The Continuity, Energy, and Momentum Equations

Plug flow is a simplified and idealized picture of the motion of a fluid, whereby all the fluid elements move with a uniform velocity along parallel streamlines. This perfectly ordered flow is the only transport mechanism accounted for in the plug flow reactor model. Because of the uniformity of o nditions in a cross section the steady-state continuity equation is a very simple ordinary differential equation. Indeed, the mass balance over a differential volume element for a reactant A involved in a single reaction may be written  

Equations 9.1-1 or 9.1-2 are of course, easily derived also from Eqs. 7.2.b-2 or 7.2.b-4 given in Chapter 7. For a single reaction and taking the reactant /I as a reference component = — 1, so that Equation 7.2.b-4 then directly 

When the volume of the reactor, V, and the molar flow rate of A at the inlet are given, Eq. 9.1-1 permits one to calculate the rate of reaction at conversion x. For a set of values (r, jc ) a rate equation may then be worked out. This outlines 

When the rate of reaction is given and a feed is to be converted to a value of, say x, Eq. 9.1-2 permits the required reactor volume V to be determined. This is one of the design problems that can be solved by means of Eq. 9.1 -2. Both aspects— kinetic analysis and design calculations—are illustrated further in this chapter. Note that Eq. 9.1-2 does not contain the residence time explicitly, in contrast with the corresponding equation for the batch reactor. E/f o s expressed here in hr m /kmol 4—often called space time—is a true measure of the residence time only when there is no expansion or contraction due to a change in number of moles or other conditions. Using residence time as a variable offers no advantage since it is not directly measurable—in contrast with V/F q. 

If there is expansion or contraction the residence time 9 has to be considered first over a differential volume element and is given by 

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The Continuity, Energy, and Momentum Equations Kinetic Studies Using a Tubular Reactor with Plug Flow... 

The reformer tube operation was simulated on the basis of a set of continuity-, energy- and momentum equations using one and two dimensional heterogeneous models. Intraparticle gradients in the rings were accounted for by the use of the generalized modulus concept. 

The governing inviscid flow equations of continuity, energy, and momentum (in conservation form) in gas region are Euler Equations (9)-(l 1) ... 

Answer by Author The procedure used to compute sonic velocities under these conditions is essentially that employed by Heinrich P] and modified by Holzman [ ] to treat the liquid density as a variable. The basic laws of continuity, energy, and momentum, and the ideal-gas equations of state are applied to a mixture in such a way as to derive an equation of motion for the mixture in terms of the properties of the gaseous and liquid constituents. The resulting equation is greatly simplified by the following assumptions ... 

The steady-state stagnation-flow equations represent a boundary-value problem. The momentum, energy, and species equations are second order while the continuity equation is first order. Although the details of boundary-condition specification depend in the particular problem, there are some common characteristics. The second-order equations demand some independent information about V,W,T and Yk at both ends of the z domain. The first-order continuity equation requires information about u on one boundary. As developed in the following sections, we consider both finite and semi-infinite domains. In the case of a semi-infinite domain, the pressure term kr can be determined from an outer potential flow. In the case of a finite domain where u is known on both boundaries, Ar is determined as an eigenvalue of the problem. 

By way of illustration, we note that in the recombination problem mentioned above the energies of the electrons are the variables in the diffusion equation, the bottleneck is the region of energies near the boundary of the continuous spectrum, and the slowness of the process is related to the small amount of energy transfer from an electron to a heavy particle in one collision. In the problem of escaping electrons in a plasma, slowness is ensured by the weakness of the electric field, the independent variable in the diffusion equation is the momentum component along the field, and the bottleneck is determined, as in the kinetics of new phase formation, by the saddle point of the integral. 

The present book is concerned with methods of predicting heat transfer rates. These methods basically utilize the continuity and momentum equations to obtain the velocity field which is then used with the energy equation to obtain the temperature field from which the heat transfer rate can then be deduced. If the variation of fluid properties with temperature is significant, the continuity and momentum equations... 

The close interrelations among mass, momentum, and energy transport can be explained in terms of a molecular theory of monatomic gases at low density. The continuity, motion, and energy equations can all be derived from the Boltzmann equation for the velocity distribution function, from which the molecular expressions for the flows and transport properties are produced. Similar derivations are also available for polyatomic gases, monatomic liquids, and polymeric liquids. For monatomic liquids, the expressions for the momentum and heat flows include contributions associated with forces between two molecules. For polymers, additional forces within the polymer chain should be taken into account. 

Energy and momentum conservation can be directly deduced from the continuity equation for the energy momentum tensor [27, 28]. For the r" resulting from (2.1) one finds... 

As the material properties have been presumed to be independent of the temperature and composition, the velocity field is independent of the temperature and concentration fields, so the continuity and momentum equations can be solved independently of the energy and component continuity equations. 

First of all, the density and all the thermodynamic coefficients are constants. Secondly, when the density and the transport properties are constants, the continuity and momentum equations are decoupled from the energy equation. This result is important, as it means that we may solve for the three velocities and the pressure without regard for the energy equation or the temperature. Third, for incompressible flows the pressure is determined by the momentum equation. The pressure thus plays the role of a mechanical force and not a thermodynamic variable. Fourth, another important fact about incompressible flow is that only two parameters, the Reynolds number and the Froude number occur in the equations. The Froude number, Fr, expresses the importance of buoyancy compared to the other terms in the equation. The Reynolds number indicates the size of the viscous force term relative to the other terms. It is mentioned that compressible flows are often high Re flows, thus they are often computed using the inviscid Euler (momentum) equations. 

The equation of continuity is of fundamental importance in balancing systems. There are several equations of continuity, dealing with the continuity of matter, momentum, energy, and other physical quantities. 

Unlike the aforementioned models, Fyhr and Rasmuson [41,42] and Cartaxo and Rocha [43] used an Eulerian-Lagrangian approach, in which the gas phase is assumed as the continuous phase and the solids particles are occupying discrete points in the computational domain. As a consequence, mass, momentum, and energy balance equations were solved for each particle within the computational domain. 

With the presence of the liquid-phase and interphase mass, momentum, and energy transport, additional source terms are added into the continuity, momentum, and scalar transport equations. As the droplets evaporate the heat of vaporization is taken from the gas phase and there is evaporative cooling of the surrounding gas. This gives rise to a sink term in the energy equation. By assuming adiabatic walls and unity Lewis number, the energy and scalar equations have the same boundary conditions and are linearly dependent [5]. 

There is also a continuity equation for the host gas, usually considered as inert relative to the processes of (2.6,7). In addition, complete macroscopic specification would require the conservation equations of energy and momentum, which would be coupled to (2.6,7). The effects of turbulence can be introduced through stochastic definitions for the macroscopic variables of the system. This approach has been used in describing "dusty gas" flows [2.8] and in analysis of gas dynamics of expansion flows with condensation [2.4]. 

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