WO2001043868A1

WO2001043868A1 - Entropy optimal operation and design of one or several chemical reactors

Info

Publication number: WO2001043868A1
Application number: PCT/NO2000/000431
Authority: WO
Inventors: Signe Kjelstrup; Dick Bedeaux; Eivind Johannessen
Original assignee: Leiv Eiriksson Nyfotek As
Priority date: 1999-12-17
Filing date: 2000-12-15
Publication date: 2001-06-21
Also published as: NO996318D0; NO996318L; AU2238501A

Abstract

The present invention concerns a method for optimisation of the production of entropy in one or more chemical reactors, where a first number of in-feeded reagents are transformed to another number of out-feeded products, and where a yield of a distinct of the out-feeded products is set up as a predetermined value (J), and a total heat Q is transferred to or from the reactor, wherein the degree of reaction X0, for reactants and intermediate products, pressure p0 and temperature, T0, or marginal conditions are known; reference profiles (reaction profile, X(x), pressure profile p(x), and temperature profile T(x) are calculated by solving conservation equations for mass, moment and energy with given marginal conditions; the first optimal set of profiles for reaction, pressure and temperature is calculated starting from the reference profiles, by Euler-Lagrange optimisation of the entropy production for the reaction(s) with Lagrange multiplier μ1 to take care of the requirement for constant production J; the second optimal set of profiles for reaction, pressure and temperature is calculated starting from the reference profiles, by Euler-Lagrange optimisation of the entropy production for the heat exchanger(s), with Lagrange multiplier μ2 to take care of the requirement of given transferred heat Q; the minimum entropy production for the total system is determined by balancing the two ideal set of profiles and reference profiles against each other according to the procedure described in own flow sheet, so that a final set of profiles is provided; the reactor is operated with the marginal conditions provided by the final set of profiles, or a set of profiles as close to this as practically possible.

Description

Entropy optimal operation and design of one or several chemical reactors

The present invention concerns continuation of known technology, a method for optimisation of entropy production in one or several chemical reactors with adjoining heat exchangers, where a first number of in-feeded reagents are transformed into a number of out-feeded products, at the same time as a certain amount of heat Q is transformed between the reactor and their surroundings. The yield of out-feeded products has a predetermined value (J). This technology has been described i Patent application NO 1998 2798. The continuation consists of detailed explanations of how heat exchangers shall be included in the optimisation.

Analysis of the patent literature

Optimisation of temperature profiles throughout reactors has previously been referred to in the patent literature. US 4,571 ,325 describes a reactor suitable for exothermic and endothermic catalytic processes wherein an optimised temperature profile is provided which leads to increased productivity. The object of the present application is not increased productivity alone, but best possible energy effective production.

Further US 4,696,735 describes a method for cooling of an exothermic reaction process in a multiphase reactor. The purpose is to provide an approximate isothermal temperature profile for the components in the reactor. A further feature is a constant temperature profile in each reactor. The present invention does not have isothermal conditions in the reactor as a target.

WO-96/16007 describes a method for obtaining a certain temperature profile along a reactor where an optimised reaction and product selectivity is obtained for an endothermic reaction. The reaction mixture is passed through a stationary reactor where the desired temperature profile is established through heat supply at the beginning of the reactor. US patent 4,589,072 is an apparatus which can determine the degree of reaction for a reaction, by means of temperature measurement or another measurement. Such an apparatus can not be used to obtain the purpose of our invention.

Norwegian patent 126 102 concerns a control system for a reaction. By measuring and controlling heat developed in the reaction, this invention can provide for that the maximum heat transfer rate of the cooling device is not exceeded, at the same time as the production equipment is optimally utilised. Optimally in this connection means that the invention provides for that the reaction proceeds with the rate which is the highest obtainable, in relation to the heat transfer rate of the cooling device. With regard to this it should be mentioned that a reaction that proceeds with maximum speed also proceeds with maximum production of entropy. Our invention will mainly operate away from maximum rate to be able to produce with less lost work for each produced unit.

