Jiang Hongbo; Liang Yanhua

(Petroleum Processing Institute, East China University of Science and Technology, Shanghai 200237)

Flow Field Simulation on Double-Ring Radial Flow Reactor

Jiang Hongbo; Liang Yanhua

(Petroleum Processing Institute, East China University of Science and Technology, Shanghai 200237)

Compared with the traditional radial fow reactors (RFRs), the double-ring RFRs possess advantages including lower pressure drop, shorter fow path and greater fow area. According to the Ergun’s equation and the continuity equation, a two-dimensional hydrodynamic model was established to describe the hydrodynamic behavior in the double-ring RFRs. The successive over-relaxation (SOR) method was applied to solve the two-dimensional hydrodynamic model. The fow assignment parameters (Ti) of mass flow in the inner channel to the outer catalyst bed and the inner catalyst bed were optimized by the Powell method. Simulations showed the trend of change in gas distribution uniformity along the axial direction and the weight hourly space velocity (WHSV) with the variation of reactor size. The model can be used to analyze the reasonability of dehydrogenation reactor design, and it can also provide quantitative reference for the design of new double-ring RFRs.

double-ring; radial fow; reactors; hydrodynamics; dehydrogenation; simulation

1 Introduction

Dehydrogenation of long-chain paraffns to mono-olefns is an important process in the production of detergents. Mono-olefin reacts with benzene to generate alkyl benzene, which is a common kind of active component in the synthesis of detergents[1]. The single-pass conversion of dehydrogenation process is low and is equal to around 12%. The dehydrogenation reaction generally happens in the radial flow reactors. Compared with the axial fow reactors, RFRs have the advantages including high space velocity, low pressure drop and possibility of using small sized catalysts[2], etc. Therefore, RFRs are mostly used in gas-solid catalytic processes, such as ammonia synthesis[3], methanol production[4], styrene production[5-7], catalytic reforming[8], etc. There are four types of RFRs, including the inward Π flow (CP Π-type), the inward Z fow (CP Z-type), the outward Π fow (CF Π-type), and the outward Z fow (CF Z-type).

The double-ring RFRs are different from the traditional radial flow reactors, because they have three channels and two catalyst beds as shown in Figure 1, where the inward fow and the outward fow take place in the same reactor. The inner catalyst bed is located between the inner channel and the central channel, while the outer catalyst bed is located between the inner channel and the outer channel. Between the main channel and the catalyst bed there is a perforated region stretching from the top to the bottom. The top and bottom of catalyst beds are enclosed by walls, denoting that the catalysts are fxed in the reactor during the dehydrogenation reaction.

The feeding inlet is on the top of inner channel. The fow enters the inner channel through its top, and then flows along the axial direction. During this process, a part of the flow passes through the inner catalyst bed, then is collected in the central channel, and finally gets out of the central channel through its bottom. The other part of the flow passes through the outer catalyst bed, then is collected in the outer channel, and finally gets out of the outer channel through its bottom. The two parts of the flow from the bottom of the outer channel and the central channel merge into one, and then leave the reactor. Therefore, the central channel and the outer channel have the same outlet pressure. Compared with the traditional RFRs, the double-ring RFRs have lower pressure drop, shorter fow path and greater fow area, so they have high value in industrial application for large quantity fows.

During the development and industrial applications of RFRs, lots of investigations on the hydrodynamic behavior of RFRs have been carried out. The numerousstudies have focused on the one-dimensional model with the assumption that the gas stream passes the catalyst bed as an ideal one-dimensional radial fow. The Bernoulli equation, the total energy equation and the modifed momentum equation are applied to describe the hydrodynamic behavior in the main channels.

However, the gas flow in RFRs is not an ideal onedimensional flow. There are also axial flows in the catalyst bed. It is necessary to study the two-dimensional model of RFRs for investigating the velocity and pressure distribution and optimizing the reactor design.

Wang[9-10]performed a new iteration strategy to solve the two-dimensional model of radial flow movingbed reactor (RFMBR), and the result of simulation showed the infuence of the ratio of center pipe crosssectional area to that of annular channel. Ma and Lan[11]have performed computational fluid dynamics (CFD) to simulate the flow field in four types of RFRs. The research showed that the flow in the outward RFRs was better than that in the inward RFRs, and the fow in the CF Π-type RFRs was better than that in the CF Z-type RFRs. The axial non-uniformity in the CP-Z type was the maximum, followed by the CP Π-type, the CF Z-type, and the CF Π-type. A. A. Kareeri, et al.[12]performed experimental investigation on the axial fow distribution of four flow types in RFRs. It was found that the ratio of the cross-sectional area of central channel to that of annular channel had great infuence on the axial uniformity.

