Modeling, Simulation and Control of Dimethyl Ether Synthesis in an Industrial Fixed-bed Reactor

Chemical Engineering and Processing 50 (2011) 85–94

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Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep

Modeling, simulation and control of dimethyl ether synthesis in an industrial fixed-bed reactor M. Farsi, R. Eslamloueyan ∗ , A. Jahanmiri School of Chemical and Petroleum Engineering, Shiraz University, Mollasadra Avenue, Shiraz, Fars 713481154, Iran

a r t i c l e

i n f o

Article history: Received 26 December 2009 Received in revised form 2 September 2010 Accepted 29 November 2010 Available online 4 December 2010 Keywords: Fixed-bed reactor Heterogeneous model DME reactor control

a b s t r a c t Dimethyl ether (DME) as a clean fuel has attracted the interest of many researchers from both industrial communities and academia. The commercially proven process for large scale production of dimethyl ether consists of catalytic dehydration of methanol in an adiabatic fixed-bed reactor. In this study, the industrial reactor of DME synthesis with the accompanying feed preheater has been simulated and controlled in dynamic conditions. The proposed model, consisting of a set of algebraic and partial differential equations, is based on a heterogeneous one-dimensional unsteady state formulation. To verify the proposed model, the simulation results have been compared to available data from an industrial reactor at steady state conditions. A good agreement has been found between the simulation and plant data. A sensitivity analysis has been carried out to evaluate the influence of different possible disturbances on the process. Also, the controllability of the process has been investigated through dynamic simulation of the process under a conventional feedback PID controller. The responses of the system to disturbance and setpoint changes have shown that the control structure can maintain the process at the desired conditions with an appropriate dynamic behavior. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Dimethyl ether as a basic chemical product is mainly consumed in industries as a solvent and propellant in various aerosol products [1]. The similarity of some physical properties of DME, liquefied petroleum gases and diesel fuel has presented DME as a clean fuel that can be used in diesel engines, power generation, and other purposes such as a substitute for LNG [2]. It can be produced from a variety of feed-stock such as natural gas, crude oil, residual oil, coal, waste products and bio-mass [3]. Although the commercially proven technology for DME production is through dehydration of pure methanol in an adiabatic fixed-bed reactor, many researchers are working on the direct conversion of syngas to DME over a dual heterogeneous catalyst [4]. DME can be produced in packed bed or slurry reactors. Although slurry reactors have the advantage of better heat removal capability by coil or jacket and good productivity and good interface contacting due to small solids particle size, they suffer from significant mass transfer resistance due to the use of inert liquid phases such as waxy liquid [5]. At present, DME is commercially produced through methanol dehydration using acidic porous catalysts in adi-

∗ Corresponding author. Tel.: +98 711 2303071; fax: +98 711 6287294. E-mail addresses: [email protected], [email protected] (R. Eslamloueyan). 0255-2701/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2010.11.013

abatic packed bed reactors [6]. Due to simplicity and lower costs, adiabatic fixed bed reactors can be the first choice for catalytic processes having low or intermediate heat of reactions [7]. However, in case of highly endothermic or exothermic reactions, the problems of reaction extinguishing or catalyst sintering may happen [8]. The fluidized bed reactors have been suggested by some researchers as a perfect reactor for DME synthesis [9,10]. Fluidized bed reactors showed the better heat removal characteristics due to freely moving catalyst particles in the bed. However, collision between catalyst particles and the reactor wall causes loss of catalyst. Simulation of packed bed reactors is not an easy task, since systems of nonlinear partial differential equations have to be solved along with nonlinear algebraic relationships [11]. Modeling and simulation of fixed-bed reactors have been done for many industrial and pilot scales reactors. For instance, Shahrokhi et al. modeled and simulated the dynamic behavior of a fixed bed reactor for methanol production from syngas and proposed an optimizer to maximization of the production rate [12]. Jahanmiri and Eslamloueyan modeled and simulated the isothermal fixed-bed reactor for methanol production in industrial scale [13]. They calculated the optimal temperature profile of the shell side to maximize methanol production. Fixed-bed DME synthesis reactors have been investigated through a number of research studies. Lee et al. modeled and analyzed DME production from syngas in a fixed-bed reactor at steady state condition [14]. Nasehi et al. simulated an industrial adiabatic

