Second law optimization of a tubular steam reformer

Transcription

1 Chemical Engineering and Processing 44 (2005) Second law optimization of a tubular steam reformer Lars Nummedal, Audun Røsjorde, Eivind Johannessen, Signe Kjelstrup Department of Chemistry, Norwegian University of Science and echnology, NO-7491rondheim, Norway Received 11 November 2002; received in revised form 29 June 2004; accepted 29 June 2004 Available online 9 September 2004 Abstract We present a numerical method that finds the path of operation that gives minimum total entropy production rate in a tubular steam reformer. he method was applied to the three main reformer reactions in a tubular plug flow reactor with pressure drop and heat exchange. he total entropy production rate was minimized subject to a given production of hydrogen, a fixed inlet pressure, a fixed total molar flow rate at the inlet, and a fixed molar flow rate of inert gas. he inlet and outlet temperatures, the outlet pressure, and the inlet mixture composition were allowed to vary. he temperature profile of the furnace gases was the control variable. Compared to a typical path of operation, we obtained a reduction of more than 60% in the total entropy production rate for the optimal path. he results suggested that a shorter reactor may perform equally well. Interestingly, the optimal path showed regions of either a constant thermal force or a constant chemical force. he new path of operation was not realistic, however, so more work is needed to realise some of the potential gain Elsevier B.V. All rights reserved. Keywords: Second law optimization; ubular steam reformer; Entropy production rate 1. Introduction he steam reformer is widely used in the industry for conversion of natural gas into synthesis gas, see e.g Rostrup- Nielsen [1]. he reformer is operated at different conditions depending on the purpose to produce the synthesis gas; for instance for methanol or ammonia production. he reformer is also used for production of hydrogen, for instance for power generation [2]. A sustainable society may in the future use hydrogen as the energy carrier. While we are waiting for production routes that use renewable energy sources only, hydrogen must be produced from natural gas. Due to their highly endothermic nature, the chemical reactions in the steam reformer require a substantial amount of added heat. he catalyst filled reactor tubes of the tubular steam reformer are therefore placed inside a large furnace. he exergy loss in this tubular reformer is high, much higher than for instance the losses in the ammonia reactor [3]. he exergy loss of the reformer in the hydrogen fired gas power plant is also large [4]. For these reasons, the tubular steam Corresponding author. el.: ; fax: address: signekj@phys.chem.ntnu.no (S. Kjelstrup). reformer is a good target for studies of exergy loss minimizations, or entropy production minimizations. Already 50 years ago, Denbigh [5] pointed to the large exergy loss in chemical reactions, and suggested that different modes of operation be sought for. hanks to large scale computers and developments of numerical methods, it has now become feasible to search for the path of operation with minimum entropy production rate in the reactor system itself, and to study the nature of this path. he trivial solution to a minimization of the total entropy production rate of a system is zero, or equilibrium. his solution is found when the system is not subject to constraints, and is not interesting. A unit that operates in the reversible limit, does not produce chemical products at any significant rate and the transfer of heat is only infinitely slow. We need to apply constraints to have a meaningful optimization. he immediate constraint to choose, that also shall be chosen here, is that of a given production. Other relevant constraints relate to the mode of operation, but we shall leave the question of finding such practical constraints for the time being. he purpose of our work is first to establish a method for entropy production minimization, and to study the nature of the solution /$ see front matter 2004 Elsevier B.V. All rights reserved. doi: /j.cep

2 430 L. Nummedal / Chemical Engineering and Processing 44 (2005) Methods of constrained optimization have been used by many authors, also to study the steam reformer. Kvamsdal et al. [6] optimized the methane conversion for changing feed flow, with a constraint on the outer reactor tube wall temperature. Multi-objective minimization studies are also in rapid progress. Rajesh et al. [7,8] simultaneously minimized the inlet total molar flow rate of methane and the outlet total molar flow rate of carbon monoxide. hey also simultaneously maximized the total molar flow rate of product hydrogen and export steam for an industrial hydrogen plant. We present here a method for minimization of the system s entropy production rate, for a given production of hydrogen, and with the temperature of the heating system as the control variable. his is a relatively simple example, but complicated and new, if one wants to calculate the local fluxes and forces in the system. Our previous work on chemical reactors has been on systems with only one chemical reaction [9 12]. We shall use irreversible thermodynamics to find the total entropy production rate of the system [13,14]. In the tubular steam reformer, there are at least three reactions that should be considered. We have therefore needed to extend and reformulate the calculation presented before [9 11]. We report here a way to find optimal operating conditions for a chemical reactor with several reactions; operating conditions that are compatible with minimum entropy production rate. Values for other properties such as reaction rates, and thermal and chemical driving forces along the reactor, follow from the optimal operating conditions and shall also be presented. We shall finally proceed to discuss the nature of the solution. We minimize the entropy production rate of a reactor that is well documented in the literature. All important parameters and expressions of the reference system were taken from Xu and Froment [15,16]. One steam reformer tube, surrounded by a heat reservoir/furnace, is studied. he temperature profile of the furnace gases is the control variable. he temperature and the flow rates of all substances, except the inert flow rate, are allowed to vary both at the inlet and at the outlet. he total inlet molar flow rate and the production of hydrogen are fixed, though. he pressure is fixed at the inlet but free at the outlet. he entropy produced by the steam reforming reactions, the pressure drop, and the heat transfer between the furnace and the reaction mixture are included, while the entropy produced by the furnace gas combustion process is not taken into account. A preliminary version of the work has been presented [17]. 2. he system he most central chemical reactions, that take place on the Ni(Al 2 O 3 ) catalyst in the steam reformer are [15,16]: CH 4 + H 2 O = CO + 3H 2 r H I > 0 (I) CO + H 2 O = CO 2 + H 2 r H II < 0 (II) CH 4 + 2H 2 O = CO 2 + 4H 2 r H III > 0 (III) Fig. 1. One reactor tube of the steam reformer. In addition to the five substances given above, nitrogen is present in the reactor as an inert. All substances are in their gaseous states under the operating conditions that are used. Coke formation and dusting are disregarded. he tubular steam reformer consists of a set of vertical catalyst-filled tubes placed inside a large furnace [1]. he role of the furnace is to heat the reaction mixture. We study one such tube surrounded by a furnace/hot reservoir. We used a plug flow reactor (PFR) model with heat transfer perpendicular to the reactor wall from the furnace gases to the reaction mixture, see Fig. 1. he reaction mixture is perfectly mixed in the radial direction with respect to temperature, pressure and chemical composition. here are also no thermal or chemical mixing processes in the axial direction. Reactor dimensions (see Fig. 1) and other parameters were taken from Xu and Froment [15,16]. Enthalpies and heat capacities were taken from Jensen [18], and Daubert and Danner [19], respectively. A reference system is calculated using the parameters in able 1. he entropy production minimization was carried out with the reference system as a bench mark. he overall heat transfer coefficient and the viscosity of the reaction mixture were kept constant. able 1 Reference reactor parameters Parameter Symbol Value Inlet temperature of reaction in K mixture Furnace gas temperature a (W c /W c,ot ) K Overall heat transfer coefficient U 100 J/Km 2 s Reaction mixture viscosity µ kg/m s Inlet total pressure P in Pa Catalyst density ρ c kg/m 3 Catalyst void fraction ɛ 0.65 Catalyst pellet diameter D p m Efficiency factor of reaction i η i 0.03 otal catalyst weight W c,ot kg Inlet methane molar flow rate F CH4,in mol/s Inlet water molar flow rate F H2 O,in mol/s Inlet hydrogen molar flow rate F H2,in mol/s Inlet carbon monoxide molar flow F CO,in mol/s rate Inlet carbon dioxide molar flow F CO2,in mol/s rate Inlet nitrogen molar flow rate F N2,in mol/s

