Plug Flow Reactor Model - PFR Modeling

PFR Modeling

The PFR is governed by ordinary differential equations, the solution for which can be calculated providing that appropriate boundary conditions are known.

The PFR model works well for many fluids: liquids, gases, and slurries. Although turbulent flow and axial diffusion cause a degree of mixing in the axial direction in real reactors, the PFR model is appropriate when these effects are sufficiently small that they can be ignored.

In the simplest case of a PFR model, several key assumptions must be made in order to simplify the problem, some of which are outlined below. Note that not all of these assumptions are necessary, however the removal of these assumptions does increase the complexity of the problem. The PFR model can be used to model multiple reactions as well as reactions involving changing temperatures, pressures and densities of the flow. Although these complications are ignored in what follows, they are often relevant to industrial processes.

Assumptions:

• plug flow
• steady state
• constant density (reasonable for some liquids but a 20% error for polymerizations; valid for gases only if there is no pressure drop, no net change in the number of moles, nor any large temperature change)
• single reaction occurring in the bulk of the fluid (homogeneously).

A material balance on the differential volume of a fluid element, or plug, on species i of axial length dx between x and x + dx gives:

= - + -

Accumulation is 0 under steady state; therefore, the above mass balance can be re-written as follows:

1. .

where:

• x is the reactor tube axial position, m
• dx the differential thickness of fluid plug
• the index i refers to the species i
• F_i(x) is the molar flow rate of species i at the position x, mol/s
• D is the tube diameter, m
• A_t is the tube transverse cross sectional area, m2
• ν is the stoichiometric coefficient, dimensionless
• r is the volumetric source/sink term (the reaction rate), mol/m3s.

The flow linear velocity, u (m/s) and the concentration of species i, C_i (mol/m3) can be introduced as:

and

On application of the above to Equation 1, the mass balance on i becomes:

2. .

When like terms are cancelled and the limit dx → 0 is applied to Equation 2 the mass balance on species i becomes

3.,

The temperature dependence of the reaction rate, r, can be estimated using the Arrhenius equation. Generally, as the temperature increases so does the rate at which the reaction occurs. Residence time, is the average amount of time a discrete quantity of reagent spends inside the tank.

Assume:

• isothermal conditions, or constant temperature (k is constant)
• single, irreversible reaction (ν_A = -1)
• first-order reaction (r = k C_A)

After integration of Equation 3 using the above assumptions, solving for C_A(x) we get an explicit equation for the concentration of species A as a function of position:

4. ,

where C_A0 is the concentration of species A at the inlet to the reactor, appearing from the integration boundary condition.