Mathematical modelling related to OptimalReactor Design

Mathematical modelling related to OptimalReactor Design (A study Report)

AbstractA reactor is the heart of a chemical process industry. So its design has a vital role. In its design, the reactor geometry is taken as the most important factor. The reactor geometry means to get the minimum reactor size (reactor volume) for the maximum yield; either it is a single reactor or the different reactor combinations of same/different sizes.In the optimization of the reactor design, we establish the rules for good reactor behavior by minimizing the overall cost and maximizing the yield and ultimately the profit by correlating different design parameters with the economic factors. In this report, some of these rules are elaborated.In the last, the major findings of the whole report are briefly discussed

Contents1.Scope of Optimization in Chemical Reactor Design12.Optimum Performance23. Different Cases2a) Optimum operation of reversible reaction in an adiabatic reactor design2b) Determining the best system for the given conversion4c) Optimum Recycle Ratio6d) Optimum Temperature progression7e) Maximizing the desired product in series reaction7f) Optimum Cycle period with Downtime8g) Optimum Reaction Temperature with Downtime9h) Minimum residence time in a PFR11i) Optimal Temperature for Staged Packed Bed reactor12j) Autocatalytic reactions with recycle14k) Optimum operation of Fermenters15l) Optimization of the Process Conditions to get maximum yield15m) Optimization of the reactor volume to get maximum yield15Summary164. Optimum Operation of Reactors18Findings20Appendix21

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1. Scope of Optimization in Chemical Reactor Design[footnoteRef:2] [2: Coulson and Richardsons Chemical Engineering series, R K Sinnott, Volume 6th, Edition 4th, Page # 486.]

A general procedure for reactor design is outlined below Collect together all the kinetic and thermodynamic data on the desired reaction and the side reactions. It is unlikely that much useful information will be gleaned from a literature search as little is published in the open literature on commercially attractive processes. The kinetic data required for reactor design will normally be obtained from laboratory and pilot plant studies. Values will be needed for the rate of reaction over a range of operating conditions: pressure, temperature, flow-rate and catalyst concentration. Collect the physical property data required for the design; either from the literature, by estimation or, if necessary, by laboratory measurements. Identify the predominant rate-controlling mechanism: kinetic, mass or heat transfer. Choose a suitable reactor type, based on experience with similar reactions, or from the laboratory and pilot plant work. Make an initial selection of the reactor conditions to give the desired conversion and yield. Size the reactor and estimate its performance. Exact analytical solutions of the design relationships are rarely possible; semi-empirical methods based on the analysis of idealized reactors will normally have to be used. Select suitable materials of construction. Make a preliminary mechanical design for the reactor: the vessel design, heat-transfer surfaces, internals and general arrangement. Cost the proposed design, capital and operating, and repeat steps 4 to 8, as necessary, to optimize the design.In choosing the reactor conditions, particularly the conversion, and optimizing the design, the interaction of the reactor design with the other process operations must not be overlooked. The degree of conversion of raw materials in the reactor will determine the size, and cost, of any equipment needed to separate and recycle unreacted materials. In these circumstances the reactor and associated equipment must be optimized as a unit.2. Optimum Performance[footnoteRef:3] [3: Chemical Engineers Handbook, Perry, Volume 1, Edition 7, Page # 7-25.]

A few specific conclusions about optimum performance can be stated: The minimum total volume of a CSTR battery for first-order reaction, and near-minimum for second order, is obtained when all vessels are the same size. An economical optimum number of CSTRs and their auxiliaries in series is 4 to 5. In a sequence of PFR and CSTR, better performance is obtained with the PFR last. Performance of reversible reactions is improved with the CSTR at a higher temperature. For the consecutive reactions A B C, a higher yield of intermediate B is obtained in batch reactors or PFRs than in CSTRs. When the desirable product of a complex reaction is favored by a high concentration of some reactant, batch or semi-batch reactors can be made superior to CSTRs. Conversion by a reversible reaction is enhanced by starting out at high temperature and ending at low temperature if equilibrium conversion drops off at high temperature. For a reversible reaction, the minimum size or maximum con-version is obtained when the rate of reaction is kept at a maximum ateach conversion by adjustment of the temperature.3. Different Casesa) Optimum operation of reversible reaction in an adiabatic reactor design[footnoteRef:4] [4: Elements of Chemical Reaction Engineering, H. Scott Fogler, Edition 3rd, Page # 486. Chemical Engineers Handbook, Perry, Edition 7thPage # 7-16,7-17.]

