WO2003008090A1

WO2003008090A1 - Method for optimisation of entropy production in chemical reactors

Info

Publication number: WO2003008090A1
Application number: PCT/NO2002/000212
Authority: WO
Inventors: Signe Kjelstrup; Dick Bedeaux; Eivind Johannessen; Lars Nummedal; Gelein De Koeijer; Audun RÖSJORDE
Original assignee: Leiv Eiriksson Nyfotek As
Priority date: 2001-06-15
Filing date: 2002-06-14
Publication date: 2003-01-30
Also published as: NO20013006L; NO20013006D0

Abstract

The present invention concerns a process for optimization/minimization of the total entropy production in one or more chemical reactors with connected heat exchanger(s), wherein a first number of products, fed to the reactor, are converted into another number of output products, where the yield of one particular product of the output products is kept at a predetermined value (J), and where the degree of conversion (Xo) for reactants and intermediates, pressure (po), and temperature (To), or boundary conditions at the inlet are known, and where reference profiles for conversion X(x), pressure p(x), and temperature T(x) are calculated by solving conservation equations for mass, momentum and energy, wherein requirements for energy balance and mass balance as a basis are fulfilled in discrete volume elements of the reactor.

Description

Method for opt i m i sa i on of ent ropy product i on i n chem i ca l reactors .

The present invention concerns a method for optimization/minimization of the total entropy production in one or more chemical reactors with connected heat exchangers), where a first number of products fed to the reactor are converted into another number of products, which are removed from the reactor, and where the yield of one of the products removed from the reactor is kept at a predetermined value (J), and where the degree of conversion for reactants and intermediates, pressure (p₀), and temperature (T₀), or boundary conditions at the inlet are known, and reference profiles for conversion X(x), pressure p(x), and temperature T(x) are calculated by solving the conservation equations for mass, energy and momentum.

Patent literature analysis

The known technology, on which this invention builds, is described in Patent application NO 1998 2798 and in Patent application NO 1999 6318.

A method for optimization of energy production by combustion of SO₂(g> in a converter has been described in NO 1998 2798 and in NO 19996318. Reference profiles were determined, and optimal sets of profiles were evaluated against each other.

The reference profiles were claimed by Fogler to be the most optimal profiles with regard to reaction. NO 19982798 took a step forward and established the first optima! set of profiles for evaluation of a reactor. Based on this qualitative conclusions about a further method for practical accomplishment of the result of the optimization could be drawn. The Patent application NO 1 99 6318 made another step forward and in addition described a quantitative method for analysis of reactor and heat exchanger together. In the present application we take another step forward in relation to known art and describe a simplified method/procedure to find the same result The similarity between the present invention and the known technology, Patent application NO 1999 6318, is that the object of the invention is the same, but the method for achieving the answer is different. The standard profiles which are determined by conservation equations for mass, energy and momentum, constitute that which we call the reference reactor also in the present invention. The invention concerns, however, a new method to find new profiles which are compatible with a lower net energy demand than that of the reference reactor.

Object of the invention

An object of the present invention is the further development of known technology to utilize knowledge about calculations of reference reactors and entropy production for one or more reactors to find a path of operation that has minimum entropy production, by specifying requirements for the production and the heat transfer in a new manner/method. Minimum entropy production is equivalent to minimum exergy loss, minimum lost work or minimum energy dissipation. We define an operation with minimum entropy production to be an energy-optimal operation forgiven requirements.

The present method is not connected to the principle of equipartition of forces, which principle is generally derived for linear transport processes (Sauar et al, 1996); especially for transport of heat and mass in distillation towers (Ratkje et al, 1995). The present invention concerns a new method of optimization to determine the energy-optimal operation of chemical reactors with heat exchangers, using irreversible thermodynamics as a starting point.

