Mass balance expression

The range in EF is from zero to unity, with a racemic value of 0.5. Enantiomer fractions are preferred to ERs, as the EF range is bounded, and a deviation from the racemic value in one direction is the same as that in the other. For example, if the (—)-enantiomer is twice the concentration as its antipode, the EF is 0.333, which is the same deviation (0.167) from a racemic EF of 0.5 as the opposite case of the (+)-enantiomer at twice the concentration as the (—)-enantiomer (EF = 0.667). The respective ERs would be 0.5 and 2. The corresponding deviations of 0.5 and 1, respectively, are not the same deviation from the racemic ER of 1. Thus, ERs can produce skewed data inappropriate for statistical summaries such as sample mean and standard error [109]. As a result, EEs are more amenable compared to ERs for graphical representations of data, mathematical expressions, mass balance determination, and environmental modelling [107, 109]. Individual ER and EF measurements can be converted [107, 108] ... 

If we do not know the pH, the logarithmic concentration diagram can also be used to give an approximate pH value. For example, find the pH of a 0.1 M maleic acid solution. Since the log concentration diagram expresses mass balance and the equilibrium constants, we need only one additional equation such as charge balance to solve the problem exactly. The charge-balance equation for this system is... 

Expressing mass balances is also an essential tool In chemical engineering, for example with mass or energy balances for open reactors. 

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-... 

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. 

Mass Balance, Continuity Equation The continuity equation, expressing consei vation of mass, is written in cartesian coordinates as... 

Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 

Energy balances differ from mass balances in that the total mass is known but the total energy of a component is difficult to express. Consequently, the heat energy of a material is usually expressed relative to its standard state at a given temperature. For example, the heat content, or enthalpy, of steam is expressed relative to liquid water at 273 K (0°C) at a pressure equal to its own vapor pressure. 

Expression of the mass balances around the bioreactor (process). 

Capture efficiency of the hood (aj is defined as the ratio of the directly captured contaminant to the amount of generated contaminant. " The directly-captured contaminant is that part of the emission captured directly into the hood, and does not include that part of the emission captured after dilution into the ambient room air. The room mass balance is determined using the conditions and expressions given in Fig. 10.107 and assuming the room air is w ell mixed ... 

It was shown in Chapter 7 that the performance of continuous crystallizers is determined by the characteristics of a feedback loop relating the output performance expressed as crystal size distribution and to the feed concentration and residence time. Thus, 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, as considered in Chapter 7 and illustrated in Figure 8.11. 

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. 

For symbolic convenience we make use of the reaction variable x, which is the decrease in concentration of reactant A in time t. Because of the reaction stoichiometry, X is also the decrease in B concentration. The mass balance expressions are... 

This is an autocatalytic reaction, in which a product of the reaction appears in the rate equation for the forward reaction. In this case the mass balance expressions are... 

From Scheme VI, the reactants follow first-order kinetics, so Ca = Ca exp (— kit) and Cb = Ceexp ( — k2t). The mass balance expression is... 

Thus, if Ca and Cb can both be measured as functions of time, a plot of v/ca vs. Cb allows the rate constants to be estimated. (If it is known that B is also consumed in the first-order reaction, mass balance allows cb to be easily expressed in terms of Ca-) The rate v(Ca) is the tangent to the curve Ca = f(t) at concentration Ca-This can be determined graphically, analytically, or with computer processing of the concentration-time data. Mata-Perez and Perez-Benito show an example of this treatment for parallel uncatalyzed and autocatalyzed reactions. 

The concentration Cq is easily found by substituting Eqs. (3-99) and (3-102) into the mass balance expression ... 

Several features of this treatment are of interest. Compare the denominators of Eqs. (3-147) and (3-149) Miller has pointed out that the form of Eq. (3-147) is usually seen in chemical applications of the steady-state approximation, whereas the form of Eq. (3-149) appears in biochemical applications. The difference arises from the manner in which one uses the mass balance expressions, and this depends upon the type of system being studied and the information available. 

The mechanism involved the overall conversion of [5] to [P], The reverse reaction is insignificant because only the initial velocity in one of the forward direction is concerned. The mass balance equation expressing the distribution of the total enzyme is ... 

In the above defined system, the over-all mass balance gives for the change of the number of moles in the gas phase the expression... 

Then the diffusion equation for the fluctuation of the metal ion concentration is given by Eq. (68), and the mass balance at the film/solution interface is expressed by Eq. (69). These fluctuation equations are also solved with the same boundary condition as shown in Eq. (70). 

Low-temperature, photoaggregation techniques employing ultraviolet-visible absorption spectroscopy have also been used to evaluate extinction coefficients relative to silver atoms for diatomic and triatomic silver in Ar and Kr matrices at 10-12 K 149). Such data are of fundamental importance in quantitative studies of the chemistry and photochemistry of metal-atom clusters and in the analysis of metal-atom recombination-kinetics. In essence, simple, mass-balance considerations in a photoaggregation experiment lead to the following expression, which relates the decrease in an atomic absorption to increases in diatomic and triatomic absorptions in terms of the appropriate extinction coefficients. 

Equations (1.1) to (1.3) are diflerent ways of expressing the overall mass balance for a flow system with variable inventory. In steady-state flow, the derivatives vanish, the total mass in the system is constant, and the overall mass balance simply states that input equals output. In batch systems, the flow terms are zero, the time derivative is zero, and the total mass in the system remains constant. We will return to the general form of Equation (1.3) when unsteady reactors are treated in Chapter 14. Until then, the overall mass balance merely serves as a consistency check on more detailed component balances that apply to individual substances. 

Based on the kinetic mechanism and using the parameter values, one can analyze the continuous stirred tank reactor (CSTR) as well as the dispersed plug flow reactor (PFR) in which the reaction between ethylene and cyclopentadiene takes place. The steady state mass balance equations maybe expressed by using the usual notation as follows ... 

Most real situations are, however, such that conditions change with respect to time. Under these circumstances, a steady-state mass balance is inappropriate and must be replaced by a dynamic or unsteady-state mass balance, expressed as... 

Chemical production rates are often expressed on a molar basis but can be easily converted to mass flow quantities (kg/s). The mass balance equation can then be expressed as... 

For a batch system, with no inflow and no outflow, the total mass of the system remains constant. The solution to this problem, thus involves a liquid-phase, component mass balance for the soluble material, combined with an expression for the rate of mass transfer of the solid into the liquid. 

For reactions involving heat effects, the total and component mass balance equations must be coupled with a reactor energy balance equation. Neglecting work done by the system on the surroundings, the energy balance is expressed by... 

Note that this could also be obtained by defining the mass balance system to include both phases and that the mass transfer rate expression, Q, now no longer appears in the balance equation. 

As discussed previously in Sec. 3.3.1.7, each additional component of the feed mixture must be expressed by a separate component mass balance equation and by its own equilibrium relationship. 

At steady-state conditions, the mass balance design equations for the ideal tubular reactor apply. These equations may be expressed as... 

This problem is similar to CONSTILL except that three components benzene, xylene and toluene are considered. Thus, as explained in Sec. 3.3.3.4, each component of the mixture may be expressed by a separate component mass balance. Using mole fractions one balance can be omitted, and replaced by the condition that the sum of the mole fractions must be equal to unity. 

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