Dynamic optimal analysis of a novel cascade membrane methanol reactor by using genetic algorithm (GA) method

Abstract

The potential of a novel cascade membrane methanol reactor (CMMR) in the presence of catalyst deactivation has been investigated by Rahimpour and Bayat [Int J Energy Res 34(15):1356–1371, 2010]. In the present paper, this novel configuration is optimized using genetic algorithm strategy, dynamically. In the first approach, optimum inlet molar flow rate, the inlet pressure of the tube and shell side for both reactors, the temperatures of feed (cooling gas) and cooling saturated water have been obtained within their practical ranges. In the second approach, a stepwise trajectory has been followed in order to determine the optimal profiles for saturated water and gas temperatures in three steps during operation. The objective of each optimization case is to maximize the methanol production rate. Here, genetic algorithms have been used as powerful methods for optimization of complex problems. The optimization results represent 19.67 and 25.63 % enhancement in the methanol production for first and second optimization approaches, respectively.

Appendices

Appendix A: Dusty gas model

It is said that commercial size CuO/ZnO/Al$_{2}$O$_{3 }$catalysts exhibit intra-particle diffusion limitations, so in the modeling of an industrial fixed-bed methanol synthesis process, internal mass transport limitation should be taken into account. Dusty gas model is widely used to describe intra-particle diffusion limitations in methanol synthesis over commercial CuO/ZnO/Al$_{2}$O$_{3}$ catalysts.

When reactants diffuse into the pores to react and form products, bulk, Knudsen and surface diffusions may take place simultaneously, depending on the size of the pores, the molecules involved in the diffusing stream, the operating conditions and the geometry of the pores. In the dusty gas model which is based on Stefan–Maxwell equations, both diffusional and convective mass transport terms are considered, and this includes the description of pressure drop over the catalyst particle resulting from the stoichiometry of reaction and accompanying convective transport of molecules. The results of sensitivity analysis show that at low pressures (up to 10 bar), Knudsen diffusion is the most important diffusing term; while at high pressures (100 bar) bulk diffusion predominates.

In this model, it is assumed that pore walls consist of giant molecules (‘dust’) which are uniformly distributed in the space. These dust molecules are considered to be dummy, or pseudo species in the mixture.

The dusty gas flux relations can be offered as follows, with a small change in the notation as to avoid conflicting with other notations used in the model:

$$\begin{aligned} \frac{N_i }{D_i^{Eff} }+\sum \limits _{i\ne j}^N {\left( {\frac{y_j N_i -y_i N_j }{D_{ij}^{Eff} }}\right)} =-\frac{P}{RT}\frac{dy{ }_i}{dr}-\frac{y_i }{RT}\left( {1+\frac{B_0 P}{\mu D_i^{Eff} }}\right)\frac{dP}{dr} \end{aligned}$$

(22)

In the above equation, $B_{0}$ is permeability of catalyst pellet, $D_{i}^{Eff}$ is effective Knudsen diffusion coefficient and $D_{ij}^{Eff}$ is effective binary diffusion coefficient which is presented by Eqs. 23 and 24 in the bellow respectively:

$$\begin{aligned}&\displaystyle D_i^{Eff} =a_p \frac{\varepsilon _s }{\tau }\frac{2}{3}\left[ {\frac{8RT}{\pi \,M_i }} \right]^{1/2}\end{aligned}$$

(23)

$$\begin{aligned}&D_{ij}^{Eff} =\frac{\varepsilon _s }{\tau }D_{ij} \end{aligned}$$

(24)

where, $a_{p}$ is the mean pore radius.

The dusty gas flux relations (Eqs. 22–24) contain three parameters: the mean pore radius, a, the ration of porosity and tortuosity factors, $\varepsilon _{s} / \tau$ and the permeability parameter, B$_{0}$. Using Darcy’s law, combines the two parameters of permeability and the mean pore radius and gives a two-parameter model using the following correlation:

$$\begin{aligned} B_0 =\frac{a_p ^2}{8} \end{aligned}$$

(25)

The reader should note that the flux relations could be rewritten for distinct components to form a set of ordinary differential equations, yielding expressions for $\frac{dy{ }_i}{dr}$. Knowing the fact that summation of all components equals to1 (i.e., $\sum _1^N {y_i } =1)$, $N-1$ ordinary differential equations should be written for the flux relations.