Consider an adiabatic reactor in which a reversible exothermic reaction is being conducted by the variation in the feed temperature; there are following two cases Using a very high feed temperature the specific reaction will be close to zero, consequently very little product will be produced. Using a very low feed temperature the specific reaction rate will be small that virtually all of the reactant will pass through the reactor without reacting.Between these two extremes, there must be an optimum feed temperature that maximizes conversion. This optimum inlet temperature is shown in the figure 1.1.

Figure 1.1:Finding the optimum feed temperature.For exothermic reactions, while the temperature and conversion increase down the length of reactor, the equilibrium conversion decreases. To determine the maximum conversion that can be achieved in an exothermic reaction carried out adiabatically, we find the intersection of the equilibrium conversion as a function of temperature with temperature-conversion relationships from the energy balance in figure 1.2.

Figure 1.2:Graphical solution of equilibrium and energy balance equations to obtain adiabatic temperature and equilibrium conversion.Variation of equilibrium conversion with temperature is shown in the figure 1.3.

Figure 1.3: Variation of equilibrium conversion with temperature for an exothermic reaction.Take an example of first order reversible chemical process.

Taking,

The optimum temperature is found as,

Where,A1 = frequency factor for forward reactionB1 = Activation energy for forward reactionA2 = frequency factor for backward reactionB2 = Activation energy for backward reactionb) Determining the best system for the given conversion[footnoteRef:5] [5: Chemical Reaction Engineering, Octave Levenspiel, Edition 3rd, Page # 133-134.]

Suppose we want to find the minimum size of two MFR in sense to achieve the specified conversion of feed. These are given in the following figure 1.4.

Figure 1.4: The graphical representation of the variables for two mixed flow reactors in series.This figure 1.5 shows that the total reactor volume is as small as possible (total shaded area is minimized) when the rectangle KLMN is as large as possible which is called the maximization of rectangles.

Figure 1.5: Graphical procedure for the maximization of the rectangles.

The area of the rectangle is according to equation 3; the area is maximized when M is at that point where the curve equals the slope of the diagonal NL of the rectangle.The optimum size ratio of the two reactors is achieved where the slope of the rate curve at M equals the diagonal NL. The best value of M is shown in figure 1.5; this determines the intermediate conversion X1 as well as the size of units needed. From this study some special cases arises: For the special case of first-order reactions equal size reactors are the best. For the reaction orders n>1, the small reactor should come first. For n0), PFR is always more efficient than MFR but a PFR will not operate at all with a feed of pure reactant, so feed must be continually primed with product. i-e PFR with some recycle must be used.The optimum recycle ratio can be found as following which a recycle for which the reactor volume is minimized or the space time. As we know that for any PFR with recycle.

Figure 1.6:Correct recycle ratio compared with recycle ratios which are too high and too low.In words, the optimum recycle ratio introduces to the reactor a feed whose 1/rA value (KL in figure 1.6) equals the average - 1/rA value in the reactor as a whole (PQ in figure 1.6).

d) Optimum Temperature progression[footnoteRef:7] [7: Chemical Reaction Engineering, Octave Levenspiel, Edition 3rd, Page # 219-220.]

It is the progression for which V/FAo for a given conversion of reactant is minimized. At any composition it always be at that temperature where the rate is maximum. The locus of the maximum rates is found by examining the r(T,C) curves of the figure 1.7 with different cases.

Figure 1.7: Operating lines for minimum reactor size.Initial feed temperatureIn the adiabatic reactor operation the temperature inside the reactor is controlled by the initial feed temperature. The following heat balance equation (point slope form) over a reactor is used to find the initial feed temperature.

