Modelling of a distillation column with operating state changes

Abstract

A method is provided for determining a state of a distillation column having multiple column stages for separating a feed fluid stream into individual fluid components. The state is determined by means of a model in a manner dependent on pressure differences prevailing between adjacent column stages. In the model, both gaseous and liquid flows between adjacent column stages are brought about by the pressure differences prevailing between adjacent column stages. A substance quantity flow characterizing gaseous flow between two column stages is given by {dot over (N)}.sub.V.Math.R.sub.V=C.sub.V.Math.Δp.sub.V. A substance quantity flow characterizing liquid flow between two column stages is given by {dot over (N)}.sub.L.Math.R.sub.L=C.sub.L.Math.Δp.sub.L. Δp.sub.V,L is a total pressure difference between two adjacent column stages. R.sub.V,L is a coefficient of resistance between two adjacent column stages and C.sub.V,L is a conductance value of flow between two adjacent column stages.

Claims

1. A method for operating a distillation column (12) having multiple column stages for separating a feed fluid stream into individual fluid components, comprising: determining a state of the distillation column from a plurality of states by means of a model in a manner dependent on pressure differences prevailing between each two adjacent column stages of said multiple column stages, wherein the plurality of states represent a complete operating range of the distillation column, wherein, in the model, both gaseous flows and liquid flows between the each two adjacent column stages are determinded by the pressure differences prevailing between the each two adjacent column stages, wherein, in the model, a substance quantity flow {dot over (N)}.sub.V which characterizes the gaseous flow between the each two adjacent column stages of said multiple column stages is given by {dot over (N)}.sub.V.Math.R.sub.V=C.sub.V.Math.Δp.sub.V, and a substance quantity flow {dot over (N)}.sub.L which characterizes the liquid flow between the each two adjacent column stages is given by {dot over (N)}.sub.L.Math.R.sub.L=C.sub.L.Math.Δp.sub.L, wherein, Δpv is a total gaseous pressure difference between the each two adjacent column stages, Δp.sub.L is a total liquid pressure difference between the each two adjacent column stages, R.sub.V is a gaseous coefficient of resistance between the each two adjacent column stages, R.sub.L is a liquid coefficient of resistance between the two each adjacent column stages, C.sub.V is a gaseous conductance value of the gaseous flow between the two each adjacent column stages, and C.sub.L is a liquid conductance value of the liquid flow between the two each adjacent column stages, on the basis of the determined state, for at least one manipulated variable influencing the state, setting a manipulated variable value influencing the state, in a manner dependent on at least one control variable, wherein the at least one control variable comprises a product quality, a product volume flow, a product mass flow or a cost variable, and operating the distillation column (12) using the determined state of the distillation column (12) using the at least one control variable.

2. The method according to claim 1, further comprising determining the state during a disappearance or appearance of the gaseous flow or the liquid flow between the each two adjacent column stages.

3. The method according to claim 2, further comprising using a regularization function to determine the state.

4. The method according to claim 2, wherein the disappearance or appearance of said gaseous flow or said liquid flow between the each two adjacent column stages comprises a change from a fluid movement to a zero flow, a change from a zero flow to a fluid movement, a change from a single-phase gas flow to a two-phase rectification or a change from a two-phase rectification to a single-phase gas flow.

5. The method according to claim 1, wherein, in the model, the state is characterized by a temperature and/or fluid component concentrations for each column stage of the each two adjacent column stages.

6. The method according to claim 1, further comprising modelling a column stage of said multiple column stages as an equilibrium stage, wherein a substance, thermal, and mechanical equilibrium of the individual fluid components flowing in the column stage and flowing away from the column stage is assumed by the model.

7. The method according to claim 1, further comprising applying a backward difference method with a predictor-corrector method for determining a change in the state with respect to time.

8. A computing unit which is set up to carry out the method according to claim 1.

9. A computer program which causes a computing unit to carry out the method according to claim 1 when said program is executed on the computing unit.

10. A machine-readable storage medium having the computer program according to claim 9 stored thereon.

11. An air separation installation, having the computing unit according to claim 8.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 schematically shows, in a block diagram, an air separation installation with distillation columns, such as can form the basis of the invention;

(2) FIG. 2 schematically shows a transformation of a distillation column with multiple stages into a model with corresponding equilibrium stages;

(3) FIG. 3 schematically shows the model of a single equilibrium stage (left: separated balance space; right: combined balance volume);

(4) FIG. 4 schematically shows a distributor in a column section;

(5) FIG. 5 shows the four possible flow states in a stage: left: pure gas flow, near left: accumulation of liquid, near right: breakthrough of liquid, right: steady flow.

(6) FIG. 6 illustrates regularization on the basis of the behaviour of a backward difference method.

DETAILED DESCRIPTION OF THE DRAWINGS

(7) The separation on which the invention is based of fluid components, such as for example the production of oxygen, nitrogen and, if desired, further substances, such as for example argon, is realized in a fluid separation installation, such as for example a conventional air separation installation, as shown for example in FIG. 1. In this case, the starting substance air is cooled down and conducted into a rectification system having a column system, which provides a sufficiently large mass transfer area.

(8) Air separation installations of the type shown are described to a large extent elsewhere, for example in H.-W. Häring (ed.), Industrial Gases Processing, Wiley-VCH, 2006, in particular Section 2.2.5, “Cryogenic Rectification”. For detailed explanations regarding construction and functioning, reference is therefore made to corresponding technical literature. An air separation installation for implementing the present invention may be formed in a wide variety of ways.

(9) The air separation installation shown by way of example in FIG. 1 has inter alia a main air compressor 1, a pre-cooling device 2, a purification system 3, a post-compressor arrangement 4, a main heat exchanger 5, an expansion turbine 6, a throttle device 7, a pump 8 and a distillation column system 10. In the example illustrated, the distillation column system 10 comprises a conventional double-column arrangement made up of a high-pressure column 11 and a low-pressure column 12, and a crude argon column 13 and a pure argon column 14.

(10) In the air separation installation 100, a feed air stream is drawn in via a filter (not shown), and compressed, by means of the main air compressor 1. The compressed feed air stream is fed to the pre-cooling device 2 operated with cooling water. The pre-cooled feed air stream is purified in the purification system 3. In the purification system 3, which typically comprises a pair of alternately operated adsorber vessels, the pre-cooled feed air stream is substantially freed of water and carbon dioxide.

(11) Downstream of the purification system 3, the feed air stream is divided into two partial streams. One of the partial streams is completely cooled down in the main heat exchanger 5 at the pressure level of the feed air stream. The other partial stream is post-compressed in the post-compressor arrangement 4 and likewise cooled down in the main heat exchanger 5, albeit only to an intermediate temperature level. Following the cooling to the intermediate temperature level, this so-called turbine stream is expanded by means of the expansion turbine 6 to the pressure level of the completely cooled partial stream, combined therewith, and fed into the high-pressure column 11.

(12) An oxygen-enriched liquid bottom fraction and a nitrogen-enriched gaseous top fraction are formed in the high-pressure column 11. The oxygen-enriched liquid bottom fraction is drawn off from the high-pressure column 11, used in part as a heating medium in a bottom evaporator of the pure argon column 14 and, in each case in defined fractions, fed into a top condenser of the pure argon column 14, into a top condenser of the crude argon column 13 and into the low-pressure column 12. Fluid evaporating in the evaporating spaces of the top condensers of the crude argon column 13 and the pure argon column 14 is likewise transferred into the low-pressure column 12.

(13) The gaseous nitrogen-rich top product is drawn off from the top of the high-pressure column 11, liquefied in a main condenser which establishes a heat-exchanging connection between the high-pressure column 11 and the low-pressure column 12, and, in fractions, is, as reflux, applied to the high-pressure column 11 and expanded into the low-pressure column 12.

(14) An oxygen-rich liquid bottom fraction and a nitrogen-rich gaseous top fraction are formed in the low-pressure column 12. The former is partly pressurized in liquid form in the pump 8, heated in the main heat exchanger 5, and provided as product. A liquid nitrogen-rich stream is drawn off from a liquid-retaining device at the top of the low-pressure column 12 and discharged as liquid nitrogen product from the air separation installation 100. A gaseous nitrogen-rich stream drawn off from the top of the low-pressure column 12 is conducted through the main heat exchanger 5 and is provided as nitrogen product at the pressure of the low-pressure column 12. From the low-pressure column 12, a stream is furthermore drawn off from an upper region, and used as so-called impure nitrogen in the pre-cooling device 2 after being heated in the main heat exchanger 5 or used in the purification system 3 after being heated by means of an electric heater.