3 L. Nummedal / Chemical Engineering and Processing 44 (2005) Reactor modelling Energy, momentum, and mole balances were written according to standard texts, [20,21]. he balance equations in differential form, the reaction rates, and the various thermodynamic properties are given below. he mole fraction of component k is expressed by its molar flow rate divided by the total molar flow rate: x k (W c ) = F k(w c ) F (W c ) where x k and F k are the mole fraction and molar flow rate of component k, and W c is the accumulated catalyst weight. F is the total molar flow rate. he accumulated weight is proportional to the position in the reactor when the cross-sectional area of the tube is constant, and the catalyst distribution is homogeneous. We shall use W c as a measure of the distance from the inlet of the tube, where = (1 ɛ)ρ c A c dz (2) where symbols are defined below Conservation equations he energy balance of a control volume of infinitesimal thickness in the axial direction of the reactor tube, for a PFR model (see Fig. 1), is d = [1/ρ c(1 ɛ)][4/d ti ]U( a ) i r i r H i k F (3) kc p,k where and a are the temperature of the reaction mixture and the furnace gases, respectively, U is the heat transfer coefficient for heat transfer between the furnace gases and the reaction mixture, ρ c and ɛ are the catalyst density and the void fraction, d ti is the inner diameter of the reactor tube, and C p,k is the heat capacity of component k. For reaction i, r H i and r i are the enthalpy of reaction and the reaction rate, respectively. he pressure drop or momentum balance in the reactor was modelled by Ergun s equation, see [20]. With a negligible change in the compressibility factor, Ergun s equation is: ( ) dp 150µ (1 ɛ) = G D p (1) 1 F P 0 D p ɛ 3 v 0 (4) A c ρ c F 0 P 0 where D p is the catalyst pellet diameter, G is the superficial mass velocity, A c is the cross-sectional area of the reactor tube, v 0 is the entering superficial gas velocity, and F 0 and F, P 0 and P, and 0 and are the total molar flow rate, the total pressure, and the temperature at the inlet and at position W c, respectively. Mole balances for all chemical substances in the reaction mixture, give for the control volume in Fig. 1 (see [21]): df CH4 = r I r III (5) df H2 O = r I r II 2r III (6) df H2 = 3r I + r II + 4r III (7) df CO = r I r II (8) df CO2 = r II + r III (9) df N2 = 0 (10) he balance equations have been formulated as functions of the molar flow rates instead of conversions. his is done for numerical reasons, and in order to ensure that the flow rates are free at the inlet and the outlet. Only three of the six mole balances are independent. he significance of these details are addressed in Section Reaction rates he reaction rates of reactions (I) (III) were given by Xu and Froment [15] ( ) k I r I = η I PH DEN 2 P CH4 P H2 O P3 H 2 P CO (11) K I ( k II r II = η II P H2 DEN 2 P CO P H2 O P ) H 2 P CO2 (12) K II ( ) r III = η III k III P 3.5 H 2 DEN 2 P CH4 P 2 H 2 O P4 H 2 P CO2 K III (13) where η i, k i, and K i are the efficiency factor, the rate constants, and the equilibrium constants of reaction i, respectively. he subscript i indicates reaction I, II, or III. P k is the partial pressure of component k, and DEN = 1 + K CO P CO + K H2 P H2 + K CH4 P CH4 + K H 2 OP H2 O (14) P H2 where K k is the adsorption constant of component k. Here K H2 O is the dissociative adsorption constant of water hermodynamic forces he thermodynamic driving forces are defined from the entropy production rate σ in irreversible thermodynamic the-

4 432 L. Nummedal / Chemical Engineering and Processing 44 (2005) ory [13,14]. his gives for the chemical reactions: rg I rg II = R ln = R ln P CO P 3 H 2 P CH4 P H2 OK I (15) P CO2 P H2 P CO P H2 OK II (16) rg III P CO2 PH 4 = Rln 2 P CH4 PH 2 2 O K (17) III where r G i is the reaction Gibbs energy (of reaction i), and R is the universal gas constant. It has been customary to use the affinity, A i, for the chemical driving force [13], where A i = r G i. We have chosen instead to use the reaction Gibbs energy, because this property is more common in the chemical engineering community. hese forces shall be referred to as chemical forces. he force for heat exchange, or the thermal force, between the furnace gases and the reaction mixture is, when integrated across the thickness of the wall: ( ) 1 = 1 1 (18) a he thermodynamic force on the flow, or the viscous flow force, is: 1 dp (19) dz where z is the Cartesian coordinate of transport. 4. he optimization problem he optimization problem consists of minimising the total entropy production rate of the chemical reactions, of the heat exchange process between the furnace gases and the reaction mixture, and of the pressure drop, subject to certain constraints. he control variable is the temperature profile of the furnace gases, a (W c ). he inlet composition and the temperature of the reaction mixture at the inlet are allowed to vary. Other parameters like the inlet pressure, the reactor geometry, and catalyst properties are fixed. At the outlet, the temperature, the pressure, and the composition are allowed to vary, but the production of hydrogen is kept constant. he minimization problem has therefore only few degrees of freedom. We formulate the optimization problem using the Euler- Lagrange method. he objective function and the constraints of the system are given in the two following subsections he objective function he total entropy production rate of the reactor is the integral of the local entropy production rate, σ [13] over the total mass of catalyst: ds irr dt = = Wc,ot 0 Wc,ot 0 + σ [ r G I r G II r G III r I + r II + r III ( ) U( a ) A c v 1 (1 ɛ)ρ c d ti dp ] (20) where W c,ot is the total catalyst mass in the reactor, and v is the superficial gas velocity. he three first terms in the equation express the entropy produced by the three chemical reactions; the next term is the entropy production rate due to the heat transfer from the furnace gases to the reaction mixture, and the last term is the entropy production rate due to viscous flow. he temperature of the reaction mixture, the pressure, the flow rates of the substances and the temperature of the furnace gases (the control variable) are functions of the accumulated catalyst weight. he total entropy production rate depends on these functions and is therefore a functional, see e.g. routman [22]: ds irr dt = ds irr ( (W c ),P(W c ),F CH4 (W c ),..., a (W c )) (21) dt his is the objective function he constraints o have a meaningful optimization, we have to include at least one constraint on the duty of the reactor (cf. Section 1). In addition, the solution must fulfil the energy, momentum, and mole balances. We divide the constraints into two classes: he first class of constraints contains the conservation equations (all the equations in Section 3.1), while the second class contains the rest. For the current case, we have four constraints in the second class: he hydrogen production of the unit, the inlet pressure, the total molar flow rate at the inlet and the flow rate of nitrogen (inert). his class of constraints can be expanded or changed with practical constraints, like on the ratio of water to carbon in the inlet and/or outlet reaction mixture. his is the topic of a future article, however. he hydrogen production in the steam reformer is: F H2,out F H2,in = FH ref 2,out F H ref 2,in (22) where F H2,in and F H2,out are the hydrogen flow rate at the inlet and outlet, respectively. Superscript ref means the value for the reference system. he three other constraints in the second class are: F,in = F ref,in (23)

5 F N2,in = F ref N 2,in (24) P in = P ref in (25) 4.3. he Euler-Lagrange functional L. Nummedal / Chemical Engineering and Processing 44 (2005) he Euler-Lagrange functional corresponding to the objective function and the constraints described in the last two sections is: Wc,ot { [ d L = σ( (W c ),P(W c ),F CH4 (W c ),..., a (W c )) + λ q (W c ) 1/ρ c(1 ɛ)4/d ti U( a ) 0 [ ( ) dp 150µ(1 ɛ) 1 F P in + λ p (W c ) G D p D p ɛ 3 v in A c ρ c F,in P [ + λ (W dfh2 0 FH2 0 c) ( r I r II 2r III ) [ dfco2 + λ FCO2 (W c ) (r II + r III ) ] ] [ dfh2 + λ FH2 (W c ) [ dfn2 + λ FN2 (W c ) 0 where j is the control volume number. Subscript m means that the variable values used were the average of the values at the neighbouring discretization points. he differential balance equations for energy, momentum, and moles, Eqs. (3) (10), (the first class of constraints) were discretized as follows: in (3r I + r II + 4r III ) ]} k F kc p,k i r ] i r H i ] [ ] dfch4 + λ FCH4 (W c ) ( r I r III ) ] [ ] dfco + λ FCO (W c ) (r I r II ) + λ H2 [(F ref H 2,1 F ref H 2,0 ) (F H 2,out F H2,in)] + λ F,in [F,in ref F ref,in] + λ FN2,in[FN 2,in F N 2,in] + λ Pin [Pin ref P in] (26) where λ H2, λ F,in, λ, and λ FN2,in P in are scalar Lagrange multipliers related to the total production of hydrogen, the total inlet molar flow rate, the inlet molar flow rate of nitrogen, and the inlet pressure, respectively. he functions λ q (W c ), λ p (W c ), λ FCH4 (W c ), λ (W FH2 O c), λ FH2 (W c ), λ FCO (W c ), λ FCO2 (W c ), and λ FN2 (W c ) are Lagrange multiplier functions related to the conservation equations. We shall proceed to solve the optimization problem numerically. he necessary changes to the formulation of the problem are given in the next section. 5. he optimization problem on discrete form j+1 j [ 1/ρc (1 ɛ)4/d ti U( a ) i = r ] i r H i k F kc p,k P j+1 P j [( ) 150µ(1 ɛ) = G D p 1 F P in D p ɛ 3 v in A c ρ c F,in P in ] m m (28) (29) In order to solve the minimization problem numerically, we converted the objective function, Eq. (20), and the first class of constraints, Eqs. (3) (10), from continuous to discrete equations. he equations were discretized into j = 1,...,n 1 control volumes. he functions used in the previous section, e.g. a (W c ) and F CH4 (W c ), became vectors a and F CH4 with n elements. he discretization of the total entropy production rate, Eq. (20), gave: ds irr dt n 1 [ r G I r G II r G III = r I + r II + r III j=1 ( ) U( a ) A c v 1 ] dp (1 ɛ)ρ c d ti ( ) (27) m F CH4,j+1 F CH4,j = [ r I r III ] m (30) F H2 O,j+1 F H2 O,j = [ r I r II 2r III ] m (31) F H2,j+1 F H2,j = [3r I + r II + 4r III ] m (32) F CO,j+1 F CO,j = [r I r II ] m (33) F CO2,j+1 F CO2,j = [r II + r III ] m (34) F N2,j+1 F N2,j = 0 (35) he second class of constraints (Eqs. (22) (25)) remained unchanged by the discretization.

