Expression of the mass balance

In a closed system with N constituents and M elements, in which R reactions take place (R N — H), numbers of moles of the individual constituents are not independent variables, as they are linked by relationships which we call mass balance equations. There are two fundamental ways of expressing the mass balance by means of constitution coefficients or of stoichiometric coefficients. We shall show that the two ways are equivalent under the assumption R = 

The mass balance equation constructed by means of the constitution coefficients has the form 

By means of stoichiometric coefficients, the mass balance equations are expressed in the form of 

Construct matrixes of constitution coefficients for the following systems, and determine their rank  

Determine the number of linearly independent reactions in each of the following systems 

---

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

Considerable simplification of the mathematical treatment is achieved by noting that Equation 6.48 is linear on the Pi, i = I, c (probabilities of propagating chain types). Out of the c equations in Equation 6.48, only c-1 are linearly independent, and so, to solve for thePi, i = 1, c, in terms of the/ , i = 1, c, an additional consistency equation, which can be seen as a dimensionless expression of the mass balance of the propagating chain types directly arising from the definition 6.47, is necessary ... 

Attempts to define operationally the rate of reaction in terms of certain derivatives with respect to time (r) are generally unnecessarily restrictive, since they relate primarily to closed static systems, and some relate to reacting systems for which the stoichiometry must be explicitly known in the form of one chemical equation in each case. For example, a IUPAC Commission (Mils, 1988) recommends that a species-independent rate of reaction be defined by r = (l/v,V)(dn,/dO, where vt and nf are, respectively, the stoichiometric coefficient in the chemical equation corresponding to the reaction, and the number of moles of species i in volume V. However, for a flow system at steady-state, this definition is inappropriate, and a corresponding expression requires a particular application of the mass-balance equation (see Chapter 2). Similar points of view about rate have been expressed by Dixon (1970) and by Cassano (1980). 

Here the pseudo-homogeneous rate r is related to the surface reaction rate r" through the area of active catalyst per unit volume of reactor. Assuming further a plug-flow regime, the integration of the mass balance equation for this simple rate expression gives an expression for CO conversion ... 

Novel general expressions were developed for the description of the behaviour of the height equivalent of a theoretical plate in various chromatographic columns such as unpacked (open capillary), packed with spherical nonporous particles and packed with spherical porous adsorbent particles. Particles may have unimodal or bimodal pore size distribution. The expression describing the mass balance in open capillaries is... 

Solutions of the mass balances with appropriate constitutive rate expressions for r(i, j), r(r, j), r(e, j), and r(rxt, j) for all components within the control volume yield the deposition rate and composition of the film. 

Both binding isotherm (Equation 8.15) and double-reciprocal plots (Equation 8.16) are expressed as a function of free ligand concentration, which is not known. By means of the mass balance on ligand,L can be related to the known total ligand concentration... 

The discussion above provides a brief qualitative introduction to the transport and fate of chemicals in the environment. The goal of most fate chemists and engineers is to translate this qualitative picture into a conceptual model and ultimately into a quantitative description that can be used to predict or reconstruct the fate of a chemical in the environment (Figure 27.1). This quantitative description usually takes the form of a mass balance model. The idea is to compartmentalize the environment into defined units (control volumes) and to write a mathematical expression for the mass balance within the compartment. As with pharmacokinetic models, transfer between compartments can be included as the complexity of the model increases. There is a great deal of subjectivity to assembling a mass balance model. However, each decision to include or exclude a process or compartment is based on one or more assumptions—most of which can be tested at some level. Over time the applicability of various assumptions for particular chemicals and environmental conditions become known and model standardization becomes possible. 

These parameters can be directly related back to the information contained in the EPM with n components (j) and m species (t). Application of the mass balance constraint equation requires that the concentration of each species must be known. Therefore, activity coefficients are computed if the ionic strength is already known from either the Davis or the extended Debye-Eluckel equation however, if ionic strength is unknown and has to be calculated, equation (5.134) can be converted to a general expression for the concentration of each species by substituting the expression for S to give... 

As it was discussed in Section 2.7, if the model of the analyte behavior in the column is defined, then this model can be applied in the mass-balance equation. The resulting solution of the mass-balance equation is the expression for the analyte retention behavior, and it is only valid in the frame of the selected model. 

Implicit in the application of the mass balance concept is the need to choose a control volume. A control volume is any closed volume, across whose boundaries we propose to account for all transport of a chemical, and within whose boundaries we propose to account for all the chemical initially present (stored), as well as all processes (sources or sinks) that produce or consume the chemical. The mass balance expression for any chemical in any control volume during any time interval can be written as... 

