Ind. Eng. Chem. Res. 1999, 38, 3051-3055
3051
The Driving Force Distribution for Minimum Lost Work in a Chemical Reactor Close to and Far from Equilibrium. 2. Oxidation of SO2 Signe Kjelstrup* and Trond Vegard Island† Department of Physical Chemistry, Norwegian University of Science and Technology, 7034 Trondheim, Norway
Our mathematical procedure for the determination of the driving force distribution in a chemical reactor that has minimum entropy production for a given production rate is applied to the oxidation of SO2 to SO3. The force of reaction that gives the minimum total entropy production is much more constant through the reactor than is a force taken from a standard textbook example. The entropy production has a peak at the entrance of the reactor. The inverse temperature plot shows that the optimal force is nearly at equal distance from the equilibrium line in the end of the reactor. Two practical ways that do not include changes in the apparatus are suggested to minimize the actual entropy production of the reactor. Reductions of 5% and 21% are obtained. The ideal result suggests that there is room for further improvements, especially if the apparatus is changed. 1. Introduction In the theoretical part (part 1 of this paper), we have given a mathematical answer to the following question: Given the output J from a chemical reactor, how can we operate the reactor so that we obtain minimum lost work in the reactor? Ideal answers were found, in the way that we were able to specify the optimal driving force of the reaction and devise a way of implementing it. The ideal operating conditions that were obtained in this manner may be impossible or difficult to use in practice because of material limitations, etc. In this article, we shall therefore study the theory when the ideal situation is not achievable. We apply the theory to a particular case and see how the ideal answers from the mathematical analysis can help us to improve reactor operation. This work is therefore a practical follow-up of the earlier theoretical article. As an example, we shall study the oxidation of sulfur dioxide to trioxide over a vanadium pentoxide catalyst:
1 SO2 + O2 f SO3 2
(1)
Sulfur trioxide is used to produce sulfuric acid, one of the most common chemicals in the world today. The reaction is strongly exothermic, and the problem, to avoid overheating in the first phase of the reaction, is well-known; see for instance Ullmann’s Encyclopedia.7 The reaction is therefore normally carried out in the industry in several steps. The problem here is, however, not to optimize a state-of-the-art industrial reactor for sulfur dioxide oxidation, but to demonstrate how the new general method of part 1 can be used to reduce the lost work in a particular reactor. A complicated multistep reactor system becomes less transparent. As an example, we have thus chosen the tubular reactor that * To whom correspondence should be addressed. Telephone/ fax: 47-7359 4179/1676. E-mail:
[email protected]. † Present address: Elkem Carbon A/S, Fiskå Verk, Kristiansand, Norway.
is described in detail in a standard textbook.1 The reaction rate of eq 1 is rather complicated, but wellknown (see below). 2. The Optimization Criterion We summarize first the formulation of the optimization criterion that was presented in the preceding paper. The entropy production rate per unit volume of a chemical reactor is3
[ AT]
σ)r-
(2)
where r is the reaction rate and -A/T is its thermodynamic driving force. Here, A is the affinity and T is the temperature. The reaction rate is normally given by rate constants and concentrations in reaction kinetics. The affinity is equal to the Gibbs energy of reaction:
A ) ∆rG
(3)
The minimization problem, formulated as a Lagrange optimization problem, is
δ A δ (x) T
∫[-rAT + λr] dx′ ) 0
( )
(4)
where λ is the Lagrange multiplier for the condition that the output J is constant and equal to
∫
J ) Ω r dx′
(5)
where Ω is the (constant) cross-sectional area of the reactor. Since A/T has a local relation to c, we can differentiate with respect to c instead of with respect to A/T and obtain from eq 4
δ δc
∫[-rAT + λr] dx′ ) 0
10.1021/ie9807452 CCC: $18.00 © 1999 American Chemical Society Published on Web 07/15/1999
(6)
3052 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999
The solution can be worked out from this expression for a particular expression for r. The result is the optimal force on the reaction along the reactor. It is not easy to control the force as such. A more practical criterion was obtained by integrating the Gibbs-Helmholz’ equation from equilibrium (eq) to the optimal (opt) state:5
(AT)
∫
)-
opt
opt
eq
∆rH d
(T1)
(7)
We have used the condition that, at equilibrium, Aeq ) 0. The equilibrium temperature for a given conversion is found by using the condition r ) 0 in the equation below, eq 8. When we know the enthalpy of reaction as a function of temperature, we can then carry out the integral. Since (A/T)opt is known from eq 6, we can then find Topt. The equation can be used as long as feeding of the reactants only takes place at the inlet. One practical version of the optimization criterion that we shall use here is, therefore, with a constant output, how can we operate the reactor so that we obtain the highest possible temperature of the cooling medium in the heat exchangers that surround the reactor? We shall see how the ideal answer to this question can help us to improve the actual situation in the direction of the ideal operation.