6 434 L. Nummedal / Chemical Engineering and Processing 44 (2005) From the equations given in this section, we see that there are 8(n 1) constraints in the first class. here are four additional constraints in the second class. he total number of constraints in the numerical optimization is therefore 8(n 1) Calculations he temperature, pressure, and molar flow rate profiles of the reference system were found by numerical integration of the energy, momentum, and mole balances, Eqs. (3) (10), over the accumulated catalyst weight. Reactor parameters and inlet conditions for the reference system were taken from able 1. All results from the reference system calculations are labelled ref. in subsequent figures. he results from the entropy production minimization shall be compared with these. he entropy production rate, Eq. (27), was then minimized subject to the discretized constraints discussed earlier. As variables in the optimization, we used all the variable vectors:, P, F CH4,..., F N2, and a. he total number of variables was therefore 9n. he minimization was performed numerically using the Matlab (R12) Optimisation oolbox function fmincon [23]. he constraints on the inlet molar flow rate of nitrogen and the inlet pressure were implemented using the upper and lower boundary functionality in fmincon. he profiles of the reference system were used as starting point in the optimization. We used a non-uniform grid of discretization points along the reactor in the numerical optimization. As the hydrogen flow rate of the optimal solution in Fig. 4 shows, the majority of the points was put close to the inlet and the outlet. his was done after initial calculations with uniform grids had shown that the profiles of the optimal solution change most rapidly close to both ends of the reactor. he optimal solution that we present in the next section, was found using a grid of 60 (n = 60) discretization points. he necessary number of discretization points was found by increasing n until the calculated minimum total entropy production rate reached a stationary value, see Nummedal et al. [11]. he entropy production rates of the reference system and of the optimal solution were calculated using the trapezoidal rule with a grid of 2000 discretization points. he extended variable vectors that were needed to do the numerical integration were found by spline interpolation. o check for consistency and errors in the optimization method, we tested it on previous results [11]. he method we present here, reproduced those results using a coarser grid of discretization points. able 2 otal entropy production rate [J/K s] of the reference and minimized system ds irr /dt ref. min. Reaction Viscous flow Heat transfer otal otal F H % Reduction 66.5 the heat transfer from the furnace gases to the reaction mixture, are shown for the reference and the minimized system in able 2. able 2 shows that the entropy production is reduced by an amazing 66.5% upon minimization. he reduction is only due to a change in the heat transfer part in the objective function. he other terms in the entropy production rate have in fact increased. he total pressure dropped almost linearly from 29 bar at the inlet to about 27 bar at the outlet (not shown) in both cases. In the minimized case the outlet pressure was slightly lower than in the reference case. he results from the reference system calculations (ref.) and the system with minimization of the total entropy production rate (min.) are presented as functions of the accumulated catalyst weight, W c, and shown in Figs All figures give details of the path of operation in the optimum state versus the reference state. he local entropy production rate is plotted in Fig. 2. he areas under the curves give the total entropy production rates in able 2. he reaction mixture and furnace gas temperatures in the reference (ref.) and the minimized (min.) system are shown in Fig. 3. We see that the inlet values of both temperatures in the reference system are raised significantly in the reactor with minimum entropy production rate. It is surprising to see that the shape of the area enclosed by the two temperature profiles for the optimized system in Fig. 3 is almost anti-symmetrical 7. Results All results are given for a hydrogen production of F H2 = mol/s in the reactors. he contributions to the entropy production from the chemical reactions, the viscous flow, and Fig. 2. he local entropy production rate as a function of the accumulated catalyst weight in the reference (ref.) and the minimized (min.) system.

