Equations mass conservation

The mass coming in through the cross-sechonal area AA per unit time at Xj is represented by 

At a steady-state condition, the mass is conserved in the space according to pwAA = pM + d/dx(pM)Ax AA The mass conservation equation is then represented by 

---

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... 

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) ... 

The Navier-Stokes equation [Eq. (1)] provides a framework for the description of both liquid and gas flows. Unlike gases, liquids are incompressible to a good approximation. For incompressible flow, i.e. a constant density p, the Navier-Stokes equation and the corresponding mass conservation equation simplify to... 

The situation is different for incompressible flow. In that case, no equation of motion for the pressure field exists and via the mass conservation equation Eq. (17) a dynamic constraint on the velocity field is defined. The pressure field entering the incompressible Navier-Stokes equation can be regarded as a parameter field to be adjusted such that the divergence of the velocity field vanishes. 

At that stage, the approximation obtained for the velocity field does generally not fulfil the mass conservation equation. In order to ensure mass conservation, corrections to the velocity and pressure field are introduced via... 

By demanding that the new velocity w field fulfils both the momentum and the mass conservation equation, the following equations for the velocity and pressure correction are derived ... 

The storage hold-up f/(r) is related to the input and output rates Fu(t) and Fd(0 by the mass conservation equation ... 

A finite correlation may replace the slip equations (6-8). With assumptions (1) and (2) and use of Eqs. (6-8)--(6-13), instead of two Eqs. (6-5), the pressure term is present only in Eq. (6-13), which may be solved separately. With assumptions (1) and (3),the phase energy equation (6-9) becomes equivalent to the phase mass conservation equation (6-3), thus reducing the order of the set. 

The mass conservation equation for the adsorber, over an increment dz of bed may be written as ... 

The energy and mass conservation equations used in the determination of the flame temperature are more conveniently written in terms of moles thus, it is best to write the partial pressure in Kp in terms of moles and the total pressure P. This conversion is accomplished through the relationship between partial pressure p and total pressure P, as given by Eq. (1.30). Substituting this expression for p, [Eq. (1.30)] in the definition of the equilibrium constant [Eq. (1.40)], one obtains... 

Since the ORR is a first-order reaction following Tafel kinetics, the solution of the mass conservation equation (eq 23) in a spherical agglomerate yields an analytic expression for the effectiveness factor... 

The gas channels contain various gas species including reactants (i.e., oxygen and hydrogen), products (i.e., water), and possibly inerts (e.g., nitrogen and carbon dioxide). Almost every model assumes that, if liquid water exists in the gas channels, then it is either as droplets suspended in the gas flow or as a water film. In either case, the liquid water has no affect on the transport of the gases. The only way it may affect the gas species is through evaporation or condensation. The mass balance of each species is obtained from a mass conservation equation, eq 23, where evaporation/condensation are the only reactions considered. 

This equation is usually referred to as the continuity equation or mass conservation equation. The source term, Sm, in the continuity equation is commonly caused by mass consumption or production from electrochemical reactions as well as mass loss/gain through phase transformation. 

Clearly, the actual pressure head in each phase depends on the fluid configuration within the pores. Hux equations for each of the three phases can be combined with mass conservation equations to derive governing transport equations. 

Because both kinetic constants k i and k 2 are exponentially dependent on the temperature, Tg, within the pellets (according to an Arrhenius form of temperature dependence) and the reactant concentration,, appears explicitly in the three mass conservation equations and also the heat balance equation, the problem must be solved numerically, rather than analytically. The boundary conditions at the pellet centre are... 

A simplified procedure for design is to assume that both tj and — AH/Cp are constant. If, then, eqn. (60) (the heat conservation equation) is divided by eqn. (59) (the mass conservation equation) and integrated, one immediately obtains... 

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

The substance being transported can be either dissolved (part of the same phase as the water) or particulate substances. We will develop the diffusion equation by considering mass conservation in a fixed control volume. The mass conservation equation can be written as... 

We will begin with domain discretization into control volumes. Consider our box used in Chapter 2 to derive the mass transport equation. Now, assume that this box does not become infinitely small and is a control volume of dimensions Ax, Ay, and Az. A similar operation on the entire domain, shown in Figure 7.1, will discretize the domain into control volumes of boxes. Each box is identified by an integer i, j, k), corresponding to the box number in the x-, y-, and z-coordinate system. Our differential domain has become a discrete domain, with each box acting as a complete mixed tank. Then, we will apply our general mass conservation equation from Chapter 2 ... 

We can see that the equations and solution technique have an added degree of complexity for bubble-water gas transfer, primarily because of the variation of pressure and because the gas control volume cannot be considered large. These are, however, simply the mass conservation equations, which should not be considered difficult, only cumbersome. Any reduction of these equations is an assumption, which would need to be justified for the particular apphcation. If transfer of a trace gas is of interest, then similar equations for the trace gas would need to be added to those provided above. 

In a frame fixed to the matrix, the mass conservation equation for the solid and melt in a one-dimensional melting column is (McKenzie, 1984 Richter and McKenzie, 1984 Navon and Stolper, 1987)... 

The third mass conservation equation needed is that for reactant B in the liquid phase. 

Now, the global rate can be estimated at any conversion, since temperature can be calculated from eq. (5.232). Then, the conversion versus reactor depth or catalyst mass can be determined from the mass conservation equation (5.228). Only arithmetic solutions of the adiabatic model are possible. 

This is a result of the assumption that the mass transfer coefficient is minimal. Then, using the mass conservation equation for ethanol in the liquid phase (eq. (3.368)),... 

Establishing the connection between Eqs. 2.30 and 2.31 and the substantial-derivative operator is facilitated by using the mass-conservation equation, which is derived formally at the beginning of the next section. For the present the result is simply stated as... 

Regardless of what other conservation equations may be appropriate, a bulk-fluid mass-conservation equation is invariably required in any fluid-flow situation. When N is the mass m, the associated intensive variable (extensive variable per unit mass) is r) = 1. That is, r) is the mass per unit mass is unity. For the circumstances considered here, there is no mass created or destroyed within a control volume. Chemical reaction, for example, may produce or consume individual species, but overall no mass is created or destroyed. Furthermore the only way that net mass can be transported across the control surfaces is by convection. While individual species may diffuse across the control surfaces by molecular actions, there can be no net transport by such processes. This fact will be developed in much depth in subsequent sections where mass transport is discussed. 

---