Mass balances cumulative

HgPh2 remaining after each rnn. Cumulated amount of products at the end of each rim Ph - Ph, Ph - Chxa, Ce designate biphenyl, phenylcyclohexane and the sum of benzene/cyclohexane respectively. Hgf and Phf correspond to the mass balance in Hg and organic componnds, respectively. 

Note that in the component mass balance the kinetic rate laws relating reaction rate to species concentrations become important and must be specified. As with the total mass balance, the specific form of each term will vary from one mass transfer problem to the next. A complete description of the behavior of a system with n components includes a total mass balance and n - 1 component mass balances, since the total mass balance is the sum of the individual component mass balances. The solution of this set of equations provides relationships between the dependent variables (usually masses or concentrations) and the independent variables (usually time and/or spatial position) in the particular problem. Further manipulation of the results may also be necessary, since the natural dependent variable in the problem is not always of the greatest interest. For example, in describing drug diffusion in polymer membranes, the concentration of the drug within the membrane is the natural dependent variable, while the cumulative mass transported across the membrane is often of greater interest and can be derived from the concentration. 

Figure 3.3 Schematic residual magma-cumulate relationships for instantaneous and average magmatic products in a linear plot of Ni and Cr concentrations. For mass balance to be obeyed, the instantaneous cumulate must lie on the tangent to the liquid line of descent at the point representing the instantaneous liquid.

At this point, the student reviewed the check clearing process in detail. Checks are issued to a supplier and an encumbrance is placed on an internal account in the chemical company the supplier cashes the check at the supplier s bank the check proceeds through a series of intermediate banks until it arrives at the chemical company s bank the issued amount is removed from the chemical company s bank account (cleared) and the check is then returned to the chemical company where the encumbrance on the internal account is removed. The system must achieve an overall mass balance such that the total or cumulative amount of money issued must eventually equal the total or cumulative amount of money cleared. 

The primary endpoint of the toxicokinetic studies is the concentration-time prohle of the substance in plasma/blood and other biological fluids as well as in tissues. The excretion rate over time and the amount of metabolites in urine and bile are further possible primary endpoints of kinetic studies, sometimes providing information on the mass balance of the compound. From the primary data, clearance and half-life can be derived by several methods. From the excretion rate over time and from cumulative urinary excretion data and plasma/blood concentration measured during the sampling period, renal clearance can be calculated. The same is the case for the bUiary excretion. 

Similarly, the cumulative copolymer composition, F, is plotted against the conversion, as shown in Figure 7.20, and calculated by a mass balance equation ... 

Treiman A. H. (1997) The parent magmas of the cumulate eucrites a mass balance approach. Meteorit. Planet. Sci. 32, 217-230. 

Mass balance can be calculated from the amount of drug remaining in the donor, washings, skin and cumulative amount in receptor. 

Understand the conventions and algebraic relations among mass-balance equations and stepwise and cumulative formation constants of complexes. 

In the equilibrium constant, the superscript asterisk indicates that the dissolution reaction is written in terms of protons. The subscript following K (zero in this case) is the number of hydroxyl ions associated with AF+ in the reaction. According to Nordstrom et al. (1990), Kq = 10. Reactions similar to (7.35) can be written in which kaolinite is dissolved to form each of the aluminum hydroxy complexes listed in mass-balance equation (7.22) for total aluminum. These reactions can be generated by successively adding the cumulative reactions for the Al-hydroxy complexes—Eqs. (7.23) to (7.26)—to Eq. (7.35). For example, adding Eqs. (7.35) and (7.23) and their log values for 25°C, we obtain ... 

If both of the components are volatile (e.g. ethanol and water), it wouldn t make sense to vaporize all of the liquid in the kettle. A mass balance can tell you that, if you vaporize the entire charge, the cumulative condensate composition will be essentially the same as the initial charge. Oh, perhaps you have removed some nonvolatile minerals which might have been in the water, but you won t enrich the ethanol concentration in this way. And... 

A simple cumulative mass balance on the cell leads to another expression for... 

