAY(2,NoComp) OF Fraction ARRAY(2) OF Pressure ARRAY(2) OF Temperature ARRAY(2) OF MolarEnergy ARRAY(2) OF MolarRate Pressure ARRAY(2) OF MolarEnergy MolarEnergy Length ARRAY(2,(Nocomp-1),(Nocomp-1)) OF Notype ARRAY(NoComp) OF MolarRatePhi EnergyRate ARRAY(2) OF MolarRate ARRAY(2,NoComp,NoComp) OF NoType ARRAY(2,NoComp,NoComp) OF NoType ARRAY(NoComp,NoComp) OF NoType ARRAY(2) OF NoType ARRAY(NoComp) OF Tempnbp NoType NoType NoType NoType ARRAY(2) OF NoType ARRAY(2) OF NoType NoType NoType efficiencies ARRAY(NoComp) OF NoType
- 97 -
Appendix A
ymv AS ARRAY(NoComp) OF Fraction Tmv AS Temperature #plug flow model calculations PFX AS DISTRIBUTION(2,NoComp,PFZx,PFZy) OF Fraction PFXint AS DISTRIBUTION(2,NoComp,PFZx,PFZy) OF Fraction PFPhi AS DISTRIBUTION(NoComp,PFZx,PFZy) OF SpecificMolarRatePhi PFPhidummy AS DISTRIBUTION(NoComp,PFZx) OF SpecificMolarRatePhi PFXavgdummy AS DISTRIBUTION(2,NoComp-1,PFZX) OF Fraction PFTint AS DISTRIBUTION(PFZx,PFZy) OF Temperature PFPhiheat AS DISTRIBUTION(PFZx,PFZy) OF SpecificEnergyRatePhi PFPhiheatdummy AS DISTRIBUTION(PFZx) OF SpecificEnergyRatePhi Xavg AS ARRAY(2,NoComp) OF Fraction Havg AS ARRAY(2) OF MolarEnergy
STREAM LiquidInlet VapourInlet LiquidOutlet VapourOutlet RefForPress
: : : : :
Fin(1), Zin(1,), Pin(1), Fin(2), Zin(2,), Pin(2), Fout(1), X(1,), P, T(1), Fout(2), X(2,), P, T(2), P
Tin(1), Hin(1) AS MainStream Tin(2), Hin(2) AS MainStream H(1) AS MainStream H(2) AS MainStream AS PStream
BOUNDARY FOR Zy := 0 TO 1 DO PFX(1,,0,Zy)-Zin(1,)=0.0 ; END FOR Zx := 0 TO 1 DO PFX(2,,Zx,0)-Zin(2,)=0.0 ; END EQUATION #---------------------------Balances------------------------------# Comp. Molar Balance in the Liquid and Comp. Overall Molar Balance FOR i := 1 TO NoComp DO $Mcomp(1,i)=-phi(i)+Fin(1)*Zin(1,i)-Fout(1)*X(1,i)+ (1+0.05*tfactor)*Ffeed*Zfeed(i) ; $Mtotcomp(i)=SIGMA(Fin*Zin(,i))-SIGMA(Fout*X(,i))+ (1+0.05*tfactor)*Ffeed*Zfeed(i) ; END # Heat Balance in the Liquid and Overall Heat Balance $Utot(1)=-phiHeat+Fin(1)*Hin(1)-Fout(1)*H(1)+ (1+0.05*tfactor)*Ffeed*Hfeed+Qtray(1) ; $Uoverall=SIGMA(Fin*Hin)-SIGMA(Fout*H)+ (1+0.05*tfactor)*Ffeed*Hfeed+SIGMA(Qtray) ; #------------------------Definitions------------------------------# Capacity relations SIGMA(X(1,))-1.0=0.0 ; SIGMA(X(2,))-1.0=0.0 ; FOR i:= 1 TO NoComp DO Mtotcomp(i)-SIGMA(Xavg(,i)*Mtotphase)=0.0 ;
- 98 -
Appendix A