The present invention is built on the method of minimizing the entropy production in one or more chemical reactors with or without heat exchangers, where feed reactants are converted to output products, and where requirements are made to the yield of a particular product J. The novel is that now requirements are made that the energy balance and the mass balance basically have to be fulfilled at any location in the reactor. The present invention thus concerns a method to optimise/minimize the total entropy production in one or more chemical reactors with their heat exchanger(s),

- where a first number of reagents fed to the reactor are converted to another number of output products, and where the yield of one of the output products is kept at a predetermined value (J), and

- where the degree of conversion, X ,, for reactants and intermediates, pressure (p₀), and temperature (To), or boundary conditions at the inlet are known, and

- where the reference profiles for degree of conversion X(x), pressure ρ(x), and temperature T(x) are calculated by solving the conservation equations for mass, energy and momentum, and

- where the requirements for energy balance and mass balance as a basis are fulfilled in discrete volume elements of the reactor, and

- where an optimal temperature profile in the longitudinal direction of the reactor is calculated, i.e. the temperature of the cooling water in a case with an exothermic reactor, possibly the flame temperature profile in the case of an endothermic reactor,

- where a first optimal set of profiles for reaction, pressure and temperature are calculated based on the reference profiles, by Euler-Lagraπge optimization of the entropy production for the reaction(s), with Lagrange multiplier λi to take care of the requirement for constant production, J;

- where a second optimal set of profiles for reaction, pressure and temperature are calculated based on the reference profiles, by Euler-Lagraπge optimization of the entropy production for the reaction(s), with Lagrange multiplier λz to take care of the requirement of mass conservation, - where a third optimal set of profiles for reaction, pressure and temperature are calculated based on the reference profiles, by Euler-Lagrange optimization of the entropy production for the reaction(s), with Lagrange multiplier λ₃ to take care of the requirement of energy conservation.

Optimal profiles in this connection is meant to be understood as the profiles which give the minimum total entropy production. Corresponding profiles for reaction and possibly pressure follow from the temperature profile if the temperature is chosen as degree of freedom. The concentration can also be used as degree of freedom. The invention mainly deals with situations wherein chemical reaction and heat exchange take place far from equilibrium. The method is characterised in that the driving force for the reaction is not constant throughout the reactor (as described in known technology), and by the fact that driving force is the same as that obtained in a different and more difficult way in the earlier Patent application 1999 6318.

The reactor operation which occurs using the novel set of profiles for the reactor instead of for example the reference set of profiles, will give less energy costs for each produced unrt For an exothermic reaction this means that the cooling water from 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 further be obtained from high temperature heat, or low temperature heat can be better utilized 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). 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 which shall occur in this place to give minimum entropy production in the total system.

The energy optimization 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 and pressure.

The importance of the invention

In practice the invention will tell how to alter 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, several smaller reactors in series, artering 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):

_ ... max a w " max

(1)

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 l Vmar. 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 more methods exist 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-minimal operation and design

The present invention uses the theory of irreversible thermodynamics as basis. This theory defines fluxes and forces in the non-equilibrium system to be investigated. The method is described by a general example as follows:

Local entropy production per unit of time and volume due to chemical reaction and heat exchange is defined by de Groot and Mazur (1962):

(2) The rate of the chemical reaction is r and -ΔG/ is the driving force for the reaction. Here ΔG is the Gibbs energy of the reaction, and T is the local temperature. The heat flux transferred to the cooler is J_q , and 7^" _a is the temperature of the coolant The symbols ε and p are the catalyst porosity and density, respectively. More than one reaction can be present. The constant a gives the relation between the reactor volume and the area of the walls of the reactor. In a tubular reactor, a~4/ά, where d is the diameter of the tube.

We see that heat that is reversibly transferred (at constant temperature) gives a zero contribution to the entropy production. Similarly, a reaction at equilibrium also gives zero contribution to the entropy production. A chemical reaction cannot proceed at equilibrium (unless in an electrochemical cell). This explains why the entropy production can be large in chemical reactors.

The total entropy production is obtained by integrating over the reactor volume and time. In the case of a stationary state tube reactor, all results can be stated integrated over the catalyst weight V.