86

M. Farsi et al. / Chemical Engineering and Processing 50 (2011) 85–94 Table 2 Equipment and catalyst parameters.

Table 1 The reaction kinetics and equilibrium constants. k = A(i)expB(i)/RT

Parameter

k0 KH2 O KCH3 OH

Parameter

A(i)

B(i)

3.7 exp10 0.84 exp−1 7.9 exp−4

−1,05,000 41,100 7050

fixed-bed reactor for DME production from methanol dehydration at steady state condition, and showed that the difference between one and two dimensional models is negligible [15]. Omata et al. studied DME production from syngas in a temperature gradient reactor for overcoming both the equilibrium limit of the reaction at high temperature and low activity of the catalyst at low temperature [16]. Then, they optimized the reactor for higher CO conversion by a combination of genetic algorithm and neural networks. Fazlollahnejad et al. investigated methanol dehydration in a bench scale adiabatic packed reactor [17]. The reactor was packed with 1.5 mm ␥-Al2 O3 pellets as catalyst and operated at atmospheric pressure. They investigated the effects of weight hourly space velocity and temperature on methanol conversion. Although many research studies have been carried out on the dynamic simulation and control of fixed bed reactors, we could not find any work in this field for DME synthesis from methanol dehydration. Fixed-bed catalytic reactors are often difficult to control due to nonlinearity, process dead time and positive feedback of feed preheating by the reactor effluent stream [18]. Jorgensen presented a good review for dynamic modeling and control of fixed bed reactors [19]. Aguilar et al. developed a robust PID control law in order to control the outlet temperature of a fluid catalytic cracking reactor [20]. The controller is synthesized using an input–output linearizing control law coupled to a proportional-derivative reduced order observer to infer on-line the unknown heat of reaction. This control strategy was robust against model uncertainties, noisy temperature measurements and set point changes. In this paper, we develop an unsteady heterogeneous model for dynamic simulation of an industrial DME synthesis reactor and design a conventional feedback control system to maintain the inlet stream temperature at the desired value. 2. Reaction kinetics The reaction equation of DME synthesis from dehydrogenation of methanol is as the following. 2CH3 OH  CH3 OCH3 + H2 O,

H298 = −23.4 (kJ/mol)

(1)

In this work, the rate expressions have been selected from the ´ c´ et al. [21]. kinetic model proposed by Berci r=

2 k · KCH

3 OH

((CCH3 OH − (CC2 H6 O CH2 O )/Keq ))

(1 + 2(KCH3 OH CCH3 OH ).5 + KH2 O CH2 O )

(2)

4

Value

Reactor Catalyst density (kg m−3 ) Catalyst void fraction Specific surface area of catalyst pellet Catalyst diameter (m) Ratio of void fraction to tortuosity of catalyst particle Reactor Length (m) Reactor diameter (m) Bed void fraction Catalyst density (kg m−3 ) Density of bed (kg m−3 ) Heat exchanger Tube side diameter (m) Tube thickness (m) Shell diameter (m) Overall heat transfer coefficient (W m−1 K−1 ) Heat exchanger length (m)

3. Process description DME production loop consists of an adiabatic packed bed reactor as well as a shell and tube heat exchanger as depicted in Fig. 1. The inlet feed to the reactor is preheated in the heat exchanger through heat transfer to the outlet products from the reactor. The reactor product is transferred to DME purification unit, and DME is separated from water and methanol. Also, the unreacted methanol is recycled to the reactor. The developed model of this process includes modeling equations for both the reactor and the feed preheater. 3.1. Reactor model In this study, a one dimensional heterogeneous model has been considered for steady state simulation of the process. The basic structure of the model is composed of heat and mass conservation equations coupled with thermodynamic and kinetic relations as well as auxiliary correlations for prediction of physical properties. In the dynamic modeling of the reactor the following assumptions have been considered. - No heat loss to the surrounding from the reactor wall - Plug flow pattern and negligible concentration and temperature variations in radial direction - Uniform temperature within each catalyst pellet (Biot number = 0.0125) Biot number is a dimensionless number defined in conduction heat transfer, and gives a simple index of the ratio of the heat transfer