(15) FIG. 2 schematically illustrates a construction of a distillation column, for example on which the invention is based, with multiple stages, such as for example the low-pressure column 12 from FIG. 1, and a transformation into a model with corresponding equilibrium stages. Conventional stages of a distillation column have a distributor, one or more stages, formed here as packings (or alternatively as trays), and a collector. The distributor receives liquid coming from above and supplies this to the (uppermost) packing such that it is distributed over the area. Conversely, gas flows upwards from the packing via the distributor. Liquid exiting from the (lowermost) packing is collected in the collector and conducted downwards. Conversely, the gas coming from below is received by the collector and conducted into the packing situated above.

(16) Overall, air (AIR) is supplied to the column, and at least oxygen and nitrogen, in each case in gaseous (GOX, GAN) and/or liquid (LOX, LIN) form, are removed from the column. Furthermore, it is normally the case that impure nitrogen (UN2) is drawn off and, if desired, noble gases such as argon, etc.

(17) According to a preferred embodiment of the invention, in the transformation of such a distillation column into a model, use is made of models 101 for distributors and models 102 for equilibrium stages, which correspond to a packing. Here, it is also possible for a collector model to be integrated into the existing distributor model by adapting the hydrodynamics. The crossover between two stages or between one stage and a distributor is in this case subjected to flow resistances R.sub.L (liquid) and R.sub.V (gas) and also conductance values C.sub.L (liquid) and C.sub.V (gas).

(18) It should be pointed out that, as an index, use is made of L for the liquid phase, V for the gas phase, n for the respective stage and j for the respective air component.

(19) Use is made according to the column system of a model for an equilibrium stage (see FIG. 3) and, in a preferred refinement, also for a distributor and collector system, a condenser and a bottom (heated/unheated). A hierarchical modelling approach allows these partial models to then be combined to form the desired overall system. The calculation of the thermal substance separation is realized via equilibrium stages which are connected one behind the other and which represent a specific height of a packing.

(20) The actual height of a packing which corresponds to precisely one equilibrium stage is also referred to as the HETP (Height Equivalent of a Theoretical Plate) value.

(21) Substance, thermal and mechanical equilibrium of the gas and liquid streams occurring in the stage are assumed. For the purpose of modelling the flow processes, unsteady conservation of mass and energy is used. Preferably, conservation of momentum is modelled in a quasi-steady manner. Overall, there are thus preferably storage terms for mass and energy but not for the momentum. Pulse waves, that is to say sound waves, are relevant on a relatively small time scale and only in exceptional cases (for example, with the estimation of a stagnation pressure).

(22) The model on which each one of the equilibrium stages 102 is based is shown on the right in FIG. 3. On the left, conventional modelling with separated balance volumes for a gas phase (NV,n) and a liquid phase (NL,n) and, on the right, the applied approach with a balance volume NZ,n. {dot over (N)}describes the molar flow across the system boundary. FIG. 3 shows how a vapor stream which rises and a liquid stream which trickles down exchange mass and energy. Here, the corresponding quantities result from the assumption of the equilibrium (rather than of a mass- and heat-transfer model).

(23) The change in the total substance quantity N.sub.Z,n with respect to time corresponds to the sum of the inflowing and outflowing fractions according to:

(24) $\frac{d}{\mathrm{dt}}{N}_{Z,n}={\stackrel{.}{N}}_{V,n-1}+{\stackrel{.}{N}}_{L,n+1}-{\stackrel{.}{N}}_{V,n}-{\stackrel{.}{N}}_{L,n}$

(25) From the conservation of energy, it follows that a change in energy is composed of a heat flow, a technical working flow and an inflowing and outflowing mass flow:

(26) $\frac{\mathrm{dE}}{\mathrm{dt}}=\frac{\mathrm{dQ}}{\mathrm{dt}}+\frac{{\mathrm{dL}}_t}{\mathrm{dt}}+\underset{\mathrm{in}}{.Math.}\frac{d}{\mathrm{dt}}\left({\stackrel{.}{N}}_{\mathrm{in}}.Math.M_{W,\mathrm{in}}\right).Math.\left(h_{\mathrm{in}}+\frac{v_{\mathrm{in}}^2}{2}+g.Math.z_{\mathrm{in}}\right)--\underset{\mathrm{out}}{.Math.}\frac{d}{\mathrm{dt}}\left({\stackrel{.}{N}}_{\mathrm{out}}.Math.M_{W,\mathrm{out}}\right).Math.\left(h_{\mathrm{out}}+\frac{v_{\mathrm{out}}^2}{2}+g.Math.z_{\mathrm{out}}\right)$