(3) This is the quantity we wish to minimize, as it leads to lost exergy (lost work) for a given production J. The production is given by

(4) Characteristic for each reaction is also a mass-balance which has to be fulfilled at any place. Take as an example the ammonia-synthesis:

2 ² 2 ^J

Then the mass balance can be written on local form as:

(5) The degree of conversion for the reaction is ξ and FHQ is the flow of hydrogen at the inlet of the reactor. The function B is defined by this equation. The energy balance for the same example on local form is:

(6) Here U is the heat transfer coefficient, L H is the enthalpy of reaction, and C_p is the heat capacity at constant pressure. The function A is defined by this equation.

The optimization is formulated in a well-known mathematical way, using the method of Euler-Lagraπge with constant multipliers functions. There is one multiplier for the constant production, namely λi. and one multiplier for the specified heat transfer. Since the driving forces are independent of each other, we obtain the following standard mathematical expression for the Lagrange function.

L= θdV+ l ~-λdy + l -S ]dV-λ{j~ \rdv) v dV J ^v dV J ' This function shall now be derived with respect to the free variable chosen. If we choose to produce more high quality heat in the cooling water, we do the derivation with respect to the temperature of the cooling water. The example given below uses this type of derivation.

The expression is complicated, but can be numerically determined, when the expressions for the reaction rates and the other variables are known as functions of accumulated amount of catalyst (Nummedal 2001). The solution procedure is described on the flow-sheet in Figure 7. The procedure starts by dividing the reactor into a finite number of volume elements. The volume elements do not need to be of equal magnitude. The expressions are then discretized, by replacing all differentials by differences. The reference reactor is first calculated (item 1, described to the right of Figure 7), thereafter the optimization follows (item 2, described in the center of Figure 7). The result is controlled by use of the opimal boundary conditions i the equations of item 1.

The equation above describes how the force should be so that the reactor shall be as entropϊ optimal (energy effective) as possible with a given production J, see Patent application NO 1998 2798. This force is not necessarily equipartitioned as for chemical reactions close to equilibrium (Sauar et al., 997)

The results from a calculation of this type shall be described in more detail in the following example of the ammonia synthesis.

The method as applied to the ammonia synthesis

In order to find minimum total entropy production in the process, that is in the reactor and connected heat exchangers, the procedure shown below is followed for the ammonia synthesis. This synthesis has a reaction and heat transfer to the surroundings (it is an exothermic reaction). The reaction equation is given above. 1) Establishment of reference profiles

Temperature, pressure and concentration profiles in the reactor with heat exchangers) are determined from the conservation equations for mass,

5 momentum and energy. The result is called the set of reference profiles.

The calculations require knowledge of inlet conditions and temperature in the heat exchanger. It is also necessary to know how the rate of reaction varies with temperature, concentration and pressure. The calculations are standard, see Fogler (1992), and his example for oxidation of SO₂. Foglers procedure is used to o calculate the reference profiles for the ammonia synthesis, using reaction kinetic information from Nielsen (1968) and thermodynamic data.

On the background of these results the entropy production, the reaction velocity, the temperature and the degree of conversion on every site in the reactor, for the s reaction and for the heat transfer are calculated. The entropy production due to pressure gradient and friction losses can here be neglected.

The reference profiles are shown together with the optimal profiles in Figures 1-6.

0 2) Establishment of optimal profiles

Temperature, pressure and concentration profiles in the reactor(s) with heat exchangers) are now determined, using the optimization method with three Lagrange parameters. The calculation is carried out with one degree of freedom in 5 the system, namely the temperature of the cooling water. All derivations have been carried out with respect to this variable. Other variables like the concentration or the distribution of catalyst are also relevant

The whole profile for the temperature of the cooling water throughout the reactor o was varied numerically, until! the entropy production had reached a stable minimum. The calculations can e.g. be performed using Matlab 5.3.1. (R11.1), optimization toolbox function fmincon by The MathWorks, Inc., or with own algorithms. The flow sheet for the calculations is shown in Figure 7. The results from the optimization are labelled "min" in the figures. The figures are compared to the reference profiles below.