pressure, p, as variables, we obtain
[
)( )
1 - X 0 0.91 - 0.5X p pSO2 X 1 - 0.0055X p0
rSO3 ) kF
(
(
)]
X (1 - X)Kp
2
(11)
where F is the mass density of the gas mixture. The affinity of the reaction and its driving force are equal to
1 A ) µSO3 - µSO2 - µO2 2 -
A ) T
( [(
-R ln
p0
1 - 0.055X () 1 -X X)(0.91 - 0.5X)
]
0.5
0 pSO p 2
)
- ln Kp (12)
where the chemical potential, µi of i, is the one for ideal gases in a mixture. We conclude by noting that r is a complicated function of p, T, and composition. It cannot be easily fit into the scheme required by irreversible thermodynamics, namely,
( AT)
3. The System
r)l-
The nonadiabatic tubular reactor that constitutes our system was described in detail.1 It consists of N ) 4631 cylindrical tubes, each of them packed with catalyst and surrounded by a constantly boiling liquid. The boiling temperature is 703 K. Sulfur dioxide and air (oxygen and nitrogen) are fed into the reactor at a total pressure, p0, in volume fractions ySO2 ) 0.11, and yO2 ) 0.10, and yN2 ) 0.79. The length of the tubes is 6.10 m, and their inner diameter is 7 mm. The empirical expression for the reaction rate accounts for the rate being limited by diffusion and by reaction kinetics:1
x [ ( )]
r ) -k
pSO2 pSO3
pO2 -
pSO3
2
(8)
pSO2Kp
The rate and the rate constant k have the dimension mol/g of catalyst, while the partial pressures, p, are given in bar. The equilibrium constant is not the thermodynamic constant, but is the one based on partial pressures, Kp (bar-0.5). The reaction rate is negligible below 670 K, and the temperature of the materials in the reactor should not exceed 880 K. The temperatures 670 and 880 K are therefore practical bounds on the real operation. By converting the expressions in Fogler1 into SI units, we have k as the following function of temperature:
97800 k ) exp - 110 ln(1.8T) + 913 T
[
]
- 11.2] [11800 T
Kp ) exp
with the coefficient l independent of -A/T. We shall therefore assume that the flux-force relationship is nonlinear, using expressions 11 and 12 for the rate and the driving force; see however ref 4. The degree of conversion and the pressure variation through the reactor, according to these equations and the balance equations for mass, momentum, and energy,1 are shown in Figures 1 and 2. The results are identical to those in the textbook:1 They agree within a numerical uncertainty of 0.5%. We see that 80% of the reaction is accomplished in the first 20% of the reactor length. The fall in pressure is almost linearly along the reactor. The minimization of the entropy production of the reactor uses the results in Figures 1 and 2 as a basis. 4. The Solution of the Optimization Problem By solving eq 6 with eqs 11 and 12, we obtain the unknown -(A/T)opt as a function of λ:
-
(AT)
opt
(10)
The volume flow into the reactor was 3590 kmol/h. By introducing the degree of conversion, X, and the
)-λ-
[
R
(9)
where T is the temperature in K. Similarly, Kp is the following function of temperature:
(13)
[
]
1 0.225 Γ + X(1 - X) (1 - 0.055X)(0.91 - 0.5X)
()
]
0 0.45pSO p 2X Γ 3 + + 2X(1 - X) (1 - 0.055X)2 p0 (1 - X)3Kp3 (14)
where 0 Γ ) pSO 2
(
)( ) (
)
0.91 - 0.5X p X 1 - 0.055X p0 (1 - X)Kp
2
This result for (A/T)opt can be combined with the condition (5) for a simultaneous determination of the two unknowns. The reactor was divided into 200
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3053
Figure 1. The degree of conversion of SO2 to SO3 as a function of position in the reactor.
Figure 3. The driving force of the reaction SO2 + 1/2O2 f SO3 as a function of position through the reactor for (a) the textbook example from Fogler1 and (b) a reactor with minimum entropy production.
Figure 2. The pressure of the gas mixture as a function of position in the reactor.
volume elements, and the combined set of equations were solved for the intensive variables, the reaction rate, the entropy production rate, the driving force in each volume element using Matlab version 4.2, and the Runge-Kutta-Fehlberg method for solving the differential equations. A more detailed procedure has been developed.2 5. Results and Discussion 5.1. The Actual and Optimal Driving Force Distributions. The driving force of the reaction that is calculated from eq 12 and the results of Figures 1 and 2 vary largely through the reactor, as shown in Figure 3. This actual force is very high at the inlet, and it passes a minimum in the center of the reactor, before it flattens out. The minimum is due to the high temperature produced from the exothermic reaction. The optimal driving force of the reaction, -(A/T)opt is compared to the actual force in the same figure. We see that the optimal force is small, around 10 J/K‚mol. It is almost constant through the reactor, about 20% higher at the inlet than at the outlet. Most of the variation occurs at the inlet. The higher value at the inlet can be connected to a larger affinity at this location. No minimum appears. The large deviation between the actual and the optimal force indicate that lost work can be saved, without changing the amount of product produced. Alternatively, more can be produced for the same lost work. The magnitude of the force is interesting. It is close to the value of the gas constant. Energies of the order 3/ kT where k is Boltzmann’s constant, or 3/ RT for a 2 2 mole of particles, are typical translational or vibrational energies. There is local equilibrium everywhere in the reactor, in the sense that the Gibbs equation and similar thermodynamic relations apply for the three indepen-
Figure 4. The entropy production rate for the reaction SO2 + 1/ O f SO as a function of position through the reactor. The 2 2 3 entropy production rate of the whole reactor is the minimum with the given production rate.
dent components. But there is no chemical equilibrium since the affinity of the reaction is non-zero (10 J/K‚ mol). For an alternative reactor design (see ref 1) we found λ around -22 J/K‚mol. This reactor was considered by Fogler to be not as good, and we see that it operates further away from equilibrium. The entropy production along the reactor that accompanies the optimal force is shown in Figure 4. The integral under the curve gives the total entropy production rate of the reactor, here, 9 J/K‚mol of SO3. The figure illustrates that the local entropy production rate varies largely, by a factor of 20. Most of the dramatic variation occurs in the first 20% length of the reactor. After 40% of the length, the reaction proceeds with an entropy production rate close to zero. Chemical equilibrium has not been reached. The small entropy production rate is therefore due to a small reaction rate. It has been maintained in the literature that the path of constant entropy production is the path that gives minimum entropy production.6 According to Figure 4, this is clearly not the case for the present optimization. Figure 3 may in principle be used to find an improved reactor operation, or a new design. The thermodynamic
3054 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 Table 1. Entropy Production and Lost Work as a Function of Operating Conditions Inlet Temperature of the Gases 778 Ka Σ (J/mol of SO3) Σexch (J/mol of SO3) Wlost (kJ/mol of SO3) P (kmol of SO3/h) reduction in Wlost (%)
reference
case 1
case 2
21.8 2.0 7.10 358 0
20.7 2.1 6.78 360 -5
16.1 2.8 5.6 375 -21
a Case 1: The first percent of the reactor is not cooled. Case 2: The temperature of the cooling water is raised from 703 to 722 K in addition to not cooling the first percent of the reactor.
Figure 5. The temperature profile of the reaction SO2 + 1/2O2 f SO3 as a function of position through the reactor for (a) the textbook example from Fogler1 and (b) a reactor with minimum entropy production.