German patent DE 3 525 248 A1 concerns a system for controlling a reaction procedure by means of temperature and concentration, both measured as a function of the time, so that use of energy and raw materials is kept on a minimum. This invention does not describe minimal use of energy by means of the second law of thermodynamic, and the method described does not contain any optimisation of the energy input either, as is the case in the present application.

European patents EP 0 861 687 A1 , EP 0 510 487 A1 and EP 0 581 273 should also be mentioned. EP 0 861 687 A1 describes a method for determination of the reaction progress of a reaction in true time, by measuring so-called swelling values. Our invention does not touch the first patent, because the swelling values do not apply to bulk phase reactions with high heat toning. EP 0 510 487 A1 concerns an automatic synthesis apparatus, and describes a control system for a reaction which proceeds in several stages. It is not the object of this invention to control a reactor, even if the invention finally can give results which can be used in a control system. The method used in EP 0 581 273 is further used to produce at least two products with given requirement for production and correlation to energy requirements, directed to economic optimisation, for given prices of energy, and not reduction in the energy requirement as such, as is the case with our method.

Fogler (1992) describes how to determine temperature, pressure and concentration profiles through a reactor which is cooled or heated by the environment, from conservation equations for mass, energy and moment, and given marginal conditions.

None of the patents discussed above concerns the object of our invention, which is minimising of the entropy production by the reactor (lost exergy) with given requirement for production. The similarity between our invention and known technology resides in that all means to control chemical reactions have to make use of temperature, pressure and concentration as variables. Which profile those variables have throughout the reactor, or as a function of time, will be specific for the purpose. Our invention therefore normally concerns different profiles from that which known technology invokes. Standard profiles, which are determined by conservation equations for mass, energy and moment, constitute that what we call the reference reactor in this invention. Our invention concerns a method of finding new profiles which are compatible with a lower net energy requirement than that of the reference reactor.

Object of the invention

Known art mentioned above does not deal with entropy optimisation at a given requirement for production of out-feeded products (J) and requirements to transferred heat (Q). Such technique is earlier only described in application NO 1998 2798. One object of the present invention is to further build on the original application. We wish to utilise information about ideal (often hypothetical) reactor operation, characterised by 1) approximately constant driving force for the reaction, and 2) constant force for heat transfer to find a balanced process where total entropy production is at minimum. Minimum entropy production is equivalent to minimum loss of exergy, minimum lost work, or minimum dissipated energy. We call it operation with minimum production of entropy, for an energy optimal operation with given requirements.

The present method is connected to the principle of equipartition of forces, which principle is derived for linear transport processes (Sauar et al, 1996); especially for transport of heat and mass in a distillation tower (Ratkje et al, 1995). It is known that all driving forces for transportation of mass and heat have to be distributed evenly throughout the process unit to achieve a minimal entropy production when the system has enough degrees of freedom to make it possible to adjust this. It was shown that this also applies to chemical reactors which operate close to equilibrium (Sauar et al, 1997). The ideal operation is often not possible because of practical limitations in the apparatus, such as lack of feeding points. It does not either have to be possible or desirable to operate close to equilibrium. A method for optimisation to determine energy optimal operation of chemical reactors with heat exchanger with starting point in irreversible thermodynamic is not earlier published by anyone. The authors of this application has explained the process of the chemical reactor in an earlier patent application and in an article (Kjelstrup and Island, 1999). The present invention concerns a method which defines such operation of one or more reactors when heat exchanger(s) belonging thereto is/are included in the optimisation.

