Balance, Continuity Equation

Mass Balance, Continuity Equation The continuity equation, expressing conservation of mass, is written in cartesian coordinates as 

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may be 

The stress has an isotropic contribution due to fluid pressure and dilatation, and a deviatoric contribution due to viscous deformation effects. The deviatoric contribution for a Newtonian fluid is the three-dimensional generalization of Eq. (6-2)  

The identity tensor 6y is zero for i and unity for i —j. The coefficient X is a material property related to the bulk viscosity, K = X + 2p/3. There is considerable uncertainty about the value of K. Traditionally, Stokes hypothesis, K = 0, has been invoked, but the validity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to 

Similar generalizations to multidimensional flow are necessary for non-Newtonian constitutive equations. 

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Example 5-6 Friction Loss in a Sudden Expansion. Figure 5-7 shows the flow in a sudden expansion from a small conduit to a larger one. We assume that the conditions upstream of the expansion (point 1) are known, as well as the areas A and A2. We desire to find the velocity and pressure downstream of the expansion (V2 and P2) and the loss coefficient, Kt. As before, V2 is determined from the mass balance (continuity equation) applied to the system (the fluid in the shaded area). Assuming constant density,... 

A stoichiometric analysis based on the species expected to be present as reactants and products to determine, among other things, the maximum number of independent material balance (continuity) equations and kinetics rate laws required, and the means to take into account change of density, if appropriate. (A stoichiometric table or spreadsheet may be a useful aid to relate chosen process variables (Fj,ch etc.) to a minimum set of variables as determined by stoichiometry.)... 

Mass Balance, Continuity Equation The continuity equation, expressing conservation of mass, is written in cartesian coordinates as 9p 9pUx 9pUy 9pUi [Pg.458]

Mass balance (continuity) equations for fixed bed adsorber ... 

Also, the depletion of oxygen from the gas phase is rather low and usually compensated by the desorption of carbon dioxide. The methodology is attractive because it permits a separation of fluid dynamics (momentum balances, continuity equations, and turbulence model) from material balance equations for the state variables of interest. Figure 8 illustrates how results from the fluid dynamic simulations (mean velocities turbulent dispersion coefficient... 

Solution In Fig. 2.7-6, the flow diagram is shown with pressure taps to measure p, and p2. From the mass-balance continuity equation (2.6-2), for constant p wherePy = P2 = P,... 

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometiy is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having bulk-flow ports and ports or interfaces through which diffusive flux can occur ... 

The continuity equation gives V2 = V AJa, and Vj = Q/A. The pressure drop measured by the manometer is pi —p2= (p — p)gA . Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 

Particle conservation in a vessel is governed by the particle-number continuity equation, essentially a population balance to identify particle numbers in each and every size range and account for any changes due to particle formation, growth and destruction, termed particle birth and death processes reflecting formation and loss of particulate entities, respectively. 

Setting f = Tout, H = Hout, and so on, specializes the integral energy balance of Equation (5.14) to a perfectly mixed, continuous-flow stirred tank ... 

In the previous discussion of the one- and two-compartment models we have loaded the system with a single-dose D at time zero, and subsequently we observed its transient response until a steady state was reached. It has been shown that an analysis of the response in the central plasma compartment allows to estimate the transfer constants of the system. Once the transfer constants have been established, it is possible to study the behaviour of the model with different types of input functions. The case when the input is delivered at a constant rate during a certain time interval is of special importance. It applies when a drug is delivered by continuous intravenous infusion. We assume that an amount Z) of a drug is delivered during the time of infusion x at a constant rate (Fig. 39.10). The first part of the mass balance differential equation for this one-compartment open system, for times t between 0 and x, is given by ... 

The information flow diagram, for a non-isothermal, continuous-flow reactor, in Fig. 1.19, shown previously in Sec. 1.2.5, illustrates the close interlinking and highly interactive nature of the total mass balance, component mass balance, energy balance, rate equation, Arrhenius equation and flow effects F. This close interrelationship often brings about highly complex dynamic behaviour in chemical reactors. 

The mass balance of soil air may be described by the classic continuity equation for compressible fluids ... 

The steady-state continuity equations which describe mass balance over a fluid volume element for the species in the stagnant film which are subject to uniaxial diffusion and reaction in the z direction are... 

To obtain an expression for tj, we first derive the continuity equation governing steady-state diffusion of A through the pores of the particle. This is based on a material balance for A across the control volume consisting of the thin strip of width dx shown in Figure 8.10(a). We then solve the resulting differential equation to obtain the concentration profile for A through the particle (shown in Figure 8.10(b)), and, finally, use this result to obtain an expression for tj in terms of particle, reaction, and diffusion characteristics. 

Nonisothermal spherical particle. The energy equation describing the profile for T through the particle, equivalent to the continuity equation 9.1-5 describing the profile for cA, may be derived in a similar manner from an energy (enthalpy) balance around the thin shell in Figure 9.1(b). The result is... 

The basis for the analysis using the SCM is illustrated in Figure 9.3. The gas film, outer product (ash) layer, and unreacted core of B are three distinct regions. We derive the continuity equation for A by means of a material balance across a thin spherical shell in the ash layer at radial position r and with a thickness dr. The procedure is the same as that leading up to equation 9.1-5, except that there is no reaction term involving (- rA), since no reaction occurs in the ash layer. The result corresponding to equation 9.1-5 is... 

To simplify the treatment for an LFR in this chapter, we consider only isothermal, steady-state operation for cylindrical geometry, and for a simple system (A - products) at constant density. After considering uses of an LFR, we develop the material-balance (or continuity) equation for any kinetics, and then apply it to particular cases of power-law kinetics. Finally, we examine the results in relation to the segregated-flow model (SFM) developed in Chapter 13. 

A material balance analysis taking into account inputs and outputs by flow and reaction, and accumulation, as appropriate. This results in a proper number of continuity equations expressing, fa- example, molar flow rates of species in terms of process parameters (volumetric flow rate, rate constants, volume, initial concentrations, etc.). These are differential equations or algebraic equations. 

The solution of this set of equations, 18.4-26 (with expression (A) incorporated) to -29, must be coupled with the set of three independent material-balance or continuity equations to determine the concentration profiles of three independent species, and the temperature profile, for either a specified size (V) of reactor or a specified amount of reaction. A nu-merical solution of the coupled differential equations and property relations is required. Equations (A), (B), and (C) in Example 18-6 illustrate forms of the continuity equation. 

This diffusive flow must be taken into account in the derivation of the material-balance or continuity equation in terms of A. The result is the axial dispersion or dispersed plug flow (DPF) model for nonideal flow. It is a single-parameter model, the parameter being DL or its equivalent as a dimensionless parameter. It was originally developed to describe relatively small departures from PF in pipes and packed beds, that is, for relatively small amounts of backmixing, but, in principle, can be used for any degree of backmixing. 

The derivation of the material-balance or continuity equation for reactant A is similar to that of equations 19.4-48 and -49 for nonreacting tracer A, except that steady state replaces unsteady state (cA at a point is not a function of t), and a reaction term must be added. Thus, using the control volume in Figure 19.15, we obtain the equivalent of equation 19.4-48 as ... 

Continuity equation from a material balance for A around the control volume,... 

The most useful form of the continuity equation (or material balance) is that which enables calculation of the amount of catalyst (W) required in terms of /a (at the bed depth x). Since... 

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