To complete the mathematical modeling of dusty gas model, the material balances and the stoichiometric relations have to be added in order to be able to describe the multicomponent reaction–diffusion problem. Since we have used the detailed Graaf kinetics for the system of methanol synthesis process, which is based on three-independent reactions, three material balances are needed to be added to flux relations. For a spherical and isothermal particle, this yields:

$$\begin{aligned}&\displaystyle \frac{d\Omega _k }{dr}=r^2r_k \;\qquad K=1,2,3\end{aligned}$$

(26)

$$\begin{aligned}&\displaystyle \frac{dy_i }{dr}=-\frac{1}{r^2}\left( {\sum \limits _{k=1}^3 {\Omega _k \sum \limits _{j=1}^5 {\nu _{kJ} F_{ij} } } }\right)\;\qquad ( {i=1, 2,\ldots ,N-1})\end{aligned}$$

(27)

$$\begin{aligned}&\displaystyle F_{{ii}} = \frac{{RT}}{P}\left( {\frac{1}{{D_{i} ^{{Eff}} }} + {\mathop {\mathop {\sum }\limits _{j = 1}}\limits _{j \ne 1}^{5}}{\frac{{y_{i} }}{{D_{{ij}} ^{{Eff}} }} - \frac{{y_{i} }}{{D_{i} ^{{Eff}} }}\left({1 + \frac{{B_{0} P}}{{\mu D_{i}^{{Eff}} }}} \right)\frac{1}{w}} } \right)\end{aligned}$$

(28)

$$\begin{aligned}&\displaystyle F_{ij} =-\frac{RT}{P}\left( {\frac{y_i }{D_{ij}^{Eff} }+\frac{y_i }{D_j^{Eff} }\left( {1+\frac{B_0 P}{\mu D_i^{Eff} }}\right)\frac{1}{w}}\right)\end{aligned}$$

(29)

$$\begin{aligned}&\displaystyle w=1+\frac{B_0 P}{\mu }\sum \limits _{i=1}^N {\frac{y_i }{D_i^{Eff} }} \end{aligned}$$

(30)

where, $F_{ii}, F_{ij}, w$ and $\Omega _{k}$ are auxiliary parameters and $v$ is the stoichiometric coefficient

The pressure drop in radial coordinate is given by:

$$\begin{aligned} \frac{dP}{dr}=-\frac{1}{r^2}\frac{RT}{w}\sum \limits _{i=1}^N \left( \frac{1}{D_i ^{Eff}} {\sum \limits _{k=1}^3 {\nu _{ik} } \Omega _k } \right) \end{aligned}$$

(31)

The boundary conditions for the set of ordinary differential equations are given by:

$$\begin{aligned}&\displaystyle \Omega _k =0\quad (k=1,\,2,\,3)\quad at\quad r=0\;( \text{ i.e.} \text{ at} \text{ the} \text{ center} \text{ of} \text{ catalyst} \text{ pellet})\end{aligned}$$

(32)

$$\begin{aligned}&\displaystyle y_i =y_{is} \,\quad \,(i=1,\,2,\,3,\,4)\,\quad at\,\quad \,r=R_p\end{aligned}$$

(33)

$$\begin{aligned}&\displaystyle P=P^{s\,}\quad at\quad \,r=R_p ) \end{aligned}$$

(34)

The effectiveness factors for methanol and water would be obtained as the following correlations based on equation A.5 and the fact that methanol is produced via reactions (37) and (38), while water is produced via reactions (38) and (39).

$$\begin{aligned}&\displaystyle \eta _{CH_3 OH} =\frac{3}{R_p^3 }\frac{( {\Omega _1^s +\Omega _2^s })}{r_1^s +r_2^s }\end{aligned}$$

(35)

$$\begin{aligned}&\displaystyle \eta _{H_2 O} =\frac{3}{R_p^3 }\frac{( {\Omega _2^s +\Omega _3^s })}{r_2^s +r_3^s } \end{aligned}$$

(36)

where, superscript s means that the variable has been calculated at the surface of catalyst pellet.

