Equations for Mass Balance

Uncharged reaction components are transported by diffusion and convection, even though their migration fluxes are zero. The total flux density Jj of species j is the algebraic (vector) sum of densities of all flux types, and the overall equation for mass balance must be written not as Eq. (4.1) but as... 

VAN AKEN et al. 0) and EDWARDS et al. (2) made clear that two sets of fundamental parameters are useful in describing vapor-liquid equilibria of volatile weak electrolytes, (1) the dissociation constant(s) K of acids, bases and water, and (2) the Henry s constants H of undissociated volatile molecules. A thermodynamic model can be built incorporating the definitions of these parameters and appropriate equations for mass balance and electric neutrality. It is complete if deviations to ideality are taken into account. The basic framework developped by EDWARDS, NEWMAN and PRAUSNITZ (2) (table 1) was used by authors who worked on volatile electrolyte systems the difference among their models are in the choice of parameters and in the representation of deviations to ideality. 

To analyze data using these two methods one must make two assumptions (1) that a sorptive entering the chamber can either be sorbed or remain in solution, and (2) the sample is perfectly mixed i.e., the concentration in the mixing chamber equals the effluent concentrations. With these assumptions, one can then develop an equation for mass balance which can be used to analyze time-dependent data using a continuous flow method (Skopp and McAllister, 1986) ... 

Essential relations describing each subsystem are overall equations for mass balance, for energy balance, for performance, and for costing in terms of performance. Presently available cost trends (17) in terms of capacity parameters (e.g. area, mass rate, power,. ..) are suitable costing equations to start with. They may be implemented to include the influence of variables such as pressure, temperature or efficiency whenever sufficient data are available. 

Another example of a problem that can be solved by intentional circular reference and iteration is the calculation of the pH of a solution of a weak acid. Combining the equations for mass balance and K, we obtain the following equation for the [H" "] of a weak acid of concentration C mol/L. 

By setting the time-derivative of each metaboUte concentration to zero under pseudo-stationary assumption, the set of differential equations for mass balance equations is converted into a system of algebraic equations (see the E-coli.txt file in the folder Chapter 13 on the attached compact disk). Each nonlinear equation contains several rate expressions and terms. The glucose impulse term, fpuise, in the mass balance equation... 

Remember In every case, be sure to check the final equation for mass balance. 

The equations for mass balance follow those of Section 11.5 ... 

Applying the Nodal Volume Conservation Equation directly to a gas node would require a full solution for the pressure dynamics in the manner described in Chapter II for a process vessel of fixed volume. However, when the nodal volume is small, the gas pressure and temperature will reach equilibrium quickly, after which time dT/dt = dp/dt = 0. Hence, from equation (18.60), dm/dt = 0. While this last equation will be fully valid only after pressure and temperature have reached equilibrium, it will be acceptable as an approximate characterization of reality at all times provided the nodal volume is small enough to allow very rapid establishment of pressure and temperature equilibria. This is the basis for modelling gas flow in networks under the assumption that steady-state equations for mass balance are valid. 

Note that this equation degenerates into the simple form dm/dt = 0 if the temperature stays constant, irrespective of the volume of the node. If, on the other hand, the nodal temperature is subject to change, the use of the steady-state mass balance will only be valid if the temperature dynamics are very fast. Temperature equilibrium will then be reached rapidly, after which time dTIdt O, and so, from (18.61), dm/dt— 0. But fast temperature dynamics require a small thermal inertia at the node, hence a small mass and hence a small volume. Accordingly the steady-state equations for mass balance will be valid for a liquid with a varying temperature only if all the network s nodes possess small volumes. 

For the separation of the liquid feed M (composition x ) into two streams of composition Xg (product, distillate) and Xf, (bottom product), the required number of theoretical plates can be determined graphically with the aid of the McCabe-Thiele method, which was used in the past because no computers were available to solve the extensive systems of equations for mass balances and equilibrium relationships. This method is no longer of practical importance, but it is an excellent didactic aid for understanding the basic principles of rectification. 

Based on these assnmptions, the following steady-state diffusion-reaction equations for mass balances of substrate concentrations within an immobilized enzyme can be written as... 

We shall now apply the methods developed in the previous chapters to model PTC reactions in liquid-liquid and solid-liquid systems, including solid-supported systems. For a more detailed account of these methods, reference may be made to the articles, among others, of Naik and Doraiswamy (1998), and Yadav and collaborators (1995,2004). The rate of the overall PTC cycle is dependent on the relative rates of the different steps in the PTC cycle. Thus, when the basic conservation equations for mass balance are written for a PTC system, the individual steps that comprise the PTC cycle must be accounted for. These steps are the ion-exchange reaction, interphase mass transfer of both inactive and active forms of the phase transfer (PT) catalyst, partitioning of the catalyst between the two phases (in liquid-liquid systems), and the main organic phase reaction. When these are considered, the normal assumption of pseudo-first-order kinetics (Equation 16.1) is no longer valid. 

For a packed bed of porous uniform adsorbents, the basic equations for mass balance and transfer rates are the same as those given in the previous chapter. 

When adsorption equilibrium is assumed in the column, a basic equation for mass balance of a single adsorbable component is given for an isothermal system as... 

For a rigorous set of differential equations for mass balance in the bed and for adsorption rate in the particle, partial differential equations are introduced for both processes, making for lengthy computation time. In order to finish numerical computation with in a reasonable amount of time it is desirable to use as simple a model as precision will allow. 

Combining the equations for mass balance, mean size and kinetics for the solids hold up gives (for j = 1)... 

It is possible to predict theoretically the mass transfer rate (or flux AT,) across any surface located in a fluid having laminar flow in many situations by solving the differential equation (or equations) for mass balance (Bird et al, 1960, 2002 SkeUand, 1974 Sherwood et al, 1975). Our capacity to predict the mass transfer rates a priori in turbulent flow from first principles is, however, virtually nil. In practice, we follow the form of the integrated flux expressions in molecular diffusion. Thus, the flux of species i is expressed as the product of a mass-transfer coefficient in phase y and a concentration difference in the forms shown below ... 

In a separation process, ion exchange resin particles are generally used in a column. A complex time-dependent differential equation for mass balance in the column has to be combined with the diffusion flux expression for a resin particle, and other appropriate boundary and initial conditions, to determine the extent of separation. It is obvious from the preceding few paragraphs that the diffusion flux expressions are difficult to handle for resin particles. For ion exchange column analysis, practical approaches therefore utilize a linear-driving-force representation of the mass flux to a resin particle (Helfferich, 1962 Vermeulen et ai, 1973) this leads to the use of mass-transfer coefficients in resin particle flux expressions. 

Under these conditions, the evolution equations for mass balance for water and surfactant concentration on the surface are as follows ... 

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