3138 + 1.33 × 10−3 T − 1.23 T

× 10−5 T 2 + 3.5 × 10−10 T

(3)

k = A(i) expB(i)/RT

(4) mol/m3 .

Ci is the molar concentration of component i in k1 , KCH3 OH and KH2 O are the reaction rate constant and the adsorption equilibrium constant for methanol and water vapor, respectively, which are tabulated in Table 1. Commercially, ␥-Al2 O3 catalyst is used in methanol dehydration reaction. Also, the characteristics of

1.905 × 10−2 2.11 × 10−3 0.615 325 4.57

the catalyst pellet and the reactor design specifications are summarized in Table 2.

where, ln (Keq ) = 086 log T +

2050 0.2 673 0.3175 × 10−2 0.066 8 4 0.5 2010 1005

Fig. 1. Process flow diagram of DME production.

M. Farsi et al. / Chemical Engineering and Processing 50 (2011) 85–94 Table 3 Feed specification of the industrial DME reactor. Parameter

dT Do  dTcold U(Thot − Tcold ) = −ug cold +  · Acold · Cp dt dz Value

◦

T ( C) P (bar) Mass flow rate (kg/s) Mass fraction (%) Methanol Water DME

A ln(Do /Di ) A 1 1 1 + i + i = U 2LKw Ao ho hi

94% 1% 5%

resistances inside and at the surface of a body. For the Biot numbers smaller than 0.1, the heat conduction inside the body is much faster than the heat convection away from the surface, and the temperature gradient is negligible inside of the body. To justify the first two assumptions, it should be notified the industrial reactor considered in this study have insulation that causes the heat loss from the reactor wall to the environment negligible in comparison to the heat release in the reactor, and also because of high Reynold’s number (about 10,000), the assumption of plug flow is reasonable. Furthermore, Nasehi et al. simulated the reactor at steady state condition by using a two-dimensional heterogeneous model with consideration of the heat loss from the insulation to environment [22]. The results of this simulation showed that the temperature difference between the reactor center and the reactor wall is at most 4 ◦ C. So, the heat loss has a negligible influence on the simulation results in one-dimensional plug flow model. The mass and energy balances for the gas and solid phase are expressed by the following equations: Gas phase:

∂T ∂T = −us g cp + hf av (Tss − T ) g cpg ε ∂t ∂z

∂Cis s = −kgi av (Ci − Cis ) + i Bs ri ∂t

(1 − ε)Bs cps

∂Ts = −hf av (Tss − T ) − i Bs (−H)ri ∂t

(1 − ε)2  · uc (1 − ε) · u2c dp = 150 + 1.75 2 3 l d ε d · ε3





Z,t=0

= Ciss , T 

Z,t=0

(14)

3.3. Auxiliary equation Auxiliary equations are used for prediction of the model parameters. In these equations i is the effectiveness factor which is defined as actual reaction rate per particle to theoretical reaction rate based on the external pellet surface concentration, which is obtained from dusty gas model calculations [25]. It can be calculated from the following equation:



i =

ri dVp



Vp . ri 

(15)

surface

In the heterogeneous model the physical properties of chemical species and overall mass and heat transfer coefficients between catalyst solid phase and gas phase must be estimated. The overall mass transfer coefficient between solid and gas phase has been obtained from the correlation proposed by Cussler [26]. (16)

1 − yi



(7)

Dim =

(8)

Binary mass diffusion coefficient has been determined by the following equation [20]:

(9)

Dij =

(17)

(yi /Dij )

10−7 T 3/2



(1/Mi ) + (1/Mj )