The present invention builds on the method for minimising the production of entropy in one or more chemical reactors with or without heat exchangers, where in-feeded reagents are transformed to out-feeded products, and where requirements are made for yield of a distinct product J. The novel is that now also requirements are made for total heat transferred to or from the reactor, Q. With this the following is taking place:

Degree of reaction, X₀, for reactants and intermediate products, pressure P_o, and temperature, T₀, are established at the entrance (marginal conditions are known)

Reaction profile, X(x), pressure profile p(x), and temperature profile T(x) are calculated throughout the reactor (are called reference profiles), by solving conservation equations for mass, moment and energy. The first optimal set of profiles for reaction, pressure and temperature is calculated starting with the reference profiles, by Euler-Lagrange optimisation of the entropy production for the reaction(s), with Lagrange multiplier λi to take care of the requirement for constant production J. The optimisation equation replaces the energy balance.

The second optimal set of profiles for reaction, pressure and temperature is calculated starting with the reference profiles, by Euler-Lagrange optimisation of the entropy production for the heat exchanger(s), with Lagrange multiplier λ₂ to take care of the requirement about given heat transferred to Q. The optimisation equation constitutes a part of the conservation equation for energy.

Minimum entropy production for the total system is determined by balancing the two ideal conditions which are provided by these two (first and second) set of profiles against each other and against the reference profiles. The result is called the set of profiles after balancing.

All four set of profiles provide each of their own sets of marginal conditions for the reactor, the last via the three first.

The invention mainly deals with situations wherein chemical reaction and heat exchange take place far from equilibrium. The method is characterised by that driving force for the reaction in the first optimal set of profiles is approximately constant (as earlier described), by the fact that driving force for the heat exchanger is constant (according to known technology), and by the fact that driving forces in the balanced situation is not constant throughout the reactor. The invention describes the calculation of four sets of profiles; first the reference profiles and the first set of profiles as earlier described, and thereafter the two new sets of profiles, called the second set of profiles, and the balanced set of profiles.

The reactor operation which occurs using the balanced set of profiles for the reactor instead of for example the reference set of profiles, will give less energy costs for each produced unit. For an exothermic reaction this means that the cooling water form the reactor can be taken out at a higher temperature, possibly that more cooling water can be taken out at the same temperature. For an endothermic reaction this means that less supplied heat is required for each unit produced, possibly that more low temperature heat can be used.

The dissipated energy (lost exergy) which resides in a chemical reaction is often very high, and has until now been considered as very difficult to minimise. This invention will make such losses of energy in industry lower. A better gain will be obtained from high temperature heat, or low temperature heat can be better utilised in the industry.

The present invention is a method for finding operating conditions and other conditions which give less loss of exergy (entropy production) with requirement for constant production of out-feeded compounds (J) and heat exchange (Q). The invention comprises one or more reactors, tube or batch reactors, to which heat exchangers belong. The method specifies among others where in the reactor the chemical production shall take place, and how much heat exchange shall occur in this place to give minimum entropy production in the total system.

The energy optimisation in the present invention gives in practice an energy saving by the fact that heat and vapour can be liberated at higher temperature (or pressure) for exothermic reactions, or in bigger amounts at the same temperature and pressure. For endothermic reactions the energy saving will cause that heat and vapour can be supplied to one or more reactors at lower temperature (and possibly pressure), or in smaller amounts at the same temperature or pressure.

The importance of the invention

In practice the invention will tell how to alter the the operating conditions for the reference profiles in direction of the balanced set of profiles. This involves advice on change of entrance conditions, cooling / heating along the reactor wall, more smaller reactors in series, extra feeding points, altering of the configuration of the units, dividing of the heat exchange in two units to be able to alter the flow pattern, etc. The absolute efficiency for exchange of energy has been described by Denbigh (1956):

(1)

~ _ W max - T o Θ w. max

Here T₀ is the surrounding temperature, and Θ is the total entropy production of the system. The maximum work, which is available for work in relation to the surroundings, is W_maχ. According to the second law of thermodynamics, the efficiency equals 1 when the process is reversible. In this case there is no entropy production. The maximum work which can be carried out by a system is also called the exergy content in the process. The equation shows how important it is to reduce the entropy production in a system.