Where,XA1 and XA2 are the initial and final conversions of a key component A.Cp is the heat capacity of the reactants.Hr is the heat of reaction.T2 is the required temperature inside the reactor.T1 is the initial feed temperature

For an exothermic reaction T1 is always lower than T2 while for an endothermic reaction T2 is greater than T1.In the adiabatic reactor in which the heat is exchanged with the surroundings, the equation 5 is modified as follows.

Where Q is the heat exchanged with the surroundings.

e) Maximizing the desired product in series reaction[footnoteRef:8] [8: Chemical Reaction Engineering, Octave Levenspiel, Edition 3rd, Page # 53-56.Elements of Chemical Reaction Engineering, H. Scott Fogler, Edition 2nd, Page # 461.]

Let the consecutive uni-molecular type first order reaction,

Where R is the desired product.The values of k1& k2 give the location and maximum conversion of R. This may be found by differentiating the equation 7,8&9.

The final result is given as,

Figure 1.8: Typical concentration-time curves for consecutive first-order reaction.

f) Optimum Cycle period with Downtime[footnoteRef:9] [9: Chemical Engineers Handbook, Perry, Volume 1, Edition 7, Page # 7-29, 7-30.]

The optimum cycle period for a first order batch reaction with a downtime of h per batch can be found as:

The number of daily batches can be found as,

The daily yield is found as,

Where,k = rate constantVr = Volume of the reactorCo = Initial molar concentrationThe ordinate of the plot is which is proportional to the daily yield. The peaks in the curve are at these values of the parameters (figure 1.9).

Figure 1.9:Plot at different values of the parameters.g) Optimum Reaction Temperature with Downtime[footnoteRef:10] [10: Chemical Engineers Handbook, Perry, Volume 1, Edition 7, Page # 7-30.]

A liquid phase reaction has the rate equation as follows,

Where, = fractional conversionCAo = 1k = Ke = The downtime is 1 h per batch. Find the temperature at which the daily production is a maximum.The reaction time of one batch is,

The number of daily batches can be found as,

Daily production is,

Maximize as a function of temperature. Equation 13 is integrated with POLYMATH for several temperatures and results are plotted. The tabulation gives the integration at 550 K. the peak value of at 550K, tb=0.6 and =0.3105.The maximum daily production is thus,

The plot is shown in the figure 1.10.

Figure 1.10

h) Minimum residence time in a PFR[footnoteRef:11] [11: Chemical Engineers Handbook, Perry, Volume 1, Edition 7, Page # 7-32.]

For a given reaction is conducted in a plug flow reactor, the rate equation is.

Where,CAo = 4k = Ke = The condition for minimum at X= 0.80, the flow reactor equation is

The plot of this equation shows the minimum to be at T=340 K in figure 1.11

Figure 1.11i) Optimal Temperature for Staged Packed Bed reactor[footnoteRef:12] [12: Chemical Reaction Engineering, Octave Levenspiel, Edition 3rd, Page # 431-432.]

For any preset number of stages the optimization of operations reduces to minimizing the total amount of catalyst needed to achieve a given conversion. Let us illustrate the procedure for two-stage operation with reversible exothermic reactions.The method of attack is shown in Fig. 1.12. In this figure we wish to minimize the total area under the versus XA curve in going from XA = 0 to XA2 = some fixed or required conversion. In searching for this optimum we have three variables which we can set at will: the incoming temperature (point Ta), the amount of catalyst used in the first stage (locates point b along the adiabatic), and the amount of intercooling (locates point c along the bc line). Fortunately, we are able to reduce this 3-dimensional search (5-dimensional for three stages, etc.) to a one-dimensional search where Ta alone is guessed. The procedure is as follows:

Figure 1.12:Optimum two-stage packed bed reactor. Guess Ta. Move along the adiabatic line until the following condition is satisfied:

This gives point b in Fig. 1.12, thus the amount of catalyst needed in the first stage as well as the outlet temperature from that stage. Especially in preliminary design it may not be convenient to use the criterion of Eq. 14. A simple alternative is a trial-and-error search. Usually two or three carefully chosen trials keeping away from low rate conditions will yield a good design, close to the optimum. Cool to point c which has the same rate of reaction as point b; thus (-ra)leaving a reactor = (-ra)entering the next reactor Move along the adiabatic from point c until the criterion of Eq. 14 is satisfied, giving point d. If point d is at the desired final conversion then we have guessed Ta correctly. If point d is not at the desired final conversion try a different incoming temperature Ta. Usually three trials will very closely approach the optimum.Figure 1.13: Sketch showing how staged packed beds can closely approach the optimal temperature progression.j) Autocatalytic reactions with recycle[footnoteRef:13] [13: Chemical Engineers Handbook, Perry, Edition 7, Volume 1, Page # 7-32.]