Mcomp(1,i)-Xavg(1,i)*Mtotphase(1)=0.0 ; Mcomp(2,i)-Xavg(2,i)*Mtotphase(2)=0.0 ; END Uoverall-SIGMA(Mtotphase*Havg)+P*Vtray=0.0 ; Utot(1)-Mtotphase(1)*Havg(1)+ P*Mtotphase(1)*PhysProp1.LiquidVolume(T(1),(P*1000),Xavg(1,))=0.0 ; Uoverall-SIGMA(Utot)=0.0 ; #--------------------Interphase Transfer-----------------------#Mass Transfer #Some equations are now plugflow FOR i:=1 TO 2 DO FOR j:=1 TO (NoComp-1) DO FOR k:= 1 TO (NoComp-1) DO IF j k THEN B(i,j,k)+Xavg(i,j)*((1/ka(i,j,k))-(1/ka(i,j,NoComp))) =0.0 ; #[((m/s)*m2)^(-1)] ELSE B(i,j,k)-(Xavg(i,j)/ka(i,j,NoComp))(SIGMA(Xavg(i,)/ka(i,j,))-Xavg(i,j)) =0.0 ; #[((m/s)*m2)^(-1)] END END END END #Liquid Diffusivities FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i = j THEN Dinf(i,j)=0.0 ; ELSE Dinf(i,j)*(PhysProp1.LiquidVolume(Tnbp(i),101325.0,Xp(,i)))^0.433* PhysProp1.LiquidViscosity(T(1),(P*1000),Xavg(1,)) -8.93E-16*(PhysProp1.LiquidVolume(Tnbp(j),101325.0,Xp(,j)))^0.267* T(1)* ((PhysProp1.SurfaceTension(Tnbp(j),101325.0,Xp(,j),Xp(,j)))/ (PhysProp1.SurfaceTension(Tnbp(i),101325.0,Xp(,i),Xp(,i))) )^0.15 =0.0 ; #[m2/s] END END END FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i < j THEN D(1,i,j)-Dinf(i,j)^((1+Xavg(1,j)-Xavg(1,i))/2)* Dinf(j,i)^((1+Xavg(1,i)-Xavg(1,j))/2)=0.0 ; #[m2/s] ELSE IF i > j THEN D(1,i,j)-D(1,j,i)=0.0 ; #[m2/s]
- 99 -
Appendix A
ELSE D(1,i,j)=0.0 ; #[m2/s] END END END END Davg(1)-(SIGMA(D(1,,)))/(NoComp^2-NoComp)=0.0 ; #[m2/s] #Vapour Diffusivities FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i < j THEN D(2,i,j)-0.010111627*(T(2)^1.75)*SQRT((1/MW(i)+1/MW(j))) /(P*1000*(DiffVol(i)^(1/3)+DiffVol(j)^(1/3))^2)=0.0 ; #[m2/s] ELSE IF i > j THEN D(2,i,j)-D(2,j,i)=0.0 ; #[m2/s] ELSE D(2,i,j)=0.0 ; #[m2/s] END END END END Davg(2)-(SIGMA(D(2,,)))/(NoComp^2-NoComp)=0.0 ; #[m2/s] #Factors Fs-(Favg(2)*(PhysProp1.VapourVolume(T(2),(P*1000),Xavg(2,)))/PlateArea) *SQRT(PhysProp1.VapourDensity(T(2),(P*1000),Xavg(2,)))=0.0 ; Lfpw-(Favg(1)*(PhysProp1.LiquidVolume(T(1),(P*1000),Xavg(1,)))/WLength) =0.0 ; tL-Zc*Zl/Lfpw=0.0 ; Zc-0.0254*(1.65+7.4803*WHeight+96.6236*Lfpw-0.5328*Fs)=0.0 ; #Liquid Mass Transfer Coefficients FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i < j THEN ka(1,i,j)-20314.732*SQRT(D(1,i,j))*(0.21313*Fs+0.15)*tL* (Favg(1)*PhysProp1.LiquidVolume(T(1),(P*1000),Xavg(1,)))=0.0 ; #[(m/s)*m2] ELSE IF i > j THEN ka(1,i,j)-ka(1,j,i)=0.0 ; #[(m/s)*m2] ELSE ka(1,i,j)-1.0=0.0 ; #[(m/s)*m2] END END END END
- 100 -
Appendix A
kaavg(1)-(SIGMA(ka(1,,))-NoComp)/(NoComp^2-NoComp)=0.0 ; #[(m/s)*m2] #Vapour Mass Transfer Coefficients FOR i := 1 TO NoComp DO FOR j := 1 TO NoComp DO IF i < j THEN ka(2,i,j)-((0.776+4.567*WHeight-0.2377*Fs+104.8366*Lfpw)/ (SQRT(PhysProp1.VapourViscosity(T(2),(P*1000),Xavg(2,))/ (PhysProp1.VapourDensity(T(2),(P*1000),Xavg(2,))*D(2,i,j)))))* (Favg(2)*PhysProp1.VapourVolume(T(2),(P*1000),Xavg(2,)))=0.0 ; #[(m/s)*m2] ELSE IF i> j THEN ka(2,i,j)-ka(2,j,i)=0.0 ; #[(m/s)*m2] ELSE ka(2,i,j)-1.0=0.0 ; #[(m/s)*m2] END END END END kaavg(2)-(SIGMA(ka(2,,))-NoComp)/(NoComp^2-NoComp)=0.0 ; #[(m/s)*m2] #Heat Transfer Coefficients ha(1)-(kaavg(1)/PhysProp1.LiquidVolume(T(1),(P*1000),Xavg(1,)))* (PhysProp1.LiquidHeatCapacity(T(1),(P*1000),Xavg(1,))/1000)* SQRT(ABS( ((PhysProp1.LiquidConductivity(T(1),(P*1000),Xavg(1,))/1000)* PhysProp1.LiquidVolume(T(1),(P*1000),Xavg(1,)) )/ (Davg(1)*(PhysProp1.LiquidHeatCapacity(T(1),(P*1000),Xavg(1,))/1000)) )) =0.0 ; #[(kJ/(s*m2*K))*m2] ha(2)-(kaavg(2)/PhysProp1.VapourVolume(T(2),(P*1000),Xavg(2,)))* (PhysProp1.VapourHeatCapacity(T(2),(P*1000),Xavg(2,))/1000)* (ABS(((PhysProp1.VapourConductivity(T(2),(P*1000),Xavg(2,))/1000)* PhysProp1.VapourVolume(T(2),(P*1000),Xavg(2,)))/(Davg(2) *(PhysProp1.VapourHeatCapacity(T(2),(P*1000),Xavg(2,))/1000))
)
)^(2/3) =0.0 ; #[(kJ/(s*m2*K))*m2] #----------------------Hydrodynamics--------------------------IF Level < Beta*WHeight THEN Fout(1)=0.0 ; ELSE Fout(1)-1.84*WLength*ABS(((Level-Beta*WHeight)/Beta))^1.5/ PhysProp1.LiquidVolume(T(1),(P*1000),Xavg(1,))=0.0 ; END IF ((Pin(2)-P)*1000g*PhysProp1.LiquidDensity(T(1),(P*1000),Xavg(1,))*Level THEN
- 101 -
)
>
0.0
Appendix A
(Pin(2)-P)*1000( (alpha*PhysProp1.VapourDensity(T(2),(P*1000),Xavg(2,))* (Fin(2)*PhysProp1.VapourVolume(T(2),(P*1000),Xavg(2,))/HolesArea)^2) +g*PhysProp1.LiquidDensity(T(1),(P*1000),Xavg(1,))*Level ) =0.0 ; ELSE Fin(2)=0.0 ; END alpha=0.05*g/(Cv^2) ; #---------------------Geometry------------------------------Mtotphase(1)*PhysProp1.LiquidVolume(T(1),(P*1000),Xavg(1,))+ Mtotphase(2)*PhysProp1.VapourVolume(T(2),(P*1000),Xavg(2,))Vtray=0.0 ; Level-Mtotphase(1)*PhysProp1.LiquidVolume(T(1),(P*1000),Xavg(1,))/ PlateArea=0.0 ; #---------------------PhysProp------------------------------H(1)=(PhysProp1.LiquidEnthalpy(T(1),(P*1000),X(1,))/1000) ; H(2)=(PhysProp1.VapourEnthalpy(T(2),(P*1000),X(2,))/1000) ; Hfeed=(PhysProp1.LiquidEnthalpy(Tfeed,(Pfeed*1000),Zfeed)/1000) ;
Tnbp=PhysProp1.BoilingPoint ; #[K] Favg(1)=(Fin(1)+Fout(1))/2 ; Favg(2)=(Fin(2)+Fout(2))/2 ; #Computation of Murphree Efficiencies SIGMA(ymv) =1.0 ; ymv*PhysProp1.VapourFugacityCoefficients(Tmv,(P*1000),ymv)X(1,)*PhysProp1.LiquidFugacityCoefficients(Tmv,(P*1000),X(1,)) =0.0 ; Emv-(X(2,)-Zin(2,))/(ymv-Zin(2,)) =0.0 ; #Plug flow model calculations FOR Zx := 0|+ TO 1 DO FOR Zy := 0 TO 1 DO FOR i := 1 TO NoComp-1 DO ( Mtotphase(1)/( (Mtotphase(1)*PhysProp1.LiquidVolume(T(1),(P*1000),PFX(1,,Zx,Zy)))+ (Mtotphase(2)*PhysProp1.VapourVolume(T(2),(P*1000),PFX(2,,Zx,Zy))) $(PFX(1,i,Zx,Zy))+