Therefore, a fourth relationship is needed. Typically, the ratio L/V can be set. Now, expressions for yi and Xi can be derived in terms of the mass balances and equilibrium relationship (see if you can derive these)... 

The fundamentals of mathematical models lie in the mass balance of the chemical in the environment, which is quantitatively expressed in terms of equilibrium and rate constants of the environmental fate processes. Incorporating these constants into a set of the mass balance equations, and solving these equations are complicated, so that computers are used frequently to reduce the time and cost. 

The interaction among the kinetic expressions and the mass balance can be conveniently expressed in the form of an information feedback loop as shown in Figure 4.8. Here it is shown how the size distribution, which dictates the area and mass of the distribution, impacts on the kinetics by altering the supersaturation, s. It is this information feedback of the system that makes control of crystallizers a challenging problem in that overshoot is easily caused. 

From the formal ion constant expression, we can write MI. ] x Ml l.l. I urlhormore. if our expor imeiits are carried out in the presence of excess ligand. C [LJ. Substituting these expressions and the mass balance expression for fM) intt) Equation 14-0, we obtain... 

The position of the [HA" and [A " lines can be determined in a similar fashion by making approximations in the following equations developed by substituting the equilibrium expressions in the mass balance and solving for [HA"] and [A ], respectively, in terms of Ct.a, [H J constants. 

In the event of unsteady-state operation, the means of incorporating the rate expression in the mass balance are still the same, but the balance becomes... 

Now, all tenns in the mass balance are expressed as volume integrals. When these three terms are moved to the left-hand side of the mass balance, given by (9-3), the following result is obtained ... 

For a number of nonlinear and competitive isotherm models analytical solutions of the mass balance equations can be provided for only one strongly simplified column model. This is the ideal model of chromatography, which considers just convection and neglects all mass transfer processes (Section 6.2.3). Using the method of characteristics within the elegant equilibrium theory, analytical expressions were derived capable to calculate single elution profiles for single components and mixtures (Helfferich and Klein, 1970 Helfferich and Carr 1993 Helfferich and Whitley 1996 Helfferich 1997 Rhee, Aris, and Amundson, 1970 ... 

This is similar to a rate-determining step situation, and is more simply derived by taking the first reaction to always be in instantaneous equilibrium, Cg/K,. Show that a new derivation of the mass balances with this basis leads to the same result as above. Note that this is a useful technique in mote complex situations of this type, when the general expression may not be possible to derive. 

Eq. (9.6-11) involves the time derivative of the average adsorbed concentration. Its form is not convenient for numerical computation, and what we will show is an alternative expression for the heat balance equation. We take the volumetric average of the mass balance equation (9.6-1) ... 

The sensitivity of headspace analysis is influenced principally by two sample-related parameters the partition coefficient of the analyte between the phases and the phase ratio of sample to vapor. Consideration of the mass balance of an analyte in a sample under headspace conditions of equilibrium allows a simple expression to be developed relating sensitivity... 

The purpose of the mass balance in a process model is to express each stream in terms of the relevant process variables B2, X, and the current density i. A fourth parameter B5, is required but is not a process variable, since it is the rate of production of product and will be fixed as a process specification. 

The designer should be aware that there is a critical reactor volume, which generally corresponds to a bifurcation point of the mass balance equations. For stable operation the reactor should be larger than this critical value. As example, for essentially first-order reaction with pure product and recycle, the feasibility condition is simply Da>. The definition of the plant Damkohler number includes reactor volume, reaction kinetics and fresh reactant feed flow rate. Similar expressions hold for more complex stoichiometry. 

The relations which express validity of the mass balance take the form of ... 

D having been obtained by addition of the absolute values of differences of the left-and right-hand sides of the mass balance equations (5.147), the J-ih equation having been divided by bj. The iteration process is stopped on validity of the condition D < Sy where is the accuracy required. A value of e = 0.0001 is generally selected. A low value of the expression D signals that values of i = 1, 2,...,iV determined from... 

To obtain an expression for the steady-state distribution of the system variables, the mass balance must now be taken over an incremental element extending from z to z + Az. This is necessary to bring the distance variable into the model. The increment Az need not worry us here because we will ultimately allow it to go to zero, thereby transforming fhe original difference equation into the more familiar form of a differential equation. This leads to the following representation of the mass balance ... 

Most authors who have studied the consohdation process of soflds in compression use the basic model of a porous medium having point contacts which yield a general equation of the mass-and-momentum balances. This must be supplemented by a model describing filtration and deformation properties. Probably the best model to date (ca 1996) uses two parameters to define characteristic behavior of suspensions (9). This model can be potentially appHed to sedimentation, thickening, cake filtration, and expression. 

---