3/2

P( ci

(18)

3/2 2

+ cj )

The overall heat transfer coefficient between solid and gas phase (hf ) has been predicted from [27]: hf Cp 

 C  2/3 p K

0.458 = εB



udp 

−0.407

(19)

The overall heat capacity in Eq. (19) depends on the temperature and composition, and is calculated using Eqs. (20) and (21):

= Tiss

3.2. Heat exchanger model

Cp(i) = a + bT + cT 2 + dT 3

(20)

Cp =

(21)



yi Cp(i)

Table 4 presents the values of a, b, c and d in Eq. (20).

There are no phase changes in the heat exchanger and this unit is small in compare with the reactor unit. The shell and tube heat exchanger used for preheating feed stream is modeled at the dynamic condition by the following equations: dT Di  dThot = −ug hot − U(Thot − Tcold )  · Ahot · Cp dt dz

w

In Eq. (16), the Sc is calculated for components along the reactor. Mass transfer diffusion coefficient for each component in the mixture has been estimated by:

   = C0 , T  = T0 , P  = P0 Boundary conditions : C  Z=0 Z=0 Z=0 Ci 

1/3   0.14

(6)

The initial and boundary conditions for these equations are given below. Also, the feed specification of the industrial DME reactor is reported in Table 3.

Initial conditions :



Cp  hi Di = 0.023(Re)0.8 k k

(13)

kgi = 103 (1.17Re−0.42 Sci −0.67 ug )

The pressure drop in the bed is calculated by Ergun equation [23]: −

1/3

(5)

Solid phase: (1 − ε)

(12)

In Eq. (12), the heat transfer coefficient in the shell side (ho ) and the tube side (hi ) are calculated using the following equations [24]:



∂C ∂Ci s ) = −us i − kgi av (Ci − Cis ∂t ∂z

(11)

The overall heat transfer coefficient in the heat exchanger is calculated from

260 18.2 50.81

Cp  ho Do = 0.33(Re)0.6 kf k

ε

87

(10)

4. Numerical solution To solve the set of nonlinear partial differential equations (PDE) obtained from dynamic modeling, the reactor length is divided into equal discrete intervals, and by using finite difference method the PDEs are converted into a set of ordinary differential equations

M. Farsi et al. / Chemical Engineering and Processing 50 (2011) 85–94 Table 5 Comparison of the steady state simulation results of the reactor with plant data.

Table 4 Parameters of heat capacity coloration.

DME CH3 OH H2 O

a

b

c × 10−5

d × 10−9

0.18 21.15 32.24

17.02 0.07 0.002

5.23 2.59 1.06

1.92 0.29 3.6

(ODE) in time domain. The set of equations are solved by 4th order Runge–Kutta [28]. Before carrying out dynamic simulation, the stationary condition of the system should be obtained through solving the governing steady-state equations. The aim of performing steady state simulation of the DME reactor is to determine the concentration and temperature profiles along the reactor at normal operation. These steady state profiles are used as the initial conditions of the unsteady state PDEs. After rearranging the modeling equations for the steady state condition, a set of ordinary differential equation is obtained. This set of equations was solved by the method of 4th order Runge–Kutta. 5. Control structure The main objective of the control system of DME reactor is to maintain the operating condition at the optimum designed state in spite of probable disturbances. Since DME is produced through an exothermic reaction in a fixed bed adiabatic reactor, the control of feed temperature is crucial for keeping the conversion at its maximum level. The proposed control structure for a typical industrial DME reactor is shown in Fig. 2. The reactor feed stream is preheated in a shell and tube heat exchanger with the hot reactor effluent. A bypass stream around this heat exchanger is manipulated to control the reactor feed temperature. In this study, a conventional feedback PID algorithm carries out the load rejection and set-point tracking tasks. 6. Results and discussion 6.1. Steady state simulation In this section, steady state simulation results of DME reactor have been presented. Assuming an adiabatic tubular reactor with plug flow pattern, the reactor has been simulated based on a one dimensional heterogeneous model. Modeling validation is done through comparing the simulation results with the industrial