7 L. Nummedal / Chemical Engineering and Processing 44 (2005) Fig. 3. he temperatures of the reaction mixture and furnace gases as functions of the accumulated catalyst weight in the reference (ref.) and the minimized (min.) system. Fig. 4. he molar flow rates as functions of the accumulated catalyst weight in the reference (ref.) and the minimized (min.) system. across a vertical axis. he inner and outer temperatures become the same at the boundaries. he molar flow rates of the reference system (ref.) and the minimized system (min.) are plotted in Fig. 4. he molar flow rate of nitrogen, F N2, was left out since it is constant (0.235 mol/s in both systems). Fig. 4 shows that hydrogen is produced in very different ways in the two reactors, even if the amount is the same. Fig. 5. he reaction rates, as functions of the accumulated catalyst weight, of reaction I and II in the reference (ref.) and the minimized (min.) system.

8 436 L. Nummedal / Chemical Engineering and Processing 44 (2005) Fig. 6. he chemical force, as a function of the accumulated catalyst weight, for reaction I and II in the reference (ref.) and the minimized (min.) system. he amount ( F H2 = mol/s) can be read from the figure as the difference between the outlet and the inlet flows. Compared to the reference reactor, the hydrogen flow into the reactor is more than ten times larger in the optimized system. he values of the input and output variables of the minimum solution are compared to the corresponding variables of the reference case in able 3. Half of the hydrogen production comes from methane and the other half from water in both reactors. he main change is that the inlet molar flow rate of methane decreased from to by the minimization; but it decreased less than the inlet molar flow rate of water. On the other hand, the inlet molar flow rates of carbon monoxide as well as carbon dioxide increased. he reaction rates of reactions (I) and (II) in the reference (ref.) and the minimized (min.) system are presented in Fig. 5. he reaction rate profiles of the first and third (not shown) reaction show similar behaviour. hey sink sharply from a high value at the inlet to a small, almost zero value. he shift able 3 Productions and molar flow rates [in mol/s] of the reference and the minimized system Variable ref. min. F CH F H2 O F CO F CO F CH4,in F H2 O,in F H2,in F CO,in F CO2,in F N2,in reaction increases slightly at the inlet before it returns to a low value. he nature of the optimal paths is most interestingly characterised by the driving forces. Fig. 6 displays the chemical force distribution for the first two reactions in both systems. We see, as in Fig. 5, that the chemical production is shifted toward the reactor inlet by the optimization. he chemical forces are very close to zero after the first 5% of the total tube length of the minimized system. In the reference system, the driving forces are also high at the reactor inlet, but they decrease more slowly, and without becoming zero. In Fig. 7, the thermal forces in both systems are plotted as functions of the accumulated catalyst weight. his figure shows most clearly the difference between the two systems. he thermal force of the minimized system is constant through the reactor except at the outlet and inlet. he value of the thermal force in the minimized system is about one sixth of what it is in the reference system. 8. Discussion Fig. 7. he thermal force, as a function of the accumulated catalyst weight, in the reference (ref.) and the minimized (min.) system. We have presented a numerical method that can be used to minimize the total entropy production rate of a chemical

9 L. Nummedal / Chemical Engineering and Processing 44 (2005) reactor with i reactions, pressure drop and external heating or cooling. he formulation is somewhat more general than what we have used before [11,17], and was tested for accuracy against these earlier results. he method was applied to the tubular steam reformer. his important reactor is endothermic and has more than one chemical reaction. We discuss the reactor results first, before we give more comments on the method he optimal path of operation able 2 and Fig. 2 give the main results of the investigation. It was possible to reduce the entropy production rate of a steam reformer tube by more than 60% compared with a typical reference system. Fig. 2 shows that the entropy production dropped rapidly from a high value at the inlet of the reactor, to a small and rather constant value. he inlet temperatures of the optimal solution put the value of about 65% saving into a negative perspective, however. A realisation of the optimal solution in practise, means that preheating is necessary of the gases that enter the reactor. his, of course means that more energy must be spent upstream of the reactor system. he number obtained, 66.5%, must therefore not be taken literally: In a final solution it must be traded off with the extra exergy