Figure 6. The inverse optimum temperature and the inverse equilibrium temperature, as a function of position through the reactor.
driving force as such is, however, not a practical variable to control. It is the intensive thermodynamic variables that can be controlled in practice. We have therefore transformed the results of Figure 3 into a version that is more suitable for control (see below). 5.2. The Optimal Temperature Profile. The transformation of the results in Figure 3 into more practical ones was done using the equations described above, eq 7. Fogler’s conversion profile was used to find a first approximation to the temperature profiles. The optimal temperature profile for this conversion profile is shown in Figure 5. We see first that the theoretical temperature profile exceeds the temperature range that the materials of the reactor can stand for a long period. The optimum solution is therefore clearly not practical for the whole reactor. This could not be concluded already from Figure 3, demonstrating that Figure 5 can give a more practical representation of the optimal solution. The optimal temperature is compared to the temperature of Fogler’s example in Figure 5. We see that the real temperature is almost everywhere below the optimal temperature. We further see that a rise in the temperature in the beginning as well as in the end of the reactor is favorable. Figure 6 gives further illustration of eq 7. The inverse optimal temperature and the inverse equilibrium temperature are plotted across the reactor. The separation between the two lines is constant in the end of the reactor, confirming the results of Figure 4 that the
reaction here works close to equilibrium (the reaction rate is close to zero and the driving force is small). The optimum solution is defined here through a constant force, consistent with Figure 3. 5.3. Operating Conditions Leading to Reduced Lost Work. The results of Figure 5 gave the approximate temperature difference between the actual and the optimal temperature everywhere in the reactor. On one hand, we saw already that the optimal temperature profile was not obtainable in the beginning of the reactor because of material limitations. On the other hand, it is in the beginning of the reactor that most of the work is lost (see Figure 4). A practical solution is therefore to raise the temperature at the inlet to the maximum temperature that is recommended for the materials, and otherwise follow the optimal profile as close as possible. In practice, it will be difficult to maintain such a continuous variation in the temperature. The realistic situation is that one has one or a few heat exchangers at his/her disposal and is able to maintain a stepwise control of the temperature. The problem of dimensioning and positioning of heat exchangers is therefore an interesting one. In this paper, we limit ourselves to show that the entropy production or the lost work can indeed be reduced without investments in additional apparatuses, by changing only the operating conditions that bring the temperature profile or the actual driving force closer to the theoretical optimum. Consider therefore a rise in temperature in the first part of the reactor. This can be obtained, for instance, by turning off heat exchange in the first part of the reactor. In case 1, we choose to do so in the first 1/100 of the reactor length. In case 2, we also raise the temperature in the last part of the reactor, by raising the temperature of the cooling fluid in the existing heat exchanger. The entropy production rate and the lost work for both cases and the reference case are shown in Table 1. When the temperature is changed in the heat exchanger, the entropy production in this part of the system will also change. The change may mean a change (positive or negative) in the lost work. For an assessment of the total situation before and after the change of the operating temperature, we therefore calculated also the lost work in the heat exchanger. The entropy production for heat exchange is
(T1)
Σexch ) h(T - Tc)∆c
(15)
where h is the heat-transfer coefficient, T is the (variable) temperature of the reactor, Tc is the temperature of the coolant, and ∆c denotes the difference between the coolant and the reactor states. Entropy production
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3055
between the reference case and that of the optimal solution. The shape of the curve resembles that of the reference case, which is not surprising, given that we have not altered the apparatus. Such changes are probably required to change the shape of the curve and obtain the gain indicated by the optimal entropy production rate (9.0 J/mol of SO3). The increase in the production rate in Table 1 points in the direction of a trade-off between reactor size and lost work. When the optimal driving force is found for a given production rate, one may consider further optimization along the axis minimum lost workproduction rate. The production rate can be further traded with reactor size. These are future topics of analysis. 6. Concluding Remarks
Figure 7. The driving force case 2 compared to results of Figure 3.