Part of the effluent from a PFR is returned to the inlet. The recycle ratio is R, fresh feed rate is F0.

The rate equation for an auto catalytic reaction in recycle reactor is written as follows

Figure 1.14k) Optimum operation of Fermenters[footnoteRef:14] [14: Chemical Reaction Engineering, Octave Levenspiel, Edition 3rdPage # 636.]

With poison-free Monod kinetics and a given feed, we have a U-shaped versus CA curve, as shown in Fig. 1.15. With this form of rate-concentration curve we should operate as follows: To reach any point between A and B run part of the feed at A in mixed flow and mix with the rest of the feed. To reach any point between A and 0 go directly to point A in mixed flow, then use plug flow beyond A. These two rules represent the key to optimum reactor behavior.

Figure 1.15:Rate concentration behavior of Monod kinetics.l) Optimization of the Process Conditions to get maximum yieldThe process of ammonia synthesis was developed by German chemist F. Haber and first used in 1933. The chemical Eq. is as follow:-

By seeing the balanced chemical equation, we can have three ways to maximize the yield of ammonia:- By continual withdraw of ammonia after intervals, the equilibrium will shift to forward direction in accordance with Le-Chateliers Principle Increase the pressure to decrease the volume of the reaction vessel. Four moles of the reactants consign to give the two moles of the products. High pressure will shift the equilibrium position to sight to give more and more ammonia Decreasing the temperature will shift it to the forward direction according to Le-Chateliers .One needs a compromise to optimize the yield and the rate. The temperature is raised to a moderate level and a catalyst is employed to increase the rate. If one wants to achieve the same rate without a catalyst, then it requires much high temperature, which lowers the yield. Hence the optimum conditions are the pressure of 200-300 atm and temperature around 673K (400 oC). The catalyst is the piece of iron catalysts emsedeled in a fused mixture of MgO, Al2O3 and SiO2. The equilibrium mixture has 35 % by volume of ammonia. The mixture is cooled by refrigeration coils until ammonia condenses (B.P = -33.4 C) and is removed. Since B.P`s of N2 and H2 are very low, they remain in the gaseous state and are recycled by pumps back into the reaction chamber.Temperature (K)KC

2007.17 x 1015

3002.69 x 108

4003.94 x 101

5001.72 x 102

6004.53 x 100

7002.96 x 10-1

8003.96 x 10-2

Table 1.1Figure 1.16

High pressure, low temperature with the use of catalyst and continual removal of NH3 will give the maximum yield of ammonia

m) Optimization of the reactor volume to get maximum yield[footnoteRef:15] [15: Chemical Reaction Engineering, Octave Levenspiel, Edition 3rd, Page # 243.]

The elementary reactions,

called the Trambouze reactions are to be run in four equal-size MFR's (mixed flow reactors), connected any way you wish. The feed is CAo = 1, the feed flow rate is = 100 liters/min. The best scheme that the computer could come up with to maximize the fractional yield of S, or (S/A) & optimize the volume of four reactors.

Figure 1.17First of all, the computer solution looks somewhat complicated from the engineering point of view. But never mind; let us proceed with our calculations. The instantaneous fractional yield, (S/A), is

Thus the best way of running these four reactors is to keep the conditions at the optimum in all four units.

Figure 1.17The volume of MFR comes from the performance equation.

Summary[footnoteRef:16] [16: Chemical Reaction Engineering, Octave Levenspiel Edition 3rd,Chapter # 10.]

