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Axial Dispersion Plug Flow Model for Methanol Dehydration Reactor


 One-dimensional heterogeneous dispersed plug flow (DPF) model is employed to model an adiabatic fixed-bed reactor for the catalytic dehydration of methanol to dimethyl ether (DME). The mass and heat transfer equations are numerically solved for the reactor. The concentration of the reactant and products and also the temperature varies along the reactor, therefore the effectiveness factor would also change in the reactor. We used the effectiveness factor that was simulated according to the diffusion and reaction in the catalyst pellet as a pore network model. The predicted distribution for the effectiveness factor was utilized for the reactor simulation. The simulation results were compared to the experimental data and a satisfactory agreement was confirmed.


Introduction
DME is a linear combination, odorless, colorless component and has no corrosive properties.
It is also not environmentally hazardous. DME, as a liquefied gas has characteristics similar to those of liquefied petroleum gas (LPG) [1][2] . It is identified as a potential diesel and cooking fuel. Its oxygen content is 34.78% and can be burned without soot emission. It has a boiling point of −25 °C, which is 20 °C higher than LPG and can be liquidized at 0.54 MPa (20 °C). DME behaves as a gas in standard conditions (0.1 MPa, 298K) 2 . DME can be produced from a variety of feed-stocks such as natural gas, crude oil, residual oil, coal, waste products and bio-mass 1 . One of the commercially processes for DME production is the catalytic dehydration of methanol. For this reaction, acidic porous catalysts such as zeolites, silica-alumina, alumina and etc. are used [3][4][5][6][7][8][9][10] . The reaction rate for this process has been mostly derived from the experiments conducted in the conditions not found in an industrial Reactor [11][12][13][14][15] . Bercic and Levec reviewed the different reaction rates and they designed some experiments to study this reaction in industrial conditions using  -alumina as the catalyst 12 . The experiments were carried out in a differential reactor (8-mm inside diameter) in a temperature range of 290-360 °C. The pressure was kept constant at 146 kPa.
The reactor was operated free of inter particle heat and mass resistances. Bercic and Levec suggested the kinetics of the reaction at this condition 12 .
They also used a laboratory scale reactor to find its conversion and temperature profile in it.
Then plug flow condition and longitude changes of concentration and temperature were considered for the reactor modeling. They considered convection and reaction terms in the mass and heat transfer equations. In order to find the effectiveness factor, the continuum model was considered for the spherical catalyst particles. They used a Rang Kutta method to solve the mass and heat transfer equations simultaneously 16 .
In the modeling presented in this paper we used the results of pore network model for the value of the effectiveness factor. In fact, since the catalyst pellet has porous structure, continuum models cannot predict its behavior precisely.
Any porous structure can be mapped into a pore network model. The pore network models have been extensively used in the last decades. The structure of the porous medium can strongly affect its characteristics. The effectiveness factor was found based on a three dimensional pore network model for the catalyst pellets 17 .
Here, we first explain the mathematical model for and the mass and heat transfer process in the reactor. And the mathematical approach for solving the equations is described as well.
After that the results are presented and discussed. In the final part a summary of the paper is presorted.

Mathematical Model
The dehydration of methanol is based on the following reversible reaction: The reaction rate   r  is considered as follows 16 : The kinetic constants and thermodynamics equilibrium constant are presented in Table 1.  The mass and energy balance equations are arranged according to diffusion, convection and reaction mechanisms in the reactor: Eq. (3) should be written for Methanol and DME and index k refers to the components. Eqs.
(2), (3) and (4) can be changed to a dimensionless form as follows: where 00 ,, The dimensionless parameters in Eqs. (5) and (6) are: The water concentration in each segment can be calculated from the total balance.
The appearance of the effectiveness factor in Eqs. (5) and (6) is due to the heterogeneity in the reactor and the mass transfer limitation in the catalysts.
We used the results of our previous study to find the effectiveness factor along the reactor 17 .
The effectiveness factor was found based on a three dimensional pore network model for the catalyst pellets. In that model pores are places where mass transfer and reaction occurs and nodes are interchange points between the pores. For more details, one can refer to the mentioned study were the effectiveness factor is calculated at different temperatures and methanol concentrations.
Eqs. (5) and (6) are nonlinear equations that can be solved subject to the following boundary conditions: The dispersion coefficient in Eq. (3) should be considered in the porous packed bed. This parameter is a function of Reynolds number and can be calculated as follows 19 : The tortuosity in Eqs. (17) and (18) is given by 20 : The axial effective thermal conductivity in Eq. (4) is determined by the following equation 21 : (20) where: where n is: The dimensionless parameters in Eq. (20) are: The SRK equation of state is used to calculate the compressibility factor 22 . Other parameters used in the simulation are presented in Table. 2.
Where k refer to different components (DME and Methanol) and i shows the grid number.
x  is the element length in the reactor and here it is considered equal to 0.01 of the reactor length these equation should be written for all the grids. Using the conditions Eqs (13)-(16), the nonlinear set of equations has to be solved in order to calculate the concentration and temperature distributions in the reactor. A try and error method is used to transform the equations into a linear form. Then the set of linear equations is solved using the LU method.
This trial method is repeated until desired precision is obtained. Fig. 1 shows the methanol concentration versus reactor length. The results are presented for different flow rates and compared to the experimental data. As the flow rate decreases, the residence time will increase and methanol has more time to be in contact with catalysts in the reactor. Therefore, the equilibrium conversion is achieved in a smaller length from the reactor from the inlet.   Fig. 3 As the inlet temperature increases, the reaction rate would also increase and the equilibrium conversion is attained in a smaller length of the reactor. Therefore, the upper limit of the temperature corresponding to the equilibrium condition is closer to the reactor inlet. The effect of inlet temperature on temperature distribution in the reactor is also shown in Fig. 4 Higher inlet temperature would increase the reaction rate and therefore the temperature would also be higher along the reactor. The final temperature due to equilibrium condition happens closer to the reactor inlet consequently. As expected DME Concentration increase along the reactor while methanol decrease. The effect of inlet methanol concentration on concentration distribution in the reactor is studied in Fig. 6 If methanol includes some water at the reactor inlet, DME conversion would strongly reduce. The methanol concentration along the reactor would be higher or in other words the DME concentration would be less as the inlet water. content of methanol increases and the necessary reactor length would increase. Therefore feed water content affects reactor length

Conclusion
Modeling and simulation for the methanol dehydration process is studied. The effectiveness factor is considered due to pore network model results for catalyst pellets. The simulations indicate that methanol inlet temperature and its flow rate affect concentration and temperature distribution in the reactor. Increase in inlet temperature and decrease in methanol flow rate causes the equilibrium condition occur in a smaller length from the reactor inlet. The presence of water in the inlet methanol would decrease the reaction rate in the reactor as well.