Loss of exergy (entropy production multiplied with the temperature of the surroundings) has been known for a long time, and was detailed described for chemical reactions by Denbigh (1956). Today there is more methods to calculate efficiency according to the second law. One of them is exergy analysis. With regard to methods for reduction of the losses, little is achieved. This is especially the case for chemical reactions. The best known method to reduce the losses is to increase the size of the processing unit, in order to increase the degree of reversibility. This results in a longer residence time. Normally this solution becomes expensive.

Method for determination of entropy optimal operation and design

The present invention is based on irreversible thermodynamics. This theory defines fluxes and forces in the non-equilibrium system which is going to be investigated. The method is described by a general example in the following.

Local entropy production per time and volume unit due to chemical reaction and heat transfer is defined by deGroot and Mazur (1962): (2)

The velocity of the chemical reaction is rand -AG/T is the driving force for the reaction. Here ΔG is Gibbs energy of the reaction, and T is the local temperature. The heat flux transferred to the cooler is J_q and T_a is the temperature in the cooling medium. More than one reaction can be present. It can be seen that reversible transferred heat (at constant temperature) gives zero entropy production. In the same way a reaction in equilibrium gives zero entropy production. A reaction cannot proceed at equilibrium (except in an electrochemical cell). That is the reason that the entropy production can be great in chemical reactors.

The total entropy production is found by integration over the volume of the reactor and over the time. If it is a stationary tube reactor, all results can be reported integrated over the volume.

(3)

It is this size which should be minimised, because it is reason for lost work when a given production J shall take place. The production is given by

(4)

Characteristic for a given reaction is also a heat toning Q, given by

(5)

Q = \j_qadV Here the constant a will give the connection between the volume of the reactor and the area of the walls. For a cylindrical reactor a = 4/d, wherein d is the tube diameter.

The optimisation is formulated by a known mathematical method, Euler- Lagranges method with constant multipliers. There is a multiplier for the requirement of constant production, which is λ-i_, and a multiplier for the specified heat transfer, λ₂. Because the driving forces are independent of each other, we get according to Kjelstrup and Island (1999):

(6)

Here δ means functionally derived. The expression is complicated, but can be numerically determined when the expressions for the reaction velocity and Gibbs energy for the reaction are known. Close to equilibrium it can be shown that it gives constant driving force for the chemical reaction in the ideal case where we have enough degrees of freedom in the reactor to achieve the condition. Further we get that the force for heat transfer shall be constant if the heat exchange is ideal (Nummedal and Kjelstrup, 2000):

(7)

1 1 _{τ τ} = --λ,

The equations are concretised by a given reaction, for example:

A → B (I)

• The reaction velocity must be known for the reaction, for example:

r = k ^■ c _A - k₂ ^■ c _B = k ^• c - k₂ ^• (\ - c) (8)

where k is velocity constant and c is concentration • Mass conservation laws states dependency between the concentrations (normalised): l - c (9)

All concentrations / partial pressures are determined as a function of the degree of reaction based on mass conservation laws.

The thermodynamic expression for the chemical force is established:

where R is the gas constant. For the sake of simplicity, an ideal mixture has been chosen, and the equilibrium constant K_eq = k₂/k-ι is inserted.

• From equation (4) and (8) the produced amount of component B is determined:

where Ω is the cross section of the reactor. It was assumed that the reactor has a constant cross section, and that the reactor is a tube type reactor. Thus it is integrated over the whole length of the reactor, L. (If the reactor is a batch reactor, it is integrated over the time t.)

• Equation (6) now can be written as:

Ω t

• The force, ΔG T, is a function of the concentration, c, the temperature, T, and the pressure, P. The same applies to the velocity of the reaction, r. A small sub-volume is conceivable where pressure and temperature are constant. This means that: (13)

where δ stands for partially derived. Vi therefore can derive in relation to the concentration.

