Mass balance solution

Algebraic analysis of this equation reveals that mass-balanced solutions exist if and only if bA = bs- Equation (9.5) can be simplified to J2 = h = J —bA. Thus, mass balance does not provide unique values for the internal reaction fluxes. In fact, for this example, solutions exist for... 

The final form of the simplified mass balance solution is... 

A recent contribution to mass-balance modeling of weathering (Bowser and Jones, 2002) utilizes a spreadsheet graphical method to interpret mass balance in watershed systems in place of the strictly numerical solution methods. Key to the approach is to solve the mass-balance equation for a 10 X 10 matrix as a function of mineral composition (specifically in terms of plagioclase feldspar and smectite compositions). Exploration by means of models that cover mineral compositional space limited the range of possible compositions and restricted the range of possible mass-balance solutions. Figure 1 (after Bowser and Jones, 2002) is an example of the spreadsheet approach applied to the Sierra Nevada ephemeral... 

Besides equilibrium constant equations, two other types of equations are used in the systematic approach to solving equilibrium problems. The first of these is a mass balance equation, which is simply a statement of the conservation of matter. In a solution of a monoprotic weak acid, for example, the combined concentrations of the conjugate weak acid, HA, and the conjugate weak base, A , must equal the weak acid s initial concentration, Cha- ... 

Write a mass balance and charge balance equations for a 0.10 M solution of NaHC03. 

Note that in writing this mass balance equation, the concentration of Ag(NH3)2i" must be multiplied by 2 since two moles of NH3 occurs per mole of Ag(NH3)2i". The second additional equation is a mass balance on iodide and silver. Since Agl is the only source of N and Ag+, every iodide in solution must have an associated silver ion thus... 

You should be able to describe a system at equilibrium both qualitatively and quantitatively. Rigorous solutions to equilibrium problems can be developed by combining equilibrium constant expressions with appropriate mass balance and charge balance equations. Using this systematic approach, you can solve some quite complicated equilibrium problems. When a less rigorous an-... 

Write charge balance and mass balance equations for the following solutions... 

Fig. 5. Mass balance in gas absorption columns. The curved arrows indicate the travel path of the solute A. The upper broken curve delineates the envelope...

Referring to Figure 2, by considering solute mass balances over n, (n — 1),. .. 2, 1 units in turn and eliminating intermediate solute mass fractions and flow rates, the amount of solute associated with the leached sohd may be calculated in terms of the composition of the sohd and solvent streams fed to the system. The resulting equation is (2)... 

The flow of hydrothermal solutions iato the oceans from hydrothermal vents, ie, springs coming from the sea floor ia areas of active volcanism, and the chemical reactions occurring there by high temperature alteration of basalts ate of significance ia the mass balance of and. Eurthermore,... 

If the source fingerprints, for each of n sources are known and the number of sources is less than or equal to the number of measured species (n < m), an estimate for the solution to the system of equations (3) can be obtained. If m > n, then the set of equations is overdetermined, and least-squares or linear programming techniques are used to solve for L. This is the basis of the chemical mass balance (CMB) method (20,21). If each source emits a particular species unique to it, then a very simple tracer technique can be used (5). Examples of commonly used tracers are lead and bromine from mobile sources, nickel from fuel oil, and sodium from sea salt. The condition that each source have a unique tracer species is not often met in practice. 

Mass Balance Constraints. Erom the schematic diagram of a continuous crystallizer shown ia Eigure 11, the foUowiag mass balance on solute can be constmcted ... 

Note that the numerator in each of the ratios in equation 7 represents the rate of solute removal from the patient. By mass balance, clearance is related to mass-transfer coefficient Kq as defined eadier in equations 3, 4, and 5, and where each of the three expressions equal rate of mass removal in g/s. 

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 ... 

The couphng equation is a vapor mass balance written at the vent system entrance and provides a relationship between the vent rate W and the vent system inlet quahty Xq. The relief system flow models described in the following section provide a second relationship between W and Xo to be solved simultaneously with the coupling equation. Once W andXo are known, the simultaneous solution of the material and energy balances can be accomplished. For all the preceding vessel flow models and the coupling equations, the reader is referred to the DIERS Project Manual for a more complete and detailed review. 

The first, and simplest, step in predicting crystallizer performance is the calculation of crystal yield. This can easily be estimated from knowledge of solution concentration and equilibrium conditions permitting calculation of the overall mass balance... 

It has been shown that an increase in crystallizer residence time, or decrease in feed concentration, reduces the working level of supersaturation. This decrease in supersaturation results in a decrease in both nucleation and crystal growth. This in turn leads to a decrease in crystal surface area. By mass balance, this then causes an increase in the working solute concentration and hence an increase in the working level of supersaturation and so on. There is thus a complex feedback loop within a continuous crystallizer, illustrated in Figure 7.11. 

Strehlow (1975) achieved a solution by conducting a mass balance over the flow field. Such a balance can be drawn up under the assumptions of similarity and a constant density between shock and flame. The assumption of constant density violates the momentum-conservation equation, and is a drastic simplification. The maximum overpressure is, therefore, substantially underestimated over the entire flame speed range. An additional drawback is that the relationship of overpressure to flame speed is not produced in the form of a tractable analytical expression, but must be found by trial and error. 

It is often experimentally convenient to use an analytical method that provides an instrumental signal that is proportional to concentration, rather than providing an absolute concentration, and such methods readily yield the ratio clc°. Solution absorbance, fluorescence intensity, and conductance are examples of this type of instrument response. The requirements are that the reactants and products both give a signal that is directly proportional to their concentrations and that there be an experimentally usable change in the observed property as the reactants are transformed into the products. We take absorption spectroscopy as an example, so that Beer s law is the functional relationship between absorbance and concentration. Let A be the reactant and Z the product. We then require that Ea ez, where e signifies a molar absorptivity. As initial conditions (t = 0) we set Ca = ca and cz = 0. The mass balance relationship Eq. (2-47) relates Ca and cz, where c is the product concentration at infinity time, that is, when the reaction is essentially complete. 

Li and Hsiao [143] provide a useful approach to the environmental problem of removing (by stripping) volatile organics from solution in a contaminated water stream by using fresh air as the stripping medium. It should be noted that a number of industrial firms perform this stripping with steam. The mass balance on the VOC component around the column (trayed or packed) as shown in Figure 8-55 uses the symbols of Reference 143. 

In transient elongational flow degradation, it was determined in the authors laboratory, by a detailed mass balance, that main chain scission accounted for >95% of the degradation in dilute solution. Any other type of depolymerization, if present, should then be of minor importance. 

For most real systems, particularly those in solution, we must settle for less. The kinetic analysis will reveal the number of transition states. That is, from the rate equation one can count the number of elementary reactions participating in the reaction, discounting any very fast ones that may be needed for mass balance but not for the kinetic data. Each step in the reaction has its own transition state. The kinetic scheme will show whether these transition states occur in succession or in parallel and whether kinetically significant reaction intermediates arise at any stage. For a multistep process one sometimes refers to the transition state. Here the allusion is to the transition state for the rate-controlling step. 

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