costs from preheating. he entropy produced by the furnace flames must also be included in the optimization, see [24]. he fact that there is a high potential for savings in the entropy production in the reformer, gives, however, hope for an overall saving. he reduction in the entropy production rate came only from the heat transfer part of the objective function. he contributions from the reaction and the viscous flow increased, rather than decreased, in the optimization. here is clearly a trade-off situation between the contributions that this method is able to weigh. he result of the trade-off is that the chemical reaction is shifted to the start of the reactor, while, apart from each end of the reactor, the reactor operates as a heat exchanger, mainly. Figs. 5 and 6 show that the chemical reactions gave their contributions to the entropy production rate in the start of the reactor. he optimal chemical driving force was very small in most of the reactor, see Fig. 6. he most interesting finding is that the optimal thermal force was finite and constant in a large part of the reactor, see Fig. 7. he Hamiltonian of this type of optimal control problem is constant [12]. When and how this condition reduces to the theorem of equipartition of entropy production [25], or equipartition of driving forces [26,27] is not yet clear. But the fact remains that the optimal system develops into a state with a constant thermal force after inlet and outlet boundary conditions are met with. In the absence of constraints on the heat transfer, also the chemical force becomes constant, in a similar optimization [28]. It is interesting to speculate on the optimal dissipative structure of process units on this background: Should we seek for designs that have constant forces? he complete shape of the optimal temperature profiles is also interesting: here seems to be a kind of anti-symmetry across the vertical axis. he outer and inner temperatures are equal at both ends because the inner temperature is free to vary at both end points. his property of the solution was explained by Johannessen and Kjelstrup [12]. hey formulated the optimization problem using optimal control theory. Equal temperatures at both ends is the direct consequence of the natural boundary conditions. his type of boundary conditions is well known in optimal control theory and variational calculus, and they apply whenever a (state) variable can vary freely at an end point. he optimal system has a higher value of the inlet flow rate of hydrogen, than the reference system has (see Fig. 4). he difference between the outlet and inlet values of F H2 are of course the same in both cases. he difference is by definition the constraint on the optimization, the hydrogen production. he increased inlet molar flow rate of hydrogen gave a lower reaction rate at the inlet compared to a case where only the inlet temperature is increased. By increasing both the inlet temperature as well as the inlet molar flow rate of hydrogen, the optimizer acquired more heat for the reactions, without getting a skyrocketing entropy production rate from the chemical reactions. he result presented here is a result of an optimization of three simultaneous processes. he entropy is produced by chemical reactions, by viscous flow and by heat exchange; and the optimizer makes a trade-off between the three phenomena. Here, it pays off to increase the entropy production rate of the chemical reactions and the viscous flow a little, since the reduction by the heat exchange process can then be so large. he fact that the reaction rates go to zero so fast, suggests that the set production can be maintained also with a shorter reactor. he entropy production from the reaction may decrease somewhat, and the contribution from pressure drop will decrease. his may save reactor material. A similar observation was made earlier [11]. It may be interesting to see how a varying reactor length affects the calculations. In their modelling of adiabatic reactors for the partial oxidation of methane to synthesis gas, [29] obtained similar looking profiles for the molar flows to the ones presented in Fig. 4. heir work did not concern any optimization. It is, however, interesting to note the similarity between the temperature profile of our optimal system and the temperature profile of the adiabatic reactor model which has no thermal contribution to the entropy production rate. he ammonia reactor [11] and the sulphur dioxide oxidation reactor [12] have been studied earlier. he ammonia reaction is exothermic, and a comparison with our current endothermic reaction system is of interest. A shape of temperature profiles, that was anti-symmetric about the vertical axis, was also observed for the exothermic ammonia reactor [9 11]. Also in the ammonia reactor, there were regions of constant driving forces. he fact that the two solutions are similar, can be taken as a support for both results being correct.