The results that show the potential saving of high quality heat are summarized in Table 1,

We see from the table that the entropy production per unit produced ammonia does not vary much from the reference reactor to the optimal reactor. It is the way that the heat transfer is done, that is crucial for the potential saving. The table shows that the potential savings are highest when heat transfer coefficients are high.

Table 1. Entropy production for reference reactor and optimal reactor for some cases with different heat transfer coefficients.

The calculations give quantitative advice on how the temperature profile of the cooling water should be in relation to the reactor temperature in order to save al the energy that is possible to save. This is shown in Figures 1-3 for the three cases described above.

Figure 1 shows the temperature profile in reactor and cooling water for the reference reactor (ref) and for the optimal reactor (min) when the heat transfer coefficient is 100 W/K m²

Figure 2 shows the temperature profile in reactor and cooling water for the reference reactor (ref) and for the optimal reactor (min) when the heat transfer coefficient is 200 W/K m² Figure 3 shows the temperature profile in reactor and cooling water for the reference reactor (ref) and for the optimal reactor (min) when the heat transfer coefficient is 400 W/K m²

The figures 1-3 show the cooling water temperature as well as the reactor temperature. We see that the temperature profiles are approximately parallel in that part of the reactor in all three cases. The size of this part of the reactor increases wrth increasing heat transfer coefficient. This can be explained by the fact that an infinite value for the heat transfer coefficient moves against a boundary limit where minimum entropy production for the reaction dominates the profile. The temperature profiles become similar to the ones we find by optimising the entropy production for the reaction as such.

Important is that the method predicts new inlet conditions for the feed gas. The optimization gives as a result that the composition at the inlet of the reference reactor has to be altered. These changes are illustrated on Figure 4, that shows the degree of conversion of hydrogen. The solution cannot be characterised by a constant driving force, neither for chemical reaction (not shown), nor for heat exchange (Figure 5). Also, the entropy production is not constant through the reactor (Figure 6).

Figure 4 shows the degree of conversion of hydrogen in the reference reactor and in the reactor with minimum entropy production. The compositions at the inlet and at the outlet are altered, but the amount of product produced is the same.

Figure 5 shows thermal driving force through the reactor in the reference reactor and in the optimal reactor.

Figure 6 shows the entropy production in the reference reactor and in the optimal reactor .

Figure 7 shows a flow chart for the solution procedure. FIGURE LIST

Figures 1-6 concern the ammonia synthesis.

The reference profiles for the ammonia synthesis are labelled ref. The profiles have been determined with data from Nielsen (1968). They have been obtained by a standard method by solving conservation equations for mass, momentum and energy with given limiting conditions. The profiles, representing known technique, are the basis for the method.

Optimal profiles for the ammonia synthesis are labelled min. The profiles have been determined by the method described above.

Figure 1. Temperature profile in reactor and cooling water for the reference reactor (ref) and for the optimal reactor (min) when the heat transfer coefficient is 100 W K m²

Figure 2. Temperature profile in reactor and cooling water for the reference reactor (ref) and for the optimal reactor (min) when the heat transfer coefficient is 200 W/K m²

Figure 3. Temperature profile in reactor and cooling water for the reference reactor (ref) and for the optimal reactor (min) when the heat transfer coefficient is 400 W K m²

Figure 4. Degree of conversion of hydrogen in the reference reactor and in the reactor with minimum entropy production. The compositions at the inlet and at the outlet are altered, but the amount of product produced is the same.

Figure 5. Thermal driving force through the reactor in the reference reactor and in the optimal reactor.