In this paper, a hydrodynamic mathematical model was established to simulate the double-ring RFR for catalytic dehydrogenation of heavy paraffins. The successive over-relaxation (SOR) method[13]was applied to solve the model. The flow assignment parameters (Ti) of mass flow in the inner channel to the outer catalyst bed and the inner catalyst bed were optimized by the Powell method[14]. Based on the simulation results, this study showed the pressure and velocity distribution in the channels and catalyst beds, as well as the relationship between the structure size, the WHSV and the gas distribution uniformity along the axial direction.

2 Mathematical Model

The mathematical hydrodynamic model of the doublering RFRs should describe the gas fow in the three main fow channels and across the distributing orifces on the perforated cylinders, and the two-dimensional fow feld of the reacting gas in two catalyst beds. The schematic diagram of the model for the double-ring RFR is shown in Figure 2, where the two-dimensional model of catalyst beds employs a cylindrical coordinate system.

Figure 2 Schematic diagram of two-dimensional model of double-ring RFRs

Compared with other reactors, the double-ring RFR is quite distinctive. The inner channel is the distributing fow channel for the gas phase, and the outer channel and central channel are the collecting channels. The inward flow passes from the inner channel to the inner catalyst bed, while the outward fow passes from the inner channel to the outer catalyst bed. A mathematical hydrodynamic model should be set up to describe the situation of fows in different channels and catalyst beds.

2.1 Controlling equations for the main flow channels

The fow in main channels (distributing fow channel and collecting flow channels) carries out the mass exchange with the outside space. This situation is called the variable mass fow. Hence, compared with the Bernoulli’s equation and the total energy equation, the modified momentum equation[15]is preferred. This paper used it to describe the fluid flow in main flow channels. The equation has the following form:

where the sign ‘±’ takes ‘+’ for distributing flow and‘-’ for collecting fow. 0 represents the inner channel, 1 represents the outer channel, and 2 represents the central channel.Kis a momentum recovery coefficient, which can be determined by the following correlations based on the experimental results[16]:

The coeffcientλis the friction coeffcient of gas fow in the main fow channels and it can be determined by the Reynolds number. The coeffcientλican be determined by Eq. (4), in whichReis the Reynolds number.

Eq. (1) and relevant parameters form the one-dimensional model. It is an ordinary differential equation set. When the one-dimensional model is established to describe RFRs, it can be solved by the Runge-Kutta method under the given initial conditions. The pressure drop of perforated cylinder can be determined by Eq. (5).

in whichζiis the resistance coefficient through the distributing orifces. This coeffcient can be calculated by the correlation based on the experimental results obtained by Wang[10].

2.2 Controlling equations for the particle beds

The two-dimensional model must provide the fields of pressure and velocity distribution in the catalyst beds which should follow the principles of momentum conservation and mass conservation. Therefore, the gas flow in the catalyst beds can be described by Ergun’s equation and the continuity equation:

wherePbiis the pressure of gas phase in the inner catalyst bed, andubiand vbiare the axial and radial velocities, respectively.Pbois the pressure of gas phase in the outer catalyst bed, anduboanduboare the axial and radial velocity, respectively.

The boundary condition must be defined to solve the equations mentioned above. Because of the presence of radial and axial air flows at the boundary, the dynamic boundary condition for the radial flow reactor is set as follows:

At the solid wall, the fow obeys the no-slip condition and the pressure gradient is zero:

In the perforated region of wall, the boundary condition is given by the pressure drop at the edge of the catalyst bed by Ergun’s equation:

All of the above equations constitute the complete mathematical model for describing the hydrodynamic behavior of the double-ring RFRs. If the reactor structure parameters and the fluid properties are known, the pressure and velocity field in the channels and catalyst beds can be calculated by the model.

