Generalized Mass Balances

The flux, assuming a dilute solution, calculated from Eq. 2.2-10, is 

This two percent difference is well within the needs of most practical calculations. Thus the dilute solution equations are more than adequate here. 

At 60 °C, the choice is less obvious because the vapor pressure is about 395 mm Hg. When we calculate the mole fraction in the same way, we find 

The dilute-solution equations underestimate the flux by a significant error of about forty percent. 

As the problems that we discuss in this chapter become more and more complex, the development of the differential equations becomes more and more tedious. Such tedium can be avoided by using the generalized mass balances developed in this section. These mass balances automatically include both steady- and unsteady-state situations. They imply the usual variety of coordinate systems, and they reflect the vectorial nature of mass fluxes. They are excellent weapons. 

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General Material Balances. According to the law of conservation of mass, the total mass of an isolated system is invariant, even in the presence of chemical reactions. Thus, an overall material balance refers to a mass balance performed on the entire material (or contents) of the system. Instead, if a mass balance is made on any component (chemical compound or atomic species) involved in the process, it is termed a component (or species) material balance. The general mass balance equation has the following form, and it can be applied on any material in any process. 

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. 

With this equation, we can now discuss a generalized mass balance equation. We still use Figure 1 to show the derivation. Based on Eq. (5), the net contribution by diffusion and convection now becomes... 

This is a generalized mass balance equation in one dimension. If diffusion and convection occur in other directions, the generalized mass balance equation becomes... 

A generalized mass balance equation in other coordinate systems is sometimes useful. In the cylindrical system, Eq. (19) becomes... 

In the previous section, we detailed diffusion equations and generalized mass balance equations. We now turn to their practical uses in the pharmaceutical sciences. Mass transport problems can be classified as steady or unsteady. In steady mass transport there is no change of concentration with time [3], characterized mathematically by... 

For simplicity, only an irreversible first-order reaction is considered. We begin with the generalized mass balance equation in rectangular coordinates with mass transfer in one dimension ... 

The negative sign before the reaction term k C,n indicates that the drug is consumed rather than being produced as in the derivation of the generalized mass balance equation. 

We solve this problem by using a generalized mass balance equation in cylindrical coordinates, centered on the disk ... 

Batch and continuous processes may also be compared by examining their governing mass-balance relations. As an elaboration of equation 1.5-1, a general mass balance may be written with respect to a control volume as ... 

For the reactor described in Figure 4.3, the general mass balance for component j at stage i is... 

A general mass balance for an arbitrary liquid-phase component in the stirred tank reactor is thus written as follows,... 

The general mass balance for each phase at nonsteady state, considering convection, mass transfer and reaction (e. g. ozone decay), can be written ... 

Generalized Mass-Balance and Design Equations for a Single-Phase Reactor... 

We first recall the generalized mass-balance equations for one-phase systems with M components. 

Equation (6.1) is the most general mass-balance equation for single phase systems. It applies to all possible mass-balance cases. 

For the sake of generality, we now develop most general mass-balance equations for a two phase system in which each phase has multiple inputs and multiple outputs and in which each phase is undergoing reactions within its boundaries. 

Refer to Figure 6.8 for a representation of the generalized mass-balance expression in a heterogeneous system. 

The generalized mass balance equation on a unit volume is ... 

Irrespective of the specific case considered, the general mass balance relationship for the system depicted in Figure 6.2 under the steady-state assumption... 

To identify air emission sources, the study team utilized a variety of measurement techniques. Table III summarizes the different techniques used to define the airborne emissions. Emissions from sewer vents, water ponds, the inactive land-farm, and oil/water separator were measured directly (Amoco/EPA, 1991c). In general, mass balance techniques are not sufficiently accurate for most inventory calculations (NRC, 1990). However, since their flow rates were small, easily measurable, and reasonably constant over time, mass balances and inlet/outlet water analyses were used to determine emissions from the cooling tower and wastewater tanks. 

P1-7a How can you convert the general mole balance equation for a given species, Equation (1-4), to a general mass balance equation for that species ... 

According to the assumptions in Section 6.2.1 the general mass balance for one component i in one differential volume dVc of the mobile phase is (Fig. 6.3)... 

After having defined the accumulation terms of the general mass balances (Eqs. 6.1-6.6), the transport and source terms are evaluated as follows. Mass transport in... 

For a feed of pure A, the general mass balance equations are... 

In the mass-balance approach an attempt is made to determine field weathering rates from element flux calculations, usually focusing on the plant-nutrient important base cations (Ca- Mg ", K, and Na+) or on silica (cf. Velbel 1985). In some studies balances are also computed for Al (Swoboda-Colberg and Drever 1992) and/or the the major anions and nitrogen species (Likens et al. 1977 Mast et al. 1990). A general mass-balance equation for the base cations (BC) might be... 

The general mass balance model for marine carbon... 

In practical terms, an indoor air quality model should provide a reasonable description of the mass balance of the test chamber experiments, trying to address factors such as material emissions, airflows into and out of the chamber and chemical/physical decay or other removal and/or transformation processes of the VOCs. VOC concentrations are increased by emissions within the defined volume of the chamber and by infiltration from external air to the chamber. Similarly, concentrations are decreased by transport via exiting chamber air, by removal to chemical and physical sinks within the chamber air, or by transformation of a VOC to other chemical forms. A general mass balance equation concerning the concentration of a VOC in a test chamber can be written in the form of one or more differential equations representing the rate of accumulation and the VOC gain and loss. This concept for a VOC concentration C (mass units/ m ) in a chamber of volume V (m ) is translated into the following differential equation ... 

Here Zi is the valence (including sign) of the th species, and m is its concentration (molality). For example, Zi would be -2 for SO "", +1 for Na", and 0 for NaCl°(aq). Next, assume that the total molal concentrations Be, of N — M — elements or atomic species are known for this system. Write N — M — I general mass balances ... 

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