Introduction

Thermal decomposition is a chemical reaction where heat is a reactant. Since heat is a reactant, these reactions are endothermic meaning that the reaction requires thermal energy to break the chemical bonds in the molecule. A classic example is the calcination process:

Calcium carbonate, , will decompose into carbon dioxide, , and calcium oxide, , when heated above at a pressure of atmosphere. For endothermic reactions in this case the heat of reaction, , is negative. This reaction is known as the common way to make quicklime, , that is an industrially important product.

This study is to model the decomposition process of calcium carbonate, and simulate its concentration field within an expanding reactor. The calcium carbonate is transported into the reactor by suspending it in water that enters the domain from the bottom, flows across a heated tube, and exits the reactor through the top. During this process the calcium carbonate particles decompose and absorb heat from the system.

To measure the effectiveness of the reactor, we will then calculate the net reduction in the concentration, denoted by , between the flow inlet and outlet : .

The mean concentration at a boundary is computed from the integral:

using NIntegrate.

The symbols and corresponding units used for this model are summarized in the Nomenclature section.

Please refer to the information provided in Heat Transfer Tutorial and Mass Transport Tutorial for a more general theoretical background for this multiphysics model.

Multiphysics Model Setup

Since this problem considers more than one physics domain, a multiphysics model is to be constructed. As a first step we look at the contributing physics domains independently. A fluid dynamics model, a heat transfer model and a mass transport model are set up to solve for the flow field , temperature field and concentration field within the reactor.

Fluid Dynamics Model

The Navier-Stokes equation (1) is used to solve for the steady state flow field within the reactor. Note that we use a 2D domain to simulate the 3D reactor, the reason for this model order reduction will be explain in the later section about the modeling domain setup.

where

is the density ,
is the viscosity ,
is the velocity field ,
is the pressure .

Heat Transfer Model

For a steady state heat transfer model the temperature distribution is described by the steady state heat equation (2). The heat convected by the fluid flow is modeled with the convective term: .

Since the thermal decomposition is an endothermic process, the heat absorbed by the reaction is modeled by the heat sink term , which depends on both the chemical reaction rate, , and the heat of reaction, :

where

is the constant pressure heat capacity ,
is the temperature ,
is the thermal conductivity ,
is the heat sink ,
is the reaction rate ,
is the heat of reaction .

Mass Transport Model

The steady state mass transport equation (3) is used to solve for the concentration field within the reactor. The mass transported by the flow field is modeled by the convective term .

The chemical reaction rate depends on both the reactant concentration and the reaction rate constant, .

Note that for decomposition reactions the reaction rate, , is negative.

The rate constant is temperature dependent according to the Arrhenius equation [4]:

Note that the standard notation of the reaction constant is denoted by , however, to avoid the conflict with thermal conductivity in this case, we use the non-standard symbol instead.

Here

is the concentration of ,
is the species diffusivity ,
is the frequency factor ,
is the activation energy ,
is the gas constant .

Even though the gas constant is involved in (5), the Arrhenius equation is not limited to gas-phase reactions.

Physics Domains Coupling

Once we set up all the contributing physics domains, we need to analyze the coupling involved in order to simulate the interaction between the different physical phenomena.

As described in the previous section, the temperature field within the reactor is computed with the heat equation (6) and depends on both the fluid flow velocity computed with the Navier-Stokes equation (7) and the concentration through reaction rate of the mass transport equation (8). Furthermore, the reaction rate itself also depends on the temperature from the heat equation (9), resulting in nonlinear, two-way relation between each physical mode.

To solve a two-way coupled nonlinear PDE system like this, a possible strategy is to first solve a simpler model with a weaker nonlinearity and use the solution of the simpler model as an initial guess for the full nonlinear model. Ignoring the heat generated by the reaction gives a good simpler model to start with. This approach is detailed in the following sections.