The cumulative residence-time function for a real tubular reactor can be derived from the general mass balance by taking into account the dispersion term (Equation 2.2-34) ... 

Water quality monitoring consists of frequent analysis of the main constituents. The required data input consists of (1) mean composition of the influent (2) mean composition of native groundwater in each layer of the target aquifer (3) native geochemistry of each layer of the target aquifer (4) the cumulative frequency curve of detention times in each model layer or flow path as derived from either separately run hydrological model or tracer breakthrough data and (5) specific information derived from the mass balance of the water phase (the reactions that are needed, how O2 and NO3" distribute over the various redox reactions, etc.). 

Diagram of mass balance in a separator M mass flow rate of solids in the feed, M mass flow rate of separated solids, M, mass flow rate of nonseparated solids, F(x) cumulative percentage oversize of feed solids, F (x) cumulative percentage oversize of separated solids, F,(a ) cumulative percentage oversize of nonseparated solids, Q. volumetric flow rate of feed suspension, U volumetric flow rate of underflow suspension, O volumetric flow rate of overflow suspension. 

The cake thickness (L ) is related to the cumulative volume of filtrate and the cake properties through a mass balance ... 

The (apparently simplest) method using the accumulation differences (11.1.8) as measured variables in the balance constraint (11.2.4) fails in practice. It has been shown also theoretically that even small systematic errors in the measured (integrated) mass flowrates cumulate and in the end, can lead to absurd values of the holdups (states of the inventories) computed (estimated) by (11.2.9). 

In classification work mass balances may be needed on a host of size ranges. When plotting cumulative data sets against size, the size used needs to be at... 

The monomer and polymer mass balance in the reactor allows cumulative and instantaneous monomer conversion to be defined, and permits computation of and [q] from light scattering, viscosity, and concentration data, mft) is the cumulative (or total) mass of monomer plus polymer in the reactor, which is the sum of initial monomer mass in the reactor and that flowed in from the external reservoir, minus the amount flowed out for ACOMP sampling. This is found from ... 

As is usual in systems of some complexity, a number of different balances can be made depending on the choice of balance space. We may choose, for example, to make an unsteady integral balance about the still, or to make a similar balance about the receiver. They can be instantaneous or cumulative in time and can involve total or component mass balances. 

Although superficially this expression resembles a mass balance, it is not clear how it is arrived at. It does not contain time or distance as an independent variable, which invariably appears when we model a compartment or what we termed a 1-D pipe. We must also rule out a cumulative balance, which is always algebraic in form. The question then arises whether a new type of mass balance formulation is required to cover this case. Fortunately, as is shown below, this does not turn out to be the case. The three mainstay formulations — compartmental, 1-D distributed, and cumulative — are able to cover tiiis case as well. 

This is beginning to look much more like the desired expression (Equation 2.20), and one final step will bring us to that goal. We invoke a cumulative mass balance, which serves to convert liquid mass L to the solid weight fraction fg. We have up to any time t... 

Two features stand out in the treatment of this problem. First and foremost, we have reinforced our confidence in the three basic mass balance formulations at the algebraic and ODE level the compart-mental, 1-D distributed, and cumulative balances. They are vindicated as a comprehensive tool kit at this level of modeling. 

To obtain an expression for the external concentration C, we apply a simple cumulative mass balance, which reads... 

In our preceding discussion of this topic, we managed to reduce the underlying model, which consists of two PDE mass balances, to a simple algebraic cumulative balance joined to an appropriate continuous spectrum of equilibrium stages. When this restriction is removed and the mass transfer resistance is brought back into play, no alternative simplifications are possible and we must return to the full PDE model. This model has been solved for a number of different equilibrium relations, most notably the linear case expressed by X = HY. The results for the latter can be expressed in terms of the following two dimensionless parameters ... 

If we assume that the entire bed attains the wet-bulb temperature after a brief start-up period, the model reduces to an ODE, which must be complemented by a cumulative mass balance to determine the drying time. We consider instead a limiting version of the full model in which the exiting airstream is assumed to be fully saturated. This asymptotic case provides us... 

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