) )*
(Favg(1)/((Level/Beta)*(PlateArea/Zl)*Zl))*PARTIAL(PFX(1,i,Zx,Zy),PFZx)+PFPhi(i, Zx,Zy)=0.0 ; END SIGMA(PFX(1,,Zx,Zy))-1.0=0.0 ; END END FOR Zx := 0 TO 1 DO
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Appendix A
FOR Zy := 0|+ TO 1 DO FOR i := 1 TO NoComp-1 DO ( Mtotphase(2)/( (Mtotphase(1)*PhysProp1.LiquidVolume(T(1),(P*1000),PFX(1,,Zx,Zy)))+ (Mtotphase(2)*PhysProp1.VapourVolume(T(2),(P*1000),PFX(2,,Zx,Zy))) ) )* $(PFX(2,i,Zx,Zy))+ (Favg(2)/(PlateArea*(Level/Beta)))*PARTIAL(PFX(2,i,Zx,Zy),PFZy)PFPhi(i,Zx,Zy)=0.0 ; END SIGMA(PFX(2,,Zx,Zy))-1.0=0.0 ; END END FOR Zy := 0 TO 1 DO FOR Zx := 0 TO 1 DO FOR i := 1 TO NoComp-1 DO (PlateArea*(Level/Beta))*SIGMA(B(1,i,)*(PFPhi(1:(NoComp-1),Zx,Zy)PFX(1,1:(NoComp-1),Zx,Zy)*SIGMA(PFPhi(,Zx,Zy))))(1/PhysProp1.LiquidVolume(T(1),(P*1000),PFX(1,,Zx,Zy)))*(PFX(1,i,Zx,Zy)PFXint(1,i,Zx,Zy)) =0.0 ; #[(mol/s)*((m/s)*m2)] (PlateArea*(Level/Beta))*SIGMA(B(2,i,)*(PFPhi(1:(NoComp-1),Zx,Zy)PFX(2,1:(NoComp-1),Zx,Zy)*SIGMA(PFPhi(,Zx,Zy))))(1/PhysProp1.VapourVolume(T(2),(P*1000),PFX(2,,Zx,Zy)))*(PFXint(2,i,Zx,Zy)PFX(2,i,Zx,Zy)) =0.0 ; #[(mol/s)*((m/s)*m2)] END
PFXint(2,,Zx,Zy)*PhysProp1.VapourFugacityCoefficients(PFTint(Zx,Zy),(P*1000),PFX int(2,,Zx,Zy))PFXint(1,,Zx,Zy)*PhysProp1.LiquidFugacityCoefficients(PFTint(Zx,Zy),(P*1000),PFX int(1,,Zx,Zy)) =0.0 ; SIGMA(PFXint(1,,Zx,Zy))-1.0=0.0 ; SIGMA(PFXint(2,,Zx,Zy))-1.0=0.0 ; END END FOR i:= 1 TO 2 DO FOR j:= 1 TO NoComp-1 DO FOR Zx := 0 TO 1 DO PFXavgdummy(i,j,Zx) INTEGRAL PFX(i,j,Zx,position) ) =0.0 ; END Xavg(i,j) - INTEGRAL (position := 0:1 )=0.0 ; END SIGMA(Xavg(i,))-1.0=0.0 ; END
- 103 -
(position
;
:=
0:1
;
PFXavgdummy(i,j,position)
Appendix A
Havg(1)=(PhysProp1.LiquidEnthalpy(T(1),(P*1000),Xavg(1,))/1000) ; Havg(2)=(PhysProp1.VapourEnthalpy(T(2),(P*1000),Xavg(2,))/1000) ; FOR Zx := 0 TO 1 DO PFPhidummy(,Zx) - INTEGRAL (position := 0:1 ; PFPhi(,Zx,position) ) =0.0 ; END Phi INTEGRAL (position )*PlateArea*(Level/Beta) =0.0 ;
:=
0:1
;
PFPhidummy(,position)
FOR Zy := 0 TO 1 DO FOR Zx := 0 TO 1 DO PFPhiheat(Zx,Zy)-(1/((Level/Beta)*PlateArea))*ha(1)*(T(1)PFTint(Zx,Zy))SIGMA(PFphi(,Zx,Zy))*(PhysProp1.LiquidEnthalpy(T(1),(P*1000),PFX(1,,Zx,Zy))/1000 )=0.0 ; PFPhiheat(Zx,Zy)-(1/((Level/Beta)*PlateArea))*ha(2)*(PFTint(Zx,Zy)T(2))SIGMA(PFphi(,Zx,Zy))*(PhysProp1.VapourEnthalpy(T(2),(P*1000),PFX(2,,Zx,Zy))/1000 )=0.0 ; END END FOR Zx := 0 TO 1 DO PFPhiheatdummy(Zx) - INTEGRAL (position := 0:1 ; PFPhiheat(Zx,position) ) =0.0 ; END Phiheat INTEGRAL (position := 0:1 ; PFPhiheatdummy(position) )*PlateArea*(Level/Beta) =0.0 ;
END #MODEL distributed_pf_tray
- 104 -