by the liquid flow in the heat exchanger was neglected. The results from this calculation are also given in Table 1. We see from Table 1 that the total entropy production rate for the reactor indeed is reduced, with the changes that we introduce in the operating conditions. The total entropy production rate of the reactor for Fogler’s conditions, 21.8, drops to 20.7 and further to 16.1 J/mol of SO3 produced. The changing conditions in the heat exchangers do not influence the trend. The lost work is obtained by adding the two entropy production rates. The production of SO3 is listed along with the operating conditions. We see that the production, P, varies between the cases. This must not be confused with the condition of constant production that is used in the proof for the optimum force, eq 5. We are here using the theoretical result to define a direction for changing operating conditions. A solution which is a compromise between theoretical and practical (and maybe economic) boundary conditions does not necessarily fulfill the the conditions set for the proof. It is interesting to see that the modest changes that are proposed in the operating conditions also lead to improvements in the production rate. The lost work given in the table is given per kmol of SO3 produced. The change in energy efficiency with this basis is 5% and 21%, in case 1 and case 2, respectively. The reduction in the lost work by taking away the cooling at the reactor entrance is 5%. The raise of the temperature of the coolant in the end of the reactor produces a more substantial gain, a 21% savings. The main conclusion is that improvements have been obtained. The savings in lost work can manifest themselves in these examples by a need for a smaller amount of cooling liquid and/or combined by a higher temperature of the coolant. In case 2, the total process is able to produce fluids at a higher temperature. The gain in the lost work can be quantified in terms of this. All numbers listed for Σ are far away from the optimal value, 9.0 J/mol of SO3, that is given by the integral of Figure 4. In other words, there is also a potential for further gain that seems larger than the ones we have obtained by modest changes. The driving force through the reactor corresponding to case 2 was calculated and compared to the reference case and the optimal case in Figure 7. We see that, in most parts of the reactor, the improved solution (case 2) has a force with value
We have just shown how one can find operating conditions for a chemical reactor that gives minimum lost work in the reactor. A specific example was studied, the oxidation of sulfur dioxide in a tubular reactor. The reaction was chosen mainly to illustrate the method of analysis. The potential for an improved energy efficiency is nevertheless significant. Whether the potential can be realized in practice is another important issue that also must be dealt with. This was not done here. With knowledge of the particular kinetics of the reaction, one can construct the optimal driving force distribution from the entropy production of the system and the constraint on the production. Knowledge of this path that gives minimum lost work is needed when one wants to make a trade-off between energy efficiency and production rate in a given reactor. Through the study of a reactor with a nonlinear flux force relation, we have also been able to examine further the validity of the principle of equipartition of forces. The minimum total entropy production rate by equipartition of forces is, according to the derivations and results of this work, not generally true when the reaction is far from chemical equilibrium. Deviations from the principle of equipartition of forces is probable in the beginning of the reactor. We shall pursue the study of other nonlinear flux force relations for chemical reactions. For certain classes of nonlinear relations the principle of the equipartition of forces may still hold (see part 1). Literature Cited (1) Fogler, H. Elements of Chemical Reaction Engineering, 2nd ed.; Prentice-Hall: New York, 1992. (2) Kjelstrup, S.; Island, T. V.; Sauar, E.; Bedeaux, D. Energioptimal drift og design av en eller flere kjemiske reaktorer. Patent Appl. NO nr. 982798, 1998. (3) Ross, J.; Mazur, P. J. Chem. Phys. 1961, 35, 19. (4) Sauar, E. Energy Efficient Process Design by Equipartition of Forces: With Applications to Distillation and Chemical Reaction. Ph.D. Thesis, Department of Physical Chemistry, Norwegian University of Science and Technology, Trondheim, Norway, 1998. (5) Sauar, E.; Kjelstrup, S.; Lien, K. M. Equipartition of forcess extensions to chemical reactors. Comput. Chem. Eng. 1997, 529534. (6) Scho¨n, C.; Andresen, B. Finite-time optimization of chemical reactions: nA ) mB. J. Phys. Chem. 1996, 100, 8843-8853. (7) Ullmann, F. Ullmann’s Encyclopedia of Industrial Chemistry, 5th ed.; VCH: Weinheim, Germany, 1995; Vol. A25.
Received for review November 23, 1998 Revised manuscript received April 15, 1999 Accepted April 17, 1999 IE9807452
