21 Basic Governing Equations

It is customary for chemical reactor engineers to start their analysis of flow processes occurring in a reactor with the formulation of species conservation equations along with the energy conservation equations. The reactor fluid dynamics is often simplified to avoid the need for solving momentum conservation equations. The model equations are often written in terms of molar concentrations of species and temperature. These equations are further simplified by assuming special conditions (such as mixing is much faster than reactions) to derive most of the commonly used reaction engineering models. As mentioned in Section 1.1, such simplifications have been extensively used to analyze the behavior of reactors. In order to further enhance our abilities to understand and control the flow processes occurring in reactors, it is necessary to provide a more rigorous treatment of momentum conservation and to consider mass, momentum and energy conservation equations simultaneously.

Any rigorous analysis of flow processes starts with the application of the universal laws of conservation of mass, momentum and energy. It may be of interest to point out that the conservation laws of momentum and energy may be derived from the homogeneity of space and time (Bird, 1998). It is very important to clearly identify and understand the implications of the underlying assumptions (both, explicit and implicit) when describing physical processes in mathematical equations. In this chapter, we will describe and discuss the basic governing equations based on these three laws for any continuous fluid. Without going into rigorous definition, an assumption of continuous fluid means that the 'mean free path' of the constituent molecules of the fluid is much smaller than the characteristic length scales of flow processes. For gases, the mean free path is of the order of 10-7 m and for liquids it is of the order of 10-10 m. Most flow processes which are of interest to reactor engineers can therefore be modeled using the continuous fluid approximation. Additional information about the governing equations may be found in Bird et al. (1960), Bird and Graham (1998) and Bird (1998). In addition to the basic governing equations developed from the universal laws, it is necessary to develop relevant constitutive equations and equations of state for the fluids under consideration to close the system of equations.

There are two approaches for deriving basic governing equations. In the Eulerian approach, an arbitrary control volume in a stationary reference frame is used to derive the basic governing equations. In an alternative, Lagrangian approach, equations are derived by considering a control volume (material volume) such that the velocity of the control volume surface always equals the local fluid velocity (Fig. 2.1). For single-phase flows, both the approaches give the same final form of the conservation equations (see Deen, 1998 for more discussion on different approaches to deriving conservation equations). These two approaches, however, offer different routes to simulate multiphase flow processes. Modeling multiphase flows and turbulent reactive flows based on these two approaches is discussed in Chapters 4 and 5 respectively. Basic governing equations for single-phase flow processes are discussed here.

2.1.1. Conservation of Mass

It is often suitable to write the mass conservation equations in terms of mass fractions of species rather than molar concentrations, especially for flow processes, where properties of the fluid may vary with composition and temperatures. The mass conservation equation for species k can therefore be written (in vector symbolism):

Stationary (a) reference frame

Flowing fluid

Stationary (a) reference frame

Control volume for differential balance moving with fluid element

Control volume for differential balance moving with fluid element

(b) Flowing fluid

FIGURE 2.1 Two approaches to deriving governing equations. (a) Eulerian; (b) Lagrangian.

(b) Flowing fluid

FIGURE 2.1 Two approaches to deriving governing equations. (a) Eulerian; (b) Lagrangian.

where t is time, p is fluid density, mk is the mass fraction of species k and U is fluid velocity. The first term of the left-hand side of this equation represents accumulation of species k in a volume element and the second term represents change in species mass fraction due to convection. The first term of the right-hand side represents the change in species mass fraction due to the diffusive fluxes, jk . Sk is the source of species k (net rate of production per unit volume). In principle, the volumetric source can be a rate of production or consumption due to chemical reactions or net exchange of species k with other phases, if present. For the non-reactive single-phase flows, source terms will be generally absent. Source term formulations for reactive flow processes will be discussed in Chapter 5. The velocity field may be known or may be obtained by solving momentum conservation equations (to be discussed later). It will be necessary to formulate equations for diffusive fluxes jk in terms of species mass fractions in order to use these equations to determine the species concentration fields in the reactor.

In general, the diffusive mass flux is composed of diffusion due to concentration gradients (chemical potential gradients), diffusion due to thermal effects (Soret diffusion) and diffusion due to pressure and external forces. It is possible to include the full multicomponent model for concentration gradient driven diffusion (Taylor and Krishna, 1993; Bird, 1998). In most cases, in the absence of external forces, it is sufficient to use the following expression for diffusive flux:

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