For the optimum reactor system, the rules for good reactor behavior for both homogenous and heterogeneous systems are summarized as follows: Rule 1. For Single Reactions To minimize the reactor volume, keep the concentration as high as possible for a reactant whose order is n > 0. For components where n < 0, keep the concentration low. Rule 2. For Reactions in Series Consider reactions in series, as shown:

To maximize any intermediate, do not mix fluids that have different concentrations of the active ingredients-reactant or intermediates. See Figure. 1.17. Rule 3. For Parallel Reactions Consider the parallel reactions with reaction orders ni

To get the best product distribution, Low CA, favors the reaction of lowest order. High CA, favors the reaction of highest order. If the desired reaction is of intermediate order then some intermediate CA, will give the best product distribution. For reactions all of the same order the product distribution is not affected by the concentration level.

Figure 1.17:(a) Plug flow (no intermixing) gives the most of all the intermediates. (b) Intermixing depresses the formation of all intermediates.

Rule 4. Complex Reactions These networks can be analyzed by breaking them down into their simple series and simple parallel components. For example, for the following elementary reactions, where R is the desired product, the breakdown is as follows:

This breakdown means that A and R should be in plug flow, without any recycle, while B can be introduced as you wish, at any concentration level, since it will not affect the product distribution. Rule 5. Continuous versus Non-continuous Operations Any product distribution that can be obtained in continuous steady-state flow operations can be gotten in a non-flow operation and vice versa. Figure 1.18illustrates this.

Figure 1.18:Correspondence between the residence time distribution of steady flow and either non-flow, batch or semi-batch systems Rule 6. Effect of Temperature on Product Distribution Given

a high temperature favors the reaction with larger E, while a low temperature favors the reaction with smaller E. Let us now see how these six rules can be used to guide us to the optimum.4. Optimum Operation of ReactorsIn reactor operations the word "optimum" can have different meanings. Let us look at two definitions which are particularly useful. Feed a stream containing reactant A to a reactor and let R, S, T,... be formed, with R being the desired product. Then by optimum We could mean maximizing the overall fractional yield of R, or

We could mean running the reactor system so that the production of R is maximized, or

For reactions in series we calculate the maximum production rate of R directly. However, for reactions in parallel we find it useful to first evaluate the instantaneous fractional yield of R, or

and then proceed to find the optimum. If unused reactant can be separated from the exit stream, re-concentrated to feed conditions and then recycled, then

Findings[footnoteRef:17] [17: Chemical Reaction Engineering by Octave Levenspiel,Edition 3rdElements of Chemical Reaction Engineering by H. Scott Fogler.Edition 2nd,Coulson and Richardsons Chemical Engineering series by R K Sinnott, Volume 6th, Edition 4th.Chemical Engineers Handbook, Perry, & Section 7, Volume 1, Edition 7.]

As for the optimal reactor design is concerned, my findings in this report are described as follows: Generally the isothermal tubular reactor (PFR) is better than the CSTR for the same conversion for all types of reactions because of its smaller volume. For Autocatalytic reactions, mixed flow reactor (CSTR) is more efficient at low conversions; (PFR) is more efficient at high conversions. For all types of Auto-thermal (in which heat may be considered to be the product which sustains in the reaction) reactions CSTR will be preferred because with PFR the reaction will die out. For all types of ordinary nth order reactions (n>0), PFR is always more efficient than CSTR but a PFR will not operate at all with feed of pure reactants. So feed must be continually primed with product that PFR with some recycle (recycle reactor) must be used. For final conversion smaller than the point of maximum rate, PFR is better than any recycle reactor. For conversions higher than the point of maximum rate, the recycle reactor with proper recycle ratio is superior to rather the PFR or CSTR.

Appendix[1]Optimum Recycle Ratio

Differentiating both side

Here differentiating under integral sign

Here

=0

Now, put

So,

While,

Differentiate with respect to R

So, we can write as:

[2] Maximization of Desired product in series Rxn:

Initial Conditions

Integration of eq., we have

Resembles with first order differential equation:

Integrating factor method applied

Here integration factor

Equation becomes

At

For maximization put

[3] Optimum Cycle with downtimeFirst order reaction ; ;

Taking Number of daily batches

Daily yield