DME exit mole flow rate (kmol/h) MeOH exit mole flow rate (kmol/h) Outlet temperature (K)

Simulation result

Plant data

Absolute error

2457 940.6 652.2

2506 937.7 644

1.95% 0.31% 1.27%

6000 DME MEOH

5000

Water

Mole Flow (kmol hr-1)

88

4000

3000

2000

1000

0

0

1

2

3

4

5

6

7

8

Length (m) Fig. 3. The profiles of mole flow rates of DME, Methanol and water along the reactor.

reactor data provided by Petrochemical Research and technology Company which is a subsidiary of National Iranian Petrochemical Company. Table 5 indicates the absolute errors of the steady-state simulation results in comparison to available data of the industrial reactor. This table shows that the proposed model has been able to predict the exit concentrations and temperature very well. Fig. 3 illustrates the molar flow rate of DME, methanol and water along the reactor length at steady state condition. The mole fraction of DME along the reactor in solid and gas phase predicted by the heterogeneous model has been compared in Fig. 4. Near the end of the reactor, where the reaction equilibrium is prevailed, the difference between gas and solid phase concentrations is vanished.

Fig. 2. Schematic of the control structure.

M. Farsi et al. / Chemical Engineering and Processing 50 (2011) 85–94

Table 6 Calculated average mass transfer and effective diffusion coefficients in the reactor.

0.5

0.41

Mole Fraction

89

DDME-mixture (m s−1 )

0.1737 × 10−6

−1

0.1745 × 10−6 0.2685 × 10−6 0.0036 0.0036 0.005

DMeOH-mixture (m s ) Dwater-mixture (m s−1 ) kDME (m2 s−1 ) kMeOH (m2 s−1 ) kwater (m2 s−1 )

0.32

0.23

Gas Phase Solid Phase

0.05

0

1

2

3

4

5

6

7

8

Length (m) Fig. 4. DME mole fraction in solid and gas phase.

Also, the predicted temperature profile along the reactor obtained from the simulation is shown in Fig. 5. The temperature approaches to equilibrium temperature due to heat generation by reaction, and remains constant. The calculated average heat transfer coefficient between gas and catalyst phase (hf ) in Eq. (19) is about 1170 W m−2 K−1 . Also, the average estimated viscosity and molar heat capacity of the reaction mixture are about 1.6 × 10−7 kg m−1 s−1 and 69.5 W mol−1 K−1 , respectively. The effective diffusion coefficients of components in the mixture and mass transfer coefficients for components are calculated along the reactor. The average effective diffusion and average mass transfer coefficients of DME, MeOH and water vapor in the mixture are presented in Table 6. 6.2. Open loop dynamic simulation The open loop behavior of the reactor has been considered in two levels. In the first level the behavior of the reactor without its preheater has been studied. The second level investigates both of the reactor and the related preheater as a heat integrated process.

Temperature (K)

680 0.14

640

600

560

520 8 50

6 40

4

Length (m)

30 20

2 0

10

Time (sec)

0

Fig. 6. Dynamic temperature profile along the reactor for 10 ◦ C step change in the inlet stream to the reactor.

To investigate the influence of disturbances on the dynamic behavior of DME reactor, the feed temperature, pressure and composition have been considered as the main probable effective loads of the system. Figs. 6 and 7 indicate the step response of the system to 10 ◦ C increase in the feed temperature. In Fig. 6, dynamic variation of the temperature along the reactor length has been illustrated in a three-dimensional diagram. Fig. 7 represents time variations of DME concentration along the reactor bed. As shown in Fig. 8, the response of the outlet temperature of the reactor for this disturbance has a time delay about 15 s, and it reaches to the new steady state point in 50 s.

660

0.5

DME mole fraction

Temperature (K)

625

590

555

0.4 0.3 0.2 0.1 0 8 6

50 40

4 520

0

1

2

3

4

5

6

Length (m) Fig. 5. Predicted temperature profile along the reactor.