The connection between Gibbs energy for the reaction and the concentration is used to characterise the local progress of the reaction. It will be incorrect to use the temperature or pressure to characterise the progress of the reaction, because temperature and pressure are not precise functions of the reaction degree.

Equation (13) has the following solution for each sub-volume:

The first part is recognised as the force (given by equation (10). The new expression for the thermodynamic force is as follows:

Equation (14) describes how the force should be to make the reactor as entropy optimal (energy effective) as possible with a given production J, see application NO 1998 2798. This force is not necessarily equipartitioned as it is for chemical reactions close to equilibrium (Sauar et al., 1997). The force stated in equation (7) is, however, always equipartitioned.

The method will be further described through an example, the methanol synthesis. The method applied to the synthesis of methanol

To find minimum for total entropy production in the equipment, both the reactor and the heat exchanger, the following method is used. As an example, calculations for the methanol synthesis are shown. This synthesis has two reactions, plus heat transfer to the environment (it is exothermic). The reactions are

CO₂ + 3 H₂ = CH₃OH + H₂O (II)

CO₂ + H₂ = CO + H₂O (III)

1. Establishment of reference profiles

Temperature, pressure and concentration profiles in the reactor(s) with heat exchanger(s) are determined from the conservation equations for mass, moment and energy. The result is called the set of reference profiles.

The calculation requires knowledge of inlet conditions and temperature in the heat exchanger. It is also necessary to know how the reaction velocity varies with the temperature, concentration and pressure. The calculation is carried out according to standard method, see Fogler (1992), and his example for oxidation of SO₂.

Foglers directions are used to calculate reference profiles for methanol synthesis, with kinetic data for the reactions from van der Bussche and Froment (1996), and with thermodynamic data. Reference profiles are shown in figure 1.

On the background of these results the entropy production on every site in the reactor is also calculated, for the two reactions and for the heat transfer, se figure 2. (Entropy production because of pressure gradient, friction loss, can be neglected). The solution is characterised by a high and varying driving force for the reaction, at the start of the reactor, see figure 3. The reactor is with other words operated far from equilibrium. 2. Establishment of profiles for ideal reaction, set of profiles 1

The first set of profiles is found according to the method in application NO 1998 2798. Temperature, pressure and concentration profiles in the reactor(s) with heat exchanger(s) are thereafter determined from the conservation equations for mass, moment and optimisation equation for the reaction, equation (4). Flow chart for the calculations is shown in figure 4. The result is called set of profiles 1 for ideal reaction. It is shown in figure 5. The profiles deviate from the reference profiles, especially with regard to the temperature. It appears that the chemical force which can be calculated from set of profiles 1 , is much more constant than in the reference reactor, see figure 6.

The calculation uses, as inlet conditions, the equilibrium condition for reaction 2, and the inlet conditions from the reference profiles which do not concern the temperature. By solving the energy balance for the ideal system, the temperature of the cooling water can be determined. The entropy production for reaction 1 is drastically reduced with as much as 76,5%, but the entropy production for the total system has increased, see figure 7.

Establishment of profiles for ideal heat exchange, set of profiles 2.

The novel method now prescribes that temperature, pressure and concentration profiles in the reactor(s) with heat exchanger(s) shall be determined from the conservation equations for mass, moment and optimisation equation for heat transport, equation (5). The optimisation equation is used to eliminate the driving force in the energy balance. The calculation uses the same inlet conditions as those for set of profiles 1 , namely the equilibrium condition for reaction 2, and the inlet conditions from the reference profiles which do not apply to the temperature. Flow chart for the calculations is shown in figure 8.

The result is called set of profiles 2 for ideal heat exchange. It is shown in figure 9. The force for heat exchange is constant, see figure 10. The calculations show that entropy production for this set of profiles also increases in relation to the reference reactor, with 14%. The relation between the different contributions to the total entropy production is, however, very different from the result in figure 7. Especially the temperature profile throughout the reactor is different for the two sets of profiles. The entropy production is, because of heat transfer, reduced by 22%.