(27) with

(28) E: energy

(29) Q: heat

(30) L.sub.t: technical work

(31) .sub.in, out: inflowing and outflowing fractions

(32) {dot over (N)}: molar flow

(33) M.sub.W: molar weight

(34) v: speed

(35) h: partial molar enthalpy

(36) g: acceleration due to gravity

(37) z: height

(38) Here, the kinetic contributions v.sup.2/2 can normally be neglected.

(39) From the conservation of momentum, it follows that:

(40) $\frac{d\underset{\_}{P}}{\mathrm{dt}}=\frac{d}{\mathrm{dt}}\underset{\tilde{V}}{\int }\rho \underset{\_}{u}\mathrm{dV}=\underset{i}{.Math.}{\underset{\_}{F}}_i$

(41) with

(42) P: total momentum

(43) ρu: momentum density

(44) F.sub.i: active force (for example gravitational force)

(45) i: running index

(46) Here, the equilibrium state contains in particular for each stage n a thermal equilibrium between gas and liquid, T.sub.n,L=T.sub.n,V, that is to say both phases have the same temperature, a substance equilibrium, f″.sub.n,j=f′.sub.n,j, that is to say the fugacity of a component j is the same in all the phases present (f′ liquid and f″ gaseous), and a mechanical equilibrium, C.Math.Δp=R.Math.{dot over (N)}, that is to say the pressure difference for a phase between two adjacent stages corresponds to the product of flow resistance and substance quantity flow. Here, for each stage, the boundary condition that the sum of the volumes of the gas V.sub.n,V and the liquid V.sub.n,L are equal to the volume of the stage V.sub.n, V.sub.n,L+V.sub.n,V=V.sub.n, is to be adhered to.

(47) On the basis of the current fluid load, a specific pressure loss and liquid content in the system are obtained, which are determined via design correlations, as described for example by Kooijman, H. A., K. R. Krishnamurty and M. W. Biddulph: A NEW PRESSURE DROP MODEL FOR STRUCTURED PACKING: 634th Event of the European Federation of Chemical Engineering. GVC—VDI-Society of Chemical and Process Engineering, Germany, 2002. These advantageously also take into consideration the stagnation of liquid at the joints of the packing layers in the case of relatively large column loads. In addition, the gaseous fluid content included in the balance results therefrom via the difference between the column and liquid volumes. It is possible via a distributor and collector system to realize inward and outward feeding of the rectification system and to improve the transverse mixing of a packing column over the total cross section (diameters of up to 5 m in large-scale installations). The model according to the invention makes it possible to simulate in closed form running-dry and overflowing of the distributor.

(48) Liquid content, and thus thermal inertia, is present in accumulated form in each distributor and collector system, which is installed above and below a packing section. These fittings are necessary for collecting and mixing the outflowing liquid, and for uniformly distributing said liquid to the packing section situated below again. Here, it is also possible for the collector model to be integrated into the existing distributor model by adapting the hydrodynamics, that is to say the collector and distributor between two stages are preferably modelled in combination.

(49) A distributor may be interpreted as a vessel which has different cross-sectional areas in which liquid can stagnate, as shown in a sectional view in FIG. 4. The entire column section is defined by the outer dashed line. The liquid runs, as identified by the solid arrows, from above through the cross-sectional areas A.sub.3 to A.sub.1, and the gas flows in the opposite direction, as identified by the dashed arrows, upwards.

(50) The uppermost cross section A.sub.3 corresponds to the inlet channel from the collector situated above or from an infeed connecting piece from the column wall. This opens out into a larger cross section A.sub.2, which distributes the liquid over the column area for the first time. In said cross section, the liquid has a free surface during normal operation and the fill level varies according to the liquid load. Further channels, which assume the cross-sectional area A.sub.1, are situated below said section.

(51) Provided on the underside of these surfaces are relatively small bores, which ensure a defined delivery of the liquid onto the packing situated below. Said bores form a constriction for the liquid flow. The gas phase flowing in the opposite direction is narrowed by the cross-sectional area A.sub.1 in order then to enter the packing layer situated above again over the entire column cross section, and for this reason a pressure loss occurs.

