Mass balance volume

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

Urea Pharmacokinetics. Pharmacokinetics summarizes the relationships between solute generation, solute removal, and concentration in a patient s blood stream. In the context of hemodialysis, this analysis is most readily appHed to urea, which has, as a consequence, become a surrogate for other uremic toxins in the quantitation of therapy and in attempts to describe its adequacy. In the simplest case, a patient is assumed to have no residual renal function. Urea is generated from the breakdown of dietary protein, accumulates in a single pool equivalent to the patient s fluid volume, and is removed uniformly from that pool during hemodialysis. A mass balance around the patient yields the following differential equation ... 

Mass balance Apphed to the control volume, the principle of consei vation of mass may be written as (Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981)... 

This is an old, familiar analysis that applies to any continuous culture with a single growth-limiting nutrient that meets the assumptions of perfect mixing and constant volume. The fundamental mass balance equations are used with the Monod equation, which has no time dependency and should be apphed with caution to transient states where there may be a time lag as [L responds to changing S. At steady state, the rates of change become zero, and [L = D. Substituting ... 

The facility wastewater monitoring program does not determine the concentration of lead and lead compounds in the scrubber discharge water, and releases to the surface impoundment (releases to land) must be calculated using mate-riai balance information. These releases to land are determined from the amount of lead removed by the scrubber (using the efficiency data provided by the scrubber manufacturer). The volume of the scrubber blowdown Is found to be 1,500 pounds per year. Enter the estimate of the amount of lead and lead compounds released to surface Impoundments in the space provided in Part III, Section 5.5.3 of the form. Because releases of lead to the surface Impoundment are greater than 999 pounds per year, you must enter the actual calculated amount in column A.2 of Section 5.5.1. The basis for the estimate of releases to the surface impoundment, entered in column B of Section 5, is mass balance calculations (code C). 

MASS BALANCE unit volume transfer (diffusion) per unit volume reaction) Empirically deiennined flux specified (3) Concen tration specified (1. 2b) Mass flux specified (2a.4) ... 

Capture efficiency can also be measured by first estimating workspace emission rates and local exhaust emissions. The local exhaust emission rate equals the duct concentration (mass/volume) multiplied by the duct flow rate (volume/time). The workspace emission rates can be calculated using appropriate mass balance models and measured ventilation rates and workspace concentrations. Capture efficiency is the ratio of duct emission rate to total emission rate (duct plus workspace). ... 

The tracer is injected into the duct at a constant rate and mixed with the flowing air. The concentration of the air-tracer mixture is measured further downstream. Assuming perfect mixing and that the air entering the test section has a zero concentration, the air volume flow rate can be calculated based on the mass balance of the tracer... 

FIGURE 1.1 Control volume for total mass balance. 

At the end of 24 hours of continuous process the system was shut down. The knowledge of flowed buffer volumes and of the optical densities inside and downstream each ultrafiltration stage allowed to estimate product distribution (see appendix for mass-balance equations and the calculation procedure). The content of each cell was recovered and ffeeze-dried in order to be stored and used for subsequent kinetic experiments. A schematic flow-sheet of the whole procedure is illustrated in figure 1. 

For the computation of compressible flow, the pressure-velocity coupling schemes previously described can be extended to pressure-velocity-density coupling schemes. Again, a solution of the linearized, compressible momentum equation obtained with the pressure and density values taken from a previous solver iteration in general does not satisfy the mass balance equation. In order to balance the mass fluxes into each volume element, a pressure, density and velocity correction on top of the old values is computed. Typically, the detailed algorithms for performing this task rely on the same approximations such as the SIMPLE or SIMPLEC schemes outlined in the previous paragraph. 

Cf, Cm, and c ui are reactant concentrations in the feed, and at the inlet and outlet of the catalyst bed, respectively and Vr is the reactor volume. The mass balance equation for mixing feed with recycle is as follows ... 

EXDET Test. A dispersant effectiveness test, named EXDET, was developed to address concerns associated with available laboratory dispersant effectiveness test procedures [160]. The EXDET procedure uses standard laboratory equipment (such as a Burrell Wrist-Action shaker) and small volumes of water, oil, and chemical dispersant. Other features include the capabilities to mass balance the dispersed and nondispersed oil and to generate replicate data for statistical analysis. 

Equation (20-70) is the unsteady-state component mass balance for fed-batch concentration at constant retentate volume. Integration yields the equations for concentration and yield in Table 20-19. 