The optimal condition

When all of the three above sets of profiles have been determined, these have to be balanced against each other to find minimum entropy production for the total system.

We define the balance factor (15)

In each position in the reactor the balance factor is defined as a linear combination of the contributions from all sets of profiles:

(16)

λ₂ = (\ - - β)λf^ence + aλ^s + βλ

The result of a certain combination where α = 0,3, and β = 0,3 is shown in figure 11. Driving force for heat transfer is known when the balance factor is known. The temperature in the cooling medium can therefore be eliminated from the energy conservation equation, in the same way as in the calculations of the set of profiles 2.

The procedure for solving the problem is given by the flow chart in figure 12. For each chosen value of α and β, the profile for the balance factor is determined. When this profile is known, a guess of a value for λi is made, and (the unknown) inlet conditions are calculated. The modified set of conservation equations is solved. If the production ol methanol is correct, the (balanced) set of profiles is calculated; if not, a new iteration is made.

By numerical interpolation in this manner, in the room delimited by the three sets of profiles, the balanced set of profiles which gives minimum entropy production for all processes together, is found. The total entropy production for the system is plotted as a function of the degree of contribution from every solution in figure 13. We see that there is a minimum in the area that occurs, not far from the reference reactor (which therefore is a reactor with good design).

Den optimal situation (the point F) is characterised in that whether the force for heat exchange or the force for the chemical reaction are constant throughout the reactor. The balanced profiles lies between the outer borders which the other sets of profiles are representing.

This is the result: That the entropy production can be reduced in relation to the reference reactor. We can state an exact saving in lost energy, but this absolute value has to be understood from the reference which has been chosen. In this case we see from the solution room which gives the balanced profiles, that the reference reactor is not far away from optimal operation. We have controlled that another choice of reference reactor does not change the coordinates for minimum in figure 13.

The set of profiles corresponding to the reactor with entropy production F, is called the balanced set of profiles. This is shown in figure 15. If the balanced profiles are followed, the reactor can be operated energetically more profitable. We see, for example, that the inlet temperature should be altered from approximately 495 to 525 °C. Analogous the composition of in-feeded product should be a little changed. The results also tell that the temperature profile in the reactor should go through a maximum. This temperature tells that heat can be recovered by a higher temperature than in the reference reactor. Optimisation of combustion of SO_2(g) according to NO 1998 2798 - a comparison

A method for optimisation of the entropy production by combustion of SO_2(g) in a converter is described in NO 1998 2798. Reference profiles and the first set of profiles were determined.

The reference profiles were by Fogler claimed to be the most optimal with regard to reaction. NO 1998 2798 proceeds and establishes the first optimal set of profiles for evaluation of the reactor. Based on this qualitative conclusions can be drawn (e.g. from earlier figure 20) about further method for practical accomplishment of the result of the optimisation. In distinction from NO 1998 2798 we proceed in this application and describe in addition a quantitative method for analysis of reactor and heat exchanger as a whole.

List of figures

Figures 1-15 concern the methanol synthesis

1. The reference profiles for the methanol synthesis. The profiles are the same as those found in the literature by van der Bussche and Froment (1996). They are obtained by solving conservation equations for mass, moment and energy with given marginal conditions. The profiles, which represent known technique, are the basis of the method.

2. Entropy production for the reference reactor from figure 1. The figure shows that the contribution to entropy production from the reaction dominates to a high degree, and is highest at the start of the reactor, where the conditions are most far from equilibrium.

3. The driving force for the methanol synthesis in the reference reactor.

The figure shows that the force is high and varying.