(52) The liquid level (in A.sub.1 and A.sub.2) has narrow, oblique hatching, the volume filled by gas (rest) has wide, oblique hatching.

(53) In principle, the modelling of the distributor is realized in a manner similar to the model approach for the equilibrium stage, and for this reason only the differences are discussed below. A constant value which is matched to empirical values or measurement data is assumed for the coefficient of resistance of the gas phase. This simplification is made owing to the small pressure loss of the slowly flowing gas phase and a constant channel geometry. As a reference value, the coefficient of resistance may be set to be approximately equal to that of an equilibrium stage. No mass transport takes place in a distributor between the gas and liquid phases since the specific phase interface is, in comparison with a theoretical equilibrium stage, smaller approximately by a factor of 1500. Consequently, balancing for the conservation of mass and energy is not possible over the entire content of the entire column section, but is realized separately for the two phases.

(54) The concentration of the outflowing gas and liquid stream is thus always equal to the concentration of the corresponding fluid content. Unlike the equilibrium stage, the changed selection of the balancing gives rise to the following situation:

(55) During a starting process from the warm, dry state, there is no liquid volume N.sub.L of the distributor model. An implicit determination of pressure via balancing of fluid-content and vessel volumes cannot therefore be realized for the liquid phase. Furthermore, this would be associated with very large gradients due to the incompressibility of liquids. Thus, for the liquid balance element, a switch is made from an isochoric consideration to an isobaric consideration. The pressure of the liquid is set to be equal to the pressure of the gas phase which is present around the distributor.

(56) With a rising liquid quantity, the liquid volume displaces a part of the gas cushion and assumes a constantly increasing volume fraction. Due to the varying balance size of the liquid element, it is additionally necessary for technical work, in the form of volume-changing work, to be performed, which is calculated according to Stephan et al.: Thermodynamik [Thermodynamics]. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. via
dLt=−p.Math.dV.

(57) Said work is performed according to the volume change by one balance element and simultaneously absorbed by the other element, since both elements have the same pressure. The change in the liquid volume is dependent on the current fill level in the distributor, and it holds that

(58) $\mathrm{dV}=A\left(H\right).Math.\mathrm{dH}=\underset{i}{.Math.}{A}_i.Math.\min \left(\max \left(\left(H-H_i\right),0\right),\left(H_{i+1}-H_i\right)\right)$
where A.sub.i=[A.sub.1,A.sub.2,A.sub.3] and H.sub.i=[0, H.sub.1,H.sub.2,H.sub.3] for the distributor geometry present.

(59) For the liquid flow, the stagnation in the vessel gives rise to a different relationship for the hydrodynamics. Neglecting frictional forces, the result for the liquid phase in the x.sub.3 direction from FIG. 4 is:

(60) $\rho L.Math.\left(u_3.Math.\frac{\partial u_3}{\partial x_3}\right)=-\frac{\partial p}{\partial x_3}+{\rho }_L.Math.g$

(61) The friction at the phase interface is not taken into consideration since no oppositely directed streams of the gas and liquid phases occur and these are separated by the metal wall of the distributor. Likewise, the internal fluid friction can be neglected owing to the stagnation of the liquid and the reduced mean speed.

(62) If the above equation is integrated in the x.sub.3 direction from the static surface of the height H to the outflow cross section on the underside, then the result is

(63) $\frac{\rho L}{2}.Math.u_3^2={\Delta p}_V+{\rho }_L.Math.g.Math.H$

(64) For the flow speed in the outflowing cross section A.sub.Hole, it follows from this that:

(65) $u_3=\sqrt{\frac{2.Math.\Delta p_V}{{\rho }_L}+2.Math.g.Math.H}\mathrm{with}\Delta p_V<0$

(66) This formulation is equivalent to the outflow formula of Toricelli with superimposed pressure difference ΔpV due to the rising gas flow.

