rxn10_femlab-fixed bed reactor

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Fixed-Bed Reactor for Catalytic Hydrocarbon

Oxidation

In this example, a process of industrial importance is discussed, that is, partial oxidation

of o-xylene in air to phthalic anhydrid (PA) in a multitube fixed-bed reactor. The totalproduction of PA is currently about 7 million lb/year, and almost all of the PA ismanufactured by the multitube fixed-bed process [1],[2].

In this process, the temperature is usually kept between 400-475oC, while the residence

time varies between 0.5-5 seconds. The catalyst of choice is usually a mix of vanadiumoxide and potassium sulfate on a silica support. The most important factor to consider for

this process is the temperature throughout the reactor. The reactions taking place in thereactor are highly exothermic, and in order to eliminate runaway conditions, the reactor

needs to be cooled. The temperature distribution will, in turn, affect the yield of thephthalic anhydride, which is to be maximized. We can control the temperature

distribution by varying tube diameter, residence time, wall temperature (cooling rate), aswell as the inlet temperature of the feed. In this example we will cover some of these

factors by setting up a detailed model of the system using the so-called two-dimensionalpseudo homogeneous model as described in the literature [2], [3], [4].

Most of the time, tubular reactors are modeled with the assumption that concentration and

temperature gradients only occur in the axial direction. The only transport mechanismoperating in this direction is the overall flow itself, which is considered to be of plug-flow

type, that is, all the fluid elements are assumed to move with a uniform velocity alongparallel streamlines. In this example, we will take a more general approach, and we will

account for variations of the concentrations and the temperature in the axial direction. Wewill also account for mixing in the axial direction, which is due to turbulence and the

presence of packing. The axial mixing is described by means of effective diffusivities andconductivities.

Model Definition

We will investigate this system by first setting up a model in 2D, where we assume thatwe have rotational symmetry. We will then introduce a few approximations, and show

that this problem can be described, to a satisfactory extent, by a time-dependent 1Dmodel. The obvious advantage with the latter approach is the much lower memory

requirement. However, it has a few limitations, one of which is that it is restricted to

simulate steady-state behavior. Furthermore, in cases where you cannot neglect axialmixing, a full 2D model is required to obtain accurate results.

The reaction kinetics of this process is of a rather complex nature but can, to asatisfactory extent, be explained by the following scheme [2], [3], [4]:
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Figure 1: Reaction paths.

A represents o-xylene, B phthalic anhydride, and C is the total amount of carbonmonoxide and carbon dioxide. Due to a very high excess of oxygen, the reactions can be

considered to be pseudo-first-order, and we can then describe the reactions kinetics asfollows:

Where y0 represents the mole fraction of oxygen, and yA0

is the inlet mole fraction of o-

xylene. Furthermore,xA is the total conversion of o-xylene,xB is the conversion of o-

xylene into phthalic anhydride, and xC represents the total conversion into carbonmonoxide and carbon dioxide.

The rate coefficients depend on temperature as described by the Arrhenius law accordingto the following expressions:

Where T0 is the inlet temperature of the reactor, and T= T- T0.

AXISYMMETRIC 2D MODEL

A schematic description of the reactor is given in Figure 2. The convective flow of gastakes place from the bottom to top and axisymmetry is assumed, which reduces the 3Dgeometry to two dimensions.


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Figure 2: Schematic representation of the reactor.

The design equations for this system can be described by [3], [4]:

Where us is the superficial velocity, g is the gas density, b is the catalyst bulk density,

ctot the total concentration, yA0

the inlet mole fraction of o-xylene, and eff the effective

thermal conductivity of the bed. We will now make use of the fact that xA =xB +xC,which means that we only need to solve for two mass balances, resulting in the following

design equations:

The rate equations can be rewritten accordingly:
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The boundary conditions for this system are as follows; see Figure 2 above for reference:

At the outlet of the reactor, we will assume that the convective part of the mass and heattransport vector is dominating.

Due to the large aspect ratio of our model geometry, we are going to scale our equations.