4. Flow chart for determination of set of profiles 1. The calculations start with reference profiles as inlet values. A value for λi is chosen. New inlet conditions are calculated. The energy balance is replaced by optimisation equation (6), and the set of equations is solved to give profiles of temperature, pressure and concentration. If the profiles do not give the requested production, the loop is gone through once again with a novel Lagrange multiplier.

5. Set of profiles 1 for ideal reaction. The figure shows which profiles the reactor get when the reaction has minimum entropy production. The profiles are compared to reference profiles. We see that there are big deviations, especially for the temperature.

6. The chemical force for ideal reaction. We see at the force varies within narrower limits than in figure 3, when the reaction has minimum entropy production.

7. Total entropy production for set of profiles 1. The figure shows that the entropy production for the reaction has become very small, while the heat exchanger which is necessary gives a higher entropy production.

8. Flow chart for determination of set of profiles 2. The calculations start with reference profiles as starting values. A value λi is chosen. Novel inlet conditions are calculated. The optimisation equation for heat-force, equation (7), is inserted into energy balance with chosen value for λ₂, and the set of equations is solved to give profiles of temperature, pressure and concentration. If the profiles do not give the requested production, the loop is gone through once again with a novel Lagrange multiplier.

9. Set of profiles 2 for ideal heat transfer. The figure shows which profiles the reactor get when the heat transfer has minimum entropy production. The profiles are compared to reference profiles. We again see that there is big deviations, especially with regard to the temperature.

10. The thermal force for ideal heat transfer. We see that the force is constant when the heat transfer has minimum entropy production.

11. Total entropy production for set of profiles 2. The figure shows that the entropy production for the reaction has grown, while the entropy production for heat transfer has become small.

12. Variation of the weight factors throughout the reactor. Example with = 0,3 and β = 0.3.

13. Flow chart for determination of (balanced) optimal set of profiles. The calculations start with reference profiles as inlet values. For every predefined combination of the parameters α and β, a profile for λ₂ is calculated from equation (22). A value for λi is specified, and inlet conditions are calculated for λ₃ = 0. The modified set of conservation equations is solved to give profiles of temperature, pressure and concentration. If the profiles do not give the requested production, the loop is gone through once again with a novel Lagrange multiplier.

14. The total entropy production as a function of the parameters α and β.

The point A represents the set of profiles from the reference reactor, the point B represents the set of profiles 1 , and the point D represents the set of profiles 2. Minimum entropy production is represented by the point F.

15. The balanced optimal set of profiles. The profiles for temperature, reaction and pressure which give minimum entropy production for the total system are shown. LIST OF SYMBOLS

A Component

B Component

ΔEx Change in exergy J/mol

ΔG Change in Gibbs' energy J/mol

J Yield kmol/h

K Equilibrium constant atm^{"0 5}

L Reactor length m

P Pressure atm

R Gas constant J/(K mol)

S Surface area r mv,²

T Temperature K u Thermal transmittance number W/(K m²) w Performed work J/mol

X Force for heat transport κ-¹

X Degree of reaction c General concentration k General velocity constant k Velocity constant mol/(g kat s atm) r Reaction velocity mol/(m³ s)

X Reactor position r Parameter

Θ Total production of entropy per produced mole J/K

Ω Cross section of the reactor r mv,² δ Functional derived η Efficiency θ Local entropy production J/(K m³ s) λ Lagrange-multiplier μ Chemical potential J/mol

P Bulk density kg/m³ d Partial derived

Subscript / ' superscript eq Equilibrium h Warm k Cold chemical Contribution from chemical reaction max Maximum n Sub-volume n varme Contribution from heat transport

0 Surroundings LITTERATURE LIST

De Groot, S.R. and Mazur, P.: Non-equilibrium thermodynamics, North- Holland, Amsterdam, 1962.

Denbigh, K. G.: The second law of chemical processes, Chemical Engineering Science, 6, 1-9 (1956).

Fogler, H.S.: Elements of Chemical Reaction Engineering, 2nd. ed., Prentice-Hall International, Inc., USA, 1992.