One-way Coupled Multiphysics Model

By ignoring the heat of reaction (i.e. , ) from the decomposition, the heat equation (10) is simplified to: . This leads to an one-way coupled multiphysics model. For clarity we have highlighted the dependent variable for each physical mode:

The dependency of the one-way coupled model can be illustrated as:

This one-way coupled multiphysics model can either be solved sequentially or in one go. Solving it sequentially would mean that each of the physics domains is solved with a separate call to NDSolve. This approach of splitting the coupled equations in chunks and solve them independently and sequentially is called a segregated solution process. In case the single physics fields can not be split easily an initial guess for the coupling variable of the first PDE is made. Then the solution of the first PDE is take to drive the next PDE and so forth. Once all PDEs have been solved, the process is started over again. This will improve the initial guess for the initial coupling variable. This procedure is executed until the solution converges.

Fully Coupled Multiphysics Model

Once we solved the one-way coupled model, we can use the solution as an initial guess for the fully nonlinear model. Here we build the fully coupled model by adding the heat of reaction .

The dependency of the fully coupled model can be illustrated as:

Note that the temperature field and the concentration field would affect with each other, and should therefore be solved as a system of PDEs in NDSolve. This solving process will be shown in the later section: "Solve the PDE Model: Fully Coupled Model Solution".

Material Parameters

To solve any of the proposed multiphysics models, we need to specify material properties of both the fluid medium and the calcium carbonate .

Note that the heat of reaction , which means the reaction is endothermic.

Domain

The expanding reactor model is bounded by two symmetric plates as shown below:

Since the depth ( direction) of the reactor is considerably more than its width ( direction) and its height ( direction) it is reasonable to neglect the variation of in the direction, so a two dimensional model is sufficient to represent the 3D reactor.

As a further step to simplify the geometry note the symmetry about the vertical () axis. It is effective to only make use of the right half of the reactor as the simulation domain . Here and denote the flow inlet and flow outlet. The wall boundary and the symmetric axis are symbolized as and , respectively.

For the sake of simplicity, we will use to denote the 2D domain in the following sections.

In order to get a good result, a finer than the default grid is used for the mesh generation. Here the maximum grid size is set to , which means that there will be about a hundred elements in the direction (height).

Flow Regime

When a simulation involves fluid flow, it is advisable to calculate the Reynolds number to determine the type of the flow. Empirically, the flow is considered as laminar when and as turbulent when . For a Reynolds number between roughly an unsteady separation of flow may form around blunt bodies [11], resulting in an oscillating vortex shedding in the flow field. Whenever this happens, the steady state fluid dynamics model will failed to converge and a steady state solution is not achievable.

The Reynolds number for a flow is calculated as:

In this model the incoming flow has an average velocity of , and the characteristic length is taken as the diameter of the heating wire: .

The parameters for the reactor are chosen so that the Reynolds number falls into the laminar flow regime, which means the chemical reactor in this case is a laminar flow reactor (LFR). One feature of the LFR is the higher residence time [12] (the time interval during which the chemicals stay in the reactor), and this is desirable for the calcination to achieve higher production rate.

Boundary Conditions

Since this multiphysics simulation contains a fluid dynamics model, a heat transfer model and a mass transport model, the boundary conditions for each physics mode are set up respectively.

Note that the boundary conditions used in both solution approaches (one-way coupling versus full coupling) are the same.

Fluid Dynamics Boundary Conditions

There are four types of the boundary conditions involved in the fluid dynamics model. At the inlet the flow velocity is set to .

On the walls of the reactor the flow velocity is set to zero to model a no-slip flow condition.

Due to the symmetry along the axis, the velocity in the direction is set to zero at the symmetry boundaries .

At the top boundary a pressure outlet boundary condition is used to model the outgoing flow. Here the outlet pressure is set equal to the ambient pressure .

Heat Transfer Boundary Conditions

There are four types of the boundary conditions involved in the heat transfer model. At the flow inlet and the tube surface , the temperatures are set to and , respectively.

The wall boundaries are assumed to be perfectly thermally insulated and are modeled by a thermally insulated boundary condition. An outflow boundary condition and a symmetric boundary condition are applied on the flow outlet and the symmetric axis , respectively.