In this case, the reactor is about 200 times its radial dimension, more specifically, 3 meterlong and 0.0127 meter in radius. By scaling the equations, we avoid excessive number of

elements and node points when setting up the mesh. The new scaled r and z-coordinates,and the new differentials for the mass balances, can be written as

In the mass balances, cis differentiated twice in the diffusion term, which implies that

the z-component of diffusion in the mass balance has to be multiplied by (1 /scalez)2

and by (1 /scaler)2 for the r-component. The convective part is only differentiated once,

and has to be multiplied by (1 /scalez). The scaling of the diffusive part of the flux canbe introduced as an anisotropic diffusion coefficient. This gives the diffusion coefficient

according to the matrix below:

Similarly for the heat balance, we get thermal conductivity according to the following

matrix:


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Results

The default plot shows the conversion of o-xylene to phthalic anhydride.

Figure 3: Temperature distribution across the tubular half plane

A plot of the temperature, see Figure 3 above, reveals that the temperature goes through a

maximum not far from the reactor inlet. This so-called hotspot is a quite common

phenomenon for a system with exothermic reactions to which cooling is applied. Also,note that the radial temperature gradients are quite significant around this hotspot.

Figure 4 shows the composition and temperature distribution in the reactor in the axialdirection. The figure shows the bulk mean conversions and the temperature profile for an

inlet temperature of 354o

C. We can see that the phthalic anhydride conversion falls offsomewhat along the tube (middle line), which is typical for consecutive reactions.

Furthermore, it can be seen that the temperature goes through a maximum, a so-called

hotspot, at Tm equal to about 30oC, not far from the inlet of the reactor.


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Figure 4: Composition and temperature versus axial coordinate in the reactor.

Figure 5: Temperature versus radial coordinate in the reactor.

In Figure 5, we can see that the radial temperature gradients are quite severe, as the

temperature along the symmetry axis is well above the mean temperature. Based on thisinformation, we can draw the conclusion that a one-dimensional model with axial mixing

would not be good enough to describe this system

Figure 6: Average dimensionless temperature vs. axial coordinate for different inlettemperatures


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The parameter study of the inlet temperature gives Figure 6 for the average temperaturein the axial direction. From this figure, we can see that the inlet temperature of the tubular

reactor does affect the axial temperature quite dramatically. A high temperature increasesthe production rate of phthalic anhydride, but it may also increase the production of

carbon monoxide and carbon dioxide. Furthermore, too high a temperature may be

detrimental to the catalyst, which means that it is very important to have good control ofthe feed temperature of the reactor.

It should also be mentioned in this context that the results in this model are in excellentagreement with the model by Froment [1]. Froments model was done in 1967 and 2D

simulations were not possible for a complex model like this. This also implied that thesemodels were limited to steady state simulations. The model presented in the first section

can be easily rewritten to a time-dependent form for use in automatic control and start-upsimulations.

References

[1] G. F. Froment, Fixed Bed Catalytic Reactors, Ind. Eng. Chem., 59(2), 18,

1967[2] SRI International Consulting, http://pep.sric.sri.com/, 2001

[3] G. F. Froment and K. B. Bischoff, Chemical Reactor Analysis andDesign, John Wiley & Sons, 1990

[4] C. N. Saterfield, Heterogeneous Catalysis in Industrial Practice, McGraw-Hill, 1991


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Model Library Chemical_Engineering_Module/Reaction_Engineering/

fixed_bed_reactor_exo

Modeling Using the Graphical User Interface

1. Start FEMLAB2. In the Model Navigator, click the Multiphysics button, and set the Space

dimension list to Axial Symmetry (2D).3. Highlight the application mode Chemical Engineering Module/Mass

balance/Convection and Diffusion. Enter Dependent variables: xb xc (spaceseparated), and Application mode name: massbal.

4. Click the Add button.

5. Select the application mode Chemical Engineering Module/Energybalance/Convection and Conduction.Name the application mode energybalandleave the dependent variables to the default T. ClickAdd.