7

8

Length (m)

30 20

2 0

10 0

Time (sec)

Fig. 7. Dynamic DME mole fraction profile along the reactor for 10 ◦ C step change in the inlet stream to the reactor.

90

M. Farsi et al. / Chemical Engineering and Processing 50 (2011) 85–94

661

0.5

DME molefraction

Temperature(K)

659

657

655

0.4 0.3 0.2 0.1 0 8

653

50

6

40

4

30

651

20

2

Length (m)

10 0

0

10

20

30

40

Time (sec)

0

50

Time (sec)

Fig. 10. Dynamic DME mole fraction profile for 0.05 step change in the methanol and water vapor mole fractions of the inlet stream.

Fig. 8. The outlet temperature profile from the reactor to 10 ◦ C step change in the inlet temperature.

652.5

651.5

Temperature(K)

The feed composition effect was studied by applying 0.05 step changes in the mole fractions of both methanol and water vapor. Figs. 9 and 10 represent dynamic responses of the reactor temperature and DME mole fraction profiles along the reactor, respectively. The response of the reactor outlet temperature to the step change in composition is shown in Fig. 11. Applying 2 bar step change in the reactor inlet pressure generates the outlet temperature response shown in Fig. 12. As can be deduced from this figure, the change in inlet pressure has no significant effect on the temperature and consequently the product composition of the reactor effluent. The above mentioned results indicate that the feed temperature and composition are the major process disturbances that should be rejected by the designed control system. In order to have a more realistic influence of inlet temperature on the reactor dynamic response, we have also studied the response of the reactor effluent temperature to a step change in the fresh feed temperature to the preheater. The reactor output temperature and DME mole fraction have been respectively shown in Figs. 13 and 14. According to these figures, increasing of the inlet feed temperature to the heat exchanger will cause an increase in the reactor feed temperature and consequently reactor outlet temperature. Due to

650.5

649.5

648.5

647.5

0

10

20

30

40

50

Time (sec) Fig. 11. The outlet temperature profile for 0.05 step change in the feed mole fraction.

652.45

652.4

Temperature(K)

Temperature(K)

680 640 600 560

652.35

652.3

652.25

520 8 6

40

4

Length (m)

2 0

0

10

20

50

30

Time (sec)

Fig. 9. Dynamic temperature profile for 0.05 step change in the methanol and water vapor mole fractions of the inlet stream.

652.2

0

10

20

30

40

50

Time (sec) Fig. 12. The outlet temperature profile for 2 step change in the feed pressure.

M. Farsi et al. / Chemical Engineering and Processing 50 (2011) 85–94

850

91

657

656

810

Temperature(K)

Temperature(K)

655 770

730

654

653

690 652

650

0

5

10

15

651

20

0

5

10

Time (min)

15

20

25

30

35

Time (min)

Fig. 13. The outlet temperature profile for 10 ◦ C step change in the fresh feed temperature to preheater.

temperature increase, conversion in the reactor decreases and this cause a new equilibrium point for the process. But, variation of the rector outlet temperature is lesser than the variation of the inlet temperature. 6.3. Closed loop process simulation Although packed bed reactors are distributed parameter systems, the output variables are usually measured at the process exit streams. In order to control the inlet feed temperature to the reactor, a feedback control loop as shown in Fig. 2 is designed based on a conventional Proportional–Integral–Derivative (PID) controllers. The PID control algorithm is usually used for the industrial control systems. In this study, the PID controller has been tuned through using Closed Loop Ziegler–Nichols method. This algorithm is one of the most common methods for tuning PID controllers. The gain of proportional controller is increased (from zero) until it reaches the ultimate gain, at which the output of the control loop oscillates with a constant amplitude. The parameters of the controller, i.e. the proportional gain, reset time, and derivative value, are derived from the

Fig. 15. Outlet temperature response for rejection of the disturbance by the closed loop control system.

ultimate gain and the sustained period of oscillation [29]. This type of tuning creates a quarter wave decay ratio which is acceptable for most systems. According to this procedure, the controller gain, reset time and derivative constant estimated in this work are 9.6, 0.56 min and 0.14 min, respectively. To compare Ziegler–Nichols’ tuning method to other tuning rules, we have tuned the reactor controller loop by using Luyben’s method [30]. This method have smaller decay ratio, but it has slower response than Ziegler–Nichols’ method. According to Luyben method, the controller gain, integral constant and derivative constant are 7.5, 2.45 min and 0.18 min, respectively. Both methods were examined through their application to the set point tracking of the reactor controller. According to this investigation, Ziegler–Nichols’ method led to better control responses, so in this study the Ziegler–Nichols’ tuning rules are applied. In this section, the dynamic closed-loop behavior of the process consisting of the reactor and preheater in response to predefined disturbances and set point variation are presented and analyzed.

0.439

0.442

0.4388

DME molefraction

DME molefraction

0.432

0.422

0.4386

0.4384

0.4382

0.412 0.438

0.402 0

5

10

15

20

Time (sec) Fig. 14. The outlet DME concentration for 10 ◦ C step change in the fresh feed temperature to preheater.

0.4378

0

5

10

15

20

25

30

35

Time (min) Fig. 16. Outlet DME concentration response for rejection of the disturbance by the closed loop control system.

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M. Farsi et al. / Chemical Engineering and Processing 50 (2011) 85–94

668

300

250

Set point

Temperature(K)

Mole flow (mol sec-1)

664 200

150

100

660

656 50

0

652 0

5

10

15

20

25

30

35

0

10

Table 7 Characteristics of the control loop response for disturbance rejection. Time (min) Rise time Peak time Settling time

4 9.7 17.3

DME molefraction

0.4382

0.4373

0.4364

0.4355

0

10

20

30

40

Time (min) Fig. 19. The variation of the outlet DME mole fraction for set point tracking.

400

320

240

160

80

Table 8 Dynamic properties of the closed loop response for setpoint tracking. Overshoot Rise time (min) Settling time (min)

40

0.439

Mole folw (mol sec-1)

6.3.2. Setpoint tracking The setpoint tracking capability of the propose control system has been investigated by applying 10 ◦ C step changes in the setpoint of the reactor inlet temperature. Figs. 18 and 19 indicate the outlet temperature and DME mole fraction from the reactor. The outlet temperature approaches to the new setpoint with acceptable speed and oscillations. In Fig. 20, the molar flow rate of the bypass stream has been represented. This figure indicates that the manipulated variable response for the servo problem has suitable dynamic behavior and can be implemented easily by an ordinary control valve. Table 8 presents the dynamic characteristics of the control loop response for the set point tracking. Table 8 shows that characteristics response of controller is acceptable and proper in industrial scale.

30

Fig. 18. The variation of the outlet temperature for set point tracking.

Fig. 17. Manipulated variable variations in case of load rejection.

6.3.1. Load rejection To observe the performance of the proposed control structure and applied PID controller for the system, 15 ◦ C step change in the temperature of the fresh feed to the preheater has been introduced to the process as the most influential disturbance. The closed loop response of the process has been presented in Figs. 15 and 16. As can be seen from these figures, the load is completely rejected after about 7 min using a PID controller. The variation of the manipulated variable, i.e. bypass flow rate, has been represented in Fig. 17. The manipulated variable response has no sever oscillation and is similar to the behavior of an appropriate control system. The dynamic characteristics of the control loop response for disturbance rejection are presented in Table 7.

20

Time (min)

Time (min)

42% 3.8 19.8

0

0

10

20

30

40

Time (min) Fig. 20. Variation of the bypass stream flow rate for set point tracking.

M. Farsi et al. / Chemical Engineering and Processing 50 (2011) 85–94

93

disturbance rejection and setpoint tracking so that it can control the reactor in a time period of 20–30 min.