(67) For the quantity flowing away out of the distributor it follows that

(68) ${\stackrel{.}{N}}_L=\frac{{\stackrel{.}{M}}_L}{M_{W,L}}=\frac{{\rho }_L.Math.u_3.Math.A_{\mathrm{Hole}}}{M_{W,L}}=\frac{{\rho }_L.Math.\sqrt{2.Math.g.Math.H+\frac{2.Math.\Delta p_V}{{\rho }_L}}.Math.A_{\mathrm{Hole}}}{M_{W,L}}$

(69) By way of the approach for the pressure-driven flow, the coefficient of resistance of the liquid is obtained via:
R.sub.L.Math.{dot over (N)}.sub.L=C.sub.L.Math.(Δp.sub.V+ρ.sub.L.Math.g.Math.H)

(70) At this point, the pressure is increased by the hydrostatic pressure of the stagnated liquid.

(71) The liquid reflux required for substance separation is provided via a condenser at the top of the column by removing heat. Situated in the bottom of the system is stagnated liquid, which can be evaporated by heat supply and thus generates a rising gas stream. Modelling for the condenser and bottom are known in the prior art.

(72) Due to incentives for the energy markets, it is also desirable to consider a complete shutdown of the rectification system. A problem for this scenario is the use, for a dynamic starting process, of design correlations determined in a steady manner. In the case of a warm starting or cold restarting process, which is schematically illustrated in FIG. 4, initially a purely gaseous flow flows through the rectification system. In this case, a change in concentration over the column height still does not occur. (The concentration profile is established analogously to the temperature profile during two-phase operation. The column is warm at the bottom and cold at the top, and, for the concentration profile, the low boiler (lowest boiling point) builds up at the top and the high boiler builds up in the bottom.) If liquid is formed in an equilibrium stage, then this accumulates first before it flows away into the stage situated below and a steady operation is then established.

(73) In order to simulate this process and thus a load-dependent outflow of liquid, the force equilibrium at a fluid film is considered. According to the condition
|F.sub.G|=|F.sub.R|+|F.sub.D|,

(74) a criterion for the breakthrough of liquid into the equilibrium stage situated below is determined. F.sub.G represents the weight force of the liquid molecules, which acts counter to the flow direction of the gas. The shear force, transmitted by the gas, onto the film is denoted by F.sub.R. Due to the pressure drop of the gas flow as a result of friction, a pressure force F.sub.D is additionally active.

(75) In order to be able to represent the operating range of a single-phase gas flow to two-phase rectification by way of a model, conditional equations are also advantageously matched. Thus, for example, the composition of the exiting gas stream from an equilibrium stage for the single-phase situation is equal to the concentration of the fluid content in the balance element. If, when biphasicity occurs, rectification takes effect, then the low boiler builds up in the gas phase and substance separation on the basis of the isofugacity condition applies.

(76) In order to improve the numerical behavior of the integrator at these switching points, all the discontinuities in the model are preferably regularized. For the purpose of illustration, reference is made to FIG. 5, where a function profile f(x) having a bend point at x′ is shown by a solid line. f(x) could describe for example the dependence of the density on the enthalpy. In the superheated (gaseous) state, this is continuously differentiable, but has a bend at the boiling point where the first liquid molecules form since the vapor fraction depends on the enthalpy in a discontinuous manner

(77) The result of regularization is sketched using circles. As a result of the smoothed function profile, no bends occur, and the Jacobian matrix does not need to be formed anew at a switching point x′. For the next time step, the prediction is closer to the sought solution with a change of equations.

(78) The adapted function profile is determined for example via a homotopy parameter λ according to
f(x)=(1−λ).Math.f(x).sub.x<x′+λ.Math.f(x).sub.x>x′,

(79) which transforms the model equations into one another. The range with deviations from the original function profile is referred to as a regularization interval and is indicated in FIG. 5 by ε.

(80) For example, it is possible according to a preferred embodiment for a quarter of a sine wave to be used for the determination of the homotopy parameter, and it holds that:

(81) $\lambda =\{\begin{array}{c}0\mathrm{for}x\le -\epsilon \\ 0.5+0.5.Math.\sin \left(\frac{\pi }{2}.Math.\frac{x}{\epsilon }\right)\mathrm{for}-\epsilon <x<\epsilon \\ 1\mathrm{for}x\ge \epsilon \end{array}$

(82) This model approach allows a simplified initialization strategy to be applied with time as a homotopy parameter. The stabilization of the model for zero flows and the possible change of the aggregate state allows the system to be initialized with a stationary gas cushion. The onset of the fluid movement and the occurring substance separation are established according to the conditions of the infeed streams. This is especially advantageous for determining a steady solution of a coupled double-column system since no concentration profiles have to be estimated beforehand.