Figure 6. Entropy production in the reference reactor and in the optimal reactor

Figure 7. Ffow chart for the solution procedure. LIST OF SYMBOLS

A Component

B Component

ΔEx Exergy change J/mol

F Flow mol/rn³

ΔG Change in Gibbs energy J/mol

J Yield kmol/h

K Equilibrium constant atm"¹-⁵

L Reactor length m

P Pressure atm

R Gas constant J/(K mol)

S Surface area m²

T Temperature K

U Heat transfer coefficient W/(Km²)

V Catalyst weight kg w Work J/mol

X Thermal driving force K-¹

X Degree of conversion c General concentration k General rate constant k Rate constant mol/(g kat s atm) r Reaction rate mol/(m³ s)

X Reactor position r Parameter

Θ Total entropy production per mole produced J/K

Ω Reactor cross sectional area m δ Functional derivative ε Catalyst porosity η Efficiency (2nd law) θ Local entropy production J/(Km³ s) λ Lagrange multiplicator μ Chemical potential J/mol

P Bulk density kg/m³ d Partially derived

Subscript/superscript eg Equilibrium a Ambient r Chemical reaction max Maximum n Subvolume n

0 Surroundings LITERATURE LIST

De Groot, S.R. and Mazur, P.: Non-equilibrium theπnodynamics, 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 equlϊbrium, not published. RA Leiden, The Netherlands, 1997.

Nummedal, Lars. Dr., Study, in preparation 2001

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, $29-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

P a t e n t c l a i m s

1. Process for optimization/minimization of the total entropy production in one or more chemical reactors with connected heat exchangers), wherein - a first number of reactants, fed to the reactor, are converted into another number of output products,

- where the yield of one particular output product is kept at a predetermined value (J), and

- where the degree of conversion (Xo) for reactants and intermediates, pressure (Po), and temperature (T₀), or boundary conditions at the inlet are known, and

- where the reference profiles for the degree of conversion X(x), pressure p(x), and temperature T(x) are calculated by solving conservation equations for mass, momentum and energy, c h a r a c t e r i s e d i n t h a t - the requirements for energy balance and mass balance as a basis are fulfilled in discrete volume elements of the reactor, and

- an optimal temperature profile in the longitudinal direction of the reactor is calculated, i.e. the temperature of the cooling water in a case with an exothermic reactor, possibly the flame temperature profile in the case of an endothermic reactor,

- a first optimal set of profiles for reaction, pressure and temperature are calculated based on the reference profiles, by Euler-Lagrange optimization of the entropy production for the reactϊon(s), with Lagrange multiplier λi for the requirement of constant production J, - a second optimal set of profiles for reaction, pressure and temperature are calculated based on the reference profiles, by Euler-Lagrange optimization of the entropy production for the reaction(s), with Lagrange multiplier λ₂ for the requirement of mass conservation,

- a third optimal set of profiles for reaction, pressure and temperature are calculated based on the reference profiles, by Euler-Lagrange optimization of the entropy production for the reaction(s), with Lagrange multiplier λ₃ for the requirement of energy conservation.

2. Process according to claim 1, characterised rn that the reactor is operated far from equilibrium, but on average closer to equilibrium with the optimal set of profiles than with the reference profiles, s

3. Process according to dalm 1 _τ characterised in that it only in exceptional cases is operated with maximum reaction rate.

o_. 4. Process according to claim 1-3, characterised in that the driving force for the reaction, defined as -ΔGT. where ΔG is the Gibbs energy for the reaction, given by chemical potentials for reactants and products, and where T is temperature, is more evenly distributed in the optimal reactor than in the reference reactor. 5

5. Process according to claim 1 , characterised in that the force for heat transfer Is constant in bigger parts of the reactor when the heat transfer coefficient Is increasing.

0 6. Process according to claim 1 , characterised in that the procedures do not depend on the presence of parallel reactions.

7. Process according to claim 1 , s characterised in that result of the optimization is independent of a different choice for the reference reactor.

8. Process according to claims 1-7. characterised in that in exothermic reactors the cooling water comes out o with a generally higher temperature, or in bigger amounts in the optimal reactors) than in the reference reactor.

9. Process according to claims 1-7, characterised in that the heating in endothermic reactor(s) is taking place s at relatively lower temperature, possibly with a lower amount of heat in the optimal reactor(s) than in the reference reactor.