668

Appendix A. Nomenclature

Temperature(K)

664

av Acold Ahot Ci Cp dp Di Do Dij Dim

660

656 activity:1.0 activity: 0.7 activity: 0.5 652

0

7

14

21

28

35

Time (min)

F hf hi

Fig. 21. Setpoint tracking response for different values of catalyst deactivation.

ho As mentioned in the modeling section, the catalyst deactivation is not included in the model because of lack of deactivation rate correlation in open literature. However, it should be noted that the worst condition for examining the capability of the control system structure would be for the case of fresh catalyst in which the reaction rate is at the highest level. Therefore, since the objective of this work is to examine the controllability characteristics of the suggested control structure, very slow catalyst deactivation cannot affect the control system structure. To illustrate the influence of the catalyst deactivation on the control system response, the process has been simulated for three levels of catalyst deactivation, and the controller responses are illustrated in Fig. 21. 7. Conclusion In this study, an industrial DME synthesis reactor from dehydration of methanol has been simulated in steady and unsteady state conditions. The reactor mathematical formulation is based on a rigorous one-dimensional heterogeneous model. Also, a feed-back control system has been designed and tuned to control the reactor inlet temperature. The comparison of the simulation results and the industrial data show that the proposed model can predict the reactor outlet temperature and concentrations with relative errors less than 2%. Some important disturbances such as fresh feed temperature, inlet reactor temperature, inlet pressure and feed composition have been applied to the process for investigating its open loop dynamic behavior. Since the deactivation rate equation of the catalyst used in this study is not available in open literature, the catalyst deactivation, which usually occurs very slowly in industrial reactors, has not been included in the modeling. In fact, the objective of this article is the investigation of controllability of the proposed control system structure for DME reactor, and because the reactor control system responds very faster (about 30 min) than the catalyst deactivation time (about 2 years), the catalyst deactivation has no significant influence on the objective of this research. Anyway, if experimental data or mathematical correlation for predicting the deactivation of DME indirect synthesis catalyst becomes available later, inclusion of the catalyst deactivation rate will be an interesting subject for future work. The dynamic characteristics of the open loop system have been used to design a PID controller for adjusting the preheater bypass flow rate in order to maintain the reactor inlet temperature at its setpoint. The suggested feedback closed loop system has shown a good performance for the cases of

k kg Keq Ki L M P r R Re Sci T u ug U V

vci yi z

specific surface area of catalyst pellet (m2 m−3 ) cross section area of cold side in heat exchanger (m2 ) cross section area of hot side in heat exchanger (m2 ) molar concentration of component i (mol m−3 ) specific heat of the gas at constant pressure (J mol−1 ) catalyst diameter (m) tube inside diameter of heat exchanger (m) tube outside diameter of heat exchanger (m) binary diffusion coefficient of component i in j (m2 s−1 ) diffusion coefficient of component i in the mixture (m2 s−1 ) total molar flow rate (mol s−1 ) gas–solid heat transfer coefficient in reactor (W m−2 K−1 ) tube side heat transfer coefficient in heat exchanger (W m−2 K−1 ) shell side heat transfer coefficient in heat exchanger (W m−2 K−1 ) rate constant of dehydrogenation reaction (mol m−3 Pa−1 s−1 ) mass transfer coefficient for component i (m s−1 ) reaction equilibrium constant for methanol dehydration reaction (mol m−3 ) adsorption equilibrium constant for component i (m3 mol−1 ) reactor length (m) molecular weight (g mol−1 ) total pressure (Bar) rate of reaction for DME synthesis (mol kg−1 s−1 ) tube diameter in heat exchanger (m) Reynolds number Schmidt number of component i temperature (K) superficial velocity of fluid phase (m s−1 ) linear velocity of fluid phase (m s−1 ) overall heat transfer coefficient (W m−2 K−1 ) volume (m3 ) critical volume of component i (cm3 mol−1 ) mole fraction of component i (mol mol−1 ) axial reactor coordinate (m)

Greek letters  viscosity of fluid phase (kg m−1 s−1 )  density of fluid phase (kg m−3 )

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