(83) In principle, for the purpose of stabilizing the column model and the representability of single-phase and two-phase flow forms, several adaptations of the basic model are expedient.

(84) For each switch of the calculation equations, use is advantageously made of a regularization function which allows a continuously differentiable crossover. For this concept, it is advantageous to define a generally valid switching variable with which the blend functions are activated. For the present simulation task, the concept of the generalized vapor fraction has proven to be particularly robust, which is explained according to Thomas below: For the vapor fraction of a flow vector, it generally holds that

(85) 0$\mathrm{vf}=\frac{h-h^{\prime }}{h^{″}-h^{\prime }}$
with the simplified notation of h′ for the condition “boiling point” and h″ for the “dew point”. In process engineering, it is normally the case that a stream which is completely liquid is characterized by the vapor fraction vf=0, while of =vf=1 holds for a single-phase gaseous substance. The precise state in the two-phase region is defined analogously via a vapor fraction 0<vf <1. For numerical reasons, it is also expedient for use to be made outside the two-phase region of an umambiguous definition of the state of the flow vector, which is continuously differentiable. For this purpose, the vapor fraction can be extrapolated beyond the phase change points, and, for the entire solution space, the following definition holds:

(86) ${\mathrm{vf}}_{\mathrm{gen}}=\frac{h\left(z,p,T\right)-h^{\prime }\left(x^{*},p,T^{\prime }\right)}{h^{″}\left(y^{*},p,T^{″}\right)-h^{\prime }\left(x^{*},p,T^{\prime }\right)}$

(87) By this formulation, vf.sub.gen assumes values greater than 1 for a superheated, gaseous stream and less than 0 for a single-phase, undercooled stream. Consequently, an imminent phase change can already be detected before the actual occurrence and, if appropriate, measures for the stabilization of the model can be enabled before the actual switching point.

(88) Use is made of regularization functions for the continuously differentiable blend of model equations of different operating states. All the functions with which the parameter λ can be varied between zero and one are referred to as regularization functions. The result of this is that a function profile according to f(x)=(1−λ).Math.f(x).sub.x<x′+λ.Math.f(x).sub.x>x′ can be defined in sections and is interblended in a specific region. The simplest function with which the parameter λ can be determined is a line equation.

(89) For the equilibrium stage and the distributor system, it is preferably possible to use the following regularizations.

(90) For the equilibrium stage, these are: phase separation for the gas phase

(91) The phase separation on the basis of the isofugacity condition is always calculated independently of the current state of the entire content in the balance element. Also, in the superheated state, virtual compositions are determined from extrapolated fugacity coefficients, which have no physical meaning. Consequently, it is necessary to blend the composition of the outflowing gas phase in the single-phase case from the equilibrium concentrations to the concentration of the volume phase. stabilization of the properties of the liquid phase

(92) For the determination of the composition of the liquid flowing away from an equilibrium stage, it is likewise necessary to make a case distinction between a dry and active packing. Owing to large gradients, with a blend of the discontinuous profile of the substance data from the gas phase to the liquid phase, smoothing at this point is not effective. stabilization of the design correlations

(93) In order to eliminate problems during the calculation of gradients for a zero flow, at this point, the transfer of the current load parameters for the calculation of the design correlations is limited. Consequently, it is possible to replace the selected design correlation for packings by any desired other correlation without having to adapt the model equations.

(94) The distributor model described above must be calculable both for dry operation and for operation with a liquid level. In comparison with the equilibrium stage, the gas and liquid phases are balanced separately. Consequently, the liquid content disappears for the single-phase case, and the fill level H in the distributor is equal to zero. The adapted outflow relationship according to Toricelli is, with the slope of the root function for H.fwdarw.0, a problem from a numerical viewpoint. The infinite slope for the function value zero leads to a singularity in the Jacobian matrix and the gradient-based solver fails. Squaring u.sub.3 and replacing the square of the liquid speed via
u.sub.3.sup.2=u.sub.3.Math.(|u.sub.3|+B.sub.1.Math.e.sup.−B.sup.2.sup..Math.u.sup.3)

(95) can remedy this. By the choice of the parameter B.sub.1, the slope du/dh|h=0 is determined such that the solution method converges. The deviation from the original function profile can be set via the selection of the parameter B.sub.2. The larger this value is, the smaller the absolute deviation.