In this chapter, we will discuss the governing equations pertaining to plug-flow conditions in an arbitrarily shaped channel, with consideration of gas and surface kinetics. The governing equations and assumptions discussed here pertain to the following Ansys Chemkin reactor models:

Tubular flow reactors have long been used throughout the chemical process industries. The tube flow configuration is a natural choice for processes that are carried out in a continuous fashion. For this reason, such reactors are usually operated at steady state. Traditional applications have included both homogeneous reactions (carried out in an empty tube) and fluid-solid heterogeneous reactions in packed beds. More recently, tubular reactors have been used extensively to deposit thin solid films via chemical vapor deposition (CVD). While this is technically a batch process with regard to the solid deposit, the reactor still operates essentially at steady state for extended periods of time. The PFR models describe the steady-state, tube flow reactor that can be used for process design, optimization, and control.

Because the general equations for chemically reacting flow involve transport phenomena in addition to kinetics and thermodynamics, rigorous reactor models are by necessity multidimensional. However, there are often practical as well as mathematical reasons for considering idealized models of reduced dimensionality. In the case of tube flow, the accepted ideal is the plug-flow reactor, in which it is assumed that there is no mixing in the axial (flow) direction but perfect mixing in the direction(s) transverse to this. It can be shown [83] that the absence of axial mixing allows the achievable reactant conversion to be maximized. Likewise, the lack of transverse gradients implies that mass-transfer limitations are absent, once again enhancing the reactor performance. Along with these practical advantages, the plug flow reactor is computationally efficient since it is modeled using first-order ordinary differential equations ( ODE’s), and no transport properties are needed.

The equations governing the behavior of a plug-flow reactor are simplified versions of the general relations for conservation of mass, energy, and momentum. [84] They can be derived most easily by writing balances over a differential slice in the flow direction , with the stipulations that (a) there are no variations in the transverse direction, and (b) axial diffusion of any quantity is negligible relative to the corresponding convective term. In this way the overall mass balance (continuity equation) for the gas is found to be

(11–1)

Here is the ( mass) density and the axial velocity of the gas, which consists of species; is the molecular weight of species , and is the molar production rate of this species by all surface reactions. The quantities and are the cross-sectional (flow) area and the effective internal surface area per unit length of material , respectively, in the reactor. Both and can change as arbitrary functions of . Equation 11–1 simply states that the mass flow rate of the gas can change as a result of generation or consumption by surface reactions on all materials in the reactor. A similar equation can be written for each species individually:

(11–2)

Here is the mass fraction of species and is its molar rate of production by homogeneous gas reactions. Such reactions cannot change the total mass of the gas, but they can alter its composition.

Turning now to the energy equation, one finds.

(11–3)

where is the specific enthalpy of species , is the mean heat capacity per unit mass of the gas, is the (absolute) gas temperature. In the right-hand summation, is the molar production rate of bulk solid species by surface reactions on material . The distinction between bulk and surface species is discussed in Chemistry—Species and Phases and Surface Chemical Rate Expressions of this manual. Equation 11–3 states that the total energy (enthalpy plus kinetic) of the flowing gas changes due to the heat flux from the surroundings to the outer tube wall (whose surface area per unit length is ) and also due to accumulation of enthalpy in the bulk solid. It is worth noting that Equation 11–3 does not involve the enthalpies of the surface site species.

The momentum equation for the gas expresses the balance between pressure forces, inertia, viscous drag, and momentum added to the flow by surface reactions. Thus,

(11–4)

where is the absolute pressure and is the drag force exerted on the gas by the tube wall, to be discussed below. The pressure is related to the density via the ideal-gas equation of state, as given in Equation 2–5 .

Since the heterogeneous production rates will depend, in general, on the composition of the surface as well as that of the gas, equations determining the site fractions of the surface species are now needed. Assuming that these species are immobile, the steady-state conservation equations are simply stated in Equation 11–5 . The surface species conservation equation is applied to every species in each surface phase contained on each surface material .

(11–5)

that is, the net production rate of each surface species by heterogeneous reactions must be zero. However, we also assume that the total site density for each surface phase is a constant. As a result, the algebraic equations represented by Equation 11–5 are not all independent, and for each phase on each material, one of the equations must be replaced by the condition,

(11–6)

In order to minimize errors, Equation 11–6 is used to replace Equation 11–5 for the species having the largest site fraction.

The system of governing equations for the reactor is now mathematically closed. However, because the residence time of the gas is often a quantity of interest, it is useful to include an equation that computes it automatically. This is simply

(11–7)

Equation 11–1 through Equation 11–7 provide a total of differential/algebraic relations involving the dependent variables , , , , , and . The functions , , , , and can all be expressed in terms of these and are obtained from calls to Gas-phase Kinetics and Surface Kinetics subroutine libraries. The quantities , , and are fixed by the reactor geometry. This then leaves only and to be addressed.

For plug-flow and related reactor models, there are many different options for handling the reactor energy balance:

(11–8)

Both and must be supplied by the user.

The viscous drag force is written in terms of a friction factor as follows:

(11–9)

The friction factor can in turn be expressed as a function of the local Reynolds number

(11–10)

where is the tube diameter (or the mean hydraulic diameter for a conduit with a noncircular cross section) and is the gas viscosity. For laminar flow () the analytical result for round tubes is

(11–11)

while for turbulent flow one can use the approximate Blasius formula,[84]

(11–12)

This approach is only approximate, especially for noncircular conduits, but viscous drag is usually of very minor importance in gas-phase reactors. In keeping with this, and in order to avoid having to calculate transport properties, the gas viscosity is computed by scaling the inlet value (supplied by the user) by and ignoring the composition dependence.

It remains to specify the initial (inlet) conditions for the reactor. Clearly, values for , , , and at should be known or easily obtainable from the problem statement, the ideal gas law, and the reactor geometry, and of course at this point. Since there are no derivatives of the in the governing equations, it might appear that no initial conditions are needed for them. However, the transient solver employed requires a consistent set of derivatives of the variables at the reactor inlet. For plug-flow simulations, this is accomplished in a separate preliminary calculation, in which a set of fictitious transient equations are solved, as given in Equation 11–13 is solved in conjunction with Equation 11–6 until steady state is reached.

(11–13)

Here is the site occupancy number for species and is the total site density of the phase in question. The initial values of for Equation 11–13 are essentially arbitrary (unless there are multiple steady states), although better guesses will lead to faster convergence.