6. Highlight the Convection and Diffusion application mode in the Multiphysicslist on the right hand of the pane.


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7. Click the Application Mode Properties button. Switch the Equation form toConservative. ClickOK

8. Repeat the procedure to set the conservative equation form for the Convectionand Conduction application mode.

9. ClickOK in the Model Navigator.OPTIONS AND SETTINGS

1. Define the following constants in the Options/Constants dialog box:NAME EXPRESSION

Deff 3.19e-7

us 1.064e-3

rhob 1300

rhog 1293

lambda 0.78e-3

cp 0.992

ctot 44.85

alpha 0.156

T0 627

deltaH1 -1.285e6

deltaH3 -4.564e6

ya0 0.00924

y0 0.208

B1 13588

B2 15803

B3 14394


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2. Define the following expressions in the Options/Expressions/ScalarExpressions dialog box:

NAME EXPRESSION

scaler 0.0127/1

scalez 3/5

A1 exp(19.837)/3600

A2 exp(20.86)/3600

A3 exp(18.97)/3600

k1 A1*exp(-B1/(T+T0))

k2 A2*exp(-B2/(T+T0))

k3 A3*exp(-B3/(T+T0))

rb ya0*y0*(k1*(1-xb-xc)-k2*xb)

rc ya0*y0*(k2*xb+k3*(1-xb-xc))

GEOMETRY MODELING

1. Click the Rectangle/Square button on the Draw toolbar and draw a rectangle ofarbitrary dimension. Double-click on the rectangle and type the values listed

below in the corresponding edit fields.

EDIT FIELD VALUE

Width 1

Height 5

r: 0

z: 0

2. Click the Zoom Extents button on the Main toolbar.PHYSICS SETTINGS

Boundary Conditions

1. Select 1 Convection and Diffusion (massbal) from the Multiphysics menu.


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2. Open the Physics/Boundary Settings dialog box.3. Enter the boundary conditions according to the following table:

BOUNDARY 1,4 2 3

Type Insulation/Symmetry Concentration Convective flux

xb0, xc0 0

NOTE: Make sure that you specify the boundary conditions both on the xb and xc tabs.

Subdomain Settings

1. Open the Physics/Subdomain Settings dialog box, and select subdomain 1.2. Start with the xb tab. Click the D anisotropic radio button. Place the cursor in the

edit field next to this button and the diffusivity matrix will appear. TypeDeff/scaler^2 in the r-diffusivity field (upper left) and Deff/scalez^2 in the z-diffusivity field (lower right).


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1. Repeat this procedure on the xc tab. The table below summarizes the completesubdomain settings:

SPECIES XB XC

D (r-direction) Deff/scaler^2 Deff/scaler^2

D (z-direction) Deff/scalez^2 Deff/scalez^2

R rb*rhob/ctot/ya0 rc*rhob/ctot/ya0

u 0 0

v us/scalez us/scalez

2. Click the Artificial Stabilization button, check the middle check box(Streamline diffusion ) with default parameters. ClickOK.

3. ClickOK in the Subdomain Settings dialog box.


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Boundary Conditions

1. Select 2 Convection and conduction (energybal) from the Multiphysics menu.2. Enter the boundary conditions according to the following table:

BOUNDARY 1 2 3 4

Type Thermal insulation Temperature Convective flux Heat Flux

q0 -alpha*T/scaler

T0 0

Subdomain Settings

3. Open the Physics/Subdomain Settings dialog box, and select subdomain 1.4. ClickPhysics tab. Click the k anisotropic radio button. Place the cursor in the

edit field next to this button and the thermal conductivity matrix will appear. Typelambda/scaler^2 in the r-diffusivity field (upper left) and lambda/scalez^2 in the

z-diffusivity field (lower right).5. The table below summarizes the complete subdomain settings:


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SUBDOMAIN 1

k (r-direction) lambda/scaler^2

k (z-direction) lambda/scalez^2

rhog

Cp cp

Q rhob*((-deltaH1)*rb+(-deltaH3)*rc)

u 0

v us/scalez

MESH GENERATION

1. Initialize the mesh.2. Refine the mesh once.

COMPUTING THE SOLUTION

Solve the problem by clicking the Solve button.

POST PROCESSING

1. Click the Plot Parameters button and go to the Surface tab.2. Select Temperature from the Expression list.

PARAMETRIC STUDY

1. Click the Solver Parameters button and select Solver: Parametric nonlinear.2. Go to Parameter area, and enter the following:

EDIT FIELD VALUE

Name of parameter T0

List of parameter values 625 626 627 628 629

3. ClickOK.4. Solve the parameterized problem by clicking the Solve Problem button.


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rxn10_femlab-fixed bed reactor

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