Appendix B: Reaction kinetics

Basically, methanol is still produced by catalytic conversion of synthesis gas (CO$_{2}$, CO, H$_{2}$ and some inert components such as argon and nitrogen) on a large scale.

The methanol synthesis reaction is exothermic and the total moles reduce as the reaction proceeds. In the methanol synthesis, three overall reactions are possible: hydrogenation of carbon monoxide, hydrogenation of carbon dioxide and reverse water–gas shift reaction, which are as follows:

$$\begin{aligned}&\displaystyle CO+2H_2 \leftrightarrow CH_3 OH\quad \Delta H_{298} =-90.55\;\text{ kJ}~\text{ mol}^{-1}\end{aligned}$$

(37)

$$\begin{aligned}&\displaystyle CO_2 +3H_2 \leftrightarrow CH_{3} OH+H_{2} O\quad \Delta H_{298} =-49.43\;\text{ kJ}~\text{ mol}^{-1}\end{aligned}$$

(38)

$$\begin{aligned}&\displaystyle CO_2 +H_2 \leftrightarrow CO+H_2 O\quad \Delta H_{298} =+41.12\;\text{ kJ}~\text{ mol}^{-1} \end{aligned}$$

(39)

Reactions (37)–(39) are not independent so that one is a linear combination of the other ones. The temperature and pressure of reaction are 495–535 K and 5–8 MPa, respectively. In the current work, the rate of expressions have been selected from Graaf et al. [41]. The rate equations combined with the equilibrium rate constants [42] provide enough information about kinetics of methanol synthesis. The correspondent rate expressions due to the hydrogenation of CO, CO$_{2}$ and the reversed water-gas shift reactions are:

$$\begin{aligned}&\displaystyle r_1 =\frac{k_1 K_{CO} \left[ {f_{CO} f_{H_2 } ^{3/2}-f_{CH_3OH} /(f_{H2} ^{1/2}K_{P1} )} \right]}{(1+K_{CO} f_{CO} +K_{CO_2 } f_{CO_2 } )\left[ {f_{H_2 } ^{1/2}+(K_{H_2 O} /K_{H_2 } ^{1/2})f_{H_2 O} } \right]}\end{aligned}$$

(40)

$$\begin{aligned}&\displaystyle r_2 =\frac{k_3 K_{CO_2 } \left[ {f_{CO_2 } f_{H_2 } -f_{H_2 O} f_{CO} /K_{P3} } \right]}{(1+K_{CO} f_{CO} +K_{CO_2 } f_{CO_2 } )\left[ {f_{H_2 } ^{1/2}+(K_{H_2 O} /K_{H_2 } ^{1/2})f_{H_2 O} } \right]}\end{aligned}$$

(41)

$$\begin{aligned}&\displaystyle r_3 =\frac{k_{2} K_{CO _{2} } \left[ {f_{CO _{2}} f_{H_{2}} ^{3/2}-f_{CH_{3} OH} f_{H_{2} O} /(f_{H_{2}} ^{3/2}K_{p2} )} \right]}{(1+K_{CO} f_{CO} +K_{CO_{2}} f_{CO_{2} } )\left[ {f_{H_{2} } ^{1/2}+(K_{H_{2} O} /K_{H_{2} } ^{1/2})f_{H_{2} O} } \right]} \end{aligned}$$

(42)

The reaction rate constants, adsorption equilibrium constants and reaction equilibrium constants which occur in the formulation of kinetic expressions are tabulated in Tables 8, 9, and 10, respectively.

Table 8 Reaction rate constants [42]

 

Table 9 Adsorption equilibrium constants [42]

 

Table 10 Reaction equilibrium constants [42]