Mass balance equation, solution

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- ... 

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... 

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

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. 

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... 

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 formal, algebraic, method. The presence of recycle implies that some of the mass balance equations will have to be solved simultaneously. The equations are set up with the recycle flows as unknowns and solved using standard methods for the solution of simultaneous equations. 

The most effective spectrophotometric procedures for pKa determination are based on the processing of whole absorption curves over a broad range of wavelengths, with data collected over a suitable range of pH. Most of the approaches are based on mass balance equations incorporating absorbance data (of solutions adjusted to various pH values) as dependent variables and equilibrium constants as parameters, refined by nonlinear least-squares refinement, using Gauss-Newton, Marquardt, or Simplex procedures [120-126,226],... 

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. 

This chapter provides analytical solutions to mass transfer problems in situations commonly encountered in the pharmaceutical sciences. It deals with diffusion, convection, and generalized mass balance equations that are presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Typical pharmaceutical problems such as membrane diffusion, drug particle dissolution, and intrinsic dissolution evaluation by rotating disks are used as examples to illustrate the uses of mass transfer equations. 

Mainly, the available models have been developed based on the fugacity approach, which use the fugacity as surrogate of concentration, for the compilation and solution of mass-balance equations involved in the description of chemicals fate. However, a new... 

We assume all reactions to be first order and irreversible within the range of the experimental conditions. The governing differential mass balance equations and their solutions have been reported [9J. The values of the constants through at 450°C are shown in Table I. A comparison of the experimental data with the theoretical predictions is shown in Figures 2 through 4 the above assumption of a first order reaction appears reasonable. 

Four mass balance equations can be written, one for each medium, resulting in a total of four unknown fugacities, enabling simple algebraic solution as shown in Table 1.5.9. From the four fugacities, the concentration, amounts and rates of all transport and transformation processes can be deduced, yielding a complete mass balance. 

As a final note, a variant of the calculation is useful in many cases. Suppose a chemical analysis of a groundwater is available, giving the amount of a component in solution, and we wish to compute how much of the component is sorbed to the sediment. We can solve this problem by eliminating the summations over the sorbed species (the over q terms) from each of the mass balance equations,... 

In order for an ion to sorb from solution, it must first move through the electrical potential field and then react chemically at the surface. To write a mass balance equation for sorption reactions (such as Reactions 10.1-10.4), therefore, we must... 

Where competitive inhibition is observed between two solutes (i.e. binding to a single, identical carrier), it is also possible to estimate carrier concentrations using a steady-state treatment [193-195], In that case, data from the competing solutes are used to generate a sufficient number of equilibrium expressions (e.g. equations (38) and (39)) and corresponding mass balance equations (e.g. equations (40) and (41)) to resolve for the total carrier concentration. 

Various munerical techniques are used to indirectly obtain solutions to large systems of equations with too many imknowns to solve explicitly. One approach is to solve the equations iteratively. This is done by first assuming that all of the anions are unbound and, hence, their free ion concentrations are equal to their total (stoichiometric) concentrations. By substituting these assumed anion concentrations into the cation mass balance equations, an initial estimate is obtained for the free cation concentrations. These cation concentrations are substituted into the anion mass balance equations to obtain a first estimate of the free anion concentrations. These free anion concentrations are then used to recompute the free cation concentrations. The recalculations are continued imtil the resulting free ion concentrations exhibit little change with further iterations. The computer programs used to perform speciation calculations perform these iterations in a matter of seconds. 

Similarity solutions of the species-mass-balance equations were assumed by Friedlander and Seinfeld for a simple photochemical-smog reaction scheme. (This scheme assumed a steady-state condition for ozone.) Demonstration runs were shown for parametric variations in the... 

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

Because we are generally able to define the chemistry of an aqueous solution containing n chemical elements by analytical procedures, n equations such as 8.48 and 8.49 exist, relating the bulk concentration of a given element mj to all species actually present in solution. Associated with mass balance equations of this type may be a charge balance equation expressing the overall neutrality of the solution ... 

For two tanks in series, solution of the mass balance equations yields the result... 

Iteration solutions were first proposed by Thiele and Geddes (Tl) in 1933. In this method, all temperatures and flows must be estimated before the solution can begin. The solution is broken into three parts first, solution of the mass-balance equations under the estimated flows and temperatures second, correction of the temperatures and third, correction of the flows. Assuming values for all temperatures and flows reduces the set of mass-balance equations shown in Table I to a linear set of equations which can be solved for the compositions at each point. Because the starting assumptions are completely arbitrary, the compositions will undoubtedly be wrong (the liquid and vapor mole-fractions will not sum to unity), and better values of temperature and flows must then be obtained for use in the next iteration. 

Many recent workers have contributed to the development of iteration solutions, especially in the method of solving the mass balance equations. Amundson and Pontinen (Al) have proposed a general method of solution through matrices. Edmister (El) has solved the equations through development of a series expression relating the amount of a component at a stage to the amount in a product. Matching relations at... 

We coiild contimje to write all these mass-balance equations as X(r) instead of C/4(r), but the solutions are not particularly instructive unless the rate expressions become so comphcated that it is cumbersome to write an expression in terms of one concentration CA. For simpHcity, we will use Ca(t) rather than X(t) wherever possible. 

As a simple example, consider the concentration versus time when a pure solvent initially in a tank (Cai = 0) is replaced by a solute at concentration Cao such us replacing pure water in a tank by a brine solution. Since there is no reaction, the mass-balance equation is... 

These look similar to series reactions, but the solution is quite different. In a PFTR or batch reactor the mass-balance equations are... 

All these arguments require a single reactant A on which to base the calculation of selectivity. For more complex situations we can stiU determine how the selectivity varies with conversion in PFTR and CSTR, but calculation of the selectivity requires complete solution of the mass-balance equations. 

We have gone about as far as is useful in finding closed-form analytical solutions to mass-balance equations in batch or continuous reactors, described by the set of reactions... 

Now, as for the unimolecular reaction, we could also write the mass-balance equations for 6a, 9b, and 9( and solve the algebraic equations for the rate in terms of partial pressures. However, for these expressions we obtain a quadratic equation whose solution is not very instructive. 

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