However, since the outflow boundary condition, symmetric boundary condition and thermally insulated boundary condition are all Neumann zero conditions, they are implicitly applied without further setup.

Mass Transport Boundary Conditions

There are four types of the boundary conditions involved in the mass transfer model. Assuming infinite supply of the species, the concentration at the flow inlet is then held at .

On the walls of the reactor the mass flux of the species is assumed to be zero, and is modeled by an impermeable boundary condition. An outflow boundary condition and a symmetric boundary condition are applied on the flow outlet and the symmetric axis , respectively.

However, since the impermeable boundary condition, outflow boundary condition and symmetric boundary condition are all Neumann zero conditions, they are implicitly applied without further setup.

Solve the PDE Model

One-way Coupled Model Solution

In the following section, the one-way coupled multiphysics model will be solved sequentially for each physics mode.

First, we build the fluid dynamics PDE to solve for the flow velocity field .

A stable solution can be found if the velocities are interpolated with a higher order than the pressure. NDSolve allows an interpolation order for each dependent variable to be specified. Here the velocities and are set to be interpolated with second order and the pressure with first order.

To inspect the fluid field within the reactor, the following visualization combines a vector plot for velocity streamlines where and a contour plot for the velocity magnitude with .

It is seen that the flow velocity decreases along the expansion region and increases around the tube.

With the flow field in hand we can proceed to the heat transfer model and solve for the steady state temperature field .

The flow enters the reactor at a temperature of and is heated as it passes by the heating tube . Note that since we ignore reaction heat generated in this case, the temperature variance is caused by the tube heating only.

Next we will solve for the concentration field of the species.

As described in the previous section, particles within the reactor are decomposing at a reaction rate .

The concentration of is held at at the inlet, and then deceases as it flows over the reactor due to decomposition. Note that the concentration drops significantly in the lower part of the reactor because of the higher reaction rate. We can verify this by visualizing the reaction rate field: .

Note that a significant decomposition occurs in the lower part of the reactor around the flow inlet. This explains why the previous plot showed a significant drop of concentration in this region.

Fully Coupled Model Solution

Making use of the one-way coupling model, we've solved for the flow velocity , temperature and the concentration in a sequential manner. Now we are ready to solve for a fully coupled model and investigate the difference.

We will add the heat of reaction that we ignored previously. This leads to a non-linear, fully coupled relation between the heat equation and the mass transport equation.

In the previous section we've shown the heat source term depends the reaction rate by:

To efficiently solve the coupled PDE model, the result from the previous step is used as an initial seed for the new solution. In fact not doing so will result in a model where the nonlinear solver does not converge. It is a general strategy to first solve a less nonlinear model and use the solution there of as an initial seed for more nonlinear model.

As we added the source term to model the heat absorption from the decomposition, the minimum fluid temperature is now below the inlet temperature .

Note that the concentration of is now higher in the upper part of the reactor. This is because the decomposition reaction has been slowed down by the lower temperature field.

To measure the effectivity of the reactor, the net of reduction is calculated between the flow inlet and outlet . For this the mean concentration at the boundary is computed with a boundary integration:

Due to the decomposition within the reactor, the concentration has been reduced by .

Nomenclature

References

1.  Tansley, Claire E. and Marshall, David P. Flow past a Cylinder on a Plane, with Application to Gulf Stream Separation and the Antarctic Circumpolar Current, Journal of Physical Oceanography. 31 (11): 3274–3283. (2001).

2.  Calvo, E. G., Arranz, M.A. and Leton, P. Effects of Impurities in the Kinetics of Calcite Decomposition, Thermochimica Acta. 170: 7–11 (1990).

3.  Patil, K., Jain, S., Gandi, R. K. and Shankar, H.S. Calcium Carbonate Decomposition under External Pressure Pulsations, AIChE Annual Meeting. 3943–3962. (2004).

4.  Fogler, H.S. Elements of Chemical Reaction Engineering, 4th Edition, Prentice-Hall Inc., New Jersey. (2006).

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