2.3 The gas distribution uniformity

To describe the gas distribution uniformity of reactor, the maximum non-uniformity of gas fow distribution along the axial direction (η) is defned as follows[11]:

wherePiandPi'are the inlet and outlet pressures in the collecting flow channel, andPjandPj'are the inlet and outlet pressures in the distributing fow channel. A smallerηimeans better uniformity performance of reactor.

3 Model Solving Strategy

During the calculation, the model coeffcient is relevant to the structure size of reactor. Table 1 presents the structure size of the double-ring RFR at a domestic petrochemical company. The inlet fuid fow of reactor is 185 990 kg/h, while the inlet temperature is 748 K and the inlet pressure is 286 000 Pa. The outlet pressure in the central channel and in the outer channel is the same, which is equal to 266 000 Pa. Upon considering the low single-pass conversion of dehydrogenation process, which is equal to about 12%, the calculation of the double-ring RFR model is based on the entrance condition of reactor. Table 2 lists the physical parameters of reactor. The method for calculation of parameter processing can be found in the literature reports[17-19].

Table 1 The structure size of double-ring radial flow reactor

Table 2 Physical parameters in dehydrogenation reactor

The double-ring RFRs are provided with two catalyst beds and three fow channels. It is more complicated thanthe traditional single-ring RFRs. But the model of the double-ring RFRs can be solved based on the method for solving the single-ring RFRs model when the fow distribution of the double-ring RFRs in its inner channel to catalyst beds is decided. The following description is related with the method for solving the single-ring RFRs.

3.1 Method for solving the single-ring RFR model

Compared to the method for solving the one-dimensional fow in the main fow channels, it is diffcult to solve the two-dimensional gas fow feld in the catalyst bed of the single-ring RFRs. There are three variables, namely the pressure, the axial velocity, and the radial velocity, which should be decided simultaneously. It can be seen from the movement of gas fow in the reactor that the gas fow field in the main flow channels can influence the twodimensional boundary condition of catalyst bed, and the reverse is also true. Furthermore, the iteration procedure is sensible to the boundary conditions. It is necessary to develop an effcient and stable iteration method.

This paper used the staggered grid to discretize the fow field, just as shown in Figure 3. The staggered grid is applicable to both the inward fow and the outward fow. By denoting the exact velocity and pressure feld of the catalyst bed asu,v,P, the estimated feld asu*,v*,P*, and then the correcting field asu',v',P', the correcting feld can be expressed as:

By using Ergun’s equation and the continuity equation, Eq. (21) is finally obtained for pressure correction. The complete process of deduction can be found in the literature report of Mu, et al.[9]

Figure 3 Schematic diagram of staggered grid in iteration

in which the coeffcientAE,AA,AB,AC,ADand source termSmare expressed as follows:

Eq. (21) is a fve-grid difference forma. Compared to the partial differential equations, it will be easy to solve. The SOR method[13]is applied to solve the two-dimensional model of the single-ring RFRs which is different from that in the reference[9]. Figure 4 is the fow diagram of the iteration process. When the catalyst bed has a modified velocity field, the next step is to update the boundary conditions of perforated region. Based on the modified velocity field of catalyst bed, the velocity fields in the main flow channels are obtained, and then the pressure felds in channels can be obtained. And the velocity and pressure drops across the distributing orifices could be obtained based on the pressure and velocity felds in the main flow channels. Before the next SOR iteration, the boundary conditions of the perforated region are updated by the modified results of the main flow channels. The above iteration procedure is repeated until the convergent results are obtained.

3.2 Strategy for solving the double-ring RFR model

The model of single-ring RFRs can be solved by the method mentioned above. This paper has tried to break the double-ring RFR into two single-ring RFRs, and then the model of double-ring RFR can be solved as two single-ring RFRs. But there are some problems in the inner channel. How much distributing mass flow of inner channel would go to the inner catalyst bed and outer catalyst bed separately? How to update the boundary condition of two catalyst beds simultaneously?

Figure 4 Flow diagram of the iteration process.

To solve these problems, the fow assignment parametersTiof mass fow in the inner channel to the outer catalyst bed and the inner catalyst bed were introduced for updating the boundary condition of two catalyst beds. This paper discretized the fow feld in the inner channel into N equidistant points from the top to the bottom. The height of channel isZ, and the distance between the adjacent points is Δz. As shown in Figure 5, the axial mass flow into A point ismiand the axial mass fow into the adjacent B point ismi+1. The axial mass fow decreases from A to B, and the reduced axial mass flow is (mi-mi+1). A part of the decreased axial flow goes to the inner catalyst bed, which can be expressed asMi,in, and the other part goes to the outer catalyst bed, which can be expressed asMi,out. As shown in Figure 5,Mi,inplusMi,outare equal to (mi-mi+1). The ratio of decreased mass flow to the inner bed is denoted asTias shown in Eq. (23), so the rest ratio to the outer catalyst bed is (1-Ti).

If the flow assignment parametersTiof mass flow in the inner channel to the outer catalyst bed and the inner catalyst bed are determined, the double-ring RFRs can be broken into two single-ring RFRs, but it is diffcult to update the boundary condition of the inner catalyst bed and the outer catalyst bed simultaneously in the doublering RFRs model. There are two velocity and pressure fields from the inner catalyst bed and the outer catalyst bed when the velocity and pressure felds of inner channel are updated. The inner channel acquires an average of two velocity felds, and then the pressure of inner channel can be obtained. The velocity and pressure drop across the distributing orifces can be obtained from the pressure and velocity felds in the main fow channels. Before the next SOR iteration of the inner catalyst bed and the outer catalyst bed, the boundary conditions of the perforated region are updated by the modified results of the main fow channels. The above iteration procedure is repeated until the convergent results are obtained.

Figure 5 Diagram of flow distribution in the inner channel to the inner catalyst bed and the outer catalyst bed

Every point along the axial direction of the inner channel has its ownTivalue, and it is difficult to determine theTivalue of all points simultaneously. By supposing that the curve ofTis smooth and continuous, a function relationship between the height of reactor andTiis assumed although the type of function is unknown. This paper has questioned if the assumedTfunction type is reasonable or not under the condition that the central channel and the outer channel should have the same outlet pressure. Once the T function is decided, the model of the double-ring RFRs can be solved

The authors have tried many types of functions to reduce the outlet pressure difference between the central channel and the outer channel. The results show that the linear function, the arcsine function and the self-defned functionT1=A,Ti=Ti-1+B((i-1)Δz)2, have better resultswhere the outlet pressure in the central channel and the outer channel is quite close. To further reduce the outlet pressure difference,Tiis supposed to have the following form:

The optimization value ofA,B,C,DandEis equal to 0.251266, 0.000176, 0.00002, 0.0001 and 0.0004, respectively. The outlet pressures in the central channel and the outer channel are 269 049 Pa and 269 227 Pa, respectively. The actual outlet pressure of the central channel and the outer channel is the same, which is equal to 266 000 Pa. Since the flow in the reactor is slightly increased along with the proceeding of dehydrogenation reaction, this phenomenon means that a higher drop of pressure exists, and a slightly higher outlet pressure of simulation is reasonable.

Figure 6 shows the hydrodynamic behavior in different channels, including theTidistribution in the inner channel, and the velocity and pressure distribution in three channels. Figure 7 is the simulated result of twodimensional fow feld in the outer catalyst bed, including the pressure and velocity distribution. Figure 8 is the simulated result of two-dimensional flow field in the inner catalyst bed, including the pressure and velocity distribution.

The gas phase pressure in two catalyst beds decreases along the radial fow direction. The pressure difference in two sides of catalyst bed is the driving force of gas phase. Along the radial flow direction, the pressure gradient of the inner catalyst bed increases with the increase in the radial velocity, and the pressure gradient of the outer catalyst bed decreases with the decrease in the radial velocity. The Figure 7 (b) of the outer catalyst bed and Figure 8 (b) of the inner catalyst bed show that the radial velocity at the top of reactor is smaller than that at the bottom of reactor, which means that the upper catalyst bed has higher utilization coefficient. This is the characteristic feature of the Z type reactor.

Figure 6 The hydrodynamic behavior in different channels

We have calculated the maximum non-uniformity of gas flow distribution along the axial direction (η), showing that the value of outer ring (η1) is 0.0305 and the value of inner ring (η2) is 0.2746, which means that the outer ring has better uniformity than the inner ring. The weight hourly space velocity (WHSV) of the outer catalyst bed and the inner catalyst bed are 30.3196 h-1and 24.8435 h-1, respectively. WHSV can reflect the contact time between the catalyst and the gas reactants. The difference of WHSV means the difference of reaction conversion between the two catalyst beds, which will lower the total reaction conversion of the double-ring RFRs. So balancingthe WHSV value between the two catalyst beds will increase the dehydrogenation conversion rate. Meanwhile, increasing the total gas distribution uniformity along the axial direction is also an optimum design goal of the double-ring RFRs.