N, and solute passage Si needed to produce desired retentate product with impurity concentrations Cj and retentate product yield Mj/Mjo. Permeate product characteristics for batch operation can be determined by mass balances using a permeate volume of Vp = Vo(l 1/X + N/X), a mass of solute i in the permeate as Mj perme e = Mjo(l — and the permeate concentration as the ratio of the... 

The steady-state condition of constant volume in the tank (dV/dt = 0) occurs when the volumetric flow in, Fq, is exactly balanced by the volumetric flow out, Fi. Total mass balances therefore are mostly important for those modelling situations in which volumes are subject to change, as given in simulation examples CONFLO, TANKBLD, TANKDIS and TANKHYD. 

Liquid flows continuously into an initially empty tank, containing a full-depth heating coil. As the tank fills, an increasing proportion of the coil is covered by liquid. Once the tank is full, the liquid starts to overflow, but heating is maintained. A total mass balance is required to model the changing liquid volume and this is combined with a dynamic heat balance equation. 

It becomes necessary to incorporate a total mass balance equation into the reactor model, whenever the total quantity of material in the reactor varies, as in the cases of semi-continuous or semi-batch operation or where volume changes occur, owing to density changes in flow systems. Otherwise the total mass balance equation can generally be neglected. 

The sign of the transfer term will depend on the direction of mass transfer. Assuming solute transfer again to proceed in the direction from volume Vl to volume V( the component mass balance equations become for volume Vl... 

Neglecting the effects of any density changes, the total mass balance then provides the relationship for the change of total volume in the mixer with respect to time. 

The steady-state mass balance, for a volume element, AV, as shown in Fig. 4.9, for reactant A, is given by... 

For a batch reactor, under constant volume conditions, the component mass balance equation can be represented by... 

Consider a differential element of column volume, AV, height AZ and cross-sectional area, Ac, such that AV=Ac AZ. Component mass balance equations can be written for each of the liquid phases, where... 

The actual volume of each phase in element AV is that of the total volume of the element, multiplied by the respective fractional phase holdup. Hence considering the direction of solute transfer to occur from the aqueous or feed phase into the organic or solvent phase, the mass balance equations become ... 

The countercurrent extraction is shown schematically in Figure 5.15. The organic course volume is denoted by O, and the corresponding aqueous flow volume is designated by A. The concentration of the extractable species are x and y in the aqueous and organic phases, respectively. There are n stages, and mass balance over them is ... 

If he selects the still pressure (which for a binary system will determine the vapour-liquid-equilibrium relationship) and one outlet stream flow-rate, then the outlet compositions can be calculated by simultaneous solution of the mass balance and equilibrium relationships (equations). A graphical method for the simultaneous solution is given in Volume 2, Chapter 11. 

We now raise a second question. If the outlet of the vessel is fed to a second tank with a volume V2 of 3 m3 (Fig. 2.8), what is the time response at the exit of the second tank With the second tank, the mass balance is... 

Consider a series of well-mixed vessels (or compartments) where the volumetric flow rate and the respective volumes are constant (Fig. 3.3). If we write down the mass balances of the first two vessels as in Section 2.8.1 (p. 2-20), they are 1 ... 

As with the stirred tank model, mass balance equations can be developed to describe mass transfer in plug flow. In this case, it is convenient to define the system as a differential cylindrical section of the tube, with length Az and volume nD2 Az/4, where D is the tube diameter. This system is fixed in space and may... 

The Navier-Stokes equations have a complex form due to the necessity of treating many of the terms as vector quantities. To understand these equations, however, one need only recognize that they are not mass balances but an elaboration of Newton s second law of motion for a flowing fluid. Recall that Newton s second law states that the vector sum of all the forces acting on an object ( F) will be equal to the product of the object s mass (m) and its acceleration (a), or XF = ma. Now consider the first of the three Navier-Stokes equations listed above, Eq. (10). The object in this case is a differential fluid element, that is, a small cube of fluid with volume dx dy dz and mass p(dx dy dz). The left-hand side of the equation is essentially the product of mass and acceleration for this fluid element (ma), while the right-hand side represents the sum of the forces... 

Fick s second law of diffusion can be derived from Fick s first law by using a mass balance approach. Consider the differential fluid element shown in Figure 4. This differential fluid element is simply a small cube of liquid or gas, with volume Ax Ay Az, and will be defined as the system for the mass balance. Assume now that component A enters the cube at position x by diffusion and exits the cube at x + Ax by the same mechanism. For the moment, assume that no diffusion occurs in the y or z directions and that the faces of the cube that are perpendicular to the y and z axes thus are impermeable to the diffusion of A. Under these conditions, the component mass balance for A in this system is... 

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