Plug flow reactor model

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The plug flow reactor (PFR) model is used to describe chemical reactions in continuous, flowing systems. One application of the PFR model is the estimation of key reactor variables, such as the dimensions of the reactor. (See Chemical reactors.). PFRs are also sometimes called as Continuous Tubular Reactors (CTRs).

Schematic diagram of a Plug Flow Reactor (PFR)

Fluid going through a PFR may be modelled as flowing through the reactor as an infinitely thin coherent "plug", where the plug is of a uniform composition travelling in the axial direction of the reactor, but with differing composition to the leading and trailing plugs. The required assumption is that as a plug flows through a PFR, the fluid is perfectly mixed in the radial direction but not in the axial direction (forwards or backwards). Each plug of differential volume is considered as a separate entity (effectively a vanishingly small, limiting to zero volume, batch reactor) as it flows down the tubular PFR, in doing so the residence time of the plug changes as does the position of the plug. The effect of this is that the residence time, $τ$ , is the same for all fluid elements as they pass a fixed axial distance in the reactor.

1 PFR Modelling
2 Operation and uses
3 Advantages and Disadvantages
4 Applications
5 See also
6 Reference & Sources

[edit] PFR Modelling

The PFR model can be derived using 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. Sometimes turbulent flow or axial diffusion is sufficient to promote mixing in the axial direction, which undermines the required assumption of zero axial mixing. However if these effects are sufficiently small and can be subsequently ignored, the PFR is a powerful modelling technique.

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, however the addition of these complications does little to aid understanding of the underlying concepts involved, as such they are not considered here.

Firstly the assumptions that are made are listed:

Assume:

plug flow
steady state
constant density (valid for most liquids; valid for gases only if there is no net change in the number of moles or drastic temperature change)
constant tube diameter
single reaction

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] = [in] - [out] + [generation] - [consumption] 1. $F i (x) - F i (x + d x) + A t d z ν i r = 0$ . ^[1]

When linear velocity, u, and molar flow rate relationships, F_i, $u = \frac{\dot{v}}{A_t} = \frac{4 \dot{v}}{\pi D^2}$ and $F_i = A_t u C_i \,$ , are applied to Equation 1 the mass balance on i becomes

2. $A_t u [C_i(z) - C_i(z + dz)] + A_t dx \nu_i r = 0 \,$ . ^[1]

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

3. $u \frac{dC_i}{dx} = \nu_i r$ , ^[1]

where C_i(x) is the molar concentration of species i at position x, A_t the cross-sectional area of the tubular reactor, dx the differential thickness of fluid plug, and $ν i$ stoichiometric coefficient. The reaction rate, r, can be figured by using the Arrhenius temperature dependence. Generally, as the temperature increases so does the rate at which the reaction occurs. Residence time, $τ$ , 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 = kC_A)

After integration of Equation 3 using the above assumptions, solving for C_A(L) we get an explicit equation for the output concentration of species A,

4. $C_A(L) = C_{Ao} e^{-k \tau} \,$ ,

where C_Ao is the inlet concentration of species A.

[edit] Operation and uses

PFRs are used to model the chemical transformation of compounds as they are transported in systems resembling "pipes". The "pipe" can represent a variety of engineered or natural conduits through which liquids or gases flow. (ie a river, pipelines, regions between two mountains, etc.)

An ideal plug flow reactor has a fixed residence time: Any fluid (plug) that enters the reactor at time $t$ will exit the reactor at time $t + τ$ , where $τ$ is the residence time of the reactor. The residence time distribution function is therefore a dirac delta function at $τ$ . A real plug flow reactor has a residence time distribution that is a narrow pulse around the mean residence time distribution.

A typical plug flow reactor could be a tube packed with some solid material (frequently a catalyst). Sometimes the tube will be a tube in a shell and tube heat exchanger.

[edit] Advantages and Disadvantages

Plug flow reactors have a high volumetric unit conversion, run for long periods of time without labor, and can have excellent heat transfer due to the ability to customise the diameter to the desired value by using parallel reactors. Disadvantages of plug flow reactors are that temperatures are hard to control and can result in undesirable temperature gradients. Maintenance is expensive for the reactors as well

[1].