Kjelstrup, S., Sauar, E., Bedeaux, D. and Kooi, H.van der: The driving force for distribution of minimum entropy production in chemical reactors close to and far from equilibrium, Not published, RA Leiden, The Netherlands, 1997.

Ratkje, S.K. and Arons, J. De Swaan: Denbigh revisited: Reducing lost work in chemical processes, Chemical Engineering Science, 50, 1551-1560 (1995).

Ratkje, S.K., Sauar, E., Hansen, E.M., Lien, K.M. and Hafskjold, B: Analysis of Entropy production Rates for Design of Distillation Columns, Industrial & Engineering Chemistry Research, 34, 3001-3007 (1995).

Sauar, E., Kjelstrup, S. and Lien, K.M.: Equipartition of Forces-Extension to Chemical Reactors, Computers Chem. Engng., 21 , s29-s34 (1997).

Sauar, E., Ratkje, S.K. and Lien, K.M.: Equipartition of Forces: A New Principle for Process Design and Optimization, Industrial & Engineering Chemistry Research, 35, 4147-4153 (1996).

Van den Bussche, K.M. and Froment, G.F. J.Catal. 161 , 1 (196)

Claims

C l a i m s

1. Method for optimisation of the production of entropy in one or more chemical reactors, where a first number of in-feeded reagents are transformed to another number of out-feeded products, and where a yield of a distinct of the out- feeded products is set up as a predetermined value (J), and a total heat Q is transferred to or from the reactor, c h a r a c t e r i s e d i n that the degree of reaction, X₀, for reactants and intermediate products, pressure p₀, and temperature, T₀, or marginal conditions are known; reference profiles (reaction profile, X(x), pressure profile p(x), and temperature profile T(x)) are calculated by solving conservation equations for mass, moment and energy with given marginal conditions; the first optimal set of profiles for reaction, pressure and temperature is calculated starting from the reference profiles, by Euler-Lagrange optimisation of the entropy production for the reaction(s) with Lagrange multiplier λ-i to take care of the requirement for constant production J; the second optimal set of profiles for reaction, pressure and temperature is calculated starting from the reference profiles, by Euler-Lagrange optimisation of the entropy production for the heat exchanger(s), with

Lagrange multiplier λ₂ to take care of the requirement of given transferred heat Q; the minimum entropy production for the total system is determined by balancing the two ideal set of profiles and reference profiles against each 5 other according to the procedure described in own flow sheet, so that a final set of profiles is provided; the reactor is operated with the marginal conditions provided by the final set of profiles, or a set of profiles as close to this as practically possible.

o 2. The method according to claim 1 , c h a r a c t e r i s e d i n that the reactor is operated far from equilibrium, but closer to equilibrium with the balanced profiles than with the reference profiles.

3. The method according to claim 1, karakterisert ved at the reactor only in exceptional cases is operated at the maximum reaction velocity.

4. The method according to claims 1-3, characterised in that driving force for the reaction, defined as -AG/T, wherein ΔG is Gibbs energy for the reaction, given by chemical potentials for reactants and products, and Tis temperature, is more evenly distributed in the optimal reactor than in the reference reactor.

5. The method according to claim 1, characterised in that the driving force in the second set of profiles for heat transfer is kept constant.

6. The method according to claim 1, characterised in that possible parallel reactions are carried out without changes in the mentioned calculation steps.

7. The method according to claim 1, characterised in that change of choice of reference reactor does not change the result of the optimisation.

8. The method according to claims 1-7 for exothermic reactor(s), characterised in that the cooling water comes out with a generally higher temperature or in larger amounts in the optimal reactor(s) than in the reference reactor.

9. The method according to claims 1-7 for endothermic reactor(s), characterised in that the heating takes place by a generally lower temperature, possibly with a lower amount of heat in the optimal reactor(s) than in the reference reactor.