10 438 L. Nummedal / Chemical Engineering and Processing 44 (2005) A valid theory According to Ross and Mazur [30] the entropy production rate of a chemical reaction is the product of its rate and its driving force, even if the rate is no linear function of the driving force. he theory is thus not limited to cases close to chemical equilibrium or close to global thermodynamic equilibrium. It has been a misunderstanding in the literature that the theory of irreversible thermodynamics applies only close to global equilibrium, see e.g. Elliott et al. [31] for a recent example. he only assumption that need be fulfilled, to use the theory, is that of local equilibrium in a volume element, meaning that we are allowed to write thermodynamic equations for that element. he theoretical method has therefore no restriction in its applications to the present case. he local values of the forces and fluxes are therefore reliable, and can be used to understand the optimal behaviour of the process unit he optimization method he Euler-Lagrange method, as formulated in the present work, can be used to find the path of minimum entropy production rate in a chemical reactor. he method allows us to introduce constraints on variables that are important for the system s function. We have here chosen certain variables, but other choices can be made, in the pursuit of practical applications. he numerical calculations were rather difficult. Previously reported procedures [11,27] broke down, when applied to the current reactor, as negative mole fractions were generated. A stable, reproducible solution to the optimization problem was found here by rewriting the mole balances, and making these functions of the molar flow rates. he molar flow rates replaced the conversions as concentration variables. he re-formulation of the mole balances removed the limitation in the procedures presented earlier [11,27]. he current formulation of the mole balances allows all molar flow rates to vary freely at the reactor inlet. he inert inlet molar flow rate must be fixed or given upper/lower boundary values, however. A test on the program was made, by letting the inlet molar flow rates of inerts vary freely. he result was a zero flow rate in the endothermic reactor, and a high inert flow rate in the exothermic reactor. hese results are expected. Inerts absorb heat, and can be used in an exothermic reactor to remove reaction heat very effectively. It is thus not surprising that the optimal situation has a high inert flow rate, since this will lower the external cooling requirement, and thus lower the entropy production rate by the heat transfer in the cooling process. In an endothermic reactor, a heat absorbing inert is not favourable, since the supplied heat preferably should heat the substances that take an active part in the reaction. he optimal situation consequently has a zero flow rate of inert. he work done so far on entropy production minimization has focused on optimization of process units [9,12,17,25,28,32]. In a process optimization, there will in general be more constraints than we have used here. Constraints stem from other parts of the process. Upstream or downstream process requirements may be included among the second class of constraints (see Section 4.2). Efforts are now being made to put two or more process units together and to optimize that system as a whole. By introducing more constraints, we expect that the regions of constant forces become smaller, and the nature of the dissipating structure becomes less transparent. 9. Conclusions We have presented a numerical method for the minimization of the total entropy production rate of a reactor system with several chemical reactions. he method was verified with data from the literature, and was shown to work for a reactor with several chemical reactions, viscous flow, and the heating/cooling process of the reaction mixture. Results were given for the tubular steam reformer, where the furnace temperature profile was the only control variable. he inlet composition and the inlet temperature of the reaction mixture were allowed to vary. he data suggests that a shorter reactor may perform equally well. Relative to a typical reference system, the total entropy production rate was smaller by about 65%. In practice, one cannot expect to reduce the lost work by this large amount, because the solution implies losses associated with (i) preheating of the reacting fluid mixture, and (ii) increase of the hydrogen flow input rate (and other less important things). he optimal solution showed interesting characteristic features. he optimal thermal force was constant in large parts of the reactor, away from the boundaries. A further understanding of the origin of this behaviour may have impact on energy efficient engineering design. Acknowledgement Nummedal and Johannessen are grateful for support from he Research Council of Norway. Appendix A. Nomenclature A c cross-sectional area of the reactor tube (m 2 ) C p,k heat capacity at constant pressure of substance k (J/K mol) C p,i products ν kc p,k reactants ν kc p,k of reaction i (J/K mol) D p catalyst pellet diameter (m) inner tube diameter of the reactor (m) d ti

11 L. Nummedal / Chemical Engineering and Processing 44 (2005) F k F k,in F F,in F H2 molar flow rate of component k (mol/s) entering molar flow rate of component k (mol/s) total molar flow rate (mol/s) total entering molar flow rate (mol/s) hydrogen production (mol/s) G superficial mass velocity (kg/m 2 s) r G i change in Gibbs energy of reaction i (J/mol) r H i enthalpy of reaction i (J/mol) i reaction index j control volume index K i equilibrium constant of reaction i K k adsorption constants of component k k component index k i rate constant of reaction i L Euler-Lagrange functional n number of steps in the optimization P total pressure (Pa) P in total pressure at the inlet (Pa) P out total pressure at the outlet (Pa) P k partial pressure of component k (Pa) R universal gas constant (J/K mol) r i reaction rate of reaction i (mol/m 3 s) r H rate of formation of hydrogen (mol/kg cat. s) ds irr /dt total entropy production rate (J/Ks) temperature of the reaction mixture (K) in entering reaction mixture temperature (K) out exiting reaction mixture temperature (K) a coolant temperature (K) (1/) integrated thermal force (1/K) U overall heat transfer coefficient (J/K m 2 s) v superficial gas velocity (m/s) v in entering superficial gas velocity (m/s) W c accumulated catalyst weight (kg) W c,ot total catalyst weight (kg) z axial position in the reactor (m) Greek symbols ɛ catalyst void fraction η i efficiency factor of reaction i λ Lagrange multiplier µ gas mixture viscosity (kg/ms) ν k stoichiometric coefficient ρ c catalyst density (kg/m 3 ) σ local entropy production rate (J/K m 3 s) References [1] J.R. 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