Conservation equations

1 Continuity. The continuity equation is a statement of conservation of mass. To nnderstand its origin, consider the flow of a fluid of density p through the six faces of a rectangular block, as shown in Fignre 5-2. The block has sides of length Axi, Ax2, and Axs and velocity components Ui, U2, and U3 in each of the three coordinate directions. To ensure conservation of mass, the sum of 

A more compact way to write eq. (5-3) is through the use of Einstein notation  

With this notation, whenever repeated indices occur in a term, the assumption is that there is a sum over all indices. Here, and elsewhere in this chapter, Ui is the ith component of the fluid velocity, and partial derivatives with respect to Xi are assumed to correspond to one of the three coordinate directions. For more general cases, the density can vary in time and in space, and the continuity equation takes on the more familiar form 

2 Momentum. The momentum equation is a statement of conservation of momentum in each of the three component directions. The three momentum equations are collectively called the Navier-Stokes equations. In addition to momentum transport by convection and diffusion, several momentum sources are also involved  

(5-6) the convection terms are on the left. The terms on the right-hand side are the pressure gradient, a source term the divergence of the stress tensor, which is responsible for the diffusion of momentum the gravitational force, another source term and other generalized forces (source terms), respectively. 

We now consider the equations governing the fluid flow, namely the equations of conservation of mass, momentum and energy. Derivations of these equations in general form can be found in many text books. These derivations will not be repeated here. 

The differential equation expressing the conservation of mass, also known as the continuity equation, is 

This is also known as the condition of incompressibility. Hence we also have Du = 0(fr D = 0) for incompressible fluid. 

However, the change of density cannot be neglected under the high level of pressure involved in processes such as injection molding. 

The equations of the conservation of momentum, also called the equations of motion, are 

A brief summary of the theoretical formulation of physicochemical processes in various regions during the laser-induced RDX ignition process [40] is given below. The model for steady-state combustion can be treated as a limiting case by neglecting all the time-varying terms. 

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The mathematical theory is rather complex because it involves subjecting the basic equations of motion to the special boundary conditions of a surface that may possess viscoelasticity. An element of fluid can generally be held to satisfy two kinds of conservation equations. First, by conservation of mass. 

These expressions are inserted in the conservation equations, and the boundary conditions provide a set of relationships defining the U and V coefficients [125-129]. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Since angular momentum is conserved, equation (A3.11.192) may be rearranged to give the following implicit equation for the time dependence of r ... 

Combining the two conservation equations gives the following stoichiometric equation between C10H20N2S4 and NaOH... 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

Flow and Performance Calculations. Electro dynamic equations are usehil when local gas conditions (, a, B) are known. In order to describe the behavior of the dow as a whole, however, it is necessary to combine these equations with the appropriate dow conservation and state equations. These last are the mass, momentum, and energy conservation equations, an equation of state for the working duid, an expression for the electrical conductivity, and the generalized Ohm s law. 

By applying the conservation equations of mass and energy and by neglecting the small pressure changes across the flame, the thickness of the preheating and reaction 2ones can be calculated for a one-dimensional flame (1). 

The physics and modeling of turbulent flows are affected by combustion through the production of density variations, buoyancy effects, dilation due to heat release, molecular transport, and instabiUty (1,2,3,5,8). Consequently, the conservation equations need to be modified to take these effects into account. This modification is achieved by the use of statistical quantities in the conservation equations. For example, because of the variations and fluctuations in the density that occur in turbulent combustion flows, density weighted mean values, or Favre mean values, are used for velocity components, mass fractions, enthalpy, and temperature. The turbulent diffusion flame can also be treated in terms of a probabiUty distribution function (pdf), the shape of which is assumed to be known a priori (1). 

Entrance effects are usually not significant industrially if L/D > 60. Below this limit Nusselt recommended the conservative equation for 10 < L/D < 400 and properties evaluated at bulk temperature... 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

Examples Four examples follow, illustrating the apphcation of the conservation equations to obtain useful information about fluid flows. 