Figure 7 Two-dimensional distribution of pressure and velocity of outer catalyst bed

Figure 8 Two-dimensional distribution of pressure and velocity of inner catalyst bed

4 Reactor Design Optimization

The double-ring RFRs are quite complicated because theTidistribution and flow field of three channelsand two catalyst beds will all change when the size of reactor is changed. This paper tried to change the inside radius (R2) of the inner catalyst bed to balance the outer ring and the inner ring, just as shown in Figure 2, which will change the catalyst loading quantity of inner catalyst bed and the radius of central channel, without changing other structure size of the double-ring RFRs. Figure 9 shows the WHSV value of two catalyst beds with different inside radiis of the inner catalyst bed. Figure 10 shows theηvalue with different inside radius of the inner catalyst bed.

As shown in Figure 9, an optimum inside radius of the inner bed does exist. Upon increasing the inside radius of the inner catalyst bed, the WHSV value of the inner catalyst bed increases and the trend of change in the outer catalyst bed is the opposite. The WHSV value becomes balanced whenR2 is roughly equal to 0.3215 m. The balanced WHSV value is about 27.3059 h-1.

Figure 9 WHSV value at different inside radius of inner catalyst bed

Figure 10 The axial non-uniformity at different inside radius of inner catalyst bed

As shown in Figure 10, the twoηvalues of gas phase axial non-uniformity of the inner ring and the outer ring are equal, whenR2 is roughly equal to 0.317 m. Although the uniformity of the outer ring is reduced, the total axial uniformity of reactor is better as compared with the current design. Upon integrating the trend of WHSV and axial non-uniformity, the superior inside radius of the inner catalyst bed is between 0.316 m and 0.322 m.

5 Conclusions

The practical strategy for solving the two-dimensional hydrodynamic mathematical model of the double-ring RFRs is established in this paper, including the method for determining the flow assignment parameters (Ti) of mass fow in the inner channel to the outer catalyst bed and the inner catalyst bed. The simulated result shows the fow characteristics in the double-ring RFR including the three main channels and two catalyst beds. The axial non-uniformity of reactor and the WHSV value difference between two catalyst beds are discussed based on the simulated fow felds.

The established model of the double-ring RFRs can provide a quantitative reference for the design of new double-ring RFRs and modification of current reactors. The optimized size of reactor will balance the WHSV of two catalyst beds and increase the axial uniformity of current dehydrogenation reactor.

Nomenclature

z— Coordinate of axial direction, m;

λ— Friction coeffcient;

r— Coordinate of radial direction, m;

L— Height of catalyst bed, m;

α,β— Parameters of Ergun’s equation,α=315.3,β=1.08;

dp— Diameter of catalyst particle, m;

K— Momentum recovery coeffcient;

ρg— Gas density, kg/m3;

ε— Porosity of the catalyst beds;

φ— Orifce ratio of main channels;

Pbi— Pressure of inner catalyst bed, Pa;

vbi— Radial velocity of inner catalyst bed, m/s;

ubi— Axial velocity of inner catalyst bed, m/s;

Pbo— Pressure of outer catalyst bed, Pa;

vbo— Radial velocity of outer catalyst bed, m/s;

ubo— Axial velocity of outer catalyst bed, m/s;

U0— Fluid velocity of inner channel, m/s;

U1— Fluid velocity of outer channel, m/s;

U2— Fluid velocity of central channel, m/s;

P0— Fluid pressure of inner channel, Pa;

P1— Fluid pressure of outer channel, Pa;

P2— Fluid pressure of central channel, Pa;

R2—Inside radius of inner catalyst bed, m;

R22—Outside radius of inner catalyst bed, m;

R11—Inside radius of outer catalyst bed, m;

R1—Outside radius of outer catalyst bed, m.

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date: 2017-01-19; Accepted date: 2017-03-13.

Liang Yanhua, Telephone: +86-21-64252816; E-mail: 13022157509@163.com.