## T

Tref where Tref is a reference temperature and Cpk is the specific heat of species k, at constant pressure. q is a flux of enthalpy. The first and second terms of the energy conservation equation represent accumulation and change of enthalpy due to convection. The third term represents change in the enthalpy due to conduction. The fourth and fifth terms represent reversible and irreversible change in the enthalpy due to pressure and viscous dissipation, respectively. The sixth term accounts for changes in enthalpy due to diffusive mass fluxes and the final term is the volumetric source of enthalpy (due to say, chemical reactions). In the energy conservation equation, the flux of enthalpy, q can be written in terms of temperature gradient as q =-kV T (2.10)

where k is thermal conductivity of the fluid.

The energy conservation equation is intimately linked to momentum conservation equations via the fourth and fifth terms. For most reacting systems, the contribution of energy released or absorbed by chemical reactions usually dominates the other terms originating from pressure and viscous effects. For highly viscous flows with low heats of reaction, it may be important to consider the viscous heating terms. An order of magnitude analysis is often used to examine the relative importance of different terms.

These equations have general applicability for any continuous medium and are valid for any co-ordinate system. Additional information about the formulations of basic governing equations can be found in Bird et al. (1960).

2.1.4. Analysis/Simplification of Governing Equations

A reactor engineer has to evaluate and analyze various terms appearing in the basic governing equations to explore the possibilities of simplifying them and tailoring them to suit the needs of the problem under consideration. It is often useful and instructive to re-write the governing equations in non-dimensional form by using characteristic reference scales (length, velocity, time, temperature and so on). The dimensionless form allows one to identify major factors governing the flow process. The choice of characteristic reference scales is obvious in simple flows (for example, average velocity, geometric length scale of the equipment). Use of such reference scales leads to a dimensionless form of governing equations containing some characteristic dimensionless numbers.

By examining these characteristic dimensionless numbers, it is possible to appreciate possible interactions of different processes (convection, diffusion, reaction and so on) and to simplify the governing equations accordingly. A typical dimensionless form of the governing equation can be written (for a general variable, $):

tr J d t \ Lr J \ L2 J V ' V t where the subscript 'r' indicates characteristic values (or reference values) used to make the governing equations dimensionless. Tr is the effective diffusion coefficient of variable $. All the symbols appearing without this subscript denote dimensionless quantities. If the characteristic time scale, tr is defined as the ratio of characteristic length (Lr ) and velocity (Ur ) scales, the dimensionless form of the equation can be rewritten:

d .. _ . / Vr UrLr d (p4>) + V ■ (pU$) = (.iL) V ■ (u) + Sk (2.12)

The dimensionless form of the equation contains one dimensionless parameter as a multiplier of the first term of the right-hand side and maybe some additional dimensionless parameters, which may appear within the dimensionless source term, Sk . Depending on the general variable, $, the effective diffusion coefficient, Tr appearing in this dimensionless number will be different, leading to different dimensionless numbers. For the species mass fraction, momentum and enthalpy transport equations, the effective diffusion coefficient will be molecular diffusion coefficient, the kinematic viscosity of the fluid and the thermal diffusivity of the fluid respectively. The corresponding dimensionless numbers are, therefore, defined as follows.

Reynolds number, Re: may be interpreted as the ratio of the convective transport to the molecular transport of momentum or as the ratio of the inertial to viscous forces:

LrUr

Examination of the relevant dimensionless numbers, with reference to specific characteristics of the problem under consideration, is useful for simplifying the governing equations. For example, for very high speed flows, the reciprocal of the Reynolds number tends to zero and it may be reasonable to ignore viscous stress terms in the momentum conservation equations. Under these conditions, momentum conservation equations reduce to well-known Euler equations. Aerospace engineers have used this simplification extensively for simulating high speed flows around complex shaped objects. On the other hand, when the Reynolds number is small (that is, when flow velocity or the size of the equipment is very small or the fluid is very viscous), the convective or inertial terms in the Navier-Stokes equations can be neglected. This approximation leads to well-known creeping flow equations.

Peclet Number, Pe: dimensionless number appearing in enthalpy or species mass conservation equations (defined for heat transfer and mass transfer, respectively). It is interpreted again as the ratio of the convective transport to the molecular transport and is defined as

If these Peclet numbers are divided by the Reynolds number, the resulting dimension-less numbers are called the Pradtl number, Pr and Schmidt number, Sc, respectively. The Prandtl number (Pr) is the ratio of momentum diffusivity and thermal diffusivity. The Schmidt number (Sc) is the ratio of momentum diffusivity and mass diffusi-vity. These five dimensionless numbers can convey very useful information about the relative contributions of convective and molecular transport and relative magnitudes of momentum, heat and mass transfer.

It must be noted that apart from these five dimensionless numbers, some additional dimensional parameters may appear in the dimensionless source terms. The source terms appearing in basic conservation equations are made dimensionless by dividing the reference source term, Sr defined as pr Ur ér

The dimensionless source term essentially represents the ratio of generation to convection. For various generation terms, several additional dimensionless numbers may be defined. For example, if the generation of momentum due to gravitational forces is considered, a dimensionless number, called as the Froude number (Fr), is defined as the ratio of convection to gravitational factors. The dimensionless numbers discussed here along with other dimensionless numbers are listed in Table 2.1 together with their physical interpretation.

Apart from analyzing the relative contributions of various transport and generation mechanisms, the reactor engineer has to use basic engineering judgement to evolve suitable simplifications of the mathematical model. For example, when the flow under consideration has a predominant direction and the variation of geometry is gradual, it is possible to use so called 'boundary layer approximations'. In such cases, the flow is influenced mainly by what is happening upstream. Special, efficient methods have been developed to solve such specific simplified forms of the equations. Various possibilities for simplification, for example ignoring the variation of fluid properties such as density, viscosity, heat capacity etc., need to be evaluated by considering the possible implications on the application of the model. Compressibility effects may be neglected when the characteristic velocity of the fluid is much smaller than the speed of sound in that fluid.

The basic governing equations, written in a form suitable for any co-ordinate system, are useful for understanding the basic concepts and significance of various terms. It is, however, necessary to rewrite these equations, after considering the possible simplifications, for a specific co-ordinate system appropriate to the problem under consideration. Cartesian and cylindrical co-ordinate systems are the most commonly used systems for analyzing flow processes in simple geometry. The basic governing equations incorporating the relevant constitutive relationships (for Newtonian fluids)

TABLE 2.1 Dimensionless Numbers

Name

Symbol Definition

Physical Significance

Reynolds number Re

Peclet number

Prandtl number Pr Schmidt number Sc

Nusselt number or Nu or Sh Sherwood number

Damkohler number Daj

Damkohler number Da2

Stanton number St

Euler number Eu

Froude number Fr

Lewis number

Weber number We

Pr Lr Vr pr CpLr Vr Lr V

CVPU Uz,

U2LP

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