Cavitation Loosely regarded as related to water hammer and hydrauhc transients because it may cause similar vibration and equipment damage, cavitation is the phenomenon of collapse of vapor bubbles in flowing liquid. These bubbles may be formed anywhere the local liquid pressure drops below the vapor pressure, or they may be injected into the hquid, as when steam is sparged into water. Local low-pressure zones may be produced by local velocity increases (in accordance with the Bernouhi equation see the preceding Conservation Equations subsection) as in eddies or vortices, or near bound-aiy contours by rapid vibration of a boundaiy by separation of liquid during water hammer or by an overaU reduction in static pressure, as due to pressure drop in the suction line of a pump. 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

In general, solutions are obtained by couphng the basic conservation equation for the batch system, Eq. (16-49) with the appropriate rate equation. Rate equations are summarized in Table 16-11 and 16-12 for different controlhng mechanisms. 

Bidispersed Particles For particles of radius Cp comprising adsorptive subparticles of radius r, that define a macropore network, conservation equations are needed to describe transport both within the macropores and within the subparticles and are given in Table 16-11, item D. Detailed equations and solutions for a hnear isotherm are given in Ruthven (gen. refs., p. 183) and Ruckenstein et al. [Chem. Eng. Sci., 26, 1306 (1971)]. The solution for a linear isotherm with no external resistance and an infinite fluid volume is ... 

Extensions When more than two conservation equations are to be solved simultaneously, matrix methods for eigenvalues and left eigenvectors are efficient [Jeffrey and Taniuti, Nonlineai Wave Pi op-agation, Academic Press, New York, 1964 Jacob andTondeur, Chem. Eng. J., 22,187 (1981), 26,41 (1983) Davis and LeVan, AJChE J., 33, 470 (1987) Rhee et al., gen. refs.]. 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

Conservation equations Expressions that equate the mass, momentum, and energy across a steady wave or shock discontinuity ((2.1)-(2.3)). Also known as the jump conditions or the Rankine-Hugoniot relations. 

It is worth investigating the time derivatives and demonstrating how to derive (9.1)-(9.4) from the more familiar forms of the conservation equations. The more familiar Lagrangian derivative djdt and d jdt are related by [9]... 

The conservation equations are more commonly written in the initial reference frame (Lagrangian forms). The time derivative normally used is d /dt. Equation (9.5) is used to derive (9.2) from the Lagrangian form of the conservation of mass... 

There are few analytic solutions to the governing equations for interesting problems. The conservation equations are typically solved approximately on digital computers. It is assumed that the sound speeds are real and the system... 

Lagranglan codes are characterized by moving the mesh with the material motion, u = y, in (9.1)-(9.4), [24]. The convection terms drop out of (9.1)-(9.4) simplifying all the equations. The convection terms are the first terms on the right-hand side of the conservation equations that give rise to fluxes between the elements. Equations (9.1)-(9.2) are satisfied automatically, since the computational mesh moves with the material and, hence, no volume or mass flux occurs across element boundaries. Momentum and energy still flow through the mesh and, therefore, (9.3)-(9.4) must be solved. 

The energy conservation equation is not normally solved as given in (9.4). Instead, an evolution equation for internal energy is used [9]. First an evolution equation for the kinetic energy is derived by taking the dot product of the momentum balance equation with the velocity and integrating the resulting differential equation. The differential equation is... 

Because we need to know how long the refined section of the bar is, it is important to describe the ramping up of the compositions in a quantitative way. We can do this by writing a differential equation which describes what happens as the zone moves from some general position x to a new position x + 8x (Fig. 4.4g). For a bar of unit cross-section we can write the mass conservation equation... 

The conservation equations developed by Ericksen [37] for nematic liquid crystals (of mass, linear momentum, and angular momentum, respectively) are ... 

For the ideal reactors considered, the design equations are based on the mass conservation equations. With this in mind, a suitable component is chosen (i.e., reactant or product). Consider an element of volume, 6V, and the changes occurring between time t and t + 6t (Figure 5-2) ... 

HOTM AC/RAPTAD contains individual codes HOTMAC (Higher Order Turbulence Model for Atmospheric Circulation), RAPTAD (Random Particle Transport and Diffusion), and computer modules HOTPLT, RAPLOT, and CONPLT for displaying the results of the ctdculalinns. HOTMAC uses 3-dimensional, time-dependent conservation equations to describe wind, lempcrature, moisture, turbulence length, and turbulent kinetic energy. 

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