14. Radial and Axial Temperature Variations in a Tubular Reactor^*

Example 14-2 Radial Effects in Packed Bed Reactor

(Example Problem by Professor Robert Hesketh, Department of Chemical Engineering, Rowan University, Glassboro, New Jersey.)

One of the common methods to produce Phthalic Anhydride is from the partial oxidation of o-xylene. This simplified reaction for the formation of phthalic anhydride from o-xylene oxidation will be represented by

In terms of symbols

Phthalic Anhydride is primarily used in plasticizers and in resins used to make boat hulls, hot tubs and synthetic marble surfaces. In the partial oxidation of o-xylene reaction there are several byproducts as well as products of combustion that are formed if the reactor is not optimized. It is desirable to avoid high temperatures in this reactor based on temperature limits on the materials (reactor walls and catalyst) as well as reducing the formation of byproducts. As you would expect high temperatures will result in the combustion of both the reactant and product resulting in the formation of CO2 and H2O. Because of the high operating temperatures, this reactor is cooled using molten salt (sodium nitrite-potassium nitrate). For this COMSOL ECRE problem we will only use approximate reaction kinetics for the overall reaction as well as make an assumption of constant overall flowrate. This assumption must be employed based on the form of the equation given in the COMSOL ECRE package which uses a constant velocity through the reactor. For safety considerations the o-xylene is diluted without to a mole fraction of 0.012 and fed to the reactor at 610K.

It is desired to make 76 metric tons of phthalic anhydride per year in a packed bed reactor 1 m in length with a 0.1 m radius.

The reaction follows the following rate law based on partial pressures

Additional Information

Construct a model of this reactor and investigate the radial and axial temperature profiles.

Run your model using values of thermal conductivity from 0.1 to 1000 J/(m K s).
Compare this model to the plug flow model for mass and heat transfer. Discuss the overall effect of the value of effective thermal conductivity of the bed on the resulting temperature profile and outlet conversion obtained.
Give a rationale as to why industry does not use tubes of 0.1m in diameter, but instead uses a smaller tube diameter of 0.0254 m.

Preliminary Calculations

To produce 76 ton/y during 350 d/y operation

For 79% conversion

Solution

In Comsol we will use a 2D axis-symmetric model in the r and z directions. This reaction takes place in a packed bed reactor where we will assume that the flow is plug flow with no axial or radial diffusion.

Assumptions

U_r= 0 Assume negligible velocity in the radial direction
U_z= constant
Neglect axial diffusion/dispersion flux with respect to convective flux
Assume constant density and mean heat capacity of the gas can be approximated by the values of air
Steady-state

Mole Balance

Starting with Equation (14-24)

Neglecting diffusion in the axial and radial directions gives the plug flow equation:

Note that the velocity U_z is constant

Rate Law

To convert the rate to per reactor volume we multiply by the bulk density of the catalyst bed, ρ_B

For the Comsol model the partial pressures need to be converted to concentration using the ideal gas law C_i= P_i/RT

with

Stoichiometry

Because of the dilute reactants, ε = 0 we also will neglect changes in volumetric flow rate with temperature and pressure

Energy Balance

Starting with the Energy Balance from equation 14-37:

The conductivity used in the energy balance will be an effective conductivity of the bed. The conduction in the axial direction is neglected resulting in the following equation:

The boundary conditions for these models will be:

Findings

Results

A surface plot of the temperatures in the reactor tube is shown in Figure E14-2.1 which shows a noticeable difference in temperatures of the fluid at the end of the reactor starting at about z = 0.7 m. For this value of thermal conductivity, the wall remains �cold� at approximately the temperature of the molten salt of 720K. This can also be seen in the cross-section plot that is shown in Figure E14-2.2. Each line on this plot represents a radial position within the reactor. In Figure E14-2.3 a cross section plot showing the temperature profile as a function of radial position and axial distance in the reactor. Each line in this figure represents an axial position in the reactor. In this plot the difference between the wall temperature and the temperature within the reactor is clearly shown. The plug flow condition, in which the temperature profile has been eliminated in the radial direction, is shown in Figure E14-2.4. This result was obtained with a thermal conductivity of 1000 J/(s m K).

Analysis: From Table 14-2.2 we observe that the removal of heat is effective in lowering the temperature within the reactor. The advantage of the molten salt cooling is that it will help to prevent a runaway reaction in the reactor tubes. As the radial thermal conductivity is increased the overall rate of heat transfer from the fluid to the reactor walls increases. As you can see from the table the conversion also decreases.

The full reaction scheme is as follows

Equations in Comsol Format:

The following is a detailed screen shot if using COMSOL Multiphysics for running this program.

The following constants and equations will need to be entered. Starting from the Model Navigator open the Non-isothermal Reactor 1 model in the folder 1-Radial Effects in Tubular Flow reactors.

Next to edit the constants and equations go to Options, Constants and enter/edit the following

Open Options, Expressions, Scalar Expressions

Now make the following edits to the energy balance by selecting Multiphysics Convection and conduction (EnergyBalance) and then from the main menu Physics, subdomain. Select Subdomain 1 and make the thermal conductivity anisotropic as shown below.

Selecting the initial tab should have T0 as an initial temperature for the solver.

Examine the boundary conditions by selecting Physics, Boundary Settings. Only one change is needed in Boundary 4. Change the variable name from Uk to Uht and Ta0 to Ta (using the notation given in the text)

The above menu shows the altered boundary condition at the wall of the reactor. Change the variable name from Uk to Uht and Ta0 to Ta (using the notation given in the text)

Notice that you can select a boundary with the mouse and the submenu will display the selected boundary condition. Select apply and then OK to move on to the next screen.

Now examine the mole balance. Select from the main menu Multiphysics, Convection and Diffusion (MassBalance). Then select from the main menu Physics Subdomain Settings. Check to see that cA is the variable for concentration and the reaction rate is given by rA and the z-velocity is uz. No edits are necessay from this menu.

Now from the main menu select Physics, Boundary Settings. Examine each of the 4 boundary conditions. Again no changes are needed from this model.

Now you are ready to solve the model. To solve the model select from the main menu Solve and then Restart since you have changed parameters and equations.

After the model has finished it will display the concentration surface plot. To examine the temperature surface plot select from the main menu Postprocessing, Plot Parameters. From this submenu choose the Surface tab and select Temperature (EnergyBalance) from the drop down menu next to Predefined quantities. Then click apply to see the plot.

To obtain the integral average temperature of the outlet stream: